Fo Research Advances

OPTICAL FIBERS RESEARCH ADVANCES

OPTICAL FIBERS RESEARCH ADVANCES

JÜRGEN C. SCHLESINGER EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2007 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Schlesinger, Jürgen C. Optical fibers research advances / Jürgen C. Schlesinger, Editor. p. cm. Includes index. ISBN-13: 978-1-60692-607-9 1. Optical communications. 2. Fiber optics. 3. Optical fibers. I. Title. TK5103.59.S35 2008 621.36'92--dc22 2007031168

Published by Nova Science Publishers, Inc. New York

CONTENTS

Preface vii

Short Communication 1

Ignition with Optical Fiber Coupled Laser Diode 3 Shi-biao Xiang, Xu Xiang , Wei-huan Ji and Chang-gen Feng

Research and Review Studies 13

Chapter 1 Evanescent Field Tapered Fiber Optic Biosensors (TFOBS): Fabrication, Antibody Immobilization and Detection

15

Angela Leung, P. Mohana Shankar and Raj Mutharasan

Chapter 2 New Challenges in Raman Amplification for Fiber Communication Systems

51

P.S. André, A.N. Pinto, A.L.J. Teixeira, B. Neto, S. Stevan Jr., Donato Sperti, F. da Rocha, Micaela Bernardo, J.L. Pinto, Meire Fugihara, Ana Rocha and M. Facão

Chapter 3 Fiber Bragg Gratings in High Birefringence Optical Fibers 83 Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski

Chapter 4 Applications of Hollow Optical Fibers in Atom Optics 119 Heung-Ryoul Noh and Wonho Jhe

Chapter 5 Advances in Physical Modeling of Ring Lasers 161 Vittorio M.N. Passaro and Francesco De Leonardis

Chapter 6 Investigation of Optical Power Budget of Erbium-Doped Fiber

187

Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto

Chapter 7 Recent Developments in All-Fibre Devices for Optical Networks

205

Nawfel Azami and Suzanne Lacroix

Contents vi

Chapter 8 Advances in Optical Differential Phase Shift Keying and Proposal for an Alternative Receiving Scheme for Optical Differential Octal Phase Shift Keying

231

M. Sathish Kumar, Hosung Yoon and Namkyoo Park

Chapter 9 A New Generation of Polymer Optical Fibers 257 Rong-Jin Yu and Xiang-Jun Chen

Chapter 10 Dissipative Solitons in Optical Fiber Systems 279 Mário F.S. Ferreira and Sofia C.V. Latas

Chapter 11 Bright - Dark and Double - Humped Pulses in Averaged, Dispersion Managed Optical Fiber Systems

301

K.W. Chow and K. Nakkeeran

Chapter 12 Dynamics and Interactions of Gap Solitons in Hollow Core Photonic Crystal Fibers

315

Javid Atai and D. Royston Neill

Chapter 13 Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier Ring Lasers

335

Byoungho Lee and Ilyong Yoon

Chapter 14 Aging and Reliability of Single-Mode Silica Optical Fibers 355 M. Poulain, R. El Abdi and I. Severin

Index 369

PREFACE An optical fiber is a glass or plastic fiber designed to guide light along its length by

confining as much light as possible in a propagating form. In fibers with large core diameter, the confinement is based on total internal reflection. In smaller diameter core fibers, (widely used for most communication links longer than 200 meters) the confinement relies on establishing a waveguide. Fiber optics is the overlap of applied science and engineering concerned with such optical fibers. Optical fibers are widely used in fiber-optic communication, which permits transmission over longer distances and at higher data rates than other forms of wired and wireless communications. They are also used to form sensors, and in a variety of other applications.

The term optical fiber covers a range of different designs including graded-index optical fibers, step-index optical fibers, birefringent polarization-maintaining fibers and more recently photonic crystal fibers, with the design and the wavelength of the light propagating in the fiber dictating whether or not it will be multi-mode optical fiber or single-mode optical fiber. Because of the mechanical properties of the more common glass optical fibers, special methods of splicing fibers and of connecting them to other equipment are needed. Manufacture of optical fibers is based on partially melting a chemically doped preform and pulling the flowing material on a draw tower. Fibers are built into different kinds of cables depending on how they will be used.

This new book presents the latest research in the field. Optical fibers, an important and promising material, have attracted more and more

attention and extended their applications to various scientific and practical aspects. In the short communication, the key role of fibers, as the carriers of information and energy in our times, was briefly summarized. Afterwards, the configuration of fiber coupled laser diode ignition system was elucidated as well as the advantages, developments and applications of this technology. Furthermore, the energy-transmitting characteristics of single-mode fibers and multi-mode ones and the key points of fiber-coupled technology were analyzed. In a practical case, the effect of the diameters of core on laser ignition, from both theory and experiments, was studied. The findings suggest that the smaller the diameters of core, the lower the ignition threshold under the same laser power. That is to say, the ignition becomes easier while using fibers with smaller core. Finally, the issue on selection of core was clarified based on the consideration of both laser power density and the endurance of fibers.

Tapered Fiber Optic Biosensors (TFOBS) are sensors that operate based on fluctuations in the evanescent field in the tapered region. In the laboratory, TFOBS are made by heat

Jürgen C. Schlesinger viii

pulling commercially-available single mode optical fibers. They have been investigated for various applications, including measurement of physical characteristics (refractive index, temperature, pressure, etc.), chemical concentrations, and biomolecule detection. In this chapter, an up-to-date review of TFOBS research is provided, with emphasis on applications in biosensing such as pathogen, proteins, and DNA detection. The physics of sensing and optical behavior based on taper geometry is discussed. Methods of fabrication, antibody immobilization, sample preparation, and detection from our laboratory are described. This chapter presents results on the non-specific response, simulation, and detection of E.coli O157:H7 and BSA. Chapter 1 will conclude with an analysis of the future direction of the Tapered Fiber Optic Biosensors.

Raman fiber amplifiers (RFA) are among the most promising technologies in lightwave systems. In recent years, Raman optical fiber amplifiers have been widely investigated for their advantageous features, namely the transmission fiber can be itself used as the gain media reducing the overall noise figure and creating a lossless transmission media. The introduction of RFA based on low cost technology will allow the consolidation of this amplification technique and its use in future optical networks.

Chapter 2 reviews the challenges, achievements, and perspectives of Raman amplification in optical communication systems. In Raman amplified systems, the signal amplification is based on stimulated Raman scattering, thus the peak of the gain is shifted by approximately 13.2 THz with respect to the pump signal frequency. The possibility of combining many pumps centered on different wavelengths brings a flat gain in an ultra wide bandwidth.

An initial physical description of the phenomenon is presented as well as the mathematical formalism used to simulate the effect on optical fibers.

The review follows with one section describing the challenging developments in this topic, such as using low cost pump lasers, in-fiber lasing, recurring to fiber Bragg grating cavities or broadband incoherent pump sources and Raman amplification applied to coarse wavelength multiplexed networks. Also, one of the major issues on Raman amplifier design, which is the determination of pump powers in order to realize a specific gain will be discussed. In terms of optimization, several solutions have been published recently, however, some of them request extremely large computation time for every interaction, what precludes it from finding an optimum solution or solve the semi-analytical rate equation under strong simplifying assumptions, which results in substantial errors. An exhaustive study of the optimization techniques will be presented.

This paper allows the reader to travel from the description of the phenomenon to the results (experimental and numerical) that emphasize the potential applications of this technology.

Fiber Bragg gratings (FBG) are a key element in optical communication devices and in fiber sensors. This is mainly due to its intrinsic characteristics, which include low insertion loss, passive operation and immunity to electromagnetic interferences. Basically a FBG is a periodic modulation of the core refractive index formed by exposure of a photosensitive fiber to a spatial pattern of ultraviolet light in the region of 244–248 nm. The lengths of FBGs are normally within the region of 1–20 mm. Usually a FBG operates as a narrow reflection filter, where the central wavelength is directly proportional to the periodicity of the spatial modulation and to the effective refractive index of the fiber. The production technology of these devices is now in a mature state, which enables the design of gratings with custom-

Preface ix

made transfer functions, crucial for all-optical processing. Recently, some work has been done in the application of FBG written in highly birefringent fibers (HiBi). Due to the birefringence, the effective refractive index of the fiber will be different for the two transversal modes of propagation. Therefore, the reflection spectrum of a FBG will be different for each polarization. This unique property can be used for advanced optical processing or advanced fiber sensing.

Chapter 3 will describe in detail this unique device. The chapter will also analyze the device and demonstrate different applications that take advantage of its properties, like multiparameter sensors, devices for optical communications or in the optimization of certain architectures in optics communications systems.

A hollow optical fiber (HOF) has a lot of interesting applications in atom optics experiments such as atom guiding and the generation of hollow laser beam (HLB). In this article the authors present theoretical and experimental works on the use of hollow optical fibers in atom optics. Chapter 4 is divided into two parts: One is devoted to the atom guide using HOFs and the other describes the atom optics researches that utilizes laser lights emanated from the HOF. First, the authors describe the electromagnetic fields inside the HOF and characterize the electromagnetic modes diffracted from the HOF. Then they describe two guiding schemes using red and blue detuned laser lights. Finally, they describe the various relevant experiments using LP01 or LP11 modes such as the generation of HLB from the HOF, funneling atoms using the diffracted fields, diffraction-limited dark laser spot, and a dipole trap using LP01 mode of the diffracted field from the HOF.

In Chapter 5, an overview on fiber ring lasers and III/V semiconductor integrated ring lasers is presented. In particular, some aspects of mathematical modelling of these devices are reviewed. In the first part of the chapter, the authors have focused our attention on the more recent theoretical and experimental studies concerning fiber ring laser architectures. Then, a complete quantum-mechanical model for integrated ring lasers is presented, including the evaluation of all the involved physical parameters, such as self and cross saturation and backscattering. Finally, the influence of sidewall roughness on either unidirectional or bidirectional regime in multi-quantum-well III/V semiconductor ring lasers is demonstrated.

In Chapter 6, the authors investigated optical power budget of an erbium-doped fiber (EDF). In addition to the output signal and amplified spontaneous emission (ASE) powers from the fiber end, lateral spontaneous emissions and scattering laser powers in the EDF were measured quantitatively by using an integrating sphere. Compared with the signal and ASE powers, it was found that considerable powers were consumed by the laterally emitting lights. As an optically undetected loss which limits power conversion efficiency (PCE) of the fiber amplifier, the effect of nonradiative decay from the termination level of pump excited state absorption (pump ESA) was estimated from decay rate analyses of the relevant levels. The nonradiative loss was comparable to amplified signal power in the EDF when pumped with a 980 nm LD. Nonradiative decay following cooperative upconversion (CUP) process is also discussed using rate equations analysis.

All-fibre components are essential components of optical networks systems. Development of such devices is of great importance to allow network functions to be performed in the glass of the optical fibre itself. Among of all fabrication techniques, the Fused Fibre Biconical Taper (FBT) technique allows optical devices with high performances. Although fibre devices are mainly based on the passive directional coupler basic structure, research is made to design components that perform complex functionalities in today optical

Jürgen C. Schlesinger x

networks systems. Recent developments on all-fibre devices in network systems are presented. Research is mainly focused on enhanced fabrication and stability of FBT fabrication technique, passive thermal compensation for stable interferometer optical structure, broadband spectral operation for multi-wavelength operations and new interferometer designs. An overview of recent fused fibre devices for optical telecommunications is presented to understand the main functionalities of these fibre devices. The limiting factors are explained in Chapter 7, to understand challenges on fibre devices development.

Optical Differential Phase Shift Keying (oDPSK) with delay interferometer based direct detection receiver was proposed as an alternative for the conventional On-Off Keying (OOK) modulation schemes. Compared to OOK, oDPSK was predicted to have a 3dB improvement in performance due to its balanced detection receiver structure. It was also predicted that due to the optical signal occupying all the symbol slots, unlike in OOK, symbol pattern dependent fiber nonlinear effects will make less of an impact on long haul optical transmission schemes based on oDPSK. Subsequent successful demonstrations of these positive attributes of oDPSK resulted in active investigations into multilevel formats of oDPSK namely, optical Differential Quadrature Phase Shift Keying (oDQPSK) and optical Differential Octal Phase Shift Keying (oDOPSK). Significant developments in theoretical models of optically amplified lightwave communication systems based on the Karhunen-Loeve Series Expansion (KLSE) method assisted such investigations. In Chapter 8, the authors discuss some of the recent advances in oDPSK and its multilevel formats that have been achieved such as proposals for receiver schematics, theoretical analysis of receiver schematics, electronic techniques to counter polarization mode dispersion induced penalties, and application of coded modulation techniques. The chapter also proposes an alternative receiver schematic for oDOPSK which can separately detect the three constituent bits from an oDOPSK symbol.

Chapter 9 describes the background to the development of Polymer Optical fibers (POFs), discusses the optical and temperature resistant properties of polymers while emphasizing the intrinsic high attenuation of them. The first generation of POFs which consists of a solid-core surrounded by cladding and transmits light by total internal reflection, is puzzled by the difficulty of high attenuation. Then, the method of using a specific structure (i.e. hollow-core Bragg fiber) to solve the problem is presented. A new generation of POFs based on the hollow-core Bragg fibers with cobweb-structured cladding can guide light with low transmission loss and high bandwidth in the wavelength range of visible to terahertz (THz ) radiation. Efficient hollow-core guiding for delivery of power laser radiation and solar radiation can be achieved by replacing the traditional polymethylmethacrylate (PMMA) with heat-resistant polymers. Lastly, this chapter concludes with a discussion of applications in diverse areas.

Chapter 10 introduces the concept of dissipative solitons, which emerge as a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Such dissipative solitons have many unique properties which differ from those of their conservative counterparts and which make them similar to living things. The authors focus our discussion on dissipative solitons in optical fiber systems, which can be described by the cubic-quintic complex Ginzburg-Landau equation (CGLE). The conditions to have stable solutions of the CGLE are discussed using the perturbation theory. Several exact analytical solutions, namely in the form of fixed-amplitude and arbitrary-amplitude solitons, are presented. The numerical solutions of the quintic CGLE include plain pulses, flat-top pulses, and composite pulses,

Preface xi

among others. The interaction between plain and composite pulses is analyzed using a two-dimensional phase space. Stable bound states of both plain and composite pulses are found when the phase difference between them is 2/π± . The possibility of constructing multisoliton solutions is also demonstrated.

As explained in Chapter 11, the envelope of the axial electric field in a dispersion managed (DM) fiber system is governed by a nonlinear Schrödinger model. The group velocity dispersion (GVD) varies periodically and thus realizes both the anomalous and normal dispersion regimes. Kerr nonlinearity is assumed and a loss / gain mechanism will be incorporated. Due to the big changes in the GVD parameter, the correspondingly large variation in the quadratic phase chirp of the DM soliton will be identified. An averaging procedure is applied. In many DM systems, an amplifier at the end of the dispersion map will compensate for the energy dissipated in that map. Here the case of gain not exactly compensating the loss is considered, in other words, a small residual amplification / attenuation is permitted. The present model differs from other similar ones on variable coefficient NLS, as the inhomogeneous features involve both time and the spatial coordinate. The goal here is to extend the model further, by permitting coupled modes or additional degree of freedom. More precisely, the coupling of fiber loss and initial chirping is considered for a birefringent fiber. The corresponding dynamics is governed by variable coefficient, coupled NLS equations for the components of the orthogonal polarization of the pulse envelopes. When the self phase and cross phase modulation coefficients are identical for special angles, several new classes of wave patterns can be found. New stationary wave patterns which possess multiple peaks within each period are found, similar to those found for the classical Manakov model. For situations where the self- and cross-phase modulation coefficients are different, symbiotic solitary pulses are studied. A pair of bright-dark pulses exists, where either or both pulse(s) cannot propagate in that waveguide if coupling is absent.

The existence and stability of gap solitons in a model of hollow core fiber in the zero dispersion regime are analyzed in Chapter 12. The model is based on a recently introduced model where the coupling between the dispersionless core mode and nonlinear surface mode (in the presence of the third order dispersion) results in a bandgap. It is found that similar to the anomalous and normal dispersion regimes, the family of solitons fills up the entire bandgap. The family of gap solitons is found to be formally unstable but in a part of family the instability is very weak. Consequently, gap solitons belonging to that part of the family are virtually stable objects. The interactions and collisions of in-phase and the π -out-of-phase quiescent solitons and moving solitons in different dispersion regimes are investigated and compared.

Chapter 13 reviews various schemes for multiwavelength fiber lasers and semiconductor optical amplifier (SOA) ring lasers. Multiwavelength fiber lasers have applications in wavelength division multiplexing (WDM) optical communication systems, optical fiber sensors and optical spectroscopy. Erbium-doped fiber amplifiers (EDFAs), Raman amplifiers and SOAs are mainly used as gain media for multiwavelength fiber lasers.

Because EDFAs are homogeneously broadened gain media, various methods have been researched to enable the multiwavelength generation. Due to the introduction of liquid nitrogen cooling, four-wave mixing, frequency shifted feedback, and so on, multiwavelength erbium-doped fiber lasers could become realized.

Jürgen C. Schlesinger xii

On the other hand, because SOA and Raman amplifiers are gain media with inhomogeneous broadening, multiwavelength generation is relatively easy. The useful features of the multiwavelength lasers are mainly dependent on a comb filter. One of the most important features of multiwavelength lasers is tunability. The tunability of wavelengths and channel spacing is required for WDM optical communication systems. Much research has been conducted to enable implementation of tunable multiwavelength fiber lasers. Various comb filters such as Fabry-Perot filters, fiber Bragg gratings, and polarization-maintaining fiber loop mirrors can be used for multiwavelength fiber lasers. The authors review several schemes for multiwavelength SOA-fiber and Raman fiber lasers in this chapter.

The optical fiber reliability in telecommunication networks has been still an issue, that’s why the question of how long an optical fibers might been used without a significant probability of failure isn’t out of interest. Much work was developed around this issue, but the optical fiber fatigue and aging process has not been yet fully understood.

The reliability of the optical fibers depends on various parameters that have been identified: time, temperature, applied stress, initial fiber strength and environmental corrosion. The major and usually unique corrosion reagent is water, either in the liquid state or as atmospheric moisture. Glass surface contains numerous defects, either intrinsic, the so-called “Griffith’s flaws and extrinsic, in relation to fabrication process. Under permanent or transient stress, microcracks grow from these defects, and growth kinetics depend on temperature and humidity. Although polymeric coating efficiently protects glass surface from scratches, it does not prevent water to reach glass fiber.

The work carried out during the last years made possible to apprehend in a more coherent way the problems of failure and rupture of fibers subjected to severe aging conditions.

In Chapter 14, some informations on the used characterization methodology for the silica optical fibers are given. In addition, Optical fibers analysis advantages, expected percussions and theoretical background are given to enlighten the potential concerned persons. The principal optical fiber test benches are described and some results are commented. Finally, final remarks are noted.

SHORT COMMUNICATION

In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 3-11 © 2007 Nova Science Publishers, Inc.

IGNITION WITH OPTICAL FIBER COUPLED LASER DIODE

Shi-biao Xiang1,2*, Xu Xiang3 , Wei-huan Ji2 and Chang-gen Feng4

1 Department of Technical Physics, Zhengzhou Institute of Light Industry, No.5 Dongfeng Road, Zhengzhou 450002, P.R. China

2 Key Laboratory of Informationalized Electric Apparatus of Henan Province, Zhengzhou 450002, P.R. China

3 State Key Laboratory of Chemical Resource Engineering, Beijing University of Chemical Technology, P.O. BOX 98, Beijing 100029, P.R. China

4 School of Mechanics and Engineering, Beijing Institute of Technology, Beijing 100081, P.R. China

Abstract

Optical fibers, an important and promising material, have attracted more and more attention and extended their applications to various scientific and practical aspects. In this article, the key role of fibers, as the carriers of information and energy in our times, was briefly summarized. Afterwards, the configuration of fiber coupled laser diode ignition system was elucidated as well as the advantages, developments and applications of this technology. Furthermore, the energy-transmitting characteristics of single-mode fibers and multi-mode ones and the key points of fiber-coupled technology were analyzed. In a practical case, the effect of the diameters of core on laser ignition, from both theory and experiments, was studied. The findings suggest that the smaller the diameters of core, the lower the ignition threshold under the same laser power. That is to say, the ignition becomes easier while using fibers with smaller core. Finally, the issue on selection of core was clarified based on the consideration of both laser power density and the endurance of fibers.

* E-mail address: [email protected]. Tel: 86-371-63557226 (Corresponding author: S. B. Xiang)

Shi-biao Xiang, Xu Xiang, Wei-huan Ji et al. 4

1. Introduction

Optical fibers as carrier of information and energy have intrigued intensive interest worldwide due to its scientific and technological significance in various practical fields. For instance, in optical communications, fibers have received tremendous attention from both experimental and theoretical aspects not only on the type of fiber materials but also on various communicating techniques [1-4], in which the most primary function of fibers is to transmit information like voice, images and videos from one place to another. A wide variety of optical fiber devices have been designed and exploited in the field of fiber-based communications, such as fiber optical amplifiers, frequency or phase modulators, planar waveguides and fiber polarizers.

Furthermore, the developments of microstructured optical fibers (MOFs) and photonic crystal fibers [5-9] enable a number of potential functionalities including tunability and enhanced nonlinearity, and extend novel fiber device applications to fiber Bragg gratings, tunable resonant filters, variable optical attenuators and nonlinear optics devices owing to their unique characteristics [10-15].

More interestingly, chemical sensors based on optical fibers have been widely explored in the past few years [16-18]. For example, sensors for gases or vapors [19-20], humidity [21-22], metallic ions, specific chemical compounds [23], viscosity [24], intensity [25] and miniature pressure [26] have been delicately designed and rapidly developed. Also, biosensors [27, 28] for enzymes, antibodies or antigens, DNA [29] and bacteria are becoming a prevailing research topic on the basis of fiber materials. They have been exhibiting promising applications in a variety of fields such as chemical analysis, biological monitoring and environmental detection. In this article, the emphasis has been highlighted on the fundamental principles and the important practice of fiber-coupled laser diode ignition.

2. Fiber Coupled Laser Diode Ignition

2.1. Brief Review on Laser Diode (LD) Ignition

Laser ignition is a kind of ignition technique, which refers to detonation or ignition of energetic materials such as solids or fluids [30-33] by laser beam. At early stage of laser ignition technique, the types of laser used for the experimental and application research are mostly Nd:YAG, Nd: GSGG, Nd: glass laser and CO2 laser [34-40]. These lasers possess the characteristics of high output power or energy, small radiation angle of light, long life-span and low price. However, the obvious disadvantages of this kind of laser are low energy conversion efficiency, in which the ratio of output light energy and input electric energy is usually lower than 3%, as well as large volume and heavy weight. With the born of LD and the naissance of LD ignition, the research and evolution of laser ignition technique come into a new era. The experimental studies for laser diode ignition began in the middle of 1980s. Ewick, Kunz, Kramer, Jungst, Merson, Glass and Roman et al have made great devotion to the field of LD ignition, of which Ewick [41] and Kunz [42] published their literatures firstly.

LD belongs to a kind of semiconductor laser stimulated by current. In LD ignition, LD is utilized as energy source, and the energy is transmitted to powders by using optical fiber, which detonates or ignites the energetic materials. This ignition configuration has the

Ignition with Optical Fiber Coupled Laser Diode 5

characteristics of safety, reliability, and strong capability of anti-interference of electromagnetism. In addition, the following advantages are also realized.

(1) It is easy for LD ignition system to realize miniaturization of apparatus due to its

small volume and light weight. (2) LD ignition system has excellent adaptability to the ambient environment because of

the input of low voltage and electric energy. (3) LD ignition system can output multi-channel laser signals by using LD arrays and

consequently control multi-point ignition through the selection to time and order of signals.

As a result, LD ignition has received extensive attention, and exhibits promising

application especially in the field of aviation and aerospace. Fig. 1 illustrates the schematic diagram of ignition system induced by laser diode. Laser

diode is employed as light source, and energy is transmitted to powders by optical fibers. The powders are ignited and subsequently exploded while enough energy is provided.

powder

fiber

fiber coupler

connectinglaser

aperture

lockdevice

Figure 1. Schematic illustration of ignition system induced by laser diode.

2.2. Optical Fiber and Fiber Coupled Technology

As a carrier to transmit laser, optical fiber plays a crucial role in LD ignition. The materials of optical fiber should possess the favorable characteristics of optical and mechanical properties as well as the characteristic of temperature. The widely used fibers are made of silica glass or plastic. The fibers can be classified into two types, one is step-index fiber and the other is grade-index one according to the distribution of refractive-index of fiber core. The refractive index of core is a constant for step-index fiber, schematically shown in Fig. 2. However, for the grade-index fiber, the refractive index of core gradually decreases outwards along the radial direction. Due to the self-focusing characteristic of the grade-index fiber, the output beam has higher energy density close to the axis of fiber. As a consequence, the laser power density can be enhanced by using the grade-index fiber.


n

y

n2 n1

Cladding

Core z

y

r

φFiber axis

Figure 2. The schematic diagram of step-index fibers.

Both theoretical analysis and experimental results indicate that the increase of power density is considerably favorable to LD ignition. That is to say, the combination of thin diameter, low attenuation, small numerical aperture and grade-index fiber is advantageous to LD ignition. Ewick and coworkers found that the threshold of ignition using grade-index fiber was decreased by around 30% than that using step-index fiber in the ignition experiments of Ti/KClO4 and CP/carbon black.

Generally, optical fibers can be classified into single-mode fibers (SMFs) and multi-mode fibers (MMFs) according to the transmission modes. SMFs exhibit excellent capability in optical communications. And the light energy transmitted by SMFs presents to be Gauss distributions, which means the more centralized energy can be obtained, and is thus favorable to LD ignition. Nevertheless, the diameter of core in SMFs is confined to a large extent.

The fiber waveguide parameters can be expressed as 2/122

21 )( nnkrV −= , where 1n and

2n are the refractive indices of the core and the cladding, respectively, and r is the core radius.

And 0/2 λπ=k is the wave number, where λ0 represents the wavelength in vacuum. Single mode operation is obtained for V <2.405, and it can be observed for the wavelength longer

than λc ( 405.20

== cV λλ ). For example, in order to obtain V =2.405 at λ =10.6 μm, a silver

halide fiber can have core diameter 2r = 8 μm while the normalized difference between the refractive indices of the core and the cladding [ 1211 /)(/ nnnnn −== ΔΔ ] should be 0.1. Alternatively, V =2.405 will be obtained for 2r = 80 μm and Δ = 0.001. In both cases one can observe a SMF operation. In typical silica SMFs, the value is of the order of 0.002. Shalem et al [43] selected to design silver halide SMFs which have 2r = 60 μm and Δ = 0.002, for which the estimated waveguide parameter was at least 10% lower than 2.405 (at λ = 10.6 μm).

To make the laser power transmitted by fiber enough high in the transmitting process, it is necessary to ensure that the fiber has enough thick core, otherwise the fiber will be inevitably damaged by laser. Based on the findings of Schmidt-Uhlig and his colleagues [44], the feasibility of transmitting 20 mW, 5 ns laser pulse from a frequency doubled Nd:YAG laser through a standard 1500 μm multi-mode optical fiber has been demonstrated. Furthermore, the experiments on the delivery of more than 20000 pulses with mean energy of 110 mJ with no damage to fiber have been performed. Consequently, multi-mode optical fibers are prevailing in LD ignition.


Commonly, the specification of optical fiber used in LD ignition is as follows, the diameter of core (2r) of 100-200 μm, numerical aperture (NA) <0.3, attenuation per kilometer dBkm<3dB and output power of fiber P ≥ 1W.

Additionally, the end-face quality of fiber is an important aspect, which includes the perpendicularity of end-face to axes, degree of levelness and cleanliness of end-face, etc. These factors have a significant effect on the transmission of laser. Surface defects and contamination not only debase the transmission efficiency but also result in the strong electric field and large thermal stress in local area under higher power density, and eventually damage fiber itself. Accordingly, the cleansing and polishing treatment to the end-face of fiber is a serious issue. Ewick adopted tailor-made polishing apparatus for fiber to polish the end-face and directly observed it by a microscope with 400× magnifications in order to determine the quality of polishing.

Besides transmitting energy, optical fiber can also be applied in coupling and splitting. The fiber-coupled technology was firstly used in the coupling between fiber and LD. LD emits the elliptic radiation, and thus the proper convergent lens is required to effectively couple laser into output fiber. The difficulty of coupling is greatly increased because of the small fiber core used in LD ignition, and the loss of coupling can even be as high as 5dB. Apparently, it is crucial to enhance the coupling efficiency at coupling sites. The second problem of this technology is the coupling between fibers, which includes two types of coupling: one is the coupling between single and single fiber (one in and one out), and the other is single and multi fiber (one in and several out). The latter is indispensable for multi-point ignition of LD [45]. The coupling between fibers is realized by linkers and commutators. To reduce the loss of coupling, it is necessary to improve the techniques of collimation, tight-fittings, and fixed-airproof. Roman and coworkers [46] used STC linkers in LD ignition to link two fibers for light in and light out, one of which has a core diameter of 100 μm and the other has an outer diameter of 140 μm. The linkers, having an attenuation of as low as 0.56dB and jack diameter of 144 μm, can operate in the temperature range of –40 to +80oC. This is a kind of linker with low attenuation, easy to manipulate, and good performance.

Further, it is also a noticeable issue on the coupling technique between fibers and ignited powders. Kramer et al [47] designed and developed two kinds of components i.e., the fiber foot and the optical window, to resolve the coupling between fibers and ignited powders. These components make LD ignition system more convenient and practical in operation. They are required not only to meet the demand of mechanical strength but also to reduce the energy loss as could as possible. The fiber foot with high mechanical strength is prepared through the following steps, firstly envelop a short fiber into glass preform within a metal shell under high temperatures, and secondly polish two ends of the shell. The advantages of the component are small cross-sections of fiber, high mechanical strength, and meanwhile fiber itself plays a role as waveguide with the characteristic of high transmission quality. However, the disadvantage is the energy attenuation caused by surface reflection and non-collimation.

The optical window is a kind of transparent solid made of glass material, which is fixed between the fiber and ignited powders. There is a little probability resulting from non-collimation. However, the material for windows can absorb laser and disperse radiation of beam, thus leading to the decrease of power density. By selecting a proper material and appropriate thickness of windows combined with the convergence method, the above-


mentioned loss can evidently be decreased. Also, they compared sapphire glass with phosphor glass as window material. When the thickness of window is the same 0.4 mm, the ignition energy for carbon black-doped powders is 3.4 mJ by using sapphire glass as window material, while the ignition energy is 2.3 mJ using phosphor glass as window, and correspondingly the ignition energy is 1.6 mJ with no window. Compared with sapphire glass window, phosphor glass window has a better performance due to its lower thermal conductivity and lower refractive index.

2.3. Experimental Study

The dependence of ignition threshold on diameter of core (2r) can be demonstrated by experimental studies and numerical calculations. The experimental setup, schematically illustrated in Fig. 3, is divided into two main parts: A for ignition part and B for testing one. T-type fiber junction separates the emitted laser into two ways, one of which directly delivers laser into photoelectric detector and then to oscillograph. The other transmits laser to ignitor, where energetic powders are ignited and shone. The time difference of two ways of light reaching the oscillograph is defined as ignition delay time ti, measured by a photoelectric detector. The threshold energy of ignition Ei can be calculated in the following formula:

Ei = Pi·ti (1)

where Pi refers to laser power imported ignitor.

Figure 3. Schematic diagram of experimental setup of laser ignition.

The maximal output power of LD used in the experiments is 1W and the wavelength is 808 nm. The laser is continuously output and is power-adjustable. The diameter of the coupled fibers is 100 μm, 200 μm and 400 μm, respectively. And the powder is Zr/KClO4 with a mass ratio of 1:1. The ignition experiments were carried out at room temperature. Fig. 4 shows the relationship of the ignition threshold and laser power with regard to three types of fibers with different diameters of core.


0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.26.06.57.07.58.08.59.09.5

10.010.511.0

2r =400μm

2r =200μm

2r =100μm

E i /mJ

P/W

Figure 4. The relationship of ignition threshold and laser power.

The three plots in Fig. 4 from bottom to top correspond to the diameter of core of 100 μm, 200 μm and 400 μm, respectively. According to the three plots, one can conclude that the ignition threshold decreases with the increasing laser power when the diameter of core has a fixed value. And the threshold increases as the diameter of core becomes thicker under the same laser power. This suggests that the ignition threshold decreases as the increasing power density under certain conditions.

3. Conclusion

Both theoretical and experimental results indicate that two issues need to be considered for the selection of diameters of core in fiber coupled laser ignition system. One is to enhance power density as could as possible i.e., to select the core with thinner diameters, which can decrease the ignition threshold. And the other is to take into account the endurance of fiber itself i.e., the fiber with too thin diameter of core is not suitable. As a result, the commonly used fibers are multi-mode fibers with diameters of core of 100-200 μm.

References

[1] Litchinitser, N. M.; Sumetsky, M.; Westbrook, P. S. J. Opt. Fiber. Commun. Rep. 2007, 4, 41-85.

[2] Li, X. Y.; Voss, P. L.; Chen. J.; Sharping, J. E.; Kumar, P. Opt. Lett. 2005, 30, 1201-1202.

[3] Hutchinson, D. P.; Richards, R. K. J. Opt. Fiber. Commun. Rep. 2007, 4, 1-11. [4] Wong, K. K.Y.; Lu, G. W.; Chen, L. K. Opt. Commun. 2007, 270, 429-432.


[5] Monro, T. M.; Ebendorff-Heidepriem, H. Annu. Rev. Mater. Res. 2006. 36, 467-495. [6] Cregan, R. F.; Mangan, B. J.; Knight, J. C.; Birks, T. A.; Russell, P. St. J; Roberts, P. J.;

Allan, D. C. Science 1999, 285, 1537-1539. [7] Knight, J. C.; Broeng, J.; Birks, T. A.; Russell, P. St. J. Science 1998, 282, 1476 [8] Benabid, F.; Couny, F.; Knight, J. C.; Birks, T. A.; Russell, P. St J. Nature 2005, 434,

488-491. [9] Talukder, A. I.; Totsuka, K.; Tomita. M. Appl. Phys. Lett. 2006, 89, 054103. [10] Eggleton, B. J.; Kerbage. C.; Westbrook, P. S.; Windeler, R. S.; Hale, A. Opt. Express

2001, 9, 698-713. [11] Villatoro, J.; Minkovich, V. P.; Pruneri, V.; Badenes, G. Opt. Express 2007, 15, 41491-

1496. [12] Hu, J.; Marks, B. S.; Menyuk, C. R.; Kim, J.; Carruthers, T. F.; Wright, B. M.; Taunay,

T. F.; Friebele, E. J. Opt. Express 2006, 14, 4026-4036. [13] Bian, S. P.; Zhang, W. J.; Kuzyk, M. G. Opt. Lett. 2003, 28, 929-931. [14] Akimov, D. A.; Schmitt, M.; Maksimenka, R.; Dukel’skii, K. V.; Kondrat’ev, Y. N.;

Khokhlov, A. V.; Shevandin, V. S.; Kiefer, W.; Zheltikov, A. M. Appl. Phys. B 2003, 77, 299-305.

[15] Skryabin, D. V.; Luan, F.; Knight, J. C.; Russell, P. St. J. Science 2003, 301, 1705-1708. [16] Wolfbeis, O. S. Anal. Chem. 2004, 76, 3269-3284. [17] Lee S. T.; George, N. A.; Sureshkumar, P.; Radhakrishnan, P.; Vallabhan, C. P. G.;

Nampoori, V. P. N. Opt. Lett. 2001, 26, 1541-1543. [18] Hassani, A.; Skorobogatiy, M. Opt. Express 2006, 14, 11616-11621. [19] Minkovich, V. P.; Monzón-Hernández, D.; Villatoro, J.; Badenes, G. Opt. Express 2006,

14, 8413- 8418. [20] Pickrell, G.; Peng, W.; Wang, A. Opt. Express 2004, 29, 1476-1478. [21] Xu, L.; Fanguy, J. C.; Soni, K.; Tao, S. Q. Opt. Express 2004, 29, 1191-1193. [22] Kronenberg, P.; Rastogi, P. K.; Giaccari, P.; Limberger, H. G. Opt. Lett. 2002, 27, 1385-

1387. [23] Santoyo, A. T.; Shlyagin, M. G.; Jimenez, F. J. M.; Oyarzabal, L. N. R. Opt. Commun.

2007, 271, 386-390. [24] Haidekker, M.A.; Akers, W.J.; Fischer, D.; Theodorakis, E.A. Opt. Lett. 2006, 31, 2529-

2531. [25] Baptista, J. M.; Marques, J. M.; Frazao, O.; Santos, S.; Santos, J. L.; Marques, M. B. Opt.

Commun. 2007, 271, 224-227. [26] Wang, X. W.; Xu, J. C.; Zhu, Y. Z.; Cooper, K. L.; Wang, A. B. Opt. Lett. 2006, 31,

885-887. [27] Emiliyanov, G.; Jensen, J. B.; Bang, O.; Hoiby, P. E.; Pedersen, L. H.; Kjær, E. M. Opt.

Lett. 2007, 32, 460-462. [28] Ye, J. Y.; Myaing, M. T.; Norris, T. B.; Thomas, T.; Baker, Jr. J. Opt. Lett. 2002, 27,

1412-1414. [29] Wang, X. W.; Cooper, K. L.; Wang, A. B.; Xu, J. C.; Wang, Z.; Zhang, Y.; Tu, T. Z.

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[33] Korepanov, V. I.; Lisitsyn, V. M.; Oleshko, V. I.; Tsipilev1, V. P. Combust. Expl. Shock Waves, 2006, 42, 94-106.

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Grand Junction, CO, 11-15 July 1988, 449-469. [37] Carleton, F. B.; Krallis, K.; Weinberg, F. J. AD-A212 342, 1989. [38] Barrows, A. W.; Forch, B. E.; Beyer, R. A.; Cohen, A.; Newberry, J. E. AD-A261 049.

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pyrotechnic seminar, 1990, 569-579.

RESEARCH AND REVIEW STUDIES


Chapter 1

EVANESCENT FIELD TAPERED FIBER OPTIC BIOSENSORS (TFOBS): FABRICATION, ANTIBODY

IMMOBILIZATION AND DETECTION Angela Leung1, P. Mohana Shankar2 and Raj Mutharasan1*

1Department of Chemical and Biological Engineering 2Department of Electrical and Computer Engineering, Drexel University,

Philadelphia, PA 19104

Abstract

Tapered Fiber Optic Biosensors (TFOBS) are sensors that operate based on fluctuations in the evanescent field in the tapered region. In our laboratory, TFOBS are made by heat pulling commercially-available single mode optical fibers. They have been investigated for various applications, including measurement of physical characteristics (refractive index, temperature, pressure, etc.), chemical concentrations, and biomolecule detection. In this chapter, an up-to-date review of TFOBS research is provided, with emphasis on applications in biosensing such as pathogen, proteins, and DNA detection. The physics of sensing and optical behavior based on taper geometry is discussed. Methods of fabrication, antibody immobilization, sample preparation, and detection from our laboratory are described. We present results on the non-specific response, simulation, and detection of E.coli O157:H7 and BSA. This chapter will conclude with an analysis of the future direction of the Tapered Fiber Optic Biosensors.

1.0. Introduction

Tapered fiber optic biosensors (TFOBS) are made from optical fibers, and, are capable of detecting specific analytes using optical responses. They have been used for the measurement of physical and chemical properties [4-8], [9-14] of biological molecules [2, 15-19] and have several applications including environmental monitoring, drug screening, clinical diagnostics, and defense. TFOBS offer many advantages including flexibility, ease of use, affordability,

* E-mail address: [email protected]. Tel.: (215) 895-2236. Fax: (215) 895-5837. (Corresponding author)

Angela Leung, P. Mohana Shankar and Raj Mutharasan 16

and ability to perform sensing using a small amount of sample. These sensors are based on the evanescent field associated with fiber, and, often are also referred to as Evanescent Field Tapered Fiber Sensors. In this chapter, the basics of TFOBS are discussed, along with an up-to-date literature review of TFOBS. Experimental methods and recent results from our laboratory are also presented.

2.0. Physics of Evanescent Field Sensing in Tapered Fibers

Optical fibers are cylindrical waveguides, and, are made of a silica core surrounded by a silica cladding. The core refractive index is higher than the cladding refractive index (RI) because it is doped with Ge. Light propagates through the core by total internal reflection (TIR). Besides the light propagating in the core, there is a small component of light, known as the evanescent field, which decays into the cladding.

Evanescent light penetration is described by its penetration depth (dp), which is the position away from the core/cladding interface at which the light decays to 1/e of its value at the core-cladding interface, and is given by:

2 2 22 sin

p

co cl

dn n

λπ θ

=−

(1)

In eqn. (1), λ is the operating wavelength of light, nco the index of the core and ncl the

index of the cladding. The angle of incidence at the core cladding interface is θ. The evanescent field in a uniform diameter fiber does not interact with the outside environment because it decays to a negligible value as it reaches beyond the cladding. This is due to the fact that in typical fibers the cladding thickness is several times that of the core. However, if the cladding is removed or the fiber is tapered down to a diameter less than the original core diameter, evanescent field can interact with the external medium affecting the transmission through the fiber.

The penetration depth in a tapered fiber depends on the local diameter of the tapered fiber, the RI of the core, and RI of the external medium. Since there is a continuous change in the diameter along the fiber in the tapered region (except in the waist), coupling of light among the modes can occur [20]. Coupling in the tapered region causes the transmission properties of the fiber to change. Presence of analytes in the tapered region can lead to RI changes in the taper. This results in changes in the coupling characteristics and causes changes in the optical throughput.

Physical characteristics of the fiber such as RI of the core and cladding, core diameter, and operating wavelength determine the number and type of modes that propagate through the fiber. The lowest order mode has the tightest confinement of the field, and hence the weakest evanescent field. As the mode order goes up, the associated evanescent field also increases. In a tapered or de-cladded fiber, the optical characteristics of the surrounding medium such as its index, absorption, etc. can affect the optical throughput.

Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 17

2.1. Wave Propagation in Absorption Sensors

The shape of the optical field in a fiber is determined by the number of modes present. Figure 1 shows a tapered fiber with a short region of constant thickness (waist) and contracting and expanding regions. The number of modes that can be supported in a fiber is determined by the V-number,

2 202co cl

aV n nπλ

= − (2)

where a0 is the radius of the core. When V<2.405, only the lowest order mode is supported, and as V increases, number of modes increase. Although in a single-mode (SM) fiber only the lowest order is supported, in the tapered region higher order modes can potentially be supported because of the larger difference in refractive index between the core and the sample (~0.12) compared to a regular fiber (~0.01). Reduction of the fiber radius increases the evanescent field strength, and enhances the interaction of the evanescent field with the analyte leading to variations in optical throughput (transmitted light).

Figure 1. Photograph of a TFOBS. The region of interest in a tapered fiber is identified by the region where Vcore<1.

2.2. Wave Propagation in Continuous Bi-conical Tapered Fibers

In our laboratory, tapered fibers were made by heat-pulling an optical fiber without removing the cladding. Unlike uniform fibers, the V-number changes along the length of a tapered fiber. When V-number becomes less than unity, the core is too small to contain the light and light guidance is determined by the original cladding which acts as the core and the external medium of RI next which serves as the cladding. The new V-number is called Vclad where the parameter, a0 in Eq. (2) is replaced by the radius of the overall fiber, b(z), and is given by

2 22 ( )( )clad cl extb zV z n nπλ

= − (3)


In tapered fibers, the V-value is generally referred to as Vclad to distinguish it from Vcore, given in Eq. (2). Note that the diameter b in eqn. (3) is a function of the location (z), indicating the existence of tapering.

2.3. Numerical Simulation of Light Transmission in a Tapered Fiber

Numerical simulation of light transmission through a tapered fiber can provide useful insight into its properties. To simplify the analyses, the simplified mode theory based on linearly polarized modes (LP) can be used to determine the transmission behavior [21]. Assuming that light enters into the fiber parallel to the axis, the only modes that are excited are the LP0m modes. The transverse components of the electrical field inside the fiber are:

( )( )

0 0

0 00 0

0 0

( ), 1( )

( ), 1

m

Xm

J U R RE r A J U

K W R RK W

≤⎧ ⎫⎪ ⎪= ⎨ ⎬>⎪ ⎪⎩ ⎭

(3)

where R is the normalized radial coordinate, r/a0, whereas U and W are constants They depend on the wavelength and RI of the core and cladding,

0

22 2 2 2

0 02

m co mU a nπ βλ

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

(4a)

0

22 2 2 2

0 02

m m clW a nπβλ

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

(4b)

where c is the speed of light and β is the propagation constant. When m=1, only the fundamental mode exists. In eqn. (3) J0(.) and K0(.) are the Bessel and modified Bessel functions of zeroth order, respectively. A is a constant determined from orthogonality principle [21]. The subscript m represents the various circularly symmetric LP0m modes that may be present in the fiber.

When Vcore<1 and Vclad>2.405, many modes are supported since the index difference between the cladding and the external medium (next) is large. As mentioned previously, tapering leads to coupling among LP0m modes [20, 22] . A simple means to visualize the taper is to model taper geometry by approximating the slopes of the taper by stepwise linear approximation. At each step ‘i’, the parameters U0m and W0m are analogous to the constants U and W in a uniform diameter fiber. They can be expressed using the local radius iρ as

0 0

22 2 2 22

m m

i i iclU nπρ β

λ⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

(5a)


0 0

22 2 2 22

m m

i i iextW nπρ β

λ⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦ (5b)

The V-number for each step is given by:

1/ 22 22i i

cl extV n nπ ρλ

⎡ ⎤= −⎣ ⎦ (6)

The values of U, W and β are calculated using the LP mode approximation [21]. The

relationship between the modal amplitudes of the LP0m modes of the ith and (i+1)th step is:

1 11 1

1 1( ) ( )

i ii iqm j zj zi i i i

m m q qm q

A E r e B E r e αβ + +−− + +

= =

=∑ ∑ (7)

where E(r) is the electric field, βm is the propagation constant on the left and αq is the propagation constant on the right. It has been assumed that E(r) are orthonormal [21]. Am is the amplitude of the modes on the left and Bq is the amplitude of the modes on the right. That is,

22

0 0( ) 1E r rdrd

πφ

∞=∫ ∫ (8)

The amplitude on the right is obtained by applying the orthogonality principle:

1 11

;

+ +− −+ = ∑∑i i i iq mj z j zi i

q m nm pqn m

B e A e Cα β (9)

1; 0

2 ( ) ( )∞ += ∫ i i

nm pq q mC E r E r rdrπ (10)

In the tapered region, light is coupled among the various LP0m modes. When Vcore=1,

power in the LP01 cladding mode is transferred to LP01 core mode and appears at the output end of the fiber. The light remaining in other modes stays in the cladding and is lost. A MATLAB® program was used to estimate the amplitude and output power. The taper geometry, wavelength and number of steps were varied to determine the resulting changes in power.

Sample simulation results, illustrated in Figure 2, show changes in transmission vs. waist diameters for two taper geometries. In Panel A, the taper geometry resembles a symmetric taper made by the fusion splicer, while in Panel B, the simulation is for a long taper similar to a heat-drawn taper. The transmission is normalized with respect to air, so that a value of 1.2 indicates a transmission increase of 20% in water compared to air. Figure 2 show that as the waist radius increases, the difference in transmission between water and air decreases. However, at intermediate values the ratio may be higher or lower than unity, particularly at smaller waist diameters. For certain values of the radius, the transmission through water is


higher than in air. At a longer wavelength (550 nm, for example) and for the same waist diameter values, the difference in transmission between air and water differ by less than 10%. To explore further, simulation was undertaken by varying the diameter of the waist in much smaller steps of 0.001 μm. These results are shown in Figure 3 for two starting diameters, 5 μm and 6.25 μm. The transmission characteristics change significantly for small changes in diameter. For example, a 5.54 μm diameter taper exhibits 30% higher transmission in water

0.60.70.80.9

11.11.21.31.41.51.6

5 9 13 17

Waist diameter (μm)

Nor

mal

ized

Tra

nsm

issi

on

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

5 9 13 17


Nor

mal

ized

tran

smis

sion

0.60.70.80.9

11.11.21.31.41.51.6

5 9 13 17


Nor

mal

ized

Tra

nsm

issi

on

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

5 9 13 17


Nor

mal

ized

tran

smis

sion

Figure 2. Transmission in water normalized with respect to air as waist diameter is altered. Top: A short symmetric taper: a= 0.425 mm, b=0.325 mm, c=0.500 mm. Operating wavelength = 470 nm. Bottom: A longer asymmetric taper: a=2.25 mm, b=0.245 mm, c=4.5 mm. Operating wavelength = 550 nm. Adapted from [3].


while a 5.58 μm waist diameter taper transmits 20% less transmission. The differences, however, become smaller for larger diameters. The example of 6.25 μm in Figure 3 shows that the changes in transmission were less than 10 %.

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.02 0.04 0.06 0.08 0.1

Change in waist diameter (mm)

Nor

mal

ized

Tra

nsm

issi

on

at 6.25 mm

at 5.5 mm

Figure 3. Transmission characteristics of a taper (a= 2.25 mm, b=0.25 mm, c=4.5 mm) in water at 470 nm as a function of change in waist diameter. Transmission is normalized with respect to transmission in air at the corresponding geometric values. Smaller starting diameter tapers show large changes in transmission for small (0.01 μm) changes is waist diameter. Adapted from [3].

These simulation results can serve as a guide to the analysis and interpretation of the experimental data on the tapered fibers. It is important to recognize that in the simulation we considered only the effect of refractive index in the waist region. In actual sensing experiments, the cells absorb at the operating wavelength, and the resulting sensor response is a complex interplay of these two phenomena. Furthermore, cells do not have homogeneous RI because the cells constitute particulate matter. Finally, the cell attachment onto the taper surface is often not uniform as we showed in our earlier report [2].

3.0. Literature Review

In this section, the applications of TFOBS for pathogen detection, toxins measurements, clinical measurements, and DNA detection are presented. In tables 1 to 3, we summarize the analytes detected, matrices in which they were detected, detection principle, basis of sensors, and detection limits.

Table 1. TFOBS For Pathogen and Toxin Measurement

Target Analyte LOD Matrix Taper Geometry Fiber Type Detection Principle References

Bacillus anthracis 3.2E5 spores/mL buffer BT Polystyrene MM Fluorescent sandwich assay [32]

Bacillus subtilis var. niger 8 x 10(4) spores/mL buffer NA (chip) NA (chip) Leaky wave (SPR) [59]

LacZ DNA in Escherichia coli 25 pM buffer Uniform MM Fluorescent intercalating agents [55]

Staphylococcus aureus Protein A 1 ng/mL ND ND MM plastic Fluorescent sandwich assay [35]

Escherichia coli O157:H7 0.016 dB/h/No, Initial number (No): 10-800 *

buffer BT MM Absorption [23]

Escherichia coli O157:H7 70 cells/mL Buffer BT SM Intensity [2]

Escherichia coli O157:H7 1 CFU/ml ground beef samples Uniform MM polystyrene Fluorescent sandwich assay [25, 26]

Salmonella 50 CFU/g

irrigation water used in the sprouting of seeds

RAPTOR – uniform Waveguide Fluorescent sandwich assay [27]

Salmonella 10(4) CFU/ml Hotdog samples


Salmonella 10(4) CFU/mL Nutrient broth TT MM Fluorescent sandwich assay [28]

Table 1. Continued


Staphylococcal Enterotoxin B Min: 0.5 ng/ml (buffer)

Buffer, human serum, urine, and aqueous extract of ham

CTT ND Fluorescent sandwich assay [33]

C. Botulinum toxin A, Pseudexin Toxin

Min: 30 pM (C. Botulinum toxin A), 60 pM (Pseudexin)

ND CTT MM Fluorescent sandwich assay [61]

Clostridium-Botulinum Toxin-A 5 ng/mL buffer TT MM Fluorescent sandwich assay [36]

E. coli lipopolysaccharide endotoxin Min: 10 ng/ml Buffer and

plasma CTT ND Fluorescent sandwich assay [38]

Ricin Concentration

Min: 100 pg/ml (buffer) Max: 1 ng/mL (river water)

Buffer, river water CTT MM (plastic clad

silica) Fluorescent sandwich assay [37]

Listeria monocytogenes 5 x 10(5) CFU/ml

frankfurter sample


Listeria monocytogenes 5.4 x 10(7) CFU/ml

Hotdog samples


Listeria monocytogenes 4.3x10(3) CFU/ml Buffer Uniform MM polystyrene Fluorescent sandwich assay [29]

Abbreviations: BT = Biconical Taper, TT = Tapered Tip, CTT = Combination Taper Tip, SM = Single Mode, MM = Multimode, ND = Not Described, * = the change in dB per hour per number of cells at inoculation, NA = Not Applicable.

Table 2. TFOBS For Biochemical Measurements


NADH, NADPH Concentration

Min: 0.2 μM (NADH), 0.5 μM (NADPH)

buffer BT SM Absorption [18]

Chinese Hamster Ovary Cell Concentration

Min: 105 cells/ml buffer BT SM Absorption [18]

Paraoxon Sub ppm buffer TT MM Chemiluminescence [62]

STAT3 ND Buffer Uniform MM Fluorescent sandwich assay [63]

Abbreviations: BT = Biconical Taper, TT = Tapered Tip, CTT = Combination Taper Tip, SM = Single Mode, MM = Multimode, ND = Not Described.

Table 3. TFOBS For Clinical Measurements


Protein A 1 μg/mL ND NA (chip) NA Leaky wave (SPR) [64]

BSA 10 fg/mL Buffer BT SM Intensity [1]

BSA 7.4 ng/mL buffer Chip (NA) Chip (NA) SPR [65]

BSA 2.5 μg/ml Buffer BT ND (plastic clad silica)

Dye-protein complex absorption [43]

Ovalbumin 2.5 μg/ml Buffer BT ND (plastic clad silica)


Table 3. Continued


Hemoglobin 2.5 μg/ml Buffer BT ND (plastic clad silica)


IgG 20 fM Buffer TT MM Fluorescent competitive assay [44]

IgG 75 pg/mL Serum and jejunal fluids diluted with buffer

BBT SM Fluorescent sandwich assay [45]

Protein C 0.1 μg/mL Buffer TT ND Fluorescent sandwich assay [16]

Protein C 0.5 μg/mL Plasma TT MM Fluorescent sandwich assay [46]

Protein C 0.5 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66]

Protein S 0.5 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66]

Antithrombin III (ATIII) 30 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66]

Plasminogen (PLG) 30 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66]

B-type natriuretic peptide (BNP) 0.1 ng/mL Plasma Uniform MM Fluorescent sandwich assay [66]

cardiac troponin I (cTnI) 1 ng/mL Plasma Uniform MM Fluorescent sandwich assay [66]

C-reactive protein (CRP) 1 μg/mL Plasma Uniform MM Fluorescent sandwich

assay [66]

Table 3. Continued


Myoglobin (MG) 75 ng/mL Plasma Uniform MM Fluorescent sandwich assay [66]

L. donovani Antibody Concentration Min: 0.244 ng/ml Serum CTT

MM (plastic clad silica)

Fluorescent sandwich assay [17]

Progesterone ng/mL Buffer ND ND Fluorescent sandwich assay [42]

Adriamycin 0.01 μg/mL blood Straight core tip MM Fluorescence quenching [49]

Cytochrome c ND Cell TT MM Fluorescent sandwich assay [50]

Cytochrome c 2.5 μg/ml buffer BT ND (plastic clad silica)


Yersinia pestis fraction 1 50 ng/mL Buffer, serum, plasma, and whole blood

BT MM Fluorescent sandwich assay [15]

cTnI 31 pM plasma TT MM quartz Nano gold particle enhanced fluorescence [67]

BNP 26 pM plasma TT MM quartz Nano gold particle enhanced fluorescence [67]

Intracellular Benzopyrene Tetrol 6.4 pM cell TT ND Autofluorescence [51] Benzo\c\phenanthridinium alkaoids ND buffer Chip NA SPR [68]

Fumonisin B1 10 ng/ml methanol/water-extracted corn TT

MM (plastic clad silica)

Fluorescent sandwich assay [69]

Myoglobin 2.9 ng/mL buffer tip MM SPR [39]

Table 3. Continued


Myoglobin 5 nmol/L buffer Uniform probe MM Fluorescent Energy Transfer [40]

Thrombin 1 nM Buffer Spheres (NA) NA Fluorescent competitive assay [48]

Thrombin 1nM Buffer Uniform MM

Coagulation of fluorescently labeled fibrinogen to unlabelled fibrinogen bound to the surface of the fibre optic

[47]

RNA pM Buffer TT SM Fluorescence [56]

DNA 70 fM Buffer Uniform MM Fluorescence [52]

interleukin-1 (IL-1), interleukin-6 (IL-6), and tumor necrosis factor-a. (TNF-alpha)

1 ng/mL Buffer and spiked cell culture medium (CCM)

ND MM Fiber-optic surface plasmon resonance (SPR)

[41]

DNA 5 nM buffer ND MM Fluorescence [53]

Abbreviations: BT = Biconical Taper, TT = Tapered Tip, CTT = Combination Taper Tip, SM = Single Mode, MM = Multimode, ND = Not Described, NA = Not Applicable.


3.1. Pathogen Detection

Escherichia coli O157:H7 [2, 23-26], Salmonella typhimurium [27, 28], Listeria monocytogenes [29-31], and Bacillus anthracis [32] are some of the pathogens which have been detected using TFOBS. Most pathogen detection studies done to date used fluorescence TFOBS [25-32], but a few of them used intensity-based TFOBS [2, 23, 24].

Ferreira et al. developed an intensity-based evanescent sensor, to be used with a 840 nm light source, to detect Escherichia coli O157:H7 growth [23]. This evanescent sensor was fabricated by chemically etching. Transmission is reduced due to light absorption by the bacteria, and the power loss is proportional to the intrinsic bulk absorption and scattering, which depends on the concentration of the bacteria. The sensitivity of this sensor was 0.016 dB / hour-No, where No is initial cell concentration and ranges from 10 to 800. Similarly, Maraldo et al. used TFOBS to detect Escherichia coli JM 101 growth on poly-L-lysine [24]. E.coli JM 101 expressing green fluorescent protein was immobilized on the poly-L-lysine coated fibers, and growth was monitored by light transmission at 480 nm. The transmission decreased exponentially with cell growth on the tapered surface. In a follow up study by Rijal et al., Escherichia coli O157:H7 (EC) was covalently bonded to the surface of a TFOBS via an antibody, and concentrations as low as 70 cells/mL was detected by changes in intensity at 470 nm [2]. Detection of EC in real samples is of great interest and was investigated by DeMarco et al. [25]. EC in seeded ground beef samples was prepared and detected by a sandwich immunoassay using cyanine 5 dye-labeled polyclonal anti-E. coli O157:H7. Light was launched at 635 nm and the fluorescence was emitted at 670 to 710 nm. Responses were obtained within 20 minutes, and E. coli O157:H7 at 3 to 30 CFU/mL were detected. A similar study was recently conducted by Geng et al., where a sandwich immunoassay was used with FOBS to detect EC in ground beef [26]. Light was launched at 635 nm and the fluorescence was emitted at 670 to 710 nm. The sensor detected 10(3) CFU/ml of pure cultured EC grown in culture broth. Artificially inoculated EC at concentration of 1 CFU/ml in ground beef was detected after 4 hours of enrichment.

Kramer et al. [27] studied the detection of Salmonella typhimurium in sprout rinse water using RAPTOR™, an evanescent fluorescence sensor developed by Research International, Monroe, Washington.. Alfalfa seeds contaminated with various concentrations of Salmonella typhimurium were sprouted, and the sprout water was measured by the instrument. Salmonella typhimurium was identified for seeds that were contaminated with 50 CFU/g. Zhou et al. [28] also used a sandwich immunoassay to detect Salmonella. Light was launched at 650 nm and the fluorescence was emitted at 680 nm. Tapered fiber tips with various geometries and treatments were studied and optimized, and Salmonella was detected at 10(4) CFU/mL.

An antibody-based sandwich fluorescence FOBS was developed by Geng et al. to detect Listeria monocytogenes [29]. Light was launched at 635 nm and the fluorescence emission was in the range of 670 to 710 nm. The sensor was specific, as shown by the significantly lower signals caused by other Listeria species or microorganisms. The LOD was 4.3x10(3) CFU/ml for a pure culture of L. monocytogenes. In less than 24 h, L. monocytogenes in hot dog or bologna was detected at 10 to 1,000 CFU/g after enrichment. Recently, Kim et al. also detected L. monocytogenes using the RAPTOR™ sensor [30]. This method achieved a LOD of 5.4 x 10(7) CFU/ml. L. monocytogenes was detected in phosphate buffered saline (PBS) by Nanduri et al. using RAPTOR™ to evaluate the effect of flow on antibody immobilization


[31]. Light was launched at 635 nm and the fluorescence was emitted at around 670 nm. It was found that both the static and the flow through mode method had a LOD of 1 x 10(3) CFU/ml. However, the effective disassociation constant and the binding valences for static modes were higher than for flow through method of antibody immobilization. The flow through mode was chosen to test real samples, and the LOD was 5 x 10(5) CFU/ml.

Bacillus anthracis, is a serious threat to national security. Tims et al. addressed the need to detect Bacillus anthracis, and achieved detection at a concentration of 3.2 x 10(5) spores/mg in spiked powders in less than 1 hour [32]. The method used was based on fluorescent sandwich assay and a polystyrene tapered fiber. The excitation wavelength was 635 nm.

3.2. Toxin Measurement

TFOBS have been used to detect toxins such as enterotoxins [33-36], ricin [37], and endotoxins [38]. Fluorescence was used for all the toxins measurements which are discussed here.

Staphylococcal enterotoxins are a major cause of food poisoning. Tempelman et al. quantified Staphyloccoccal enterotoxin B (SEB) in a fluorescent sandwich immunoassay on a fiber optic biosensor [33]. A 635 nm diode laser was used to excite the labeled antibody. The fluorescence level was measured and gave a detection limit of 0.5 ng/mL. Shriver-Lake et al. used an array biosensor to detect SEB at a LOD of 0.5 ng/mL in buffer and six different types of food samples [34]. Staphylococcus aureus is the only species which produces protein A and was detected by Chang et al. using a fluorescent sandwich FOBS at a LOD of 1 ng/mL [35]. Excitation of this sensor was at 488 nm. Similar to SEB, Clostridium botulinum toxin A was detected by a fluorescent sandwich FOBS at 5 ng/mL [36]. A light source at 514 nm was used in this case.

Narang et al. reported a sandwich fluorescent TFOBS ricin detection in buffer and in river water [37]. The light source was 635 nm. Antibody to ricin was immobilized onto tapered fiber surface using silanization and avidin-biotin linkage. The avidin-biotin method had a higher sensitivity and wider linear dynamic range. The response of the avidin-biotin sensor was linear in the range of 100 pg/mL to 250 ng/mL. The LOD for ricin in buffer solution was 100 pg/mL, and in river water it is 1 ng/ml. At concentrations greater than 50 ng/ml, there was a strong interaction between ricin and avidin due to the lectin activity of ricin. This interaction was reduced for fibers coated with neutravidin or with the addition of galactose.

James et al. developed a method to detect lipopolysaccharide (LPS) endotoxin, which is the most powerful immune stimulant and causes sepsis [38]. LPS from E. coli was detected at a LOD of 10 ng/mL using fluorescent FOBS based on the competitive assay. Polymyxin B was used as a recognition molecule and was covalently immobilized onto the surface of the probe. Fluorescent labeled LPS was introduced to the fiber and attached to the Polymyxin B. Unlabeled LPS was then introduced and competed with the labeled LPS for the binding sites on the Polymyxin B. As LPS concentration increases, fluorescence decreases.


3.3. Clinical Measurements

Most clinical measurements done with TFOBS used proteins as analytes. Notable examples include cardiac markers [39, 40], cytokines [41], and hormones [42]. Investigators have detected model proteins using TFOBS in order to characterize TFOBS’ potential. Preejith et al. detected model proteins using fiber optic evanescent wave spectroscopy [43]. They immobilized Comassie Blue on a multimode fiber surface using a porous glass coating. Comassie Blue normally absorbs at 467 nm, but it forms a dye-protein complex with the protein when exposed to an acidic environment, and such a complex absorbs at 590 nm. The protein concentration is inversely proportional to the output power at 590 nm, because increase in protein concentrations causes the evanescent absorption to increase. Calibration curves were obtained for BSA, hemoglobin, ovalbumin, and cytochrome c in the range of 0 to 20 μg/mL. In our laboratory, BSA was recently detected at 10 fg/mL in stagnant condition using intensity-based TFOBS [1]. Tromberg et al. detected antibody to IgG at 20 fM on a fluorescent FOBS tip using a competitive assay [44]. Light was launched at 488 nm and the fluorescence was emitted at 520 nm. Rabbit IgG was immobilized on the fiber tip, and exposed to fluorescein isothiocyanate (FITC) labeled and unlabeled anti-IgG. The response was inversely proportional to the amount of unlabeled anti-IgG, because the unlabeled anti-IgG displaced the labeled one. Hale et al. developed a fluorescent optical fiber loop sensor to detect antibody to IgG [45]. The sensor was used with a two-step sandwich assay. IgG was labeled with the fluorescent dyes fluorescein isothiocyanate or tetramethyl rhodamine. Antibody to IgG was detected at 75 pg/mL with this method.

Deficiency in Protein C (PC), if left untreated, may result in thrombotic complications, and, thus presents an important clinical challenge. Spiker et al. detected PC at 0.1 μg/mL in buffer using a sandwich fluorescent fiber optic sensor [16]. Real-time detection of PC in plasma is an important challenge in the clinical setting. Convective flow plays a vital role in the transport of PC in a viscous medium such as plasma. Tang et al. who examined PC detection in plasma with fluorescent sandwich FOBS and obtained a detection limit of 0.5 μg/mL [46].

Cardiac markers myoglobin (MG) and cardiac tropinin I (cTnI) can be measured to predict the occurrence of myocardial infarction, because they are released from cardiac muscles when they are damaged. A fiber-optic SPR sensor was developed by Masson et al. to detect MG and cTnI at 3 ng /mL [39]. A direct fluorescence FOBS was also used to detect myoglobin at 5nM [40]. An excitation wavelength of 425 nm was used to excite the Cascade Blue-labeled antibody, which was entrapped in the sensing element and fluoresces at 425 nm. Fluorescence quenching occurred when myoglobin attaches to the labeled antibody. Recently, Tang et al. developed a fiber-optic multi-analyte system which simultaneously quantifies two groups of multi-biomarkers related to cardiovascular diseases (CVD): anticoagulants (protein C, protein S, antithrombin III, and plasminogen) for deficiency diagnosis; and cardiac markers (B-type natriuretic peptide, cardiac troponin I, myoglobin, and C-reactive protein) for coronary heart disease diagnosis.

Garden et al. detected thrombin at 1 nM using fluorescent FOBS [47]. Excitation was at 495 nm and emission was at 520 nm. Unlabeled fibrinogen was first attached to the FOBS surface. Then, coagulation of solution phase fluorescently labeled fibrinogen to unlabelled fibrinogen bound to the surface was observed. Lee et al. detected thrombin at 1 nM using a


fluorescent FOBS immobilized with an antithrombin DNA aptamer receptor [48]. The aptamer was immobilized on the surface of silica microspheres, which were distributed in microwells on the distal tip of an imaging fiber that was coupled to a modified epifluorescence microscope system. Another set of microspheres was prepared with a different oligonucleotide to measure the non specific binding. The distal end of the imaging fiber was incubated with fluorescein-labeled thrombin (F-thrombin), and the non-labeled thrombin was detected using the competitive method.

Progesterone was found to have evidence of carcinogenicity based on animal studies. Progesterone can be found in various surface waters commonly used for drinking water. In a study by Tschmelak et al., a fluorescence FOBS was immobilized with a labeled-antibody and used successfully to detect progesterone at concentrations lower than ng/L [42].

A fluorescent tip FOBS was used to measure adriamycin (ADM) at 10 ng/mL in vivo in a blood vessel [49]. A polymeric fluorescent D-70 membrane with pore sizes of 1-2 μm was immobilized on the fiber tip. Fluorescence was quenched by ADM present in the blood and the fluorescence signal was measured by a photomultiplier tube (PMT) at a wavelength of 530 nm.

The protein cytochrome c is involved in apoptosis and was detected by a sandwich fluorescent nanobiosensor fabricated by Song et al. [50]. δ-Aminolevulinic acid (5-ALA), a photodynamic therapy (PDT) drug, was activated by a He-Ne laser at 632.8 nm to induce apoptosis in MCF-7 human breast carcinoma cells. When mitochondria are damaged by PDT, cytochrome c is released into the cytoplasm; therefore cytochrome c concentration is an indication of apoptosis. Results indicate that 5-ALA PDT-treated cells had a much higher fluorescence signal, pointing to high cytochrome c concentrations in the treated cells.

Yersinia pestis is an etiologic agent of plague. A sandwich fluorescent FOBS devised by Cao et al. was used to detect Yersinia pestis Fraction 1 antigen at a limit of 5 ng/mL [15]. The light source was a 514 nm argon ion laser. This system detected 50 – 400 ng/mL of protein in serum, and the results were in excellent agreement with ELISA results.

Nath et al. developed a fluorescent FOBS to detect L. donovani specific antibodies [17]. The sensor was made by de-cladding an optical fiber so that the evanescent wave propagated outside the tapered region. The sensor was used with a 488 nm light source. Cell surface protein of L. donovani was immobilized covalently on the sensing region. Then, the sensor was incubated with patient serum for 10 minutes, followed by incubation with goat anti-human IgG tagged with FTIC, which excites at 525 nm. The amount of L. donovani specific antibodies in the patient serum was proportional to the fluorescence. There were no false positive results from leprosy, tuberculosis, typhoid, and malaria serum.

Cullum et al. detected benzo[a]pyrene tetrol (BPT) at 6.4 ± 1.7 E pM in mammary carcinoma cells using a sandwich fluorescent fiber-optic nanosensor tip [51]. BPT is a metabolite of benzo[a]pyrene. Using a 325 nm light source, the authors were able to calibrate the sensor and obtain an unknown concentration by observing the level of fluorescence. This technique is useful for cancer screening since carcinogens bind to DNA and form substances such as BPT.

Three cytokines related to chronic wound healing are interleukin-1 (IL-1), interleukin-6 (IL-6), and tumor necrosis factor-α (TNF-alpha) [41]. A fiber-optic SPR sensor was modified with antibodies at the surface, and detected these proteins with LOD of 1 ng/mL in buffered saline solution and spiked cell culture medium (CCM).


3.4. DNA Hybridization

Kleinjung et al. detected DNA hybridization at 3.2 attomoles (70 fM) using a fluorescent multimode FOBS with 13-mer probe attached to the de-cladded core [52]. The complementary strands were labeled and detected when introduced to the sensor. This sensor was able to distinguish between matching sequences, single nucleotide mismatch, and mismatch caused by additional deviations.

Zeng et al. examined the interfacial hybridization kinetics of oligonucleotides immobilized onto silica using a fluorescent FOBS that was excited at 632 nm [53]. A dT20 DNA probe was used as recognition molecules, while target fluorescein-labeled non-complementary DNA (ncDNA) dT20 and fluorescein-labeled dA20 were detected. The target DNA concentrations were 5 nM to 0.1 μM. The response of the sensor fit the second order Langmuir model.

Molecular beacons (MB) are oligonucelotide probes that fluoresces upon hybridization with target DNA or RNA molecules [54]. Liu et al. immobilized MB on a fluorescent FOBS and determined the effects of ionic strength and target DNA concentration on hybridization kinetics. Using an excitation wavelength of 514 nm, they found the LOD was 1.1 nM of DNA. The sensor showed selectivity by distinguishing between 100 nM of ncDNA, 100 nM of one-base mistmatch, and 100 nM of cDNA [54].

3.4.1. Pathogen Detection via DNA A fluorescent FOBS was developed by Almadidy et al. to detect short sequences of oligonucleotides that identify E. coli microbial contamination [55]. DNA probes were first immobilized to silica surface via a silane reagent. Then, stepwise synthesis of oligonucleotides by the β-cyanoethyl-phosphoramidite protocol took place on the surface. The sensor was exposed to both complementary (cDNA) and non-complementary (ncDNA) 20-mers, as well as genomic DNA from E.coli. The cDNA and ncDNA were introduced at a concentration of about 1.7 nM, whereas genomic DNA was introduced at 1.7 pM to 170 pM. Fluorescent intercalating dye was used to detect hybridization. Quantities as low as 100 fM were detected using this method.

Pilevar et al. detected Helicobacter pylori total RNA using a fluorescent FOBS that had probes immobilized on its surface [56]. IRD-41 is a near-infrared fluorophore which is excited by 785 nm light. Real-time hybridization measurement of IRD 41-labeled oligonucleotide at various concentrations to the surface bound probes was performed. Complementary DNA at lower than nM concentration was detected. Sandwich assays were performed with Helicobacter pylori total RNA, and results showed that this sensor could detect H. pylori RNA in a sandwich assay at 25 pM.

4.0. Methods

4.1. Fabrication

Corguide fibers (Corning Glass Works, NY, attenuation at 1300 and 1500 nm of 0.36 and 0.26 dB km−1, respectively) with a core diameter of 8 μm and total diameter of 125 μm were used in all the fabrication methods described here. The fabrication methods commonly used


in our laboratory are chemical etching, heat pulling by flame, and heat pulling by fusion splicer.

4.1.1. Chemical Etching Chemical etching using hydrofluoric acid (HF) is one of the simplest ways to create tapers with a step change in radius. Acrylic (Plexiglas) was used to construct the etching reactor because HF does not attack most plastic materials. In order to monitor the etching, a spectrofluorometer was used to detect the transmission through the fiber as etching took place. This instrument has a compact 75 W Xenon arc lamp (Ushio Inc., Japan) coupled to a monochromator and a PMT (model R1527P in housing 710, PTI Inc.) coupled to a monochromator.

The plastic sheathing of the fiber was removed by immersing the fiber in acetone (Fisher Scientific) for 15–20 min followed by mechanical removal with a fiber optic stripper (NO-NIK). A fiber optic cleaver (NO-NIK) was used to make a clean-cut fiber tip so as to enhance the efficiency of light collection into the fiber.

HF (Fisher Scientific, Philadelphia) at a concentration of 49.5 wt.% was used. Two hundred microliter of HF was introduced into the reaction chamber. Once HF was injected, the transmission was monitored at 350 nm. When the diameter of the fiber was etched to a certain fraction of the initial diameter, the etching process was stopped by first removing the HF and then washing the chamber twice with 5 N NaOH as rapidly as possible. The fiber was then immersed in a 200 mL of 0.1 N NaOH bath for 60 min to stabilize the fiber. If this step was not carried out, any remaining HF would have continued to etch the fiber until it dissolved completely. It was found that the length of the etching time needed at room temperature was about 40 min.

4.1.2. Heat Pulling Using a Manual Propane Torch Heat pulling using a micro-propane torch is another method of obtaining tapers. The apparatus for this method is illustrated in Figure 4. The polymeric sheathing of a 30-35 cm long fiber was removed similarly as in the chemical etching method. The fiber was then mounted on the apparatus with two paper clips of identical weights (2.8 g) on either ends to provide tension to the fiber. A micro-flame (Model 6000, Microflame, Inc., MN) was positioned such that the fiber was approximately one third distance from the top end of the visible end of the flame, and the flame was removed as soon as the paper clip touched the stop. The ends of the tapered fiber were cleaved using a fiber-optic cleaver (NO-NIK) to give clean cut ends. The fiber was placed in an optical fiber holder to be used in the experiments. The dimensions of a fiber were measured after taking micrographs of the taper using an IMT-2 optical microscope (Olympus, Japan) equipped with a video camera (Cohu Corp., Japan) linked to a computer. The dimensions were measured in the Scion Image software (Scion Corp., MD) after a calibration was performed according to the microscope objective used.


Fiber

Heat

4 mm

2 mm

MagnificationMicroflame torch

Weight

Fiber Tapering stand

Fiber

Heat

4 mm

2 mm

MagnificationMicroflame torch

Weight

Fiber Tapering stand

Figure 4. Some of the fibers discussed in this paper were fabricated using a simple fiber tapering device. Fiber without sheathing was mounted on a stand and two pieces of weights were attached to the ends to provide tension required for tapering. The fiber was then heated with a flame while carefully monitoring the diameter of the taper. When the desired diameters were reached, the two ends of the fibers were clipped and the tapered fiber were placed on an optical fiber holder to be used in the experiments. Adapted from [2].

4.1.3. Heat Pulling Using a Fusion Splicer The polymeric sheathing was removed over a distance of 5 cm at the center and both ends of the fiber. The fiber was cleaned with isopropanol and the ends were cut clean using a fiber cleaver (Ericsson EFC 11-4). The fiber was inserted into the programmable fusion splicer (Ericsson FSU975), where electric current was applied via a pair of electrodes for up to 60 seconds while the taper was pulled automatically. Various current levels (3-13 mA) and pull times (2-30 s) were used to produce fibers of varying taper diameters and lengths. A micrograph of the fiber was taken via a camera inside the fusion splicer, and the dimensions were measured in the Scion Image software (Scion Corp., MD).

4.2. Optical Characterization of Tapers

4.2.1. Preliminary Characterization Using Water The preliminary characterization method used in our laboratory for determining the evanescent field strength is the comparison of the transmission in water to that in air. The reason for the choice of these two media was that they provide the most difference in refractive index that is expected to be present in the waist region for biological samples. If a


taper exhibited little or no transmission change going from air to water, its transmission was not expected to change significantly at the presence of dilute analyte solutions.

4.2.2. Characterization in the Visible Range Using E.coli JM101 Characterization in the visible range was performed on fusion spliced and torch heat-drawn tapers using E.coli JM101 (ECJ) as the analyte. Tapers that showed little or no transmission change in water compared to air, also showed no or low response to the presence of ECJ suspensions. Both symmetric and asymmetric tapers of small and large waist diameters had this behavior.

Several tapers exhibited a significant transmission difference (~50%) in water compared to air. These tapers also showed little or no change in the presence of the ECJ. Small RI changes due to the presence of ECJ suspension were not sufficient to produce the transmission changes, resulting in poor sensitivity. Any impact on the light through the fiber was already saturated from change due to water itself, such that the presence of ECJ suspension had little further impact. Other tapers that had this characteristic property, but were of smaller waist diameter showed weak sensitivity. Most of such tapers were symmetric tapers. On the other hand, the asymmetric tapers showed lower relative transmission through water, but allowed further modulation in transmission from the presence of ECJ.

We compared relative transmission at 470 nm for fusion splicer tapered fibers. In general, there were two types of responses. In the first type, the transmission increased or decreased monotonically, as ECJ concentration increased. In the second type, an initial increase for low cell concentration is followed by a decrease at higher cell concentration. There were tapers which also showed an increase in transmission at low ECJ concentrations, followed by decrease at intermediate concentrations and then an increase at 7 million cells/mL.

Heat drawn tapers have typically a much longer convergent, waist and divergent sections, each on the order of millimeters. Similar to the fusion spliced tapers, HD tapers showed two basic characteristics. In one case, tapers showed a decrease in transmission as concentration increased. In the other case, tapers showed a slight increase and then a decrease in transmission for higher concentrations. At low concentration, the RI of cellular suspension influenced transmission response, and caused the increase in transmission. At higher concentrations, the evanescent light absorption by the cells dominated the response. These results suggest that torch-drawn tapers have excellent potential as biosensors.

4.2.3. Characterization in the RI Range Using Glucose Solutions In order to characterize the tapered fibers in the IR region, we measured transmission properties under various RI fluids in the tapered region, using an experimental setup similar to Figure 5 but in flow condition. The transmissions at 1310 and 1550 nm were monitored and recorded simultaneously using a spectrum analyzer and LabView program. Once the transmission stabilized in air, de-ionized (DI) water was flowed in at 0.5 mL/min. Glucose solutions of various concentrations were then flowed past the taper, with de-ionize (DI) water flowed in to rinse out the taper in between glucose solutions.


Ando AQ-6310B Spectrum AnalyzerAnritsu GB5A016 1550nm Laser

sensor

ThorlabsTED200 Temperature controller

FC-FC adapter

ThorlabsLDC202 Laser diode controller

Constant temperature water bath or incubator

Reservoir

Ando AQ-6310B Spectrum AnalyzerAnritsu GB5A016 1550nm Laser

sensor

ThorlabsTED200 Temperature controller

FC-FC adapter

ThorlabsLDC202 Laser diode controller

Constant temperature water bath or incubator

Reservoir

Figure 5. Experimental setup at 1550 nm in stagnant condition.

4.3. Antibody Immobilization

The most commonly used antibody immobilization method in our laboratory was adapted from Hermanson [57] with modification for the fiber surface and geometry. Prior to immobilization, the taper was cleaned with 1 M hydrochloric acid for 30 minutes, sulfuric acid for 10 minutes, and 1 M sodium hydroxide for 10 minutes. The sample holder and taper were rinsed several times with de-ionized water between cleaning steps. The cleaning procedure produced reactive hydroxyl groups on tapered surface. The surface was then silanylated with 3-aminopropyl-triethoxysilane (APTES; Sigma-Aldrich) in de-ionized water for 2-24 hours. The fiber was then dried overnight in a vacuum oven at 40oC, or in a regular oven at 75oC. The APTES reaction creates amine groups at the surface, which can further react with carboxylic groups in the antibody to form a peptide bond. The polyclonal antibody to BSA (anti-BSA; Sigma Catalog # B1520) contains carboxyl groups which were activated using 1-ethyl-3-(3-dimethylaminopropyl)-carbodiimide (EDC; Sigma-Aldrich) and stabilized by sulfo-N-hydroxysuccinimide (Sigma-Aldrich). EDC converts carboxylic groups into reactive unstable intermediates which are susceptible to hydrolysis. However, Sulfo-NHS replaces the EDC, resulting in a more stable reactive intermediate which catalyzes reaction with amine groups. To prepare the antibody, 0.4 mg of EDC and 1.1 mg of sulfo-NHS was added to each mL of antibody solution and the reaction was left on for 30 minutes at room temperature. Then, 1.4 μL of 2-mercaptoethanol was added to quench the EDC. This intermediate was added to the silanylated tapered fiber surface and covalent bonding was carried out at room temperature for 2 hours, in stagnant condition. At the end of antibody immobilization, Hydroxylamine was added to regenerate the carboxylic groups of the antibody. Transmission through the fiber was recorded during antibody immobilization and is shown in Figure 6.


0

0.5

1

1.5

22.5

3

3.5

4

4.5

5

0 20 40 60 80

time (min)

Cha

nge

in T

rans

mis

sion

(dB

)

0

5

10

15

20

25

30

35

40

Tem

pera

ture

(C)Temperature

Transmission

Figure 6. Transmission change vs. time for antibody immobilization at 1550 nm. Temperature was held constant at 30 oC ± 0.5 oC as indicated. Adapted from [1].

Alternatively, the antibody can be activated via carbohydrate groups. For this protocol, 1 mg/ml of antibody was dissolved in PBS and protected from light. Then, 100 μL of 0.1 M NaIO4 solution was added to antibody and allowed to react for 30 minutes. The silanized fibers were exposed to the solution for 2 hours. Then, 10 μL of NaCNBH3 was added for 30 minutes to reduce the Schiff Base to a second amine.

Another possible method of functionalization which is currently under investigation is the use of Protein G with gold. In this method, the taper was first coated with a 1:1000/v:v Polyurethane/Toluene mixture and dried overnight. The taper was then coated with 10 to 100 nm of gold using Denton Vacuum Desk IV® system. After gold coating, the taper was enclosed in the fiber holder by epoxy. During the first step of immobilization, Protein G was flowed into the sample chamber and left there in stagnant condition for 90 minutes. The sample chamber was then rinsed thoroughly with PBS, and antibody was flowed in and left there for 90 minutes. Then, the chamber was rinsed thoroughly prior to using it in a detection experiment.

4.4. Sample Preparation

All biological samples were prepared as per the instructions of the manufacturer using solutions of 0.1% Sodium Azide in PBS as the solvent. Usually a bulk solution of antibody is made and then aliquots of 4 mL are dispensed into sterilized scintillation vials. The vials are then stored at -30 C freezer until use. The antibody vials were for single use and were disposed at the end of the experiment. As for the analytes such as BSA and E.coli, they were prepared in bulk in sterile centrifuge tubes each holding a maximum of 50 mL. Each tube contained one concentration of analyte, and they were all stored at 4oC. The tubes were placed back refrigerated at 4oC after each use.


4.5. Detection

4.5.1. E.coli O157:H7 in Stagnant Condition E.coli O157:H7 (EC) was detected using a wavelength of 470 nm in stagnant conditions. Tapers were fabricated using heat pulling by torch or fusion splicer. The surfaces of the tapers were functionalized with antibody to E.coli O157:H7 using APTES and carboxylic linkage. The taper was exposed to various concentrations of pathogen, and showed transmission changes as the antigen attached.

0.97

0.99

1.01

1.03

1.05

1.07

0 5 10 15 20 25

Time, min

Norm

alize

d Tr

ansm

issio

n

E coli 0157:H7 at 1e06/mL

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0.88

27 29 31 33 35 37 39

Time, min

Nor

mal

ized

Tra

nsm

issi

on

Release of E coli 0157:H7

Figure 7. Detection and release of 1 million cells/mL of EC on antibody immobilized tapered fiber. EC detection and release experiment were performed on a 8.8 μm diameter TFOBS. After attachment (top panel), release buffer (glycine-HCl/ethylene glycol buffer, pH 1.7) was injected into the chamber to release EC (bottom panel).


An EC stock solution (7x109 cells/mL) was prepared as per the vendor’s (KPL) rehydration protocol in 10 mM PBS at pH 7.4. Lower concentrations (7x107 cells/mL, 7x105 cells/mL, 7x103 cells/mL, and 70 cells/mL) were prepared in PBS (pH 7.4) by serial dilution. 150 μL of each sample was injected into the sample chamber. After EC attachment, the sample was removed and the chamber was loaded with either HCl/PBS buffer at pH of 2.3 or Glycine-HCl/ethylene glycol (1:1 v/v) buffer at pH 1.7.

The response due to attachment and release of EC cells are shown in Figure 6. Immediately upon addition of the 1E6 cells/mL of EC sample, there was a rapid increase in transmission due to the RI change of the medium. Subsequently, a gradual and exponential decrease in transmission occurred due to EC attachment. Cells change the RI surrounding the fiber and absorb light from the evanescent field. When the attachment reached equilibrium, no further light is absorbed and the transmission remained constant.

The antigen attached to the sensor may be released by altering the pH as the antibody-antigen binding is pH-dependent. The response due to release was equal in magnitude and opposite in direction, as shown in Figure 7. This change occurred because cells released into the bulk are too far away from the taper surface to influence light transmission. When cells released reached equilibrium, the transmission reached a constant value.

1

1.05

1.1

1.15

1.2

1.25

1.3

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1.4

1.45

0 10 20 30 40Time, minutes

Nor

mal

ized

Tra

nsm

issi

on a

t 470

nm

PBS

7000 #/mL

70 #/mL

Figure 8. The response at 470 nm due to different concentrations of EC cells. Adapted from [2].

Intuitively, one would imagine that the transmission change would be directly proportional to the concentration. However, results show that the magnitude of the change is inversely proportional to the pathogen concentration, as shown in Figure 8. In addition, the response for this experiment was an increase in transmission, contrary to the experiment shown in Figure 7. The cause of this is not entirely clear, but we believe that it is due to the combined effects of evanescent absorption and scattering of the evanescent light. As cells


cover the taper surface, the evanescent light is absorbed by the cells in proportion to the surface coverage. On the other hand, the RI is increased due to cell attachment. If the sample were homogeneous, increase in refractive index tends to increase transmission through the core due to reduction in the evanescent field. Hence, cell attachment result in transmission increase or decrease.

0.92

0.94

0.96

0.98

1.00

1.02

1.04

1.06

0 1000 2000 3000 4000

Time, S

Nor

mal

ized

Tra

nsm

issi

on

50% Wild

70% Pathogen30% Wild

100% Wild

50% Pathogen50% Wild

Figure 9. Effect of pathogenic and non-pathogenic EC mixture. Experiments were performed on a 9.5 μm diameter TFOBS. When 0% pathogen (100% wild strain JM101) was injected around the taper, there was no significant transmission change through the taper. When a solution containing an EC and the wild strain is added to the solution, EC bind to the antibody thus resulting in a decrease in transmission through the fiber. As the concentration of EC is increased to 50% and 70%, there is a greater binding of pathogen to the antibody on the surface and thus greater change in transmission occurs. Adapted from [2].

In order to evaluate specificity, the response to non-pathogenic E. coli was measured. Stock solution containing EC was mixed with a wild strain of E. coli (JM101) in volumetric proportions of 0%, 50% and 70%. The total bacterial count was 7x107 cells/mL. The detection experiments were carried out in the same manner as with pure EC. The sensor showed good selectivity to the pathogenic antigen as shown in Figure 9.

It is useful to obtain the kinetics EC attachment on antibody-immobilized surfaces. The immobilization and detection responses show exponential behavior, similar to the adsorption process often referred to as Langmuir kinetics. The Langmuir kinetics model can be expressed as [58]:

1 o b sk teθ −= − (11)


where θ ( )10 ≤≤ θ is the fractional coverage of the reactive sites at time t. The

parameter, obsk , is the observed binding rate constant, which depends on the bulk concentration of the reactant. We hypothesize that the transmission is indicative of attachment, and express the Langmuir model as follows:

( ) ( )( )1 bkC tI I e −∞Δ = Δ − (12)

where ( )IΔ is the transmission change at time, t , ( )∞ΔI is the steady state transmission change, and Cb is the bulk concentration. Taking the natural log on both sides of Eq. (12) we obtain:

( ) ( )

( )ln b

I IkC t

I∞

∞

⎛ ⎞Δ − Δ= −⎜ ⎟⎜ ⎟Δ⎝ ⎠

(13)

-4

-3

-2

-1

0

0.E+00 1.E+08 2.E+08 3.E+08 4.E+08 Cb t, # bacteria-min/mL

ln[(ΔI

* – Δ

I)/ Δ

I* )]

Slope = 9.2 E-7 min-1 (#/mL)-1

Figure 10. Calculation of rate of attachment (slope k) for EC.

The above suggests that the characteristic rate constant k during initial time can be determined from a plot of the left hand side versus Cb* t in Eq. (13). Figure 10 is an example of a graph displaying Eq. (13). The kinetic constant (k) was found to be in the range of 4x10-9 min-1 (pathogen/mL) -1 to 7x10-9 min-1 (pathogen/mL) -1.

4.5.2. BSA in Stagnant Condition Although we were able to detect E.coli O157:H7 at 470 nm, the sensitivity of the sensors was limited due to the diameter in relation to the wavelength. Because the fibers are very fragile, we are unable to fabricated tapers that are less than 5 μm in diameter. However, at 470 nm the penetration of the evanescent field is limited. Also, cells are relatively large compared to the evanescent field generated at 470 nm. On the other hand, according to Eq. (1), there are reasons to believe that the evanescent field would be larger at a longer wavelength. Therefore,


detection of BSA was detected similarly to EC but performed mostly using near-IR wavelengths. We first reported the use of a 1550 nm laser with TFOBS to monitor the real-time attachment of BSA to the antibody-immobilized surface [1]. While cuvette measurements established that BSA was non-absorbing at 1550 nm, antibody-immobilized TFOBS showed transmission changes at bulk concentrations of 10 fg/mL of BSA. The experimental setup for near-IR detection is shown in Figure 5.

Solutions of BSA from 10 fg/mL to 1mg/mL were prepared. After antibody was immobilized, it was rinsed with PBS, and 200 μL of BSA was injected into the sample chamber. Only one concentration was used in each experiment of attachment and release. After attachment, the BSA was removed and the sensor was rinsed with PBS. Then, PBS adjusted to a pH of 2 by H2SO4 was added to release the BSA. The acidic PBS weakens the binding of BSA to the antibody because it changes the conformation of the protein. After this the fiber surface was regenerated with the cleaning sequence, followed by modification by APTES.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25 30 35

Time (minutes)

Cha

nge

in T

rans

mis

sion

(dB

)

Attachment

Release

Figure 11. BSA attachment and release of 10 pg/mL sample. The attachment response was obtained when the tapered region was first exposed to 10 pg/mL of BSA. Data was collected for 30 minutes, rinsed with PBS, and then the tapered region was exposed to pH2 PBS for BSA release. The transmission changes due to attachment and release are in opposite direction and have approximately the same magnitude. Adapted from [1].

When BSA was injected into the sample holder, transmission decreased due to change in surface refractive index caused by the presence of BSA. When BSA was replaced by low pH PBS, transmission increased back almost to the starting value. Similar experiments were performed with many tapers using different concentrations, and we conclude that experimental results are reproducible with the different fibers. The results of the attachment and release of 10 pg/mL of BSA are shown in Figure 11.

Multiple step attachment experiments was performed on three TFOBS with up to five different solutions of BSA, with concentration ranging from 100 fg/mL to 10 ng/mL. The experiments were initially performed with a starting concentration of 100 fg/mL. In the last


experiment, shown in Figure 12, the initial concentration was set at 10 fg/mL. The BSA solutions were added in order from the lowest to the highest concentration, with removal of each sample after collection of data for up to 40 minutes. The transmission decreased as a function of time as BSA attached to the antibody. The steady state transmission for each concentration also decreased.

-34.3

-34.1

-33.9

-33.7

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-33.3

-33.1

-32.9

-32.7

50 100 150 200 250 300

Time (minutes)

Tran

smis

sion

(dB

)

19

21

23

25

27

29

31

Tem

pera

ture

(oC

)

Temperature10 fg/mL

100 fg/mL

1 pg/mL

Transmission

Figure 12. Semi-batch staircase experiment showing attachment of BSA from10 fg/mL to 10 pg/mL. Temperature was maintained at 30 oC ± 0.5 oC using an incubator. The BSA solutions were added sequentially from lowest to highest upon removal of the previous solution. The purple line with peaks represents transmission through the fiber. The peaks correspond to time instants when the samples were introduced. The dotted line at the bottom represents the trend exhibited by the steady state transmission with respect to time. Adapted from [1].

Like the EC results, transmission change is not linearly proportional to concentration. We believe that the reason for this is that at low concentrations, the surface of the fiber is not saturated with the antigen BSA. An estimate of the antibody/antigen surface coverage can be made with a few simplifying assumptions, and it was suggested that the concentration required for saturation is less than 4 ng/mL.

Transmission changes are caused by the evanescent field interaction with the surface layer of antigen. Once the concentration approaches ng/mL levels, the surface is saturated with BSA. Additional BSA molecules would attach on top of the surface layer. However, the evanescent field magnitude decays away from the surface. Therefore the effect of BSA on top of the first layer results in much smaller changes. In addition, the condition for immobilization varies from one experiment to another. It is possible that nonlinearity was observed because at the lowest concentration, the bulk refractive index is approximately the same as that of PBS, and the BSA molecules on the fiber surface act as isolated points of high refractive index. When the concentration increases to saturation point, the fiber surface is covered with a layer of BSA which has a higher refractive index than PBS.


5.0. Conclusion

It is seen that TFOBS have several advantages in in terms of detection, including sensitivity, selectivity, ease of use, affordability, ability for remote sensing, and small sample volumes. They have been used for many applications such as pathogen detection, medical diagnostics based on protein or cell concentration, and detection of DNA hybridization.

As far as the sensor physics is concerned, intensity-based sensors have been used to a limited extent in cell detection. On the other hand, fluorescence based TFOBS are widely used for protein and DNA detection because amplification is a convenient tool, and often necessary to achieve low LODs. In addition, SPR is commonly used for protein characterization and has also been used for the detection of DNA hybridization. The ng/mL LOD of SPR makes it suitable for many medical applications. While fluorescence is very selective, its LOD is higher than SPR’s. In addition, fluorescence requires multiple steps for the preparation of the sensor or the sample.

In terms of target analytes, one possible area of growth is the use of SPR or intensity-based TFOBS protein and DNA detection. Another application may be drug screening using TFOBS. Because of recent concerns of homeland security, there will likely be a significant push for research in bio-threat detection. Pathogen detection also remains important in maintaining a safe environment and food supply. Clinical applications of TFOBS will likely be important as medical professionals seek convenient methods to diagnose diseases.

TFOBS have been used as intensity-based sensors in our laboratory. We have used three methods of fabrication: step-etching using hydrofluoric acid, heat pulling by flame, and heat pulling by fusion splicer. The sensing ability of TFOBS was characterized by measuring the transmission in water, E.coli JM101 solutions, and glucose solutions. TFOBS were functionalized with antibodies using covalent bonding or surface coating with gold and Protein G.

TFOBS was used in our laboratory to measure E.coli O157:H7 in stagnant condition. One surprising finding was that concentration had an inverse effect on the transmission. TFOBS was shown to be selective to the pathogens. BSA was detected at 10 fg/mL in stagnant condition at 1550 nm. Transmission data was fitted to the Langmuir absorption model to determine the attachment rate.

As TFOBS evolves, new efforts will be focused on enhancing the sensitivity and selectivity. Improved surface chemical modification and stability of the recognition molecule can increase the sensitivity and robustness of TFOBS, especially for intensity-based TFOBS because it is the most sensitive when molecules are bound to its surface. As was shown in this chapter, there is a solid foundation of work to support the use of TFOBS and a wide variety of applications. Given its promising advantages, it is likely that TFOBS will remain a popular choice for detection in the future.

Acknowledgements

This work was supported through the National Science Foundation Grant # CBET-0329793, “Ultra Sensitive Continuous Tapered Fiber Biosensors for Pathogens and Bioterrorism Agents”.


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[65] Wu, S.Y., Ho, H.P., Law, W.C., Lin, C.L., Kong, S.K. (2004) Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration. Optics Letters 29: 2378-2380.


[66] Tang, L., Ren, Y.J., Hong, B., Kang, K.A. (2006) Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases. Journal of Biomedical Optics 11: 21011-21011 to 21011-21010.

[67] Hong, B., Kang, K.A. (2006) Biocompatible, nanogold-particle fluorescence enhancer for fluorophore mediated, optical immunosensor. Biosensors & Bioelectronics 21: 1333-1338.

[68] Minunni, M., Tombelli, S., Mascini, M., Bilia, A., Bergonzi, M.C., Vincieri, F.F. (2005) An optical DNA-based biosensor for the analysis of bioactive constituents with application in drug and herbal drug screening. Talanta 65: 578-585.

[69] Thompson, V.S., Maragos, C.M. (1996) Fiber-optic immunosensor for the detection of fumonisin B-1. Journal of Agricultural and Food Chemistry 44: 1041-1046.


Chapter 2

NEW CHALLENGES IN RAMAN AMPLIFICATION FOR FIBER COMMUNICATION SYSTEMS

P.S. André1,2, A.N. Pinto1,3, A.L.J. Teixeira1,3, B. Neto 1,2, S. Stevan Jr.1,3, Donato Sperti1,3,4, F. da Rocha1,3,

Micaela Bernardo2,5, J.L. Pinto1,2, Meire Fugihara1,3, Ana Rocha1,2 and M. Facão2

1Instituto de Telecomunicações, Aveiro Portugal 2Departamento de Física, Universidade de Aveiro, Aveiro, Portugal

3Departamento de Electrónica, Telecomunicações e Informática, Universidade de Aveiro, Aveiro, Portugal

4Università Degli Studi di Parma, Parma, Italy 5Portugal Telecom Inovação SA, Aveiro, Portugal

Abstract

Raman fiber amplifiers (RFA) are among the most promising technologies in lightwave systems. In recent years, Raman optical fiber amplifiers have been widely investigated for their advantageous features, namely the transmission fiber can be itself used as the gain media reducing the overall noise figure and creating a lossless transmission media. The introduction of RFA based on low cost technology will allow the consolidation of this amplification technique and its use in future optical networks.

This paper reviews the challenges, achievements, and perspectives of Raman amplification in optical communication systems. In Raman amplified systems, the signal amplification is based on stimulated Raman scattering, thus the peak of the gain is shifted by approximately 13.2 THz with respect to the pump signal frequency. The possibility of combining many pumps centered on different wavelengths brings a flat gain in an ultra wide bandwidth.

An initial physical description of the phenomenon is presented as well as the mathematical formalism used to simulate the effect on optical fibers.

The review follows with one section describing the challenging developments in this topic, such as using low cost pump lasers, in-fiber lasing, recurring to fiber Bragg grating cavities or broadband incoherent pump sources and Raman amplification applied to coarse wavelength multiplexed networks. Also, one of the major issues on Raman amplifier design,

P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 52

which is the determination of pump powers in order to realize a specific gain will be discussed. In terms of optimization, several solutions have been published recently, however, some of them request extremely large computation time for every interaction, what precludes it from finding an optimum solution or solve the semi-analytical rate equation under strong simplifying assumptions, which results in substantial errors. An exhaustive study of the optimization techniques will be presented.

This paper allows the reader to travel from the description of the phenomenon to the results (experimental and numerical) that emphasize the potential applications of this technology.

1. Introduction

The deployment of optical communication systems through long haul networks required the development of transparent optical amplifiers, for replacement of the expensive and limitative optoelectronic regeneration. The increasing distance between amplification sites saves amplification huts reducing by this way the investment and operational cost in the network management.

The first choice for transparent optical amplification pointed out to the Erbium Doped Fiber amplifiers (EDFA), which was a mature technology by the beginning of the last decade of the XX century. However, the growing demand in terms of transmission capacity has been increasing dramatically, fulfilling the entire spectral band of the EDFA, and wideband amplifiers are now required. Raman fiber amplifiers (RFA) have emerged as a key technology for the optical networks.

In lumped amplified systems (using for example EDFAs) the amplification modules are placed every 40~50 km of span. This module amplifies back to the initial power level, the transmission signal attenuated during propagation. The distance between amplifiers is determined by the span loss, by the limit imposed from the maximum admissible power allowed in the fiber without inducing nonlinear effects and by the minimum acceptable power that avois a degradation of the signal-to-noise-ratio.

The use of Raman amplification allows the confinement of the signal inside the limits imposed by the nonlinearities and of the signal-to-noise-ratio degradation resulting from higher span distances. This advantage of the distributed (Raman) over lumped amplification is illustrated in figure 1.

Figure 1. Distributed and lumped amplification signal evolution.

New Challenges in Raman Amplification for Fiber Communication Systems 53

The distributed amplification scheme can be used to cover very long span links or to increase the distance of ultra-long haul systems.

Raman fiber amplifiers are based on the power transfer from pump(s) signal(s) to information carrying signals (usually described as probes) due to stimulated Raman scattering (SRS) which occurs when there is sufficient pump power within the fiber. Since the gain peak of this amplification is obtained for signals downshifted approximately by 13.2 THz (for Silica), relative to the pump frequency, to achieve gain at any wavelength we need to select a pump whose frequency complies with this relation. In this way, it is possible to optimize the number of pumps to obtain a wide and flat gain [1-3]. However, it is necessary to bear in mind, that due to the pump-to-pump interaction, the shorter wavelength pumps demand more power to be effective [4,5].

From a telecommunications point of view, the pump wavelengths must be placed around 1450 nm because the signal wavelengths used on the so called 3rd transmission window are centered around 1550 nm and the maximum gain occurs for a Stokes frequency shift of 13.2 THz.

The RFA had become attractive just after the development and commercialization at a reasonable cost of a key component: the high power pump laser [6]. Typically a high power laser for Raman amplification, provides an optical power of 300 mW, launched over an optical fiber, which for a standard single mode fiber (SMF) is equivalent to a power density of 3.75 GW/m2. This high power injected into the fiber, especially when multipump lasers are utilized, imposes new concerns in terms of safety.

Therefore, the use of RFAs requires the utilization of automatic power reduction or automatic laser shut down systems to prevent the hazard of high power leakage from the optical cables or service cabinets. Also, as the optical power rises, the nonlinear effects, such as the fiber fuse effect, start to become relevant. This effect has threshold intensity of 10~30 GW/m2 and it is responsible by a catastrophic destruction of the fiber core. This destruction once started propagates in direction to the optical source, resulting also in the destruction of the pumping laser [7]. For operating wavelengths of 1550 nm, the fuse effect power threshold is ~1.5 W for SMF fibers, while for dispersion shift fibers (DSF) this power is reduced to ~1.2 W [8]. This effect is also responsible by the damage of the optical connectors interface [8].

In terms of implemented systems, several architectures have been proposed, based in all Raman or hybrid Raman/EDFA amplification [10]. The use of bidirectional Raman amplification has also been reported for long reach access networks. Experimental results have shown the feasibility of systems with symmetric up-and-downstream signals with bitrates up to 10 Gb/s, supported by distributed Raman amplification over 80 km of fiber [11]. Field transmission experiments have been reported with 8 × 170 Gb/s over 210 km of single mode standard fiber, achieving spectral efficiency of 0.53 bit/s/Hz [12].

As the traffic increases, wavelength division multiplexing (WDM) arises to enlarge the transmission capacity. This, in turn, requires flexible and broadband architectures which reinforces the interest in Raman amplification. Nowadays, WDM exists in two formats: Dense WDM (DWDM) working at C and L spectral windows, allocating a maximum of 150 channels spaced by 0.8 nm [13], and Coarse WDM (CWDM) working at O, E, S, C, and L spectral windows, allocating a maximum of 18 channels spaced by 20 nm [14]. The DWDM solution is extensively used in long haul systems, sending as much information as possible. CWDM is a good solution whenever less information is transmitted over short distances in a


less expensive way than DWDM. As CWDM works with far apart signals, it can make use of uncooled distributed feedback (DFB) lasers [15,16] needing multiplexing components with flexible tolerances. However, as the channels in CWDM systems are far apart, optical amplification is still a matter of concern. Traditional EDFA bandwidth (20~40 nm) cannot support the full band of CWDM channels [17].

Other technical solution to amplification of signals is the semiconductor optical amplifier, which presents a low saturation power (around 13 dBm) when compared with other fiber based amplifiers, but with a signal-to-noise ratio degradation quite considerable. A good solution for the amplification of both DWDM and CWDM relies on Raman amplifiers. A wide and flat spectral gain profile is achievable thanks to the combination of several pumping lasers operating at specific powers and wavelengths. The composite amplification is determined from the mutual interactions among the pump and signal wavelengths. Gain spectra as large as 100 nm were obtained using multiple pumps. Emori et al. have presented an experimental Raman amplifier with a 100 nm bandwidth using a WDM laser diode unit with 12 wavelengths ranging from 1405 to 1510 nm, whose maximum total power was equal to 2.2 W [4, 18]. Therefore, a gain equal to 2 dB is obtained over a 25 km SMF link and a 6.5 dB gain using a 25 km DSF link, both with 0.5 dB of maximum ripple. Kidorf et al. provided a mathematical model to implement a 100 nm Raman amplifier using low power pumps with maximum power of each pump equal to 130 mW [14]. They used 8 pumps from 1416 nm to 1502 nm along 45 km of SMF, obtaining a gain around 4 dB with a maximum ripple equal to 1.1 dB.

The growing maturity of high pump module technologies is providing competitive solutions based on Raman amplification and currently many alternative techniques are being developed to overcome the ordinary one pump and dual pumping methods [19, 20]. In particular, we report here two major techniques. First, the use of low power pumping lasers provides gain comparable to the ordinary one pump Raman amplification. This technique is especially interesting for combining commercial and low cost lasers [21]. The second particular technique corresponds to an evolution of the cascaded Raman amplification. Actually, a sixth order cascade Raman amplifier was recently proposed [22]. In the cascade Raman amplification, the pump power is downshifted in frequency by using a pair of fiber Bragg gratings (FBG) placed in spectral positions multiples of 13 THz, from the pump frequency. In a particular case, the generation of the fiber pump laser is obtained by using only one passive reflector element and distributed reflectors over the long optical fiber, established by a nonlinear fiber intrinsic effect called Rayleigh backscattering.

The enlargement of the bandwidth of Raman amplifiers is also achieved using incoherent pumping instead of multi-pump schemes [23-27]. Vakhshoori et al. proposed a high-power incoherent semiconductor pump prototype that uses a low-power seed optical signal, coupled into a long-cavity semiconductor amplifier. It was achieved 400mW of optical power over a 35nm spectral window [27]. A 50 nm bandwidth amplifier was obtained with an on/off gain equal to 7 dB. It was also demonstrated that the use of six coherent pumps is less efficient, in terms of flatness, than the use of two incoherent pumps [24]. The signal wavelengths were comprised between 1530 nm and 1605 nm and the transmission occurs over 100 km of optical fiber. Another advantage of using incoherent pumping is the reduction of nonlinear effects, such as Brillouin scattering, four wave mixing of pump-pump, pump-signal and pump-noise [28].


RFAs have become a crucial component for the implementation of fiber optic communication systems [9]. An exponential increase on the product distance × capacity of the transmission experiments on optical communication systems was observed in the last decade. The majority of these experiments, especially since the year 2000, have employed RFA as amplification technology [9]. This survey attempts to cover the most recent aspects in the field of Raman amplification for fiber communication systems.

2. Theoretical Description of Raman Scattering

In 1928 Raman scattering was discovered independently and almost simultaneously by two research groups, one working in India and lead by Sir C. V. Raman [29], and the other by G. S. Landsberg and L. I. Mandelstam working in Russia [30]. In 1930, the Nobel committee distinguished Sir C. V. Raman for his discovery of the molecular scattering of light and since then this effect has been known as the Raman effect.

Raman effect is a scattering effect of light. Light scattering occurs as a consequence of fluctuations in the optical properties of a medium. In optical fibers three types of scattering effects are relevant: Rayleigh, Brillouin and Raman scattering.

Rayleigh scattering is an elastic process, i.e., the incident and the scattered photon have the same energy, therefore the same frequency. Rayleigh scattering in fibers couples light from guided modes to unguided ones leading to optical attenuation. Indeed, in modern fibers operating in the near infrared, Rayleigh scattering is the major source of attenuation, as absorption is practically negligible. In fact, Silica lattice and electronic resonances are in the mid infrared and in the ultra-violet, respectively. Therefore in the near infrared, fibers operate, essentially in an off-resonance regime, apart from impurities, which in nowadays fibers are reduced to an extremely low level [31]. However, besides the off-resonant interaction with bound electrons, optical waves also interact with molecules inside Silica fibers, through scattering.

Raman and Brillouin scattering are both inelastic processes, i.e., the incident and scattering photons have different energies. The energy lost by the incident field is stored into the medium in the form of vibrational energy, named phonons. Indeed, the origin of both Raman and Brillouin scattering effects resides in the interaction of light with these vibrational states (phonons). In the Brillouin scattering low frequency vibrational states are involved, usually referred as acoustic phonons. In the Raman process high frequency vibrational states are presented, named as optical phonons.

Raman scattering can occur in two distinguished forms: Spontaneous Raman Scattering, and Stimulated Raman Scattering (SRS).

In the spontaneous form, Raman scattering occurs when the incident field interacts with vibrational modes, mainly excited by thermal effects, of the molecules constituting the medium. From this interaction, it can result another optical phonon, with frequency Ω , and a down shifted optical photon with frequency 0= −ΩSν ν , or a up shifted photon of

frequency 0= +ΩAν ν and in this case an optical phonon is annihilated, υ0 is the frequency

of the incident signal. As the frequency Ω is related to the normal vibrational modes of the molecules constituents of the medium, by analyzing the scattered light, information about the medium can be retrieved. This is the main idea behind Raman spectroscopy, a widely used


technique for materials characterization. In amorphous materials, like Silica, Ω can assume a value belonging to a broad spectral range, starting from zero and going up to 40 THz. Experimentally both down shifted and up shifted frequencies waves have been observed and have been named as Stokes and anti-Stokes, respectively.

Stimulated Raman Scattering was discovered by E. J. Woodbury and W. K. Ng, almost accidentally in 1962, when working with a Ruby laser [32]. They observed a strong spectral line not coincident with any spectral line of the fluorescence spectrum of Ruby. To understand this process let us assume that an incident photon is scattered by an optical phonon in the medium, and in this process a down shifted photon and an optical phonon are created. We can see that we have two ways of creating phonons, the scattering process and the thermal mechanism. If the intensity of the incident light is small, the rate of phonons created by scattering is low and due to thermal equilibrium the density of phonons in the medium is unchanged, and therefore the medium maintains the same optical properties. If the intensity of light is increased above a certain threshold, the optical properties of the medium can be changed in a way that the scattering process is enhanced [33]. In this situation, the incident light stimulates the scattering process and we are in the presence of Stimulated Raman Scattering. Through this positive feedback the scattering process can be enhanced by several orders of magnitude. Due to the Bosonic nature of the photons, this process can indeed provide gain. The photon emission process by a scattering center, it can be stimulated by the presence of another photon, and this stimulated emission is the origin of the gain. The term emission is used in this context in a quite abusive way because there is no absorption to a real state, but this process can be treated considering that the scattering photon is initially absorbed to a virtual state and after re-emitted.

If we consider that the decay from the virtual states only occurs spontaneously, the Stokes power grows linearly with the pump power. In the other way, if we consider that the decays from the virtual states must be triggered by another photon, the Stokes power grows exponentially with the pump power. Off course, in reality both spontaneous and stimulated emission occurs. If the photon that triggers the stimulated emission is part of a signal we are in the presence of optical gain, which can be beneficial for optical communication systems [34]. If this photon was initially generated by spontaneous emission we are in the presence of amplified spontaneous emission noise which is usually considered as harmful, at least for telecommunications purposes. The spontaneous emission process always leads to an excess of noise in the system.

The optical gain provided by the Raman process can be completely characterized by the Raman-gain coefficient ( )ΩRg , which is related with the imaginary part of the third-order

nonlinear susceptibility. The characterization of the amplified spontaneous emission process requires, besides the Raman-gain coefficient ( )ΩRg , another coefficient named noise

spontaneous emission factor ( )Ωspn . However, it turns out that another source of noise must

be also considered to characterize the noise in Raman amplifiers. This source of noise arises from Rayleigh scattering. Most of the Rayleigh scattered photons are lost through non-guided modes, but some of them are coupled to the counter-propagating mode. Those photons can be amplified and through another Rayleigh scattering process can appear as extra-noise at the amplifier output. This effect is usually named as double Rayleigh scattering and will be described in more detail in section 4.3.


3. Modeling of Raman Amplifiers

The implementation of RFA, using an optical fiber as gain medium, requires that the pump and information signals must be injected into the same fiber. A basic scheme for a RFA architecture is displayed in figure 2. The signal and pump waves are launched into the optical fiber (the gain medium) by a coupler, so, that stimulated Raman scattering can occur. Since the SRS effect occurs uniformly for all the orientations between pumps and signals, Raman amplifiers can work both in forward and/or backward pumping configuration.

Figure 2. General scheme for a distributed Raman amplifier. For simplicity the optical isolators used to protect the pumps and signals sources, were omitted.

The model for power evolution in Raman amplifiers assuming a multipump multisignal configuration is often based on an unified treatment of channels, pumps and spectral components of the amplified spontaneous emission (ASE). The major interactions can be reasonably drawn by considering the pump-to-pump, signal-to-signal and pump-to-signal power transfer, attenuation, Rayleigh back scattering, spontaneous Raman emission and its temperature dependence. Other effects, such as noise generation due to spontaneous anti-Stokes scattering, polarization and nonlinear index are neglected, but they can reach considerable importance in certain regimes of transmission. It must be noted that signal channels and Raman pumps are treated as fields at single frequencies, so ignoring the interactions due to the spectral shape of signals and pumps.

In a general approach, the power evolution of pumps, signals and ASE (in forward and backward directions), with time along the fiber distance is given by the following set of coupled differential equation [35]. For Np pumps, Ns probe signals and NASE spectral components for ASE, the system is formed by Np+Ns+2NASE equations.

( ) ( )

( ) ( )[ ] ( ) ( )[ ] ( ) ( )

( ) ( )[ ]( ) υηυ

γυηυυυα

Δ++Γ+

+⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ+Γ−+−++−

=∂

∂∂

∂±

−+−

=

±

+=+=

−+−+−

=

±±

∑

∑∑∑

jijj

i

jjii

iii

m

ijijiji

m

ijjjij

j

ijj

i

jjii

i

i

i

tzPtzPgh

tzPtzPghtzPtzPgtzPtzPg

ttzP

VztzP

1,,

),(,12,,,,

,1,

1

1

11

1

1

m

m

(1)

Vi is the frequency dependent group velocity. The ± signs stand for the forward or

backward propagating waves, being αi and γi the coefficients of attenuation and Rayleigh of


the ith wave at frequency υi. h and kB are the Planck and Boltzmann constants, respectively, and T is the fiber absolute temperature. The phonon occupancy factor is given by:

( ) 1

1exp−

⎥⎦

⎤⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

Tkh

B

jiij

υυη (2)

The frequencies υi are numbered by their decreasing value (the lower order corresponds

to the higher frequency). Thus, the terms in the summation in expression 1, from j=1 to j=i-1 cause amplification since the wave i is receiving power from the lower order waves (with higher frequency). For the same reason, the terms in the summation from j=i+1 to j=m originate depletion. For mathematical convenience the gain spectrum was divided into slices of width Δυ, spanning the range over which ASE spectral components are significant.

The terms that contain a product of powers describe the coupling via stimulated Raman Scattering, being its strength determined by the Raman gain coefficient of the fiber, gij obtained by equation 3.

( )

eff

jiRij A

gg

Γ−

=υυ

(3)

where Aeff is the effective area of the fiber and the factor Γ is a dimensionless quantity comprised between 1 and 2 that takes into account the polarization random effects. The achieved gain, as well as the slope of the gain spectrum, depends on the transmission fiber [36, 37]. In figure 3, two Raman gain coefficient spectra are displayed, showing the different strengths of the Raman coupling of a SMF fiber and a dispersion compensating fiber (DCF).

-30 -20 -10 0 10 20 30-4

-3

-2

-1

0

1

2

3

4

Ram

an g

ain

coef

ficie

nt (W

-1km

-1)

pump-signal frequency difference (THz)

SMF DCF

Figure 3. Raman gain coefficient spectra for two germanosilicate fibers: Single mode fiber (SMF) and dispersion compensating fiber (DCF), for a pump wavelength of 1450 nm.


As a matter of fact, the small effective area of the DCF (15 μm2) is determinant for its higher Raman gain coefficient when compared to the SMF (80 μm2) or when compared with DSF fibers (50 μm2). Those spectra also show peaks that are broader than those presented by crystalline materials, since the amorphous nature of Silica allows a continuum of molecular vibrational frequencies.

To obtain a steady-state power distribution, the time derivative in equation 1 is settled equal to zero, and the set of equation takes the form of expression 4.

[ ] [ ] ( )

[ ]( ) υηυγ

υηυυυ

α

Δ++Γ++

+⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ+Γ−+−++−

=±

−+−

=

±

+=+=

−+−+−

=

±

∑

∑∑∑

jijji

jjiiii

im

ijijiji

m

ijjjij

j

ijj

i

jjii

i

PPghP

PghPPgPPg

dzdP

1

12

1

1

11

1

1

m

(4)

In spite of the simplification, the modeling is still computationally intensive, especially

for the situation of backward or bidirectional pumping. In those situations, the mathematical problem that describes the power evolution of pumps and signals along the fiber is a boundary value problem (BVP) which is more difficult to solve than the initial value problem (IVP) in the forward pumping scheme. An immediate approach to the numerical solution of such problem is the shooting method [38]. There are other allowable numerical methods, such as relaxation methods, or collocation methods [39]. Generally, shooting methods are faster than relaxation ones. In shooting methods, we choose values for all the dependent variables at one boundary, solve the system of ordinary differential equation (ODE) as an IVP and verify if the obtained values on the other boundary are consistent with the stipulated values (boundary conditions) [40]. Then, the parameters are repeatedly changed using some correction scheme until this goal is attained. The selection of the correction scheme is crucial for stability and efficiency of the resulting algorithm. An other variant of the shooting method, we can guess boundary values at both ends of the domain, integrate the equation to a common midpoint and repeatedly adjust the guessed boundary values so that the solution tends to the same value at the middle point. This adjustment task is usually performed by the Newton-Raphson method.

Recently, some shooting algorithms with different correction schemes for the design of Raman fiber amplifiers have been proposed in order to improve convergence of the solutions even for larger fiber lengths [41]. This scheme is obtained by modifying the numerical method used to perform the IVP integration (fourth order Runge-Kutta, Runge-Kutta-Felhberg, etc). Other approaches to solve the equation 4 propose a shooting method to a fitting point using a correction scheme based on a modified Newton approach. Therefore, by introducing the Broydens rank-one method into the modified Newton method, the algorithm becomes more efficient and stable. This happens because the intensive numerical calculations of the Jacobi matrix are substituted by simpler algebraic calculations [41].

The use of projection methods such as collocation, gives a continuous approximation of the solution as a function of the fiber length. The basic idea is to approximate the BVP solution by a simpler function that represents an approximation.


Nevertheless, the Raman equations (equation 4) are also solvable through semi-analytical methods, using the average power analysis (APA) presented by Min et al. [42]. The amplifier is split into n small segments, in order to avoid the position dependency of the powers of equation 4. The equations are then solved analytically in each segment, considering as input conditions the outputs provided by the solution on the previous segment. Equations 5 to 8 show how the powers are iteratively computed. The output pump/signal power at each section end is given by:

),( υzGPP inout

±± = (5)

being G(z,υ) the section gain,

[ ]zBAzG Δ−+−= ))()()((exp),( υυυαυ (6) The constants, A(υ) and B(υ) are obtained through:

j

m

ijji

j

i

jij

PgB

PgA

∑

∑

+=

−

=

=

=

1

1

1

)(

)(

υ

υ (7)

The optical power term in each section can be substituted by its length averaged values

given by:

))(ln(1)(

υυG

GPP in−

= ± (8)

For a RFA, the net gain is usually defined as the ratio between the signal powers at the

end and at the beginning of the fiber link, as defined in equation 9:

)0()(

==

=zP

LzPG

signals

signalsnet

(9)

The so-called on/off gain is another useful quantity that measures the increase in signal

powers at the amplifier output when the pumps are turned on, as follows:

off pumpswith )(on pumpswith )(

/ LzPLzP

Gsignals

signalsoffon =

== (10)

The numerical issues due to the backward pumping can be surpassed by assuming that

the pump inputs are located at the same fiber end that the signal inputs. Therefore, the pump equations are integrated reversely as if they were backward by multiplying them by (−1). A guessed initial input is necessary to perform the integration, but the algorithm is able to adjust it using an optimization routine that adjust the initial input until the output at the fiber end


reaches the real backward pump input. The use of the APA approach has shown a reduction of two orders of magnitude in the computation time, being the obtained results in agreement with the ones resulting from traditional numerical methods.

To demonstrate the numerical resolution of the steady-state Raman propagation equations, we assume the scheme in figure 4, where three bidirectional pumps (two backward and one forward) and four probe signals are considered. The counterpropagated pumps have power levels set equal to 0.1W, working at 1450 nm and 1460 nm, respectively. The copropagated pump is working at 1470 nm with an output power also equal to 0.1W. The forward pumping signal are then injected into 40 km of SMF fiber and combined with 4×1000 GHz spaced C band probe signals with an initial optical power equal to 1μW. The spatial evolution of pumps and probe signals are displayed in figure 4.

0 10 20 30 400.6

0.7

0.8

0.9

1.0

1.1

1.2

Prob

e P

ower

(μW

)

Fiber length (km)

1530 nm 1538 nm 1546 nm 1554 nm

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10 1450 nm 1460 nm 1470 nm

Pum

p P

ower

(W)

Fiber length (km)

Figure 4. Spatial evolution of two counterpropagated pumps, one copropagate pump and four probe signal along a 40 km SMF fiber span amplifier. Probe signals evolution (top) and pump signals evolution (bottom).


The implementation of equation 4 also allows the calculation of the total noise for each signal (forward and backward ASE) within the amplifier, whose spatial evolution for this system can be followed in figure 5.

0 10 20 30 401

2

3

4

5

6

7

8

9To

tal N

oise

Pow

er (n

W)

Fiber length (km)

1530 nm 1538 nm 1548 nm 1554 nm

Figure 5. Spatial evolution of the total noise (forward and backward ASE) power along 40 km SMF fiber span.

The noise figure of an optical amplifier amounts the degradation of the signal to noise ratio (SNR) when the signals are amplified. The most important source of noise in optical amplifiers is ASE, which, for Raman amplifiers is due to spontaneous scattering. Assuming that the signals are initially as noiseless as possible, and that their degradation is due to signals spontaneous beat noise produced by ASE, the noise figure, in linear units, is given by equation 11 [36]:

net

ASE

GhLzP 11)(2NF ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

Δ=

≈+

υυ (11)

where hυ is the photon energy and PASE

+ is the forward ASE measured over the reference bandwidth Δυ. The first term corresponds to the noise from the signal spontaneous beating and the second one to shot noise.

Another quantity, named effective noise figure, accounts the noise that a discrete amplifier placed at the end of an unpumped fiber link would need to have the same noise performance that a distributed Raman amplifier. In decibel units, the effective noise figure is computed using:

( )dBdBdB

eff LNFNF α−= (12)


Typically, WDM systems allocate a large number of channels spaced over wide bandwidths. Considering the previous system but doubling the pump powers and using 64 probe signals (instead of 4), we obtained the spectra of the gain and noise figure which are plotted in figure 6.

1530 1540 1550 1560 1570 15800

2

4

6

8

10

Signal Wavelengths (nm)

Net

Gai

n (d

B)

-2.0

-1.5

-1.0

-0.5

0.0

Effective N

oise Figure (dB)

Figure 6. Net gain and effective noise figure spectra for a system with two counterpropagated pumps, one copropagate pump and 64×100 GHz probe signals along 40 km SMF fiber span.

As depicted in this section and despite some remaining numerical issues, the modeling of a multipump Raman amplifier anticipates many valuable applications for WDM systems, namely the broadband gain. It is important to notice that gain spectra as wide as 100 nm are achievable and that the gain value can be kept quite constant by an appropriate tailoring of the amplifier architecture. This procedure involves solely the proper dimensioning of the pump power levels and operating wavelengths, as discussed more extensively in section 4.

Another interesting feature of RFA is the noise performance. The ASE noise in RFA is intrinsically low (as suggested by the negative effective noise figure presented above). The reason relies in the fast relaxation of the optical phonons, the absorption of signal photon to the upper virtual state is extremely small. The inversion of population is almost complete.

4. Challenges in Raman Amplification

4.1. Gain Profile Optimization

One of the most impressive features of Raman fiber amplifiers is assuredly the possibility to achieve gain at any wavelength, by selecting the appropriate pump wavelength. Therefore, it is possible to operate in spectral regions outside the Erbium doped fiber amplifiers bands over a wide bandwidth (encompassing the S, C and the L spectral transmission bands).


Nevertheless, some studies have been reporting that despite the Raman gain dependence is essentially due to the pump-signal frequency difference; there is also some weaker dependence on the pump absolute frequency [43]. However, since a deeper study of this topic is beyond the scope of this work, we will not consider it in the gain optimization.

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Figure 7. Numerical simulation of broadband Raman amplifier gain. Bars show backward input pump powers and wavelengths. Ticker line show 14×400 GHz probe signals optimized net gain and thin lines the gain contribution of each individual pump. The simulation was carried out through 25 km of SMF fiber.

A flat spectral gain profile is achievable with the combination of several pumping lasers operating at specific powers and wavelengths. The Raman gain created by pumps at different frequencies is slightly shifted from each other to partly overlap and form a composite gain. When the pump powers and frequencies are properly chosen, this wide gain can also be considerably flat. Another important feature to take into account when designing a flat gain scheme, is the strong Raman interaction between the pumps, since the higher frequency pump is responsible for the amplification of the lower frequency signals, more pumping power is needed, as some will also be transferred to the lower frequency pumps. This interaction between pumps also affects the noise properties of the amplifier. However, some novel pumping schemes have been recently proposed in order to prevent those unwanted effects: copumping, time dependent Raman pumping, higher order pumping and broad-band pumping [44].

Typically, laser diodes with output powers in the 100-200 mW range can be used in a multipump scheme. This scheme is normally composed of a set of laser diodes operating in the 14XX nm region, whose spectral width is narrowed and stabilized by a FBG. Optical couplers are used to combine and depolarized them, in order to suppress the polarization


dependent gain. The multipumping allows bandwidth upgradeability by the addition of new laser diodes. Theoretically, the larger the number of pumps the better the gain ripples. Nevertheless, there are economic issues that prohibit the use of an arbitrary number of pumps. For this reason, we have to find a balance between the system performance and the cost of amplification.

Optimization of the gain spectrum has been widely performed making use of several global search methods, such as neural networks [45], simulated annealing [46] and genetic algorithm (GA) [47]. During the search process, the pump powers and frequencies are directly substituted into the system of propagation equations to calculate de gain profile. Depending on the speed of the numerical method used to integrate the system of equations, the amount of numerical computations involved can be considerably large and the optimization inevitably time consuming. Those solutions are not suitable for practical applications where the real optimal solution must be provided in a short time.

However, some alternatives can be found by replacing the usual intensive numerical integrations with simpler algebraic calculations using the APA method while integrating the Raman propagation equations.

Another simple but important issue when using a global optimization method relies in a proper dimensioning of the search domain. Using the APA method, all the inputs are located at the same fiber end, even for the counter pump situations. Therefore, the pump power inputs are chosen by presuming a typical propagation profile. By this way, it is advisable to try lower power values for the higher frequency pumps and higher power values for the lower frequency pumps (the opposite happens at other fiber end). Regarding to the optimization of the pump frequencies, it is advisable to divide our spectral range into the number of pumps and then shift those values by 13 THz.

A second approach to speed up the search of the optimal pump configuration uses the genetic algorithm (GA) method only to search the pump frequencies and a quadratic programming method to solve the power integral [48]. The search domain of the GA method is by this way reduced to a half, enabling faster convergence.

Another approach combines GA with the Nelder-Mead search. This so called hybrid GA can be useful in certain situations for the purpose of saving some function evaluations and consequently to perform the optimization in the least time possible [49]. The hybrid GA follows the routine depicted in Figure 8. Firstly, the initial population, as well as the other GA operators are dimensioned: selection, crossover and mutation. The selection, together with the crossover, is responsible for the bulk of GA processing power. The mutation is an operator that plays a secondary role in the GA. Since, the genetic operators can be performed by different methodologies, it is important to choose the ones that are more adequate to the problem we are dealing with, in order to improve the GA search procedure [50]. It must be noted that if the search space is not large, it can be searched exhaustively and the best possible solution will be probably found. The maximum number of allowed generations is also an important feature because, when carefully chosen, it can save a large number of function evaluations. The Nelder-Mead method uses a simplex in a n-dimensional space, characterized by the n+1 distinct vectors that are its vertices. At each step of the search, a new point in or near the current simplex is generated. The function value at the new point is compared with the function values at the vertices of the simplex and one of the vertices is replaced by the new point, giving a new simplex. This step is repeated until the diameter of the simplex is less than the specified tolerance


Figure 8. Scheme of the hybrid GA implementation.

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Figure 9. Total simulation time for a hybrid GA against the number of generations for a population size of 50 individuals. The line is a visual guide.

By determining properly the right moment to switch from one method to another, it is possible to reduce the simulation time to a half when compared to simple GA. This result can be observed in figure 9. The normalized total simulation time (τ) against the number of generations is plotted. Here, the reference is the slowest simulation, the one that use 40 generations, identified by the letter B. The best situation (tagged as A) tooks a simulation time


equal to about a half of the time of the worse situation and it was attained with 17 generations. An heuristic explanation relies on the intrinsic nature of the GA. We verify that for small number of generations (bellow 15) the GA time is small but the system reaches a worse fittest solution. Thus, the Nelder-Mead method needs more time to reach a desirable solution. When the number of generations increases the GA reaches a best solution but the needed computation time increase accordingly.

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Figure 10. Power evolution of optimized pumps along 20 km of SMF (lines). The geometric shapes stand for the used experimental values.

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Figure 11. Experimental (arrows) and simulation (line) on/off spectral gain for the 20 probe signals and 7 counter propagated pumps, over 20 km of SMF fiber.


In order to enlighten the conclusions provided by the hybrid GA algorithm, a laboratorial implementation was carried out to test the optimization results. A Raman amplified system with 20 km of SMF fiber, 20 probe signals and 7 backward pumps was implemented. Since the pump wavelengths are already settled, only the optimization of the power levels is needed. The simulation used the stochastic uniform method for selection, the scattered crossover method and the uniform mutation. A population of 50 individuals and a number of generations equal to 35 were considered. The spatial evolution of the pumps signals optimized values are displayed in figure 10 jointly with the pump signal experimental values. In figure 11, the optimized and experimental on/off gain spectra are presented.

This is a good agreement between the optimization modeling and the experiment. The maximum ripple attained by the optimization is 0.41 dB being the experimental maximum ripple equal to 0.23 dB. The mean square deviation between simulation and experimental results is equal to 0.0036. Indeed, a flat gain over a wide bandwidth (~80 nm) was attained, using seven pumps with a total input power equal to 453 mW.

4.2. Raman Amplification Using Multiple Low Power Lasers

One of the main issues in Raman amplification is related to the stability of the high power lasers, the costs and the need for efficient cooling. To go around these problems, the usual solution is the use of several pump signals, what results in added advantages, like high, flat and wide-gain bandwidth [51-53].

The technology evolution allowed that high power pumps are nowadays commercially available, although some problems still limited [54]. The pressure on optical components prices, lead to the creation of CWDM standards [55]. This is reflected specially on price dropping of uncooled lasers with relatively high powers (>10mW). The price to pay is wavelength wondering, however, neither for CWDM nor for Raman, wavelength stability is not a stringent requirement, allowing simple control. With this technology the possibility of achieving Raman gain by combining multiple of these low power lasers was successfully implemented [21].

Teixeira et el proposed the use of an array of low cost lasers to achieve wideband Raman amplification, providing both experimental and simulation results [21]. In this work a counterpropagating topology was implemented, using 40 C band lasers with 20 mW output power spaced by 0.8 nm (1533 nm – 1557 nm). These lasers were combined using a multiplexer, bringing up a total power of more than 200mW (23 dBm). This power is enough to generate SRS. Several fibers were tested: True Wave and dispersion compensating. To characterize the gain profile, an array of 40 L-band 0.8nm spaced probe lasers (1565 nm -1605 nm) with a total optical power of 1 mW was used. Figure 12 (a) shows the implemented setup. Figure 12 (b) presents the simulation results for the implemented system to four different pumping configurations. The first curve corresponds to the traditional approach, where one high power pump (23.6 dBm) at a single wavelength (located at 1530 nm) is used. In the second case, three lasers spaced by 0.8 nm starting at 1530 nm having total power of 23.6 dBm were multiplexed. The results for the two above pumping configurations are approximately equal, having only a wavelength shift of 0.8 nm as expected due to the average pumps wavelength difference. Similar simulation was experienced considering 40 lasers, each with 7.6 dBm (after the multiplexer), resulting in a total power of 23.6 dBm. In this case, the


gain curve appears even smothered and the peak shifted by ~16 nm; the peak gain is similar, however a small enhancement on the 3 dB gain bandwidth was obtained and the gain profile smothered. In order to explore the advantages of the methodology (gain flatness), the power distribution for the pumps was optimized to reach an equalized gain, while maintaining the same total pump comb. The peak gain was decreased at the expense of an increased flattened profile.

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Figure 12. a) Implement setup for the simulation and experimental systems, b) simulated Raman gain profiles for several sets of pumping configurations, with 23.6 dBm of total power. PM demotes an optical power meter, OSA denotes an optical spectrum analyzer and MUX is an optical multiplexer.

Due to limitations on available probe and pump signals, the experimental implemented system only can scope part of the spectral bands used in simulation. The gain is only measurable when the pump powers go above 10 dBm. A maximum of 3 dB net gain was achieved in the L band for full pump power, 23.6 dBm, as displayed in figure 13 b). Also, in the same figure, a minimum of 2dB gain over more than 30nm, with 1dB ripple, was achieved without any power distribution optimization. The results have demonstrated the effectiveness of the technique to achieve Raman amplification.


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Figure 13. a) Pump to pump Raman effect; b) experimental Raman gain achieved for several values of the pump power, with probes at 0 dBm.

In figure 13 (a) it is illustrated the pump to pump effect which is commonly occurring in dense WDM systems. This effect starts to be noticeable above 10 dBm of total power and is evident for 7.6 dBm per channel. This phenomenon can be harmful due to uneven distribution of power during transmission, however, if correctly considered can be used to obtain beneficial extra gain in the system, if a pre equalization is also implemented .

4.3. Raman Amplification Using Rayleigh Backscattering

Raman amplification pumping can also be achieved by recurring to the traditional methods of shifted gain [19, 20]. In these methods, several FBG reflector pairs are used to generate resonant cavities in the maximum of the Raman gain spectrum. Thus, with a Ytterbium laser operating in the vicinity of 1090 nm, where it exhibits its maximum efficiency, it is possible to generate pumps in the E band, as demonstrated by Papernyi et al, where a set of 6 FBG reflector pairs were used to generate pumping in the E-band [22]. The latter amplifies the C band, where the probe signals transmission usually occurs.

The main penalties of traditional Raman amplification are associated with intrinsic nonlinear phenomena such as nonlinear refraction and Rayleigh backscattering, since it is required to use high powers and long fiber spans. This last effect occurs when a fraction of scattered light is backreflected towards the launch end of the optical waveguide. This reflection is called single Rayleigh backscattering (SRB). Part of this scattered light is also backreflected in the forward direction and it is called double Rayleigh backscattering (DRB), as shown in figure 14 [56]. SRB and DRB can be controlled by actuating properly on the fiber drawing process or by a correct power design [57]. The Rayleigh backscattering has been studied, modeled and characterized by many authors [56-59]. It is known that the process results from multiple reflections of light inside the fiber and therefore spontaneous and unstable lasing can occur [60]. However, this phenomenon has been observed as an impairment to signal transmission [61, 62].


Figure 14. Simple Rayleigh backscattering (SRB) and double Rayleigh backscattering (DRB) representations over an infinitesimal length of fiber.

Recently, a method that, up to some extent, allows the control of this phenomenon was reported [56,63]. With the possibility of controlling the SRB and DBR effect, novel applications can be drafted. One suggestion is the use of this effect to generate distributed resonant cavities, which will degenerate in lasing if enough gain is achieved. These are achieved with the help of only one end FBG set [63]. This is advantageous when compared to the previously described methods to obtain cascaded Raman amplification, since it needs only one FBG set, minimizing the need for identical FBG to be used and tuned at different sites which can be not colocated.

In order to demonstrate the application of this technique to control SRB and DRB, the experimental system reported in figure 15 was implemented. A Raman pump in the E-band, at 1428 nm, was coupled to the transmission fiber, with controllable power up to 1.5 W. A circulator was used to protect the laser from back reflections and, simultaneously, to allow the measurements of the back reflected power spectrum. Two different scenarios were observed: the FBGs are absent between the fiber and the coupler; and the setup was complete as described in figure 15. These two scenarios target to show the controlling effect achieved by the FBGs.

A set of three FBG with wavelengths centered at: 1520 nm, 1531.6 nm, 1535.6 nm all having 95% reflectivity, were placed after the pump and act as reflective elements.

In a first setup, a 14 km DCF fiber with dispersion parameter equal to -1393 ps/nm and Raman coefficient of 3.05 x 10-3 m-1W-1 was used as transmission medium.

Figure 15. Experimental setup for the double shifted Raman experiments; WDM denotes a band coupler and Att denotes an optical attenuator.

Considering the first scenario, where no FBGs were present, the common Raman effect in the C band was observed, figure 16 a) for a pump power of 350 mW. When the power of


the pump was increased to 600 mW, Rayleigh backscattering spontaneous lasing effect is observed, as displayed in figure 16 b). This effect presents random behavior, both in wavelength and power, being the spectrum time dependent.

In a second scenario the FBGs were present, control of the random process generated by the SRB and DRB was achieved and the lasing was stabilized in the FBGs wavelengths. In this situation a virtual cavity was established, formed by the FBG and the Rayleigh backscattered light. To generate more than one laser in the C-band a set of cascaded wavelength mismatched FBGs were used. These gratings are responsible for a multipeak frequency dependent reflection back into the fiber of the amplified spontaneous emission and DRB light from the fiber. This, in conjunction with the FBG, create resonant cavities, which generate stable wavelength constant lasing actions, from now on called as FBG-DRB lasing.

Due to the different reflectivities of the FBGs and the Raman gain profile, different lasing powers for each configuration occur in the C-band. Whenever the power of the generated lasers in the C-band is high, cascaded Raman effects will occur that generate gain in the far L and U-band. The FBG-DRB lasing and consequent stabilization process with the simultaneous L-U band spontaneous emission is reported in figure 16 c), where a pump power of 1.2 W was used [63].

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Figure 16. Transmission spectra for 14 km DCF fiber: a) Spontaneous ASE for a pump of 300mW; b) spontaneous lasing for a pump of 600mW; c) C-Band FBG-DRB lasing and far L and U-band Raman generated ASE for a 1.2 W pump.

The results show a 38 nm flattened ASE bandwidth in the U-band, generated by the FBG-DRB. By introducing a copropagating probe at 1625 nm, a gain of 10 dB was measured for an E-band pump power of 1 W.

In a second setup, different optical fibers were tested in order to compare the pump power laser threshold. A 14 km long DCF fiber, a 50 km long DSF fiber and a 50 km long non zero dispersion shift fiber (NZDSF) were used [60]. Figure 17 shows the different lasing thresholds and curve shapes resulting from the intrinsic differences between the optical fibers. From figure 17, it can be observed that this process is more efficient in the DCF fibers, where the threshold power is 350 mW, while for the NZDSF fiber is 650mW and 1W for the DSF fiber.


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Figure 17. Depletion of the E-band pump and peak power of the C band lasers as a function of the pump power for several fiber types; from left to right: DSF, NZD and DCF.

Usually, the simulation of Raman amplification as convergence and stability problems, especially for high pump powers, has reported in previous sections. The simulation of the lasing effect with high pump power has similar difficulties. To avoid such problems the solving method for the differential equation system is simplified to an analytical method based on the transfer matrix (APA) as proposed in section 4.1. Inside these fiber slices, the parameters are considered to have small variations and the solution of the equation system is obtained by stabilization after multiple passes along the length of the fiber [64]. The initial solution uses an analytical approach that was based in the undepleted case. The approach to the pump depletion is included in the attenuation of the pump.

In each fiber slice, the Rayleigh backscattering is calculated at the boundary and this backscattering power is added to the signals in the same direction and wavelength, that also suffer amplification and depletion.

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Figure 18. (a) Optical power density spectra for 14 km DCF fiber from E-band to U-band; (b) Output power evolution of the lasing effect of the FBG-DRB at 1520 nm.


Figure 18 (a) presents the simulation results of such algorithm for 29.03 dBm of pump power. The evolution of the power densities from E-band to U-band spectra is shown for a long fiber span. The E-band pump signal suffers depletion in the long propagation fiber. This pump works as a seed of the C-Band FBG-DRB lasing, which generate the L-U-Band Raman gain.

Since the process of lasing is not stable, the simulation process presents a slow stabilization, but, the boundary powers over the FBG are quickly stabilized. Figure 18 (b) presents a comparison of the threshold laser power obtained by experimental and semi-analytical methods. The output peak power of the FBG-DRB signal at 1520 nm is related with the input pump power.

As observed for the DCF fiber, stable multiple laser actions were achieved for moderate pump powers (350 mW) for both simulation and experiment.

4.4. Amplification with Incoherent Pumps

A technique to increase the bandwidth and decrease the spectral ripple of RFA is available with incoherent pump lasers. A Raman amplifier with incoherent pumps can be modeled as a multipump Raman amplifier. In such case, the spectrum of the incoherent pump is well approximated by a large number of pumps of infinitesimal spectral width and whose power sum equals the integral power of the incoherent pump. Therefore, the theoretical model used for incoherent pump schemes is based on the model, previously presented, for coherent multipump configurations.

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Figure 19. Pump spectrum for the incoherent pump.

An incoherent pump spectrum, as displayed in figure 19, with 10 nm FWHM, can be approximated by 100 pumps of infinitesimal spectral width, having an aggregate power equal to the integral power of the incoherent pump. The incoherent pump here considered was


obtained from a high power FBG (Fiber Bragg Grating) laser, from which the stabilization grating was removed [65].

To evaluate the advantages of this technique, the Raman on/off gain and the noise figure were measured for coherent and incoherent pumping over 40 km of SMF fiber. The probe signal combo consists of 13 channels, with 1 mW power, spaced by 100 GHz over the 1546-1556 nm spectral region. Both co-propagating and counter-propagating architectures were considered. The coherent pumping source was a high power FBG laser with a wavelength of 1490 nm. In both cases the pump power was 290 mW.

The results of Raman on/off gain and effective noise figure are shown in figure 20. The relatively low on/off gain is due to the fact that the pump wavelengths have not been optimized for this signal band.

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The incoherent pumping gain slopes are 0.015±0.008 dB/nm and 0.017±0.004 dB/nm for co and counter propagation configurations, respectively. For coherent pumping, the gain slopes are 0.042±0.01 (co-propagation) and 0.052±0.005 dB/nm (counter-propagation). Such results show that the incoherent pumping configuration presents a flatter gain.

The noise figure is approximately the same for coherent and incoherent pumping in the counter-propagating configuration. However, in the co-propagating case, the noise figure is considerably lower for coherent pumping.

In agreement with previous works [23-26], these results indicate that the incoherent pumping technique can be used to decrease the spectral ripple of the Raman gain.

4.5. Raman in CWDM Systems

Another important challenge is the deployment of RFA for access networks, namely for CWDM networks. Since CWDM systems require large bandwidths to guarantee the transmission of a reasonable number of channels, spaced by 20 nm, wide band Raman amplifiers are well suited for this purpose.


The Raman amplifier bandwidth can be enlarged by using multiple pumps. Optimization of the number of pumps and their wavelengths enables the large needed gain spectra and that could be placed in any range of wavelengths used in optical communications.

The design of an amplifier that fits more than two CWDM channels can be achieved, with the following procedure. The number of channels to be transmitted is determined in order to define the required bandwidth. The optical fiber characteristics impose a minimum to the required gain, and finally the number of pumps as well as their characteristics are decided. The scheme of figure 21 illustrates the important issues to be considered to design a multi-pumped Raman amplifier for a CWDM system.

Since the number of CWDM channels and the length of the link as well as its losses are defined, the minimum required gain to compensate the transmission losses and the minimum bandwidth to transmit all the required channels may be determined using the rectangle shown in figure 21. The gain has to be high enough to compensate the losses caused by the optical fiber and the bandwidth should be large enough to support all the transmitted channels. The purpose is to obtain a spectrum that encloses this rectangle.

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Figure 21. Design concerns for a multi-pumped Raman amplifier for CWDM systems.

Another point is to guarantee that the maximum deviation between the values of the designed and the needed gain as smallest as possible. The curved line in figure 21 represents the obtained spectrum after optimization of pumps characteristics. The ripple represents the maximum deviation cited above.

Another concern in designing the spectrum is to make it flat, with all the channels at the same level, in order to avoid reception constrains.

The multipumped Raman amplifier can be designed using a set of coupled nonlinear equations as equation 4. Solving the coupled equations for one signal and one pump, may be simplified when pump depletion is ignored. This approximation is valid because the pump power is higher than the signal power, Pp » Ps [66]. However, whenever multiple pumps are used this simplification cannot be used due to the interaction between pumps which enhances the effect of depletion due to the higher powers involved.


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Figure 22. Example of a multi-pumped Raman amplifier applied to CWDM systems.

Figure 22 shows an example of an optimized Raman amplifier spectrum applied to CWDM systems. It was designed to transmit five probe channels at 1490 nm, 1510 nm, 1530 nm, 1550 nm and 1570 nm. The transmission link is based on 80 km SMF fiber, with 0.23 dB/km losses, which implies a 18.4 dB gain with a minimum spectral bandwidth of 80 nm.

The graph in figure 22 is the result of a forward pumping configuration. The number of pumps used in this example was six. The continuous line represents the gain spectrum obtained with the six pumps the arrows represent the transmitted probe channels.

The bandwidth is 100 nm, 20 nm larger than the minimum required bandwidth, in order to guarantee that all the signals are amplified. The gain is around 18.4 dB with a maximum deviation between the designed and needed gain equal to 1 dB, and a maximum gain deviation for each channel being 0.9 dB.

The six pumps used are centered at 1380 nm, 1393 nm, 1405 nm 1428 nm, 1444 nm, and 1468 nm with powers of 450 mW, 200 mW, 330 mW, 160 mW, 45 mW, and 55 mW, respectively. The pump of lower wavelength needs the highest power due to the interactions between pumps: The lower wavelength pump loses energy to the higher wavelengths, causing its depletion.

This optimization scheme was verified experimentally, with 3 probe channels CDWM system, pumped with 3 pump signals at 1470 nm, 1490 nm and 1510 nm. For the optimization the hybrid GA algorithm, previously presented, was used. This pump allocation problem is less exigent, in terms of ripple, than for a DWDM system, since the probe signal are far apart.

The implemented scenario consists of a 40 km SMF fiber, with a counterpropagating pump scheme and a 7 dB gain target. The optimized pump powers were 128.1 mW, 65.0 mW and 146.9 mW, respectively. The maximum gain excursion was 0.002 dB and 0.12 dB for the simulation and experimental systems, respectively.

Experimentally, we can observe that the Raman amplification improves the eye opening penalty of a signal transmitted along a fiber link allowing a good reception at the end of the


transmission path. To illustrate this behavior, the eye diagram of a signal after a 40 km link is shown in figure 23.

Figure 23. Comparison between eye diagrams with and without Raman amplification.

Figure 23 illustrates a real case where there is a signal centered at 1567 nm and two pumps centered at 1508.8 nm, one in forward configuration and another in backward configuration. The powers of the pumps are chosen to be 100 mW each.

The eye openings are given in Volt and the gain was obtained using the on/off definition (equation 10). Using the relation between power and voltage, P=V2/R, the on/off gain becomes GVoltage = 10log10(Vwith pump/Vwithout pump).

It is notorious that an increase of the eye opening obtained when both pumps are turned on. The scale is the same to all the graphs in figure 23 to allow comparisons of the eye opening amplitude. The eye opening to the bidirectional configuration is higher due to the higher pump power, while the forward and backward systems have 100 mW, the bidirectional system uses 200 mW. The respective gains of the eye openings are 1.94 dB, 2.35 dB, and 2.98 dB to the forward, backward and bidirectional systems, respectively. The results show a higher gain for the counter propagating situation.

5. Conclusion

Raman fiber amplifiers are a technological key component that fulfill the challenging strict requirements of the beginning of this century, enabling applications not feasible with conventional EDFAs.


In this contribution, we have discussed the origin of Raman scattering and the critical properties for system design, such as pumping allocation, cascade pump and broadband amplification for multiple CDWM networks. It was also presented solutions that provide that gain, such as the use of low power pumps or incoherent pumps.

These issues are, in the authors point of view, the relevant questions and challenges associated with Raman amplification on communication systems.

Acknowledgments

This work was supported by the POSC program, financed by the European Union FEDER fund and by the Portuguese scientific program. The authors also greatly acknowledge the ARPA (POSI/EEA-CPS/55781/ 2004) and TECLAR (POCI/A072/2005) projects and to FCT and ALBAN scholarship program.

References

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Chapter 3

FIBER BRAGG GRATINGS IN HIGH BIREFRINGENCE OPTICAL FIBERS

Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski

Instituto de Telecomunicacoes, polo de Aveiro, Aveiro, Portugal

Abstract

Fiber Bragg gratings (FBG) are a key element in optical communication devices and in fiber sensors. This is mainly due to its intrinsic characteristics, which include low insertion loss, passive operation and immunity to electromagnetic interferences. Basically a FBG is a periodic modulation of the core refractive index formed by exposure of a photosensitive fiber to a spatial pattern of ultraviolet light in the region of 244–248 nm. The lengths of FBGs are normally within the region of 1–20 mm. Usually a FBG operates as a narrow reflection filter, where the central wavelength is directly proportional to the periodicity of the spatial modulation and to the effective refractive index of the fiber. The production technology of these devices is now in a mature state, which enables the design of gratings with custom-made transfer functions, crucial for all-optical processing. Recently, some work has been done in the application of FBG written in highly birefringent fibers (HiBi). Due to the birefringence, the effective refractive index of the fiber will be different for the two transversal modes of propagation. Therefore, the reflection spectrum of a FBG will be different for each polarization. This unique property can be used for advanced optical processing or advanced fiber sensing.

The chapter will describe in detail this unique device. The chapter will also analyze the device and demonstrate different applications that take advantage of its properties, like multiparameter sensors, devices for optical communications or in the optimization of certain architectures in optics communications systems.

1. Introduction

The development of the fiber optical technology was an important step in the revolution of global communications and in information technology. One of these developments happened

Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 84

in the 70’s with the first optical fibers with low attenuation [1], a feature that enabled long- distance communication with high bandwidth. The intrinsic optical bandwidth of the optical fibers has also allowed the propagation of different simultaneous channels, allowing the transmission of data at Tbit/s rates [2]. In these systems, in addition to transmission and amplification, it is often necessary to do all-optical processing to the signal. This is due to the inherent advantages of the optical processing, relative to the optic-electric-optic processing, like the higher flexibility to operate at different bit rates and modulation formats and also at the higher bandwidth. The evolution of the fiber optical technology has also enabled the development of devices for all optical processing. In this way, the insertion loss is reduced and the processing quality improved. One of the factors contributing to all-fiber optical processing devices was the discovery of the photosensitivity in optical fibers. It was documented for the first time in 1978 by Hill et al. [3] and led to the development of fiber Bragg gratings (FBG).

A FBG is, generally speaking, a periodic perturbation, along the longitudinal axis, of the refractive index in the fiber core. The production of the refractive index perturbation is done optically in a photosensitive fiber. With the current techniques, it is possible to produce fiber Bragg gratings with different optical properties, which can be designed according to the desired optical processing. In addition to the high flexibility in the production of gratings with custom amplitude and phase responses, the compatibility with common transmission fiber also reduces the insertion loss and decreases the production costs.

The application in optical sensors is also a large potential market for FBG. Their intrinsic low immunity to electromagnetic interference, high dynamic range, passive operation, resistance to corrosion and the possibility of multiplexing hundreds of sensors have made FBG a quite interesting sensor for different applications including medicine, civil, aeronautics or biomechanics. Their properties enable the measurement of temperature and also deformation with extremely high resolution. Nevertheless, it can also be used to measure other parameters using indirect measurements [4-7]. The high potential of these devices has also induced the creation of several companies dedicated to the production and installation of fiber sensors.

There are already good references for the study of FBGs [8,9]. The purpose of this chapter is not to study in detail these devices, but to describe a special case when a fiber Bragg grating is written in high birefringence fibers (HiBi FBG). These special gratings have unique polarization properties that give them exclusive capabilities for optical communications. This is due to the possibility of applying a different optical processing for different polarization components of the signal being transmitted.

HiBi FBGs are also quite interesting for multiparameter sensors, due to their response to temperature variations and deformation. Sensors capable of measuring simultaneously several physical parameters have increased in importance in today’s technological world. In particular, there are various applications of such sensors in civil, mechanical, biomedical or aeronautical engineering, where measurements of different parameters are required [10]. Engineering structures are an example of an application area for the multiparameters sensors, where strain sensing can lead to better understanding about their lifetime and failure. Such knowledge can be critical for some applications like smart skins for airplanes and aeronautical vehicles.

Fiber Bragg Gratings in High Birefringence Optical Fibers 85

2. Fiber Bragg Gratings

A FBG is an optical device produced within the core of a standard optical fiber (figure 1). Basically, it is a periodic modulation of the core refractive index formed by exposure of a photosensitive fiber to a spatial pattern of ultraviolet light. The length of a FBG is dependent on its application, but it generally varies between a few millimeters to a few centimeters.

Fiber Bragg grating

ΛFiber

Figure 1. Scheme of a Fiber Bragg grating written in an optical fiber.

The periodic modulation of the refraction index generates a resonant condition at the Bragg’s wavelength (λB) which is given by the Bragg’s condition:

2B effnλ = Λ

(1)

where neff is the effective refraction index of the fiber and Λ is the modulation period. Therefore, when a FBG is illuminated by a broadband source, a spectral band centered at λB will be reflected back. The reflection function can be determined using the coupled mode theory [11-14], since it is difficult to determine analytically. The exception is the uniform FBG, where it is possible to calculate the reflectivity in an analytical way. Considering a uniform periodic modulation of the refractive index, with amplitude Δn, the reflection coefficient of the grating can be given by

( ) ( )( ) ( )

sinhsinh cosh

LL i L

κ ϕρ λ

δ ϕ ϕ ϕ−

=+

(2)

where L is the length of the FBG, the propagation constant mismatch, δ , is given by

2 effnπ πδ

λ= −

Λ, (3)

2 2ϕ κ δ= − , and κ is the coupling constant given by


nπκ η

λΔ

= (4)

where η is the overlap integral and can be approximated as η≈1 for single mode fibers with step index.

The reflectivity is given by

( )

( )

22

22

2

sinh

cosh

LR

L

ϕρ

δϕκ

= =−

(5)

and the phase by

( )( )

Imarctan

ReR

ρφ

ρ⎡ ⎤

= ⎢ ⎥⎣ ⎦

(6)

Figure 2 shows the calculated reflectivity and the phase of a uniform FBG with L =5 mm

and Δn = 2x10-4 as given by the above equations.

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 1-4

-2

0

2

4

Pha

se (

rad)

Δλ [nm]

Δλ [nm]

Ref

lect

ivity

Figure 2. Reflectivity and phase of a uniform FBG. Parameters: L=mm and Δn= 2x10-4.

If the period changes linearly with the length of the grating, the FBG is said to have a linear chirp. Figure 3 shows the simulation of the reflectivity and group delay of a linear chirped FBG. The simulation method is based on the coupled mode theory.


z

1546 1547 1548 1549 1550 1551 1552 1553 1554-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Wavelength [nm]

Ref

lect

ivity

(dB)

-50

0

50

100

150

200

250

300

Gro

up d

elay

[ps]

Figure 3. Reflectivity and group delay of a linear chirped FBG.

3. High Birefringence Fibers

In an ideal monomode fiber, with a perfect cylindrical core, and with uniform diameter, the fundamental propagation mode is a degenerated combination of two orthogonal propagation modes. However, in real fibers, that degeneration does not exist. In fact, small variations of diameter in the fiber’s core generate a birefringence in the optical fiber. The birefringence can also be a result of an anisotropic stress in the fiber. The local birefringence, B, in each position of the fiber, is defined as

( )x y f x yB n n C σ σ= − = − (7)

where xn and yn are the mean refractive index of the orthogonal polarization modes, σx and

σy are the main stress on the polarization axes and Cf is the photoelastic constant of the fiber. In monomode silica fibers Cf is around 3.08 x10-6 mm2/N for wavelengths near 1500 nm, while B is typically B ≈ 10-7. Due to this small birefringence value, the two polarization components of the light propagating in the fiber have a propagation velocity very similar. Therefore, small environmental perturbations will lead to an energy coupling between one polarization to another. As a result, a linearly polarized light will rapidly evolve to a random polarization. This situation can be avoided with high birefringence fibers. In these fibers, the core has an anisotropic stress, which is generated due to the geometric properties of the fiber. Due to the photoelastic effect, the stress induces a birefringence in the core. Typical values are B ≈ 10-4 [15]. Due to the high birefringence, the propagation constant is different for the two orthogonal propagation modes, which means that the coupling between both transversal propagation modes is far lower as compared to standard fibers. Therefore, the higher the birefringence, the easier will be for a linearly polarized light, propagating in one of the


orthogonal modes, to maintain its state of polarization. Due to this feature, HiBi fibers are also known as polarization maintaining fibers. Figure 4 shows the main structure of the most common HiBi fibers.

Fast axis (Y)

Slow axis (X)

air

PANDA IEC Bow-Tie

Elliptical Core Side-Hole D-Shaped Elliptical Core

Figure 4. Schematic of the transversal section of some of the most known HiBi fibers.

The PANDA (Polarization-maintaining AND Attenuation-reducing), IEC (Internal Elliptical Cladding) and Bow tie fibers have anisotropic glass structures around the core, with a Poisson coefficient different from the rest of the fiber. These structures create the anisotropic stress in the core, which produces the birefringence. The Side-Hole, the Elliptical Core and the D-Shaped Elliptical Core fibers have an elliptical core to generate the birefringence, aided by two air structures, in the case of the Side-Hole or by the shape of the cladding, in the case of the D-Shaped elliptical core. The main axes of the HiBi fibers are designated as fast axis (Y) for the lower refraction index and slow axis (X) for the higher refraction index.

Coherence Length

If a linearly polarized light propagates in a monomode fiber, with a polarization angle of 45º, relatively to the main axes of the fiber, both orthogonal polarization modes will be excited with equal power. If the fiber has a constant birefringence, the mismatch, ФHB(z), between the


orthogonal polarization components will change as a function of the propagation distance on the fiber, z, and it’s given by

( ) ( )HB x yz zβ βΦ = − (8)

where βx and βy are the propagation constants in the X and Y axes respectively. The mismatch will change periodically with the fiber, leading to a change in the state of polarization from linear to elliptical and back again to linear (figure 5).

? =0 ? =π/2 ? =π ? =3π/2 ? =2π

Figure 5. Evolution of the state of polarization in a birefringence fiber.

The spatial periodicity of the evolution of the state of polarization is designated as coherence length (LB). It is determined by the birefringence of the fiber and can be expressed as

/BL Bλ= (9)

where λ is the operating wavelength. Typical coherence lengths for HiBi fibers are in the millimeter scale [16].

4. Fiber Bragg Gratings Written in HiBi Fibers

HiBi fibers can have two linear polarization modes with refractive indexes nx and ny for the slow and fast modes respectively. When a FBG is written in one of these fibers, the periodic modulation will be the same for the two orthogonal polarization modes; however since the effective refraction index is different for the two polarizations, the Bragg wavelength will also be different for each mode. Consequently, expression (1) can be rewritten for the two orthogonal modes:

2 , X,Yi in iλ = Λ = (10)

where λi are the Bragg wavelengths for each polarization mode.


The wavelength difference between the two reflection peaks, ΔλHB, can be calculated by

2 2

HB x y

x yn n

λ λ λΔ = −

= Λ − Λ (11)

The reflectivity of a HiBi FBG will be given by the linear sum of the reflectivity of the

two polarization components, i.e. R(λ)=Rx(λ)+Ry(λ). Rx and Ry are the reflectivity for each polarization given by

( )

( )

22

22

2

sinh

cosh

ii

ii

LR

L

ϕρ

δϕκ

= =−

, i=x,y (12)

where

2 i

inπ πδ

λ= −

Λ, i=x,y (13)

and

2 2i iϕ κ δ= − , i=x,y (14)

Figure 6 shows a simulation, using the previous model, for the reflectivity of a HiBi FBG

with birefringence of B = 3.2 × 10-4.

1547.5 1548.0 1548.5 1549.00.0

0.2

0.4

0.6

0.8

1.0Polarization x

Ref

lect

ivity

Wavelength [nm]

Polarization y

Figure 6. Reflectivity of a simulated HiBi FBG. Simulation parameters: B=3.2 × 10-4, Λ=535 nm, L=10 mm.


If the HiBi FBG is illuminated with light having the two orthogonal components, the reflection spectrum will have those two peaks at orthogonal polarizations. This feature can be very important in some applications, namely in optical communications, as it will be confirmed further in this chapter.

The production of HiBi FBGs uses the same techniques as the ones used in regular FBGs. The only difference will be in the utilization of photosensitive HiBi fiber. Generally it is used a hydrogenated HiBi fiber.

Table 1 shows the dimensions of the anisotropic glass structures around the core of some HiBi fibers obtained through the photographs of the transverse section. The table also displays the main characteristics of HiBi fibers obtained from the manufacturers data sheet.

Table 1. Characteristics of different HiBi fibers. The structures of the HiBi fibers were obtained by microphotography.

Fiber type Commercial provider

Wavelength (nm)

Core diameter

(μm)

Cladding diameter

(μm) Intrinsic stress-applying region

IEC (FS-PM-6621) 3M 1300 8 125 Ellipse: Major axis: 75 μm; Minor

axis: 30 μm Bow tie (F-SPPC-15) Newport 1550 8 125

From core center to extremity of bow tie lobe: 18.4 μm

Bow tie (HB-1500G) Fibercore 1550 8 80

From core center to extremity of bow tie lobe: 16.5 μm

PANDA (SM-13-P-7) Fujikura 1300 8 125

From core center to opposite extremity of side cylinder: 41 μm; Diameter of side cylinder: 32 μm

The reflection spectra for gratings written in the above fibers are shown on figure 7, where the plots of the best-fitted bands are also presented [19]. All the gratings were produced with the phase mask technique. The estimated length of the grating is 10 mm.

From these spectra it can be seen the effect of the intrinsic birefringence of the HiBi fibers. The IEC fiber has the higher birefringence, corresponding to larger spectral splitting between both polarizations bands, while the bow tie fiber presents the lowest birefringence.

1546.0 1546.5 1547.0 1547.5 1548.0

0.0

0.2

0.4

0.6

0.8

1.0

I

NO

RM

ALI

ZE

D IN

TE

NS

ITY

WAVELENGTH (nm) Figure 7. Continued on next page.


1548.0 1548.5 1549.0 1549.5

0.0

0.2

0.4

0.6

0.8

1.0

I

NO

RM

ALI

ZE

D IN

TE

NS

ITY

WAVELENGTH (nm)

1548.0 1548.5 1549.0 1549.5

0.0

0.2

0.4

0.6

0.8

1.0

I

NO

RM

ALI

ZE

D IN

TE

NS

ITY

WAVELENGTH (nm)

Figure 7. Reflection spectra of Bragg gratings written in different HiBi fibers: IEC ( ∇ ), Panda ( Ο ) and bow tie ( Δ ). The continuous line represents the simulated best fit.

Table 2 shows the best-fit parameters obtained in the simulation process. From the fit it is also possible to obtain the values of birefringence of the HiBi fibers.

Table 2. Parameters of FBGs written in HiBi fibers obtained for the best fit for the experimental data.

HiBi Fiber Bands λ (nm) neff kL Λ (nm) B IEC (FS-PM-6621)

λY

λX

1546.57 1547.29

1.44539 1.44606

1.7212 1.7196

535 6.7 x 10-4 @ 1550 nm

PANDA (15P8)

λY

λX 1548.39 1548.82

1.44709 1.44750

1.7172 1.7162

535 4.1 x 10-4 @ 1550 nm

Bow tie (SPPC-15)

λY

λX 1548.61 1548.95

1.44730 1.44762

1.7167 1.7159

535 3.2 x 10-4 @ 1550 nm

The Bragg wavelength peaks of the optical spectrum for both polarizations can change

with temperature and strain. Therefore, considering a HiBi FBG under a temperature variation of ΔT and under a strain aligned with the main axes of the fiber ΔεX, ΔεY and ΔεZ, the


resultant wavelength shift, Δλx and Δλy of both wavelength peaks, λx and λy, can be expressed as

( ) T

nTn)](pp[

2n

XYZ12X11

2X

ZX

X Δ⎥⎦

⎤⎢⎣

⎡ ∂∂+α+εΔ+εΔ+εΔ−εΔ=

λλΔ

(15)

( ) T

nTn)](pp[

2n

YXZ12Y11

2Y

ZY

Y Δ⎥⎦

⎤⎢⎣

⎡ ∂∂+α+εΔ+εΔ+εΔ−εΔ=

λλΔ

(16)

where p11 and p12 are the components of the photoelastic tensor and α is the thermal expansion coefficient of the fiber, α= 0.55×10-6 K-1 [17]. For a fiber based on germanium

and silica, p11=0.113, p12=0.252 and the thermo-optic coefficient is ( )

nTn ∂∂

=8.6×10-6 [18].

Figure 8 shows schematically the effect on the reflection spectrum of a HiBi FBG when it is under temperature variations, under transversal strain or longitudinal strain. The effect of temperature variations or longitudinal strain in the reflection spectrum is equivalent to a translation in the wavelength. On the other hand, when under a transversal strain, the peak separation will change. This difference can be used in multiparameter sensors as it will be discussed further in this chapter.

Wavelength

Pow

er

Wavelength

Pow

er

Increasing longitudinal stress and/or temperature

Increasing transversal strain

Figure 8. Evolution of the reflection spectrum of a HiBi FBG when under a longitudinal stress, temperature variation or transversal strain.


4.1. Characterization of Bragg Gratins Written in High-Birefringence Fiber Optics

4.1.1. Transverse Strain The sensitivity of HiBi FBG to transversal strain can be characterized using a mechanical set-up, like the one shown in figure 9. The transversal load is applied using a micro scratch mechanical system. The system uses an arm to apply a load with a precision of 0.1 N. A grating written in HiBi fiber was placed between two plates having a length of 13 mm. The apparatus arm applies the load to the upper plate. The transverse loads were made for several orientations of the birefringence axis with respect to the direction of the applied load through two fiber rotators. Figure 9 also shows the optical system used to analyze the FBG reflection spectrum. Optical spectra were recorded using an amplified spontaneous emission (ASE) of an erbium doped fiber amplifier as light source, an optical circulator and conventional optical spectrum analyzer (OSA).

Circulator

Figure 9. Set-up for the characterization of HiBi FBGs under a transversal load. The detail shows the transverse section of a HiBi fiber oriented along the angle ϕ.when subjected to applied force F. ASE: Broadband optical source (amplified spontaneous emission); OSA: optical spectrum amplifier.

Figure 10 (a), (b) and (c) shows an example of the reflection spectra of a FBG written in a IEC HiBi fiber as a function of an applied load of 0°, 45° and 90°, respectively.

The results show that, if a load is applied to one of the main axes, fast or slow, it leads to a change in the wavelength of the spectral band associated with the orthogonal axis, while the band associated with the correspondent axis will show a smaller variation. For the applied load angle of 45º both polarization bands present similar evolution. The figure also shows, for different applied load angles (ϕ), the evolution of the peak wavelength of each reflection band with the transverse strain applied to the sample. The strain calibration points in the spectra deformed areas were obtained by identifying and measuring local maximum, minimum and inflexion points. The band split that occurs in some of the spectra is due to a phase shift induced by the applied load. The resulting complex structure is known to be responsible for spectral changes of FBG subject to mechanical stress [9].


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Figure 10. Left: Changes in the spectral response of a FBG written in an IEC HiBi fiber when subjected to an applied load oriented along the angle ϕ: (a) 0° (X-axis); (b) 45º and (c) 90° (Y-axis). Right: Peak position of each band as a function of the applied load. The lines represent the linear best fit for the experimental data.

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Figure 11. Curves of peak sensitivities of the FBG in IEC HiBi fiber as a function of the applied load angle.


Figure 11 shows the wavelength sensitivity curves obtained for both polarization bands. The graph also displays the periodic evolution of the bands as a function of the applied load angle.

Identifying and measuring the reflection peaks as a function of the applied load can be used to obtain the calibration line for each polarization band. The respective slopes can be evaluated and, from them, the dependence of the Bragg wavelength position with the strain can be obtained. Table 3 shows some measurements obtained with FBGs written in IEC and PANDA fibers.

Table 3. Slopes and strain sensitivities of FBGs written in IEC and PANDA HiBi fibers as a function of the direction of applied load (module values). Both fibers have a

diameter of 125 μm.

X- polarization band Y- polarization band

HiBi Fiber Angle of applied

load Slope

(nm/N/mm) Strain sensitivity

(pm/με) Slope

(nm/N/mm) Strain sensitivity

(pm/με)

ϕ = 90° 0.51 7.02 0.07 1.02 IEC ϕ = 0° 0.02 0.29 0.11 1.55

ϕ = 90° 0.46 3.78 0.02 0.24 PANDA ϕ = 0° 0.01 0.11 0.13 2.80

4.1.2. Longitudinal Strain The Bragg wavelength dependence with the longitudinal strain can be measured by gluing one extremity of the fiber in a holder, while the other is glued to a translation stage, which applies a known deformation using a calibrated micrometer.

Figure 12 shows the reflection spectra of a FBG and peak position of each band, written in an IEC fiber as a function of longitudinal strain.

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Figure 12. Left: Changes in the spectral response of a FBG written in IEC HiBi fiber when subjected to a longitudinal strain. Right: Peak position of each band as a function of the longitudinal strain. The lines represent the linear best fit for the experimental data.


Both bands show the same behavior, which is an increase of peak wavelengths as the strain increases. The slopes and the Bragg wavelength sensitivity to longitudinal strain are given in table 4. The obtained ratios between strain and applied load were 758 με/N (X-axis) and 755 με/N (Y-axis).

Table 4. Slopes and longitudinal strain sensitivity of a FBG written in an IEC HiBi fiber.

Bands Slope (nm/N) Longitudinal strain sensitivity (pm/με) X – polarization 1.44 1.9 Y - polarization 1.51 2.0

4.1.3. Temperature The temperature dependence of the reflection bands of HiBi FBGs can be characterized using a cooling/heating system. Figure 13 shows the evolution of the reflection bands and peak position of each band of a Bragg grating written in an IEC fiber as a function of temperature.

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Figure 13. Left: Changes in the spectral response of a FBG written in an IEC HiBi fiber when subjected to different temperatures. Right: Peak position of each polarization band as a function of the temperature. The lines represent the linear best fit for the experimental data.

Table 5 shows the temperature sensitivity values for a FBG written in IEC, PANDA and bow tie HiBi fibers.

Table 5. Slopes of temperature for FBGs written in HiBi fibers.

Slope (pm/°C) HiBi fiber X – polarization band Y – polarization band

IEC 125 μm 6.76 6.71 PANDA 125 μm 3.28 3.40 Bow Tie 125 μm 10.93 11.12 Bow Tie 80 μm 8.02 8.46 The results show that there are quite large variations between the sensitiveness to

temperature for different HiBi fibers. The values changed between ~ 3 pm/°C for the PANDA fiber to ~ 11 pm/°C for the IEC fiber. There are also differences in the


coefficients between polarization bands of the same fiber. For example, in the PANDA fiber this difference is 0.24 pm/°C. These results can be used for simultaneous measurements of temperature and longitudinal strain with only one FBG in a HiBi fiber. This approach, along with others, will be described in the next section.

5. Application in Multiparameter Sensors

FBG sensors are generally based on a unique grating written in a standard fiber optic. The wavelength shift in the reflection spectrum may be used to measure a single component of strain or temperature variation, but not both simultaneously. An adequate measurement of both temperature and strain requires a suitable sensor with a differential sensitivity between parameters. HiBi FBGs can be used as sensors to simultaneously measure one component of transverse strain, temperature and/or longitudinal strain. As it was shown previously in this chapter there are differences in the calibration coefficients of both polarization bands, which can be used to simultaneous measure the temperature and longitudinal strain with only one HiBi FBG. Since the variations of temperature or longitudinal strain causes both bands to shift, and the variation of strain causes asymmetric spectral response in the polarization bands depending of the direction of the applied load, allows the FBG in the HiBi fiber to measure simultaneously transverse strain and temperature or transverse strain and longitudinal strain.

Several types of optical sensors using FBG written in HiBi fibers, which simultaneously measure longitudinal strain and temperature have been proposed and demonstrated [20-24]. Some of the methods include the recording by a CCD camera of the LP01 and LP11 spatial modes [22], using a HiBi FBG partially exposed to chemical etching [20] or by using a quasi-rectangular HiBi fiber to increase the birefringence [21]. In those works, only the longitudinal strain component was measured in simultaneous with temperature. But, there are many applications where it is desirable to determine the transverse strain components in addition to longitudinal strain. Several techniques based in HiBi FBG have already been reported for transverse strain sensing [19, 25-29]. However, when a transverse strain is applied to a HiBi FBG, depending of the fiber orientation relatively to the applied load, the separation of the two Bragg wavelengths can be quite low, so it becomes impossible to resolve the two peaks. To overcome this problem, it can be used an interrogation system capable of detecting independently and simultaneously the two orthogonally polarized signals reflected from the HiBi FBG [26].

There are many applications where it is necessary an ultra small sensor to measure simultaneously components of transverse strain, longitudinal strain and temperature. The use of two superimposed Bragg gratings in HiBi fiber have been described in the literature like potential sensors for monitoring four parameters, two components of transverse strain, longitudinal strain and temperature. [30-34].

5.1. Simultaneous Measurement of Transverse Strain and Temperature

The change in the Bragg wavelength of a HiBi FBG, for each polarization, due to a temperature change ΔT and a transversal strain Δε, is given by


X XX T

Tλ λλ ε

ε∂ ∂

Δ = Δ + Δ∂ ∂

(17)

Y YY T

Tλ λλ ε

ε∂ ∂

Δ = Δ + Δ∂ ∂

(18)

were ∂λX/∂T and ∂λY/∂T are the temperature coefficients and ∂λX/∂ε and ∂λY/∂ε are the transverse deformation coefficients.

Expressions (17) and (18) can be rearranged and written in matrix form in order to calculate the transverse strain and temperature, given the measured wavelength shifts for each polarization band:

1 X

Y

T λε λ

− ΔΔ ⎡ ⎤⎡ ⎤= Κ ⎢ ⎥⎢ ⎥Δ Δ⎣ ⎦ ⎣ ⎦

(19)

where K is a matrix given by

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

=

ελλ

ελλ

YY

XX

,T

,TK (20)

A simultaneous measurement of transverse strain and temperature can be obtained by

determining the coefficients of K, which are determined with previous characterization. Two examples of simultaneous measurement of these parameters are shown in the table 6

for an IEC fiber and table 7 for a PANDA fiber. The results were obtained using the values of the Bragg wavelength changes for both polarizations bands.

Table 6. Simultaneous measurements of temperature and transverse strain using a FBG written in a IEC HiBi fiber. The set values were determined by the experimental system

equipment. Direction of applied load: 0º. [19].

Set values 12 °C 23 °C 31 °C 46 °C

11.8 °C 24.6 °C 31.5 °C 45.5 °C 61 µε 65 µε 71 µε 70 µε 75 µε

12.0 °C 25.4 °C 32.6 °C 46.3 °C 76 µε 69 µε 79 µε 73 µε 81 µε

12.7 °C 26.8 °C 34.0 °C 48.0 °C 91 µε 76 µε 83 µε 78 µε 79 µε


Table 7. Simultaneous measurements of temperature and transverse strain using a FBG in written in a PANDA HiBi fiber. The set values were determined by the experimental

system equipment. Direction of applied load: 90º.

Set values 7 °C 21 °C 22 °C 40 °C 53 °C

7 °C 20 °C 23 °C 39 °C 51 °C 11 µε 12 µε 12 µε 9 µε 11 µε 10 µε





5.2. Simultaneous Measurement of Transverse Strain and Longitudinal Strain

For the measurement of the longitudinal (ΔεZ) and transverse (ΔεX or ΔεY) strain, the equations can also be written in matrix form, given the measured wavelength shifts for each polarization band:

1

,

Z X

X Y Y

ε λε λ

−Δ Δ⎡ ⎤ ⎡ ⎤

= Κ⎢ ⎥ ⎢ ⎥Δ Δ⎣ ⎦⎣ ⎦ (21)

where K is now given by:

,

,

,

,

X X

Z X Y

Y Y

Z X Y

K

λ λε ε

λ λε ε

∂ ∂⎡ ⎤⎢ ⎥∂ ∂⎢ ⎥=⎢ ⎥∂ ∂⎢ ⎥∂ ∂⎣ ⎦

(22)

Table 8 shows an example of simultaneous measurements of longitudinal and transversal

strain obtained using the wavelength changes of both polarizations bands.


Table 8. Simultaneous measurements of longitudinal and transverse strain using an FBG written in a IEC HiBi fiber. The set values were determined by the experimental

system equipment. Direction of applied load: 90º.

Set values 0 µε 9 µε 14 µε

1 µε 8 µε 13 µε 83 µε 64 µε 66 µε 68 µε

1 µε 8 µε 13 µε 167 µε 160 µε 161 µε 154 µε

0 µε 7 µε 12 µε 251 µε 240 µε 246 µε 244 µε

1 µε 8 µε 13 µε 335 µε 320 µε 316 µε 313 µε

5.3. Simultaneous Measurement of Transverse Strain, Longitudinal Strain and Temperature

Two superimposed Bragg gratings can be written in high birefringence fiber optics to measure simultaneously temperature, transverse and longitudinal strain.

This section demonstrates the use of a pair of Bragg gratings written in high birefringence fiber optics to measure, simultaneously, three physical parameters [31]. The Bragg gratings are superimposed in the same position of the fiber optic, in order to behave as a single sensor with reduced dimension.

5.3.1. Superimposed Bragg Gratings

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Figure 14. Optical reflection spectrum of two superimposed Bragg gratings written in HiBi IEC fiber [31].


Figure 14 shows an optical reflection spectrum of two gratings recorded at the same fiber position. The two FBG were written with different periods in an IEC HiBi optical fiber with 125 µm diameter. The figure shows the polarization bands (Y-polarization and X-polarization) of each pair. Their relative intensity is not the same as the optical source was not flat along the full wavelength range.

The superimposed HiBi FBGs were characterized by longitudinal, transversal strain and temperature. The measurements of transversal load were made with the fiber oriented with the fast or slow birefringence axis in the direction of the applied load.

Figure 15 shows the dependence of the peak position of each reflection band against the transversal strain applied to the sample (load applied along the Y-axis direction). The best-fitted lines are not parallel; their slopes are different depending on the polarization band. This asymmetric behavior can be used to distinguish the effects of longitudinal and transversal strain acting upon the grating pair.

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Figure 15. Dependence of the peak wavelength on transverse strain for the reflection bands [X ( ) and Y ( )] of the two superimposed FBGs written in an IEC HiBi fiber. Direction of applied load: 90º. The lines represent the linear best fit to the experimental data [31].

The behavior of the reflection bands, when the sensor is under longitudinal strain, is the same for both gratings. The temperature dependence of the reflection bands of the both FBGs has also approximately the same behavior, which is an increase in the wavelength with an increase of temperature.

Table 9. Slopes of temperature, longitudinal and transverse strain for the two superimposed FBGs in an IEC HiBi fiber. Direction of applied transverse load: Y-axis [31].

Polarization bands Slopes

Y1 X1 Y2 X2 ∂λ/∂T (pm/°C) 8.4 7.8 7.8 7.5 ∂λ/∂ε Y (pm/με) 0.08 4.02 0.19 4.11 ∂λ/∂ε Z (pm/με) 1.3 1.39 1.39 1.36


The corresponding slopes of temperature, longitudinal and transversal strain for both polarization bands, for the best-fitted lines of superposing FBGs in IEC HiBi fiber, are given in table 9.

5.3.2. Simultaneous Measurements The change in the Bragg wavelength of the reflection spectrum of the both FBGs, due to a temperature change ΔT, a transverse strain (ΔεX or ΔεY) and longitudinal strain ΔεZ, for each polarization, is given by

1 1 11 ,

,

X X XX X Y Z

X Y Z

TT

λ λ λλ ε εε ε

∂ ∂ ∂Δ = Δ + Δ + Δ

∂ ∂ ∂ (23)

1 1 11 ,

,

Y Y YY X Y Z

X Y Z

TT


∂ ∂ ∂Δ = Δ + Δ + Δ

∂ ∂ ∂ (24)

2 2 22 ,

,

X X XX X Y Z

X Y Z

TT


∂ ∂ ∂Δ = Δ + Δ + Δ

∂ ∂ ∂ (25)

2 2 22 ,

,

Y Y YY X Y Z

X Y Z

TT


∂ ∂ ∂Δ = Δ + Δ + Δ

∂ ∂ ∂ (26)

where ∂λX1/∂T, ∂λX2/∂T, ∂λY1/∂T and ∂λY2/∂T are the temperature coefficients, ∂λX1/∂εX,Y, ∂λX2/∂εX,Y, ∂λY1/∂εX,Y and ∂λY2/∂εX,Y are the transversal deformation coefficients, and ∂λX1/∂εZ, ∂λX2/∂εZ, ∂λY1/∂εZ and ∂λY2/∂εZ are the longitudinal deformation coefficients.

Equations (23) to (26) can be rearranged and written in matrix form, in order to calculate the transverse, longitudinal strain and temperature, given the measured wavelength shifts for each polarization band. In this way, the calculation of the three parameters being measured can be made using the following (the choice of reflection bands was arbitrary):

1

, 1

2

Y

X Y X

Z Y

TK

λε λε λ

−

Δ Δ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥Δ = Δ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ Δ⎣ ⎦ ⎣ ⎦

1 (27)

where K is assembled from the several sensitivities for temperature and deformation:

1 1 1

,

1 1 1

,

2 2 2

,

K

Y Y Y

X Y Z

X X X

X Y Z

Y Y Y

X Y Z

T

T

T

λ λ λε ε

λ λ λε ε

λ λ λε ε

⎡ ⎤∂ ∂ ∂⎢ ⎥

∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂

= ⎢ ⎥∂ ∂ ∂⎢ ⎥

⎢ ⎥∂ ∂ ∂⎢ ⎥∂ ∂ ∂⎢ ⎥⎣ ⎦

(28)


After a previous characterization, in order to obtain K, the temperature, longitudinal and transversal strain components can be simultaneously measured. Some of the obtained results with the grating pair described above are given in table 10.

Table 10. Simultaneous measurements of temperature, transverse and longitudinal strain using two superimposed FBGs in IEC HiBi fiber. The set values were determined

by the experimental system equipment [31].

167 με 251 με Set values

15 °C 45 °C 15 °C 45 °C 12 °C 42 °C 12 °C 37 °C 13 με 10 με 16 με 10 με 12 με 117 με 99 με 228 με 177 με 16 °C 33 °C 16 °C 43 °C 18 με 16 με 16 με 24 με 22 με 141 με 132 με 252 με 187 με 16 °C 36 °C 18 °C 43 °C 32 με 29 με 21 με 32 με 32 με 116 με 139 με 236 με 178 με

5.4. Bragg Gratings in Reduced Diameter High Birefringence Fiber Optics

Bragg gratings written in reduced diameter high birefringence fiber optics can also be used for multiparameter sensing. Changes in the stress profile of HiBi fibers due to reduced diameter can modify the response of a FBG sensor system to strain or temperature optimizing the simultaneous measurement of those parameters. Chemical etching can be a good tool to reduce the fiber diameter. The changes in the birefringence properties of HiBi fibers as a function of fiber diameter can be analyzed using fiber samples chemically etched in hydrofluoric acid (HF), while the optical spectra of pre-recorded gratings are measured [34].

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EXPOSURE TIME (min)

Figure 16. Diameter of an IEC HiBi fiber as a function of the exposure time. HF concentration: 20% [34].


The diameter of the fibers during the etching can be measured by having several samples of the fiber in the acid. The samples are removed successively from the acid, rinsed in distilled water, dried, and then measured under a microscope with a calibrated scale.

The evolution of the diameter, as a result of etching, for an IEC fiber is presented in figure 16. HF acid was diluted to 20 % (parts per volume) in order to reduce the velocity of chemical etching and to increase the sampling points along the process. Figure 17 shows the changes in the transversal section of the IEC fiber, with 125 μm of diameter (left) and after etching (right), with 86 μm of diameter. The internal elliptical cladding can be observed in these photographs. The major axis of the ellipse has approximately 75 μm. The etched IEC fiber shows a higher asymmetry on the borders close to the axes along the major axis of the internal elliptical cladding.

Figure 17. Microphotographs of the transverse section of an IEC HiBi fiber. Left: standard HiBi fiber with 125 μm of diameter. Right: etched HiBi fiber with 86 μm of diameter [34].

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Figure 18. Left: evolution of the reflection bands of a FBG written in an IEC HiBi fiber as a function of the etching time. Right: peak position of the polarized bands (Y-polarized (∇) and X-polarized (Δ))as a function of the fiber diameter. The lines represent the linear best fit for the experimental data. HF concentration: 20 % [34].


Figure 18 (left) illustrates the optical reflection spectra of the FBG in IEC fiber, obtained as a function of HF exposure time. After 36 minutes of exposition time, the optical spectrum had a single band, which means that, the fiber birefringence was almost zero. That is a consequence of the stress release due to the etching.

Figure 18 (right) shows the changes in the peak position of the reflected polarized bands as a function of the IEC fiber diameter. The different slopes for the X and Y polarized bands can be related to asymmetric changes of the internal stress applied by the internal elliptical cladding.

The evolution of the birefringence, as a function of the diameter, can be seen in figure 19.

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RE

FRIN

GE

NC

E

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Figure 19. Calculated birefringence of the IEC HiBi fiber as a function of diameter. HF concentration: 40 % (∇) and 20 % (Δ) [34].

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Figure 20. Left: evolution of the polarized bands of a FBG written in a bow tie HiBi fiber as a function of the etching time. Right: peak position of polarized bands (Y-polarized (∇) and X-polarized (Δ)) as a function of the fiber diameter. The lines represent the linear best fit for the experimental data. HF concentration: 20 % [34].

A similar characterization can be made to other types of HiBi fibers. For example, figure 20 (left) shows the effect of chemical etching in the optical spectrum of a Bragg grating written in a bow tie fiber. The etching rate is lower and it is possible to observe that the two polarization bands collapse. Initially both bands show a trend to longer wavelengths on their


peak position, as the diameter changes from 100 µm to 65 µm (figure 20 (left)). Further etching now causes the X polarized band to shift sharply to shorter wavelengths, until both bands collapse when the diameter reaches approximately 40 µm. This value agrees with the intrinsic stress-applying region dimensions, where the distance between the boundaries of the two internal side-lobes is approximately 37 μm.

Figure 21 shows the birefringence for a bow tie fiber as a function of the diameter. The results show that IEC and bow tie fibers have vanishing birefringence for diameters that are close to the value of the maximum dimension of the stress-applying region.

BIR

EFR

ING

EN

CE

DIAMETER (μm)

Figure 21. Measured birefringence of bow tie HiBi fiber as a function of diameter. HF concentration: 20 % [34].

5.4.1. Reduced Diameter for the Simultaneous Measure of Transverse Strain and

Temperature A FBG in an etched HiBi fiber can be applied as a sensor to simultaneously measure the transverse strain and temperature. Once again, a previous calibration of the different sensitivities must be made. The temperature and transverse strain coefficients for an etched IEC fiber is shown in table 11. It also displays the coefficients for a non-etched bow tie fiber with a similar diameter.

Table 11. Slopes of temperature and transverse strain of a FBG written in etched IEC and non-etched bow tie HiBi fibers [34].

Temperature Transversal strain Fiber (diameter) ∂λx/∂T (pm/°C) ∂λy/∂T (pm/°C) ∂λx/∂ε (pm/με) ∂λy/∂ε (pm/με)

Etched IEC (82 μm)

7.00 6.90 0.7 (X -axis) 3.4 (Y -axis)

2.23(X -axis) 0.1 (Y -axis)

Bow tie (80 μm)

8.02 8.46 0.02 (X -axis) 1.2 (Y -axis)

1.16 (X -axis) 0.3 (Y -axis)


The results of simultaneous transversal strain and temperature measurements obtained with matrix K and the values of XλΔ and YλΔ of the reflection spectra are displayed in table 12.

Table 12. Simultaneous measurements of temperature and transverse strain using etched FBGs in IEC HiBi fiber (diameter of 82 μm). The set values are determined by

the experimental system equipment. Direction of applied load: 90º [34].

Set values 16 °C 26 °C 36 °C 46 °C 56 °C 15 °C 28 °C 33 °C 44 °C 56 °C 33 με

37 με 37 με 39 με 42 με 42 με 17 °C 29 °C 36 °C 47 °C 56 °C 48 με




80 με 100 µε 94 με 91 με 85 με The errors obtained using a FBG in normal and reduced diameter HiBi fibers as a sensor

are of comparable magnitude, but the dynamic range for strain measurements with the later ones is almost doubled as compared to the former sensors. This fact is important for technological applications where FBG can be tailored to attend a specific measurement range.

6. Applications to Optical Communications

All optical processing devices are becoming a key element in the next generation of optical communication systems, since they play a critical role in pulse formatting, spectral shaping and optimized all-optical routing and switching. These devices don’t have the typical bottleneck associated to the optical-electrical-optical conversion and the majority is transparent to modulation format and bit-rate. FBGs are quite interesting for these applications, due to their low insertion loss and due to the avoidance of the decoupling of the signal outside the fiber. Moreover, the production technology is now in a mature state, which enables the design of gratings with custom made transfer functions, crucial for all-optical processing. Some advanced processing can be made if the transfer function is different for the two transversal modes of propagation in the fiber. This can be achieved by a HiBi FBG. One of the devices that take full advantage of the optical processing capabilities of the HiBi FBG is the orthogonal pumps source [35-37], which can be used in all optical wavelength converters [38, 39]. A tunable PMD compensator can also be developed based on the polarization processing properties of these special gratings [40, 41]. Also, a tunable microwave-photonic notch filter that makes use of a time delay element based on tunable HiBi chirped FBG has been demonstrated [42, 43] In addition, the interference due to laser


coherence, typical in those micro-wave photonic filters was also reduced due to the polarization properties of the HiBi FBGs.

The following sections describe some example application of HiBi FBG in optics communications.

6.1. Optical Delay Line for PMD Compensation

In a linearly chirped grating, written in a HiBi fiber, each position of the grating will reflect two wavelengths at orthogonal polarizations (figure 22). This means that the group delay of these gratings is a combination of two linear functions, one for each polarization, with the same slope (DFBG) and shifted by ΔλHB:

( )( ) ( )

y FBG

x FBG HB

D b

D b

τ λ λ

τ λ λ λ

= +

= − Δ + (29)

where b in (29) is a constant.

Therefore, the relative group delay induced by a linearly chirped FBG written in a HiBi fiber (∆τ=τx-τy) is calculated using the following expression

2

FBG

FBG

DD B

τ λΔ = − Δ≈ − Λ

(30)

Expression (30) shows that the dynamic tuning of the induced PMD can be made by

adjusting the birefringence of the fiber, which can be done by applying a transversal stress in the fiber, as shown before in this chapter.

1546 1548 1550 1552 1554-50

-40

-30

-20

-10

0

y polarization x polarization

Wavelength [nm]

Ref

lect

ivity

[dB

]

0

50

100

150

200

250

300

ΔλHB

Gro

up d

elay

[ps]

Δτ

Figure 22. Reflectivity and group delay of a linearly chirped HiBi FBG for both transversal propagation modes [47].


6.1.1. Compensation Using a Linear Chirp As can also be observed in expression (30), it is also possible to tune the PMD by adjusting the dispersion of the grating. That can be done using different methods [44-46]. One of them is by using thermal gradients to induce a linear chirp to a uniform FBG. Let us consider a uniform HiBi FBG put in a thermal contact with metal substrate. By applying different temperatures to the substrate, different linear temperature gradients will be generated. This gradient will induce a linear chirp to the FBG, due to thermo-optic and photoelastic effects. By changing the temperature gradient, the dispersion will also change, inducing a tunable differential delay line [47]. Figure 23 shows the experimental results of the evolution of ∆τ as a function of the applied temperature gradient to a 24 mm uniform HiBi FBG.

10 20 30 40 50

-50

0

50

100

Δτ [p

s]

ΔT[ºC]

Figure 23. Relative group delay as a function of the applied linear gradient to a uniform HiBi FBG with 24 mm length.

Therefore, the presented device can be included in a PMD compensator as a tunable optical relative group delay line.

6.1.2. Compensation Using a Nonlinear Chirp Let us now suppose that we have a HiBi FBG with a quadratic chirp. The group delay is now composed by two parabolic functions (one for each polarization) shifted by ΔλHB. If the grating is tuned by temperature or longitudinal stress, the relative induced delay between the orthogonal polarizations, for a specific wavelength will change [40]. Figure 24 shows a simulation of a quadratic chirped FBG, with a length of 25 mm, written in a HiBi fiber with birefringence B = 5x10-4. For a tuning of 4.5 nm in the central wavelength, the relative group delay at 1550 nm changed from 41.6 ps to 12.1 ps. In this way, with this method, it is possible to do small corrections in the relative group delay.

The advantages of this method are its tuning simplicity and the flexibility in the operation range. However, the technique needs a FBG with a nonlinear chirp, which is quite complex to produce. It is generally produced with a custom made phase mask with a nonlinear chirp.


1544 1546 1548 1550 1552 1554 15560

50

100

150

200

250

Δτ = 12.1 ps

Gro

up d

elay

[ps]

Wavelength [nm]

Δτ = 41.6 ps

Tuning

Figure 24. Simulation of the group delay of a HiBi FBG with quadratic chirp. Line: Y polarization; dots: X polarization [47].

6.2. Tunable Multiwavelength Linear Polarized Fiber Lasers

Fiber lasers have different applications in sensors and telecommunications due to their reduced linewidth, power and spectral profile. Like other lasers, fiber lasers need two components: a gain medium and a resonant cavity. For a fiber laser operating around 1550 nm, it is generally based on an optical pump with 980 or 1480 nm of wavelength, an erbium-doped fiber and an optical filter. The gain is obtained from the amplified spontaneous emission due to the optical pump.

Generally, fiber optical lasers based on an optical ring with erbium-doped fiber don’t enable the generation of more than one laser line [48,49]. This is a consequence of the fact that erbium is a medium with homogeneous gain at room temperature, resulting in strong mode competition, which induces laser instability. A method was proposed to reduce the homogeneity of the fiber by cooling the fiber to 77 K [50, 51]. However, by obvious reasons, it is not very practical. Other methods used special fibers like the elliptical core fibers [52] or the twincore fibers [53].

PC

Circulator

Two Tunable HiBi FBG

Output

Optical Pump (980 nm) EDF

WDMc

Figure 25. Diagram of a multiwavelength fiber laser based on HiBi FBGs. EDF: Erbium doped fiber; PC: Polarization controller; WDMc: WDM optical coupler;


Another way to reduce the homogeneity of the fiber is to use different laser lines operating at different longitudinal modes. For this kind of implementation, HiBi FBGs can have an important role, since they will reflect two wavelengths at orthogonal polarizations. An implementation method for a tunable laser with up to four laser lines is depicted in figure 25.

-60

-50

-40

-30

-20

-10

0

-60

-50

-40

-30

-20

-10

0

1530 1535 1540 1545 1550 1530 1535 1540 1545 1550

-60

-50

-40

-30

-20

-10

0

xx

yy

x

x

xx

x

y y

yPo

wer

[dB

m]

Wavelength [nm]

Figure 26. Optical spectra at the output of the fiber laser with different operation modes. The operating laser lines are at a linear polarization (x or y).

The two tunable HiBi FBGs enable the selection of 4 different wavelengths. By tuning the polarization controller (PC) inside the optical cavity, it is possible to select the appropriate laser lines. Figure 26 shows some of the possibilities that can be achieved with just two HiBi FBGs.

One of the advantages of this technique is its ability to generate two laser lines at orthogonal polarizations (see last spectrum of figure 26). Therefore, it can be used as two orthogonal pumps in a polarization insensitive wavelength converter [38].

6.3. Optical Networks Architectures Using HiBi FBG for Performance Improvement

6.3.1. Optical Code Division Multiple Access Metro optical code division multiple access (OCDMA) networks can benefit from the polarization multiplexing, since two users using codes in the same time-wavelength chip can be given orthogonal polarizations to operate, therefore reducing interference. One of the


implementation techniques is the “polarization assisted OCDMA with HiBi FBG” [54]. The technique uses the polarization properties of the HiBi FBG along with a special code generation scheme to improve the performance of OCDMA based networks. The coders are based on HiBi FBGs. To implement the suggested polarization assisted OCDMA, each HiBi FBG will reflect a pair of wavelengths λiλj, which are consecutive and cross polarized. In the case of the proposed method, a set of three of these HiBi FBGs, spaced by the fiber length needed for achieving the corresponding time chip spacing, results in two subsequent codes. Here, λi corresponds to a X polarized reflection and λj to a Y polarized one. This allows two consecutive user spreading sequences to share the same encoder. An implementation example is depicted in figure 27 .

User A

PBC

Circ HiBi FBG

λ24,25 λ3,4 λ11,12

X

User B

Y

Encoder

HiBi fiber X

Y

A

B

Figure 27. Schematic of the proposed encoder implementation showing the use of the polarization to encode simultaneously two users with different wavelengths at orthogonal polarizations. Legend: PBC: polarization beam combiner; Circ: optical circulator [54] (© 2006 IEEE).

X Y

λ11,12 λ3,4 HiBi fiber Circ

PBS

Decoder

HiBi FBG

λ24,25

Figure 28. Schematic of the proposed implementation for the decoder based on HiBi FBG. Legend: PBS: polarization beam splitter; Circ: optical circulator [54] (© 2006 IEEE).

Each bit of information from users A and B is a wavelength comb which includes at least the wavelengths of the correspondent code (or a standard modulated broadband source can be used). Both bit sequence signals are multiplexed using a polarization beam combiner, thereby


ensuring that they enter the encoder at the correct orthogonal polarizations. The encoder is based on three HiBi FBG reflecting the wavelengths λ3, λ11 and λ24 for X polarization and λ4, λ12 and λ25 for Y polarization. To achieve networking operation many such encoders need to operate simultaneously, and due to the properties of the technique, the number of encoders needed reduces to almost half.

The decoder can be imprinted with standard FBG, since each user has its own code. However, for reduced user interference and to reduce the number of decoders needed, it can be based on HiBi FBG like the one exemplified in figure 28.

The decoding process is similar to the encoding, where the HiBi FBG correlates two codes simultaneously. Afterwards a polarization beam splitter is used to separate both users. If no polarization maintaining fiber is used in the transmission link between the encoder and the decoder, the former must be preceded by a polarization rotator to ensure correct polarization coupling to the receiver. The polarization rotator can be automatically controlled by the receiver of one of the users using simple electronics. Even if no alignment of the polarization is made between non adjacent users, on average, only half the power will induce interference since the decoder will process only one of the two available polarizations. In the same way, the sensitivity to heterodyne crosstalk is also reduced since the power of the adjacent user, generated by the same encoder, is orthogonally polarized. In opposition to other coding/decoding techniques, like the ones based on arrayed waveguide gratings and optical delay lines, the proposed coder/decoders are quite compact, simple to use and have low insertion losses. On the other hand, since the gratings can have a length down to 1-2 mm and still have a high reflectivity, the time slots can be as low as a few picoseconds which can be considered enough for the majority of applications.

6.3.2. Radio over Fiber In radio over fiber systems (RoF), using the same central station to transmit to different local stations, one can use frequency interleaving to improve the bandwidth efficiency, exploiting the unused band between the carrier and data when high modulation frequencies are used with single side band (SSB) format. However, frequency interleaving also increases the bit error rate (BER), due to the interference of adjacent carriers. This drawback can be minimized if polarization multiplexing is used, i.e., the carriers and data are at orthogonal polarizations (figure 29).

Interference: same polarization(40 GHz spacing)

Traditional Implementation Implementation with HiBi FBG

Data of interest

Interference: bothpolarizations

(20 GHz spacing)

o o o o o o o o X Y X Y X Y X Y o o o o o o o o x y x y x y x y

Figure 29. Diagram of the concept of interleaving using polarization multiplexing between carriers and data.


The implementation of this concept can be made using a HiBi FBG filter at the transmission which creates the SSB format and, at the same time, selects one polarization for the carrier and the orthogonal one for the data. At the local station another HiBi FBG removes the selected channel with reduced interference, since the interference will only be made by the data of the adjacent channels, which are at higher wavelength spacing and with lower power, relatively to the adjacent carriers. This technique has an impact on the overall performance of the system since the bandwidth efficiency can be improved without increasing the BER [55].

7. Conclusion

This chapter described some of the characteristics and functionalities associated with HiBi FBG. Their anisotropic behavior, relative to stress and/or strain, make them well suited for multiparameter sensors, including temperature, transversal strain and longitudinal strain. Moreover, their polarization processing capabilities also give them an interesting potential for different applications in optics communications. These applications include the development of new devices, like multiwavelength fiber lasers or in the optimization of certain architectures, like OCDMA. Some of the applications for sensing and optical communications were described but many more are yet to come.

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In: Optical Fibers Research AdvancesEditor: Jurgen C. Schlesinger, pp. 119-159

ISBN 1-60021-866-0c© 2007 Nova Science Publishers, Inc.

Chapter 4

APPLICATIONS OF HOLLOW OPTICAL FIBERS

IN ATOM OPTICS

Heung-Ryoul Noh1 and Wonho Jhe2∗

1Department of Physics and Institute of Opto-ElectronicScience and Technology, Chonnam National University,

Gwangju 500-757, Korea2School of Physics and Astronomy,

Seoul National University, Seoul 151-742, Korea

Abstract

A hollow optical fiber (HOF) has a lot of interesting applications in atom opticsexperiments such as atom guiding and the generation of hollow laser beam (HLB).In this article we present theoretical and experimental works on the use of hollowoptical fibers in atom optics. This article is divided into two parts: One is devotedto the atom guide using HOFs and the other describes the atom optics researches thatutilizes laser lights emanated from the HOF. Firstly, we describe the electromagneticfields inside the HOF and characterize the electromagnetic modes diffracted from theHOF. Then we describe two guiding schemes using red and blue detuned laser lights.Finally, we describe the various relevant experiments using LP01 or LP11 modes suchas the generation of HLB from the HOF, funneling atoms using the diffracted fields,diffraction-limited dark laser spot, and a dipole trap using LP01 mode of the diffractedfield from the HOF.

PACS: 32.80.Pj, 42.50.Vk, 39.25.+k, 32.80.-t, 03.75.Be

1. Introduction

For the last two decades, there has been much progress on atom optics that manipulatesatoms by using laser lights [1, 2, 3, 4, 5, 6, 7, 8, 9]. This field includes the studies suchas focusing, reflecting, diffracting, and guiding atoms. In particular, it is an atom guidethat provides a high spatial resolution of atom manipulation. So far several types of atom

∗E-mail address: [email protected]. (Corresponding author)

120 Heung-Ryoul Noh and Wonho Jhe

guides using the optical, magnetic means, and using hollow optical fibers (HOFs)have beendemonstrated. Of these, guidance of atoms by using HOFs is widely investigated, since itenables atoms to be controlled very accurately and guided over a long distance at highspatial accuracy without much atomic loss. A hollow optical fiber has many interestingapplications in sensor [10] and harmonic generation [11], and optical communications [12,13, 14, 15]. In this article, we describe the application of the HOF in atom optics.

The basic principle of an optical atom guide is that atoms in laser beams are eitherattracted to or repelled from the regions of high intensity laser light, depending on the signof the laser frequency detuning with respect to the atomic resonance. For two level atoms,the optical dipole potential due to the guiding laser is described by [16]

U(r) =~∆

2ln

1 +I(r)/Is

1 + 4∆2/Γ2

, (1)

where∆ = ωL − ω0 is the laser detuning from the atomic resonance,Is is the saturationintensity, I(r) is the guiding laser intensity, andΓ is the spontaneous decay rate of theupper level. When the laser frequency is tuned slightly below the atomic resonance, orred-detuned, atoms are attracted to the high-intensity regions. One the other hand, atomsare repelled from the high-intensity regions when the light is tuned above the resonance, orblue-detuned.

One way to guide atoms is to launch the laser light inside the hollow region of the hollowfiber and tune the laser frequency to the red side of the atomic resonance as proposed byOl’shanniet al. [17]. This guiding scheme was successfully realized by Rennet al. at JILA[18]. In that study, a red-detuned laser was coupled to the lowest-order grazing incidencemode inside the glass capillary. The second method is to introduce the laser light into theglass core of the hollow fiber and tune the laser frequency to the blue side of the atomicresonance. This method was first proposed by Savageet al. [19, 20] and later by Jheetal. [21] for different waveguide configurations. This guiding scheme was demonstrated forthe capillary fiber by Rennet al. [22] and then subsequently demonstrated for the micron-sized hollow optical fiber by Itoet al. [23]. A similar experiment was later performed byWorkurkaet al. [24] and also by Dallet al. [25, 26].

In addition to the atom guidance, the HOF has another application in atom optics, whichis the generation of laser beams diffracted from the facet of the HOF. The first use of theHOF for the generation of a hollow laser beam (HLB) by imaging the field distribution ofLP01 mode was demonstrated by Yinet al. [27]. Such hollow laser beams have been usedfor atom guidance [28], atom fountains [29], and atom traps [30]. The characterizationof the output-field distribution of the hollow optical fiber was described in detail [31, 32].The hollow laser beams made by a combination of two orthogonal LP11 modes without animaging lens can be used for funneling and guiding atoms [32]. They can also be used forgeneration of the diffraction-limited dark laser spot [33]. Furthermore the diffracted outputof the LP01 mode has a bright focused spot, which can be used for a tight optical dipole trapwhen a red-detuned laser is used [34].

This article is organized as follows: In the next section, we characterize the electromag-netic fields inside the HOF and the field distributions diffracted from it. In Secs. 3. and 4.,the atom guidance by using the red and blue-detuned laser lights are presented, respectively.The experimental works with diffracted LP11 modes from the HOF are described in Sec.

Applications of Hollow Optical Fibers in Atom Optics 121

5., and the discussions on the applications of diffracted LP10 modessuch as atom guiding,atom fountain, crossed HLB trap, and single optical dipole trap follow in Sec. 6. The finalsection presents the summary of the work.

2. Characteristics of Electromagnetic Field for a Hollow OpticalFiber

2.1. Electromagnetic Field Modes Inside the Hollow Optical Fibers

We describe the electromagnetic field modes for both inside and outside the hollow opticalfibers. The electromagnetic fields inside HOF are given in the first subsection, and then thediscussion on the electromagnetic fields diffracted from HOF follows in the next subsection.The schematic diagram of the HOF with the hollow diameter of2a and the core thicknessd ≡ b − a is shown in Fig. 1 [20, 35]. Since the difference of the refractive indices ofcore and hollow region is not small, the weakly-guiding approximation seems to be notapplicable. However, this approximation proved to be well applicable for the HOF [36, 37].Therefore, instead of discussing the cumbersome vectorial approach, we will use the scalartheory for the analysis of electromagnetic fields modes. A capillary fiber composed of ahole of radiusa and outer glass part is also used in the guiding experiment. The discussionson the guiding modes for the capillary fiber with red-detuned laser beam will be given inSec. 4.1..

Figure 1. The schematic diagram of the hollow optical fiber. The diameter of the hollowregion and the thickness of the cylindrical core are2a andd, respectively. The refractive in-dices of the hollow, the core, and the cladding are 1,n1, andn2, respectively. The thicknessof cladding can be taken to be infinite.

In the cylindrical coordinate(r, θ, z), the longitudinal component of the electric field


Ez(r, θ) withEz(r, θ, z, t) = Ez(r, θ) exp[i(ωt−βz)] satisfies the Maxwell equation givenby

∂2Ez

∂r2+

1

r

∂Ez

∂r+

1

r2∂2Ez

∂θ2+

(

k2n2 − β2)

Ez = 0 , (2)

whereω, β, k andn arethe angular frequency, the propagation constant, the wave number,and the refractive index, respectively. The solution of Eq. (2) is given by

Ez(r, θ) =

AIν(vr) sin(νθ + φ) (r < a),(BJν(ur) + CNν) sin(νθ + φ) (a ≤ r ≤ b),DKν(wr) sin(νθ + φ) (r > b),

(3)

whereJν andNν (Iν andKν) are the (modified) Bessel functions of the first and the sec-ond kind of orderν, respectively,n1 (n2) is the refractive index of the core (cladding),andφ is a phase constant. The constantsA, B, C, andD can be determined by the con-tinuity conditions ofEz at r = a andb. The quantityu =

√

k2n21 − β2 is thetransverse

propagation constant, whereasv =√

β2 − k2 andw =√

β2 − k2n22 arethe transverse

attenuation constants. The magnetic longitudinal components,Hz, can be obtained by re-placing sin(νθ + φ) in Eq. (3) with cos(νθ + φ). The transverse components (Er, Eθ,Hr, andHθ) can be derived from these longitudinal components [38, 39]. The solutions ofEq. (3) are called HEνµ (EHµν) modes, whenEz andHz have the same (different) signs[40]. The dispersion equations describing the light propagation modes of the HOF can thenbe derived from the secular equation obtained by applying the boundary conditions to thetangential componentsEz, Eθ, Hz, andHθ [39]. The explicit expressions of the dispersionrelation are presented in Eqs. (5)-(7) of Ref. [35].

Let us assume the silica-glass HOF with2a = 7µm, d = 3.8µm, andn2 = 1.45,in which the core is germanium doped with the relative refractive index difference∆n =(

n21 − n2

2

)

/2n21 = 0.0018. We also consider the guiding of rubidium atoms with the wave-

lengthλ = 780 nm of D2 line. From the dispersion equation, one can find that six propa-gation modes, TE01, TM01, HE11, HE21, HE31, and EH11 can be excited at the wavelengthof 780 nm (Fig. 2). The lowest mode is the HE11 mode, whereas the TE01, TM01, andHE21 modes exhibit almost the same dispersion curves, so that they form the second groupof propagation modes. In the same way, EH11 and HE11 modes consist of the third group.

When the refractive indices of the core and the cladding are nearly the same in a step-index solid fiber, the weakly guiding approximation is generally used [39, 41]. In thiscase, since one of the tangential field components is far larger than the other orthogonaltransverse or longitudinal components, the guided mode can be approximately describedonly by the dominant transverse component, for which the ”linearly polarized” LPlm-modedescription is usually employed [38, 39, 41] wherel is the azimuthal mode number andmis the radial mode number. In case of an HOF, despite the large difference of the refractiveindex between the core and the hollow region, one can still use the LPlm modes due tothe relatively small intensity in the hollow region. The transverse component for an LPlm

mode is given in the same form as the longitudinal one [Eq. (3)] and thus we use it as aguided mode in the following calculations due to its simplicity over the traditional modedescription method.

From the continuity conditions at the boundariesr = a andr = a + d, one can derive


the following simple dispersion equation describing the LP modes as [42][

Jm(ua)

Im(va)− u

v

J ′

m(ua)

I ′m(va)

] [

Nm(ub)

Km(wb)− u

w

N ′

m(ub)

K ′

m(wb)

]

=

[

Nm(ua)

Im(va)− u

v

N ′

m(ua)

I ′m(va)

] [

Jm(ub)

Km(wb)− u

w

J ′

m(ub)

K ′

m(wb)

]

. (4)

The dispersion equation in Eq. (4) yields the LPm1 modes(m = 0, 1, 2, · · · ) for thehollow fibers considered here. In fact, from the numerical analysis of Eq. (4), one observesthat three LP modes can be excited in the 7-µm hollow core at the wavelength of 780nm. Figure 2 shows the dispersion curves with respect to several lower modes. The solidcircles indicate three LP modes: LP01, LP11, and LP21 mode. Comparing with the the exactnumerical results described above, one can find that: (i) LP01 mode is approximately equalto HE11 mode, (ii) LP11 mode is made up of TE01, TM01, and HE21 modes, and (iii) LP21mode consists of EH11 and HE31 modes. It should be noted that in Fig. 2(a), the pointswhere each dispersion curve intersects the horizontal axis represent the cutoff frequencies.According to Eq. (4), the 7- and 2-µm hollow-core fibers become multi-moded at 780 nm,whereas the 1.4- and 0.3-µm hollow fibers become single-moded.

Figure 2. (a) Dispersion curves of the propagating modes at the wavelength of780 nm inthe 7-µm hollow optical fiber with the core thickness of 3.8µm and the relative refractiveindex difference of 0.18%. The cross-sectional intensity profiles for (b) LP01 mode, (c)LP11 mode, and (d) LP21 mode (Figure from Ref. [35]).

Figures 2(b-d) show the cross-sectional mode-patterns of the 7-µm hollow fiber; (b), (c),and (d) present the LP01, LP11, and LP21 modes, respectively. Note that one can selectivelyexcite one of these LP modes by careful alignment of the incident angle of the laser beam.


These CCD camera images show that the LP01 modeis suitable for guiding atoms: LP01mode has no nodes around the cylindrical inner wall of the hollow core, as shown in Fig.2(b).

0 2 4 6 8 10 12 14

0.00

0.05

0.10

0.15

0.20

0.25

CladdingCoreHollow

Inte

nsi

ty(a

rb.unit)

r (mm)

LP01

LP11

Figure 3. The radial intensity distributions of LP01 (solid curve) and LP11 (dotted curve)modes inside the HOF.

The intensity distributions of LP01 and LP11 modes along the radial direction are shownin Fig. 3 as solid and dotted curves, respectively. In Fig. 3, the both the diameter of the holeand the core thickness are typically assumed to be 4.5µm. While the intensity distributionof LP01 mode is azimuthally symmetric as shown in Fig. 2(b), that of LP11 mode hasan angular dependencesin2 θ as shown in Fig. 2(c). Therefore the real two-dimensionalintensity distribution of LP11 mode should be obtained by multiplying the angular factorsin2 θ to the function in Fig. 3. The guided LP01 mode produces the blue-detuned opticalevanescent fields on the core-vacuum interface, which then generates the optical potentialbarrier so that atoms can be guided in the dark hollow region of the core.

2.2. Characterization of Diffracted Fields from a Hollow Optical Fiber

In the previous subsection, we have described the electromagnetic field modes inside thehollow optical fibers. The various applications using the inside modes will be described inSecs. 3. and 4. In this subsection, we will provide a theory of the diffracted laser lights froman HOF. As was mentioned in the previous section, we will adopt a simple scalar approachfor the calculation.

Since we know the electric field, Eq. (3), on the facet of the HOF (z= 0), we cancalculate the diffraction pattern at(x, y, z) using the Huygens-Fresnel integral [43]

E(x, y, z) =z

2π

∫ ∫

E0(x0, y0)

(

1

ρ+ ik

)

eikρ

ρ2dx0dy0, (5)

where(x0, y0) is the coordinate of a source point,ρ =[

z2 + (x− x0)2 + (y − y0)

2]1/2

,andE0(x0, y0) is the electric field at the fiber facet. In the near-field region wherez


(kb2/2) ' 200 µm, using the Rayleigh-Sommerfeld theory [44, 45], one can calculate thediffraction pattern without any approximation onρ as in the Fresnel diffraction calculation.For a given LPlm mode represented byE0

lm atz = 0, we then obtain

Elm(x, y, z) = E0lm(x0, y0) ∗ h(x, y, z) (6)

where

h(x, y, z) =

[

exp

(

−ikz√

1 + (x/z)2 + (y/z)2)]

×[

−iλz(

1 + (x/z)2 + (y/z)2)]

−1(7)

and ‘∗’ denotes a two-dimensional convolution integral.Eq. (6) becomes a normal product in the transformed space,

Ulm(ξ, η; z) = U0lm(ξ, η) ×H(ξ, η; z) (8)

whereUlm(U0lm) is the Fourier transformation ofElm(E0

lm), and

H(ξ, η; z) = exp[

−ikz√

1 − λ2(ξ2 + η2)]

. (9)

The main task is to calculate the diffracted field in the transformed space and thenconvert the results to real space by an inverse transformaton. In the cylindrical coordinates(r,θ) and (ζ,ψ) in each space, the source field in the transformed space is given as:

U0lm(ζ, ψ) = 2πil sin(lψ + φ0)

∫ +∞

0r0E

0lm(r0)Jl(2πζr0)dr0

≡ U0lm(ζ) sin(lψ + φ0), (10)

whereE0lm(r0, θ0) ≡ E0

lm(r0) sin(lθ0 + φ0). SubstitutingU0lm into Eq. (8), we inverse-

transformUlm to obtain

Elm(r, θ, z) = 2π(−i)l sin (lθ + φ0)

×∫

∞

0U0

lm(ζ) exp[−ikz√

1 − λ2ζ2 ]Jl(2πζr)ζdζ. (11)

Here,U0lm is given analytically by

U0lm(ζ) = 2πil×

[+aA

(

ζJ1+l(aζ)Il(av) + vI1+l(av)Jl(aζ))

v2 + ζ2

−aB

(

ζJ−1+l(aζ)Jl(au) − uJ−1+l(au)Jl(aζ))

u2 − ζ2

+bB

(

ζJ−1+l(bζ)Jl(bu) − uJ−1+l(bu)Jl(bζ))

u2 − ζ2

−aC

(

ζJ−1+l(aζ)Nl(au) − uN−1+l(au)Jl(aζ))

u2 − ζ2


+bC

(

ζJ−1+l(bζ)Nl(bu) − uN−1+l(bu)Jl(bζ))

u2 − ζ2

−bD

(

ζJ1+l(bζ)Kl(bw) − wK1+l(bw)Jl(bζ))

w2 + ζ2] (12)

whereζ ≡ 2πζ. Using these results, one can calculate the general profile of the diffractedbeam at any positionz. The numerical results of the radial intensity distributions for smallz’s are presented in steps of 5µm in Fig. 4. Figure 4(a) explains how the LP01 modediffracts in free space near the HOF: the two peaks (which represents a cross-section ofring-shaped mode) atz = 0 diminish away while an additional central peak grows up. InFig. 4(b), one can see that the peaks of LP11 also diminish whereas another pair of peaksgrow. Nevertheless, there still does exist a dark column along the central axis.

-10-50510

LP01

distance(um)010 r (mm)0

70

z

(mm)

(a) LP

-10-50510

LP01

distance(um)-10-50510

LP01

distance(um)010 r (mm)0

70

z

(mm)

(a) LP

-10-50510

LP11

distance(um)010 r (mm)

0

70

z

(mm)

(b) LP

-10-50510

LP11

distance(um)-10-50510

LP11

distance(um)010 r (mm)

0

70

z

(mm)

(b) LP

Figure 4. Development of the radial intensity distributions due to the diffraction of(a) LP01

and (b) LP11 mode near the facet of HOF (z= 0 at the facet).

We now describe the experimental verification of theoretical calculated results for thedevelopments of LP10 and LP11 modes using a simple imaging technique. In wave optics,the light propagating through a lens from a source plane to a screen can be described byequation (4.3-18) of reference [38], which shows that the field distribution at the screen isidentical to the original distribution within a magnification constantm ≡ −do/di when(di

−1 + do−1 − f−1) = 0, wheredi is the distance between the lens and the source field,

do, the distance between the lens and the screen, andf , the focal length of the lens. Thiscondition is easily realized in experiments whendi = f do. Therefore, one can observethe intensity distribution ofz = 0 at the screen far enough from the lens by locating the lensat a distance of the focal length from the tip of the HOF. In addition, one can observe theintensity distribution (diffraction pattern) of an arbitraryz by moving the lens byz towardthe screen.

The experimental results are shown in Fig. 5. We measured the intensity distributionof z = 0 at the screen 100 cm away from the lens with a focal length of 4.3 mm, andrepeated the measurement at several values ofz by moving the lens. As shown in Fig.5, the experimental results show good agreement with the theoretical results. The outputintensity distribution of LP01 mode atz = 100µm has a maximum at the center with the


10 mm

(a) LP (z=0 m)01 m (b) LP (z=150 m)01 m

(c) LP (z=0 m)11 m (d) LP (z=150 m)11 m

-20 -10 0 10 20

LP01

(z=150 mm)

Inte

nsi

ty(a

rb.u

nit)

Radial Distance (mm)

(e)

Figure5. Output intensity distributions of the LP01 mode ((a) atz = 0; (b) atz = 150 µm)and the LP11 mode ((c) atz = 0; (d) atz = 100 µm), and (e) beam profile of the output ofthe LP01 mode atz = 150 µm in a radial direction. The solid curve in the graph representsthe fitting curve obtained from the calculation.

full width at half maximum(FWHM) of 7.6µm although it is in the shape of a ring insidethe HOF. In case of LP11 mode, on the other hand, both the dark center and the angularnode line are preserved continuously even after propagation into free space.

3. Atom Guidance by Hollow Optical Fiber with Red-DetunedLaser

In this section, we describe the atom guidance experiment by a hollow optical fiber with thered-detuned Gaussian laser lights. This scheme was suggested by Ol’Shaniiet al. [17], andexperimentally realized by Rennet al. for the hollow glass capillary using the red-detunedGaussian laser beam [18, 46]. We first describe the experiment performed by Rennet al.,and then the parametric excitation experiment will be briefly discussed [47].

Let us briefly summarize the theory of atom guidance by the red-detuned laser in thehollow capillary fiber which is composed of a hole of radiusa and a glass with the refractiveindex ofn. The thickness of the surrounding glass is assumed to infinite compared to radiusof the hole. In the cylindrical coordinate(r, θ, z), the electric field of the lowest guidedEH11 mode is given by

~E(r, z, t) = eE0(r)ei(ωt−βz) , (13)

wheree is a unit transverse vector, andE0(r) is given by

E0(r) =

AK0(γa)J0(χr) (r < a),AJ0(χa)K0(γr) (r > a),

(14)

whereJν (Kν) is the first (second) kind of the (modified) Bessel functions of orderν, A isa normalization constant,χ2 = k2 − β2, γ2 = β2 − n2k2, andk = 2π/λ .


The valueχ canbe determined by the following characteristic equation,

γJ0(χa)K1(γa) + χJ1(χa)K0(γa) = 0 , (15)

Whenka 1, Eq. (15) can be simplified as

J0(χa) =iχ

2k

n2 + 1√n2 − 1

J1(χa) . (16)

χa is 2.405 + 0.022i for a 40-µm hollow-core diameter capillary atλ = 780 nm. Theimaginary part ofβ is the attenuation coefficient of the mode amplitude and is given by

Im(β) =(χa

2π

)2 λ2

2a3

n2 + 1√n2 − 1

, (17)

for ka 1. For the 40-µm-diameter capillary fiber, the attenuation length,[Im(β)]−1, was6.2 cm. Figure 6 shows the intensity profile of EH11 mode. As can be seen from Fig. 6 andEq. (14), the intensity profile of the laser beam undergone by the atoms inside the hollowregion is approximately given byI(r) = I0J

20 (χr), whereI0 is the peak laser intensity.

When the guiding laser is red-detuned to the atomic resonance frequency, the atoms canbe guided along the capillary hole owing to the attractive dipole force exerted by the laserlight.

-40 -30 -20 -10 0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Glass GlassHole

Inte

nsi

ty(a

rb.unit)

r (mm)

Figure 6. Radial intensity profile of EH11 modeinside the capillary fiber with the innerdiameter of 40µm.

The experiment of Rennet al. [18] consisted of two separate vacuum chambers con-nected by a 3.1-cm-long capillary fiber with an outer diameter of 144µm and capillary-corediameter of 40µm (Fig. 7). The guiding light from a Ti-Sapphire laser was coupled intothe EH11 mode. They observed a 50% coupling efficiency and an attenuation loss of 7%per cm of fiber length for the EH11 mode. The guided atoms leaving the capillary fiber aresurface ionized on a heated Pt or Re wire in the second chamber. Then the resulting ionsare detected with a channeltron electron multiplier and recorded with a pulse counter.


Figure 7. Experimental apparatus.

Figure 8. Guided atom signal versus laser detuning. The intensities of the guidinglaser areI0 = 3.6 × 103 W/cm2 (upper curve) andI0 = 1.3 × 103 W/cm2 (lower curve) (Figurefrom Ref. [18]).


Figure 8 shows the detuning dependence of the number of guided atoms for twolaserintensities. The zero point of the detuning (δ) was taken to be the average transition fre-quency of the5S1/2 − 5P3/2 multiplet of 85Rb and87Rb. The guiding signal rose to halfmaximum atδ ' −2 GHz and full maximum atδ ' −3 GHz. For the laser detuningslarger than a few GHz, the guided signal decreased approximately asδ−1. With increasingintensity the signal increased and the position of maximum flux shifts to larger negativedetunings.

Figure 9. Guided atom flux vs laser detuning from resonance at several laserintensities. Thefiber lengths are 3.1 and 6.2 cm for (a) and (b)−(d), respectively, and the inner diameter is40µm (Figure from Ref. [46]).

In a subsequent paper, Rennet al. reported on the improved results for the dependenceof the guided atom signals on the laser detuning for several laser intensities [46]. Figure 9shows the typical dependence of guided atom flux on the laser detunings. At an intensityof ∼ 0.6 MW/m2 [Fig. 9(a)], the curve follows the similar trend as those shown in Fig. 8:The flux reaches a maximum near a detuning of about -1 GHz and then falls off rapidly forlarger detunings. As the intensity increased, as shown in Figs. 9(b)−9(d), they observeda substantial flux at significantly higher detunings and found a dip formed in the profileat intermediate detunings. As a function of increasing intensity the dip grows deeper andbroader [Figs. 9(c)−9(d)]. They found that the formation of the hole was attributed toviscous dipole forces which heat the atoms to larger transverse energies than can be guided[46].

Recently Hayashiet al. reported on the parametric excitation of laser guided Cs atoms


in an HOF [47]. Periodic modulation of the optical potential achieved by the intensity mod-ulation of the red-detuned guiding laser resonantly enhances the amplitude of the atomicmotion. Eventually the atoms hit the inner wall of the HOF and absorbed on the surface.Thus the parametric resonance causes a reduction in the number of atoms coming out of theHOF.

The experimental apparatus consists of two separate vacuum chambers connected by a3-cm long borosilicate glass capillary having an outer diameter of 2 mm and a hollow-corediameter of 86µm. One chamber serves as a Cs oven, which contains a Cs ampoule andis heated to 310 K to produce Cs vapor at 1023 Pa. The other chamber is a high vacuumdetection chamber having a background pressure of 1026 Pa. Guided atoms leaving thecapillary outlet are surface-ionized on a heated Pt wire. The generated ions are detected bya Channeltron electron multiplier and counted by a pulse counter. A Ti:sapphire laser witha wavelength of 852 nm was used to guide Cs atoms. For parametric excitation, the laserintensity is modulated by an acousto-optical modulator (AOM).

Figure 10. Dependence of the count rate of Cs atoms on the modulation frequencyof thelaser intensity (Figure from Ref. [47]).

Figure 10 shows the count rate of Cs atoms as a function of the modulation frequencyof the laser intensity. The modulation depth is set at 0%, 10%, 40%, and 60%. The guidinglaser is -15 GHz red-detuned from D2 resonance line. As the modulation depth is increased,the guided atom flux lowers appreciably. For 10% modulation, a broad resonant structureappears in the parametric excitation spectrum. As the modulation frequency is increased,the number of guided Cs atoms starts to decrease around 100 Hz, takes its minimum at 5


kHz, and then grows for higher frequencies. Above 100 kHz, the flux recovers to its originallevel. For 40% modulation, parametric excitation occurs over a much wider frequencyrange. Still, the dip of the spectrum is centered around 5 kHz. When the modulation depthis increased to 60%, severe guiding loss takes place over the whole frequency range, andthe resonant structure disappears.

4. Atom Guidance by Hollow Optical Fiber with Blue-DetunedEvanescent Light

4.1. Atomic Guide by Glass Capillary Fiber

In this section, the atom guidance experiments by the blue-detuned laser beams throughcapillary fibers or micron-sized hollow optical fibers are presented. First, in this subsection,we will describe the guiding experiments by means of hollow capillary fibers, performedby two groups Rennet al. [22] and Baldwinet al. [25] for the Rb and metastable He atoms,respectively.

Rennet al. [22] at JILA reported the first atom guidance with the blue-detuned evanes-cent waves. They used a hollow glass capillary whose hollow-core diameter (2a) was 10µm and outer diameter was 77µm. In the absence of the laser field, atoms are supposed tobe attracted to the glass surface due to the long-range van der Walls forces. At low inten-sities, the optical repulsive potential is weaker than the van der Waals potential, which isgiven by [48, 49]

UvdW = − 3

4f

n2 − 1

n2 + 1

~Γ

(kr)3, (18)

wheref is the oscillator strength,k is the wave number,n is the refractive index, so that thetotal potential is always attractive. Therefore, one can define the threshold intensity abovewhich atoms feel the net repulsive potential, which is the position of3/(2k) apart from thewall.

The experimental setup consisted of two vacuum chambers that were connected by thehollow capillary fiber like the red-detuned case. A blue-detuned high-power laser with 500mW was focused on the annular region of the fiber end and a weaker, red-detuned escortlaser with 10 mW was focused in the hollow region where it was coupled to the EH11

grazing incidence mode. The escort laser facilitates atom loading into the capillary guide,which might be hampered by the scattered light near the core facet. Figure 11 shows theenhancement of the atomic flux due to optical guidance by a 6-cm-long glass capillary withthe 20-µm core diameter. With the red-detuned escort laser alone, 200 atoms/s were guidedin the fiber. When the evanescent light was excited on the glass interface, on the other hand,the flux increased at least by a factor of 3 at the optimum guide-laser detuning of +3 GHzand for the escort laser detuning of -1.6 GHz. When the escort laser detuning was decreasedto -9.4 GHz, the number of atoms injected into the evanescent guide was also reduced. Onthe other hand, tuning the escort laser to the blue side of the resonance inhibited injectionof atoms into the guide and thus completely suppressed the guided atom flux.

They also measured the intensity dependence of the atomic guidance signal. Unlike thered-detuned guidance discussed in the previous subsection, there seems to exist threshold


Figure 11. The dependence of the guided atomic flux on the guide-laser detuning inthepresence of the red-detuned escort laser. The detuning of the escort laser is -1.6 GHz (-9.4GHz) for the upper (lower) curve (Figure from Ref. [22]).

behaviour near the intensity of about 6 MW/m2, and the flux increases roughly linearlyabove this value. However, it was not straightforward to account for this behaviour in termsof the cavity potential because the hollow diameter was rather large, unlike the micron-sizedHOFs where the van der Waals interaction near the core wall becomes significant (refer tonext section for quantitative discussions of the cavity quantum electrodynamic effects).

The JILA group also demonstrated evanescent-light guidance of laser cooled87Rbatoms in the hollow-core fibers [50] using the similar apparatus. The flux of an atomicbeam generated from the source chamber by the low velocity intense source (LVIS) method[51] was approximately∼ 109 atoms/s and the brightness was∼ 1013 atoms/(sr·s). Thetransverse velocity of LVIS was in the rangevt = 8.0±1.5 cm/s and its longitudinal veloc-ity was measured to bevr = 10.0± 2.0 m/s. The fibers used in their experiment were glasscapillary tubes with 100-µm inner and 160-µm outer diameter. They have guided atomsby using several fibers of different lengths varying from 17 to 30.5 cm. In the 30.5-cm-long fiber, in particular, the atomic flux of7 × 104 atoms/s was measured and5.9 × 105

atoms/s was the highest flux obtained in the 23.5-cm-long fiber. The transverse velocity ofthe guided atoms was found to be9.4 ± 1.7 cm/s.

Dall et al. demonstrated guidance of metastable helium atoms by blue-detuned evanes-cent waves in hollow capillary fibers [25]. A bright helium atomic beam in the metastable23S1 state is generated by the nitrogen-cooled discharge source. The atomic beam isinitially collimated in transverse directions by using a diode laser operating at 1083 nm

500

400

300

200

100

0-4 -2 0 2 4 6 8

Guide Laser Detuning(GHz)

Flu

x

(Hz)

Escort Laser Detuning-1.6 GHz-9.4 GHz


(23S1 → 23P2). Then the atoms are cooled longitudinally by a second laser using a Zee-man slower, where the longitudinal velocity is decreased from∼ 900 m/s down to∼ 100m/s. Finally, the atoms are compressed in two-dimensional magneto-optical trap by a sepa-rate laser, which results in the beam flux of up to 1010 atoms/s over 1 cm2 area. The guidedatoms are detected by the channeltron detector. To couple the light into the capillary fiber,one end of the capillary is optically polished at 45 angle so that the focused guiding lasercan be coupled from the side as discussed in Section 4.2. They have used three types of cap-illaries: square-section capillary (350µm wide, 49µm hole), round-section capillary (110µm wide, 40µm hole), and further round-section capillary (150µm wide, 10µm hole).With the input light power of 23 mW, the coupling efficiency exceeds 70% for the squarecapillary, whereas it is as low as 10% for the 110/40µm thin-walled capillary.

Figure 12 shows the guided atom signals with respect to the copropagating guide-laserdetuning for the 350/49µm capillary with the guide-laser power of 15 mW (the saturatedabsorption signal is also recorded for frequency reference). As can be seen, the guidingsignal increases rapidly and decreases gradually as the detuning is increased. Figure 12(b)shows the effect of reducing the laser intensity to∼ 1/3 with respect to that shown inFig. 12(a). The reduced width of the transmitted signal is consistent with the decreaseof the evanescent-wave potential. The signal in Fig. 12(b) can be well fitted with the∆−2

dependent function rather than∆−1, as expected from the expression of the dipole potential.Note that the rapid decline of the transmitted signal with respect to∆ in Fig. 12(b) was notdue to capillary curvature. Rather, it was attributed to the rapid change of the area of thebright regions associated with the speckle pattern (formed by the multimode guide light) inthe capillary. They also performed the atomic guidance experiment in the 110/40µm and150/10µm round capillaries and obtained qualitatively similar results with respect to thecase of the square 350/49µm capillary [25].

4.2. Atomic Guide in Micron-Sized Hollow Optical Fiber

In a series of works, the Japan-Korea collaboration has reported on optical guidance ofthermal rubidium atoms by the blue-detuned evanescent waves induced in the micron-sizedHOFs. Figure 13 shows the schematic diagram of the experimental setup [23]. In thevacuum chamber, a rubidium atomic beam from the hot oven, well collimated by severalapertures, is introduced into the HOF that is coaxially placed behind a holed mirror. Thefiber has the hollow diameter of 7µm (2µm), core thickness of 3.8µm (4µm), and lengthof 3 cm. The typical incident atomic flux was of the order of 106 atom/s.

The transmitted Rb atoms through the hollow fiber were detected by a channel electronmultiplier, via two-step photoionization detection with two overlapping lasers; a diode lasertuned to the Rb D2 line and a high-power Ar-ion laser at the wavelength of 476.5 nm (theionization energy is 4.177 eV above the 5S1/2 ground state). The condition for efficient two-step photoionization of the ground-state atom is given byPi ∼ P0 (σ0/σi) wherePi is thelight intensity required for ionizing atoms in the excited state andσi is the ionization cross-section from the excited state to the ionization level [52]. For the Ar-ion laser intensity of0.5 GW/m2, assuming the resonant excitation cross-sectionσ0 = (3/2π)λ2 = 3×10−9 cm2

for the5S1/2 → 5P3/2 transition [53] andσi = 2.5 × 10−17 cm2 [54], the photoionizationefficiency of about 30% was estimated.


Figure 12. (a) The transmitted atom signal (dots) and saturated absorption signal(solidcurve) as functions of the laser detuning for the square 350/49µm capillary in the copropa-gating configuration. The inset shows a detailed view at small detunings. (b) Same as(a) butwith laser power reduced by∼ 1/3. The dashed curve shows the∆−2 fit for the far-detunedblue wing (Figure from Ref. [25]).


Figure 13. Schematic of the experimental setup.

Figure 14 shows the typical photoionization spectrum of the85Rb atoms guided bythe 7-µm HOF over the distance of 3 cm as a function of the frequency detuningδ ofthe guiding laser with respect to the 5S1/2, F = 3 upper ground state. As expected, theguided atomic flux is greatly enhanced in the blue-detuning region. The foot level of thephotoionization signal extends to the detuning over +20 GHz. The broken line in Fig. 14shows the background transmission level that was obtained without the guiding laser, whichcame from those atoms ballistically flying through the hollow fiber [23]. By comparisonof the maximum guided atomic flux with the background transmission, the atomic-guideenhancement factor was found to be about 20. From a similar experiment with the 2-µmhollow fiber, in which the axis of the optical fiber was slightly tilted against that of the Rbatomic beam, a higher enhancement factor of 80 was obtained [55].

Figure 15 shows the novel characteristics of atomic isotope separation achieved in the7-µm hollow fiber. The upper curve of Fig. 15 shows the case where the guide laser isblue-detuned for both isotopes. In this case, both isotopes can be guided in the hollowfiber. On the other hand, the lower curve of Fig 15 shows the case where the guide laser isblue-detuned for87Rb atoms but nearly red-detuned for85Rb atoms. As is clear, the87Rbatoms are guided by the hollow fiber, while the transmission of the85Rb atoms is greatlysuppressed, which represents the interesting feature of an in-line atomic-state filter.

In a subsequent experiment Itoet al. investigated a novel atomic guiding scheme inwhich the guide light beam is coupled to the hollow core sideways at a 45 angle via totalinternal reflection near the edge [56]. There are several advantages for this method: First,the scheme can easily remove unwanted light scattering such as the propagating modesinside the hollow region. Second, this scheme prevents the leak of the incident light nearthe entrance of the hollow fiber, so that heating or optical pumping of atoms near the fiberentrance can be minimized. Moreover, this scheme also enables one to couple the guidelight from the rear direction. Recently, Fatemiet al. reported a side-coupling methodfor the mutitimode HOF using embedded microprism [57]. Microprisms embedded into amultimode, double-clad hollow fiber, allow laser light to be coupled into the fiber at multiplelocations along the length of the fiber.

In the atom guidance through HOF, in addition to the repulsive optical dipole interac-tion, the attractive cavity quantum electrodynamic (QED) interaction is also induced be-


Figure 14. Two-step photoionization spectrum of the85Rb atoms in the5S1/2, F = 3 stateguided by the 7-µm hollow fiber over a distance of 3 cm (solid line). The broken line showsthe background transmission level without the guide laser (Figure from Ref. [23]).

Figure 15. In-line spatial separation of85Rband87Rb in the 7-µm hollow optical fiber. Theupper curve shows the case where a guide laser is blue detuned for both isotopes while thelower curve shows the case where a guide laser is blue-detuned for87Rb but red-detunedfor 85Rb (Figure from Ref. [23]).


tween the dielectric wall and the nearby atoms [48, 58, 59]. Moreover, whenthe cavitypotential exceeds the optical potential near the surface, atoms will be attracted to the innerwall and will be lost. Therefore, one can expect to observe the threshold behaviour of theatomic transmission in the cylindrical dielectric cavity [49, 60, 61, 62, 63].

Figure 16. Two-step photoionization signal of87Rb atoms guided by the 1.4-µm hollow-core optical fiber as a function of the guide-laser power (Figure from Ref. [65].

Figure 17. Atomic transmission signal of the guided87Rbatoms in the 0.3-µm hollow fibernear the low-intensity threshold region (Figure from Ref. [64]).

Ito et al. have measured the threshold guide-laser intensities where the atomic trans-mission starts increased [55, 64]. The hollow-core diameter is only of the order of or lessthan the resonant wavelength of 780 nm for the87Rb D2 transition, and consequently theinduced cavity effects are much significant so that the small threshold intensities are noweasily measurable even in the 300-nm hollow fiber. Figure 16 shows experimental data of


photoionization signal due to the atoms guided in the 1.4µm fibernear the threshold [65].As can be seen, the threshold behaviour in the atomic transmission is clearly observed at thesmall power of 125µW. The laser power can be equivalently expressed in terms of the pureoptical potentialUop(r = a) normalized to the mean transverse kinetic energyKav = mv2

tr

of the guided87Rb atoms (mis the atomic mass andvtr is the transverse root-mean-squarevelocity of 87Rb atoms). Then the threshold intensity for the case of 1.4µm fiber corre-sponded to the ratio ofUop(r = a)/Kav = 2.5. When the total potential including thecavity potential is considered, on the other hand, the ratio of the total potential to the trans-verse kinetic energy becomes approximately one, as observed in the case of plane surface[66]. The same experiment is also done in the slightly larger HOF with the core diame-ter of 2.0µm. The results show very similar threshold behaviours and the correspondingthreshold laser-power is 40µW, which is only 30% of the value for the 1.4-µm case. Theatomic guidance experiment was also performed in the much smaller 0.3-µm hollow fiber.From the atomic transmission data near the threshold region presented in Fig. 17, one canobtain the threshold guide-laser power of about 2.6 mW. This laser power is much largerthan those for the other two cases of larger HOFs. Note that the larger values of the thresh-old intensities for the smaller hollow-core fibers can provide the direct manifestation of thecavity QED effects in the cylindrical cavity.

Figure 18. Spatial distribution of Rb atoms guided by the 7-µm hollow fiber (Figure fromRef. [55]).

Ito et al. reported on the possibility of fabricating an arbitrary pattern by using theatomic waveguide with the hollow optical fiber, which may lead to a novel lithographictechnique of optically controlled atom deposition [55]. The experimental setup is similarto that in Fig. 13. Figure 18 shows the surface-ionization signal of Rb atoms guided bythe 7-µm hollow optical fiber over the distance of 3 cm. The spatial distribution of the


Figure 19. Two-step photoionization spectrum of the87Rb atoms in the5S1/2, F = 2 stateguided through the 1.4-µm hollow fiber at a low oven temperature (Figure from Ref. [55]).

guided atomic flux is obtained by the cross-sectional scan of the hot-wire at the distanceof 12 mm downstream from the exit facet of the hollow fiber. The guide-laser frequency isblue-detuned at the optimal value of +3 GHz with respect to the85Rb,5S1/2, F = 3 upperground state. The FWHM of the spatial distribution shown in Fig. 19 is 20µm. Consideringthe quantum efficiency of 0.9 of the channeltron and the cross section of the hot wire, theguided Rb fluxΦ is measured to be 105 atom/s above the background level.

5. Experiments with Diffracted LP11 Modes

5.1. Generation of a Hollow Laser Beam Diffracted from a Hollow OpticalFiber

In this subsection, we describe the method of generation of hollow laser beams (HLBs)by means of diffracted LP11 modes from an HOF. Although a doughnut-shaped divergentbeam generated directly from an HOF is needed, for the simultaneous realization of anatomic funnel and an HOF atomic guiding, however, the calculations and experimentalresults in section 2.2. reveal that such an HLB cannot be produced in a simple method.These problems can be solved by using one of the TE01, TM01, and HE21 modes whichresult in LP11 modes. All of these modes are, like the LP01 mode, ring-shaped inside theHOF, and, in addition, each output beam forms a doughnut-shaped hollow beam since theyare only the superposition of the output-field pair of the LP11 modes (For example, for TE01mode, the pair is shown in the upper row of Fig. 20(a)). Consequently we have superposedtwo LP11 modes instead of exciting the desired mode directly. In this combination, oneLP11 mode has a polarization and a node line orthogonal to those of the other mode.


TE01+HE21 TE01-HE21

TM01+HE21 TM01-HE21

APPhalf-wave

PBS1M1

M2PBS2

4X

HOF

40X

CCD

SCREEN

plate

camera

Screen

Figure20. Generation of an HLB by a proper superposition of LP11 modes: (a) diagram forfour degenerate configurations of LP11 modes and (b) a sketch of the experimental setup.LD, APP, M, and PBS, in (b) stand for laser diode, anamorphic prism pair, mirror, andpolarizing beam splitter, respectively.

The experimental setup is presented in Fig. 20(b). If one wants a specific mode to beexcited dominantly in a multimode fiber, the transverse distribution of the incident lightshould resemble that of the mode as much as possible and, in particular, its polarizationshould be also matched [67]. A half-wave plate just after the laser makes it possible tobalance a relative intensity ratio of one mode to the other, which allows generation of amore symmetric mode in an azimuthal direction. The resulting combined beam in frontof the fiber may look like a single linearly-polarized beam with its plane of polarizationrotated 45 with respect to the horizontal plane, but one should note that each beam can beadjusted separately, which was important in exciting modes different from each other.

The first two pictures in Fig. 21, which were obtained by blocking one of the two opticalpaths, represent the patterns of perpendicular modes atz = 0 before they are merged, and


5 mm

(a) (b) (c)

Figure 21. Superposition of two orthogonal LP11 modes. Transverse intensity profilesat z = 0 are shown: (a) and (b) before, and (c) after superposition. Polarization andangular variation of the corresponding electric field can be described by, for example, (a)x sin(θ + φ) and (b)y cos(θ + φ).

the last one shows their combined pattern, which is similar to that of LP01 mode. Figure 22shows its output intensity distribution atz = 250µm. The peak-to-peak distance is about17µm and the dark spot size is about8.2µm. We have checked its azimuthal isotropyby measuring the beam profiles along eight different radial axes and they showed gooduniformity within the maximum error of about 7%.

-20.0 -10.0 0.0 10.0 20.0

10 mm

Radial distance ( m)m

Inte

nsit

y (

arb

. u

nit

)

(a) (b)

Figure 22. HLB made of the diffracted output of the superposed mode. (a) CCD images ofthe intensity distribution (z = 250 µm) and (b) its profile in a radial direction.

Let us now discuss briefly on the application of this beam to an atomic funnel. Figure23 shows a possible configuration of our overall atom guiding system. It consists of threemain parts: a pyramidal or axicon mirror trap [68, 69], a HLB atomic funnel, and an HOFevanescent-wave atomic guiding.85Rb atoms trapped in a pyramidal or an axicon mirrortrap are pushed through a small hole by the power imbalance of the laser light along themirror axis, and guided inside the HLB. With the HLB converging into the hollow regionof the HOF, they are finally guided through the HOF. The guiding laser light is coupledinto the core of the HOF from the side by the total reflection at the glass-vacuum interfacewhich is ground and polished at an angle of 45 as employed in reference [56]. The cold


Pyramid

Mirror

Trap

Atomic Funnel

made by

Dark Hollow BeamHollow-core

Optical

Fiber

Evanascent

Field

Blue-detuned

Laser Beam

Trapping Beam

Channeltron

Figure 23. Set-up of a novel atomic guiding system. The guiding beam is launchedinto thefiber from the side, and channeltron is used to detect the guided atoms.

atom funnel by employing similar apparatus with the red-detuned Gaussian funneling beamrather than a blue-detuned HLB was recently realized [70, 71].

5.2. Diffraction-Limited Dark Laser Spot Produced by a Hollow OpticalFiber

Shin et al. reported on the generation of the diffraction-limited HLB having submicron-sized dark spot by using the diffracted LP11 mode from the HOF [33]. From the practicalpoint of view, leakage of light from the cladding region on the fiber facet is the main obstacleto obtaining the good optical quality of diffracted HLB since a short fiber is generally usedfor atom optical experiments, where contamination due to the leaked light is inevitable. Toproduce an ideal HLB with HOF, it is required to block the cladding-guided light so thatunwanted scattering can be avoided. For this purpose, a microsphere is employed as anevaporation mask for the core of HOF.

Figure 24(a) shows the image of the field distribution of the LP01 mode on the fiber facetobserved by the imaging method [27]. The scattered light on the cladding surface is clearlyobserved. They blocked the cladding-mode light by selectively metal-coating the claddingon the fiber’s facet with a microsphere used as an evaporation mask. They first attach the


Figure 24. Intensity distribution of the LP01 modeon the HOF-end facet imaged on thescreen for (a) the normal HOF and (b) the metal-coated HOF. (c) scanning electron micro-scope (SEM) image of the microsphere attached to the HOF center after metal evaporation.(d) Enlarged SEM picture near the hollow-core region with the sphere removed.

microsphere on the fiber center masking the hollow core, evaporate thin film of aluminum,and then remove the microsphere afterwards. A 20-µm-diameter microsphere is used toeffectively mask the hollow core having the diameter of2a+ 2d ' 12.3µm. In Fig. 24(c),the scanning electron microscope (SEM) image shows that the microsphere is positionedwell on the fiber center. Once it is attached to the fiber, the van der Waals force holds thesphere tightly in its position. The more-detailed SEM picture of the facet after removingthe microsphere, i.e. the metal-coated HOF facet, is presented in Fig. 24(d). Fig. 24(b)shows that the cladding-mode light is completely blocked by the described procedures.

They measured the beam profiles and the dark hollow size of the diffracted LP11 modeas shown in Fig. 25. In Fig. 25(a), one can observe the beam-propagation characteristicsof the freely-diffracting HLB: whereas the bright ring on the fiber-end facet is diminished,the new peaks are developed from the center at around 30µm, maintaining the dark regionalong the axis (the inner peaks diverge with a diffraction angle of 40 mrad). In particular,the dark spot is preserved along the central axis even when HLB is spontaneously focuseddue to diffraction. The experimental results are in good agreement with the numericalsimulations obtained by the Rayleigh-Sommerfeld theory, as also shown in Fig. 25(a).Fig. 25(b) shows the radial intensity distributions atz = 15, 20, 25, 30µm, respectively.Note that the central dark region nearz = 30 µm is slightly contaminated, which may beassociated with the intensity imbalance of two peaks, a slight excitation of LP01 mode, orthe resolution limit of imaging system.

To estimate the size of the dark hollow region, they fitted the measured profile for agivenz with two independent Gaussian curves and define the radius of dark spotRmax asa half the distance between the central maxima of each curve. As shown in Fig. 26, the


Figure 25. Characteristic dimensions of the dark hollow region, measured in termsof thedark-spot radiusRmax and the half-widthw, which are in good agreement with the numer-ical simulation.

0 20 40 60 80 1000

1

2

3

4

5

hole radius ofhollow fiber

(mm

)

z (mm)

Rmax

(simulation)

Rmax

(experiment)

w

Figure 26. (a) Experimental and numerical results of the radial-intensity profilesof thediffracted LP11 mode measured in steps of 5µm from the HOF-end facet (z= 0). (b)Intensity profiles measured near the focus atz = 15, 20, 25, 30µm.


minimum radius of dark spot is about 2µm which is similar to the hollow radiusa itself.As an alternative definition characterizing the dark spot, on the other hand, we also havefitted the dark region profile between the two curves with an inverted Gaussian curve, andestimate the half width of dark spotw as the half width at half maximum of the singleinverted Gaussian. In this way, as in Fig. 26, we obtain that the smallest value ofw is lessthan 1µm (about 0.8µm aroundz = 35 µm).

6. Experiments with Diffracted LP10 Modes

In this section, we describe the various experiments performed by using the hollow laserbeams produced by a hollow optical fiber. First, we describe the micro-imaging methodto generate HLBs and discuss the experiments of atom guiding, atom fountain guided byan HLB, and crossed atom trap. Finally, as an application of Diffracted LP10 modes, thediscussions on the optical dipole trap is presented.

6.1. Micro-Imaging Method for Hollow Fiber Modes

Yin et al. [27] obtained an HLB by using a micro-collimation technique for the outputbeam of a micron-sized hollow optical fiber. The principle of this method is very simple:for a fiber waveguide consisting of a hollow cylindrical core, some low-order modes can beguided in the hollow core, such as the LP01, LP11, LP21, and LP31 mode [35]. Therefore,when one uses a microscope objective with a short focal length to image the output intensitydistribution at the facet of a hollow fiber, a simple HLB can be obtained.

The inner and outer diameter of the hollow-core of the fiber was 7µm and 14.6µm,respectively and the outer diameter of the cladding of the fiber was 123.4µm. The relativerefractive index difference,∆n = (n2

1 − n22)/(2n

21) = 0.0018 andn2 = 1.45, wheren1

andn2 are the refractive index of the core and the cladding, respectively. The numericalaperture is about 0.124. Figure 27 shows the relationship between the dark spot size (DSS)and the propagation distanceZ of the dark HLB. It can be observed that (i) the DSS ofthe dark hollow beam collimated by a M-20×objective is about 50µm atZ = 100 mmand about 100µm atZ = 500 mm, and (ii) the relative divergent angle in the near field ofHOF is about6.5 × 10−5, whereas the divergent angle in the far field is4.0 × 10−4. If oneuses an HOF having a slightly larger hollow-core, an HLB with a smaller DSS and betterpropagation invariance may be obtained.

The HLB generated by the micro-imaging method was used to couple into the HOFto increase the coupling efficiency by Takamizawaet al. [72]. In order to eliminate theundesirable preinteraction before atoms enter the hollow region, they used an annular beamwhose shape and diameter are approximately the same as those of the core. The use ofannular beam also makes it feasible to excite the LP01 mode with a relatively low powerof ∼10 mW to reflect atoms. The evanescent wave produced with the LP01 mode hasa cylindrical shape around the inner-wall surface without nodes, and consequently it issuitable for the atom guidance.

A Gaussian light beam coupled to the HOF excites LP01 mode. The output light beamfrom the HOF, which has an annular shape just after exiting the HOF, is collimated andrecoupled to another HOF with two convex lenses. Then, annular light beam with the same


Figure 27. The relationship between the DSS and the propagation distanceZ of theHLBmeasured with (a) M-20× and (b) M-40× objective lens.

intensity profile as that of LP11 mode is reproduced at the focal point behind the secondlens. The typical values of the focal length and the distance of the two lenses are 25.2 mmand 500 mm, respectively. The coupling efficiency was increased to 75% when the annular-shaped beam was used compared to 59%, which was the maximum coupling efficiencywhen a Gaussian beam was coupled to the HOF [72]. The hole diameter, core thickness,and the length of the used HOF was 7µm, 3.8µm, and 3 m, respectively.

6.2. Atom Guiding with Hollow Laser Beams

Xu et al. performed optical guiding of trapped cold atoms by a hollow laser beam producedby micro-collimation and micro-imaging technique as discussed in the previous subsection[28]. The atomic guiding direction was downward along the gravity (+z direction), whereasthe HLB propagated along the−z direction (counterpropagating scheme) or along the+zdirection (copropagating scheme). A Ti:sapphire laser was used as the guiding laser sourcewith a maximum output power of 1.8 W. It was coupled to the core of HOF with a couplingefficiency of about 30%. The typical HLB power used for guiding atoms was 250 mW.They used a micron-sized HOF that has a hollow diameter of 4µm, core thickness of 2µm, and length of 25 cm. In both guiding schemes, they obtained the identical radius of themaximum-intensity ringρm(z) that varies linearly with the distancez [ρm(z) = ρm(0) −


αz, whereρm(0) = 1.4 mm is the value at the trap center (z=0) andα = 1.27(4) × 10−3].They used a standard vapor-cell magneto-optical trap (MOT) of85Rb atoms. The num-

ber of trapped atoms was typically2 × 107 and the trap diameter was about 1 mm so thatthe loading efficiency of the trapped atoms into HLB was 98%. By time-of-flight measure-ment, the temperature of atoms in the MOT was found to be about 140µK, which wasfurther cooled down to 16µK by the polarization-gradient cooling. After the sub-Dopplercooling, the cooling and repumping lasers were blocked by mechanical shutters, and theHLB was simultaneously introduced to the atoms to guide their gravitational falling. Thenumber and the temperature of guided atoms were detected by observing the probe-inducedfluorescence with a photomultiplier. The probe laser beam was placed horizontally at 105mm below the trap center.

0.00 0.05 0.10 0.150.0

0.5

1.0

1.5

2.0

2.5

Free

16 GHz10 GHz

6 GHz2 GHz

1 GHz

Guid

ed

Ato

mFlu

x(a

rb.units

)

Time of Flight (s)

0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.240.0

0.5

1.0

1.5

2.0

2.5

Free

16 GHz10 GHz

6 GHz

Guid

ed

Ato

mFlu

x(a

rb.units

)

Time of Flight (s)

Figure 28. Typical TOF signals of atoms guided by a single HLB. (a) In the copropagatingscheme, the laser detuningδ2 is 1, 2, 6, 10, and 16 GHz, respectively. (b) In the counter-propagating scheme,δ2 is 6, 10, and 16 GHz, respectively.

Figure 28 shows time-of-flight signals of guided cold atoms in both guiding schemesat various laser detunings with respect to the5S1/2, F = 2 → 5P3/2 transition line. Forcomparison, the detected signal without the HLB for the freely falling atoms is also shown.In particular, it is observed that the number of atoms guided by the copropagating HLB is


about 20-fold enhanced with respect to that without the HLB at 2 GHz detuning.In thiscase, the guided atoms also become accelerated along the+z direction due to the increasedradiation pressure at small detunings [Fig. 28(a)]. In the counterpropagating case, on theother hand, the guided atoms are decelerated as the detuning is decreased [Fig. 28(b)].

-4 -2 0 2 4 6 8 10 12 14 160

10

20

30

40

50 Copropagating

Counterpropagating

(b)

(a)

Guid

ing

Effic

iency

(%

)

Detuning (GHz)

Figure 29. Guiding efficiency as a function of the detuningδ2 in the copropagating (a) aswell as the counterpropagating (b) scheme. The solid curves represent numerical simulationresults.

Figure 29 presents experimental and numerical guiding efficiencies versus detuning inthe copropagating (a) as well as in the counterpropagating (b) scheme. Note that in numer-ical simulation, the HLB was assumed, to a good approximation, as the lowest Laguerre-Gaussian (LG10) mode given by

I(ρ, z) =4P

πw2

ρ2

w2exp

(

−2ρ2

w2

)

, (19)

whereP is the laser power,w = w0

√

1 + ξ(z/zR)2 is the beam waist at distancez, w0

is the beam waist atz = 0, andzR = πw20/λ is the Rayleigh length. It can be observed

that at small detuning, atoms are most efficiently guided in the copropagating scheme (forexample, the maximum guiding efficiency is about 50%at the detuning of 2 GHz). Onthe other hand, the counterpropagating guiding is generally less efficient as found in Fig.29. However, for large detunings, both schemes provide similar guiding efficiencies andthe maximum efficiency of 23%is obtained at 10 GHz detuning in the counterpropagatingscheme.

6.3. Atom Fountain with Hollow Laser Beams

The development of an atomic fountain based on laser-cooled atoms [73, 74] has createdprospects for an improved accuracy and stability of frequency standards. In such a clock,one approach to solve the line shift due to cold collisions is to use laser light for guiding theupward launched atoms [75]. This is because the guiding can enhance the number of atomswhich come back into the microcavity without increasing the atomic densities.


Optical guiding of an atomic fountain by using a cylindrical HLB was demonstratedbyKim et al. [29]. The generated HLB by using micro-imaging method was collimated bythe objective lens and propagated downwards toward the center of the Rb MOT. The powerof the guiding laser was 250 mW and the beam waist was 3 mm. The HLB was nearlycollimated in order to remove the dipole force of the guiding direction, which can causebroadening of the spatial distribution of guided atoms. With an intensity of 3 mW/cm2 ineach beam, the typical diameter of an atomic cloud in the MOT was about 1 mm, and thenumber of trapped atoms was typically 2× 107. Cold atoms were then launched upwardsin a rather simple way by varying rapidly the vertical magnetic field resulting in the atomicZeeman shift. After 1-ms acceleration, the detuning of the laser beams was changed from−2.5 Γ to−70 Γ, lowering the atomic temperature to 33.7µK in the frame moving upwards.A typical launching velocity of ascending atoms was 1.4 m/s and the atoms were launchedup to 10 cm.

The number of guided atoms was detected by observing the fluorescence with a photo-multiplier tube, which was induced by a horizontally placed probe laser at 10.5 cm belowthe center of the MOT. They observed that 0.5% of the launched atoms were detected with-out the HLB. On the other hand, a tenfold enhancement of the HLB-guided atomic fountainwas clearly obtained without appreciable heating. In Fig. 30, the line (a) is the time-of-flight (TOF) signal of atoms that are launched without the HLB, while the other line (b)is the TOF signal with the HLB at a detuning of 19 GHz. From this TOF signal, one candeduce the guiding efficiency of atoms and the temperature. Without the HLB, the temper-ature was about 33.7 (±2.1)µK and about 34.4 (±1.7)µK with the HLB.

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450.0

0.2

0.4

0.6

0.8

1.0

P=250 mWd

2=19 GHz

vlaunch

=1.4 m/s

HLB off (a)

HLB on (b)

Num

berofA

tom

s(a

rb.units

)

Time of Flight (s)

Figure 30. TOF signals in the HLB-guided atomic fountain experiment. The line (a)is forthe case without the HLB whereas the line (b) is for the case with the HLB.

To characterize the enhancement due to the guiding HLB, they introduced the enhance-ment factor, defined as the ratio of the number of atoms guided with the HLB to that withoutHLB. In Fig. 31, the line with filled squares shows the relationship between the enhance-ment factor and the detuning measured with respect to the5S1/2, F = 2 → 5P3/2 transitionline. The inset shows the enhancement of the guiding efficiency for larger detuning. Theline with empty circles shows how the number of scatterings, or the heating, is changed with


0 5 10 15 20

0

10

20

30

40

0

500

1000

1500

2000

0 50 1000

10

20

30

40

Num

ber

of

Scatte

ringsE

nhancin

gF

acto

r

d2

[GHz]

Enhancing FactorNumber of Scattering

Figure 31. Dependence of the enhancement factor (•) andthe number of scatterings () onthe detuningδ. The inset shows the enhancement factor for larger detuning.

the detuning. They observe that for small detuning, the enhancement factor is more than 35,but there is serious heating. As the detuning increases, the enhancement factor as well asthe heating decrease. Note that the number of scatterings decreases more rapidly (∼δ−2)than the enhancement factor. At a detuning of 19 GHz, the enhancement factor is over 10and an atom experiences spontaneous emissions about 40 times during the launching andfalling processes. According to the calculation, however, they found that the heating due tospontaneous emissions was not so serious.

In order to reduce a loss of atomic coherence, an HLB with a large detuning may beused. For example, if a 15-W Ar-ion laser is used for a tenfold enhancement of guidingefficiency, the average rate of spontaneously scattered photons is calculated to be 10−3 Hz.While the number of atoms being guided in the fountain is increased, the HLB introducesinhomogeneous energy shifts of the ground-state hyperfine levels. In a trap based on a sheetof blue-detuned light supporting against gravity, a Stark shift of 270 mHz is obtained for4-s trapping time, which is larger than the line shift due to cold atom collisions [76]. Onepossibility for reducing the light shift in the HLB is to use a much higher-order Bessel beamfor the HLB or to use the evanescent waves of a hollow optical fiber. Since the evolutionof atoms in an HLB depends on the shape of the HLB, it is suggested that the ensemble-averaged heating and the light shift will be changed with the shape of the HLB [77]. Inprinciple, if the potential of the HLB is square, then atoms in the HLB may not feel anyscattering or light shift.

6.4. Crossed-HLB Trap of Rb

Xu et al. constructed a blue-detuned optical dipole trap by intersecting two horizontal,cylindrical HLBs at a right angle in the center of a Rb MOT [30]. The polarizations of thebeams were chosen to be orthogonal in the crossed region in order to suppress standing-wave effects. The detuning of HLBs was 20 GHz from5S1/2, F = 3 → 5P3/2, F

′ = 4transition line of85Rb and the trap depths were about 10µK in thex-direction and 90µKin the z-direction, respectively. They estimated that 60% of the atoms in the MOT were


initially loaded in the HLB trap.

0 20 40 60 80 1000

20

40

60

80

100

Experimental DataSimulation Results

Tra

ppin

gE

ffic

iency

(%)

Trapping Time (ms)

Figure 32. The trapping efficiency of a crossed-HLB trap as a function ofthe trapping time.The two horizontal HLBs have a power of 200 and 400 mW, the detuning of each HLB is20 GHz, and the initial temperature of atoms is 16µK.

The number and the temperature of the trapped atoms could be deduced by the TOFmeasurements. They found that the temperature of the trapped atoms was about 7µK andthis value was close to the minimum height of potential barrier of 10µK, where about105

atoms stayed inside the trap for 100 ms and the lifetime of trapped atoms was about 20ms as shown in Fig. 32. Figure 32 also shows the trapping efficiency as a function of thetrapping time. For comparison, the simulation results are also shown as the solid line inFig. 32 [78]. Since the detuning was much larger than the splitting between the hyperfine-structure levels of the excited state, the three-level interaction mode was quite good for thesimulation.

6.5. Optical Dipole Trap

Shin et al. proposed a three-dimensional, microscopic, and diffraction-limited far-off-resonance optical dipole trap (DFORT) for neutral atoms operating in the Lamb-Dickeregime, which is produced by employing the diffracted LP01 mode of an HOF [34]. DFORTprovides, in particular, a large trap volume so that it can be loaded with a large number ofcold atoms. Moreover, the 3D DFORT can be also operated as an elongated 1D opticaltrap. Such a microscopic and tight optical trap can be realized with moderate laser powerand a simple experimental setup. As the LP01 mode propagates in free space, its initialannular intensity distribution, which is represented by the two peaks on the HOF exit facetdiminishes away while the central bright peak starts to appear. The resultant generationof a tightly focused bright spot near the fiber facet can be qualitatively understood as thediffraction of the plane wave (i.e., the uniform LP01 mode) by a ring-shaped aperture andthe subsequent constructive interference near the central axis. Note that the intensity alongthe axial direction first increases to a maximum and then gradually decreases down to zero[Fig. 33(b)], and this axial asymmetry of the intensity gradient may be useful for efficient


loading of the nearby precooled atoms into DFORT. Note also that the axial intensitygradi-ent is larger than that of a typical focused Gaussian beam and the spot is even more tightlyfocused in the transverse radial direction [Fig. 33(b)].

r/l

z/l

r/l-10 0 10

0.0

0.5

1.0

r/l

z/l

z/l0 200 400

Figure 33. (a) Typical diffraction profile of the LP01 modeof HOF whena = 4λ andd = 3.5λ. (b) The contour plots of the produced optical potential. A, the radial distributionof DFORT atz = z0 = 50.9λ, B, axial distribution fromz = 0 to z = 600λ.

As a specific application, they consider the87Rb atoms and the laser powerP = 100mW at the wavelengthλ = 800 nm, which is far detuned from the D1 and D2 resonancelines. Then they obtain a cigar-shaped DFORT having the following basic parameters: themaximum trap depthU0 = 7.9 mK, the axial trap frequencyfz = 4.9 kHz, the radial trapfrequencyfr = 125 kHz, the trap volumeVtrap ' 10−8 cm3, and the scattering rate atthe trap centerΓsc = 2π × 310 Hz. In this case, the Lamb-Dicke parameters in the axialand the radial direction, defined by the ratio of the corresponding ground-state sizea0i tothe laser wavelength,ηi = 2πa0i/λ (i = z, r), are given byηz = 0.86 andηr = 0.18,respectively. For133Cs atoms atλ = 900 nm, on the other hand, the trap depth isU0 = 7.0mK, the oscillation frequencies arefz = 3.3 kHz andfr = 842 kHz, the Lamb-Dickeparameters areηz = 0.75 andηr = 0.15, and the scattering rate isΓsc = 2π × 317 Hz.


The DFORT has the unique feature of a tightly focused trap with a large trap volumeandconvenient loading and cooling of the precooled atoms, and may be also operated as anelongated one-dimensional optical trap.

7. Conclusion

We have reviewed various atom optical experiments using the hollow optical fibers. Thedetailed study on characteristics of electromagnetic fields inside and outside the HOF ispresented. We review the neutral atom guidance by means of red- and blue-detuned laserbeams, and the applications of the laser beams emanating from the HOF. In particular,an atomic guide in small hollow-core fibers has interesting applications. For example, byusing submicron-sized hollow fibers, it is possible to carry cold atoms at arbitrary points ona substrate for precise control of atomic deposition. By using bent hollow fibers, an opto-gravitational trap [79] and an atom-laser cavity [80] can be also constructed. Furthermore,the nonlinear dynamics has been theoretically studied for the guided atoms inside HOF withthe potential depth periodically modulated [81]. Moreover, cold atoms can be manipulatedby a sharpened, nanometric optical fiber tip or trapped by an evanescent field induced nearthe tip [82], just like an optical tweezer is used for manipulating nanometric particles [83].In particular, this may lead to the possibility of single atom manipulation so that atomic-scale crystal growth can be realized.

Cold atom guidance by hollow fiber can be also applicable to crystal growth of siliconon the atomic scale with a near-field optical device if a laser of 252 nm wavelength isavailable. The cavity QED effect in the near-field region is also an interesting subject:hollow fiber or sharpened fiber with an induced optical near-field can be a unique toolfor experimental study of the cavity QED effects by measuring the atomic deflection andcomparing it with the dipole forces [64]. Recently, the scope of the atomic guidance throughthe HOF is extended: As well as cold atoms, the micro- or nano-sized particles were guidedthrough the HOF [84]. Furthermore, in stead of hollow optical fibers, a photonic band-gapfiber (PBG) is also used for guiding atoms [85] or Bose-Einstein condensed atomic sample[86].

In the second part of this article, we have investigated the characteristics of the outputintensity distribution of each LPlm mode in HOF using the diffraction theory, and comparedthem with experiments. We have observed that the LP01 or LP11 mode itself cannot satisfythe basic requirement for an atomic funnel since the former does not support a dark columnalong the central axis while the latter causes loss of atoms due to the line of nodes. Toovercome these limitations, two LP11 modes were excited separately with their node linesand polarizations orthogonal to each other and then combined at an even fraction. Theresultant mode has an annular intensity distribution nearly similar to that of LP01 and itsoutput forms a divergent doughnut-shaped beam with the minimum dark spot of a fewµm,and this HLB can provide a deep optical potential for an atomic funnel which focuses atomsfrom their source onto a micron-sized hollow region of an HOF with a high efficiency,and the evanescent field associated with the corresponding mode can allow atoms guidedthrough an HOF. Moreover, the microscopic dark spot may be useful for the increase of theatomic density in the optical funnel trap suggested in reference [87, 88]. They can also beused for generation of the diffraction-limited dark laser spot [33]. Small focused dark spot


of HLB may be useful in atom optical experiments such as atomic lens, atom trap, andatomswitch.

The hollow laser beam generated by imaging the guided laser output of the hollow fiberhas been used for atom guidance [28], atom fountains [29], and atom traps [30]. Further-more, the diffracted output of the LP01 mode has a bright focused spot, which can be usedfor a tight optical dipole trap when a red-detuned laser is used [34].

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Chapter 5

ADVANCES IN PHYSICAL MODELINGOF RING LASERS

Vittorio M.N. Passaro1* and Francesco De Leonardis2**

1Photonics Research Group, Dipartimento di Elettrotecnica ed Elettronica,Politecnico di Bari, via Edoardo Orabona n. 4, 70125 Bari, Italy

2Photonics Research Group, Dipartimento di Ingegneria dell’Ambiente e per lo SviluppoSostenibile, Politecnico di Bari, viale del Turismo n. 8, 74100 Taranto, Italy

Abstract

In this chapter, an overview on fiber ring lasers and III/V semiconductor integrated ringlasers is presented. In particular, some aspects of mathematical modelling of these devices arereviewed. In the first part of the chapter, we have focused our attention on the more recenttheoretical and experimental studies concerning fiber ring laser architectures. Then, acomplete quantum-mechanical model for integrated ring lasers is presented, including theevaluation of all the involved physical parameters, such as self and cross saturation andbackscattering. Finally, the influence of sidewall roughness on either unidirectional orbidirectional regime in multi-quantum-well III/V semiconductor ring lasers is demonstrated.

Keywords: Semiconductor Ring lasers, Fiber Ring Lasers, Multi quantum well, Modeling

Introduction

Nowadays, fiber ring lasers are typical fiber devices obtained by placing a fiber amplifierinside a cavity to provide optical feedback, so involving a number of linear and non linearphysical effects. Moreover, semiconductor ring lasers are of great interest for monolithic andintegrated optoelectronic applications. Since these lasers do not require the use of cleavedfacets or difficult Bragg gratings for optical feedback, they can be conveniently integrated soreducing the occupation area. Moreover, the semiconductor ring lasers offer new capabilities

* E-mail address: [email protected]** E-mail address: [email protected]

Vittorio M.N. Passaro and Francesco De Leonardis162

in the travelling-wave unidirectional oscillation. Unidirectional operation is desired because itoffers the advantages of enhanced mode purity (high side-mode suppression ratio), reducedsensitivity to feedback and higher single beam power. Unidirectional ring lasers are used fortelecommunications systems, feedback laser diodes, multi-wavelength and all-optical flip flopoperation.

In this work we present both fiber and semiconductor ring lasers, with some recentadvances on their physical models. The chapter is substantially divided in two parts. The firstpart is essentially a review of fiber ring laser architectures, their relevant physical effects,technologies and main applications, including erbium doped devices, and continuous waveand mode-locking operations. In the second part, we focus on integrated architectures ofsemiconductor ring lasers and their physical models. In particular we investigate the influenceof some technological parameters (ring sidewall roughness, ring radius) over thebackscattering coefficient influencing the operation regime and performance of anysemiconductor ring laser. The theory of the physical model based on quantum mechanicalapproach is briefly summarized and some numerical results and simulations applied to multi-quantum-well III/V semiconductor ring lasers are presented. Here how either unidirectional orbidirectional regime is related to the ratio between the injection current and the backscatteringcoefficient value is shown. Finally simulation results relevant to an architecture of the ringlaser with asymmetrical output coupler to obtain a stable unidirectional regime are presented.

In general, our model is based on four differential equations in total, two coupledequations for the counter-propagating modes, one rate equation for the carriers injected in thelaser active region and one equation describing the phase difference dynamics between thetwo modes. Both the self- and cross- saturation effects and the backscattering effect over bothmicroscopic and macroscopic scale have been taken into account, where with microscopicscale we mean the effect of the backscattered wave on the gain medium, while asmacroscopic scale the effect of the backscattering on the Maxwell's equations is considered.By our model we can evaluate all the physical coefficients of the rate equations by means of afull quantum mechanical analysis. In fact, starting by the energy band information of theMQW structure and assuming typical values for the time constants of the electron scattering,our model is able to evaluate the linear and non linear gain coefficients by using a compactdensity matrix formalism. Then, these parameters are put inside the rate equations to calculatethe dynamic evolution of the oscillating modes in the ring resonator. By the study of thisdynamics, the investigation of any ring laser operating regime, bidirectional or unidirectional,should be possible.

Fiber Ring Lasers: An Overview

Fiber ring lasers have attracted tremendous interest because of their many importantapplications in fiber-optic test and measurement [1], fiber communications [2], fiber sensorsystems [3], and high-resolution spectroscopy. This is mainly due to their features, such aswide tunable range, narrow linewidth, and tuning at high speed allowing fast componentcharacterization. However, despite of these advantages, the fiber ring lasers present a numberof drawbacks essentially related to the possibility of multimode operation, which induces thebeat-noise generated as a result of beating effect between the lasing longitudinal modes andcould severely limit the lasers applications.

Advances in Physical Modeling of Ring Lasers 163

Differents methods to realise a fiber laser have been proposed in recent years. All theexperimental architectures can be classified in two main groups: fiber lasers based onnonlinear effects, and fiber lasers based on doped fiber amplifiers. Basically, both ring andFabry-Perot geometries have been used to realise the resonant cavity in which a piece ofnonlinear or doped fiber as active medium must be inserted. Nonlinear effects play animportant role in high power fiber lasers. They are induced by the large amount of powerdensity in the small area fiber core. Among the nonlinear effects, the Stimulated RamanScattering (SRS) and the Stimulated Brillouin Scattering (SBS) have shown significantpotentials.

The fiber-Raman lasers not only have a lower threshold compared with the single passSRS, but also they can be tuned over a wide frequency range (10 THz). Fig. 1 schematicallyshows a typical fiber Raman laser [4].

FiberPump

Pump

M1

M2

PrismLens Lens

Figure 1. Schematic architecture of a tunable fiber Raman laser.

A piece of single-mode fiber is placed inside a Fabry-Perot cavity formed by the partiallyreflecting mirrors M1 and M2. The cavity provides a resonant, wavelength-selective feedbackfor Stokes light generated inside the fiber by means of the SRS effect. The intracavity prismis needed to tune the laser wavelength by spatially dispersing various Stokes wavelengthswhich can be selected by rotating the mirror M2. It is evident that the laser thresholdcorresponds to the pump power at which the Stokes amplification during a round trip is largeenough to balance the cavity losses. As mentioned in [4], a threshold level of about 1 W canbe obtained using a fiber length of about 10 m. By adding separate mirrors for each order ofStokes waves, the fiber-Raman laser can be operated at several wavelengths simultaneously,each of which can be independently tuned by tuning the mirrors [5]. Several experimentshave demonstrated that it is relatively easy to achieve synchronization in fiber-Raman lasers.The reason is that the laser can select a particular wavelength satisfying the synchronous-pumping requirement among the wide range of possible wavelengths near the peak of theRaman-gain spectrum. Moreover, the laser wavelength can be tuned by simply changing thecavity length. This technique is referred to as time-dispersion tuning to distinguish it fromprism tuning based on spatial dispersion provided by the prism. Thus, synchronously pumpedfiber –Raman lasers have attracted attention for generating ultrashort optical pulses. Inaddition, if the Raman pulse falls in the anomalous group velocity dispersion (GVD) regimeof the fiber, the soliton effects can create pulses with widths of about 100 fs or less.

Similar to the SRS case, the Brillouin gain in optical fibers can be used to make fiber-Brillouin lasers by placing the fiber inside a cavity. To this aim, both ring cavity and Fabry-Perot geometries have been used. Fiber-Brillouin can lead to obtain a threshold input powerof 0.56 mW by using an all-fiber ring resonator, as shown schematically in Fig. 2 [4]


Fiber-Brillouin lasers consisting of a Fabry-Perot cavity exhibit features which arequalitatively different from those of a ring cavity configuration. The difference arises fromthe simultaneous presence of forward and backward propagating components associated withthe pump and Stokes waves. The simultaneous presence of many equi-spaced spectral lines inthe output of a fiber-Brillouin laser indicates the possibility of obtaining ultrashort opticalpulses if the laser can be mode-locked. Thus, an intra-cavity modulator could be useful torealise the mode-locking process [4].

However, Brillouin backscattering can be the limitation of the output power for narrow-band signals; and Raman scattering can generate a frequency shift which decreases the pumppower and signal power. For these reasons, a great attention has been reserved to fiber lasersbased on doped fiber amplifier.

PumpLaser

Spectrum Analyzer

LensDirectional

Coupler

Fiber ring

Polarizationcontroller

Figure 2. Schematic architecture of a fiber-Brillouin ring laser.

The erbium-doped fiber amplifier (EDFA) was invented by D.N. Payne and co-workersin 1987 [6]. The EDFA commercialization made long haul optical communicationsinexpensive and reliable and optical fiber became the standard for long-haultelecommunication systems. However, in situations where high-power lasers are needed, anEDFA does not work very well because the high power density damages the fiber. Nd-dopedfiber lasers (NDFL) and Yb-doped fiber lasers (YDFL) were developed for scaling outputpower of fiber lasers. Neodymium can be pumped at 808 nm to get good absorption, whileytterbium can be pumped at 975 nm. Both of these elements can emit light at around 1060 nmwith slightly different energy transition mechanisms. Initially, YDFLs attracted less attentionwith respect to NDFLs essentially because Nd3+ has the advantage of a four-level pumpingscheme, while Yb3+ works with a three or quasi four-level scheme. A four level laser systemtends to easier lasing because it has a lower threshold. However, ytterbium offers higherpower conversion efficiencies and larger output powers. In fact, even if its energy levelstructure and the re-absorption effect make the threshold pump power relatively high,ytterbium does not have self-quenching effects [7]-[8] as neodymium, and it can have ahigher ion concentration. Moreover, ytterbium can be more efficient due to the small quantumdefects.

Several theoretical works have been presented in literature to model and design fiber ringlasers. Generally speaking, these models are mainly focused on the study of the gain medium(in case of doped fiber lasers) and on nonlinear effects that origin in the ring cavity. In the


former case, the mathematical modelling is constituted from a system of rate equationsdescribing the time evolution of the carrier population and photon densities. These two kindsof rate equations contain complete information about the dynamics of the laser system. Sincethey are non linear coupled equations, a straightforward analytical solution is not feasible. Anumerical solution of these set of equations is presented in [9]. They can be analyzed undercertain approximations corresponding to real physical situations and an analytical solution ispossible in some cases [9].

A generalization of the mathematical model to the case of multi-frequency erbium-dopedfiber ring lasers employing a periodic filter and a frequency shifter has been also presented[10]. In this work, the rate equation systems describing the laser dynamics are coupled withthree spatial differential equations related to the pump propagation inside the fiber ring andboth forward and backward propagating amplified spontaneous emission (ASE) powers. Aniterative solution of the rate equations and propagation equations for two counter-propagatingASE powers and pump has been implemented using a fourth-order Runge–Kutta routine inwhich appropriate boundary conditions were been imposed at beginning and end of the activefiber, according with the experimental setup.

More recently, a steady-state model has been proposed to analyse and design a quasi-continuous wave injection-seeded ring laser [11]. As it is known, frequency stabilized laserdiodes can achieve a high degree of wavelength stability and are well-suited as masteroscillators for spectroscopic lidar systems. However, diode lasers alone do not have sufficientpower to provide for a high signal-to-noise ratio in these systems. One method to efficientlyscale the system power is to use the stabilized laser as a seeder in an injection-locked fiberring laser.

Two factors can limit the performance of seeded ring lasers. First, the spectral overlapbetween the seed laser and a ring laser cavity mode. The use of a low-finesse ring cavity hasbeen shown to relax this spectral requirement making the seeding process relatively easy. Thesecond issue is that the laser self oscillation can dominate the seeding process. Self-seedingcan be overcome through the incorporation of intracavity filters or using sufficient seedpower. The latter condition has been assumed in a model proposed in literature [11].

However, it is worth noting that the above cited mathematical models do not requireheavy computational efforts, essentially because they do not include optical nonlinear effects,that can originate in the ring fiber in high power regime. Generally, the nonlinear effects infiber ring lasers can be modelled by means of the nonlinear Schroedinger equation (NLS) [4].In particular, the NLS equation represents a powerful mathematical tool not only to design thefiber-Raman or fiber-Brilluoin lasers [4], but also to analyse the wave behaviour of an opticalfiber exhibiting a weak Kerr nonlinearity. The NLS has been extensively investigated in thiscontext, with particular emphasis given to the robust and stable soliton solutions that resultfrom a fundamental balance between linear dispersion and cubic nonlinearity [4]. Thus,soliton pulses are ideal carriers for transmitting optical data. For applications for whichpolarization effects are important, one must consider a system of coupled NLSs, that isgenerally not integrable. Despite the increased numerical complexity, in order to managesoliton like solutions it is not only useful to optimise the design of long-haul communicationssystems [12], but also to develop efficient optical fiber ring lasers [13]-[16]. For the ring laserconfiguration, Kerr nonlinearity of the birefringent optical fiber generates a nonlinear rotationof the polarization state that depends on the pulse intensity. Then, the insertion of a passivepolarizer provides an effective intensity filter that stabilizes, or mode locks, a propagating


pulse by periodically attenuating all components of the pulse that are not aligned with thepolarizer. Simple devices such as these have been shown experimentally to generate stableand robust soliton-like pulse trains, that can be used for a wide variety of telecommunicationspurposes [13]-[17].

Recently, an advanced mathematical model has been proposed in literature [18] for fiberring lasers. This model includes in the standard NLS equations two key modeling elementsthat describe the mode-locking dynamics: the nonlinear polarization rotation induced bycross-phase modulation, and the polarization control through the passive polarizer. Inparticular, the theoretical model consists of the coupled NLSs with periodic perturbations thatare due to the polarizer. In deriving this model, the contributions from continuum radiationhave been neglected. This is due to the filtering function of the passive polarizer in the mode-locking process. Although a residual amount of radiation is expected from the periodicity ofthe cavity, it remains negligible in comparison with the energy in the localized mode-lockedpulse. In addition, a source of amplitude fluctuations arising from the interplay betweennonlinearity and polarization control has been considered. In conclusion, the mathematicalmodel proposed in [18] represents a useful theoretical tool to examine the underlying mode-locking mechanism of the fiber ring and to describe the systematic and predictable amplitudefluctuations that result from this interaction.

As mentioned above, the main drawback in the fiber ring laser is related to the multimodelongitudinal behaviour. This disadvantage has induced many research groups to investigateexperimental solutions and search different configurations to optimise the fiber ring laserperformance. A number of studies have been presented in literature to reduce the beat noise,and to realise a fiber ring laser with single-longitudinal-mode (SLM) operation. For example,a compound-ring cavity to reduce the beat-noise has been proposed [19] where, due to theshort ring-length, the dual-coupler fiber ring acts as a small free spectral range (FSR) etalonfilter and, combined with the tunable optical bandpass filter, selects one longitudinal mode.However, this proposed solution needs PZT to accurately control the length of the cavity.

Recently, the research on SLM operation using saturable absorbers has been reported toovercome the limited spectral width of the mode selection filters [19]. Additionally, thelongitudinal lasing mode become unstable when the fiber ring cavity and the filter are highlysensitive to the temperature drift and other external disturbances. To prevent the unstableoperation, the fiber lasers of various schemes have been reported so that length of tunablefilter and fiber ring cavity were stabilized [20]-[21], or a saturable absorber was used tosuppress multimode operation [22]-[24]. As it is known, a large free spectrum range FSR isalso required to facilitate the SLM operation. In this sense, an architecture based on S-bandring lasers has been recently proposed [25]. Fig. 3 shows the proposed configuration andexperimental setup of an S-band erbium fiber laser with a triple-ring cavity resonator.

The architecture is constituted by the main ring (Ring-1), coupled with two rings havingdifferent lengths (Ring-2, and Ring-3) by means of two 50:50 optical couplers (C). The Ring-1 is composed of an S-band EDFA module, a 90:10 optical coupler (C1) to couple out theoptical beam, a fiber Fabry-Perot tunable filter (FFP-TF), and two polarization controllers(PCs). It is worth noting that the presence of two PCs lead to align the state of polarization ofthe Ring-1 cavity to guarantee a SLM oscillation. In addition, the FFP-TF is necessary notonly to determine a lasing wavelength, but also it serves as a mode-restricting component toprovide the first restriction on the possible laser modes.


C

980 nmPump LD

W W

C C

C1

PCPC

FFP-TF

S-Band EDFA ModuleOutput

Ring 3 Ring 2

Ring 1

EDF

Isolator

Figure 3. Experimental setup of an S-band erbium fiber laser with a multiple-ring cavity structure.

Finally, the S-band amplifier, constituted by three isolators, a pump laser at 980 nm, andtwo erbium doped fibers (EDF) with a saturated output power of 16.1 dBm at 1498 nm, has adepressed-cladding design and a power-sharing 980 nm pump laser to generate EDF gainextension effect [26].

The behaviour of this architecture is based on Vernier effect. In fact, indicating withFSRm and FSRs the free spectral range of the main and sub-ring cavities, respectively, thevalue of effective FSR becomes the least common multiple number of both FSRm and FSRs.As a result, the mode suppression can be achieved and governed by the length of the main-ring and sub-ring cavities. Thus, the laser mode oscillates only at a frequency thatsimultaneously satisfies the resonant conditions of main cavity and all the sub-ring cavities.Due to the combination of a FFP-TF with a triple-ring cavity, a SLM selection in this fiberlaser is successfully achieved. The polarization state of proposed laser should be maintainedby adjusting the PCs as that of the Ring-1.

The experimental results performed by means of an optical spectrum analyzer (OSA)with a 0.05 nm resolution [25], indicate that an output power larger than 0 dBm, a powerstability less than 0.05 dB, a wavelength variation less than 0.02 nm and a side-modesuppression ratio (SMSR) larger than 54.3 dB / 0.05 nm can be obtained over an operatingrange of 1481 to 1513 nm.

As mentioned before, one of the main problems considered in the fiber ring laser is thebeat-noise. Recently, a novel method to suppress the beat-noise fiber ring laser using a Fabry–Pérot laser diode (FP-LD) has been proposed in [27]. As reported there, the beat-noise offiber ring lasers is primarily in the low-frequency region of about 10 MHz due to the longring cavity length, which is typically of the order of several tens of meters. Thus, basically aninjected FP-LD can act as a high-pass filter to suppress the low-frequency beat-noise of fiberring lasers [28]. This is mainly due to the fact that the FP-LD has a fast carrier recovery rate


(1 ns) and experiments the gain saturation effect. Fig. 4 shows the experimental set-up used[27].

Pump

Fiber Fabry-PerotFilterFP-LD

Circulator 1

Circulator 2

Output

3 dB Coupler

Angled Splices

Fusion SpliceBi-EDF

A

B

12

3

3

2

1

Figure 4. Configurations of the highly polarized, low beat-noise, tunable fiber ring laser.

The architecture is constituted by an Bismuth oxide-based EDF (Bi-EDF) [29] pumpedby one 1480-nm semiconductor laser diode via Port 1 of an optical circulator (circulator 1),which fairly exhibits a flat pass-band in the wavelength range from 1460 to 1630 nm. Thelength of the Bi-EDF was of 51.4 cm. In the experiments, the refractive index of the Bi-EDFcore and cladding were 2.03 and 2.02, while the diameters of core and cladding were 3.9 and124.7 m, respectively. The erbium concentration in the Bi-EDF was 6500 wt ppm and Boronand Lanthanum are co-doped in the Bi2O3 -based fiber to increase the pump efficiency. TheBi-EDF peak absorption measured at 1480 and 1530 nm were 141 and 219 dB/m,respectively. Both ends of Bi-EDF were first angle spliced to high numerical aperture fiber(Nufern 980-HP fiber) before splicing to Port 2 (SMF-28 fiber) of Circulator 1 and to Port 3(SMF-28 fiber) of Circulator 2, providing better mode field diameter matching. The splicingloss attained was less than 0.2 dB for the angled splices. The angled splices reduce thereflection in the laser cavity to less than 60 dB.

The set-up presents also a large FSR fiber Fabry–Pérot (FFP) filter employed to tune thelaser wavelength and a double-channel planar-buried hetero-structure FP-LD with cavitymode spacing and threshold current of 1.12 nm and 10.9 mA, respectively. The opticalbandwidth and FSR of FFP filter were about 50 pm ( 6.25 GHz) and 110 nm, respectively.Thus, since the gain bandwidth of Bi-EDF is less than 110 nm [14], only one wavelength inthe fiber ring laser cavity is excited. The FP-LD was biased at 12 mA, slightly above thethreshold current, to realize the low beat-noise laser output. Finally, a 3-dB fused fiber taperwas included in the laser cavity to provide the laser output.


The experimental results [27] show that the configuration described in Fig. 4 leads toobtain high performance, including a fiber ring laser wavelength tuning from 1536.82 to1570.72 nm in 1.12 nm steps, with a maximum output power of about +3 dBm. Thepolarization degree and extinction ratio of the laser output are about 99% and 60 dB,respectively. Finally, the beat-noise, with and without FP-LD, was dramatically reduced by50 dB.

In parallel to the amount of research focused to realise SLM fiber ring laser, a number ofstudies has been proposed about multi-wavelength erbium-doped fiber lasers (MW-EDFLs).They have attracted considerable interest for potential applications in optical test andmeasurement, and optical wavelength-division-multiplexing communication and sensingsystems [30]. Compared with compact semiconductor-based lasers, EDFLs are competitivebecause of their all-fiber structure, as well as their capacity to provide high power andultrashort pulse width [31]. They can be used in applications that require multiple wavelengthsources, with small equal-wavelength spacing, a large number of peaks within a broad band,and high output uniformity across the channels. These requirements pose a very challengingtask for building a cost-effective multi-wavelength EDFL for continuous-wave or pulsedoperation. Previously, due to the homogeneously broadened gain property, many MW-EDFLswere developed with wavelength spacing larger than homogeneous linewidth of about 3.5nm,to overcome gain competition. Many different approaches have also been explored fordeveloping MW-EDFLs, including use of polarization or spatial hole burning, use ofindependent gain media, frequency shifting, and phase modulation [32]-[36]. However,recently a novel room-temperature-operated MW-EDFL with wavelength spacing less thanthe homogeneous broadening linewidth, based on interchannel four-wave mixing (FWM), hasbeen proposed [37]. The EDF gain-clamping effect is compensated by the parametric fourwave mixing (FWM) between multi-wavelength channels in a highly nonlinear fiber sectionthat is inserted into the fiber ring cavity and it is based on highly nonlinear photonic crystalfiber (HNL-PCF). Finally, sampled-fiber Bragg grating (SFBG) is used in input to one port ofthe circulator (see Fig. 5).

EDFA

Sampled FBGCirculator

10% OutputPolarization controller

HNL-PCF

Output coupler

Figure 5. Schematic diagram of the multi-wavelength erbium-doped fiber laser.


The multiple wavelength operation has been initiated, by adjusting the polarizationcontroller and a specific sampled fiber Bragg grating (SFBG), with 0.8nm wavelengthspacing. The architecture uses the inter-channel four-wave mixing-induced dynamic gain-flattening mechanism to stabilize the output. Thus, the FWM processes created by means ofthe HNL-PCF suppress the EDFL homogeneous line broadening and stabilize the multiplewavelength oscillation. By tuning the intracavity polarization controller and then the FWMefficiency, the number of concurrent lasing wavelengths can be changed from two to five, andthe peak power differences for the main oscillation wavelengths are less than 2.0 dB.

Channel spacing of 0.5nm operation of 10GHz dual-wavelength mode locking with thehelp of fiber birefringence, has been obtained with cavity structure with 60 m HNL-PCF [37].In addition, a supermode suppression ratio higher than 60dB, and a time-bandwidth productsranging from 0.39 to 0.41, have been measured.

Semiconductor Ring Lasers

Nowadays, multi-quantum-well (MQW) semiconductor ring lasers are of great interest formonolithic and integrated optoelectronic applications [38]. Since these lasers do not requirethe use of cleaved facets or difficult Bragg gratings for optical feedback, they can beconveniently integrated so reducing the occupation area. Moreover, the semiconductor ringlasers offer new capabilities in the travelling-wave unidirectional oscillation. Unidirectionaloperation is desired because it offers the advantages of enhanced mode purity (high side-mode suppression ratio), reduced sensitivity to feedback and higher single beam power.Unidirectional ring lasers have been used for feedback laser diodes [39]. Other importantapplications of semiconductor ring lasers include multi-wavelength [40] and all-optical flipflop operation [41-42]. A number of experimental studies have been also carried out toanalyse the operation regimes of the semiconductor ring lasers [43-45].

The main drawback of the semiconductor ring lasers could be represented by the instableregime of behaviour (switching between unidirectional and bidirectional). Thus, this physicalsituation needs an accurate physical model to individuate the design criteria in order to realisestable unidirectional regime.

In this sense, recently it has been experimentally demonstrated that GaAs-AlGaAs ringlasers show either bidirectional or unidirectional regimes depending on the injection current[45]. In particular, bidirectional operation reveals that just above threshold the ring laseroperate in a regime where the two counter-propagating modes are continuous waves. As theinjected current is increased, a new regime appears where the intensities of the counter-propagating modes undergo alternate sinusoidal oscillations. Finally, for injection currentslarger than a critical value, the unidirectional regime appears in a stable way. Therefore, thecontrol of the unidirectional operation is of fundamental importance in order to use thesemiconductor ring laser as an integrated source. In this chapter, we propose a very accuratephysical model to analyse the operating regimes of a MQW semiconductor ring laser.Similarly, a physical model has been recently presented to analyse the operating regimes ofany MQW semiconductor ring laser [46], as a strong generalization of the models previouslyproposed in literature. In fact, it can be applied to a generic MQW semiconductor structure,not only to a two level gas system [47]–[48] (for gas laser), and it considers the macroscopiceffect of the backscattering non included in the model proposed in [47]–[48]. In addition, the


model in [46], differently by [45], leads to calculate all the physical parameters without anysemi-empirical approximation, taking into account the backscattering effect on the gainmedium. It is based on four differential equations in total, two coupled equations for thecounter-propagating modes, one rate equation for the carriers injected in the laser activeregion and one equation describing the phase difference dynamics between the two modes.

Anyway, in the following section the theory of the physical model proposed in [46] isbriefly summarized with the aim to introduce some recent advances of this numerical model.In particular we have investigated the influence of some technological parameters (ringsidewall roughness, ring radius) over the backscattering coefficient influencing the operationregime and performance of any MQW semiconductor ring laser.. Then, we present somenumerical results and simulations applied to a GaAs-AlGaAs MQW semiconductor ring laser.In that section we will show as either unidirectional or bidirectional regime is related to theratio between the injection current and the backscattering coefficient value, i.e. technologicalconsiderations have been summarized. Finally, simulation results relevant to an architectureof the ring laser with asymmetrical output coupler to obtain a stable unidirectional regime arepresented.

Theory

The theory is based on the semi-classical interaction between radiation and matter. Then, theatomic systems are modelled as quantum phenomena while the electromagnetic (e.m.) field isclassically described by the Maxwell’s equations. In particular, the electric dipole er operatorrelates the system quantum-mechanical description with the polarization P of the mediumused as a source of the e.m. field.

Assuming a predominant single transverse mode as electric field inside the ring laser, wecan write:

( ) ( ) ( )( ) ( )E , F . .n nj t tn

n

t E t e c cω φ+= +∑r r (1)

where c.c. indicates the conjugate complex terms, ( )nE t is the electric field amplitude, nω is

the angular pulsation of the optical mode inside the MQW ring cavity and ( )n tφ is the time-dependent phase of the electric field. In general, the field function ( )F r can be written as

( ) ( , ) ( )F r G x y U z= , where ( , )G x y and ( )U z indicate the transverse and longitudinal profileof the electric field, z representing the propagation direction (curvilinear coordinate). Thesubscript n takes into account all possible longitudinal modes in the ring cavity. With thisrepresentation for the field, the polarization P induced by the gain medium is given by:

( ) ( ) ( )( ) ( )P , F .n nj t tn

nt P t e ccω φ+= +∑r r (2)

where ( )nP t is the complex, slowly-varying component of the polarization of the n-thlongitudinal mode. The wave equation for the time evolution of the electric field is given by:


2 22

0 02 2 2

1t v t t

μ μ∂ ∂ ∂−∇ + + = −

∂ ∂ ∂J E PE (3)

where v is the velocity of the electric wave in the ring resonator. The second term is includedto take into account the cavity losses. In particular the current density is expressed in terms ofthe fictional conductivity σ=J E , where the conductivity σ is assumed as the sum of twocontributions, 1σ and 2σ . The former includes all kind of optical losses (propagation loss,radiation loss, leakage loss, etc..) of the n-th longitudinal mode. It is related to the qualityfactor nQ of the cold cavity by means of the relationship ( )1 n nQσ εω= , where ε is thepermittivity of the cavity. The latter contribution is included to take into account the effect ofthe backscattering induced by the cavity sidewall roughness. According to [49], thebackscattered wave induces a contribution to the fictional conductivity given by: 2 bσ ε= ,where b is the backscattering rate. This rate coefficient is related to the amplitudereflectivity R due to the backscattering as ( )eff effb cR n R= π , being c the free-space lightvelocity, effn the effective index of the optical wave in the ring cavity and effR the ringeffective radius.

With the aim to study the dynamics of the two counter-propagating modes, we canparticularise Eqs. (1)-(2) to only two modes, one clockwise (CW, as mode 1) and the othercounter-clockwise (CCW, as mode 2). Under slowly-varying amplitude and phaseapproximations, extensively used in the laser dynamics modelling, the wave equation (3)produces the following set of equations:

( ) 1 2 11 1 2 2 1

1 1

1 1cos Im2 2 2

E E bE PQω ω ω

ψ δω ε

= − − + + (4)

( ) 2 1 22 2 1 1 2

2 2

1 1cos Im2 2 2

E E bE PQω ω ω

ψ δω ε

= − − − + + (5)

( )1 2 21 1 2 1 1

1 1 1

1 1Re sin2 2

EP b

E Eω ω

φ ψ δ ωε ω

= − − + +Ω − (6)

( )2 1 12 2 1 2 1

2 2 2

1 1Re sin2 2

EP b

E Eω ω

φ ψ δ ωε ω

= − − − + +Ω − (7)

where 2 1ψ φ φ= − and nΩ are the eigen-frequencies of the cold cavity eigen-modes. Eqs.(4)-(7) are the Lamb's self consistency equations and take into account the effects related to thegain medium and the macroscopic effect of the backscattering. The system (4)-(7) can besolved when the polarization vector is known.

This model leads to evaluate the linear and nonlinear terms of P by performing quantumcalculations applied to the MQW semiconductor structure. By this way, it is possible tocalculate all physical parameters, involved in the ring laser dynamics, starting from the


physical description of the MQW energy band without any semi-empirical approximation. Toevaluate the P vector we use the density matrix formalism [50] to write:

( ),

Tr( ) ba ab ab bab a

P n M n M Mρ ρ ρ= = +∑ (8)

where Tr is the transposte of matrix and ρ is the density matrix operator, given by:

aa ab

ba bb

ρ ρρ

ρ ρ⎡ ⎤

= ⎢ ⎥⎣ ⎦

being aaρ the probability to have an electron in a state of a BV (valence band) sub-band,

bbρ the probability to have an electron in b state of a BC (conduction band) sub-band,

abρ and baρ the probabilities to have a transition between a and b levels or b and a levels,respectively (it holds *

ba abρ ρ= ). M is the dipole moment operator (formed by the electron-hole pair relevant to two sublevels of the same order, one in BC and the other in BV) in theform of a 2x2 matrix as:

00

ab

ba

MM

M⎡ ⎤

= ⎢ ⎥⎣ ⎦

abM , baM have been calculated as in [36]. In Eq. (8), n denotes the electron total densityincluded in the conduction and valence sub-bands which must verify the followingrelationship for MQW semiconductor lasers:

( ) ( ) ( )(0) (0)

,g

bb aa cv ba c ba v ba bab a W

n g W f W f W dWρ ρ∞

⎡ ⎤− = −⎡ ⎤⎣ ⎦⎣ ⎦∑ ∫ (9)

being ( )cv bag W the state density, W W Wba b a= − the energy difference between level b and

a , fc ( fv ) the Fermi-Dirac distribution function at level b and a , respectively.By solving the continuity equation as proposed in our previous work [46], we derive the

elements of the density matrix and, then, the polarization vector as a sum of one linear andone non linear terms. By substituting the components of the polarization vector obtained bythe quantum-mechanical analysis in Eq. (4)-(5), considering only two modes, one clockwise(CW, 1) and the other counter-clockwise (CCW, 2), rearranging the equations in terms of themode intensities 2

1 1I E= and 22 2I E= , and assuming 1 2ω ω= , we obtain the rate equations

for any semiconductor MQW ring laser:

( ) ( ) ( )11 1 1 1 12 2 1 2 1 1 12 2 1 22 2 cos tot

dII I I I I I I b I I

dtα β θ ξ η δ= − − − + + Ψ + (10)


( ) ( ) ( )22 2 2 1 21 1 1 2 2 2 21 1 1 22 2 cos tot

dII I I I I I I b I I

dtα β θ ξ η δ= − − − + + Ψ − (11)

with ( )2 112totψ ψ π δ δ= − + − , ( )1 2

12

δ δ δ= + and:

4

2i

i iiii iiiiMω

βε

= N I

4

2i

ij iijj iijj ijji ijji ijij ijijMω

θε

⎡ ⎤= + +⎣ ⎦N I N I N I

4

2i

i iiij iiij iiji iiji ijii ijiiMω

ξε

⎡ ⎤= + +⎣ ⎦N I N I N I

4

2i

ij ijjj ijjjMω

ηε

= N I 1,2i = and j i≠

where ( ) ( ) ( ) Im nqkm

g

jnqkm cv ba c ba v ba qkm ba

W

g W f W f W je dW∞

− Ψ= −⎡ ⎤⎣ ⎦∫I C ,

* *( )I

n q k mVnqkm

n

F r F F F dr=∫N N is a real quantity (field overlapping integral,

*( ) ( )n n nF r F r dr= ∫N ), which is not zero just in the active region volume, and 2

nn n

n

gQω

α = −

is the net linear gain of MQW structure, being ng the gain of the MQW structure. Theparameters nqkmΨ and qkmC are defined in [31].

The introduction of the variable δ in Eq. (10)-(11) avoids the explicit dependence on theindividual scattering phases 1δ and 2δ , and, then, all physical quantities will depend only onthe average value δ . The equation system (10)-(11) has a stable solution as totψ = 0 when

ψ π= and 1 2δ δ= . The coefficients included in the model, i.e. iβ , ijθ , iξ and ijη , represent

the self-saturation coefficient, the cross-saturation coefficient, the self interference coefficientand the cross interference coefficient, respectively. They are responsible of the modecompetition phenomenon [46]. Then, by manipulating Eq. (6)-(7), we obtain:

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

2 1 2 1 2 1 2 2 1 1

1 221 1 12 2 1 2 21 12 21 1 12 2

2 1

2 1

1 2

sin sin2

tot

tot tot

dI I

dtI I

I I I I I II I

I IbI I

ψω ω γ γ ν ν

ν ν χ χ σ σ

ψ δ ψ δ

= Ω −Ω − − − − + −

⎛ ⎞+ − + − + −⎜ ⎟⎜ ⎟

⎝ ⎠⎛ ⎞

− + + −⎜ ⎟⎜ ⎟⎝ ⎠

(12)


with

2i

i iiii iiiiRω

νε

= N

2i

ij iijj iijj ijji ijji ijij ijijR R Rω

νε⎡ ⎤= + +⎣ ⎦N N N

2i

ij iiij iiij iiji iiji ijii ijiiR R Rω

χε⎡ ⎤= + +⎣ ⎦N N N

2i

ij ijjj ijjjRω

σε

= N

where 1Ω and 2Ω are the cold cavity frequencies for CW and CCW mode, respectively, 1ω ,

2ω are the laser beam frequencies, 1γ , 2γ are the pulling effect coefficients, 1ν , 2ν are the

self-pushing effect coefficients, 12ν and 21ν are the cross-pushing effect coefficients. The

terms 12χ , 21χ , 12σ , and 21σ are nonlinear coefficients induced by the nonlinearcontribution in the polarization vector. Moreover, it holds:

( ) ( ) ( ) 4 Re nqkm

g

jnqkm cv ba c ba v ba qkm ba

W

R M g W f W f W je dW∞

− Ψ= −⎡ ⎤⎣ ⎦∫ C

and

( ) ( ) ( ) ( )

( )

2

22

2

( )

2I

g

V i baii cv ba c ba v ba ba

m Wi ba

in

F r dr W WM g W f W f W dW

hW W

ωγ

ε

τ

∞ −= −⎡ ⎤⎣ ⎦

⎛ ⎞/− + ⎜ ⎟

⎝ ⎠

∫∫N

being inτ the electron average relaxation time.In order to complete the ring laser model, we have introduced the classical rate equation

for the injected carriers, in the form:

( ) ( )

( ) ( )

20 eff2 3

1 1 1 1 12 2 1 2 1 1 12 21

20 eff

2 2 2 2 21 1 1 2 2 2 21 12

2ε ndN J= -AN-BN -CN - I g -β I -θ I - I I ξ I +η Idt ed hω

2ε n- I g -β I -θ I - I I ξ I +η I

hω

⎡ ⎤⎣ ⎦

⎡ ⎤⎣ ⎦

(13)

where oε is the vacuum dielectric permittivity, d is the active region thickness, J is the lasercurrent density, n is the refractive index, A, B, C are the leakage recombination coefficient,the bimolecular recombination coefficient and the Auger coefficient, respectively. Eqs. (10)-(13) are the coupled differential equations for the two counter-propagating modes inside anyMQW ring laser.


It is worth noting that the self and cross interference coefficients are equal to zero only inabsence of any backscattering (i.e. R = 0) inside the laser cavity. In this ideal case Eqs (10)-(11) include only the self- and cross-saturation coefficients and, then, the only stable regimeis the unidirectional one. In the following section we will show that the backscattering effectcan be also responsible of a stable bidirectional regime, depending on R values.

Since the backscattering effect depend on technological effects like the ring sidewallroughness, the goal of our model is also to evaluate the operating regime of the MQW ringlaser related to the statistical information about this sidewall roughness. The sidewallroughness or boundary imperfections can have two significant effects: 1) energy scatteringtowards the radiation field, reducing the total quality factor of the optical mode; 2) powerredistribution between the two counter-propagating waves influencing the operational regimeof the ring laser. Differently from [45], where the backscattering effects are estimated in anempirical way, we calculate the R coefficient and the influence of the scattering losses bymeans of an analytic approach [52]. Thus, the sidewall imperfections can be described by arandom function having a Gaussian distribution for its self-correlation function as:

( )2' 2 ' 2( ) exp /c ccorr s s s s Lσ ⎡ ⎤− = − −⎢ ⎥⎣ ⎦ (14)

where s is the curvilinear coordinate, cL is the correlation length and cσ is the standarddeviation of the roughness correlation function.

We have analyzed the scattering loss due to sidewall imperfections by using the volumecurrent method [52]. This method consists of the calculation of the current density associatedto the sidewall roughness profile and, thus, solving the Maxwell’s equations in presence ofthis current source. Since the wave electric field has only components along the r and φdirections in the plane of the ring (for TE polarization), we concentrate our attention on thedominant component Eφ , because it has a peak at the ring edges. This component isinfluenced by the ring cross-section and radius. We have used a mixed numerical techniquebased on both effective index, conformal mapping and Wentzel-Kramers-Brillouin (WKB)methods to take into account these influences [52]. Thus, the electric field of the wavetravelling along the ring excites an additional contribution to the current density in the regionsas perturbed by the presence of roughness, and this contribution will induce the fieldradiation, calculated by the azimuth component of the vector potential φA . The determination

of vector potential has been executed by evaluating the free-space Green’s function. Then, wehave determined the power density radiated by means of the radially directed Poynting vector.Both types of radiations have been considered, i.e. tunneling radiation and phase-matchedradiation. However, for large resonator radii (>50 μm), we have found the phased-matchedradiation as the only significant contribution to the scattering loss. Now, we have evaluatedthe scattering-related quality factor of the ring resonator, scattQ . It is defined as the ratiobetween the stored energy and the power lost by scattering per each round trip, in the form:

2 exp( / 2)1 exp( )

eff eff scatt effstoredscatt

lost scatt eff

n R RPQ

P Rπ π α π

λ α π−

= =− −

(15)


by which the scattering coefficient scattα has been calculated as:

212 sinh eff eff

scatteff scatt

n RR Q

πα

π λ−⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

(16)

and the total quality factor of the ring cavity has been estimated as:

42 1 exp( / 2)

1 1 exp( )eff eff total eff

ntotal eff

n R RQ

R

π π η α πλ η α π

− −=

− − − (17)

where the parameter η is the coupling efficiency between the ring laser and an output buswaveguide and the total optical losses in the ring laser is given by

total scatt prop bend leakα α α α α= + + + , being scattα the sidewall roughness scattering loss, leakα theleakage loss to the substrate, propα the propagation loss and bendα the curvature-inducedbending loss. The leakage loss to the substrate is negligible by introducing in the ring laserstructure a buffer layer. The bending loss coefficient can be considered negligible due to thestrong confinement in the ring resonator and to its large radius (>50 µm). Thus the opticallosses are mainly dominated by propα and scattα .

Finally, the current induced by the sidewall roughness is a field source inducing a powertransfer between the two counter-propagating waves with a backscattering amplitudereflectivity R given as [52]:

( ) ( )2 222 20 02 / 4 expeff rib c eff cR R n t F k n Lφπ ε ωδ π σ ⎡ ⎤= −⎢ ⎥⎣ ⎦

(18)

where 2nδ is the ring-air relative permittivity change, ribt is the ridge overall height of thering cavity, Fφ is the azimuth component of the normalized electric field travelling inside thering cavity. Therefore, the backscattering coefficient is related by our model to the ring lasertechnological parameters, i.e. 2nδ , cL , cσ , ribt , effR .

Numerical Results

The numerical simulations have been performed by considering a standard GaAs-AlGaAsMQW ring laser structure. The active region is constituted from one (or more) GaAs wellssandwiched between two Al0.2Ga0.8As waveguide regions, each 100 nm thick. The p-type andn-type Al0.4Ga0.6As cladding layers are 1.0 and 1.5 μm thick, respectively. A top GaAs caphas been also included. A total optical loss of 25 cm-1 has been taken into account. We haveassumed a ring radius of 200 μm (small ring), a rib width of 2 μm and considered thebackscattering coefficient as a parameter. By this way we can show as the physical operationof the MQW ring laser depends on the relationship between the injection current andbackscattering coefficient. Anyway, it is possible to calculate the backscattering coefficient


starting from the statistical information of the ring sidewall roughness taken fromexperimental data, as explained by Eq. (18).

Analysing the stationary solution of Eqs. (10)-(11), the relationship 1 2 0,δ δ π+ = must besatisfied. In particular 1 2 0δ δ+ = gives rise to a condition of minimum stimulated energy,whereas 1 2δ δ π+ = gives rise to an instable condition of maximum stimulated energy andcan be discarded.

We have investigated the impact of the output coupler configuration [46] on the operatingcharacteristics of the semiconductor MQW ring laser. We have assumed an evanescent fieldcoupler, a very common element in integrated optics. However, one of the main problemswith these couplers is the sensitivity of the coupling efficiency η to the changes of couplerdimensions, particularly the coupling gap. Such variations make it difficult to accuratelyobtain a given coupling efficiency, a high reproducibility and good stabilization of the ringlaser operating regime. Fig. 6 shows the stationary regimes of 1I and 2I versus the couplingefficiency.

Figure 6. Intensities of both beams and phase difference versus the coupling efficiency.

In this simulation we have assumed cL =0.07 µm, cσ = 0.012 µm and an injection currentof I =100mA. The plot shows that the operating regime of the ring laser becomesbidirectional by increasing the coupling efficiency, starting from an unidirectional condition.In fact, for a coupling efficiency ranging from 5% to 16%, the quality factor of the ringresonator (see Eq. 17) assumes relatively large values, so inducing the current I to be wellgreater than the threshold, thI . A dominant beam in the ring cavity grows due to the mode


competition effect, as induced by the self- and cross-saturation coefficients iβ , ijθ . For acoupling efficiency larger than 16%, a significant part of the optical power leaves the ringresonator and this induces the threshold current to be close to I = 100mA. In this case, it isnot possible for only one of the counter-propagating beams to be completely extinguished. Infact, the backscattering effect between the beams always induces radiation travelling in theopposite direction.

Fig. 7 shows the stationary regimes of 1I and 2I versus the correlation length cL fordifferent values of the standard deviation cσ , by assuming an injection current value ofI =100mA (one well) and 1 2 0,δ δ π+ = . As usual, mode 1 designates the CW solution andmode 2 the CCW one, respectively.

Figure 7. Intensities of both beams versus the correlation length for various standard deviations of ringsidewall roughness function.

The plot shows that for a value of cσ smaller than a critical value, depending of theinjection current ( ,c thσ =0.0047 µm in this case), the MQW ring laser works in an

unidirectional regime without any dependence on cL . This means that for each value of cLthe backscattering coefficient is too small to compensate the mode competition effect inducedby the self- and cross- saturation coefficients iβ , ijθ (see Eqs. (10)-(11)). Therefore, adominant beam in the ring cavity grows while the backscattering effect will always maintain avery weak wave travelling in the opposite direction, being orders of magnitude below thedominant beam. The dominant beam can be randomly either the CW or CCW beam,


depending mathematically on the initial conditions or, physically, on the local optical lossesinside the ring cavity. For values of roughness standard deviation cσ > ,c thσ , there exists a

range for cL where the MQW ring laser shows a bidirectional operating regime. In particular,this range increases with increasing cσ .

Since the self-correlation function of the sidewall roughness is described as a Gaussianfunction (see Eq. (14)), this means that exists a range of cL , close to the peak of the Gaussianshape, where the backscattering coefficient compensates the mode competition effect. In therange of cL where the regime is bidirectional, it is possible to observe a maximum in the plot.This maximum occurs where the peak of the Gaussian self-correlation function is situated.

The previous discussion is also confirmed in Fig. 8, which shows the intensities of CWand CCW beams versus the injection current in the stationary condition, for different valuesof the statistical parameters of the sidewall roughness.

Figure 8. Intensities of both beams versus the laser current for various roughness correlation lengths.

The plot again shows that the operation regime of the MQW semiconductor ring laserdepends on the relationship between ( cL , cσ ) and the injection current. If the current is closeto the threshold, i.e. thI I≈ , the beams hold the same intensity (bidirectional condition). Forcurrents well larger than thI , a dominant beam in the ring cavity grows inducing theunidirectional regime. In Fig.8 we have assumed cσ =12 nm and used different values of cL ,larger than the position of the Gaussian peak. We can observe that the range of injectioncurrents where the MQW ring laser works in a bidirectional regime increases by increasing


the cL value. Figs. 7 and 8 are particularly important because they lead to an estimation of theMQW ring laser operation regime related to the etching step of the fabrication process. Infact, by performing a number of measurements of the etching profile of ring sidewallsobtained on different samples, it is possible to extract the statistical (Gaussian) information onthe sidewall roughness function and, then, give theoretical predictions by our model on thelaser working regime.

Fig. 9 shows the intensities of CW and CCW beams versus the effective ring radius fordifferent values of injection current in case of cL =0.07 µm and cσ = 0.012 µm. It is possibleto observe that the operating regime of the MQW ring laser is influenced by the ring cavitysizes. In fact, for each value of the injection current the graph shows that the operationalregime converts from unidirectional to bidirectional by increasing the ring radius. Thisbehaviour depends on the circumstance that the density current decreases by increasing thering radius and, therefore, the intensity of the dominant beam is reduced. In this condition thebackscattering effect is not negligible and the beams hold the same intensity. The curvesstopped for an appropriate value of the ring radius, depending on the injection current. Forring radii larger than this value, the MQW ring laser remains under the threshold.

Figure 9. Intensities of both beams versus the effective ring radius for various laser currents.

On the basis of our results, one additional component has to be included in thearchitecture of the MQW ring laser to favour only one circulating direction over the other, i.e.to achieve an unidirectional regime also for injection current values where the ring lasershould be bidirectional. The solution consists of an output coupler including a grating [46].


Then, we have investigated the influence of the grating reflectance gR and the output coupler

reflection and transmission coefficients, cR and cT respectively, over the laser behaviour.It is clear that the presence of the grating strengthens the CW solution with respect to the

CCW one. A number of parametric simulations by varying gR and cR show that, at constant

injection current I =100 mA, correlation length cL =0.07 µm and standard deviation cσ = 12nm (the value for bidirectional regime of the MQW ring laser without output coupler), it ispossible to realise a purely unidirectional condition for gR > 0.9 and cR =10%. It is alsopossible to obtain an unidirectional condition with a value of gR < 0.9, but it still needs to

increase cR .

Conclusion

In this chapter a short review on ring fiber lasers and recent advances of a highly detailedphysical model of MQW semiconductor ring lasers is presented. In particular, solutions tomultimode operation in fiber ring lasers and high performance in multi-wavelength erbium-doped fiber lasers have been described. Moreover, the operation regimes of GaAs-basedsemiconductor ring lasers have been demonstrated and related to the physical coefficientsincluded in the model. Different behaviour of the ring laser can be observed as a function ofinjection current, ring radius and statistical information of the ring sidewall roughness asdetermining the backscattering coefficient. Results relevant to output coupling by a gratingare discussed to guarantee a unidirectional regime in any condition. Thus, this chapter putsinto evidence as integrated MQW ring lasers are promising candidates to realise sources withenhanced mode purity, reduced sensitivity to feedback and higher single beam power.However, in particular applications such as optical test and measurements, opticalwavelength-division-multiplexing communications systems and sensing, the fiber ring laserslead to obtain high performance essentially due to their wide tunable range and very narrowlinewidth.

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Chapter 6

INVESTIGATION OF OPTICAL POWER BUDGET OF ERBIUM-DOPED FIBER

Hideaki Hayashia, b, Setsuhisa Tanabea and Naoki Sugimotob

a Graduate School of Human and Environmental Studies, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan

b Research Center, Asahi Glass Co.,Ltd., Kanagawa-ku, Yokohama 221-8755, Japan

Abstract

We investigated optical power budget of an erbium-doped fiber (EDF). In addition to the output signal and amplified spontaneous emission (ASE) powers from the fiber end, lateral spontaneous emissions and scattering laser powers in the EDF were measured quantitatively by using an integrating sphere. Compared with the signal and ASE powers, it was found that considerable powers were consumed by the laterally emitting lights. As an optically undetected loss which limits power conversion efficiency (PCE) of the fiber amplifier, the effect of nonradiative decay from the termination level of pump excited state absorption (pump ESA) was estimated from decay rate analyses of the relevant levels. The nonradiative loss was comparable to amplified signal power in the EDF when pumped with a 980 nm LD. Nonradiative decay following cooperative upconversion (CUP) process is also discussed using rate equations analysis.

1. Introduction

With the spreading and popularization of Internet and broadband communication, larger data traffic and higher processing speed are required in optical telecommunication systems. To meet these demands, optical fiber communication networks have developed rapidly. By using transmission fibers, metropolitan area networks (MANs) in inner-city has been established for several years as well as long-haul networks [1]. There are two method of increasing the information capacity in a single fiber; one is time division multiplexing (TDM), and the other is wavelength division multiplexing (WDM) [2]. The TDM is a technology of increasing the bit rate. On the other hand, the WDM is a technology of coupling optical signals of different wavelengths in the same fiber. The transposable

Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 188

bandwidths are represented as the TDM speed times the number of wavelengths in the WDM system.

In the optical fiber networks, optical amplifiers are one of the key components. The optical signals decay due to the background losses of the transmission fibers (typically 0.2 dB/km at around 1550 nm). The insertion losses of optical add-drop multiplexer (OADM) components also decrease the signal intensity, thus the amplifications of the signals are necessary at every few tens kilometers. For the optical amplifiers in the WDM systems, it is required that as many optical signals with different wavelengths as possible are amplified at the same time. As a practical amplification medium, erbium doped fibers (EDFs) have been extensively studied due to their excellent gain operation around 1.5 μm in the loss minimum window of transmission silica fiber [3, 4]. Figure 1 shows loss spectrum of a transmission silica based fiber and amplification bandwidths of EDFs. Since the development of an efficient silica-based erbium doped fiber amplifier (EDFA) in 1987 by a research group of University of Southampton [5], considerable research efforts have been made to improve the efficiency and broaden the bandwidths. To extend the bandwidths from conventional C-band (1530-1565 nm) to L-band (1570-1610 nm) in the WDM system, several glass hosts for EDFA such as fluoride, tellurite, Bi2O3-based, and multi component antimony silicate (MCS) have been proposed since the later half of the 1990’s [6-9].

1400 1500 1600Wavelength (nm)

Tran

smis

sion

loss

(dB/

km)

0.2

0.3

0.4

0.5

C-band L-band

Silica EDF for

Bi2O3-based EDF for

C-band: 1530-1565

L-band: 1580-1605

3 dB down bandwidth

C-band: 1530-1565

C+L-band: 1535-1610

Extended L-band: 1545-1620

Figure 1. Loss spectrum of a transmission silica based fiber and the bandwidths of Silica EDFs and Bi2O3-based EDF. Here “bandwidth” means 3 dB down bandwidth.

Silica based EDFs have been installed in the actual optical network system and practically played a critical role. However, their power conversion efficiency (PCE) is limited to 50-55% when pumped with 980 nm LD at present (i.e. ER-1090 amplifier by Sumitomo Electric Industries, Ltd. or HP980 amplifier by OFS). Since pump LD cost represents a significant proportion of the total amplifier cost, increasing the PCE is a concern for the amplifier development. Although the PCE is one of the most important factors for the amplifier design, it is not perfectly understood what limits the PCE in the EDFs. Except for

Investigation of Optical Power Budget of Erbium-Doped Fiber 189

emissions or loss origins that can be evaluated at the output end of the fiber (i.e. amplified signal, amplified spontaneous emission (ASE) or splice loss), considerable optical power budget of the EDFs is not clear. For example, lateral spontaneous emission of 1550 nm band has not ever been evaluated quantitatively although there is a report that the lateral emission spectra have been measured to calculate the cross section [10]. In order to optimize the PCE and amplifier performance, understanding of overall optical power budget of the EDFs is essential.

In this study, we constructed a novel evaluation system for measuring lateral emissions from the EDF by using an integrating sphere. We used a Bi2O3-based EDF (BIEDF) for the evaluation due to its potential for high performance amplifier [8, 11-13]. The lateral emissions such as spontaneous emission, upconversion emission, scattering light of laser diode (LD), and the scattering light of signal or ASE were measured quantitatively as well as the in-situ data results of the gain properties as a fiber amplifier. The variations of the lateral emissions with signal wavelength, signal power, or pump power were investigated. In addition, we estimated the effect of other nonradiative decay processes that follow pump excited state absorption (pump ESA) or cooperative upconversion (CUP). To investigate the nonradiative decay from the termination level of the pump ESA, the luminescence decay of the 550 nm band was measured. The effect of the CUP is then discussed theoretically using rate equations and optical propagation equations. Finally, we present the optical power budget of the BIEDF and clarify what decreases the PCE in the amplifier.

2. Background

The configuration and principle of an EDFA is shown in Fig. 2. An EDFA is composed of pump lasers, WDM couplers that couple input signals with pumping lights, isolators that prevent the reflection of output signals, and an EDF as an amplification medium. The EDF can be operated as a laser for the signal wavelength ranging in the band around 1550 nm, by utilizing a pump beam of a LD at the wavelength of 980 nm or 1480 nm [14, 15].

The 4f energy diagram of Er3+ ion and the main transitions involved in the three level laser operation are shown in Fig. 3. The ground state absorption (GSA) cross-section of the Er3+ ion exhibits a peak at 980 nm, and the Er3+ ions are excited from the ground 4I15/2 level to the 4I11/2 level. They decay to the metastable 4I13/2 level immediately, and the stimulated emission from the 4I13/2 level to the 4I15/2 level takes place. In addition to these transitions, the following transitions are accounted in this study: the quantum noise due to the ASE; the CUP via two photons in the first excited level of 4I13/2. Energy transfers from one Er3+ ion to other, and then the remaining excited Er3+ ion rapidly decay back to the 4I13/2 level; 1550 nm-band spontaneous emission (1550 nm-SE); the pump ESA from the 4I11/2 level to the 4F7/2 level; the upconversion emission around 550 nm-band (550 nm-SE); Nonradiative transitions (NRs) between the 4F7/2 level and the 4I13/2 level.

The PCE of optical amplifiers is calculated using following expression: PCE(%) = ( (PsOUT – PsIN) / PpIN) × 100, (1)


where PsOUT, PsIN, and PpIN are output signal power, input signal power, and launched pump power, respectively (Unit: W).

980nm

1480nm

Pump LD Pump LDEr-doped fiber

Signal

hν

WDM coupler

Isolator

Ener

gy

Er3+ ions at ground state

Excited ions in active-center

Figure 2. Configuration and principle of Er-doped fiber amplifier.

0

5

10

15

20

Ener

gy (×

103 cm

-1)

Er3+

4I13/2

4I15/2

4I11/2

4I9/2

4S3/2

4F9/2

2H11/2

4F7/2

980 nmAmp.Sig.

NR

CUP1550nm-SE

550 nm-SE

Signal

ASE

ESA

GSA

0

5

10

15

20

Ener

gy (×

103 cm

-1)

Er3+

4I13/2

4I15/2

4I11/2

4I9/2

4S3/2

4F9/2

2H11/2

4F7/2

980 nmAmp.Sig.

NR

CUP1550nm-SE

550 nm-SE

Signal

ASE

ESA

GSA

Figure 3. 4f energy diagram of Er3+ ion and the relevant transitions.

3. Preparation and Gain Characteristics of BIEDF

The glass preform containing Bi2O3 and SiO2 as main constituents was prepared using a conventional melting method. For the fiber core composition, 0.5 mol% of Er2O3 was added to the glass batch. Single mode EDF (cladding diameter of 125 μm) with plastic coatings was then fabricated. The core diameter of the BIEDF was 3.9 μm. The refractive index of the core and the numerical aperture (NA) of the fiber at 1550 nm were 2.03 and 0.20,


respectively. A BIEDF of 16 cm length was fusion-spliced to high NA fibers (Nufern 980HP) using a commercial fusion-splicer. The insertion loss of the spliced BIEDF at 1310 nm was 0.61 dB. By using a cutback method, the propagation loss of the BIEDF at 1310 nm was estimated to be 0.77 dB/m. Accordingly, the average splice loss per point was estimated to be 0.24 dB. Angled cleaving and splicing were applied to suppress the reflection due to the large difference of refractive induces between the BIEDF and the silica fibers [12]. It was confirmed that pig-tailed BIEDFs passed Bellcore (Telcordia) GR-1221 CORE qualification test [16]. Gain and noise figure profiles of the 16 cm BIEDF are shown in Fig. 4. The BIEDF was pumped with 140 mW by forward direction at 980 nm. The gain of the BIEDF reached 18.8 dB at 1535 nm in case the input signal power was –10 dBm.

1520 1540 1560 1580 1600 16200

5

10

15

20

25

Wavelength (nm)

Gai

n an

d N

oise

Fig

ure

(dB

)

GainNF

Figure 4. Gain and NF profiles of the 16 cm BIEDF. Launched pump: 140 mW forward at 980 nm; Input signal: –10 dBm at 1535 nm.

4. Lateral Emission Properties of BIEDF

4.1. Lateral Emission Measurement

Experimental setup for evaluating the lateral and fiber-propagating emission powers is shown in Fig. 5. The BIEDF of 16 cm length was coiled with 6 cm diameter and set in an integrating sphere (10inch: Model LMS-100s, Labsphere Inc.). The input and output end of high NA silica fibers were connected with instruments through small hole (5mm diameter) of the integrating sphere. The splice points were set just outside of the sphere. It was then pumped with a LD (FITEL) by forward direction at the wavelength of 980 nm. The pump power and temperature of the LD were controlled with a LD-driver (Model 525, Newport Corp.) and a temperature-controller (Model 325, Newport Corp.), respectively. A tunable laser (Model TLS210, Santec Corp.) was used for a single-channel signal source, and then the pump light and the signal light was coupled using a WDM coupler/Isolator (WDM/ISO). The spontaneous emissions and scattering lights laterally emitted from the BIEDF were detected with two kinds of fiber multi-channel CCDs with Si and InGaAs detectors. Each CCD


coupled proprietary spectrometer. The maximum wavelength range of the visible spectrometer (Model USB-2000, Ocean Optics Inc.) and the near-infrared spectrometer (Model NIR-512, Ocean Optics Inc.) were 350-1000 nm and 900-1700 nm, respectively. A premium-grade fiber with 1mm core (Model QP1000-2-VIS/NIR, Ocean Optics Inc.) was used to link the CCD and the output port of the sphere. For the spectral calibration, a standard halogen lamp (Model SCL-600, Labsphere Inc.) was used. The lamp was set at the center of the sphere and driven at 2.60A with a current-regulated DC stabilized power supply (Model PAN-5A, Kikusui Electronics Corp.). The absolute powers of total radiant flux of lateral emissions were then calculated. At the same time, the output spectra of fiber-propagating signal and ASE were detected with an optical spectrum analyzer (OSA: Model MS9780A, Anritsu Corp.) with 1 nm resolution.

First, we measured the spectral power distribution of various emissions. The pump power, the input signal power, and the signal wavelength dependences of the emissions were then investigated.

Integrating sphere

CCD/Spectrometer

PC

BIEDFOSA

WDM/ISO

Pump LD Device Under Test

Splice pointSignal source

Figure 5. Experimental setup for evaluating the lateral and fiber-propagating emission powers of the BIEDF. Basically the splice points were set outside of the integrating sphere.

4.2. Spectral Power Distribution

First, we show absolute power spectrum of lateral emissions and output emissions from the fiber end (Fig. 6). The ordinate represents spectral power distribution of radiant flux. The upconversion emission around 520 nm (2H11/2 → 4I15/2) and 550 nm (4S3/2 → 4I15/2), scattering light of LD around 980 nm, spontaneous emission of 1550 nm band, and ASE were detected by the two kind of multi-channel CCD which was connected with integrating sphere. We can also see weak emission around 660 nm that is related to the pump ESA process [17, 18]. The spectral shapes of the upconversion emission and 1550 nm band spontaneous emission were approximately identical with those in bulk glass [8]. When 100 mW of pump power and 0 dBm of signal power at 1530 nm were input, the optical powers of the upconversion emission, the LD scattering, and the 1550 nm band spontaneous emission, were 0.2 mW, 0.2 mW, and 3.1 mW, respectively. Here the splice points were set outside of the integrating sphere. In the case that the splice points were set inside of the sphere, the scattering of pump LD was increased to 4.3 mW, and 1.8 mW of the scattering light of the amplified signal was


detected by the CCD. The optical powers of amplified signal at 1530 nm and ASE band that detected by the OSA were 11.9 mW and 0.2 mW, respectively.

When the splice points were set outside of the integrating sphere, the sum of emission powers detected by the OSA and the CCDs were 12.1 mW and 3.5 mW, respectively. On the other hand, when the splice points were set inside of the sphere, the optical powers of the LD and signal scattering lights increased. The differences should represent the scatterings at the splice points. That is, the LD and signal scattering lights at the splice points result in the power losses of 4.1 mW and 1.8 mW, respectively.

500 1000 150010-2

10-1

100

101

102

103

104

Wavelength (nm)

Spec

tral p

ower

dis

tribu

tion

(μW/n

m)

550nm-SE

980nm-Scat.

ASE

1550nm-SE

Amp.Sig.

Figure 6. Spectral power distribution of various lateral emissions and amplified signal from the BIEDF. Launched pump and input signal power were 100 mW and 0 dBm, respectively. The splice points of the BIEDF were set outside of the integrating sphere. 550 nm-SE = 550 nm band spontaneous emission; 980 nm-Scat. = Scattering light of the LD at 980 nm; 1550 nm-SE = 1550 nm band spontaneous emission; Amp.Sig. = Amplified signal at 1530 nm; ASE = Amplified spontaneous emission.

4.3. Signal Wavelength Dependence

Figure 7 shows the signal wavelength dependences of the optical powers of the lateral emissions and the fiber propagating emissions. The launched power of the pump LD at 980 nm was fixed to 100 mW. Input signal power was set to 0 dBm. The right axis in the figure shows the gain of the output signal (square plots, unit: dB). In the wavelength range from 1530 nm to 1560 nm that corresponds to the C-band, we can see that the signal gains more than 10 dB were obtained with the BIEDF of only 16 cm length. The optical power of the 1550 nm band spontaneous emission was larger than that of the ASE in the entire C-band region. As for the ASE, the spontaneous emission of 1550 nm band, and the scattering light of the 980 nm LD, the optical powers of their emissions showed negative correlations with that of the amplified signal at measured wavelengths. The correlation of the ASE was the strongest among these emissions. These results indicates that more powers are consumed for


the output signal power in the C-band by lowering the ASE and lateral powers in vain, which is desirable as a fiber amplifier.

On the other hand, the optical power of the upconversion emission around 550 nm showed weak positive correlation. In other words, the upconversion emission power was large at the wavelength that the output signal power was large. This suggests that the upconversion emission is populated by the signal photons. In addition to the pump ESA, signal ESA using the signal photons can also occur. The initial level of upconversion emission, 4S3/2, would be populated by the signal photons through pump ESA. It can be said from the above correlation that the effect of the input or output signal wavelength on the signal ESA process is smaller than that of the output signal power.

1500 1520 1540 1560 1580102

103

104

Wavelength (nm)

Opt

ical p

ower

(μW

)

Amp.Sig. ASE 0.55u-SE 0.98u-Scat. 1.55u-SE Si

gnal

gai

n (d

B)

10

0

-10

Figure 7. Signal wavelength dependence of optical powers of various emissions in the BIEDF. Launched pump and input signal power were100 mW and 0 dBm, respectively. The splice points of the BIEDF were set outside of the integrating sphere. 550 nm-SE = 550 nm band spontaneous emission; 980 nm-Scat. = Scattering light of the LD at 980 nm; 1550 nm-SE = 1550 nm band spontaneous emission; Amp.Sig. = Amplified signal; ASE = Amplified spontaneous emission.

4.4. Signal Power Dependence

The signal power dependences of the optical powers of various emissions are shown in Fig. 8. The launched pump power was set to 100 mW, and input signal wavelength was fixed to 1530 nm. The ASE, the spontaneous emission of 1550 nm band, and the scattering light of the 980 nm LD decreased with increasing the input signal power. Even in the small signal region, the lateral emission power was larger than the ASE at the same 1550 nm band. The lateral 1550 nm spontaneous emission was larger than the amplified signal when the input signal power was smaller than -20 dBm. On the other hand, the upconversion emission around 550 nm increased with the input signal power. This positive correlation also suggests the existence of the signal ESA process using the input and amplified signal photons, because the output signal power of a fiber amplifier increases with increasing the input signal power.


-30 -20 -10 0102

103

104

Input signal power (dBm)

Opt

ical p

ower

(μW

)

Amp.Sig. ASE 0.55u-SE 0.98u-Scat. 1.55u-SE

Figure 8. Input signal power dependence of optical powers of various emissions in the BIEDF. Launched pump power were 100 mW. The splice points of the BIEDF were set outside of the integrating sphere. 550 nm-SE = 550 nm band spontaneous emission; 980 nm-Scat. = Scattering light of the LD at 980 nm; 1550 nm-SE = 1550 nm band spontaneous emission; Amp.Sig. = Amplified signal; ASE = Amplified spontaneous emission.

4.5. Pump Power Dependence

101 102

100

101

102

103

104

Excitation power (mW)

Opt

ical

pow

er (μ

W)

Amp.Sig. ASE 0.55u-SE 0.98u-Scat. 1.55u-SE

Figure 9. Pump power dependence of optical powers of various emissions in the BIEDF. Input signal power was 0 dBm. The splice points of the BIEDF were set outside of the integrating sphere. 550 nm-SE = 550 nm band spontaneous emission; 980 nm-Scat. = Scattering light of the LD at 980 nm; 1550 nm-SE = 1550 nm band spontaneous emission; Amp.Sig. = Amplified signal; ASE = Amplified spontaneous emission.


Figure 9 shows the pump power dependence of the optical powers of various emissions. All the emission species increased with the pump power, and the dependence of the upconversion emission around 550 nm was nearly 2 and obeying quadratic law. This means that the emission occurs as a result of the pump ESA or the CUP, each of which is due to a two-photon process. The pump power dependence of the lateral 1550 nm spontaneous emission was small and almost saturated under the pump power of larger than 60 mW. This indicates that there exist sufficient photons at the 4I13/2 level even when the excitation power is very small.

5. Nonradiative Loss

Other than lateral emissions described above, various nonradiative decay processes can be considered; deactivation by hydroxyl group in glass, nonradiative decay which is related to the pump ESA, the decay which is related to the CUP, and the multiphonon relaxation from the 4I11/2 level. Among these origins, the effect of hydroxyl groups was neglected here because this BIEDF was sufficiently dehydrated during the fabrication [19, 20].

5.1. Pump ESA Process

5.1.1. Lifetime Measurement of Er3+: 4S3/2 Level To analyze the effect of the nonradiative decay from the termination level of the pump ESA, luminescence decay of 550 nm band was measured, and then the quantum efficiency of the Er3+:4S3/2 level was calculated from the measured lifetime [21]. Second harmonic of Nd: YVO4 laser at 532 nm (Model J80-H10-532QW, Spectra Physics) was used as a pump source. The pump power was adjusted to 1 W, and the pump light that was modulated into pulses (Repetition: 15000 Hz; Pulse width: 13 ns) was incident on the optically polished Er-doped Bi2O3-based glass sample (18×15×3.5 mm in size). The luminescence of 550 nm band of the

0 1 2 3 4[×10-5]

Inte

nsity

(arb

.uni

t)

Ti me(s)

Excitation: 532 nmPower: 1 WMonitering: 550 nmSl it: 8 mm

τf=2.7 μs

Figure 10. Luminescence decay curve of the Er3+: 4S3/2 level in the Bi2O3-based glass. Circle plots represent measured data, and solid line represents single exponential fitting of these data.


glass sample was monochromated (Model 1681B, Spex) and detected with a photomultiplier (Model 1424M, Spex) that 0.8 kV of voltage was applied. The signal was collected using a sampling oscilloscope (500 MHz; Model TDS520, Tectronix Corp.), and the lifetime was determined by least square fitting of the obtained decay curve with exponential functions.

Measured luminescence decay curve of 550 nm band (4S3/2 → 4I15/2) is shown in Fig. 10. Sharp peak that can be observed near zero of the decay time must be the scattering of the pump LD, because the monitoring wavelength is relatively close to the pumping one. After excluding the effect of the LD scattering, the lifetime of the 4S3/2 level was determined to be 2.7 μs by using single exponential function.

5.1.2. Nonradiative Losses Following Pump ESA By using the obtained lifetime value, we discuss decay from the 4S3/2 level as a result of the pump ESA process. For simplicity, we assumed that all the photons that were excited to the 4F7/2 levels relax nonradiatively to the 4S3/2 level. This assumption will be valid because the energy gap between the 4F7/2 and the 4S3/2 is narrow (750 cm-1) [22].

Generally quantum efficiency of an emission, η, is written as follows: η = A × τf = A / (A + W), (2)

where A is spontaneous emission probability, W is nonradiative transition probability, and τf is fluorescence lifetime. We calculated the A coefficient from the Judd-Ofelt analysis (3100 s-1) [23-25]. Accordingly, the quantum efficiency was estimated to be 0.8%.

The nonradiative energy loss from the 4S3/2 level to the 4I11/2 level (unit: W), PNR (4S3/2→ 4I11/2), can be then expressed as follows:

NRP (4S3/2→4I11/2) = RP (4S3/2→4I15/2) / η ×ΔE (4S3/2→4I11/2) / ΔE ( 4S3/2→4I15/2), (3) where PR (4S3/2→4I15/2) is upconversion emission power, ΔE is energy gap between two 4f levels. The nonradiative energy loss from the 4I11/2 level to the 4I13/2 level, PNR (4I11/2→4I13/2), can be considered separately.

NRP (4I11/2→4I13/2) = [PL － RP (4S3/2→4I15/2) + NRP (4S3/2→4I11/2) －PR (4I11/2→4I15/2)] × EΔ (4I11/2→4I13/2) / EΔ ( 4I11/2→4I15/2). (4) Here PL is launched pump power, PR (4I11/2→ 4I15/2) is the optical power of 1000 nm

emission band. By using these expressions described above, PNR (4S3/2→ 4I11/2) and PNR (4I11/2→ 4I13/2)

were calculated to be 13 mW and 31 mW, respectively. Although the visible upconversion luminescence power was only 0.2 mW, we have to count the nonradiative decay from the 4S3/2 level due to low η.


5.2. Coorpelative Upconversion Process

5.2.1. Calculation of Cup Process We can estimate the effect of the CUP process using the rate equations analysis. Figure 11 shows the 4f energy level diagram of Er3+ ions and transitions used for the analysis. When a BIEDF is pumped with a 980 nm LD by forward direction, the time dependence of populations can be expressed as follows [26-28]:

( ) ( ) 441331221121322121211 / NANRCNNRRNWRAdtdN ++++−++= , (5)

2

233211222121212 2)(/ CNNWNRNWRAdtdN −++++−= , (6) ( ) 2

244333134321133 / CNNWNRRWNRdtdN ++++−= , (7) 334443414 )(/ NRNWAdtdN ++−= , (8)

where N1, N2, N3, and N4 represent the population of the 4I15/2, 4I13/2, 4I11/2, and 4F7/2 levels, respectively. For simplicity, we neglected the intermediate levels between the 4I11/2 and the 4F7/2 levels, and assumed that all the photons pumped at the 4F7/2 level via the pump ESA transit nonradiatively to the 4S3/2 level. Total Er3+ ion number density for the calculation was set to 1.54 × 1026 m-3, which corresponded to 0.5 mol% of Er2O3. R21, R12, R31, R13, and R34 are radiation transition rate between these levels that are calculated from absorption and emission cross sections (σs

e, σsa, σp

e, σpa, and σESA, respectively). A21 and A41 represent

spontaneous emission probabilities that are calculated by the Judd-Ofelt analysis [25]. Nonradiative transition probability, W43, is calculated in the way described in Section 5.1.2. W21 and W32 can be also estimated from the lifetime measurements of the bulk glasses in the same way as described in Section 5.1.1. The fiber length and the numerical aperture were set to 16 cm and 0.20, respectively. C represents cooperative upconversion coefficient. Here we assumed homogeneous distribution of Er3+ ions in the glass and homogeneous upconversion process [29-31]. Mode field diameter at 980 nm and at 1530 nm were set to 4.2μm and 6.3 μm, respectively.

The signal and pump lightwaves propagating along the fiber (Is and Ip) are expressed as the following set of ordinary differential equations [26-28].

dIs / dz = (σs

e N2 – σsa N1) Γs Is – αs Is (9)

dIp / dz = – (σp

a N1 – σpe N3 + σESA N3) Γp Ip – αp Ip (10)

Γs and Γp are overlap factor at the signal wavelength and pumping wavelength, respectively. αs and αp are parameters that represent the intrinsic fiber background loss at the signal and pumping wavelength, respectively. Here we assumed that the αs were identical with αp, and treated them as fitting parameters. We applied the Quimby’s assumption that σESA is equal to 2σp

a [32]. Although spontaneous decay was accounted for, the ASE was neglected since the


input signal power was sufficiently large (0 dBm) and the fiber length was sufficiently short. Splice loss from the BIEDF to high-NA silica fiber was set to 0.24 dB/point.

Assuming a steady state condition (the time derivatives to be zero), the set of differential equations were numerically integrated using the fourth order Runge-Kutta method with an initial condition at the input end of the fiber (z=0 m) [27]. The parameters used for numerical calculations are shown in Table 1. Input signal and launched pump powers were set to 1 mW (0 dBm) and 100 mW, respectively. By using these calculations, we obtained the relationship between the output signal power and the CUP coefficient.

Table 1. Parameters used for numerical calculations.

Parameter Symbol Value UnitSpontaneous emission rate A 21 250 s-1

A 41 3100 s-1

Nonradiative decay rate W 32 69 s-1

W 32 3.30×104 s-1

W 43 3.70×105 s-1

Signal emission cross section at 1530 nm βse 7.41ﾗ 10-25 m2

Signal absorption cross section at 1530 nm βsa 8.19ﾗ 10-25 m2

Pump emission cross section at 980 nm βpe 3.06ﾗ 10-25 m2

Pump absorption cross section at 980 nm βpa 2.36ﾗ 10-25 m2

Overlap factor at 980 nm βs 0.82Overlap factor at 1530 nm βp 0.52Er3+ ion density β 1.54ﾗ 1026 m-3

Cooperative upconversion coefficient C Fitting parameter m3/sBackground loss β Fitting parameter

N10

5

10

15

20

Ener

gy (×

103

cm-1

)

Er3+

4I13/2

4I15/2

4I11/2

4I9/2

4S3/24F9/2

4F7/2

R13

R34

A41

Amp.Sig.

R21

W21

W32

W43

CN2

N3

N4

A21 R12R31

Figure 11. 4f energy diagram of Er3+ ion and the transitions used for the rate equations analysis.


5.2.2. Effect of Cooperative Upconversion Here we estimate the effect of the cooperative upconversion (CUP) using rate equations analysis as described in the former section. Figure 12 shows the variation of calculated output signal with the CUP coefficients. The difference between the output power at a given CUP coefficient and the output at zero of the coefficient (value at y-intercept) represents energy loss via the CUP process. The calculations were performed for three values of α. For any α, the output power decreased exponentially with increasing the CUP coefficient.

Snoeks et al. reported that the value of the CUP coefficient was 3.2 ×10-24 m3/s in a soda lime silicate glass that was doped with 1.4×10-26 m-3 of Er3+ ions [31]. When we assume that the CUP coefficient of the BIEDF (1.54×10-26 m-3 of Er3+ ion number density) is same as that of the soda lime silicate, the curve of α = 4 seems reasonable. In this case, the effect of the CUP process results in approximately 10 mW. If we decrease the Er concentration in glass, the CUP will be reduced because the CUP coefficient is a function of the Er3+ ion density [33].

0 0.5 1 1.5 2[×10-23 ]

05

10152025303540

Out

put p

ower

(mW

)

CUP coeffi cient (m3/s)

α=0α=4α=8

Soda li mesilicate

Figure 12. Variation of signal output power with the CUP coefficient in the BIEDF doped with 1.54 × 1026m-3 of Er3+ ions. Plots represent calculation data, and solid lines are exponential fitting of these data. Dashed line represents literature value for a soda lime silicate glass doped with 1.4 × 1026m-3 of Er3+ ions (C = 3.2 × 10-24m3/s) [31].

6. Energy Budget of BIEDF

The optical power budget of the BIEDF that has been clarified in this study is shown in Table 2. Here the launched pump power and the input signal power were 100 mW and 1 mW (0 dBm), respectively. The insertion loss of 0.61 dB corresponds to 13.1 mW. The output signal power at 1530 nm and the sum of lateral emissions and scattering powers were 11.9 mW and 9.4 mW, respectively. It can be said that considerable powers were consumed by the lateral emissions and scatterings in the BIEDF.


Taking into account the output signal, the ASE, the lateral emissions, and the insertion loss, 65% of total power (65 mW) was not detected either by the CCDs or the OSA. The power of the nonradiative decay from the termination level of the pump ESA to the 4I11/2 level was estimated to be 13 mW. That from the 4I11/2 level to the 4I13/2 level was 31 mW. Approximately 10 mW can be attributed to the nonradiative decay following the CUP. We can say that nonradiative decays above also affect the decrease of the PCE in the BIEDF. Even counting all sources of loss described above, however, we could not identify approximately 11% of total launched power. A possible reason is that we underestimate nonradiative losses at present. For precise estimation of the pump ESA effect, high measurement accuracy of the very weak upconversion luminescence is necessary. For the CUP effect, we will have to consider the clustering of the Er3+ ions and resulting pair induced quenching [15, 34, 35].

Table 2. Energy budget of the BIEDF when pumped with 100 mW of launched power.

Emission species and source of loss mW Amplified signal 12 Insertion loss (splice loss+background loss) 13 980 nm LD scattering (at splice point) 4.1 980 nm LD scattering (w/o splice point) 0.2 Signal scattering (at splice point) 1.8 Amplified spontaneous emission 0.2 1550 nm band spontaneous emission 3.1 550 nm band upconversion emission 0.2 Nonradiative decay from the 4S3/2 to the 4I11/2 13 Nonradiative decay from the 4I11/2 to the 4I13/2 31 Nonradiative decay following CUP aprx.10 Unidentified aprx.11Launched pump power: 100 mWInput signal power: 1 mW

7. Conclusion

We have analyzed optical power budget of an erbium-doped amplifier (EDF). Lateral spontaneous emissions and scattering laser powers in a Bi2O3-based EDF (BIEDF) were evaluated quantitatively by using an integrating sphere. Comparing with amplified signal, it was clarified that considerable power was consumed by the laterally emitting lights. While the LD scattering, the signal scattering, and the 1550 nm band emission powers decreased with increasing input signal power, the lateral 550 nm emission power increased. In the same way, among the lateral emissions, only 550 nm band showed positive correlation with the spectrum of the output signal. These results suggested that the upconversion emission was promoted by the signal ESA.


As a result of decay rate analysis, it was revealed that the nonradiative power loss related to the pump excited state absorption (pump ESA) was comparable with the output signal power because the quantum efficiency of the initial level of the upconversion emission was only 0.8%. In addition, as a result of rate equations analysis, it was suggested that the effect of nonradiative decay following the cooperative upconversion (CUP) was not negligible when Er3+ ion density was an order of 10-26 m-3.

These analyses performed in this study can be applicable for not only a BIEDF but also commercial silica based EDF that the power conversion efficiency (PCE) is usually limited to 50-55%, other rare earth-doped amplifiers or lasers. The measurement system using an integrating sphere is also useful to analyze the lateral emissions from waveguide amplifiers in which precise control of their structures is necessary.

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23, 274-276. [8] Tanabe, S.; Sugimoto, N.; Ito, S.; Hanada, T. J. Lumin. 2000, vol. 87-89, 670-672. [9] Goforth, D. E.; Minelly, J. D.; Ellison, A. J. G.; Trentelman, J. P.; Samson, B. N.

Technical Digest of Optical Amplifiers and their Applications 2000, Nara, 2000, pp. OTuA4-1.

[10] Zech, H. IEEE Photon. Technol. Lett. 1995, vol. 7, 986-988. [11] Sugimoto, N. J. Am. Ceram. Soc. 2002, vol. 85, 1083-88. [12] Ohara, S.; Sugimoto, N.; Ochiai, K.; Hayashi, H.; Fukasawa, Y.; Hirose, T.; Nagasima,

T.; Reyes, M. Opt. Fiber Technol. 2004, vol. 10, 283-295. [13] Hayashi, H.; Sugimoto, N.; Tanabe, S. Opt. Fiber Technol. 2006, vol. 12, 282-287. [14] Barnes, W. L.; Laming, R. I.; Tarbox, E. J.; Morkel, P. R. IEEE J. Quantum Electron.

1991, vol. 27, 1004-1010. [15] Prudenzano, F. J. Lightwave Technol. 2005, vol. 23, 330-340. [16] Bell Communications Research, Generic requirements GR-1221-CORE 1994, Issue 1. [17] Auzel, F. Chem. Rev. 2004, vol. 104, 139-173. [18] Sun, H.; Xu, S.; Dai, S.; Wen, L.; Zhang, J.; Hu, L.; Jiang, Z. J. Non-Cryst. Solids 2005,

vol. 351, 288-292. [19] Hayashi, H.; Sugimoto, N.; Tanabe, S.; Ohara, S. J. Appl. Phys. 2006, vol. 99, 093105. [20] Hayashi, H.; Sugimoto, N.; Ochiai, K.; Ohara, S.; Fukasawa, Y.; Tanabe, S. Extended

Abstract of International Congress on Glass XX, Kyoto, 2004, pp. O-14-028. [21] Tanabe, S.; Hayashi, H.; Hanada, T.; Onodera, N. Opt. Mat. 2002, vol. 19, 343-349.


[22] Miyakawa T.; Dexter, D.L. Phys. Rev. B 1970, vol. 1, 2961-2969. [23] Judd, B.R. Phys. Rev. 1962, vol. 127, 750-761. [24] Ofelt, J.S. J. Chem. Phys. 1962, vol. 37, 511-520. [25] Tanabe, S.; Photonics Based on Wavelength Integration and Manipulation; IPAP

Books: Tokyo, 2005, vol. 2, pp. 101-112. [26] Becker, P. C.; Olsson, N. A.; Simpson, J. R. Erbium-Doped Fiber Amplifiers; Academic

Press: NY, 1999, pp 153-195. [27] Komukai, T.; Yamamoto, T.; Sugawa, T.; Miyajima, Y. IEEE J. Quantum Electron.

1995, vol. 31, 1880-89. [28] Khoptyar, D.; Jaskorzynska, B. J. Opt. Soc. Am. B 2005, vol. 22. 2091-2098. [29] Myslinski, P.; Nguyen, D.; Chrostowski, J. J. Lightwave Technol. 1997, vol. 15,112-120. [30] Bilxt, P. IEEE Photon. Technol. Lett. 1991, vol. 3, 996-998. [31] Snoeks, E.; van den Hoven, G. N.; Polman, A.; Hendriksen, B.; Diemeer, M. B. J.;

Priolo, F. J. Opt. Soc. Am. B 1995, vol. 12, 1468-74. [32] Quimby, R.S. Appl. Opt. 1991, vol. 30, 2546-52. [33] Gapontsev, V.P.; Platonov, N.S. Materials Science Forum 1989, vol. 50, 165-222. [34] Nilsson, J.; Bilxt, P.; Jaskorzynska, B.; Babonas, J. J. Lightwave Technol. 1995, vol. 13,

341-349. [35] Masuda, H.; Takada, A.; Aida, K. J. Lightwave Technol. 1992, vol. 10, 1789-99.


Chapter 7

RECENT DEVELOPMENTS IN ALL-FIBRE DEVICESFOR OPTICAL NETWORKS

Nawfel AzamiInstitut National des Postes et Télécommunications, Madinat Al Irfane,

Rabat-Instituts, Rabat, Morocco.Suzanne Lacroix

Ecole Polytechnique de Montréal, Laboratoire des fibres optiques,Montréal, Québec, Canada

Abstract

All-fibre components are essential components of optical networks systems.Development of such devices is of great importance to allow network functions to beperformed in the glass of the optical fibre itself. Among of all fabrication techniques, theFused Fibre Biconical Taper (FBT) technique allows optical devices with high performances.Although fibre devices are mainly based on the passive directional coupler basic structure,research is made to design components that perform complex functionalities in today opticalnetworks systems. Recent developments on all-fibre devices in network systems arepresented. Research is mainly focused on enhanced fabrication and stability of FBTfabrication technique, passive thermal compensation for stable interferometer opticalstructure, broadband spectral operation for multi-wavelength operations and newinterferometer designs. An overview of recent fused fibre devices for opticaltelecommunications is presented to understand the main functionalities of these fibre devices.The limiting factors are explained to understand challenges on fibre devices development.

Introduction

The fibre is not only the choice transmitting medium for high speed long-haultelecommunication. It is also currently used in sensing networks applications and morerecently in quantum information systems. Components are key elements of such networks.All-fibre devices and their full compatibility with the transmission medium make themparticularly attractive to perform operations such as multiplexing, routing, or filtering with

Nawfel Azami and Suzanne Lacroix206

low insertion loss, low polarization mode dispersion, and ease of interconnection. In thischapter, new developments of all-fibre components for optical networks systems arepresented.

A number of techniques have been developed to fabricate all-fibre components. Amongthem, the fusion-tapering or, in short, the FBT technique is extensively used especially for thefabrication of 2x2 couplers. Because of their intrinsic low loss, they offer the possibility ofhigh power handling (such as in all-fibre lasers), as well as individual photon manipulation(such as quantum information processing in quantum key distribution and quantumcomputing.) Most devices and components considered herein are however firstly designedfor use in standard telecommunication networks. In the first part of this chapter, FusedBiconical Taper (FBT) fabrication technique is described as well as the basic designsstructures of all-fibre devices.

A basic optical fibre communication system is presented in Fig. 1. An electrical signalfrom the data source is fed into the optical transmitter, which is contains a laser or an LED.The modulated light from the transmitter is launched into the fibre and transmitted to thereceiver via a demodulator. The receiver consists of a light detector with appropriateamplification and noise filtering. In a digital system a decision gate is also included. Opticalfibres prove economic when good use can be made of the bandwidth that they offer. OpticalWavelength Division Multiplexing (WDM) and Dense WDM (DWDM) systems have beendeveloped to perform multi channels propagation in a single optical fibre. Development ofstable multiplexers/demultiplexers is of great importance to combine wavelength channels inthe optical fibre. These types of multiplexers can also be used as demodulators whenDifferential Phase Shift keying modulation is used. Designs of all-fibre wavelengthmultiplexers/demuliplexers are usually complex since they require techniques for thermalcompensation of the wavelength channel drift. Moreover the sinusoidal spectral response ofbasic structures such as tapered fibre couplers or Mach-Zehnder interferometers is notappropriate. A flattened spectral response is more appropriate since it allows minimizinginsertion loss even when the carrier wavelength drifts. It also reduces crosstalk betweenadjacent channels. The second part of this chapter is dedicated to new developments on stableWDM/DWDM. In particular, passively temperature-independent all-fibre devices techniquesand new design of flat top multiplexers are presented.

During the last twenty years, interest in communicating by sending signals along opticalfibres has grown enormously. This interest lies in the very high capacity of transmission inoptical fibres, the very low attenuation of the signal during the propagation, as well as thehigh performances of Erbium doped fibre amplifiers (EDFA) and Raman amplifiers.Development of these amplifiers allows achievement of multi-channel lightwave systemswith high bit rates performances. For silica fibres, the attenuation is quite small, particularlyin the C-band, between 1525 nm and 1570 nm. Erbium doped optical fibres demonstratedhigh performances on amplification of signals with low noise. However, multi-channelsystems need additive components when compared to single-channel communicating systems.As an example, Erbium gain non-uniformity causes power divergence of WDM channels,limiting the system performances. Gain flattening filters (GFF) and Dynamic gain equalizersmodules are requested to flatten the amplifiers gain. Development of such devices using FBTfabrication technique is presented in the third part.

Raman amplifiers have also proved their high capacity of achieving high gain with lownoise. Distributed Raman amplification is very attractive since it allows amplification of

Recent Developments in All-Fibre Devices for Optical Networks 207

signals in the transmission fibre. Because of the high pump power needed in Ramanamplification, all-fibre components are of more benefit thanks to their high optical powerhandling. Multi-channel signal systems need multi-wavelength pump lasers when Ramanamplification is used. For this reason, and for system reconfiguration agility, widebanddevices are of particular interest for Raman amplifiers. In the last part, new developments onlarge bandwidth all-fibre devices for Raman amplification are presented.

Figure 1. Schematic optical fibre network configuration.

1. Fused Biconical Taper

Fibre optic couplers either split optical signals into multiple paths or combine multiple signalson one path. The number of input and output ports, expressed as an NxM configuration,characterizes a coupler, N representing the number of input fibres, and M the number ofoutput fibres. Fused couplers can be made in any configuration, but the simplest is the 2x2symmetric directional coupler, which is the equivalent in guided optics of a beam splitter inbulk optics. Although the most frequent components are 2x2 couplers, tapered single fibresare also a basic component of interest in themselves and, as such, are studied in the following.

1.1. Manufacturing

The fusion-tapering manufacturing technique consists in fusing laterally two (or more) fibrestogether using, as an heat source, a micro-torch, an oven or a CO2 laser. Depending on thefusion duration, one obtains a cross section with a degree of fusion ranging theoretically fromzero (for unfused fibres) to 1 (corresponding to a circular cross section theoretically obtainedafter an infinite duration). From a practical point of view, the degrees of fusion usually rangebetween 0.5 and 0.7. Example cross sections are shown in Fig. 2.

0.005 0.25 0.5 0.75 1

Figure 2. Cross sections of 2x2 symmetric fused fibre couplers. The degrees of fusion and indicatedbelow each cross section. Note the deformation of the cores as the fusion increases.


As shown schematically in Fig. 3, the fused structure is then stretched so as to create abiconical structure until the desired profile or the desired response is obtained. Tools andrules for the elaboration of recipes to design specific components are given in Ref. [1,2]. Eachtapering recipe includes several tapering segments, usually no more than 3 or 4, except forcomplex concatenated structures, such as Mach-Zehnder interferometers. Apart from the fibrelocal temperature, each segment is characterized by four main parameters, namely, the pullingspeed, the elongation, the flame position, and the effective width the flame or length of thehot zone during the process. Temperatures range is typically between 1450 ±50 °C but mayreach 1700 °C. The ends of the tapered structure are usually pulled apart at equal and oppositespeeds relative to the centre of the heat source. As a result, the tapered structure is symmetricso that the slopes of the down-taper transition and of the up-taper transition regions areidentical. Pulling speeds, typically of the order of millimetres per minute, are usually constantfor a given segment and depend on the fibre temperature. For a given temperature, it isadjusted so that the fibre neither breaks nor sags during the process. The final elongation ofthe component for a given segment determines the end of this particular segment. During thetapering process, diagnostics are made: the shape of the device is controlled through abinocular microscope; its optical transmissions (in both arms) are recorded at a givenwavelength as a function of elongation or for a whole range of wavelengths using abroadband light source and an Optical Spectrum Analyzer (OSA) as a detector.

Broadband

lightsource

Diagnostics(OSA)

Heat source(flame, CO2 laser, oven)

Streching motors

Figure 3. Manufacturing of a 2x2 coupler using the FBT technique.

1.2. Adiabaticity Concept

The slopes of the longitudinal structure largely determine the behaviour of the component.The propagation along a tapered fibre is said to be adiabatic whenever the fibre transmissionis not affected by the taper slope [2]. This is only possible for gentle slopes. In contrast, whenthe slopes are abrupt, transfer of power to higher mode may occur. This is, from a generalpoint of view, undesirable for couplers as this causes power leakage. An adiabaticity criterionis derived for every particular structure, whether a single fibre, or a coupler made of two ormore fused fibres. The fused fibres may be identical to create a symmetric structure or not, inthe more general case of asymmetric couplers. The adiabaticity criterion provides the upperlimit normalized slope that a structure may have for an adiabatic behaviour. Details of thecalculation and graphical representation of adiabaticity criteria are given in Ref. [1,2]. Formost structures made of standard 125 µm diameter fibres, the limit slope is of the order of 10-

3 µm-1. While adiabaticity is usually required for couplers, non-adiabaticity of tapered single


fibres can be used to design a variety of all-fibre spectral filters (see Section 3.1). Theirprinciple of operation is overviewed below.

1.3. Non-adiabatically Tapered Fibres

When one tapers a fibre with steep slopes, e.g. heating over a zone of the order of a fewmillimetres, one observes oscillations in the transmitted power as a function of elongation, ata given wavelength. For a given elongation, these oscillations are also present as a function ofwavelength. As explained in more details below, this is the result of the alternation of localmodes coupling and beating effects along the tapered structure.

In the downtaper region, as the fibre diameter decreases, the fundamental LP01 core modeexpands in the cladding. When the diameter is reduced by a factor of 2 or more, thefundamental mode becomes guided by the cladding-air interface. The mode is said to be “cutoff” as a core mode: it becomes a cladding mode. If the slope is steep, some power istransferred to other cladding modes (LP02, LP03, ...) by coupling effects.

In the central region, where the slopes are small, the adiabaticity criterion is again obeyedand the excited LP0m modes, all of them being cladding modes, accumulate phase differencesthrough the beating effect.

While arriving on the uptaper region, mode coupling again occurs before the power isfinally recovered in the core. Depending on the relative phase of the excited LP0m modes,(therefore on the wavelength and on the elongation) power may be partially or totallyrecovered in the core. All the power, which is not recovered in the core, is in the claddingmodes and possibly trapped by the protective jacket of the fibre, thus lost.

This process of coupling-beating-coupling thus confers to a tapered fibre an oscillatorybehaviour according to the various parameters affecting the modal phase differencesaccumulated mostly in the beating region. The LP01 and LP02 modes are responsible for themain oscillation. Higher order modes (LP03, ...) possibly superimpose to it smaller amplitudeand larger frequency oscillations. For a pair of modes, e.g. LP01 and LP02, the wavelengthresponse is essentially sinusoidal and it is exploited to design spectral filters, such as those toflatten the Erbium doped fibre gain described in Section 3.1.

1.4. Transfer Matrices of 2x2 FBT Symmetric Couplers

In the following, the principle of operation of adiabatic 2x2 FBT symmetric couplers isoverviewed. For a coupler made of individual guides in close proximity, the power transferfrom branch to branch is usually analyzed in terms of coupling between the modes of theindividual guides. However, in the case of FBT couplers, it is necessary to call forsupermodes. For a 2x2 symmetric coupler (the only coupler considered herein for the sake ofsimplicity), these are referred to as SLP01 and SLP11, respectively. They are the fundamentaland the first asymmetric modes of the superstructure, i.e., the fused structure. Note that theadiabaticity criterion, which is supposed to be obeyed, refers to these supermodes. Thisconcept of supermodes is essential for the following reasons.

For the power transfer to occur, the fused structure is tapered down to a diameter suchthat the cores no more play their guiding roles.


As the cores are reduced, the fields spread out of the cores and the guiding process isensured by the cladding-air interface.

As a result, individual guides can no more be identified in the fused and tapered region,where the power transfer occurs.

The transfer of power is then described in terms of a beating phenomenon between thefist two supermodes SLP01 and SLP11, which are equally excited at the entrance of the couplerwhen light is launched only one of the entrance branch. Due to their different propagationconstants, they accumulate a phase difference along the structure. Whenever they are inphase, the power is retrieved in the main branch, while, whenever they are out of phase, thepower is retrieved in the secondary branch. An intermediate phase difference valuecorresponds to a branching ratio between 0 and 1. More quantitatively, a coupler ischaracterised by its transfer matrix

Mα = eiα cosα isinαisinα cosα

⎡

⎣ ⎢

⎤

⎦ ⎥

(1)

where α is an average common propagation phase, which is, for this reason often omitted. Itis defined as

2α = ∫0

L

(β01 + β11)dz(2)

and 2α is the accumulated phase difference between both supermodes along the length of thecoupler

2α = ∫0

L

(β01 − β11)dz(3)

β01 and β 11 being, in these formulas, the propagation constants of the supermodes SLP01 andSLP11, respectively. Note that these propagation constants are wavelength dependent, whichconfer the coupler a spectral dependence. The transfer matrix relates the amplitudes in thetwo exit branches to those in the entrance branches. For example, an excitation in a singlebranch corresponds to the entrance vector

1

0

⎡

⎣ ⎢

⎤

⎦ ⎥

and thus to an exit vector

eiα cosα isinαisinα cosα

⎡

⎣ ⎢

⎤

⎦ ⎥

1

0

⎡

⎣ ⎢

⎤

⎦ ⎥ = eiα cosα

isinα⎡

⎣ ⎢

⎤

⎦ ⎥

Note the i factor, corresponding to a π/2 phase factor between both branches, which isunusual referring to the analogy between a fibre coupler and a beam splitter.


The corresponding intensity transmissions in main and secondary branches, respectivelylabelled 1 and 2, are

T1 = cos2 α =1+ cos2α

2

T2 = sin2 α =1− cos2α

2 (4)

A 50%/50% splitter, also referred to as 3 dB coupler, is thus a coupler havingα=π/4+pπ/2, with p integer (usually null, to ensure small spectral dependence).

Due to the lack of circular symmetry of the guiding structure, the couplers are inherentlypolarisation dependent. The cross section has two symmetry axes x and y, which define theprincipal polarisation axes. The coupler transmissions must more generally be written as asuperposition of transmissions in each polarisation

T1 = T1x + T1y = ηcos2 αx + (1−η)cos2 α y

T2 = T2x + T2y = ηsin2 αx + (1−η)sin2 αy (5)

where η and (1-η) are the proportions of power launched in the x and y polarisations,respectively. However, strongly fused couplers are virtually polarisation insensitive inasmuchtheir waist is not too small. This is the case of most 3dB and other standard beam splitters, thepolarisation dependence of which is ignored.

The transfer matrices are very useful tools to predict the responses of more complexstructures made of concatenation of several couplers. The simplest one is the Mach-Zehnder(MZ) interferometer made of two concatenated couplers, which may be different, thuscharacterised by α ≠α’. Such an interferometric structure is sketched in Fig. 4.

Figure 4. All-fibre Mach-Zehnder structure. The phase difference between the arms ϕ=β1L1−β2L2 isrealised through a length difference L1−L2 and/or a propagation constant differenceβ1−β2 =2πν(n1−n2)/c.

For a phase difference ϕ between the two MZ arms, the transmission column vector(containing individual guide amplitudes) may be calculated by using matrix products asfollows


⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎥⎦

⎤⎢⎣

⎡− 0

1cossinsincos

.0

0.'cos'sin'sin'cos

2

2'

αααα

αααα α

ϕ

ϕ

α

ii

ee

ei

ie i

i

ii

(6)

The intensity transmissions in each branch are then easily derived to be

T1 =1

2(1+ cos2α cos2α '−sin2α sin2α 'cosϕ)

T2 =12

(1− cos2α cos2α '+sin2α sin2α 'cosϕ)(7)

The different parameters α, α’, and ϕ give a flexibility to design a variety of differentcomponents with specific functionalities. As an example, the DWDM components areexamined in Section 2. In this case, one has the couplers parameters α=α’=π/4, which arealmost wavelength independent over the range of interest.

2. Stable Wavelength Division Multiplexer All-Fibre Devices

WDM Mach-Zehnder Interferometers (MZI) are extensively used as multiplexer,demultiplexer, add-drop modules, and in many other applications. For most of theseapplications, a control of the thermal dependence of the refractive index is required. In orderto simplify the description of the MZI transmission as a function of the optogeometricalparameters of the two fibres (Fig. 4), let us suppose an ideal MZI with no loss and an infiniteisolation by using 3dB couplers. Using eq. 7, the transmittivity from port 1 to port 2 can bewritten as:

⎥⎦⎤

⎢⎣⎡ −

π= )(2cos)( 2211

2 LnLnc

T νν(8)

where ν is the signal frequency, c is the light velocity, L1 and L2 are the lengths of fibres 1and 2 in the central zone respectively, and n1 and n2 their effective indices. The Free SpectralRange (Δν) and the pth transmission peak frequencies νp are given by:

( ) ( )22112211 2.

2

LnLncp

LnLnc

p −=

−=Δ νν

(9)

Inter-channel spectral distance Δν is then induced by fibres with different refractive indexprofiles or/and different lengths. MZIs are known for their narrow band capabilities. For thispurpose, they must be stable over a range of environmental conditions, such as temperature,within a defined range in case of temperature variations. However, the refractive indices orthe optical path lengths of the two connecting fibres of the device between the two couplers


usually vary with temperature. If the thermooptic coefficients (i.e. the temperaturedependence of their refraction indices) of the two fibres are not equal or if the optical paths ofthe two fibres are not equal, the temperature variations cause variations in the differentialphase shift. Consequently, the channel spacing of the device, defined as the wavelengthseparation between the transmission peaks of two adjacent channels, as well as the peakwavelengths and the pass-band, become unstable. This would cause significant problems forWDM applications, due to the small separation between channels. In the next section, resultsof temperature-independent all-fibre MZI manufactured with the FBT technology arepresented.

The thermal dependence of an optical fibre can be expressed with the aid of the thermo-optic coefficient, which describes the change of the index of refraction with the change oftemperature dn/dT. If both arms in the central zone are equal (L1=L2=L), the thermal shift ofthe MZI transmission peaks is given by:

dνp

dTνp−

1Δn

dn1

dT

dn2

dT−

⎛⎜⎝

⎞⎟⎠

⋅1L

dLdT

⋅+⎡⎢⎣

⎤⎥⎦

⋅(10)

where Δn= n1 – n2 and L is the length of the central zone of the MZI. The thermal expansioncoefficient for silica (L-1.dL/dT) is about 5.10-7 °C-1. The contribution of the thermalexpansion of silica fibre to the thermal shift of the MZI transmission peaks is nearly0.75 pm/°C for a transmission peak at 1.55 μm. Thermal expansion of silica is usuallyneglected in Mach-Zehnder interferometer structures and is not an issue for thermalcompensation. However, thermal expansion of silica is of great importance in taperedcouplers design because of the impact on supermodes propagating index.

2.1. Passive Thermal Compensation Using UV Treatment

It is well known that photosensitive fibre hydrogenation may produce large refractive indexchanges if the fibre is exposed to UV radiation [3,4]. This process has been extensively usedfor fabrication of Bragg gratings and balanced MZI. More recently, it has been shown thathydrogenation of an optical fibre followed by UV exposure can control the thermaldependence of the refractive index. This may be used in a device, such as an all-fibre MZI. Inthe following section, the process applied to one fibre-arm of the MZI is presented. Thisprocess, applied before the fabrication of the MZI, consists of hydrogenation and expositionto UV radiation. The optical fibre is put in a pressure chamber, filling the chamber withhydrogen at a suitable pressure (about 1800 psi) and left there for a period of time suitable toachieve the desired photosensitivity (about 12 hours). This process produces an increase inthe index of refraction of the fibre, which becomes n+dn [3,5,6]. Thereafter, thephotosensitive fibre is exposed to UV radiation. As is mentioned in ref. [6], such an exposurecan lead to a further increase of the fibre refractive index. It has been recently found that onecan control or adjust the thermal dependence of the optical fibre by controlling the UVexposure time of the photosensitive fibre [7]. Moreover, it has been discovered that thechange of the thermal dependence provided by this method remains constant even though theindex of refraction is further changed, for example by exposing the fibre to heat. Thus, by


heating the fibre to a temperature greater than 800 °C, one can bring down the fibre index ofrefraction back to the value n, without affecting the adjusted thermal dependence.

As a demonstration, all-fibre MZIs were fabricated using the treated fibre (fibre 1) and adissimilar fibre (fibre 2) with different refractive indices (Fig. 5). Thermal dependences of thefibres are also different. Fig. 6 illustrates the change of the thermal dependence of the all fibreMZI as a function of the UV exposure time of the hydrogenated fibre (Corning SMF-28™).The thermal dependence of the MZI does not change during the first 5 minutes of exposure toUV radiation. The change in thermal dependence then starts to occur gradually and continuesmore steeply as shown in Fig. 6. Between 10 and 25 minutes of exposure, the thermaldependence change is essentially linear. As is shown in Fig. 6, the reproducibility of thethermal dependence is good. The small variability of the thermal dependence may be due tovariations in the fabrication process of the MZI (for example small variations of thetemperature from device to device) and also to variations in the final free spectral range of theMZI that have been tested (20nm ± 1nm).

3 dB 3 dB

Fibre 1

Fibre 2

Figure 5. All-fibre Mach-Zehnder interferometer with different fibres.

-20

-10

0

10

20

300 5 10 15 20 25 30 35

Time of exposure to UV (min)

Ther

mal

dep

ende

nce

(pm

/C)

Figure 6. Experimental thermal dependence of all fibre MZI as a function the UV exposure time of oneof its fibre arms.


2.2. Passive Thermal Compensation Using Specific Dopants in Fibres

The cores of silica fibres are usually doped with Germanium to increase the refractive indexwith respect to the undoped cladding. However, many other dopants can be used to controlthe refractive index, such as Fluor or Phosphorus. Concentrations of such dopants in the fibrecore or cladding have direct impact on the effective index of the fibre and on the temperaturedependence of this effective index. The adjustment of the composition with dopants can takeplace in the core of the fibre or in the cladding or both. It has been demonstrated that the typeof dopant used and its concentration can be selected to control the thermal wavelength drift ofa MZI to about 1-2 pm/°C accuracy within a desired temperature range which is generallybetween about -35°C and +85°C [8].

2.3. Flat-top WDM Devices

WDM optical systems allow multi-channels communication in a single optical fibre. Achannel is spectrally characterised by a wavelength and a width. Channel spacing in WDMsystems is constantly decreasing and can be as low as 25 GHz. In many cases, sinusoidalspectral response of multiplexers/demultiplexers is not appropriate, especially in long hauloptical networks where tight specifications on insertion loss, crosstalk and differential groupdelay are required. Moreover, fluctuations of the signal laser wavelength may induce lossfluctuations in Dense Wavelength Division Multiplexing (DWDM) systems when asinusoidal transmission device is used. Flat-top spectral responses are preferred because theyallow minimizing the crosstalk between adjacent channels, the fluctuations of channel loss,and the differential group delay. Thin films DWDM devices can be easily designed to meet

Figure 7. Typical spectral response of flat top interleaver with three cascaded couplers.


flat top channel specifications. However, all-fibre DWDM devices are still attractive becauseof their low polarization mode dispersion. In their usual two couplers Mach-Zehnderinterferometers configuration, only sinusoidal spectrum can be achieved. Flat-top channelspacing multiplexer has been implemented by three cascaded couplers of different couplingratios linked by two differential delays [9,10,11]. A non sinusoidal spectrum is obtained byusing three couplers instead of the usual two, while the second differential delay is exactlytwice the first one. Cascading wavelength insensitive couplers allow constant isolation andinsertion loss over more than 100 nm. As an example, Fig. 7 shows typical optical spectra ofa 100 GHz spacing DWDM interleaver consisting of three cascaded couplers.

2.4. All Fibre Optical Add Drop Module

Optical Add-Drop Modules (OADMs) are key devices for optical networks. OADMs are theaccess points to the optical network and allow adding or droping wavelengths at differentsites along the network. The most usual all-fibre design used a balanced MZI with twoidentical Fibre Bragg Gratings (FBGs) embedded in the two MZI arms [12]. Optical signalsare launched into port 1 (Fig. 8). The 3dB coupler splits the input power evenly into the twoMZ arms. Only those signals carried at the Bragg wavelength get reflected by the FBGs andreturn back into the first 3dB coupler. Whenever the optical paths of both reflected waves arebalanced, all the wavelengths over the bandwidth of interest are phase-matched and all theoptical energy is transferred into port 4 with little energy returning back to the bar path (seeeq. 7 with α=π/4 and φ=0). The port 4 becomes the drop-port, at which signals at the Braggwavelength of the FBGs get filtered out from other channels. Signals carried at wavelengthsother than the Bragg wavelength transmit through the FBGs and merge into the second 3dBcoupler. Similar to the reflected one, all the transmitted waves over the wavelength span ofinterest are phase-matched under a balanced MZ structure and most of the energy is carriedinto port 3. Port 3 then becomes the pass-port, through which signals outside of the FBGreflection band are transmitted. Port 2 can then be used as the add-port, into which othersignals carried at the Bragg wavelength are launched. Those additional signals get reflectedby the FBGs, carried through the cross path arm of the second 3dB coupler, and join port 3without interfering with each other

The most common fabrication method approach is that a MZI is made first and the FBGpair is then written on the established interferometer [13-15]. Another approach for whichavailable FBGs are integrated into a MZ interferometer has also been demonstrated [16].

3 dB 3 dB1

4 λg

λg

λg-dropp

λg-addIN

OUT

2

3

Figure 8. Balanced all-fibre Optical Add Drop Module.


2.5. All Fibre Differential Phase Shift Keying Demodulator

The differential phase shifted keying (DPSK) modulation has been attracted great attentionfor its application for dense wavelength division multiplexing (DWDM) transmission, sinceDPSK with optical Mach-Zehnder interferometer (MZI) demodulation provides severaladvantages over the conventional intensity modulation detection [17]. Optical differential-phase shift keying (DPSK) is a modulation format that offers high receiver sensitivity, hightolerance to major nonlinear effects in high-speed transmissions, and high tolerance tocoherent crosstalk [18,19]. In DPSK, data information is carried by the optical phasedifference between successive symbols. As an example, a Conventional DPSK (CDPSK) usesphase difference in the set (0,π) [20].

Figure 9. DPSK demodulator. (a) successive symbols with π-phase difference. (b) successive symbolsin phase.

For direct detection of DPSK signal (by conventional intensity detectors), a DPSKdemodulator is used to convert the phase-coded signal into an intensity-coded signal. Fig. 9illustrates demodulation of a DPSK optical signal using 1-bit-time-unbalanced Mach-Zehnderinterferometer designed with 3 dB couplers (also called delay line interferometer). Theincoming differential phase-shift keying optical signal is first split into two equal-intensitybeams in two arms of a Mach Zehnder, in which one beam is delayed by an optical pathdifference corresponding to 1-bit time delay. After recombination, the two beams interferewith each other constructively or destructively, depending on the optical phase differencebetween adjacent bits. Using eq. 7 with α=α’=π/4 (3 dB couplers) and ϕ the phase differencecorresponding to 1 bit time, one can easily show that the resultant interference intensity oftwo adjacent bits in phase is directed in port 3 (output port), while the resultant interferenceintensity of two adjacent bits having π-phase difference is directed in port 2. The resultantinterference intensity is the intensity-keyed signal in output port 3. The all-fibre Mach-


Zehnder interferometer design demonstrated low insertion loss, low polarization dependentloss and low polarization dependent isolation over a wide spectral band by using 3dBwavelength insensitive couplers [21]. Tunable all fibre DPSK demodulators are also veryattractive since they allow re-configuration of wavelength channels. In an all-fibre structure,phase tuning is typically achieved by applying an electrical voltage to one arm of the MZ thathas been metallized to this aim. The refractive index of the metallized arm change because ofthe thermo-optic effect, which allows tuning of the phase difference ϕ.

3. Developments on All-Fibre Devices for Erbium Amplifiers

Erbium-based optical amplifiers have been developed during the 1980s to replace theexpensive and complex electronic repeaters. The advantage of Erbium-doped fibre amplifiers(EDFAs) lies in the practical issues related to coupling losses, polarization insensitivity, highgain, low noise, and capability to regenerate several channels simultaneously. However,EDFAs need components for their integration in optical networks. As an example, EDFAsusually incorporate a gain equalizer filter to flatten the gain spectrum. Because of their highperformance Erbium amplifiers are also used in a two-stage configuration. The mid-stageallows incorporating many devices to optimize the network performance, such as a chromaticdispersion compensation fibre, a Polarization Mode Dispersion (PMD) compensation module,a dynamic gain compensator, a gain flattening filter, and add-drop modules…Fig. 10illustrates a basic configuration of a two-stage EDFAs.

Pump/Signal combiner

EDFA

Isolator

EDFA

Mid-stage

GFF DGC

pump pump

Figure 10. Two-stage EDFA basic configuration.

3.1. Fibre Gain flattening Filters

All-fibre amplifiers are commonly used in telecommunication networks to amplify signals ona wide bandwidth. Filters are required to flatten the non-uniform gain of EDFAs or Ramanamplifiers. Fixed gain flattening filters (GFFs) flatten the gain profile of optical amplifiers byselectively removing excess power. These filters are often fabricated using short- or long-period fibre gratings. However, efficient gain flattening filters can also be achieved with acascade of tapered fibres [22]. As discussed in Section 1.3, abruptly tapered fibres allow thecoupling between the fundamental mode and several cladding modes. The controlled taperprofile is used for tailoring the filter loss spectrum. Tapered fibres show a sinusoidal spectral


response. They can thus be combined to create any spectral filter as in a Fourier series(Fig. 11).

GFF are static filters used to flatten the amplifier spectrum gain for a given gain shape.Hence, GFFs are designed for given operating conditions, such as total signal power, pumppower, number of channels, and temperature conditions. When these parameters vary, thegain shape of the amplifier also varies, and the GFF is no longer able to flatten the amplifierspectrum.

Figure 11. Spectral response of a gain flattening filter for erbium-doped amplifiers made byconcatenation of four tapered fibre filters.

3.2. Dynamic Gain Tilt Compensation

In this section, we focus on recent development of the EDFA gain control using all-fibredevices. An optical amplifier may not always operate at the gain value for which the gainflatness is optimized. Many factors contribute to this sub optimal operating condition: span-loss variation, input channel count change, and spectral tilt due to stimulated Ramanscattering. As a result, the amplifier gain is tilted, and such tilt can have significant impact onthe system performance. Generally, spectral gain flatness of an EDFA due to change ofoperating conditions is characterized by the Dynamic Gain Tilt parameter (DGT). DGT(dB/dB) is defined as the gain variation at wavelength λ,when the gain variation at a referencewavelength λ0 is 1 dB.

)()()(

0λλλ

GGDGT

ΔΔ

=(11)

The DGT is a characteristic function of erbium ions and do not depend on fabricationtechniques or opto-geometrical parameters of the fibre. It is then an efficient parameter tocharacterize the variation of the gain in EDFAs. It can be easily shown that the DGT is afunction of the absorption and emission Giles parameters:


)()()()()(00 λαλ

λαλλs

s

ggDGT

++

=(12)

In the case of C-band EDFAs, the dynamic gain-tilt (DGT) is the main factor of the gainflatness deterioration, especially in the case of 980 nm pumping. Another important amplifiercontrol function is to maintain the output signal level per channel. The dynamic control of theper-channel output power in the EDFA is important to avoid SNR degradation. Many designshave been proposed for dynamic gain tilt compensation. Variable optical attenuator (VOA) isthe most common device used for gain tilt compensation [23]. However, the large insertionloss of the VOA deteriorates the signal to noise ratio and/or the power conversion efficiencyof the amplifier. Automatic power control (APC) scheme and a variable attenuation slopecompensator (VASC) demonstrate better performances than the VOA [24]. The APC isemployed in the first EDFA stage and the VASC in the mid-stage does not change itsinsertion loss in spite of the attenuation slope change. In reference [25], an all-fibre Mach-Zehnder interferometer with appropriate couplers is presented. The design allows dynamicgain tilt compensation by only changing the isolation of the interferometer while the centrewavelength remains unchanged. The all-fibre structure allows high optical performanceincluding low insertion loss, low polarization dependent loss and low polarization modedispersion. The gain slope tuning is made using the thermo-optic effect while the device stillpassively insensitive to external temperature variations. For illustration, this all-fibre device ispresented in the following.

Figure 12. Mach-Zehnder interferometer with metallised fibre in one arm.

A MZ can be used in a linear spectral region to compensate the gain tilt of an EDFA. TheMZ couplers are identical and wavelength dependent such that 0 dB insertion loss is realizedat λ0 (1520 nm) and 3dB is realized at λ1(1580 nm). The MZ is then characterized by aminimum insertion loss at λ0 and a maximum insertion loss at λ1. One of the two branches ofthe MZ is metallised to allow phase tuning between the two arms of the interferometer(Fig. 12). Applying an electric voltage allows to increase the temperature, and thus changethe refractive index of the fibre. As a result, phase changes occur between the two arms viathermo-optic effect, allowing the change of the slope in the 1530-1570 nm spectral band. Thephase change has only an impact on the isolation at λ1, while central wavelengths stayunchanged. Figure 13 shows the spectral response of the MZs for different phase differencesinduced between the two branches. Initial configuration is such that the isolation of thecomponents is 0 dB. A flat transmission near 0 dB over the C-band is realised for no appliedvoltage. We focus on the 1540-1565nm spectral band where the EDFA has a linear DGT. The


maximum insertion loss at 1540nm is 1 dB for –5 dB gain tilt. The deviation from linearity isless than ±0.25 dB. The polarization dependent loss (PDL) is less than 0.3dB. A maximum of3 Volts allow –5 dB tilt between 1540 nm and 1565 nm. The response time, defined by thecharacteristic time allowing a change of the attenuation slope from 5 dB to 5/e dB was 210ms.

-25

-20

-15

-10

-5

01500 1550 1600

Wavelength (nm)

dB

-6

-4

-2


dB

-25

-20

-15

-10

-5

01500 1550 1600

Wavelength (nm)

dB

-6

-4

-2


dBFigure 13. (a) Transmission of MZI for different applied voltage. V=0, 1, 2.25, 3 and 3.6 Volts. (b)Zoom of the 1540-1565 nm range.

Two concatenated MZIs are required to allow positive and negative slope compensationas well as a constant average insertion loss for any slope. The first MZI has opticalcharacteristics presented in the previous section. The second MZI has a complementarytransmission (couplers with 0 dB at λ1 and 3 dB at λ0). ). A great advantage of this all-fibreDSC is that a very low electrical power is needed to compensate the gain tilt. A maximumtotal electrical power of 250 mW is needed due to the low thermal conductivity of silica.

4. Developments on All-Fibre Devices for Raman Amplifiers

Raman Fibre Amplifiers (RFAs) are of great interest for the development of long distance,high capacity WDM systems. Their main advantages are their low noise, wide amplificationbandwidth and saturation characteristics. RFAs have also the advantage that the opticalamplification occurs in the transmission fibre transmission itself. RFAs differ in principlefrom EDFAs as they utilize the stimulated Raman scattering effect to create optical gain.However, RFA suffer from polarization dependent gain (PDG). A solution to reduce PDG isthe use of pump laser with low degree of polarization (DOP). One can scramble the state ofpolarization of the pump with the aid of a depolarizer. Experimental and theoreticalinvestigations have been reported on the statistical properties of PDG [26-30]. These reportsshow that the PDG is linked to the PMD of the fibre. Fig. 14 illustrates the basic design of anoptical fibre system using Raman amplification. In a RFA, a strong pump laser provides gainto signals at longer wavelengths through stimulated Raman scattering. One of the majorattractions of Raman amplification is that it can be used over a very wide wavelength rangeby multiplexing together different pumps wavelengths.


Figure 14. Raman amplification configuration.

Polarization beam combiners (PBC) are key component of diode-pumped Ramanamplifiers. They allow combining the output of two pumps operating at the same wavelengthbut in different polarization modes into a single fibre, thereby doubling the effective poweroutput. PBCs can also be used in EDFAs, typically to combine 1,480 nm pumps in the secondstage. The combination of the two linear-orthogonal polarizations having the same power,allows in addition a complete depolarization of the output pump. Moreover, polarizationcombiners allow protecting the system from the failure of any single laser. RFAs may also besubject to dynamic reconfiguration of the pumps lasers wavelengths. Broadband polarizationcombiners can be necessary to ensure system reconfiguration. Optical depolarizers play also akey role by scrambling the state of polarization of wave pumps that are not doubled.

4.1. Broadband All-Fibre Polarization Combiners

The fibre optic coupler made by the FBT technology has been one of the most widely useddevices in optical fibre systems. Other than the most common function of optical powersplitting, such couplers may have other applications. In particular, they can be designed tooperate as optical polarization beam splitter/combiners [31,32]. For example, opticalnetworks use optical polarization beam splitters in their PMD compensator modules, andpump depolarizers in Raman amplifiers. A large bandwidth is usually specified in case ofmulti-channel lightwave systems. Ideally, a bandwidth as large as the complete optical band(S+C+L) is suitable. However, all-fibre polarization beam splitters suffer from a narrowbandwidth, which limits the spectral operating wavelength range to a few nanometers. Aspectral width of 17 nm for –15 dB extinction ratio has been reported [33]. A wider spectralwidth of 38 nm for –15 dB extinction ratio has been demonstrated using a weakly fusedcoupler design [34]. Recently, an all-fibre polarization beam splitter/combiner has beenreported on a very wide band of more than 200 nm for –15 dB extinction ratio [35]. In thefollowing section a brief description of this device is presented.

An all-fibre MZI design is used for which specific couplers are designed. The firstcoupler is an all-fibre polarization beam combiner (PBC) and the second coupler is an all-


fibre WDM coupler. The structure is schematized on Fig. 15. Input fibres (fibres 1 and 4) arepolarization-maintaining fibres (PMF). The central zone of the MZI (fibres A and B) andoutputs (fibres 2 and 3) are standard single mode fibres (SMF). In regard to Fig. 15, the x-polarization is coupled into port 4 and the y-polarization is coupled into port 1. Experimentaltransmissions from input ports 1 and 4 to output ports A and B are illustrated in Fig. 16 forboth polarizations. By using transfer matrix formalism, the transmitivity of the MZI is givenby

)().().( ,, λφλ YX

PBCMZWDMYX MMMM = (13)

where the transfer matrix of the WDM and the PBC are given in section 1.4.Let us note αx,y(λ) and αw(λ) the phase difference between the symmetric and anti-

symmetric super-modes of the fused fibre PBC for x- and y-polarizations, and the WDMcoupler respectively. These parameters are defined in section 1.4 (eq. 2). The central zone ofthe interferometer structure is characterized by a φ-phase shift between the two branches ofthe MZI (fibres A and B). It has been shown that, If αw(λ)=αx(λ) and φ=π then the devicetransfer matrix for x- and y-polarizations becomes wavelength independent. The spectraldependence of the PBC can be counterbalanced by a π-phase between the two MZI armswhile using a WDM coupler having the same spectral dependence transmitivity as the PBC.

Figure 15. Wideband polarization combiner design.

Fig. 17 shows typical experimental spectral transmissions of the interferometer structure.The non-perfect extinction ratio at the input ports (PM fibres 1 and 4) induces polarizationbeating and spectral ripples on the device transmission. The extinction ratio at the PM fibresinputs is estimated to be –30 dB. For such an extinction ratio, the ripples are only observed onthe isolation ports. The insertion loss is lower than 0.2 dB in the 1440 nm-1550 nm-


wavelength range. Insertion loss increases near 1440 nm because of the water absorptionpeak, which induces excess loss, and increases above 1550 nm because the spectraltransmissions shift between the PBC and the WDM coupler degrades the isolation. More than200 nm spectral width for –15 dB extinction ratio is achieved.

Figure 16. Experimental transmissions of PBC. 1 and 4: input ports; A and B: output ports.

-30

-25

-20

-15

-10

-5

01400 1450 1500 1550 1600

Wavelength (nm)

dB X(4-3)

Y(1-3)

X(4-2), Y(1-2)

-30

-25

-20

-15

-10

-5

01400 1450 1500 1550 1600

Wavelength (nm)

dB X(4-3)

Y(1-3)

X(4-2), Y(1-2)

1 and 4: input ports; 2 and 3: output ports.

Figure 17. Experimental spectral transmissions of the interferometer structure.

4.2. Stable All-Fibre Depolarizers

Passive depolarizers are used to scramble the State Of Polarization (SOP) of an incominglight source, reducing the mutual coherence between the orthogonal polarization componentsof the light source. Highly birefringent (Hi-Bi) fibres are usually used to depolarize wide-band sources but are not suitable for narrow-band sources because of the long length required.Recently described passive devices are based on incoherent fibre ring structures, using acascaded fibre ring design [36] or a dual fibre ring design [37]. These designs allow


depolarizing any SOP by using directional couplers in a fibre ring scheme and by adjustingthe ring birefringence. Recently a new technique for the single-mode fibre depolarizer basedon polarization combiners using a linear design (as opposed to ring design) has beenproposed. The use of a linear design allows a one-way propagation of the two orthogonalpolarization components, which make the device very stable. This new design provides lowloss, low polarization dependent loss and high depolarization of light for any input SOP. Theassembly of such a device is made using power light detection that makes the integrationdevice easier than the degree of polarization optimization technique.

Figure 18. All-State of Polarization all-fibre depolarizer design.

The all-SOP all-fibre depolariser linear design presented in ref [38] is a combination oftwo polarisation combiners (PC) and a 2x2 directional coupler (Fig. 18). An optical phasedelay (delay1) is induced between the waves propagating in the two branches A and B. Apolarisation rotator device is used to realise half-π rotation of the light wave SOP propagatingin fibre B. Interference occurs at the coupler since the SOPs at the inputs of the coupler areparallel. The average intensities at the outputs of the 2x2 coupler are equal if the delayinduced is much greater than the coherence length of the light source. A second phase delay(delay2) is induced between the waves propagating in A and B fibres. A half-π rotator deviceis used such that the SOP of the wavelength propagating in A and B fibres are orthogonal andaligned with the eigen axis of PC2. To increase the polarisation scrambling at the output of thedepolariser, the condition on equal average intensities IX and IY has to be satisfied. If theoptical delay induced by delay1 is much greater than the coherence length of the source and if


a 3 dB transmission coupler is used then the average intensities for x and y-polarisations at theinput of PC2 are equals for any SOP at the input. Also, the two orthogonal components x andy are completely uncorrelated when the delay lengths (ΔL1 and ΔL2) as well as the differencelength (ΔL1-ΔL2) are greater than the coherence length of the laser

DOP less than 4.5% is obtained for coherence length less than 1 m. The linear designeliminates the re-circulations in PC or coupler fibre occurring in a fibre ring configuration.The loss is then given by the sum of the losses of each subcomponents. Low loss (1.5 dB),and low polarization dependent loss (0.5 dB) over 1400–1500 nm spectral range, for allSOPs, and for a 0–70°C temperature range, has been demonstrated with this type of design.By using a wide band PC, the depolariser presented is made achromatic over a 100 nmspectral band (Fig. 19). The one passage light propagation in symmetrical branches (identicalfibres) makes it very stable. Although the PM fibre is often temperature sensitive the smalllength of half wave PM fibre used as a rotator device (1.8 mm) keeps the SOP thermallystable. The DOP variation obtained is +/-2% and loss variation is ±0.05 dB for a 0°C to 70°Ctemperature range. In addition, the all silica-fibre structure allows the depolarisation of anylaser with a coherence length lower than the loop length and permits high power handling. Bydesign, this depolariser allows 2 inputs and 2 outputs, for each input corresponds an output.

Figure 19. Maximum degree of polarization (for any input SOP) versus wavelength.

Conclusion

Fused Biconical Taper fibre devices have shown great integration in today optical networks.From basic directional coupler to cascaded Mach-Zehnder interferometers designs, all fibrecomponents are used to perform many functionalities in all parts of the optical network. In thetransmission part, temperature-independent WDM and DWDM interleavers with flat-topspectrum are attractive because of their very low chromatic dispersion, differential groupdelay, and polarization dependent loss. In the amplification part, all-fibre devices based onfused biconical taper fabrication technique demonstrated high potential to multiplex pumpsand signals. Cascaded tapered fibres and cascaded couplers demonstrated their capability tocorrect the amplifier gain non-uniformity being respectively used as Gain Flattening Filters


and dynamic gain compensators. For Raman amplification, broadband polarization combinersand depolarizers have been developed to limit gain and channels power fluctuations whilehigh power handling and wide spectral band are ensured, which allow broadbandmultichannel amplification and amplifiers re-configuration. Access to the network can beperformed anywhere along the fibre by using all-fibre optical add-drop modules whilepolarization mode dispersion is minimized compared to circulators based design. Many otherfunctionalities can be performed by all-fibre devices.

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[24] H. Nakaji, M. Kakui, H. Hatayama, C. Hirose, H. Kurata, and M. Nishimura, “Superiornoise performance and wide dynamic range Erbium-doped fiber amplifiers employingVariable Attenuation Slope Compensator”, IEICE Trans. Electron, 2001, Vol. E84-C,pp 598-604.

[25] N. Azami, “All-Fiber Dynamic Gain Slope Compensator”, Optics Communications,2003, Vol. 230/4-6 pp 325-329.

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[28] Q. Lin and G. P. Agrawal, “Statistics of Polarization-dependant gain in fiber-basedRaman amplifiers”, Optics Letters, 2003, Vol. 28, No 4, pp 227-229.

[29] E. Son, J. Lee, and Y. Chung, “Gain variation of Raman amplifier in birefringent fiber”,Optical Fiber Conference (OFC), 2003, Paper TuC5.

[30] N. Azami, “Characterization of Polarization Dependent Gain in Raman fiberamplifiers”, Optics Communications, 2003, Vol. 230/1-3, pp 181-184.

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[32] T. Bricheno and V. Baker, “All-fiber polarization splitter/combiner”, IEE Electron. Lett,1985, Vol. 21, pp 251-252.

[33] M. Eisenhamm, E. Weidel, “Single mode fused biconical coupler optimized forpolarization beam splitting”, J. of Lightwave Technol., 1991, Vol. 9, No 7, pp 853-858.

[34] C.W. Wu, T.L. Wu, H.C. Chang, “A novel fabrication method for all-fiber, weaklyfused, polarization beamsplitter”, IEEE Photonics Technol. Lett., 1995, Vol. 7, No. 7, pp786-788.

[35] N. Azami, A. Villeneuve, F. Gonthier, “All-Fiber Wide-Band Polarization BeamCombiner”, IEE Electron. Letters, 2003, Vol. 40, No 17, pp 1043-1044.

[36] P. Shen, and J.C. Palais, ‘Passive single-mode fiber depolarizer’, Appl. Opt., 1999, Vol.38, pp 1686–1691.

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Chapter 8

ADVANCES IN OPTICAL DIFFERENTIAL PHASE SHIFTKEYING AND PROPOSAL FOR AN ALTERNATIVE

RECEIVING SCHEME FOR OPTICAL DIFFERENTIALOCTAL PHASE SHIFT KEYING

M. Sathish Kumar1a, Hosung Yoon2b and Namkyoo Park1c,1Optical Communication Systems Lab, School of EECS, Seoul National University,

Seoul, Korea, 151-742,2Network Infra Laboratory, Korea Telecom, Daejeon, Korea, 305-811,

Abstract

Optical Differential Phase Shift Keying (oDPSK) with delay interferometer based directdetection receiver was proposed as an alternative for the conventional On-Off Keying (OOK)modulation schemes. Compared to OOK, oDPSK was predicted to have a 3dB improvementin performance due to its balanced detection receiver structure. It was also predicted that dueto the optical signal occupying all the symbol slots, unlike in OOK, symbol pattern dependentfiber nonlinear effects will make less of an impact on long haul optical transmission schemesbased on oDPSK. Subsequent successful demonstrations of these positive attributes ofoDPSK resulted in active investigations into multilevel formats of oDPSK namely, opticalDifferential Quadrature Phase Shift Keying (oDQPSK) and optical Differential Octal PhaseShift Keying (oDOPSK). Significant developments in theoretical models of opticallyamplified lightwave communication systems based on the Karhunen-Loeve Series Expansion(KLSE) method assisted such investigations. In this chapter, we discuss some of the recentadvances in oDPSK and its multilevel formats that have been achieved such as proposals forreceiver schematics, theoretical analysis of receiver schematics, electronic techniques tocounter polarization mode dispersion induced penalties, and application of coded modulationtechniques. The chapter also proposes an alternative receiver schematic for oDOPSK whichcan separately detect the three constituent bits from an oDOPSK symbol.

a E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]

M. Sathish Kumar, Hosung Yoon and Namkyoo Park232

1. Introduction

Single mode optical fibers with their enormous bandwidth of around 15THz and extremelylow attenuation of 0.2dB/Km in the 1550nm window offer immense promise as a viablemedium to realize high bit rate long distance data transmission systems. Over the years, therehas been a remarkable growth in the data transmission rates achieved with one of the mostrecent experiments claiming 14Tbps over a distance of 160Km [1]. As the transmissiondistance and signaling rates increase, certain problems inherent to the optical fiber mediumsuch as attenuation, chromatic dispersion, nonlinear effects, and Polarization ModeDispersion (PMD) start to crop up which inhibits increments in the link distance and datatransmission rates.

Developments in the area of optical fiber amplifiers have resulted in mature technologiessuch as doped fiber amplifiers and fiber Raman amplifiers to overcome the attenuation limits.Also, developments in optical fiber device technology such as fiber gratings, both long periodand Bragg, and high degree of control over refractive index profiles of core and cladding havehelped in identifying feasible and effective methodologies to counter chromatic dispersion.Combating the ill effects of fiber nonlinearities and PMD still continue to be challengingproblems primarily due to their statistical nature.

More recently, alternate modulation techniques such as optical Differential Phase ShiftKeying (oDPSK) [2] a bi-level version of optical differential phase modulation and opticalduobinary signaling have been proposed and actively investigated upon. Optical duobinaryschemes are based on the principle of introducing controlled inter symbol interference so thatcompared to On-Off Keying (OOK), for a given data transmission rate, the bandwidth of theoptical signal propagating through the fiber is reduced. This obviously has an advantage overOOK in that spectral width dependent signal distorting mechanisms such as chromaticdispersion and PMD will have less of an impact. A tutorial discussion on duobinary signalingschemes could be found in [3]. In optical differential phase modulation, irrespective ofwhether it is bi-level or multilevel, the phase of the optical field during the current signalinginterval is modulated relative to its phase in the previous signaling interval. The detection ofthe data at the receiver side at any particular signaling interval is hence dependent on thephase of the received optical signal in the previous interval. It is worth to note that opticaldifferential phase modulation was under investigation during the late eighties and earlynineties while coherent optical communication systems were aggressively explored [4].However, the idea of optical differential phase modulation as most of the recent publicationsconcentrate on is based mainly on an interferometric delay line based direct detectiontechnique and will be the one discussed in this chapter.

In comparison to OOK, oDPSK provides a 3dB performance improvement [2]. The 3dBimprovement offered by oDPSK can easily be translated to an increase of approximately15Km in the transmission length or a reduction in signal intensity dependent nonlinear effectssuch as stimulated Raman scattering, four wave mixing, cross phase modulation, etc.Moreover, since oDPSK has all the bit slots occupied by optical intensity, unlike in OOK, bitpattern dependent undesirable impacts of fiber nonlinear effects also get alleviated.

While oDPSK transmits one bit per signaling interval, its multilevel versions, namelyoDQPSK and oDOPSK, transmit two and three bits respectively per signaling interval.Obviously, for a given bit rate and pulse format (Return to Zero (RZ) or Non Return to Zero

Advances in Optical Differential Phase Shift Keying… 233

(NRZ)), the oDQPSK and oDOPSK schemes can provide an increment in the spectralefficiency by factors of two and three respectively over OOK and oDPSK. This is significantin that as with optical duobinary transmissions, oDQPSK and oDOPSK due to their reducedbandwidth requirements will be more immune to spectral width dependent signal distortingmechanisms. Over and above these advantages, oDQPSK and oDOPSK carry with them moreor less the same other advantages that oDPSK has over OOK. However, the price that needsto be paid for this improved spectral efficiency of the multilevel versions of oDPSK is aninferior error rate performance.

This chapter aims at providing a review of some of the advances that have been achievedin the domain of optical differential phase modulation schemes such as proposals for receiverschematics, theoretical analysis of receiver schematics, electronic techniques such asequalizers to counter PMD induced penalties, and application of coded modulation techniquessuch as Trellis Coded Modulation (TCM) [5]. The chapter also proposes an alternativereceiver schematic for oDOPSK which can separately detect the three constituent bits from anoDOPSK symbol.

2. Transmitter and Receiver Schematics for oDPSK and oDQPSK

Figure (1) shows the transmitter and receiver schematics for oDPSK along with the resultantone dimensional signal space diagram and constellation in the inset. The coherent optical fieldemitted by the laser is phase modulated by a suitable optical phase modulator which is drivenby differentially encoded NRZ data. The phase modulated output is passed through a pulsecarver to obtain RZ optical pulses which are subsequently transmitted through the fiber. Apulse carver is essentially a Mach Zehnder Modulator (MZM) complimentarily driven by asinusoidal clock [2] [6]. The phase modulator can be a simple optical phase modulator or aMZM [2].

At the receiver side, the optically amplified signal is first passed through an optical bandpass filter and then through a delay line interferometer with one arm of interferometerintroducing a time delay of T where T is the signaling interval. The constructive anddestructive port outputs of the delay line interferometer are used to illuminate a pair ofidentical photo detectors to facilitate optoelectronic conversion. The difference of the twophoto detector outputs is then passed through an electrical post detection filter whose outputis sampled at appropriate time instants once every signaling interval to obtain y as shown infigure (1). The obtained sample y is compared with a threshold of zero to obtain the estimatesof the transmitted binary data. It may be noted that the constructive port output (the top outputin figure (1)) effectively feeds in duobinary modulated optical signal to the photo detectorwhile the destructive port feeds in alternate mark inversion modulated optical signal [2][7][8]. As mentioned in [2], it is this balanced detection using a pair of photo detectors thateffectively gives a 3dB advantage for oDPSK over OOK.

The concepts of multilevel optical differential phase modulation schemes such asoDQPSK and oDOPSK are in effect a two dimensional extension of oDPSK. In thesemultilevel versions of oDPSK, inphase and quadrature components of the optical carrier arephase modulated independently, combined and then transmitted. The inherent orthogonalitybetween the inphase and quadrature components of the optical carrier enables unambiguousidentification of the modulating data at the receiver side.


Figure 1. Transmitter and receiver schematics for oDPSK. The inset shows one dimensional signalspace diagram.

When it comes to receiver schematic for oDQPSK, there is no much of an option andmore or less the same technique as the one used in electrical communication systems [3] isemployed. However, oDOPSK opens up a range of options for detection. As such, wededicate two separate sections later on for discussing the receiver schematics for oDOPSK.The rest of this section will discuss the transmitter and receiver schematic for oDQPSK.

Figure (2) shows the transmitter and receiver schematic for oDQPSK. Comparing thisschematic with that of oDPSK as given in figure (1), it can be readily observed that thetransmitter schematic is in effect a parallel concatenation of two oDPSK transmitters. Theincoming laser field is split into two equal parts in terms of power and passed through parallelphase modulators ensuring that the split optical fields have a relative phase shift of π/2between them. This is to separate out the inphase and quadrature components of opticalcarrier. The inphase and quadrature phase modulated optical fields are then combined andguided through an optical fiber towards the oDQPSK receiver. At the receiver side, thereceived signal is split into two equal parts and passed through parallel concatenated delayinterferometers. Compared to the delay interferometer setup depicted in figure (1) withregards to oDPSK, the difference here is that the arms of the delay interferometers introducephase shifts that have to be such that the absolute value of the phase difference is π/2. Morespecifically

2/21 πθθ =− (1)


Figure 2. Transmitter and receiver schematics for oDQPSK. The inset shows the signal space diagramwith θ1 and θ2 as pπ/4 and qπ/4, such that p and q are odd integers satisfying the condition given inequation (1).

The post detection electronic processing to extract the data bits can be simplifiedconsiderably by selecting θ1 and θ2 as pπ/4 and qπ/4, such that p and q are odd integerssatisfying the condition given in equation (1), and using an electronic precoding circuit at thetransmitter side as reported in [9]. The precoder is designed to satisfy the following Booleanexpressions (for p=1, q=-1).

)()()(.)( 11111211 −−−−−− ⊕⊕+⊕⊕= kkkkkkkkk IbIQIbIQI (2.a)


))(()(.)( 12111111 −−−−−− ⊕⊕+⊕⊕= kkkkkkkkk IbIQIbIQQ (2.b)

Here Ik and Qk are the NRZ data driving the top and bottom phase modulators respectivelyin figure (2) during the kth signaling interval and b1k and b2k are the two binary data bitswhich constitute the kth transmitted oDQPSK symbol and which are subsequently detecteddirectly from the outputs y1 and y2 respectively.

With the parameters selected as given in the above paragraph and with the appropriateprecoder identified through equations (2.a) and (2.b), it is possible to obtain the informationbit sequence carried by the inphase and quadrature components of the optical carrier from abi-level detection of the outputs y1 and y2 respectively [9]. The resultant signal space diagramand constellation when θ1 and θ2 is pπ/4 and qπ/4 with p and q as discussed above is alsoshown in figure (2) in the inset. It may be noted that when one of the phase shifts of the delayinterferometers of the receiver is set as |π/2| and the other as 0, the orientation of the signalvectors get rotated by π/4 from what it was when θ1 and θ2 were pπ/4 and qπ/4 with p and qas discussed above. This has an obvious disadvantage in that the outputs y1 and y2 will nowbe three valued as opposed to binary valued when θ1 and θ2 were pπ/4 and qπ/4 such that pand q are odd integers satisfying the condition given in equation (1). It can be inferred fromthe signal space diagram and the discussion above that the receiver schematic for oDQPSKwith θ1 and θ2 as pπ/4 and qπ/4 in effect treats the inphase and quadrature components of theoptical carrier as two separate independent oDPSK channels.

3. Transmitter and Receiver Schematics for oDOPSK

Figure (3) depicts a possible transmitter schematic for oDOPSK [10]. The idea is based onthe fact that an oDOPSK signal comprises essentially of two oDQPSK signals having arelative phase offset of π/4 between them [3]. The differentially encoded data b1 and b2 drivetwo parallel phase modulators as it was in the case of oDQPSK transmitters. However, thedifferentially encoded data b3 brings about a phase shift of 0 or π/4 in the signal propagatingthrough the last phase modulator. This effectively rotates the signal constellation by 0 or π/4radians.

Unlike in oDQPSK, wherein a direct mapping of the ideas from electrical communicationsystems was followed to arrive at possible receiver schematics, for oDOPSK, the majordriving factors in identifying receivers have been optimized performance as well as ability toextract the three constituent bits directly through a bi-level detection of samples. This has ledto suggestions of receiver schematics for oDOPSK which employ more than two delayinterferometers. Before venturing into a review of such receiver schematics, it is worth to notethat in principle it is possible to extract the eight distinct symbols that comprise the oDOPSKsymbol set using a receiver schematic like the one used for oDQPSK. This should be quiteelementary to understand since the signal space diagram of oDOPSK is two dimensional andto uniquely represent a point in a two dimensional space, only two coordinates are required.Those two coordinates could be obtained readily from a schematic exactly like the oDQPSKreceiver by treating the outputs y1 and y2 as multilevel [11].


Figure 3. Transmitter schematic for oDOPSK.

Figure (4) depicts two receiver schematics proposed in [10] and [11] for oDOPSK. Incase of figure (4.a), each interferometer output, after passing through the electrical postdetection filter, is sampled once every signaling interval and treated as a bi-level sample.

(a) (b)

Figure 4. Receiver schematics for oDOPSK; (a) is after [11] and (b) after [10].

To be more specific, the absolute value of the samples obtained (y1 to y4) are disregardedand only their numerical signs are taken into consideration. From the fact that in oDOPSK thephase difference between successive symbols can take on only values which are integermultiples of π/4 mod 2π, table (1) can readily be formulated from the receiver schematic


shown in figure (4.a). Table (1) clearly shows that for each value of phase difference there isa unique combination of the numerical signs of y1 to y4 which easily enables the identificationof the transmitted symbol and subsequent decoding of the symbol to its respective binarytriplet. However, it should also be noted that out of 24 possible combinations, only 23 aremade use of and the inevitable redundancy that exists throws up the option of error correctionusing maximum likelihood estimation techniques [11] [12].

Table 1. Relation between polarity of detected samples and phase of oDOPSK symbolsfor receiver shown in figure (4.a)

Phase difference y1 y2 y3 y4

0 + + - -π/4 + - - -

π/2 - - - -

3π/4 - - - +

π - - + +

5π/4 - + + +

3π/2 + + + +

7π/4 + + + -

Coming to the receiver schematic given in figure (4.b), the samples obtained from thedelay interferometers outputs (y1, y2) and the XOR logic block output (y3) are treated as bi-level as was in the case of figure (4.a). We defer discussions on this receiver schematic for alater section wherein we discuss an alternative receiver schematic for oDOPSK.

4. Error Rate Performance Evaluation

As is well known, optical amplifiers have become essential components of current state-of-the-art long haul fiber optic communication systems. Consequently, current fiber opticcommunication systems are essentially Amplified Spontaneous Emission (ASE) noise limited.As such, in performance evaluations, usually the sole noise source taken into account is theASE. An accurate method to arrive at the Characteristic Function (CF) of the detectedoptically preamplified signal using the Karhunen Loeve Series Expansion (KLSE) method andsubsequent evaluation of probability of error through saddle point integration was reported in[13] and modified later in [14] for computational efficiency. Though the KLSE methodreported in [13] and [14] was implied for OOK systems, the method could easily be modifiedto use it for oDPSK as well as its multilevel versions as suggested in [2][15][16]. The majoradvantage of the KLSE method compared to others such as those developed in [17] is that theKLSE method could be used to evaluate the error rate performance for arbitrary pulse shapesand can account for the pre and post detection filter transfer functions along with other linearimpairments of the fiber medium such as chromatic dispersion and PMD [14].


In the following subsections we first discuss the essence of KLSE method for error rateevaluation of optically amplified lightwave communication systems and then the error rateperformance.

4.1. Essence of the KLSE Method for Error Rate Evaluation

Consider the general setup of a direct detection fiber optic communication receiver as shownin figure (5). Let the received optically amplified signal be

)()()( twtste += (3)

where s(t) and w(t) stand respectively for the desired signal and ASE noise sample function.The ASE noise is a complex Additive White Gaussian Noise (AWGN) process with a twosided Power Spectral Density of NO W/Hz. As the first step in deriving the KLSE method, itcan be shown that [13]

∫ ∫∞

∞−

∞

∞−

−= '))'(2exp()(),'()'(21 * dfdftffjfEffKfEy kk π (4)

where yk is the detected sample as shown in figure(5), E(f) is the Fourier transform of thereceived signal e(t), tk is the time instant at which the post detection filter output is sampledand K(f’,f) is as given below

Figure 5. General setup of a direct detection fiber optic communication receiver.

)'()'()(),'( * fHffHfHffK oeo −= (5)

with Ho(f) and He(f) standing respectively for the transfer function of the optical and electricalfilters.

Rewriting equation (4) above as

∫ ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∞→−∞→

b

a

b

akk

bak dfdfftjfEffKtfjfELty ')2exp()(),'()'2exp()'(

21 * ππ (6)


and by treating a and b as finite but sufficiently large in magnitude so as to cover severaltimes the filter bandwidths [13], the inner integral in equation (6) becomes effectively theright hand side of a Fredholm integral equation of second kind which can be representedformally as [18]

∫=b

ammm dffffKf )(),'()'( φλφ (7)

with φm(f) and λm being its eigenfunction and eigenvalue respectively and K(f’,f) acting as theHermitian kernel.

Based on the fact that the eigenfunctions form a complete set of orthonormal basisfunctions in the interval a<f<b, )2exp()( kftjfE π can be expressed as a series expansion ofthe basis functions as

∑ +=m

mmmk fNSftjfE )()()2exp()( φπ (8)

In equation (8) above the coefficients Sm correspond to the desired signal and Nm to theASE noise. It can be shown that Nm will have both its real and imaginary parts as Gaussianrandom variables with zero mean and variance No/2 W/Hz each [13][14]. Substitutingequation (8) into (6) and making use of the definition of the Fredholm integral equation asgiven in equation (7), it can readily be shown that

∑+

=m m

mmk

NSy

λ

2

21

(9)

Since the real and imaginary parts of Nm are mutually uncorrelated and Gaussiandistributed [13] [14], the CF of yk will be products of CF of non central chi squaredistributions each with single degree of freedom. This CF of yk will be as given below

∏⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛ −⎟⎟

⎠

⎞⎜⎜⎝

⎛−

=Φm o

m

m

m

o Nj

SjN

j2

2exp

21

1)(2

ξλ

ξ

λξ

ξ (10)

The probability density function (pdf) of yk can be obtained by inverting the CF. From thepdf so obtained, the probability of error can be evaluated. Since the CF given above cannot beinverted in a straightforward manner, saddle point integration method is used [14].

The alteration of the above discussed KLSE method so as to make it suitable forevaluating the error rate performance of oDPSK and its multilevel versions is ratherstraightforward in that all it needs to be done is to incorporate appropriately the transfer


functions of the delay line interferometers into the Hermitian kernel. Thus for oDPSK and itsmultilevel versions, each of the detected outputs (yk) when expressed in the form of equation(6) will have its Hermitian kernel as

)()'(2

))2(exp())'2(exp()'(),'( * fHfHfTjTfjffHffK ooe ⎟⎠⎞

⎜⎝⎛ +−++

−=θπθπ (11)

where θ is the phase shift introduced by the corresponding delay interferometer and T is asgiven earlier.

Treating the signal s(t), i.e. the data sequence transmitted by the transmitter, as a periodicsignal of sufficiently large period facilitates the usage of the FFT/IFFT algorithms whichmakes the computational process more efficient [14]. Also, by treating the data sequence asperiodic and by representing it using appropriate pseudorandom sequence such as the de-Bruijn sequence [16], it is possible to estimate the impacts of fiber induced inter symbolinterference caused due to chromatic dispersion and PMD. To incorporate these into theKLSE model, the fiber transfer function has to be included in the Hermitian kernel. Thetransfer function of the fiber which incorporates the chromatic dispersion is well known [14]and the principal states model of Poole [19] facilitates rather easy incorporation of the PMDeffects into the fiber transfer function [16].

4.2. Error Rate Performance

Figure (6) shows the Bit Error Rate (BER) performance for oDPSK, oDQPSK and oDOPSKas a function of the Optical Signal to Noise Ratio (OSNR) in a back-to-back condition. Thereceiver schematics assumed in these evaluations are the ones discussed through figures (2)and (3) for oDPSK and oDQPSK respectively and the one discussed through figure (4.a) foroDOPSK. The optical filter was modeled as a first order Gaussian filter with its 3dBbandwidth as three times the baud rate and the electrical post detection filter was modeled asa fifth order Bessel filter with 3dB bandwidth as equal to the symbol rate. From here on, werefer to the 3dB bandwidth as just the bandwidth. The OSNR was evaluated with unpolarizedASE noise within a reference spectral width of 0.1nm. The bit rate was taken as 40Gbps forall three cases. The binary data sequence was modeled as pseudo random sequences of length31, 63 and 511 respectively for oDPSK, oDQPSK and oDOPSK. These results as well as theothers to be presented later in this chapter are for RZ pulse formats. We restrict ourselves tothe RZ formats because in general as per the results in [11], [15] and [16], the RZ format givesa better error rate performance compared to the NRZ formats.

Though the above results in back-to-back condition give a very good picture of therelative error rate performance of oDPSK, oDQPSK and oDOPSK schemes, it has to be notedthat in actual long haul systems, signal distorting mechanisms like PMD will have to be dealtwith. Electronic post detection equalization techniques such as Linear Equalizers (LE) [3]have been proposed as viable technologies to deal with PMD induced pulse broadening [20].

In the following we discuss the performance of the above discussed schemes in thepresence of first order PMD with and without the use of LE. We begin with a very brief


review of LE schemes. More details could be found in standard digital communication textbooks such as [3].

Figure 6. Bit Error Rate performance for oDPSK, oDQPSK and oDOPSK as a function of the OSNR inback-to-back condition.

Figure 7. A five tap Linear Equalizer.


The LE is essentially a transversal filter as shown in figure (7). The tap delay elementsshown as square boxes in figure (7) introduce a delay of Ts which can be equal to thesignaling interval or a fraction of that. The tap weights ck with which each delay elementoutput is multiplied is constantly updated using a tap updater algorithm called as the LMSalgorithm. Details of the LMS algorithm are beyond the scope of this chapter and could befound in [3]. The sequence vk stands for the received distorted symbol sequence. The tapupdater (the LMS algorithm) is in turn fed by an error signal ek as shown in figure (7). This

error signal is derived from the difference between the equalized output kI and its nearest

information symbol kI as estimated by the decision circuit. It should be easilyunderstandable from this figure that the tap updater algorithm works towards minimizing themagnitude of this error signal which drives it.

It may be noted that to estimate the kth information symbol, the LE discussed above hasto depend not only on the kth received symbol vk but also on two before that and two afterthat.

To incorporate the LE into the KLSE method of error rate evaluation so as to theoreticallyevaluate the performance improvement achievable with the use of LE in the presence of PMDinduced pulse broadening, the LE is modeled as an electrical filter with transfer function [15]

∑−=

−=2

2)2exp()(

kskLE fkTjcfH π (12)

where, it is assumed that the LE has five taps as in figure (7).

Figure 8. OSNR penalty of equalized and unequalized oDQPSK and oDOPSK systems.


The improvement in performance in oDQPSK and oDOPSK systems by the use of a fivetap LE in the presence of first order PMD effects is depicted in figure (8). The OSNR penaltyto attain a reference BER of 10-12 is plotted as a function of Differential Group Delay (DGD).The OSNR penalty for a particular DGD value was computed as the difference between theOSNR sensitivity in back-to-back condition (DGD=0ps) and the OSNR sensitivity at theDGD value of interest. The bit rate for all the considered systems was taken as 40Gbps andthe tap delay Ts of the LE was set as the symbol duration. The receiver optical and electricalfilter bandwidths were optimized to provide maximum instantaneous DGD tolerances. It hasbeen reported in [15] that the maximum instantaneous DGD tolerance was provided when thereceiver optical and electrical filter bandwidths were optimized for a reference DGD of 25pswhich turns out to be 1/2 and 1/3 times respectively of the symbol duration for oDQPSK andoDOPSK considered herein. What we mean by optimization of filter bandwidths formaximum DGD tolerance is as follows. When the filter bandwidths were optimized formaximum OSNR sensitivity at a DGD value of 25ps, the OSNR penalty of the system toinstantaneous DGD values, especially around and beyond DGD values of 20ps, was lesserthan what it was when the filter bandwidths were optimized for maximum OSNR sensitivityin a back-to-back condition [15]. The optimum filter bandwidths for a reference DGD of 25pswas found to be Bo = 3.5, Be = 0.8 times the baud rate for oDQPSK without equalizers andBo = 3, Be = 0.6 times the baud rate for oDQPSK with LE where Bo and Be are the opticaland electrical filter bandwidths respectively. For oDOPSK, Bo and Be were found to be 3.4and 0.9 times the baud rate and 3.7 and 0.7 times the baud rate respectively for unequalizedand equalized systems.

Two important observations can be made from figure (8). The use of LE results in lowerOSNR penalty for both the systems and the oDOPSK system has lower OSNR penaltycompared to oDQPSK for a given bit rate. Reason for the latter is the fact that the oDOPSKsymbols, at the same bit rate as supported by oDQPSK, are broader in time domain.

Before moving on to the next section, we would like to mention that Decision FeedbackEqualizers (DFE) could as well be used in place of the LE to electronically compensate forPMD induced pulse distortions. Results of OSNR penalty reduction in the presence of firstorder PMD by the usage of DFE is presented in [15]. A four tap DFE with two feed forwardfilter taps and one feed back filter tap was considered in [15]. The results in [15] show thatimprovements over LE occur only at relatively larger values of DGD that are above 50ps forthe oDOPSK systems whereas for oDQPSK systems, there was practically no difference inthe OSNR penalty between the LE and DFE systems.

5. Application of Coded Modulation Techniques

As the demands on data transmission rate and distance increase, one of the prime candidatesthat can assist meeting these requirements is coded modulation technique. Coded modulationtechnique is the name given to such error control techniques which combine coding andmodulation into a single operation unlike in conventional error control coding schemeswherein coding and modulation are treated as two different operations. One of the codedmodulation techniques called as TCM has been successfully employed in electricalcommunication systems and will be discussed here from an application view point tooDQPSK systems.


(a)

(c)

(b)

(d)

Figure 9. The concept of TCM; (a) is a rate ½ convolutional encoder with one of the input bits driving itand the other connected directly to the output, (b) is the resultant trellis diagram, (c) is the partitioningof oDOPSK constellation and (d) is the trellis with the state transitions marked with the transmittedoDOPSK symbol selected based on the partitioning shown in (c).


One of the first reports of TCM applied to optical differential phase modulation systemswas [21] wherein TCM was applied to oDPSK systems by incorporating a rate 1/2, constraintlength 3 convolutional encoder into the modulation process. Later, in [22] the idea wasextended to application of TCM to oDQPSK systems. TCM was essentially an invention ofUngerboeck [5] in the eighties and ever since has grown considerably and has found usefulapplications in electrical communication systems. For the benefit of readers who are notfamiliar with TCM techniques, we provide a brief overview.

TCM is best explained with an example. Consider the setup depicted in figure (9.a). Wehave a rate 1/2 convolutional encoder [3] [5] whose input is driven by one of two parallelbinary data streams. The other binary data stream appears as it is at the output of the system.At the final output of the system, there are three parallel binary data streams (two from therate 1/2 convolutional encoder and one from the direct output). Thus, in a way, the combinedsetup can be viewed as a rate 2/3 convolutional encoder with its trellis structure as given infigure (9.b). During each signaling interval, depending on the combination of the threeparallel binary data streams, one of 23 possible signal points from an octal signal set such asthat of the oDOPSK is selected for transmission. The key to the success and effectiveness ofTCM is the way the signal point to be transmitted is selected. A heuristic approach towardsthis, as explained in [5] [23], is to partition the constellation of the chosen modulation schemeprogressively so that the minimum Euclidean distance after each level of partitioning is morethan what it was before. The partitioning of the oDOPSK constellation based on this heuristicguideline is as shown in figure (9.c). The two outputs from the convolutional encoder electthat partitioned subset that has two signal points in it and the output that is directly connectedto the input selects the final signal point to be transmitted. It should be obvious from theabove discussion as to how the coding and modulation steps are interconnected and areinseparable in a TCM system. Figure (9.d) shows the trellis diagram of the TCM schemediscussed above with the transmitted signal points labeled on each transition.

To complete this example on TCM, we explain briefly the demodulation/decodingoperations at the receiver. Having received a noisy sequence of oDOPSK symbols of aparticular length from a TCM based transmitter, the receiver computes the Euclidean distancebetween the received symbol sequence and all other possible symbol sequences of length thesame as that of the received sequence. Due to the usage of a convoutional encoder to selectthe transmitted symbols, only certain particular symbol sequences would only be validtransmitted sequences. For example in the present system, with the initial state of theconvolutional encoder as ‘00’, a look at the trellis structure shown in figure (9.d) readily tellsthat the sequence a-g-b cannot be a valid transmitted sequence of length three. It is this factthat arms the TCM system with its error correction capability. The sequence that is nearest interms of Euclidean distance with the received noisy sequence is estimated as the actualtransmitted sequence. This method of calculating the Euclidean distance and taking decisionsbased on the Euclidean distance is derived from the principle of maximum likelihoodsequence estimation and from the assumption that the noise corrupting the transmittedsymbols are statistically independent and Gaussian in distribution [3]. However, in the case ofnon Gaussian noise as well, if the noise corrupting the symbols is statistically independentand the statistical distribution of the noise is nearly Gaussian, Euclidean distance basedestimation is justified.

The maximum likelihood sequence estimation and the resultant Euclidean distance basedestimation can be carried out efficiently using the Viterbi algorithm [3] [5]. An error in the


decoding stage occurs when the Viterbi algorithm selects a wrong path in the trellis. This canhappen if the Euclidean distance between the received sequence and a valid sequence otherthan the actual transmitted sequence is less than the Euclidean distance between the receivedsequence and the actual transmitted sequence. Due to the manner in which the oDOPSKsymbol sequence to be transmitted is selected, namely the partitioning of the constellation andthe involvement of a convolutional code, the minimum Euclidean distance between any twogiven sequences will be equal to the maximum Euclidean distance (between diametricallyopposite symbols) in the oDOPSK constellation [5]. Now since the dominant error event inthe decoding stages of conventional error control coding scheme is free distance or minimumHamming distance respectively for convolutional codes and block codes, it is understood thatthe dominant error event in the decoding of TCM will be the minimum Euclidean distanceerror which, in the present example, will be the event of erroneous estimation betweendiametrically opposite symbols in the oDOPSK constellation [3] [5]. Under high OSNRscenarios such as the ones often encountered in present day systems due to the usage ofoptical amplification technologies, the probability of such an error event will thus very wellapproximate and provide a lower bound on the Symbol Error Rate (SER) of TCM encodedsystems. It is worth to note that though the basic modulation scheme involved in this exampleis oDOPSK, from the data transmission viewpoint the system is an oDQPSK system sincethere are only two information bits carried per signaling interval. In further discussions in thischapter, we refer to the above discussed TCM system as oDQPSK-TCM system.

An important point that needs to be mentioned here with regards to the receiver foroDQPSK-TCM is that out of the two possible receiver schematics discussed in section 3 foroDOPSK, due to the fact that the multiple outputs from the receiver are not statisticallyuncorrelated and hence statistically dependent, they are not the ideal ones for TCMdemodulation [24]. Due to this reason, we use a receiver scheme as was depicted in figure (2)with θ1= π/4 and θ2 = 3π/4. The procedure for SER evaluation with such a receiver schematicunder the assumption of the minimum Euclidean distance error event dominating significantlythe error events in the decoding process is reported in [22] and references therein and is quitestraightforward.

Figure (10) depicts SER performance of oDPSK, oDQPSK and oDQPSK-TCM system asa function of OSNR. The OSNR was calculated as earlier with unpolarized ASE noise within areference spectral width of 0.1nm. The optical and electrical filter parameters have beenretained the same as those for the results depicted in figure (6) of the previous section.However, in these results, the signaling rate is taken as 20Gbaud for all the three caseswhereby the oDPSK system supports a bit rate of 20Gbps and the oDQPSK and oDQPSK-TCM systems support a bit rate of 40Gbps. Also, a symbol error is said to have occurred inoDQPSK if any one or both the bits that constitute a symbol is received erroneously

As can be seen, the application of TCM improves the performance of an uncodedoDQPSK system to that of the binary oDPSK system at identical OSNR values. Moreimportantly, for a symbol rate of 20Gbaud, the oDQPSK-TCM supports an information bitrate of 40Gbps at the same performance as a binary oDPSK system that supports 20Gbpswhile consuming the same bandwidth. Thus the application of TCM effectively doubles thebandwidth efficiency without having to tradeoff SER.


Figure 10. SER performance of oDPSK, oDQPSK and oDQPSK-TCM system as a function of OSNR.

Figure 11. OSNR penalty for increasing values of DGD for oDQPSK-TCM, uncoded oDQPSK withoutequalizer and uncoded oDQPSK which uses LE.


Figure (11) shows the OSNR penalty of uncoded oDQPSK with and without LE discussedin the previous section and that of the unequalized oDQPSK-TCM for increasing values ofinstantaneous Differential Group Delay (DGD). The results are referenced to the OSNRsensitivity of unequalized oDQPSK optimized in back-to-back condition for a SER of 10-12

[15]. The equalized uncoded oDQPSK as well as the unequalized oDQPSK- TCM wereoptimized for a reference DGD of 25ps as was done in the last section with optical andelectrical filter bandwidths as 3.5 and 0.8 times the baud rate respectively for unequalizedoDQPSK and 3 and 0.8 times the baud rate respectively for oDQPSK-TCM. These resultsbring out the robustness of oDQPSK employing TCM in the presence of first order PMD. Itcan be observed that the coding gain due to TCM over uncoded, unequalized oDQPSK isretained throughout the range of DGD values considered in this figure.

6. Alternative Receiver Schematic for oDOPSK

In this section, we discuss an alternative receiver schematic for oDOPSK which needs onlytwo delay interferometers in contrast to the four required by the schematics depicted in figure(4). The receiver discussed in this section is very closely associated with an earlier onereported in [25].

The maximum likelihood detection principle [3] as applicable to oDOPSK can be readilyexplained with the help of figure (12) wherein we depict the receiver schematic discussedearlier through figure (2) with θ1= π/8 and θ2 = 5π/8 along with the resultant oDOPSKconstellation and signal space diagram. The term φk in this diagram stands for the phasedifference between the received oDOPSK symbols in the present and the previous intervals.This phase difference can take on discrete values ranging from 0 to 7π/4 in steps of π/4.

Now having received y1 and y2 during a particular signaling interval, classical detectiontheory works on the principle of hypothesis testing as given below

Figure 12. Receiver schematic with θ1= π/8 and θ2 = 5π/8 and associated oDOPSK constellation andsignal space diagram.


)/()/( ykP

s

k

ysP><

(13)

where P(a/b) is the conditional probability of event a given event b and s , k , and y arerespectively the vectors [s1 s2], [k1 k2] and [y1 y2] with the first entry within the parenthesisstanding for the inphase component and the second entry for the quadrature component. Thevectors s and k are two different signal points in the signal space diagram. It may be notedthat what the above equation does in principle is it compares the probability of the transmittedsymbol being s or k given that the vector y was received. This hypothesis test is conductedover all symbols and the transmitted symbol is estimated as the one that has the highestprobability of being transmitted given that the vector y is received.

Now if the noise that corrupts the transmitted signals is AWGN, and the two componentsof the received vector y namely y1 and y2 are statistically independent (which they will be ifthey are uncorrelated and have Gaussian distribution), the hypothesis test given in the lastequation can be rewritten as

222

211

222

211 )()()()( ykyk

s

k

ysys −+−<>

−+− (14)

which is in fact a Euclidean distance comparison in the signal space between the receivedvector y and the two signal points s and k . Since all the eight signal points in oDOPSKconstellation are equidistant from the origin, equation (14) can be rewritten as

0)()( 222111

s

k

ksyksy><

−+− (15)

The numerical sign of y1 and y2 can be readily made use of to identify the quadrant inwhich the transmitted symbol is most likely to be. Thus, while assigning binary triplets to theeight different symbols of the oDOPSK constellation, if two of the bits in these triplets aremade to tally with the quadrant in the signal space diagram where the symbol lies, those twobits could be readily detected by detecting the numerical sign or in other words a bi-leveldetection of y1 and y2. As per the signal space diagram and constellation shown in figure (12),starting from the point φk=7π/4 in the first quadrant, the triplets could be assigned in ananticlockwise manner in the order (111), (110), (010), (011), (001), (000), (100) and (101). It


may be noted that this assignment is in agreement with gray encoding rules according towhich closest points in the signal constellation have to differ by only one bit.

To identify the third bit which separates the two signal points within a quadrant, werevert back to equation (15) and analyze as follows. Within the same quadrant, (s1-k1) and (s2-k2) would be of same magnitude and the only difference if any would be in the numericalsigns. Thus, while conducting the test as given in equation (15) above to estimate the mostprobable transmitted symbol among two symbols from the same quadrant, what matters is notthe magnitude of the differences (s1-k1) and (s2-k2) but their numerical signs. Thus equation(15) can be rewritten for symbols within the same quadrant as

02211

s

k

yy><

Γ+Γ (16)

with Γ1 and Γ2 being the numerical signs of (s1-k1) and (s2-k2) respectively.With reference to the signal space diagram shown in figure (12) and the concepts

presented above, the following with regards to estimating the third bit can readily be arrivedat

Ist quadrant

Γ1 = +, Γ2 = -, if s = 7π/4 and k = 3π/2; therefore, 0

1

0

21 ><

− yy and 21 yy + = +

IInd quadrant

Γ1 = -, Γ2 = -, if s =π and k = 5π/4; therefore, 0

0

1

21 ><

+ yy and 21 yy − = -

IIIrd quadrant

Γ1 = -, Γ2 = +, if s = 3π/4 and k = π/2; therefore, 0

0

1

21 ><

− yy and 21 yy + = -

IVth quadrant

Γ1 = +, Γ2 = +, if s = 0 and k = π/4; therefore, 0

1

0

21 ><

+ yy and 21 yy − = +


From the above we note that either (y1-y2) or (y1+y2) is the deciding factor and when oneis the deciding factor the other has a constant numerical sign. From this observation, the fourdifferent decision rules given above can be combined into a single decision rule for estimatingthe third bit as given below.

0

1

0

))(( 2121 ><

+− yyyy (17)

The above equation suggests that a binary decision on the sum and difference of y1 and y2

followed by an XNOR operation on those decisions can readily provide an estimate of thethird bit. In fact, this is the receiver schematic suggested in [25] with a minor variation in thatthe XNOR is replaced by the XOR apparently due to the swap in positions of 0s and 1s in thethird bit of the triplet as compared to what it is herein.

The receiver schematic depicted in figure (4.b) also works as per the same principle asdiscussed above. The two inputs to the XOR are effectively (y1-y2) and (y1+y2) [25]. Also,with an appropriate precoding of the binary data as given in [10], it is possible to directlyobtain the three constituent data bits from the detected binary levels of y1, y2 and the product(y1-y2)(y1+y2).

Further, if equation (17) is rewritten as

0

1

0

)( 22

21 >

<− yy (18)

it becomes obvious that the decisions can be taken depending solely on the difference of theabsolute values of y1 and y2. The conversion of the detected samples to their absolute valuescan be achieved in effect by considering the fact that the detected analog samples y1 and y2

are in fact dealt with in the receiver electronics in the digital domain through an analog todigital converter. More the resolution of the analog to digital converter better will be theresultant digital representation of the detected analog voltage. This is the methodology used inalmost all the electrical soft decision decoding receivers [3]. In an analog to digital converter,it is possible in principle to identify the numerical sign of the digitally converted sample andas such it is possible to alter that numerical sign. Thus, if the detected sample y1 or y2 isnegative, its numerical sign can be altered and the following decision rule can be applied todetect the third bit.

0

1

0

)( 21 ><

− yy (19)


The advantage of this receiving scheme is that the dependence of the decision makingvariable on the sum as well as difference of y1 and y2 is removed and is now dependent onlyon the difference between y1 and y2. This is of course at the cost of an additional electronicoperation of changing the numerical sign of the detected samples. It may also be noted that amere change in numerical sign does not alter the pdf of the detected samples. The completeschematic representation of this receiver is as given in figure (13).

Figure 13. Schematic representation of an oDOPSK receiver which employs only two delayinterferometers.

The BER or probability of error for this receiver schematic can be readily arrived at as

BER = ( P(y1>0/ b1= 0)+P(y1<0/b1 = 1)+ P(y2>0/b2=0)+P(y2<0/b2=1)+P((y1-y2)>0/b3=0)+ P((y1-y2)<0/b3=1))/6 (20)

where b1, b2 and b3 stand for the constituents of the binary triplets assigned to the eightsymbols of the oDOPSK constellation and are assumed to take on logic levels 1s and 0s withequal probability. It has to be specially taken note of the fact that y1-y2 as it appears in theabove equation is after the numerical signs of both y1 and y2 are converted to positive.

The above BER can be computed as discussed earlier in section (4) using the KLSEmethod. However, while computing the last two probabilities in equation (20) that involvesthe difference of y1 and y2, the CF of y1-y2 is required. This can be obtained by replacing theHermitian kernel in the procedure outlined in section (4) by the difference of the Hermitian


kernels of the two arms of the delay interferometer after appropriately accounting for thepower splitting at the front end of the receiver after the optical filter.

(a) (b)

Figure 14. BER performance for the oDOPSK receiver structure depicted in figure (13).

Figure (14.a) and (14.b) shows the BER as a function of OSNR. Figure (14.b) zooms on toa portion of the BER curve. These figures show the individual BER for b1, b2 and b3 as well asthe average BER. The optical and electrical filter bandwidths while evaluating these resultswere taken as 3 times and 1 time the baud rate respectively. The bit rate was set as 40Gbps. Itcan be noted that while the BER curves for b1 and b2 overlap as they should due to identicalvariances and same average means for y1 and y2, the BER of b3 is slightly more than that of b1

and b2. This is obviously due to the fact that the variance of y1-y2 is twice the variance of y1

and y2.Before we conclude this chapter, it needs to be mentioned that we have not included a

performance evaluation of the oDOPSK receiver schematic shown in figure (4.b). This is dueto the fact that the BER evaluation of that receiver schematic using the accurate KLSE methodbecomes a cumbersome problem due to the fact that the XOR output is a function of bothy1+y2 as well as y1-y2. This renders the BER evaluation an exercise of double integration ofthe joint pdf of y1+y2 and y1-y2. The BER performance of that receiver using other techniquesof evaluation has appeared in [10] and its inclusion here is not suitable for a fair comparisonwith the results reported in this chapter since all the results in this chapter have beenevaluated using the KLSE method.

7. Conclusion

This chapter has presented a detailed review of optical differential phase modulation schemestouching upon oDPSK, oDQPSK and, oDOPSK. Various proposed receiver schematics forthese modulation schemes were presented along with their error rate performance. Anintroduction to the KLSE method for error rate evaluation of the discussed systems was also


presented. Use of electronic equalization methods in the form of LE was discussed and theresultant improvement in OSNR penalty to achieve a target BER of 10-12 was discussed.Coded modulation techniques in the form of TCM which can improve error rate performanceof optical differential phase modulation schemes were introduced and a comparison was madebetween uncoded oDQPSK system and oDQPSK systems that use TCM. An alternativereceiver schematic for oDOPSK which needs only two delay interferometers was presentedalong with its error rate performance.

While oDOPSK, oDQPSK and, oDOPSK with electronic equalizers and/or TCM schemeshold a lot of promise for realizing future long haul optical transmission systems, the mainstumbling block that can arise towards the successful application of post detection electronicprocessing based performance enhancers such as equalizers and TCM is the speed of theelectronic hardware. In this regards, some developments in electronic hardware as reported in[26] and [27] holds some promise.

References

[1] Sano, A.; Masuda, H.; Kisaka, Y.;Asawa, S.;Yoshida, E.;Miyamoto, Y.;Koga, M.;Hagimoto, K.;Yamada,; Furuta, T.;Fukuyama, H. ECOC 2006. Cannes, (Post DeadlinePaper)

[2] Gnauck, A. H.; Winzer, P. J. J. Lightwave Tech. 2005, Vol. 23, 115-129[3] Proakis, J. Digital Communication. McGraw Hill, 2001[4] Betti, S.; Marchis, G. D.; Iannone, E. Coherent Optical Communication Systems. Wiley,

New York, 1995.[5] Ungerboeck, G. IEEE Commun. Magazine. 1987, Vol.25, 5-11[6] Winzer, P.J.; Dorrer, C.; Essiambre, R. J.; Kang, I. IEEE Photon.Technol.Lett. 2004,

Vol 16, 1379-1381[7] Penninckx, D.; Bissessur, H.; Brindle, P.; Gohin, E.; Bhakhti, F. Proc. ECOC 2001.

Amsterdam, The Netherlands, 2001, 456-457[8] Wei, X.; Liu, X.; Chandrashekar, S.; Gnauck, A. H.; Raybon, G.; Leuthold, J.; Winzer,

P. J. Proc ECOC 2002. Copenhagen, Denmark, 2002, paper 9.6.3.[9] Griffin, R.A.; Carter, A.C. OFC 2002. 367-368.[10] Kim, C.; Li, G. Optics Express. 2004, Vol.12, 3415-3421.[11] Yoon, H.; Lee, D.; Park, N. Optics Express. Vol.13, 371-376[12] Yoon, H.; Lee, D.; Park, N. Ninth Optoelectronics and Communications Conference and

Third Conference on Optical Internet (IEEE Communication Society). 2004. paper14C3-4.

[13] Lee, J. S.; Shim, C.S. J. Lightwave. Tech. 1994, Vol.12, 1224-1228.[14] Forestieri, E. J. Lightwave Tech. 2000, Vol. 18, 1493-1503[15] Yoon, H.; Kim, N.Y.; Park, N. IEEE Photons. Tech. Letters. 2005, Vol. 17, 2577-2579[16] Wang, J.; Kahn, J.M. J. Lightwave Tech. 2004, Vol. 22, 362-371[17] Marcuse, D. J. Lightwave Tech. 1990, Vol. 8, 1816-1823[18] Cohen, H. Mathematics for Scientists and Engineers, Prentice Hall, 1992.[19] Poole, C.D.; Wagner, R.E. Electron. Lett. 1986, Vol. 22, 1029-1030 [20] Haunstein, H. F.; Wolfgang, S. G.; Dittrich, A.; Sticht, K.; Urbansky, R. J. Lightwave.

Tech. 2004, Vol. 22, 1169-1182


[21] Buelow, H.; Thielecke, G.; Buchali, F. Optical Fiber Communication (OFC) , LosAngeles, CA 2004, Paper WM5

[22] Kumar, M. S.; Yoon, H.; Park, N. IEEE Photons. Tech. Letters. 2007, Vol. 19,1245-1247

[23] Ungerboeck, G. IEEE Commun. Magazine. 1987, Vol.25, 12-21[24] Wang, Jin-Der.; Chung, H. Y. IEEE Trans. Commun. 1990, Vol. 38, 1549-1556[25] Han, Y.; Kim, C.; Li, G. Electron. Lett, 2004, Vol. 40, 1372-1373[26] Mizuochi, T.; Ouchi, K.; Kobayashi, T.; Miyata, Y.; Kuno, K.; Tagami, H.; Kubo, K.;

Yoshida, H.; Akita, M.; Motoshima, K. OFC 2003, PD 21, Atlanta, 2003[27] Dawid, H.; Fettweis, G.; Meyr, H. IEEE Trans. VLSI Systems. 1996, Vol. 4, 17-31


Chapter 9

A NEW GENERATION OF POLYMER OPTICAL FIBERS

Rong-Jin Yu and Xiang-Jun Chen School of Information Science and Engineering, Yanshan University,

Qinhuangdao 066004, China

Abstract

This chapter describes the background to the development of Polymer Optical fibers (POFs), discusses the optical and temperature resistant properties of polymers while emphasizing the intrinsic high attenuation of them. The first generation of POFs which consists of a solid-core surrounded by cladding and transmits light by total internal reflection, is puzzled by the difficulty of high attenuation. Then, the method of using a specific structure (i.e. hollow-core Bragg fiber) to solve the problem is presented. A new generation of POFs based on the hollow-core Bragg fibers with cobweb-structured cladding can guide light with low transmission loss and high bandwidth in the wavelength range of visible to terahertz (THz ) radiation. Efficient hollow-core guiding for delivery of power laser radiation and solar radiation can be achieved by replacing the traditional polymethylmethacrylate (PMMA) with heat-resistant polymers. Lastly, this chapter concludes with a discussion of applications in diverse areas.

1. Introduction

The optical fiber is a circular dielectric waveguide that guides optical information and energy. The first theoretical study of wave propagation on circular cylindrical dielectric structures was made by Hondros and Debye [1]. It was not until 1966 that the proposed to use circular glass fibers as optical transmission lines was made [2]. Since then, the attenuation of more than /1000dB km for silica fiber is reduced to about 0.2 /dB km by advanced purifying and manufacturing processes. The optical fiber is generally made from glasses, polymers or other materials. Compared to glass materials, polymeric materials have the shortcoming of intrinsic high attenuation while having the advantage of great flexibility. The elastic limits of polymeric materials are high. It is this high elasticity which allows the fabrication of tough and flexible POF with diameters of one millimeter and above. However, inorganic glasses with low elastic limits dictate that remarkable flexibility may only be observed in the optical

Rong-Jin Yu and Xiang-Jun Chen 258

fibers of very small diameter, typically 125 mμ . The inherent brittleness of glass requires that a more elastic polymer coating be applied to the very fragile core-cladding structure to protect its surface and prevent the growth of Griffiths cracks and consequent fracture.

For a long time in the past, the optical fibers consist of a solid-core surrounded by cladding, and transmit light by total internal reflection, with the addition of a few hollow waveguides with inner metallic or dielectric coatings to provide high reflectivity. The transmission losses of a solid-core fiber, including solid-core photonic crystal fiber, are larger than (at least equal to) the losses of the fiber-core material itself. Thus, first generation POFs which consist of a solid-core surrounded by cladding are facing the difficulty of high attenuation.

One feasible way of solving the problem is to use a hollow-core fiber instead of the solid-core fiber. But the hollow-core fibers do not get any special attention and exploitation due to the very large success of low-loss high-index core (solid-core) silica fibers by total internal reflection and their extensive use in various fibers for transmitting light. Until a few years ago, the conventional wisdom in most books related to the fiber had been that confined and lossless propagation in fibers is accomplished by total reflection from the dielectric interface between the core and the cladding, or must make use of the concept of total internal reflection to save light inside the core of the optical fiber. In fact, very early in the developmental history of optical fibers, the idea of using Bragg reflection in a cylindrical fiber to obtain lossless confined propagation in a core with a refractive index lower than that of the cladding medium was proposed in the year 1978 [3]. Up to the late 1990s, the hollow-core Bragg fibers [4] and hollow-core photonic band gap fibers [5] were demonstrated. Both experimental and theoretical investigations confirm that transmission losses by using hollow-core Bragg fibers are dramatically lower than the losses of its constituent materials. It seems that hollow-core Bragg fibers are a suitable structure that can very effectively confine the transverse leakage of guided wave.

Once adopting the hollow-core Bragg structures, surmounting the shortcoming of high attenuation and taking the advantage of great flexibility of POFs, a new generation of POFs can guide light with low transmission loss and high bandwidth in the wavelength range of visible to THz radiation. Efficient hollow-core guiding for delivery of power laser radiation and solar radiation can be achieved by replacing the traditional PMMA with heat-resistant polymers. We expect that a new generation of POFs will find many applications in diverse areas, and is irreplaceable for some applications.

2. Development History of POFs

The first report of POF was in 1960’s, which was almost the same as the invention of silica-base optical fiber. In 1966, Du Pont invented the first POF named “Crofon” that was of step-index (SI) type composed of PMMA core surrounded by a partially-fluorinated polymer cladding. In 1975, Mitsubishi Rayon commercialized the first SI POF whose trade name was “Eska” and Asahi Chemical and Toray soon followed in 1970’s. The POF market was originally dominated by these three major companies who have been manufacturing SI type POFs composed of PMMA core [6]. In the more than forty years history of POFs, efforts were made to improve the performance of attenuation, bandwidth and temperature resistance of POFs. In the mean time, the work of Kaino (NTT’s Ibaraki Laboratories), Koike (Keio

A New Generation of Polymer Optical Fibers 259

University) and their co-workers is very important in the advancement of technology of POFs in the area of the reduction of attenuation and the realization of high bandwidth.

PMMA, which has been the most typical core material for POF, has large attenuation due to the intrinsic absorption loss of carbon-hydrogen stretching vibration. Theoretical attenuation limit of PMMA based POF is around 100 /dB km at 0.65 mμ wavelength, while the attenuation abruptly increases from near infrared to infrared region. The intrinsic absorption loss can dramatically decreased by using a polymer which has no carbon-hydrogen bonding in it. The work of Kaino and co-workers has brought about the reduction of the losses of PMMA-D8-core fibers to 20 /dB km at 680nm [7]. By substituting all hydrogen bonding in polymer molecules with fluorine, remarkable low attenuation of 10 /dB m was achieved by the perfluorinated (PF) polymer based graded-index (GI) POF. The first PF polymer based GI-POF was reported in 1994. In 2000, PF polymer based GI-POF named “Lucina” was commercialized from Asahi Glass Co., using a PF polymer named CYTOP [8]. However, the fibers reduce attenuation at the expense of increased cost. The deployment of both deuterated and fluorinated POFs is limited by the high cost of the fiber materials.

Considering the transmission bandwidth of POFs, commercially available POFs have been of the SI type with a numerical aperture ( NA ) of about 0.5, whose bandwidth of transmission is limited to about 5MHz Km⋅ . For a SI-POF, quite a degree of improvement in its bandwidth can be achieved by reducing NA of the POF. For the low NA SI-POF ( NA= 0.31), its bandwidth is 160MHz at 100m ( 200MHz> at 50m ), which is currently commercialized. It meets the requirement of standardization of 156 /Mb s , transmission 50m approved by the ATM forum in May 1997. It is a common knowledge that the main limitation on the bandwidth of multimode optical fibers is modal dispersion, which means that different optical modes propagate at different velocities and the dispersion grows linearly with length. One way to overcome the modal dispersion is to use single mode (SM) POF. The first SMPOF was reported in 1991, which was successfully prepared by the interfacial-gel polymerization technique [9]. In the fiber, the core diameter was 3 15 mμ− and the attenuation of the transmission was about 200 /dB Km at 652nm wavelength. Another way to solve the problem for POFs with large cores is to use multi-layer step-index (ML-SI) POF [10], multi-core step-index (MC-SI) POF [11], or GI-POF [12]. In ML-SI POFs, the core region is composed of several layers with different refractive index. This concentric multi-layer structure decreases modal dispersion compared to conventional SI type POF and a data rate as high as 500 /Mb s for 50m transmission is achieved experimentally. MC-SI POF has a low numerical aperture (0.25) and a core region composed of 19 cores of small-core. By reducing the core diameter, not only modal dispersion but also bending loss is decreased. A data transmission at 500 /Mb s for 50m is also achieved by the MC-SI POF. For GI POF, the refractive index of the fiber core is graded parabola-like from a high index at the fiber core center to a low index in the outer core region. For the GI POF produced by the interfacial-gel polymerization method, its bandwidth measured is 3GHz for a fiber length of 100m . A low-loss PF polymer based GI POF has been developed and PF polymer based GI POF is able to transmit a data rate of 10 /Gb s or higher because of its material dispersion property [13].

For the temperature resistance of POFs, the high-temperature performance of a polymer is limited by its glass transition temperature ( gT ). For PMMA, gT is about 105. Maximum

operating temperature for PMMA-core SI-POFs is 80. Ziemann et al. [14] had carried out a


accelerated aging test for the fibers from the three leading POF manufacturers. The results of the test show that for all fibers and wavelengths ( 650nm , 590nm , 525nm and 520nm ) the estimated possible operating temperature for 20 years use is over 70. Some applications, such as in automobiles, aerospace environments and transmitted power demand performance at temperature in excess of 80. The gT of polycarbonate (PC) is around 170. The use of

PC and partially fluorinated PC as core material enables temperatures of up to 115 and 145, respectively. The gT of polyethersulfone (PES) is about 225, maximum operating

temperature of PES is 197. Polyimide material has even more high operating temperature (316). The attenuation of these high temperature resistant polymers is generally larger than that of PMMA, therefore making the polymers useless in fabricating the fibers, but using the polymers (such as polyimide) as the coating of high temperature resistant silica fiber.

In a word, in the forty years development of POFs, there is no better position in both performance (especially attenuation) and cost comparing with silica glass fibers. Thus, first generation POFs have limited their penetration in important market-segments, and are only suited to ornament, illumination, sensors and short-distance data transmission applications.

3. Hollow-Core Fibers

Hollow-core fibers reported to date in the literature can generally be classified into four types: (1) those in which the refractive index of the cladding is greater than that of the core, (2) those in which inner wall coating has high reflectivity, (3) hollow-core photonic bandgap fiber, and (4) hollow-core Bragg fiber.

3.1. Those in Which the Refractive Index of the Cladding Is Greater Than That of the Core

As is known to all, waveguiding is achieved in conventional solid-core fibers due to the total internal reflection from the interface between the core with the refractive index coren and the

cladding with the refractive index ( > )clad core cladn n n . For the hollow-core fiber in which the refractive index of the core is lower than that of the cladding, the propagation of light is achieved by the regime of grazing incidence and is accompanied by radiation losses (leaky guide). In fact, this hollow-core fiber is a capillary tube, as shown in Fig.1. The coefficient of optical losses in the hollow fibers scales as 2 3/aλ , where λ is the radiation wavelength and a is the core radius of the fiber. Thus, most of applications are performed by using the hollow-core fibers with large inner radii and short length. For example, a 10cm -long and 150 mμ -diameter hollow-core fiber filled with argon gas is used on extreme ultraviolet (EUV) light generated through the process of high-harmonic up-conversion of femtosecond laser [15].


Figure 1. Cross section of the hollow-core fiber with clad coren n> (leaky guide).

3.2. Those in Which Inner Wall Coating Has High Reflectivity

Figure 2. Cross section of two hollow-core fibers with inner wall coatings.

For hollow-core fiber in which inner wall coating has high reflectivity as shown in Fig.2, the fibers are metallic, glass or polymer tubes with inner metallic or (and) dielectric coatings provided with high reflectivity in the wavelength range of interest. Among them, the hollow-core fiber whose inner wall material has a refractive index less than one is referred to as an attenuated total reflectance (ATR) guide. In general, the 1n < or ATR guides are made of sapphire, 2GeO or some special 1n < oxide glass. The idea of an 1n < structure originated from Hidaka, et al. in 1981 [16]. To be useful for laser transmission, the ATR guides must have the region of anomalous dispersion, where n is less than 1, fall within some useful laser wavelength range. The first 1n < guides studied by Hidaka, et al. focused on glass tubes made from lead and germanium doped silicates. By adding heavy ions to silica glass, he was able to shift the infrared edge to longer wavelengths so that the 1n < region of anomalous dispersion occurred within the 2CO laser wavelength band. Gregory and Harrington [17] pointed out that sapphire or 2 3Al O has 1n < from 10 to 16.7 mμ and, in addition, it has a

very small k value of 0.05 at 10.6 mμ . This means that the theoretical loss is very low (less than 0.1 /dB m for a 1000 mμ -bore tube) for this material. But sapphire has a high modulus. Therefore, it cannot be bent to small diameters. These hollow-core fibers are an attractive alternative to solid-core infrared fibers. These fibers have losses as low as 0.1 /dB m at 10.6 mμ and may be bent to radii less than 5cm . For applications in high-power laser delivery, the fibers have been shown to be capable of transmitting up to 2.6kW of 2CO laser power [18]. They also have usage in both temperature and chemical fiber sensor applications. Recently, hollow polycarbonate tubing with inner Cu coating is used on broadband THz

hollow-core

cladding (glass)

hollow-core

n<1 material

structural tube

hollow-core

structural tube

metallic filmdielcetric film


transmission, and the lowest loss of 3.9 /dB m was obtained from a 3mm -bore fiber at 158.51 mμ [19].

3.3. Hollow-Core Photonic Bandgap Fiber

In 1999, the hollow-core photonic bandgap fiber have been experimentally demonstrated for the first time [5]. For hollow-core photonic bandgap fibers as shown in Fig.3, they are optical fibers with cladding made of fused silica or polymer incorporating arrays of air holes. The

Figure 3. Cross section of hollow-core photonic bandgap fibers.

core is formed by omitting several unit cells of material from the cladding. The “holey” cladding has a two-dimensional photonic bandgap that can confine light to the core for wavelengths around a minimum-loss wavelength. For example, the transmission losses of hollow-core plastic (PMMA) photonic bandgap fibers can be decreased by an order of magnitude with nine rings of air holes in comparison with conventional plastic fibers according to our numerical analysis by using multipole method [20]. Xu, et al. [21] analyzed the loss of an air-core silica glass photonic bandgap fiber and demonstrated that it is possible reduce the transmission loss to a level below /0.01dB km , with eight rings of air holes at 1.53 mμ . Experimentally, the lowest loss reported in hollow-core photonic bandgap fibers with fused silica is 1.2 /dB km at 1620nm [22]. The ultimate limit to the attenuation of such fibers is determined by surface roughness due to frozen-in capillary waves. The attenuation of 1.2 /dB km at 1620nm already appears to be dominated by this mechanism. On the other hand, under consideration of fiber design, Roberts, et al. [22] pointed out that “the reduction in reported attenuation from 13 /dB km to 1.7 /dB km (and now 1.2 /dB km ) was partly from enlarging the core from 7 to 19 unit cells, reducing F by a factor of 3/ 2(19 7)/ 4.5≈ . F could be further reduced by enlarging the core to 37 cells for example. However, this would be accompanied by propagation of more higher-order core modes, increased bending loss and closer spectral packing of surface modes. Since these are already apparent in our 19-cell hollow-core photonic crystal fiber (HC-PCF), we expect 37-cell HC-PCFs to be of very limited practical value.” In view of this, it is very difficult for hollow-core photonic bandgap fibers to further reduce the attenuation and achieve single mode at the same time. The hollow-core fiber needs a cladding structure that can confine the transverse leakage of guided wave more effectively. Until now, it seems that hollow-core Bragg fiber is a suitable selection.


3.4. Hollow-Core Bragg Fiber

So far, there are three classes of hollow-core Bragg fibers: (a) “OmniGuide” fibers with very large cladding indices contrast [23], (b) ring-structured hollow-core fibers with a single material [24, 25], (c) cobweb-structured hollow-core fibers with a single material and a certain number of

supporting strips [26, 27].

(a) OmniGuide fiber (b) Ring-structured fiber (c) Cobweb-structured fiber

Figure 4. Cross section of hollow-core Bragg-fibers.

In Bragg fibers, the hollow-core is surrounded by a 1-dimensional Bragg reflector consisting of alternating layers of high- and low-index materials. The cladding of “OmniGuide” hollow-core fiber consists of two (solid) materials with different indices as shown in Fig.4(a), that of ring-structured hollow-core fiber consists of a single material in which rings of holes are used to define the low-index layers as shown in Fig.4(b), and that of cobweb-structured hollow-core fiber consists of a single material in which air layers are used as the low-index layers as shown in Fig.4(c). These Bragg fibers made a breakthrough in fiber’s transmission losses less than the absorption losses of the material. The attenuation, which is lower than the material loss, has been observed in “OmniGuide” fibers and ring-structured fibers. The transmission losses of “OmniGuide” hollow-core fibers are dramatically lower than the losses of its constituent materials [28]: “Recent data show the losses of 0.65 /dB m for such fiber, when made of a material with losses of 30000 /dB m . Thus, its structure suppresses the losses of constituent materials by a factor of more than 45000.” But, it is difficult to find two materials they have larger indices contrast, similar thermal and mechanical properties, as well as compatible processing technique in realizing the structure. Therefore, only two material combinations have been demonstrated: Te ( n =4.6) in combination with a polymer ( n =1.59) [4], and 2 3As Se ( n ~2.8) in combination with PES ( n ~1.55) [29]. For ring-structured hollow-core fibers, the attenuation, which is lower than the material (PMMA) loss, has been observed in the infrared ( 1120nmλ > ), specifically at 1390nm wavelength, at which the transmission loss is only 40 /dB m compared with the 420 /dB m material loss [30].

We proposed a modified cladding structure, i.e. a hollow-core Bragg fiber with cobweb- structured cladding [26]. The structure uses a single dielectric material and may solve the problem of structural support by using a certain number of supporting strips. The supporting strips are always symmetric in the cross-section and use the same dielectric material as alternating layers. Our research shows that the field profiles are slightly deformed due to the introduction of supporting structure. Although a small fraction of power is leaked out as a


result of the introduction of supporting structure, properly selected parameters of supporting structure will keep the loss at a low level, neglecting the presence of the supporting strips. The number and width of supporting strips should be as small as possible, generally, m (number of supporting strips) = 6~12 and sw (width of supporting strips) = 3/3 ~ 0/λ λ , where λ is the operating wavelength of the fibers.

In comparison to “OmniGuide” fibers, the feasibility of cobweb-structured fibers is greatly improved. For ring-structured fibers, the refractive index of low-index layers in the cladding is between high-index (host material) and 1 (air). As a result, the cladding indices contrast of ring-structured fibers is smaller than that of cobweb-structured fibers. As far as the ability to confine the transverse leakage of guided wave is concerned, the ring-structured fibers are smaller compared to the cobweb-structured fibers. In order to compare the confinement losses of hollow-core ring-structured Bragg fiber with hollow-core cobweb-structured Bragg fiber, we make the design of analogous structure. Argyros et al. [25] have presented the design that supports a single-polarization, circularly symmetric nondegenerate mode in an air-core ring-structured Bragg fiber. The design presented has 0.403i mμΛ = ,

0.578e mμΛ = , 0.355d mμ= and core radius( or ) = 2.89 mμ , giving / 0.83id Λ = . The host material was assumed to be lossless with a refractive index of 1.49 (corresponding to PMMA material). When N (number of rings in cladding) = 9, the confinement losses of the 01TE

mode (lowest-loss mode) and 02TE mode (second-lowest-loss mode) are about 0.83 /dB m

and 57.14 /dB m , respectively. The ratio of the loss of the 02TE mode to the loss of the 01TE

mode reaches approximately 70. In our design, the same parameters: 2n (PMMA) = 1.49,

cor (core radius) = 2.89 mμ , 2d (thickness of high-index layers) = 0.243 mμ , 1d (thickness of

low-index layers) = 0.335 mμ and N (number of alternating layers in cladding) = 9, as well as 1 1n = are used. The host material was also assumed to be lossless. The calculated results

show that the least-loss wavelength of the 01TE mode is located at 0.72 mμ . The confinement

losses of the 01TE mode and 02TE mode at 0.72 mμ wavelength are 55.32 10 /dB m−× and 32.97 10 /dB m−× , respectively. The ratio of the loss of the 02TE mode to the loss of the 01TE

mode reaches approximately 56. Thus it can be seen that the confinement loss of the 01TE mode in the hollow-core cobweb-structured Bragg fiber is reduced by 15600 times in comparison to that of the air-core ring-structured Bragg fiber.

These hollow-core Bragg fibers not only can reduce unwanted material properties, such as absorption, scattering, dispersion and nonlinearity to a large extent, but also can act as a modal filter [3]. Sterke et al. [31] found that such Bragg fibers can be guaranteed to be effectively single-moded. Johnson et al. [23] presented their work of “how the lowest-loss

01TE mode can propagate in a single-mode fashion through even large-core fibers, with other modes eliminated asymptotically by their higher losses and poor coupling, analogous to hollow metallic microwave waveguides.” The single-mode operation of the Bragg fibers is achieved through asymptotic way during the transmission of guided waves, i.e. the number of modes in large-core Bragg fibers causes the change as follows, at the beginning, the transmission with multimode is followed by a few modes, and then the transmission becomes


single moded at last. Thus the single mode is achieved in a certain length range of the fiber. Moreover, Bassett and Argyros [32] presented a method for calculating the single-mode length range: “The individual modes are characterized by two lengths, 1%L at which the

transmitted power in that mode is reduced to 1%, and 0.01% 1%2L L= , at which the power is

reduced to 0.01%. We characterize each fiber as a whole by two lengths, max 1%L L= for the

best guided mode, and 0.01%smL L= for the second best guided mode. We consider the

usefully single moded for lengths between smL and maxL .”

4. Hollow-Core Bragg Fiber with Cobweb-Structured Claadding

The refractive index profiles of hollow-core Bragg fiber with cobweb-structured cladding, together with those of ring-structured and “OmniGuide” hollow-core Bragg fibers are shown in Fig.5 for comparison. The parameters of the fiber with cobweb-structured cladding are

cor (hollow-core radius), 1n (=1, air), 2n (high-index), 1d (thickness of air layers),

2d (thickness of high-index layers), 2 1( / )d dη = , 1 2( )d dΛ = + , N , m and sw , where N is

the number of alternating layers in cladding, m and sw are the number and the width of the supporting strips, respectively.

(a) cobweb-structured fiber (b) ring-structured fiber (c) “OmniGuide” fiber

Figure 5. Profiles of refractive index for hollow-core Bragg fibers.

In cylindrical waveguides, modes can be labeled by their ‘angular momentum’ integer m ; the ( , , )z t ϕ dependence of the modes is given by ( )j z t mfe β ω− + . In the hollow-core fiber with cobweb-structured cladding the modes will be affected by the supporting strip. Because the supporting strip is periodic in ϕ , the modes can be written as ( ) 2 /j z t m j n

ne eβ ω ϕ π ϕ φ− + ∑ , where

n is integer, φ is the periodicity of supporting strip in ϕ direction. The effective wavevector /k m rϕ = in the ϕ direction goes to zero for r →∞ . So the bandgap of this structure is the

same as “OmniGuide” Bragg fiber in Ref. [23] and purely depends on rk and β as long as

sw is small enough. For designing hollow-core Bragg fiber with cobweb-structured cladding, some important

structural parameters related to the permitted normalized frequency range of the 01TE mode, and their varying rule were analyzed by using a plane wave expansion method [27]. The


lowest-loss mode in Bragg fiber is 01TE mode. The simulated results for hollow-core Bragg

fiber with cobweb-structured cladding show that leakage losses of 02TE mode for the fibers

with 0.05η = , 2 0.25d mμ= , 2 1.49n = , 4N = and different core radii ( 10cor mμ= and

50 mμ ) at 0.65 mλ μ= are 65.4 10× and 38.7 10× times larger than those of 01TE mode,

respectively. Thus, the permitted frequency range of 01TE mode is of especially interest. The most commonly used material in POF is PMMA, its refractive index is 1.49. Using PMMA as the high-index material of the Bragg reflection layers, the first two TE modes in the Bragg reflection layers are calculated with the plane wave expansion method [33]. Figure 6 shows the mode index of the first two TE modes in the Bragg reflection layers. 01TE mode is the fundamental mode in the hollow core. Its mode index must be below 1 and approach to 1. The frequency range formed by two intersecting points ( P and Q ) of the two TE mode curves

and the air line ( 1effn = ) is approximately the permitted frequency range of 01TE mode in

hollow core. For 0.01η = and 0.05, 1n (air), 2 1.49n = (PMMA), we can see from Fig.6 that this kind of structure can guide light in the hollow core over a wide frequency range. Different η have a strong effect on the permitted normalized frequency range of the 01TE mode. For 0.01η = , normalized frequency can achieve the range from 2.91 to 45.76, while for 0.05η = , normalized frequency is within the range 1.34 to 9.5. The permitted normalized frequency range of 01TE mode shrinks more than 5 times as η changes from 0.01 to 0.05. In order to figure out the influence of the structural parameter η of Bragg reflection layers on the permitted normalized frequency range of 01TE mode, the permitted normalized frequency

range of 01TE mode with different η at a fixed 2n (1.49) was calculated. The results are listed in Table 1.

Figure 6. Permitted normalized frequency range of TE01 mode for Bragg fiber with η =0.01, 0.05. The two curves indicate the first two TE modes in the Bragg reflection layers, and the solid line is the air line ( n =1) [27].


Table 1. The permitted normalized frequency range of 01TE mode vs different η at a

fixed 2n (1.49) [27]

η 0.5 0.4 0.3 0.2 0.1 0.08 0.06 0.05 0.04 0.02 0.01

Q point value 1.36 1.59 1.96 2.72 4.98 6.11 8.00 9.5 11.76 23.07 45.76

P point value 0.57 0.60 0.65 0.75 0.99 1.09 1.24 1.34 1.49 2.07 2.91

/λΛ range ( ~ )Q P 0.79 0.99 1.31 1.97 3.99 5.02 6.76 8.16 10.27 21.00 42.85

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

η

n2=1.49

upper limit of d2′

lower limit of d2′

Figure 7. Range of normalized high index layer thickness ( 2 2d d λ′ = ) vs. η [27]

Figure 8. Permitted normalized frequency range of TE01 mode vs. 1d [27]

In regard to the range of allowed values of 2d , we define 2 2 /d d λ′ = as the normalized high-index-layer thickness, where λ is the operating wavelength of the fiber. The upper and lower limit of 2d ′ can be obtained by means of the Q and P point values for each η in Table 1. Take 0.05η = as an example, the upper limit ( Q point value) of the permitted normalized frequency range of 01TE mode is 9.5, which means 1 2( ) / 9.5d d λ+ = .

Substituting 1 2 / 0.05d d= into it, we can obtain 2 2 / 0.4524d d λ′ = = . The relationship

between 2d ′ and η is shown in Fig.7. From Fig.7, we can see that the values of 2d ′ for the


upper line are approximately 0.45, indicating that the maximum thickness of 2d cannot go

beyond 0.45λ . In general, 2d takes 0.4 ~ 0.3λ λ . The minimum thickness of 2d ′ decreases when decreasing η .

Figure 9. Permitted normalized frequency range of TE01 mode as a function of 2n for η = 0.05 [27].

In regard to the relationship between the 1d and permitted normalized frequency range of

01TE mode, the permitted normalized frequency range of 01TE mode with 2 0.25d mμ= and

2 1.49n = at different 1d is illustrated in Fig.8. One obvious feature of Fig.8 is that the

permitted normalized frequency ranges of 01TE mode and the corresponding thickness 1d of

air layer are approximately a linear relationship. Thus, so long as the thickness 1d of air layer

increases at a fixed 2d , the normalized frequency range broadens.

In regard to the relationship between 2n and permitted normalized frequency range of

01TE mode, a series of 2n ranging from 1.45 to 5.8 at a fixed η (0.05) are calculated, as

shown in Fig.9. The permitted normalized frequency range of 01TE mode increases when 2n decreases. Most of polymers have the refractive index smaller than 1.8. Therefore, they are advantageous as the materials of the fiber with a large transmission frequency range.

In regard to the tolerance of the parameters, we take a dielectric material PMMA as an example. The design objective is a hollow-core fiber to use as optical fiber communication in the wavelength range from 0.65 mμ to 1.65 mμ . The design parameters are 0.05η = ,

2 0.25d mμ= and 2 1.49n = . Its normalized frequency / λΛ is in the range from

5.25/0.65=8.1 to 5.25/1.65=3.2, all within the permitted normalized frequency range of 01TE

mode (9.5-1.34) as shown in Fig.6(b). If 2d has a error of 2 20%d ± in the production

process, this corresponds to 2 0.2d mμ= and 0.3 mμ . For 2 0.2d mμ= , the normalized frequency range is from 5.2/0.65=8 to 5.2/1.65=3.15. This is within the permitted normalized frequency range of 01TE mode (11.76-1.49) as shown in Table 1 for 0.04η = . For

2 0.3d mμ= , the normalized frequency range is from 5.3/0.65=8.15 to 5.3/1.65=3.21. This is


almost within the permitted normalized frequency range of 01TE mode (8.00-1.24) as shown

in Table 1 for 0.06η = . If 1d has a error of 1 25%d ± , this corresponds to 1 3.75d mμ= and

6.25 mμ . For 1 3.75d mμ= , the normalized frequency range is from 4/0.65=6.15 to

4/1.65=2.42. This is within the permitted normalized frequency range of 01TE mode (7.21-1.18) for 0.067η = . For 1 6.25d mμ= , the normalized frequency range is from 6.5/0.65=10

to 6.5/1.65=3.94. This is within the permitted normalized frequency range of 01TE mode (11.76-1.49) for 0.04η = . Finally, polymers are considered to have different refractive indices for the same material, due to different molecular weight or polymerization condition. If the index of PMMA has a variation of 2 0.02n ± , which corresponds to 2 1.47n = and 1.51,

then the permitted normalized frequency range of 01TE mode are (9.74-1.38) and (9.27-1.31), respectively. They are essentially consistent with the normalized frequency range (9.5-1.34) as originally designed for 2 1.49n = and 0.05η = .

The confinement loss and transmission loss for hollow-core Bragg fiber with cobweb-structured cladding were modelled by using an asymptotic formalism [34]. Many results show that the fibers with only 3-4 alternating layers in cladding can achieve the low confinement loss and transmission loss, and the confinement and transmission losses decrease with increasing the hollow-core radius ( cor ). In order to achieve both low loss and wide

wavelength range, fiber design should adopt smaller 2d value and lager 1d value, besides

increasing cor and N .

5. Functional Exploiting of Hollow-Core Bragg Fiber with Cobweb - Structured Cladding

With the appealing properties described above, the possibility of using hollow-core Bragg fiber with cobweb-structured cladding for transmitting the information and delivering the laser energy was analyzed.

5.1. Fibers for Use in Optical Communications from Visible to near Infrared Region

Today, the capacity of optical fiber communications has expanded gigabits per second into terabits per second, enough to meet the current traffic demand due to the explosive growth of data transfer and internet services. Large-capacity and long-distance optical fiber communication trunk line has been installed in many countries. The next big step will be extending the network from fiber-to-the-curb into every building and home.

In the area of fiber to the home (FTTH) or fiber to the premises (FTTP) application, passive optical networks (PON), especially ethernet passive optical networks (EPON) and gigabit ethernet passive optical networks (GEPON), are generally preferred for home fiber connections. Usually, the transmission bandwidth and transmission distance required for the


networks are 100MHz - 10GHz and 100 -10m km , respectively. Therefore, the fibers with lower loss, higher bandwidth and cheaper cost are in demand. People have been trying to find materials and methods to meet those requirements and POF is one of the major approaches being explored in addition to silica glass single-mode fiber and multi-mode hard plastic clad fiber (HPCF).

The simulated results for hollow-core Bragg fibers with cobweb-structured cladding had proved that depending on the modal-filtering effect, they may realize the transmission of

01TE single-mode or a few modes, thus achieving the transmission of higher bandwidth (GHz ) [35].

Figure 10. Absorption loss spectrum of PMMA [36].

A fiber design for use in optical communication from visible to near infrared region is presented. The fiber parameters are 2 1.49n = (PMMA), 1 1n = , 2 0.25d mμ= , 1 5d mμ= ,

75cor mμ= and 3N = . According to absorption loss spectrum of PMMA [36] as shown in Fig.10, we calculate the transmission losses of the fiber. The absorption losses of PMMA at the wavelengths of 0.65 , 0.85 , 1.3 and 1.55 mμ are about 100 /dB km , 32.5 10 /dB km× ,

42.5 10 /dB km× and 47.8 10 /dB km× , respectively. The transmission losses of 01TE mode at

these wavelengths are 43.9 10 /dB km−× , 34.3 10 /dB km−× , 0.13 /dB km and 0.80 /dB km , respectively. The results show that after inevitable factors (material purity, imperfection and nonuniformity of fiber structure and existence of supporting strips) being considered, the transmission losses of the fiber are still very low.

Thus, by using an inexpensive material (PMMA), it allows the fibers to meet the needs of the transmission distance and bandwidth for EPON and GEPON, and to realize the wavelength division multiplexing (WDM).


5.2. Fibers for Use in THz Waveguiding

The THz radiation, whose frequency range is about 0.1 10THz− , has important applications in spectroscopy, imaging, space science and information transmission. To date, progress in THz wave generation and detection techniques has been enormous. However, most of the present THz systems rely on free space propagation due to the absence of low loss waveguides and transparent materials in the THz region. The waveguides constructed with some metals suitable for microwave guides or some dielectrics (such as silica) suitable to optical waveguiding have very high losses for THz wave. Even if for high-resistivity silicon, the most common material for the passive devices in the THz technology, its absorption coefficient is of the order of 10.04cm− . In recent years, THz waveguides have been fabricated from some dielectrics (such as sapphire, plastics) except from metals such as Cu, brass, and stainless steel. The loss coefficients of high-index core (solid-core) photonic crystal fibers using high-density polyethylene (HDPE) [37] and polytetrafluoroethylene (Teflon) [38] are less than 10.5cm− ( 0.1 3THz− ) and approximately 10.12cm− , respectively. Hollow polycarbonate waveguides with inner Cu coatings for broadband THz transmission have been reported [39]. The lowest loss 3.9 /mdB ( 10.00898cm− ) was obtained from a 3mm core diameter fiber at 158.51 mμ wavelength. Recently, a simple subwavelength-diameter ( 200 mμ ) plastic (polyethylene) wire, similar to an optical fiber for guiding a THz wave has

been reported as well [40]. Its attenuation constant is reduced to less than 10.01cm− in the frequency range near 0.3THz .

A fiber design for use in THz waveguiding is presented. The structural parameters of fibers (A, B, C) are as follows: 9cor mm= , 2 1.52n = , 1 1n = , 2 25d mμ= , 1 500d mμ= and

3N = (fiber A); 12cor mm= , 2 1.52n = , 1 1n = , 2 70d mμ= , 1 1050d mμ= and 3N = (fiber

B); 16cor mm= , 2 1.52n = , 1 1n = , 2 150d mμ= , 1 2250d mμ= and 3N = (fiber C). The host material was assumed to be lossless with a refractive index of 1.52 (corresponding to HDPE material). The confinement loss as a function of wavelength for 01TE , 02TE , 01TM and

02TM modes is shown in Fig.11. The lowest-loss mode is 01TE mode. The confinement loss

of the 01TE mode at the least-loss wavelength is 81.13 10 /dB km−× at 83.5 mμ (fiber A), 72.15 10 /dB km−× at 233 mμ (fiber B), and 75.05 10 /dB km−× at 500 mμ (fiber C).

Figure 11. Confinement loss as a function of wavelength for 01 02 01, , ,TE TE TM and 02TM modes.


Figure 12. Transmission loss as a function of wavelength for 01 02 01, , ,TE TE TM and 02TM modes.

Then, we attempt to take the calculation of losses further by including the material absorption. Based on the absorption spectra of HDPE in wavelength range 50 1200m mμ μ− [41], the transmission losses of three hollow-core fibers (A, B, C) with cobweb cladding are calculated. The calculated results are shown in Fig.12. The data in Fig.12 show that the transmission losses of 01TE mode for fiber A in the wavelength range of

26 05 0m mμ μ− are below 5.5 /dB km . The lowest loss is 0.63 /dB km (corresponding to loss coefficient 6 11.45 10 cm− −× ) at 90 mμ . The transmission losses of 01TE mode for fiber B in the wavelength range of 2 50 4 00 m mμ μ− are below 5.0 /dB km . The lowest loss is 2.0 /dB km at 270 mμ . The transmission losses of 01TE mode for fiber C in the wavelength range of 14 020 00m mμ μ− are below 5.6 /dB km . The lowest loss is 2.09 /dB km at 560 mμ .

The above transmission losses were taken into account only the absorption spectra of the material (HDPE). In fact, certain spectral region in the THz waves may not be available for signal transmission due to the strong absorption of water present in the constituent materials and air-core for the polymer fibers [42]. Therefore, while using hollow-core polymer Bragg fiber with cobweb-structured cladding in transmitting light through air-core, it is very important to eliminate the water from the constituent material and avoid moist air in the environment during fabrication and storage.

5.3. Fibers for Infrared (IR) Applications

IR optical fibers may be defined as fiber optics transmitting wavelengths greater than approximately 2 mμ . IR fibers can be useful for the medical, industrial, civil, and military arenas. For example, they are used in surgical applications by transmitting 2CO laser radiation (10.6 )mμ and :Er YAG laser radiation (2.94 )mμ . When used as fiber sensors, IR fibers are generally used either to transmit blackbody radiation for temperature measurements or as an active or passive link for chemical sensing, achieving non-contact temperature monitoring and remote spectroscopic chemical sensing. The application in the industrial arena includes welding and cutting. Scanning near-field microscopy by using high-quality single-mode and multimode IR fiber-tapered tips can obtain 20-nm topographic resolution and about 200-nm optical resolution for a variety of samples. IR fibers are also used for military applications including anti-aircraft missile defense. The development of infrared fiber optics


began in the year 1960. The first IR fibers were fabricated in the mid-1960’s using arsenic-sulphur glasses [43]. So far, there are four classes of infrared fibers:

(i) fluoride, germanate, tellurite or chalcogenide glass based solid-core fibers; (ii) crystalline silver halide solid-core fibers; (iii) hollow-core fibers in which inner wall coatings have high reflectivity; and (iv) solid-core photonic crystal fibers and hollow-core photonic bandgap fibers. The optical-loss values of the sulfide based chalcogenide glass fibers at the Naval

Research Laboratory have been reduced to only 0.1 to 0.2 /dB m in fiber lengths of about 500m by using improved chemical purification and better fiber fabrication techniques [44]. The optical losses of crystalline silver halide solid-core fibers by an extrusion process have been reduced to lower than 50 /dB km in a broad IR region from 9 to 14 mμ and lower than 1 /mdB in the region from 3 to 20 mμ [45]. The losses of rectangular hollow waveguides with 1- m -long and 1 1mm mm× cross-section by first depositing thin-film coatings of 2PbF on phosphor bronze strips and then soldering four of these phosphor bronze metal strips together are as low as 0.1 /dB m at 10.6 mμ [46]. Photonic crystal fibers for the middle infrared were fabricated by multiple extrusions of silver halide crystalline materials [47]. These fibers are composed of two solid materials: the core consists of pure AgBr (n=2.16) and the cladding includes AgCl (n=1.98) fiberoptic elements arranged in two concentric hexagonal rings around the core. IR transmissive As-S glass and As-Se glass triangular photonic band gap fiber structures were theoretically modeled [48]. From numerical simulations, Pottage et al. [49] discovered a new type of air-line bandgap that is of considerable importance in the design of practical hollow-core photonic bandgap fibers made from high-index glass (n≥2.0) for guidance in the mid/far-IR. A silica based hollow-core photonic bandgap fiber in which fiber-core diameter is 40 mμ (nineteen capillaries were omitted from the centre of the stack to form the core), the overall outside diameter is 150 mμ and the nearest-neighbor hole spacing is around 7 mμ , has been fabricated [50]. The peak of the bandgap is at 3.14 mμ with a typical attenuation of 2.6 /dB m . By further optimization of the structure, modeling suggests that a loss below 1 /dB m should be achievable.

The design is a hollow-core Bragg fiber with cobweb-structured cladding for the mid-IR region. In the wavelength region between 100 mμ and 1 mμ , many longitudinal and rotational resonances of molecules are present in almost all substances, especially the long-chain polymers [2]. Polymers such as teflon and polyethylene show relatively strong absorption at

11000cm− (10 mμ ). The absorption coefficient α at 10 mμ wavelength is about 1100cm− for

teflon and about 150cm− for polyethylene [16]. A fiber design for use in infrared is presented. The structural parameters of fibers (A, B) are as follows: 1500cor mμ= , 2 1.4d mμ= ,

1 30d mμ= , 1 1n = , 2 1.37n = (teflon) and 3N = (fiber A); 1200cor mμ= , 2 2.8d mμ= ,

1 28d mμ= , 1 1n = , 2 1.55n = (PES) and 3N = (fiber B). The absorption coefficient of the

host material (teflon) is 1100cm− (corresponding to absorption loss 74.343 10 /dB km× ). The calculated results are shown in Fig.13(a). The data in Fig.13(a) show that the transmission


loss of 01TE mode for fiber A in the wavelength range of 2.8 mμ to 10.6 mμ are below

39.5 /dB km . The lowest loss is 1.47 /dB km at 3.9 mμ . The absorption loss of the host

material (PES) is 73 10 /dB km× . The calculated results are shown in Fig.13(b). The data in Fig. 13(b) show that the transmission loss of 01TE mode for fiber B in the wavelength range

of 8 mμ to 13 mμ are below 30.9 /dB km . The loss for the 10.6 mμ wavelength of 2CO laser is 18.9 /dB km .

Figure 13. Transmission losses of the mid-IR region for fibers (A, B).

The numerical results show that despite the strong absorption of the polymers in the mid-IR region, the transmission losses of the fibers are lower by comparison with those of other IR fibers reported in the literature. And the polymer fibers have an advantage over other fibers in flexibility.

5.4. Circular-Polarization-Maintaining Single-Mode Fibers

Standard single-mode fibers support two degenerate, orthogonally polarized modes ( 11HE mode). Random imperfections in the fiber structure and external forces on the fibers can create asymmetries that break the polarization degeneracy, resulting in polarization mode dispersion and polarization fading in interferometers. Conventional polarization-maintaining fibers (highly birefringent fibers) and some single-polarization single-mode photonic crystal fibers supported a linear polarization mode. The fibers require accurate alignment of the birefringence axes of the two fibers when coupling, splicing and some sensing applications are considered. Therefore, in the year 1980, Jeunhomme and Monerie [51] have suggested the design of a circular-polarization-maintaining single-mode fiber cable . Recently, Argyros et al. [25] have presented the design that supports a single-polarization, circularly symmetric nondegenerate mode in an air-core ring-structured Bragg fiber.

We presented the design that supports a circular-polarization-maintaining single mode in a hollow-core and cobweb-structured cladding Bragg fiber. The structural parameters of the fiber are 10cor mμ= , 1 1n = , 2 1.585n = (PC), 2 0.21d mμ= , 1 2.1d mμ= and 3N = . The intrinsic losses of the host material (PC) are 166 /dB km at 650 656nm− and 224 /dB km at 764nm [52]. The calculated results show that the transmission losses of 01TE mode (lowest


loss mode) are 0.226 /dB km at 650 656nm− and 0.170 /dB km at 764nm , those of 02TE mode (second-lowest loss mode) are 3.513 /dB km at 650 656nm− and 1.848 /dB km at 764nm . The ratio of the loss of the 02TE mode to the loss of the 01TE mode is 15.54 ( 650 656nm− ) and 10.87 ( 764nm ). In accordance with the research reported in Ref.32, the fiber is single moded for lengths between 11.4km and 88.5km ( 650 656nm− ), and 21.7km and 117.6km ( 764nm ).

We expect that this type of hollow-core Bragg fibers with circular-polarization-maintaining single-mode and low-losses will find many applications, such as gyroscopes, current sensors and coherent communication systems.

6. Applications of Hollow-Core Bragg Fiber with Cobweb-Structured Cladding

A new generation of POFs has the advantages of both low-cost and high-performance in terms of attenuation, bandwidth and flexibility. It will find many applications in diverse areas and increases market acceptance.

As respects information transmissions, the new generation of POFs can guide the light of visible to terahertz radiation, and can be applied to optical fiber communications and optical fiber sensing, such as LANs, specially FTTH, THz wave fiber communications. It can also be used as an active or passive links for chemical sensing and remote spectroscopic chemical sensing, a variety of physical quantity sensing as well as medical diagnostics including noninvasive blood glucose monitoring and detection of tumors. As respects delivery of power laser radiation and solar radiation, hollow-core Bragg fibers with cobweb-structured cladding can deliver solar radiation into darkroom, be used for indoor illumination, replacing former guided light tube or solid-core polymer fiber. Efficient hollow-core guiding for delivery of power laser radiation (10.6 mμ 2CO laser, 2.94 mμ :Er YAG laser, etc) can be achieved by replacing the traditional PMMA with heat-resistant polymers, and can be used for medical therapy and processing including micro-processing and material processing.

By using gas-filled hollow-core Bragg fibers with cobweb-structured cladding, it is possible to obtain the EUV light generated through the process of high-harmonic up-conversion of femtosecond laser and ultrahigh efficiency laser wavelength conversion by pure stimulated rotational Raman scattering, as well as to use laser light to levitate and guide particles through the hollow-core fiber, etc.

Circular-polarization-maintaining single-mode low-loss fibers and high-strength, flexibility and resistance to shock fibers will provide the possibilities for some new applications. These fibers will stimulate further progress, both in fiber and allied systems technologies.

The new generation of POFs based on hollow-core Bragg fiber with cobweb-structured cladding will find many applications and is irreplaceable for some applications such as THz wave low-loss transmission.


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Chapter 10

DISSIPATIVE SOLITONS IN OPTICAL FIBER SYSTEMS

Mário F.S. Ferreira and Sofia C.V. LatasDepartment of Physics, University of Aveiro, 3800-193 Aveiro, Portugal

Abstract

We introduce the concept of dissipative solitons, which emerge as a result of a doublebalance: between nonlinearity and dispersion and also between gain and loss. Such dissipativesolitons have many unique properties which differ from those of their conservativecounterparts and which make them similar to living things. We focus our discussion ondissipative solitons in optical fiber systems, which can be described by the cubic-quinticcomplex Ginzburg-Landau equation (CGLE). The conditions to have stable solutions of theCGLE are discussed using the perturbation theory. Several exact analytical solutions, namelyin the form of fixed-amplitude and arbitrary-amplitude solitons, are presented. The numericalsolutions of the quintic CGLE include plain pulses, flat-top pulses, and composite pulses,among others. The interaction between plain and composite pulses is analyzed using a two-dimensional phase space. Stable bound states of both plain and composite pulses are foundwhen the phase difference between them is 2/π± . The possibility of constructingmultisoliton solutions is also demonstrated.

1. Introduction

Solitary waves have been the subject of intense theoretical and experimental studies in manydifferent fields, including hydrodynamics, nonlinear optics, plasma physics, and biology [1]-[5]. In fact, the history of solitons dates back to 1834, the year in which James Scott Russellobserved that a heap of water in a canal propagated undistorted over several kilometres [6].However, the term “soliton” was coined only in 1965, to reflect the particle-like nature ofsolitary waves that remain intact even after mutual collisions [7]. Such waves correspond tolocalized solutions of integrable equations such as the Korteveg de Vries and nonlinearSchrödinger equations. In these circumstances, solitons were usually attributed only tointegrable systems. However, the concept of soliton was subsequently broaden to include alsothe localized solutions of non-integrable systems.

Mário F.S. Ferreira and Sofia C.V. Latas280

Concerning the field of nonlinear optics, one can distinguish between temporal andspatial solitons [8]. Spatial optical solitons are beams of light in which nonlinearitycounteracts diffraction, leading to a robust structure which propagates without change ofform. Such structures will play a major role in the future in the field of all-optical processingand logic. Temporal solitons, on the other hand, represent shape invariant (or breathing)pulses, formed by a balance between nonlinearity and dispersion. It is believed that temporalsolitons will play a major role in future all-optical high-capacity transmission systems [9][10].

Until now, the main emphasis has been given to the well-known conservative solitonsystems, where only the diffraction or dispersion needs to be balanced by the nonlinearity.However, a new field has emerged in the last few years concerning the formation of solitonsin systems far from equilibrium [11]. These solitons are termed dissipative solitons or auto-solitons and they emerge as a result of a double balance: between nonlinearity and dispersionand also between gain and loss. Such dissipative solitons have many unique properties whichdiffer from those of their conservative counterparts. For example, except for very few cases[5], they form zero-parameter families and their properties are completely determined by theexternal parameters of the optical system. They can exist indefinitely in time, as long as theseparameters stay constant. However, they cease to exist when the source of energy or matter isswitched off, or if the parameters of the system move outside the range which provides theirexistence.

Even if it is a stationary object, a dissipative soliton shows non-trivial energy flows withthe environment and between different parts of the pulse. Hence the dissipative soliton is anobject which is far from equilibrium and which presents characteristics similar to a livingthing. In fact, we can consider animal species in nature as elaborate forms of dissipativesolitons. An animal is a localized and persistent “structure” which has material and energyinputs and outputs and complicated internal dynamics. Moreover, it exists only for a certainrange of parameters (pressure, temperature, humidity, etc.) and dies if the supply of energy isswitched off. The same analogy can be applied to individual organs within an animal, sinceeach maintains its shape and function over time.

Many non-equilibrium phenomena, such as convection instabilities, binary fluidconvection and phase transitions, can be described by the complex Ginzburg-Landau equation(CGLE) [12]-[14]. In the field of nonlinear optics, the CGLE can describe various systems,namely optical parametric oscillators, free-electron laser oscillators, spatial and temporalsoliton lasers, and all-optical transmission lines [9][15]-[27]. In these systems there aredispersive elements, linear and nonlinear gain, as well as losses. In some cases, the CGLEadmits a multiplicity of solutions for the same range of system parameters. This reality againresembles the world of biology, where the number of species existing in the sameenvironment is trully impressive.

In this chapter we will discuss the cubic-quintic CGLE and the characteristics of some ofits solutions. In Section 2 we present the CGLE and in Section 3 the perturbation approach tosolve this equation is discussed. Some analytical and numerical solutions of the CGLE arepresented in Sections 4 and 5, respectively. Finally, Section 6 summarizes the mainconclusions.

Dissipative Solitons in Optical Fiber Systems 281

2. The Complex Ginzburg-Landau Equation

In one of the forms used in nonlinear optics, the cubic-quintic complex Ginzburg-Landauequation (CGLE) can be written as [5][19]-[27]:

qqqqiqqqiT

qiqiqqT

qDZqi 442

2

22

2

2

2νμε

∂∂βδ

∂∂

∂∂

−+++=++ (1)

where Z is the propagation distance or the normalized number of round trips, T is the retardedtime, q is the normalized envelope of the electric field, β stands for spectral filtering ( β >0),δ is the linear gain or loss coefficient, ε accounts for nonlinear gain-absorption processes(for example, two-photon absorption), μ represents a higher order correction to the nonlineargain-absorption, and ν is a higher order correction term to the nonlinear refractive index.The parameter D is the group velocity dispersion coefficient, with 1±=D , depending onwhether the group velocity dispersion (GVD) is anomalous or normal, respectively.

The CGLE is rather general, as it includes dispersive and nonlinear effects, in bothconservative and dissipative forms. It is known in many branches of physics, including fluiddynamics, nonlinear optics and laser physics.

Equation (1) becomes the standard nonlinear Schrödinger equation (NLSE) when theright-hand side is set to zero. When this does not happen, Eq. (1) is non-integrable, and onlyparticular exact solutions can be obtained. In the case of the cubic CGLE ( 0==νμ ), exactsolutions can be obtained using a special ansatz [28], Horota bilinear method [29] orreduction to systems of linear PDEs [30]. Concerning the quintic CGLE, the existence ofsoliton-like solutions in the case 0>ε has been demonstrated both analytically andnumerically [5][20][26][31]. Exact solutions of the quintic CGLE, including solitons, sinks,fronts and sources, were obtained in [32], using Painlevé analysis and symbolic computations.

It must be noted that Eq. (1) can not be used as it stands to describe the behaviour offemtosecond optical pulses. For such ultrashort pulses, some higher-order nonlinear anddispersive effects must be taken into account, which results in additional terms to be added tothe right-hand side of Eq. (1) [33]-[38].

3. Results from the Soliton Perturbation Theory

Assuming that D=+1 and that all the other coefficients in the right-hand side of Eq. (1) aresmall, we can use the adiabatic soliton perturbation theory [9][34][39][40] to evaluate thedynamical evolution of the soliton parameters the amplitude η and the frequency κ , withwhich the one soliton solution is given by:

[ ] [ ]⎭⎬⎫

⎩⎨⎧ −−+−−+= σκηκθκηη iZZZiTZiZTZhZZTq 22 )()(

2)(exp)()(sec)(),( (2)


Applying the perturbation procedure, we get the following set of ordinary-differentialequations:

5322

1516

34

3122 μηεηκηβηδηη

++⎟⎠⎞

⎜⎝⎛ +−=

dZd

(3)

κβηκ 2

34

−=dZd

(4)

As can be seen from Eq. (4), the soliton frequency approaches asymptotically to 0=κ(stable fixed point) if 0≠η . The stable fixed points for the soliton amplitude, on the otherhand, are given by minimums of the potential function φ defined by:

ηφη

dd

dZd

−= (5)

Considering the Eq. (3), we have the following expression for the potential function:

( ) 642

4582

61)( μηηεβδηηφ −−+−= (6)

For the zero-amplitude state to be stable, the potential function must have a minimum at0=η , in addition to a minimum at 0≠= sηη . These objectives can be achieved if the

following conditions are verified [20]:

0<δ , 0<μ , 2/βε > , 4815 sμηδ > (7)

We can verify from the above conditions that the inclusion of the quintic term in Eq. (1)is necessary to have the double minimum potential.

The stationary value for the soliton amplitude can be obtained from Eq. (6) and is givenby:

μδμεεεε

η8

5/24)(5)(5 22 −−−−−= ss

s (8)

where 2/βε =s for small values of β . However, the result given by Eq. (8) can be

generalized for arbitrary values of β using sε given by [20][26][41]:


2

2

921413

2 βββε

+−+

=s (9)

From Eq. (8) it can be seen that a stationary amplitude 1=sη occurs when thecoefficients satisfy the relation:

08)2(515 =+−+ μβεδ (10)

The discriminant in Eq. (8) must be greater than or equal to zero for the solution to exist.For given values of β , μ , and ε , the allowed values of δ to guarantee a stable pulse

propagation must satisfy the condition 0min ≤≤ δδ , where

( )μεεδ

245 2

mins−

= (11)

When 0=δ , the peak amplitude is found to achieve a maximum value:

( )μεεη s−

−=45

max (12)

For 0=μ and sεε = the peak amplitude becomes arbitrary.

On the other hand, for given values of β , μ , and δ , the minimum value of allowedεbecomes

5/24min δμεε += s (13)

Considering the last condition in Eq. (7) or, alternatively, from Eq.s (8) and (13) we findthat there is a minimum value for the peak amplitude, given by:

4min 815μδη = (14)

Fig. 1 shows the potential function given by Eq. (6) when the relation (10) is satisfied for3.0=β , 5.0=ε , 25.0−=μ (curve a), 34375.0−=μ (curve b) and 5.0−=μ (curve

c). Curves a and b present a minimum at 1=η and 0=η since they satisfy the conditions(7), corresponding to negative values of the linear gain ( 05.0−=δ and 1.0−=δ ,respectively). However, curve c has no minimum at 0=η , since the corresponding value ofδ is positive ( 033.0=δ ).


Figure 1. Potential φ versus soliton amplitude when the relation (10) is satisfied for 3.0=β ,

5.0=ε , 0=ν , 5.0−=μ (curve a), 34375.0−=μ (curve b) and 25.0−=μ (curve c).

Figure 2. Phase portrait of Eq.s (3) and (4) corresponding (A) to curve c and (B) to curve b of Figure 1.


Fig. 2 illustrates the stability characteristics of the stationary solutions using the phase-plane formalism. Fig. 2a corresponds to curve c in Fig. 1, and we observe that, in this case,soliton propagation can be affected by background instability due to the amplification ofsmall-amplitude waves. The steady-state solution shows a limited basin of attraction. Forexample, initial conditions with 7.0=iη and 1±=iκ evolve toward the trivial solution

0=sη of Eq.s (3) and (4). For these initial conditions, the nonlinearity is not sufficientlystrong to balance dispersion, and the pulse disperses away. The dashed curves in Fig. 2a giveapproximate limits between different basins of attraction. From a perturbation analysis of Eq.s(3) and (4) around 0=η , one can show that these curves cross the 0=η axis at

33.0±=cκ . Thus, waves weak initial amplitudes grow up to 1=sη if 33.0<iκ . In this

case, soliton propagation can be severely affected by the background instability. Fig. 2bcorresponds to curve b in Fig. 1, and we can see that, in this case, the background instabilityis avoided, since the small-amplitude waves are attenuated, irrespective of their frequencyκ .Besides the stable stationary point at 1=sη , we note, in this case, the existence of anotherstationary point at 5.0≈sη , which is unstable.

This simple approach shows that, in general, the CGLE has stationary soliton-likesolutions, and that, for the same set of equation parameters, there may be two of themsimultaneously (one stable and one unstable). Moreover, this approach shows that solitonparameters are fixed.

4. Exact Analytical Solutions

Several types of exact analytical solutions of the CGLE have been obtained considering aparticular ansatz [5][26]. However, due to restrictions imposed by the ansatz, these solutionsdo not cover the whole range of parameters. In the following, we will assume a stationarysolution of Eq. (1) in the form:

[ ] ZiTaidTaZTq ω−= )(lnexp)(),( (15)

where a(T) is a real function and d, ω are real constants.

4.1. Solutions of the Cubic CGLE

The cubic CGLE is given by Eq. (1) with 0==νμ . Inserting Eq. (15) in this equation weobtain the following solution for a(T):

a T A h BT( ) sec ( )= (16)

where


222

32

)2( BddBA β+−

= (17)

ββδ

−+=

ddB 2 (18)

and d is given by

)2(2)2(8)21(9)21(3 22

βεβεεβεβ

−−++±+

=d (19)

On the other hand, we have

( )( )2

2

241

dddd

βββδω

+−+−

−= (20)

The solution (16)-(18) is known as the solution of Pereira and Stenflo [28]. Although theamplitude profile of the solution (16)-(18) is an hyperbolic secant as in the case of the NLSEsolitons, two important differences exist between the CGLE and the NLSE solitons. First, forCGLE pulses the amplitude and width are independently fixed by the parameters of (1),whereas for NLSE solitons A=B. The second difference is that the CGLE solitons are chirped.

The solution given by Eq.s (16)-(18) has a singularity at d d− + =β β 2 0 , which takes

place on the line )(βε s in the plane ( , )β ε defined by Eq. (9). For a given value of β , the

denominator in the expression for B in Eq. (18) is positive for sεε < and negative for

sεε > . Hence, for solution (16)-(18) to exist, the excess linear gain δ must be positive for

sεε < and negative for sεε > . In the last case, both numerical simulations and the solitonperturbation theory show that the soliton is unstable relatively to any small amplitudefluctuations [20][26]. On the other hand, for 0>δ and sεε < the solution (16)-(18) isstable, since after any small perturbation it approaches the stationary state. However, thebackground state is unstable in this case, since the positive excess gain also amplifies thelinear waves coexistent with the soliton trains. The general conclusion is that either thesoliton itself or the background state is unstable at any point in the plane ),( εβ , whichmeans that the total solution is always unstable.

The stationary value of the pulse width 1/B can be significantly reduced by a convenientchoice of the system parameters [42]. In fact, it can be verified from Eq.s (11) and (12) that,for a given value of the filter strength β , as the nonlinear gain coefficient approaches the

value sε given by Eq. (9), the amplitude A increases to infinity and its width 1/B tends to


zero. This singularity can be used in soliton lasers to vary the pulse parameters by a smallvariation of the material parameters.

If β and ε satisfy the Eq. (9) and δ = 0, a solution of the cubic CGLE with arbitraryamplitude exists, given by [5][26]:

)(sec)( DThCTa = (21)

where C is an arbitrary positive parameter and C/D is given by:

( ) ( )( )14132

141419222

222

−+

−+++=

βββββ

DC

(22)

We have also

d =+ −1 4 1

2

2ββ

(23)

22

241 Ddββω +

−=

Figure 3. Simultaneous propagation of four arbitrary-amplitude solitons with with amplitudes 2, 1.5, 1,and 0.5, for 0=δ , 2.0=β 0== μν and sεε = .


It can be verified that the cubic CGLE becomes invariant under the scale transformation

Dqq → , DTT → , ZDZ 2→ when δ = 0 . This is the reason for the existence of thearbitrary-amplitude solitons. On the ther hand, we can see that the limiting value of theamplitude-width product A/B for the fixed-amplitude solitons coincide with the value C/D onthe line (22) [20]. This shows that arbitrary amplitude solitons can be considered as a limitingcase of fixed amplitude solitons when δ → 0 . However, the arbitrary amplitude solitons havestability properties different from those for fixed amplitude solitons. In fact, arbitraryamplitude solitons are stable pulses, which propagate in a stable background because δ = 0 .This feature is illustrated in Fig. 3, which shows the simultaneous propagation of four stablesolitons with amplitudes 2, 1.5, 1, and 0.5, for 0=δ , 2.0=β and sεε = .

4.2. Solutions of the Quintic CGLE

Considering the quintic CGLE and inserting Eq. (15) in Eq. (1), the following generalsolution can be obtained for 2af = [5][26]:

( )TffffffffTf

212121

21

2cosh)()(2)(

α−−+= (24)

where

223 dd ββμα−−

= (25)

and d is given by Eq. (19). The parameters 1f and 1f are the roots of the equation:

0)41(3

)2(238

222

22 =

+−−

+−

++− dd

fd

fdd ββ

δβεβ

βν

(26)

and the coefficients are connected by the relation:

012

316222

2412 22

=⎥⎦

⎤⎢⎣

⎡+

−−−

+⎥⎦

⎤⎢⎣

⎡−

−−+ dd

βεβεβμβ

βεβεεβν (27)

One of the roots of Eq. (26) must be positive for the solution (24) to exist, while the othercan have either sign.

When the two roots are both positive, the general solution given by Eq. (24) becomeswider and flatter as they approach each other. These flat-top solitons correspond to stablepulses, whereas the solution (24) is generally unstable for arbitray choice of parameters. If

21 ff = , the width of the flat-top soliton tends to infinity and the soliton splits into two


fronts. The formation and stable propagation of a flat-top soliton will be demonstratednumerically in Section 5.

If β and ε satisfy the Eq. (9) and δ = 0, a solution of the quintic CGLE with arbitraryamplitude exists, given by:

[ ] ( )( ) ( )TPS

PdTaTf2cosh2

413)()(2

2

+−+

==εβ

β(28)

where P is an arbitrary positive parameter and

( ) ( ) Pdd

dS 2

2222

234192βββμεβ

−−+

+−= (29)

ββ

2141 2 −+

=d

(30)

Pdββω

241 2+

−=

When 0→μ , the solution (28) transforms to the arbitrary-amplitude solution of thecubic CGLE, given by Eq. (21)-(22).

5. Numerical Solutions

Due to restrictions imposed by the ansatz, the analytic solutions of the quintic CGLEpresented above do not cover the whole range of parameters and almost all of them areunstable. To find stable solutions in other regions of the parameters, different approximatemethods [41], a variational approach [43]-[45], or numerical techniques must be used.

As shown by the perturbative analysis presented in Section 3, the parameter space wherestable solitons exist has certain limitations. We must have 0>β in order to stabilize thesoliton in frequency domain. The linear gain coefficient δ must be zero or negative in orderto avoid the background instability. The parameter μ must be negative in order to stabilizethe soliton against collapse. Concerning the parameter ν , it can be positive or negative.

Stable solitons can be found numerically from the propagation equation (1) taking as theinitial condition a pulse of somewhat arbitrary profile. In fact, such profile appears to be oflittle importance. For example, Fig. 4 illustrates the formation of a fixed amplitude soliton ofthe cubic CGLE starting from an initial pulse with a rectangular profile. It must be noted that,in this case, the linear gain is positive but relatively small ( 003.0=δ ) and the solitonpropagation remains stable within the displayed distance.


In general, if the result of the numerical calculation converges to a stationary solution, itcan be considered as a stable one, and the chosen set of parameters can be deemed to belongto the class of those which permit the existence of solitons. In the following we show someexamples of stable soliton solutions found with this method.

Figure 4. Formation of a fixed-amplitude soliton solution of the cubic CGLE starting from an initialpulse with a rectangular profile of amplitude Ao = 0.7 (a) and Ao = 1.0 (b), when 003.0−=δ ,

2.0=β , and 09.0=ε .

Figure 5. (a) Evolution of the peak amplitude and (b) the final pulse profile when 01.0−=δ ,150.=β , 20.=ε , 0=ν , 13750.−=μ (dashed curves) or 40.=ε , 38750.−=μ (full

curves), considering an input pulse )T(hsec)T,(q =0 .

Fig. 5 shows (a) the evolution of the peak amplitude and (b) the final pulse profileobtained numerically from Eq. (1), assuming an input pulse with a sech profile andconsidering the following parameter values: 01.0−=δ , 150.=β , 0=ν , 20.=ε ,

13750.−=μ (dashed curves) or 40.=ε , 38750.−=μ (full curves). When inserted in

Eq. (8), these values provide a stationary amplitude 1=sη . This prediction of the


perturbation theory, as well as the stability of the stationary solution are confirmed by thenumerical results of Fig. 5.

For small values of the parameters in the right-hand side of Eq. (1) the stable solitonsolutions of the CGLE have a sech profile, similar to the soliton solutions of the NLSE, andcorrespond to the so-called plain pulses (PPs). However, rather different pulse profiles can beobtained for non small values of those parameters. As an example, Fig. 6 illustrates theformation and stable propagation of a flat-top soliton, starting from an initial pulse with asech profile. The following parameter values were considered: 1.0−=δ , 5.0=β ,

66.0=ε , 01.0−==νμ .

Figure 6. Formation and evolution of a flat-top soliton, considering an input pulse)T(hsec)T,(q =0 , for 1.0−=δ , 5.0=β , 66.0=ε , 01.0−==νμ .

Fig. 7 shows (a) the amplitude profiles and (b) the spectra of a plain pulse, as well as oftwo composite pulses (CPs). The following parameter values were considered: 01.0−=δ ,

5.0=β , 03.0−=μ , 0=ν , 5.1=ε (plain pulse), 02.=ε (narrow composite pulse)and 52.=ε (wide composite pulse). Fig. 7c illustrates the formation and propagation of thewide composite pulse starting from the plain pulse solution represented in a) and b). Acomposite pulse exhibits a dual-frequency but symmetric spectrum (Fig. 6b) and can beconsidered as a bound state of a plain pulse and two fronts attached to it from both sides [5].The “hill” between the two fronts should be counted as a source, because it follows from thephase profile that energy flows from the centre to the CP wings.

If one of the fronts of a CP is missing one has a moving soliton (MS) [5]. The MS alwaysmoves with a velocity smaller than the velocity of the front for the same set of parameters. Infact, the front tends to move with its own velocity but the soliton tends to be stationary, due tothe spectral filtering. The resulting velocity of the MS is determined by competition betweenthese two processes.

Increasing slightly the nonlinear gain coefficient and keeping the values of the otherparameters equal to those used in Fig. 7 the stationary wide composite pulse shown in Fig. 7cis lost and a non stationary expanding structure appear, as illustrated in Fig. 8.


Figure 7. (a) Amplitude profiles and (b) spectra of a plain pulse and of two composite pulses when01.0−=δ , 5.0=β , 03.0−=μ , 0=ν , 5.1=ε (plain pulse), 02.=ε (narrow composite

pulse) and 52.=ε (wide composite pulse). Figure 7c illustrates the formation and propagation of thewide composite pulse, starting from the plain pulse solution.

Figure 8. Nonstationary expanding structure obtained from an initial plain pulse when 01.0−=δ ,5.0=β , 03.0−=μ , 0=ν and 183.2=ε .


Pulsating and exploding soliton solutions of the CGLE were also observed recently [46].Pulsating solitons correspond to fixed solutions in the same way as the stationary pulses andcan be found when the parameters of the CGLE are far enough from the NLSE limit. On theother hand, exploding solitons appear for a wide range of parameters of the CGLE andoriginate from soliton solutions which remain stationary only for a limited period of time.Following the explosion, there is a “cooling” period, after which the solution becomes“stationary” again. This is a periodic phenomenon, like other phenomena occurring in thenature.

It can be verified that different stable stationary solutions of the quintic CGLE can existsimultaneously for the same set of parameters [5][19]. This can be understood consideringthat solitons, fronts and sources are elementary building units which can be combined to formmore complicated structures. In more complex systems, the number of solutions may be veryhigh. This reality again resembles the world of biology, where the number of species is trullyimpressive.

6. Soliton Bound States

After finding the conditions for the existence of stable solitary-pulse solutions of the CGLEequation, the next natural step is to consider their interactions and, in particular, thepossibility of the existence of bound states of these pulses [19][25][47]-[52]. In fact, theproblem of soliton interaction is crucial for the transmission of information. In the case ofHamiltonian systems, the interaction between the pulses is inelastic. Energy exchangebetween the pulses is one of the mechanisms that makes the two-soliton solutions of thesesystems unstable, even when such stationary solutions do exist. The situation is ratherdifferent for dissipative systems. In this case, all solutions are a result of a double balance:between nonlinearity and dispersion and also between gain and loss. Moreover, the propertiesof dissipative solitons are completely determined by the external parameters of the opticalsystem.

For given values of the CGLE parameters, the amplitude and width of its solitonsolutions are fixed. As a consequence, during the interaction of two solitons, basically onlytwo parameters may change: their separation r and the phase difference, φ , between them.These two parameters provide a two-dimensional plane in which we may analyze of pulseinteraction, namely the formation of bound states, their stability and their global dynamics[19][25][51][52]. This reduction in the number of degrees of freedom is a unique feature ofsystems with gain and loss. In the case of Hamiltonian systems, the amplitudes of the solitonscan also change, which can affect the stability of the possible bound states.

In order to analyze numerically the soliton interaction in the 2-D space provided by theseparation, r, and phase difference, φ , between the two solitons, Eq. (1) can be solved withan initial condition

)exp()2/()2/()( 00 φirTqrTqTq ++−= (31)


where 0q is the stationary solution obtained numerically from Eq. (1) when the values of itsparameters are specified. Initial condition (31) with arbitrary values for r and φ will result ina trajectory on the interaction plane. Bound states will be singular points of this plane.

Figure 9. Trajectories on the interaction plane showing the evolution of two plain pulses for01.0−=δ , 5.0=β , 5.1=ε , 0=ν , and 03.0−=μ . We have X )cos(φr= and

Y )sin(φr= .

Fig. 9 shows an example of a numerical simulation of an interaction between the twosolitons on the interaction plane, considering the following parameter values: 01.0−=δ ,

5.0=β , 5.1=ε , 0=ν , 03.0−=μ . This figure indicates that, for the given set of

parameters, there are at least four singular points. The points 3P and 4P are saddles andcorrespond to unstable bound states. In these states, the phase difference between the solitonsis zero or π . In addition, there are two symmetrically located stable foci (points 1P and 2P ),which correspond to stable bound states of two solitons with a phase difference 2/πφ ±=between them. The stationary pulse separation in these bound states is 62.1≈r .

As a consequence of its asymmetric phase profile, the two-soliton solution correspondingto the stable bound states 1P and 2P in Fig. 9 moves with a constant velocity. The directionof motion depends on the sign of φ . An example of stable propagation of a two-solitonbound state with a phase difference of 2/π between the pulses is given in Fig. 10.


Stable bound states of two CPs, with a phase difference 2/πφ ±= between them, canalso be observed. This is illustrated in Fig. 11, which shows the stable propagation of a boundstate of two composites pulses with a phase difference 2/π . The following parameter valueswere assumed: 01.0−=δ , 5.0=β , 0.2=ε , 0=ν , 3.0−=μ . In contrast with thebehaviour of the plain pulse bound state shown in Fig. 10, the CP bound state moves at thegroup velocity.

Figure 10. Propagation of a bound state of two plain pulses with a phase difference of 2/π betweenthem.

Figure 11. Propagation of a bound state of two composite pulses with a phase difference of 2/πbetween them.


The two-soliton solution can be assumed as the building block to construct various multi-soliton solutions. An example is given in Fig. 12, corresponding to a four-plain pulsesolution, with a phase difference of 2/π between adjacent pulses. As observed in the case ofthe two-PP solution, multisoliton solutions formed by plain pulses move with a constantvelocity along the T axis.

Figure 12. Four-plain pulse solution with a phase difference of 2/π between adjacent pulses. Thedash-dotted (full) lines in (b) correspond to the initial (final) phase profiles.

Figure 13. Five-plain pulse solution and the correspondent phase profiles. The dash-dotted (full) lines in(b) correspond to the initial (final) phase profiles of the pulses in (a).

Multisoliton solutions formed by plain pulses with zero velocity can be obtained bychoosing appropriately its phase profile. Fig. 13a illustrates the evolution of a five-solitonsolution whose initial phase profile is given by the dash-dotted line in Fig. 13b. This phaseprofile evolves during the propagation, and achieves a final profile given by the full curve inFig. 13b. In spite of some oscillations, this multisoliton bound state remains relatively stable


and propagates with zero velocity. The final phase profile shown in Fig. 13b correspondsindeed to a stationary stable solution. From Fig. 13 we can infer that a zero velocity multi-soliton solution formed by plain pulses must present a symmetric and concave phase profile,such that the temporal displacement of half of the structure is balanced by the oppositedisplacement of the other half. These solutions can be the basic building blocks for morecomplicated structures.

7. Conclusion

The concept of dissipative solitons was explained in this chapter. In fact, this concept is wide-ranging and provides a new paradigm for the investigation of phenomena involving stablestructures in nonlinear systems far from equilibrium. Here, we have considered the particularcase of nonlinear optical fiber systems with gain and loss, which can be described by thecubic-quintic complex Ginzburg-Landau equation (CGLE). These include spatial andtemporal soliton lasers, parametric amplifiers and optical transmission lines. However, themodel can also be applied in other fields of physics.

The conditions to have stable solutions of the CGLE were discussed using theperturbation theory. Several exact analytical solutions, namely in the form of fixed-amplitudeand arbitrary-amplitude solitons, were presented. The numerical solutions of the quinticCGLE include plain pulses, flat-top pulses, and composite pulses, among others. We used thetwo-dimensional phase space (distance-phase difference) to analyze the dynamics of the twosoliton system. We have found stable bound states of both plain pulses and composite pulseswhen the phase difference between them is 2/π± . Two-composite pulses bound states havezero velocity, which is in contrast with the behaviour of the bound states formed by planepulses. As a consequence of the existence of two-soliton bound states, three-soliton and othermultisoliton bound states also exist. In particular, we have shown the possibility ofconstructing stable bound states of multiple plain pulses with zero velocity by choosingappropriately the phase profile of the whole solution.

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Chapter 11

BRIGHT - DARK AND DOUBLE - HUMPED PULSESIN AVERAGED, DISPERSION MANAGED OPTICAL

FIBER SYSTEMS

K.W. Chow† and K. Nakkeeran‡

†Department of Mechanical EngineeringUniversity of Hong Kong, Pokfulam, Hong Kong

‡School of Engineering, Fraser Noble Building, King’s collegeUniversity of Aberdeen, Aberdeen AB24 3UE, UK

Abstract

The envelope of the axial electric field in a dispersion managed (DM) fiber sys-tem is governed by a nonlinear Schrodinger model. The group velocity dispersion(GVD) varies periodically and thus realizes both the anomalous and normal dispersionregimes. Kerr nonlinearity is assumed and a loss / gain mechanism will be incorpo-rated. Due to the big changes in the GVD parameter, the correspondingly large vari-ation in the quadratic phase chirp of the DM soliton will be identified. An averagingprocedure is applied. In many DM systems, an amplifier at the end of the dispersionmap will compensate for the energy dissipated in that map. Here the case of gain notexactly compensating the loss is considered, in other words, a small residual ampli-fication / attenuation is permitted. The present model differs from other similar oneson variable coefficient NLS, as the inhomogeneous features involve both time and thespatial coordinate. The goal here is to extend the model further, by permitting coupledmodes or additional degree of freedom. More precisely, the coupling of fiber loss andinitial chirping is considered for a birefringent fiber. The corresponding dynamics isgoverned by variable coefficient, coupled NLS equations for the components of theorthogonal polarization of the pulse envelopes. When the self phase and cross phasemodulation coefficients are identical for special angles, several new classes of wavepatterns can be found. New stationary wave patterns which possess multiple peakswithin each period are found, similar to those found for the classical Manakov model.For situations where the self- and cross-phase modulation coefficients are different,symbiotic solitary pulses are studied. A pair of bright-dark pulses exists, where eitheror both pulse(s) cannot propagate in that waveguide if coupling is absent.

302 K.W. Chow and K. Nakkeeran

1. Introduction

Transmission of information (voice, video, and data) over distances (short, moderate, long,and ultra-long) is a common requirement in the past, present and future. Carrier communi-cation of information using the electromagnetic waves is the best technology for high-speedtransmission. Out of different frequency bands in the electro-magnetic wave spectrum, op-tical regime has various advantages. Optical fibers are commonly used in optical commu-nication for channelling the light pulses for digital transmission. Both linear and nonlinearoptical effects in fibers play vital roles in determining the dynamics of pulse propagation.The field of nonlinear optics has blossomed and is undergoing a new revolution in recentyears. The nonlinear optical response is now a key element for new emerging technologies.This is particularly true for soliton and other types of nonlinear pulse transmission in opti-cal fibers/nonlinear materials, since this form of light propagation can be used to realize thelong-held dream of very high capacity dispersion-free communications. In the recent past,it has been proved beyond doubt that solitons do exist not only in optics but also in manyother areas of science. Solitons that exist in optics called “optical solitons” have been draw-ing a greater attention among the scientific community, as they seem to be right candidatesfor transferring information across the world through optical fibers.

Nonlinear pulse propagation in a long-distance, high speed optical fiber transmissionsystem can be described by the (perturbed) nonlinear Schrodinger equation (NLSE). NLSEincludes linear dynamics due to group velocity dispersion of the pulse, and nonlinear mech-anism due to the Kerr effect [1]. Much research efforts on the development of such a systemhave been made with the intention to overcome or control these effects [2, 3]. In this di-rection, recent numerical studies [4–6] and experiments [7] have shown that a periodicdispersion compensation seems to be the most effective way for improving the optical trans-mission system. The main purpose of dispersion management is to reduce several effects,such as radiation due to lumped amplifiers compensating the fiber loss [8,9], resonant four-wave mixing [10,11], modulational instability [12], jitters caused by the collisions betweensignals [13], and the Gordon-Haus effect resulting from the interaction with noise [14], alsoto decide a desired average value for the dispersion [12].

Basically, dispersion-management technique utilizes a transmission line with a periodicdispersion map, such that each period consists of two types of fiber, generally with differentlengths and opposite group-velocity dispersion (GVD) [4]. Lakoba has proved the non-integrability of the system equation governing the pulse propagation in dispersion-managed(DM) fibers [15]. As analytical solution for DM solitons is not available, researchers haveso far utilized the Lagrangian method to study the dynamics of DM solitons [4]. Veryrecently we have developed a complete collective variable theory for DM solitons whicheffectively includes the residual field due to soliton dressing and radiation [16]. Manyworks have reported on fitting a Hermite-Gaussian ansatz function for the oscillating tailsof the numerical stationary solution (fixed point) of the DM solitons [4, 17–19]. Fromnumerical studies [5, 6] of DM fiber line, the pulse is deformed from the ideal soliton, hasa chirp and requires an enhanced power for the average dispersion. Meanwhile Kumar andHasegawa [20] have obtained a new nonlinear pulse (quasi-soliton) by programming thedispersion profile such that the wave equation has a combination of the usual quadraticpotential and the linear potential.

Bright - Dark and Double - Humped Pulses... 303

The envelope of the axial electric field in a DM fiber system is governed by a NLSmodel. The GVD varies periodically and thus realizes both the anomalous and normaldispersion regimes. Kerr nonlinearity is assumed and a loss / gain mechanism will beincorporated. Due to the big changes in the GVD parameter, the correspondingly largevariation in the quadratic phase chirp of the DM soliton will be identified. An averagingprocedure is applied [21].

In many DM systems, an amplifier at the end of the dispersion map will compensate forthe energy dissipated in that map. Here the case of gain not exactly compensating the lossis considered, in other words, a small residual amplification / attenuation is permitted.

The present model differs from other similar ones on variable coefficient NLS [22], asthe inhomogeneous features involve both time and the spatial coordinate. The goal here isto extend the model further, by permitting coupled modes or additional degree of freedom.More precisely, the coupling of fiber loss and initial chirping is considered for a birefrin-gent fiber [23]. The corresponding dynamics is governed by variable coefficient, coupledNLS equations for the components of the orthogonal polarization of the pulse envelopes.When the self phase and cross phase modulation coefficients are identical for special angles,several new classes of wave patterns can be found.

The first result will be a stationary wave pattern which possesses multiple peaks withineach period, similar to those found for the classical Manakov model [24].

Another new result is the family of symbiotic solitary pulses, and this novel finding isapplicable to configuration where the self phase and cross phase modulation coefficients aredifferent. Indeed the constraints imposed on these coefficients extend or generalize resultsobtained earlier in the literature. As a simple example, a pair of bright - dark pulses existswhere each individual wave guide separately will only admit bright solitons. This couplingnonlinearity is truly remarkable. As a second example, bright or dark solitons are allowedto propagate in waveguides which would otherwise prohibit their existence.

2. Double-Hump Bright - Dark Periodic and Solitary Pulses

We consider the averaged, dispersion management system for coupled waveguides:

i∂A

∂z+

∂2A

∂t2+ (AA∗ + σBB∗)A + iβA + β2t2A = 0, (1)

i∂B

∂z+

∂2B

∂t2+ (σAA∗ + BB∗)B + iβB + β2t2B = 0. (2)

A, B are the complex envelopes of the axial electric fields, z is the distance and t is theretarded time. The quantity β measures the quadratic phase chirp, and the residual gain orloss is specifically selected to match this parameter. The parameter σ represents cross phasemodulation coefficient arising from the coupling. The derivation of system (1, 2) from thefirst principle of averaging over a dispersion map can be found in our earlier work [21].

System (1, 2) can be solved exactly by several techniques, but we shall focus on the Hi-rota bilinear method. As the description has been given in our earlier work, the presentationhere will be brief. The quadratic phase factor or chirp is first isolated as


A = exp

(iβt2

2

)ϕ, B = exp

(iβt2

2

)ψ. (3)

The reduced governing equations for the auxiliary variables ϕ and ψ will be free ofquadratic terms in time. To search for the special modes of optical pulses, we furtherconstrain the wave pattern to be expressed as

ϕ =g exp(−iΩ1)

f, ψ =

G exp(−iΩ2)f

. (4)

G, g and f are dependent variables for the bilinear operation with the restriction that f isreal. Typically they are combinations of exponential functions for solitary pulses but ellipticfunctions for periodic patterns. The phase factors Ω1, Ω2 are functions of the distance (z)only. They will have their derivatives determined in the bilinear equations, and hence theythemselves are readily recovered by quadrature.

The resulting bilinear equations are then

f

[(D2

t + 2iβtDt +∂Ω1

∂z+ 2iβ

)g · f

]+ g(−D2

t f · f + gg∗ + σGG∗) = 0, (5)

f

[(D2

t + 2iβtDt +∂Ω2

∂z+ 2iβ

)G · f

]+ G(−D2

t f · f + σgg∗ + GG∗) = 0. (6)

They are solved by using rather straightforward differentiation formulas developed fromfirst principles. D is the bilinear operator, with its definition and properties described morefully in Appendix A.. For periodic wave patterns, Hirota derivatives of theta functions canbe simplified by identities involving products of theta functions (Appendix B.).

As illustrative examples, the simplest periodic wave pattern will be given by the choice,

g = A0θ1(t[h1(z)]), G = B0θ3(t[h1(z)]), f = θ4(t[h1(z)]). (7)

The theta functions are Fourier series with exponentially decaying coefficients and theclassical Jacobi functions can be expressed as ratios of theta functions. The amplitudeparameters, A0, B0, isolated here for convenience will also be functions of z. The distancedependent wave number function h1(z) will render the period of the pattern to change withlocation, and the precise form is determined by forcing the odd Hirota derivatives to vanish.The loss / gain factor is not arbitrary as it has to match the precise forms of the functionsA0, B0. Theta functions will be convenient in the intermediate calculations. However, theJacobi elliptic functions are preferred in the final expressions, as they can be easily handledby most established routines in computer algebra.

A summary on existing results will be instructive:

1. When the cross phase modulation coefficient, σ, is arbitrary, the wave system willpermit periodic patterns in terms of a single elliptic function. The long wave limitwill, not surprisingly, yield solitary bright or dark pulses.


2. When σ is constrained to be unity, there are other varieties of solutions. In particular,one class of wave patterns can be expressed in terms of products of elliptic functions.The physical implication is that the intensity will display two, or perhaps more, peaksper period.

The goal of this section is to derive still further new wave patterns by choosing productsof elliptic functions as the starting point of these calculations, while still assuming the crossphase modulation coefficient, σ, to be one. The motivation comes from the choice of wavepatterns for the case of coupled nonlinear Schrodinger models with constant coefficients.Proceeding along the lines of reasoning just described will yield

A =√

6r√

1− k2

√1− 2c

√1− k2

[c− dn2(rte−2βz)√

1− k2

]exp

[iβt2

2− 2βz − iΩ1

], (8)

B =√

6 rk sn(rte−2βz)dn(rte−2βz)√1− 2c

√1− k2

exp

[iβt2

2− 2βz − iΩ2

]. (9)

Ω1 =r2 exp(−4βz)

4β

[6c2(1− k2)

1− 2c√

1− k2+

2√

1− k2

c

], (10)

Ω2 =r2 exp(−4βz)

4β

[6c2(1− k2)

1− 2c√

1− k2− 2(5− 4k2)

], (11)

r is a free parameter and represents the wave number at the initial location (z = 0). Thequantity c will be one of the roots of the quadratic equation

3c2 − 2c

[√1− k2 +

1√1− k2

]+ 1 = 0, (12)

k is the modulus of the elliptic function. Waveguide B will generally exhibit two peaks perperiod. Waveguide A will degenerate to a dark solitary pulse with multiple peaks in thelong wave period. Figures 1a, 1b illustrate this pattern.

3. A Generalized Model with Different Dispersion Coefficients

In many applications involving wave propagation along different channels or waveguides,the optical pulses will experience different measures of group velocity dispersion. Hencethe coefficients of the second derivative terms of the coupled NLS equations will generallybe different. Remarkably a special model system will still permit analytical progress, andwe shall consider pairs of bright - dark solitons in this model. Generally bright (dark) soli-tons occur for the conventional NLS model in the anomalous (normal) dispersion regimesrespectively. However, due to the special nonlinear effects in coupled NLS systems, thesebright / dark solitons can occur in the appropriate waveguide which are otherwise prohib-ited in the single mode NLS. They have sometimes been termed ‘symbiotic solitons’ in theliterature.

In optical physics, such waves have indeed been studied for configurations associatedwith conventional NLS with Kerr nonlinearity [25], intra-pulse stimulated Raman scattering


-10-5

05

10

1.0

0.5

0.00

0.5

1

1.5

2

t [Norm. Unit]z [Norm. Unit]

|A|2 [N

orm

. Uni

t]

(a)

-50

5

1.0

0.5

0.00

0.02

0.04

0.06

0.08

0.1

t [Norm. Unit]z [Norm. Unit]

|B|2 [N

orm

. Uni

t]

(b)

Figure 1. Evolution of the periodic solution (8) and (9) for the parameters β = 0.05, r = 1,k = 0.9 and c = 1.61.

[26], quasi-phase matched parametric oscillator [27], second harmonic generation [28], andthree-wave solitons [29]. In other systems, symbiotic solitons also occur in phenomenaconnected with Bose - Einstein condensates [30], discrete systems [31], multi-dimensionalNLS by separation of variables [32], and quadratic, nonlinear media with loss and gain [33].

More precisely, we shall consider

i∂A

∂z+ δ

∂2A

∂t2+ (AA∗ + σBB∗)A + iβA +

β2t2A

δ= 0, (13)

i∂B

∂z+

∂2B

∂t2+ (σAA∗ + BB∗)B + iβB + β2t2B = 0. (14)


Here A and B are again complex envelopes but the first waveguide is permitted to havea dispersion coefficient δ (positive or negative) relative to waveguide B. The chirp factors,however, must be modified to

A = exp

(iβt2

2δ

)ϕ, B = exp

(iβt2

2

)ψ.

A periodic pattern is obtained earlier in the literature as

A = rkQ1sn(rte−2βz) exp

[−2βz +

iβt2

2δ− ir2e−4βz

4β(Q2

1 + δ(1− k2))

], (15)

B = rQ2dn(rte−2βz) exp

[−2βz +

iβt2

2− ir2e−4βz

4β(Q2

2 − k2)

], (16)

Q1 =

√2(δ − σ)σ2 − 1

, Q2 =

√2(σδ − 1)σ2 − 1

. (17)

The restrictions are either

δ > σ if σ > 1, (18)

or

δ < σ if σ < 1, (19)

The long wave limit is

A = rQ1 tanh(rte−2βz) exp

[−2βz +

iβt2

2δ− ir2e−4βzQ2

1

4β

], (20)

B = rQ2sech(rte−2βz) exp

[−2βz +

iβt2

2− ir2e−4βz(Q2

2 − 1)4β

]. (21)

(20) represents a dark soliton, and propagates in the anomalous regime if δ is positive, while(21) remains a bright soliton in the anomalous dispersion regime.

The contribution in the present work is to document another set of periodic / solitarywave pattern which relieves or compensates the constraints imposed by (18) and (19). Fur-thermore, for some parameter regimes, one can achieve a pair of symbiotic solitons withbright (dark) solitons propagating in the normal (anomalous) dispersion regimes respec-tively. Following reasoning similar to the previous sections, we can attain another set ofwave profiles by exchanging the choice of elliptic functions in (15) and (16):

A = rR1dn[rt exp(−2βz)] exp

[−2βz +

iβt2

2δ− iΩ1

], (22)

B = rkR2sn[rt exp(−2βz)] exp

[−2βz +

iβt2

2− iΩ2

], (23)


where the parameters R1, R2 are

R1 =

√2(σ − δ)σ2 − 1

, R2 =

√2(1− σδ)σ2 − 1

, (24)

and the requirement of real square roots dictates that

δ >1σ

if σ < 1, (25)

or

δ <1σ

if σ > 1, (26)

(25) and (26) are different from (18) and (19). Ω1, Ω2 are angular frequency parametersgiven by

∂Ω1

∂z= −r2 exp(−4βz)

[δ(2− k2) +

2σ(1− σδ)σ2 − 1

], (27)

∂Ω2

∂z= −r2 exp(−4βz)

[1− k2 +

2σ(1− σδ)σ2 − 1

]. (28)

The long wave limit is even more instructive. On taking the limit k −→ 1, where in general(sn, cn, dn) become (tanh, sech, sech) respectively, one now has

A = rR1sech[rt exp(−2βz)] exp

[−2βz +

iβt2

2δ− iΩ10

], (29)

B = rR2 tanh[rt exp(−2βz)] exp

[−2βz +

iβt2

2− iΩ20

]. (30)

where Ω10, Ω10 are the long wave (k −→ 1) limits of Ω1 and Ω2 respectively.For σ < 1, both waveguides are in the anomalous dispersion regimes (as δ > 1). For

σ > 1, δ can be either positive or negative. In particular, negative values of δ here will implythat waveguide A is in normal dispersion regime. Remarkably, a bright (dark) soliton nowpropagates in the normal (anomalous) dispersion regime respectively. These phenomenaare quite contrary to the well known results.

4. Conclusions

A class of periodic and solitary waves has been studied for a system of coupled envelopeequations. This system can model averaged, dispersion managed systems where the residualgain / loss in each cycle of the dispersion map has been carefully chosen. Waves with mul-tiple peaks per period or symbiotic pairs of solitary pulses are obtained analytically. Theywill enhance our capability in modeling such systems and strengthen our understanding inthis and similar optical systems.


Several aspects still allow rooms for future work and expansions. In particular, config-urations where both waveguides are in the normal dispersion regime have not been workedout in details yet, although the same physics is expected to hold true qualitatively.

One issue which has not been addressed is the stability of these wave patterns. Numer-ical simulations of perturbed wave trains must be pursued. Recent works and experiencehave indicated that stability will probably still prevail in some parameter regimes. Theprecise elucidation will await further efforts.

Acknowledgement

Partial financial support has been provided by the Research Grants Council through the con-tract HKU 7123/05E. KWC and KN wish to thank The Royal Society for their support in theform of an International Joint Project Grant. KWC and KN are very grateful to Prof. JohnWatson for his valuable support for this research collaboration. KN also wishes to thank theNuffield Foundation for financial support through the Newly Appointed Lecturer Award.

A. Hirota Bilinear Operator

The Hirota operator for any two functions f and g is defined as [34, 35]

Dmx Dn

t g · f =(

∂

∂x− ∂

∂x′

)m (∂

∂t− ∂

∂t′

)n

g(x, t)f(x′, t′)∣∣∣∣x=x′,t=t′

, (31)

and the properties in association with differentiation of exponential functions are espe-cially striking (m, n are constants):

Dx[exp(imx)g · exp(inx)f ] = [Dxg · f + i(m− n)gf ] exp[i(m + n)x], (32)

D2x[exp(imx)g · exp(inx)f ] = [D2

xg · f + 2i(m− n)Dxg.f − (m− n)2gf ]× exp[i(m + n)x]. (33)

Most existing works on the Hirota operator focus on the case of constant wave number orfrequencies. The important point in this work is to extend Hirota derivatives to the case oftime or space dependent wavenumbers.

The bilinear identities for Hirota derivatives, even for the case of variable wave number,can be obtained from simple, straightforward differentiation. Examples are:

Dz exp[tξ1(z) + ξ2(z)] · exp[tη1(z) + η2(z)]= t[ξ′1(z)− η′1(z)] + ξ′2(z)− η′2(z) · expt[ξ1(z) + η1(z)] + ξ2(z) + η′2(z), (34)

Dz exp(a) ·m(z) exp(b) = m

[Dz exp(a) · exp(b)− 1

m

∂m

∂zexp(a + b)

]. (35)


B. Theta Functions

The theta functions θn(x), n = 1, 2, 3, 4 in terms of the parameter q (the nome) are definedby [36–38]:

θ1(x) = 2∞∑

n=0

(−1)nq(n+1/2)2 sin [(2n + 1)x] , (36)

θ2(x) = 2∞∑

n=0

q(n+1/2)2 cos [(2n + 1)x] , (37)

θ3(x) = 1 + 2∞∑

n=1

qn2cos (2nx) , (38)

θ4(x) = 1 + 2∞∑

n=1

(−1)nqn2cos (2nx) , 0 < q < 1. (39)

Basically they are Fourier series with exponentially decaying coefficients. Relationshipsbetween the theta and elliptic functions are:

sn(u) =θ3(0)θ1(z)θ2(0)θ4(z)

, cn(u) =θ4(0)θ2(z)θ2(0)θ4(z)

, dn(u) =θ4(0)θ3(z)θ3(0)θ4(z)

, (40)

z =u

θ23(0)

, k =θ22(0)

θ23(0)

. (41)

Arguments of the theta and elliptic functions are related by a scale factor. The modulus ofthe elliptic functions, k, is related to the theta constants by (41).

Theta functions possess a huge number of identities involving addition and subtractionof arguments:

θ3(x + y)θ3(x− y)θ22(0) = θ2

4(x)θ21(y) + θ2

3(x)θ22(y), (42)

θ4(x + y)θ4(x− y)θ22(0) = θ2

4(x)θ22(y) + θ2

3(x)θ21(y), (43)

Such identities can be proven by re-arranging terms of the multiple sums [37]. By con-sidering the leading and quadratic terms in the Taylor series of y in identities of the form(42-43), one obtains

θ′′4(0)θ4(0)

− θ′′3(0)θ3(0)

= θ42(0),

θ′′4(0)θ4(0)

− θ′′2(0)θ2(0)

= θ43(0),

θ′′3(0)θ3(0)

− θ′′2(0)θ2(0)

= θ44(0). (44)

D2xθ3(x) · θ3(x) =

2θ′′2(0)θ23(x)

θ2(0)+ 2θ2

3(0)θ24(0)θ2

4(x), (45)

D2xθ4(x) · θ4(x) = 2θ2

3(0)θ24(0)θ2

3(x) +2θ′′2(0)θ2

4(x)θ2(0)

. (46)

Hence formulas for Dxθm · θn, D2xθm · θn can be developed for m, n integers using this

line of reasoning.


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Chapter 12

DYNAMICS AND INTERACTIONS OF GAP

SOLITONS IN HOLLOW CORE PHOTONIC

CRYSTAL FIBERS

Javid Atai and D. Royston NeillSchool of Electrical and Information EngineeringThe University of Sydney, NSW 2006 Australia

Abstract

The existence and stability of gap solitons in a model of hollow core fiber in thezero dispersion regime are analyzed. The model is based on a recently introducedmodel where the coupling between the dispersionless core mode and nonlinear surfacemode (in the presence of the third order dispersion) results in a bandgap. It is foundthat similar to the anomalous and normal dispersion regimes, the family of solitons fillsup the entire bandgap. The family of gap solitons is found to be formally unstable butin a part of family the instability is very weak. Consequently, gap solitons belongingto that part of the family are virtually stable objects. The interactions and collisionsof in-phase and theπ-out-of-phase quiescent solitons and moving solitons in differentdispersion regimes are investigated and compared.

1. Introduction

Gap solitons (GSs) were originally introduced in Ref. [1]. Recent years have witnessed anupsurge of research activity on gap solitons in various areas of physics such as nonlinearoptics and Bose-Einstein condensation (BEC). In optics, a nonlinear dispersive mediumwhose spectrum contains one or more forbidden bands can support gap solitons. An ex-ample of such a system is a fiber Bragg grating (FBG). The periodic variation of lineardielectric constant in an FBG leads to a photonic band structure. The linear cross couplingbetween the counter-propagating waves results in a large effective dispersion (5 to 6 ordersof magnitude larger than the dispersion of standard optical fiber) [2, 3]. For sufficientlyhigh light intensities, the large Bragg grating induced dispersion may be counterbalancedby Kerr nonlinearity resulting in a gap soliton.

Significant theoretical [3–6] and experimental [7–10] efforts have been directed to-ward understanding and characterizing GSs in periodic media. In particular, it has been

316 Javid Atai and D. Royston Neill

shown that GSs in an FBG form a two-parameter family of solutions [4]. It has alsobeenshown that approximately half of the soliton family is stable against oscillatory pertur-bations [11–13]. Experimental activities in this area have focused on generating zero-velocity (quiescent) GSs due to their potential applications in optical buffers and storageelements. To date, GSs with a velocity of 0.23 of the speed of light in the fiber have beenobserved [14]. GSs have been studied in more sophisticated systems such as in the pres-ence of higher order dispersion [15], quadratic nonlinearity [16], cubic-quintic nonlinear-ity [18], dual core fibers with FBGs [17] and in waveguide arrays [19] and photonic crystalfibers [20].

Since their demonstration in 1996 [21], photonic crystal fibers (PCFs) have been thesubject of extensive research due their interesting and peculiar properties. PCFs are spe-cially designed optical fibers with many microstructured air holes running along the fiber’slength. They can be divided into two main categories depending on the mechanism of lightguidance, namely the solid core and hollow core PCFs. Solid core PCFs are similar toconventional optical fibers in that they guide light through total internal reflection. On theother hand, in hollow core PCFs (HC-PCFs) the microstructured cladding surrounding thehollow core creates a photonic bandgap that guides the light [22–24].

Introduction of atomic or molecular gases into the core of HC-PCF results in efficientnonlinear optical interactions due to strong confinement of light in the core region. Somerecent results include demonstrations of generation of stimulated Raman scattering (SRS)in hydrogen [25], and electromagnetically-induced transparency (EIT) [26, 27]. They havealso been utilized in delivery of high energy pulses [28–30] and in soliton lasers [31].

In Ref. [34] a model for pulse propagation in HC-PCF based on experimental [32] andnumerical [33] results was proposed. The model took into account the coupling of a lineardispersionless mode propagating in a gas-filled core with a nonlinear dispersive surfacemode propagating in silica. In Ref. [35] a simpler model was considered where the secondand third order group velocity dispersion terms were absent. The model contained a linearloss term which accounted for the power leakage from the core to the cladding. In bothmodels a bandgap opens in the system’s spectrum. The models in Ref. [34, 35] belong toa general class of models that give rise to wavenumber bandgap [36, 37]. A wavenumberbandgap arises as a result of the coupling between the co-propagating waves (in this casethe core and surface modes). On the other hand, a frequency bandgap (e.g. the above-mentioned bandgap structure in a FBG) is due to the coupling of counterpropgating waves.

The stability of GSs in the model of Ref. [34] has been investigated in both anoma-lous [38] and normal [39] dispersion regimes. It is shown that, strictly speaking, GSs inboth anomalous and normal dispersion regimes are unstable. However, due to the fact thatinstability is weak in a part of the soliton family, the GSs belonging to that part of the familyare “virtually” stable objects. In addition, an important finding reported in Ref. [39] is thatGSs in the normal dispersion are far more stable than their counterparts in the anomalousdispersion.

In this article, we will first investigate the existence and stability of gap solitons in aHC-PCF in the special case when the second order group velocity dispersion is negligible.The model, which is based on the model of Ref. [34], and the characteristics of the bandgapand soliton solutions will be discussed in Section 2.. In Section 3. stability of quiescent andmoving GSs will be presented and their stability will be compared with that of GSs in the

Dynamics and Interactions of Gap Solitons... 317

anomalous and normal dispersion regimes. In Section 4. we will investigate and comparetheinteractions of quiescent GSs and collision dynamics of moving solitons in the normal,anomalous and zero dispersions. In particular we will analyze the effect of initial phasedifference and separation on the outcome of collisions and interactions. The results aresummarized in Section 5..

2. The Model and Gap Soliton Solutions

The system of equations governing the propagation of the above-mentioned surface andcore modes in the zero GVD are based on the model introduced in Ref. [34]. In the normal-ized form it reads:

iuz − icuτ + iγuτττ + |u|2u + v = 0, (1)

ivz + icvτ + u = 0, (2)

whereu andv are the amplitudes of the surface and core waves, respectively,z andτ arethe propagation distance and reduced time andc represents the group velocity mismatchbetween the modes. The coefficients of Kerr nonlinearity and the linear coupling betweenthe modes have been scaled to unity. Therefore, there are two free parameters in the modelnamelyγ andc.

It should be noted that whenγ = 0 Eqs. (1) and (2) reduce to the model of Ref. [35].However, as was pointed out in Refs. [34,38,39], due to the small temporal width of solitonsand that the carrier wavelength may be close or exactly equal to zero GVD point, the thirdorder dispersion needs to be present. Undoing the rescalings and using a typical value of|β3| = 0.2 ps3/km the soliton’s width is found to be in the range of 100-300 fs. This valueof β3 corresponds to a normalized value ofγ ≈ 0.3. Also,∆z = 1 and∆τ = 1 correspondto ranges 1-10 cm and 30-100 fs in physical units.

In order to determine the linear spectrum of the system, we substitute(u, v) ∼exp(ikz− iωτ) into the linearized Eqs. (1) and (2). This results in the following dispersionrelation:

2k± = −ω3γ ±

√

(ω3γ + 2ωc)2 + 4. (3)

By definition we setc > 0 in which case the wavenumber bandgap exists forγ > 0.Straightforward analysis of Eq. (3) shows that the bandgap is−1 < k < +1 provided that

c2 ≥1

4. Thesolid curves in Fig. 1 represent the branches of the dispersion relation (3) for

c = 1 andγ = 0.3 with the bandgap being−1 < k < 1.In the gap, soliton solutions to (1) and (2) were sought in the form of

u(z, τ), v(z, τ) = U(τ), V (τ) exp(ikz). Substituting this ansatz into (1) and (2)results in a set of equations for complex functionsU(τ) andV (τ). These equations can besolved numerically using the relaxation method. It is found that, similar to the anomalousand normal dispersion regimes, the family of gap solitons completely fill the bandgap, and


−5 −2.5 0 2.5 5

ω

−10

−5

0

5

10

k

Figure 1. Dispersion diagram corresponding toc = 1 andγ = 0.3 for quiescent gap solitons(solid lines) and moving ones withδ = 0.7 (dashed lines). The bandgap for quiescentsolitons is−1 < k < 1. The bandgap for moving solitons is−0.77 < k < 0.78.

|U(τ)| and|V (τ)| are always single-humped. As is shown in Fig. 2, the real and imaginaryparts ofU(τ) andV (τ) are even and odd functions ofτ , respectively.

The GSs in the model of Eqs. (1) and (2), similar to their counterparts in the anomalousand normal dispersion regimes [34, 39], satisfy Vakhitov-Kolokolov criterion [40]. Thiscriterion states that a necessary condition for the stability of solitons against nonoscillatory

perturbations with purely real growth rates isdE

dk> 0 whereE is the energy of the soliton

family and is given by:

E(k) =

∫

+∞

−∞

(

|U(τ ; k|2 + |V (τ ; k|2)

dτ. (4)

However, the soliton family or part thereof may be unstable against oscillatory pertur-bations. A finding of Ref. [39] was that the energy of the soliton family in the normaldispersion regime is considerably lower than that of the anomalous dispersion regime. Asis shown in Fig. 3, the energy of solitons in (1) and (2) is less than that of the anomalouscase and greater than the normal dispersion case. Based on the results of Ref. [39] onemay conjecture that the solitons in the zero dispersion regime are more stable than theircounterparts in the anomalous dispersion and less so compared with the ones in the normaldispersion regime. This issue will be considered in the next section.

Moving solitons can be obtained by rewriting Eqs. (1) and (2) in the boosted referenceframe through the coordinate transform(z, τ) −→ (z, τ − δz) whereδ is the velocity shift.The dispersion relation of the transformed system of equations is given by:


−20 −10 0 10 20

τ

−2

−1

0

1

2

Im(U)

Re(U)

(a)

−20 −10 0 10 20

τ

−2

−1

0

1

2

3

4

Im(V)

Re(V)

(b)

Figure 2. The real and imaginary parts of theU(τ) andV (τ) for a quiescent gap solitonwith c = 1 andγ = 0.3 andk = 0.

2k± = −(ω3γ + 2ωδ) ±

√

(ω3γ + 2ωc)2 + 4. (5)

The bandgap defined by Eq. (5) varies withδ. The dashed curves in Fig. 1 displaythe branches of Eq. (5) forδ = 0.7. The bandgap for moving solitons exists in the range


− 0.9 − 0.6 − 0.3 0 0.3 0.6 0.9

k

0

80

160

240

320

Ener

gy

AnomalousNormalZero

Figure 3. The total energy of gap solitons withc = 1 andγ = 0.3 in the anomalousdispersion (see Ref. [34,38]), the normal dispersion (see Ref. [39]) and the zero dispersionregime (Eqs. (1) and (2)).

δmin < δ < c whereδmin is negative and can be obtained numerically. It is also found thatthe bandgap defined by (5) is completely filled with soliton solutions all of which satisfyVK criterion. In addition, similar to the case of quiescent solitons, the energy of movingsolitons in this model is found to be greater than that in the normal dispersion and less thanthat of moving GSs in the anomalous dispersion regime.

3. Stability of Solitons

In this section we investigate the stability of GSs in this model by means of direct numer-ical simulations and linear stability analysis. Evolution of GSs were simulated by numeri-cally solving Eqs. (1) and (2) using the symmetrized split-step Fourier method. Absorbingboundary conditions were implemented in order to attenuate any radiation that reaches theboundaries of computational window. To seed any inherent instability in the system, theGSs found by the above-mentioned relaxation algorithm were initially perturbed asymmet-rically and then propagated. It is found that, the GSs in the model of (1) and (2), like theircounterparts in the anomalous and normal dispersion regimes, are unstable against oscilla-tory perturbations. But, in a part of the GS family the instability is weak and as a resultsolitons may propagate for long distances before the instability is manifested. As a conse-quence, the GSs belonging to this part of the family can be considered as being “practically”stable

A key result of Ref. [39] was that GSs in the normal dispersion regime are significantlymore stable than their counterparts in the anomalous dispersion. Moreover, it was con-


jectured that the higher degree of stability of GSs in the normal dispersion regimewas, atleast in part, due to the fact that their total energy was considerably smaller than GSs in theanomalous dispersion. Based on this conjecture and since the total energy of GSs in Eqs.(1) and (2) is greater (smaller) than those in the normal (anomalous) dispersion (see Fig. 3),one expects the GSs in the zero dispersion to be more stable than those in the anomalous andless stable compared to the GSs in the normal dispersion regime. Our simulations corrobo-rate this prediction. A comparison between the propagation of GSs in different dispersionregime is provided in Fig. 4.

To quantify the degree of instability of GSs in this model, we have utilized a linear sta-bility analysis to calculate the instability growth rates for small perturbations. Substitutingthe following perturbed soliton solution

u (z, τ) , v (z, τ) = Uδ (τ) + f(τ) eσz, Vδ (τ) + g (τ) eσz ekz (6)

into the boosted equations (see Section 2.) and linearizing, we arrive at the following eigen-value problem:

Ay = σy (7)

whereUδ (τ) andVδ (τ) are the soliton solutions corresponding to velocityδ andf (τ) andg (τ) are the eigenmodes of the small perturbations andσ is the corresponding complexeigenvalue.y = [f, f?, g, g?]T , and

A =

−ik + 2i |Uδ|2 + Df iU2

δ i 0

iU?2δ ik − 2i |Uδ|

2 + Df? 0 −ii 0 −ik + Dg 00 −i 0 ik + Dg?

.

with Df = Df? = (c + δ) ddτ

− γ d3

dτ3 , andDg = Dg? = − (c − δ) ddτ

. In the aboveexpressions, asterisk represents complex conjugate.

The eigenvalue problem posed by Eq. (7) can be solved using standard numerical tech-niques. The results of the stability analysis are summarized in Fig. 5 as graphs ofRe(σ) vs.k for c = 1, γ = 0.3 andδ = 0, 0.25 and 0.5. Since the instability growth rate for all thecases is positive the GSs are formally unstable. However, one observes that the growth ratesfor a part of the family, particularly toward the lower edge of the bandgap, are very small. Inthis case, the instability will only be observable after a long propagation distance. Solitonsexhibiting this character are therefore “virtually stable” objects. We have adopted the defi-nition of Ref. [39] for virtual stability and quasi-stability. That is, for a soliton to be virtually

stable it must remain stable for at least300Znonlin(whereZnonlin ∼1

|Uδ|2) andfor it to be

quasi-stable it must remain stable for propagation distances50Znonlin < z < 300Znonlin.There are a number of noteworthy features in Fig. 5. Firstly, we note that the growth

rates in the zero dispersion regime are greater than those in the normal dispersion region (c.f.Fig. 5 in Ref. [39]) and smaller than those in the anomalous dispersion regime (c.f. Figs. 3and 4 in Ref. [38]). This is consistent with the results of the direct numerical simulationsshown in Fig. 4. Secondly, increasingδ gives rise to larger growth rates, particularly for


solitons near the upper edge of the bandgap. Nevertheless, varyingδ doesnot have anappreciable effect on the border of stable and quasi-stable regions. The weak dependenceof boundary of stable and unstable regions on the velocity of solitons has also been reportedfor GSs in a FBG [11,13].

− 60 − 40 − 20 0 20 40 60

τ

48

(a)

z

0− 60 − 40 − 20 0 20 40 60

τ

(b)

z

0

250

− 60 − 40 − 20 0 20 40 60

τ

1600

0

(c)

z

Figure 4. Examples of propagation of asymmetrically perturbed quiescent gapsoliton corre-sponding tok = −0.4, c = 1, andγ = 0.3 in (a) anomalous dispersion, (b) zero dispersionand (c) normal dispersion. In (c) the initial perturbation causes the soliton to acquire a smallvelocity. In this figure and all others below, only the u-component is shown.


− 0.8 − 0.4 0 0.4 0.8

k

0

0.1

0.2

0.3

0.4

Re(

σ)

Stable Quasi-Stable

(a)

− 0.8 − 0.4 0 0.4 0.8

k

0

0.1

0.2

0.3

0.4

0.5

Re(

σ)

Stable Quasi-Stable

(b)

− 0.8 − 0.4 0 0.4 0.8

k

0

0.2

0.4

0.6

0.8

1

Re(

σ)

Stable Quasi-Stable

Unstab

le

(c)

Figure 5. Instability growth rate of GSs in the model of Eqs. (1) and (2) withc = 1 andγ = 0.3 versusk for (a) quiescent gap solitons, (b) moving gap solitons withδ = 0.25 and(c) moving gap solitons withδ = 0.5. In the “Stable” region, the solitons propagate forlong distances i.e.z & 300Znonlin without any conspicuous instability development. In the“Quasi-Stable” region, instability occurs in the range50Znonlin < z < 300Znonlin.


4. Interactions and Collisions of Solitons

In view of nonintegrability of the model, the collision dynamics and interactions betweenthe solitons may be quite complex. In Refs. [38, 39], the collisions between in-phase GSsin the anomalous and normal dispersion regimes were considered. Moreover, in [39] theinteraction of in-phase andπ-out-of-phase quiescent GSs in the normal dispersion regimewas investigated and it was shown that in the case ofπ-out-of-phase solitons the outcomeof interaction depends onk and the initial separation of solitons.

In this section we will investigate the interaction of quiescent solitons in the anomalousand zero dispersion regimes. In addition, the collisions of in-phase andπ-out-of-phasemoving GSs in anomalous, normal and zero dispersion regimes will also be studied.

4.1. Interactions of Quiescent Solitons in Anomalous and Zero Dispersions

The interaction of GSs in zero and anomalous dispersion regimes was simulated by propa-gating two identical quiescent solitons belonging to the “Stable” regions with a time sepa-ration of∆τ and a phase difference of∆φ.

It is found that, irrespective of dispersion regime, when∆φ = 0 (i.e. GSs are in-phase),the solitons alwaysrepel each other. This behavior was reported for the GSs in the normaldispersion regime (see [39]) .

In the case of∆φ = π, the interaction of solitons becomes dependent on∆τ andk.These interactions can be divided into three types, denoted here as Types A, B, and C. Inthe Type A interactions the pulses initially attract each other and collide without mergingand then bounce back. An example of this type of interaction in the zero dispersion regime isshown in Fig. 6(a). In the Type B interactions the pulses attract and temporarily merge andform a “lump” which subsequently disintegrates into two separating solitons with differentamplitudes and velocities. Fig. 6(b) displays an example this type of interaction. In the TypeC interactions the pulses repel each other resulting in two separating pulses with differentamplitudes and velocities (Fig. 6(c)).

It should be noted that the velocity and amplitude of resulting solitons in the TypeC interaction as well as the interaction between in-phase solitons depend on the degree ofinitial overlap of solitons. If the solitons are initially weakly overlapping then the differencebetween the velocities and amplitudes of the eventual moving solitons will be small (see forexample Fig. 7 and Fig 8(b) in Ref. [39]). On the other hand, increasing the initial overlapbetween solitons (i.e. reducing∆τ ) leads to generation of solitons whose velocities andamplitudes differ considerably.

Fig. 7 displays the regions in the plane of (∆τ,k) where the types A, B and C in-teractions occur in zero and anomalous dispersion regimes. A noteworthy feature in Fig.7(a) is that in the zero dispersion the boundary between the types C and A is very weaklydependent on the initial separation of solitons.

4.2. Collisions of Moving Solitons

In Refs. [38, 39] it was shown that in the anomalous and normal dispersion regimes thecollisions between in-phase counterpropagating solitons belonging to the “Stable” region


− 60 − 40 − 20 0 20 40 60

τ

50

0

(a)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(b)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(c)

z

Figure 6. Examples of interaction of quiescent gap solitons in the zero dispersionregimewith c = 1, γ = 0.3, and∆φ = π. (a)k = −0.64, ∆τ = 10; (b) k = −0.64, ∆τ = 8; (c)k = −0.9, ∆τ = 8.

with δ = ±0.5 are always elastic. In particular, It was also found the relative collisioninduced loss of energy is≈ 0.1%.

In this section the effect of phase and velocity shift on the collisions will be considered.First, we consider the collisions between GSs with initial velocities±0.25. As shown inFig. 8, in-phase GSs in different dispersion regimes withδ = ±0.25 bounce off each otherelastically. On the other hand, as is displayed in Fig. 9, theπ-out-of-phase solitons withδ = ±0.25 collide and merge temporarily and form a “lump” which then quickly breaks


8 9 10 11 12

∆τ

− 0.9

− 0.8

− 0.7

− 0.6

− 0.5

k

A

A

B

C

(a)

8 9 10 11 12

∆τ

− 0.9

− 0.8

− 0.7

− 0.6

k

A

B

C

(b)

Figure 7. Regions of different types of interaction in the plane of (∆τ , k) for (a) the zerodispersion and (b) the anomalous dispersion.

up into two separating solitons. In addition, the collisions do not generate any noticeableradiation.

Figs. 10 and 11 show that the collisions of in-phase andπ-out-of-phase solitons withδ = ±0.5 in different dispersion regimes . In this case, regardless of the initial phase


− 60 − 40 − 20 0 20 40 60

τ

50

0

(a)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(b)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(c)

z

Figure 8. Examples of collisions of moving gap solitons withk = −0.6, c = 1, γ =0.3, δ = ±0.25 and∆φ = 0. (a) Anomalous dispersion; (b) zero dispersion; (c) normaldispersion.

difference, the solitons collide and form a lump which breaks up into two solitons whichtravel at almost the same velocity as the initial solitons. The effect of the initial phasedifference is that in the case of∆φ = π the emerging solitons have different velocity shiftscompared to those with∆φ = 0.


5. Conclusion

In this article, we have characterized the gap soliton solutions in a recently introduced modelin the absence of the second order dispersion. Similar to the anomalous and normal disper-sion regimes, the family of GSs in this case is found to be formally unstable but in a part ofthe family the instability is very weak and the solitons belonging to that part of the familyare therefore virtually stable. Interactions of quiescent solitons and collisions of moving

− 60 − 40 − 20 0 20 40 60

τ

50

0

(a)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(b)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(c)

z

Figure 9. Examples of collisions of moving gap solitons withk = −0.6, c = 1, γ = 0.3,δ = ±0.25 and ∆φ = π. (a) Anomalous dispersion; (b) zero dispersion; (c) normaldispersion.


− 60 − 40 − 20 0 20 40 60

τ

50

0

(a)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(b)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(c)

z

Figure 10. Examples of collisions of moving gap solitons withk = −0.6, c = 1, γ =0.3, δ = ±0.5 and∆φ = 0. (a) Anomalous dispersion; (b) zero dispersion; (c) normaldispersion.

solitons in zero, anomalous and normal dispersion regimes are analyzed. Depending on theinitial separation and the wavenumber, the solitons may either attract and bounce, attractand merge temporarily and break up into separating solitons, or repel each other. We alsofind that the outcome of the collisions of moving solitons depends on the initial phase andthe velocity shift. In all dispersion regimes, whenδ = 0.25, the in-phase solitons collideand bounce off each other elastically whereas theπ-out-of-phase solitons collide and forma lump which subsequently disintegrates into two separating solitons. On the other hand,


− 60 − 40 − 20 0 20 40 60

τ

50

0

(a)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(b)

z

− 60 − 40 − 20 0 20 40 60

τ

50

0

(c)

z

Figure 11. Examples of collisions of moving gap solitons withk = −0.6, c = 1, γ =0.3, δ = ±0.5 and∆φ = π. (a) Anomalous dispersion; (b) zero dispersion; (c) normaldispersion.

whenδ = 0.5, the solitons always collide elastically and two separating solitons emerge. Inthis case, the velocity shifts of the emerging solitons depend on the initial phase difference.


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[19] D. Mandelik, H.S. Eisenberg, Y. Silberberg, R. Morandotti and J.S. Aitchison, “Band-Gap Structure of Waveguide Arrays and Excitation of Floquet-Bloch Solitons”,Phys.Rev. Lett. 90, 053902 (2003); D. Mandelik, H.S. Eisenberg, Y. Silberberg, R. Moran-dotti and J.S. Aitchison, “Gap Solitons in Waveguide Arrays”,Phys. Rev. Lett. 92,093904 (2003).

[20] A. Ferrando, M. Zacares, P. Fernandez de Cordoba, D. Binosi and J. Monsoriu, “Spa-tial soliton formation in photonic crystal fibers”,Opt. Express 11, 452-459 (2003); D.Neshev, A.A. Sukhorukov, B. Hanna, W. Krolokowski and Y.S. Kisvshar, “ControlledGeneration and Steering of Spatial Gap Solitons”,Phys. Rev. Lett. 92, 093904 (2004).

[21] J.C. Knight, T.A. Birks, P.St.J. Russell, D.M. Atkin, “All-silica single-mode fiber withphotonic crystal cladding”,Opt. Lett. 21, 1547-1549 (1996).

[22] T.A. Birks, P.J. Roberts, P.St. J. Russell, D.M. Atkin, T.J. Shepherd, “Full 2-D pho-tonic bandgaps in silica/air structures”,Electron. Lett. 31, 1941-1943 (1995).

[23] R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St. J. Russell, P.J. Roberts, D.C.Allan, “Single mode photonic band gap guidance of light in air”,Science 285, 1537-1539 (1999).

[24] P.St. J. Russell, “Photonic crystal fibers”,Science 299, 358-362.

[25] F. Benabid, G. Bouwmans, J.C. Knight, P.St.J. Russell, F. Couny, “Ultra-high effi-ciency laser wavelength conversion in gas-filled hollow core photonic crystal fiber bypure stimulated Raman scattering in molecular hydrogen”,Phys. Rev. Lett. 93, 123903(2004).

[26] S. Ghosh, J. Sharping, D.G. Ouzounov, A.L. Gaeta, “Resonant optical interactionswith molecules confined in photonic bandgap fibers”,Phys. Rev. Lett. 94, 093902(2005).

[27] F. Benabid, P.S. Light, F. Couny, P.St.J. Russell, “Electromagnetically-induced trans-parency grid in acetylene-filled hollow-core PCF”,Opt. Express 13, 5694 (2005).

[28] D.G. Ouzounov, F.R. Ahmad, D. Mller, N. Venkataraman, M.T. Gallagher, M.G.Thomas, J. Silcox, K.W. Koch and A.L. Gaeta, “Generation of Megawatt optical soli-tons in hollow-core photonic band-gap fibers”,Science 301, 1702-1704 (2003);


[29] F. Luan, J.C. Knight, P.St.J. Russell, S. Campbell, D. Xiao, D.T. Ried, B.J.Mangan,D.P. Williams, P.J. Robert, “Femtosecond soliton pulse delivery at 800 nm wavelengthin hollow-core photonic bandgap fibers”Opt. Express 12, 835-840 (2004).

[30] W. Gobel, A. Nimmerjahn, and F. Helmchen, “Distortion free delivery of nanojoulefemtosecond pulses from Ti:sapphire laser through a hollow-core photonic crystalfiber” Opt. Lett. 29, 1285-1287 (2004).

[31] H. Lim and F.W. Wise, “Control of dispersion in a femtosecond ytterbium laser by useof hollow-core photonic bandgap fiber”,Opt. Express 12, 2231-2235 (2004).

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[33] K. Saitoh, N. A. Mortensen, M. Koshiba, “Air-core photonic band-gap fibers: theimpact of surface modes”,Opt. Express 12 (2004) 394.

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Chapter 13

MULTIWAVELENGTH OPTICAL FIBER LASERSAND SEMICONDUCTOR OPTICAL AMPLIFIER

RING LASERS

Byoungho Lee* and Ilyong YoonSchool of Electrical Engineering, Seoul National University

Gwanak-Gu Sinlim-Dong, Seoul 151-744, Korea

Abstract

We review various schemes for multiwavelength fiber lasers and semiconductor opticalamplifier (SOA) ring lasers. Multiwavelength fiber lasers have applications in wavelengthdivision multiplexing (WDM) optical communication systems, optical fiber sensors andoptical spectroscopy. Erbium-doped fiber amplifiers (EDFAs), Raman amplifiers and SOAsare mainly used as gain media for multiwavelength fiber lasers.

Because EDFAs are homogeneously broadened gain media, various methods have beenresearched to enable the multiwavelength generation. Due to the introduction of liquidnitrogen cooling, four-wave mixing, frequency shifted feedback, and so on, multiwavelengtherbium-doped fiber lasers could become realized.

On the other hand, because SOA and Raman amplifiers are gain media withinhomogeneous broadening, multiwavelength generation is relatively easy. The usefulfeatures of the multiwavelength lasers are mainly dependent on a comb filter. One of the mostimportant features of multiwavelength lasers is tunability. The tunability of wavelengths andchannel spacing is required for WDM optical communication systems. Much research hasbeen conducted to enable implementation of tunable multiwavelength fiber lasers. Variouscomb filters such as Fabry-Perot filters, fiber Bragg gratings, and polarization-maintainingfiber loop mirrors can be used for multiwavelength fiber lasers. We review several schemesfor multiwavelength SOA-fiber and Raman fiber lasers in this chapter.

* E-mail address: [email protected]. Tel: +82-2-880-7245, Fax: +82-2-873-9953

Byoungho Lee and Ilyong Yoon336

1. Introduction

The realization of the laser has made many applications possible. Among those applications, alight source for optical communication systems is one of the most important applications. Aswavelength-division-multiplexing (WDM) optical communication systems have becomemore developed, multiwavelength light sources have also been widely researched. In the firststage, multiwavelength lasers could be made as a simple structure consisting of the array oflasers and a multiplexer [1, 2]. However, there have been difficulties with these lasers such aslarge insertion loss and bulky size. Therefore, multiwavelength fiber lasers using a single gainmedium are desired. There are many possible gain media for optical communication such aserbium-doped fiber (EDF), semiconductor optical amplifier (SOA), and stimulated Ramanscattering (SRS). In this chapter we review a wide variety of multiwavelength fiber lasersemploying a single gain medium.

Because EDF is a homogeneously broadened gain medium, a laser using EDF normallylases at a single wavelength. Various methods have been researched to enable the multiplewavelength generation, such as the introduction of liquid nitrogen cooling, four-wave mixing,frequency shifted feedback, and so on. For the multiwavelength EDF laser (EDFL), schemesto suppress mode competition are a main subject.

On the other hand, Raman amplifiers and SOAs are inhomogeneously broadened gainmedia. Therefore, multiwavelength generation is relatively simple compared with an EDFL.Many methods have also been proposed to implement multiwavelength lasers using thesetechnologies. One of the useful characteristics of multiwavelength fiber lasers is tuningcapability. The tunability of lasing wavelengths and channel spacing is required for WDMoptical communication systems. Therefore, much research has been conducted for theimplementation of tunable multiwavelength fiber lasers. For multiwavelength SOA-fiber andRaman fiber lasers, the schemes for tuning and these lasers’ characteristics are main subjects.We classify and review many schemes for the design of tunable fiber lasers in this chapter.

2. Multiwavelength Fiber Lasers Using EDFA

The most challenging difficulty of an EDF amplifier (EDFA) for a multiwavelength laser isthat the EDFA is a homogeneously broadened medium. In a homogeneously broadenedmedium, all atoms in the excited state have the same gain spectrum. Therefore, when a laseremploys a homogeneously broadened gain medium, only the wavelength which has thelargest net gain (gain minus cavity loss) can survive. The other wavelengths decay due toloss. When a number of wavelengths are in a cavity, each channel experiences modecompetition. However, because Er3+ ions are surrounded by a glass host, the interaction withthe silica and other dopants leads to some degree of inhomogeneous broadening contribution.Therefore, the schemes for the multiwavelength EDFL involve increasing of thecompetitiveness of weak wavelengths or decreasing of the homogeneously broadenedlinewidth of the EDFA.

Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 337

2.1. Cavity Loss Balancing

The first multiwavelength operation of an EDFL was demonstrated in 1992 [3]. The cavitylosses of the lasing wavelengths are carefully controlled to suppress single channel lasing asshown in Fig. 1. The cavities of lasing wavelengths are separated and the losses of cavities arecontrolled independently so that many wavelengths can lase. This is equivalent to the net gainflattening. There is no dominant wavelength due to the flattened net gain. However, thismethod requires a careful control of cavity losses. Thus it can be easily expected that lasersemploying this scheme are relatively unstable and sensitive to environmental conditions.

Gain medium

W

D

M

PC

FLM

FLM

λ1, λ2, …, λ8

λ1

λ2

λ8

(a)

W

D

M

λ1

λ8

W

D

M

Gain medium

Polarizer

Isolator

λ1, λ2, …, λ8

(b)

Figure 1. (a) An eight-channel laser configuration based on a linear cavity. (b) An eight-channel laserconfiguration based on a ring cavity (PC: polarization controller, FLM: fiber loop mirror, WDM:wavelength division multiplexer) [3].


2.2. Liquid Nitrogen Cooling

When an EDFA is cooled, the homogeneous linewidth of the EDFA is narrowed. Spectralhole burning and homogeneous linewidth were measured as a function of temperature in Ref.[4]. The homogeneous linewidth was measured as 1.3 nm at 61 K. It was shown that thehomogeneous linewidth exceeded 11.5 nm at room temperature.

For a multiwavelength application of an EDFA, there was other research to make aninhomogeneously broadened EDFA by liquid nitrogen cooling [5]. In Ref. [5], the main ideawas the suppression of dynamic crosstalk between adjacent channels. The liquid nitrogencooling made an 11 dB suppression of crosstalk. The channel spacing was 4 nm and thehomogeneous linewidth was measured as ~1 nm.

Comb filter

Isolator

Doped fiber

77K

75:25 couplerOutput

WDM coupler

Pump

Figure 2. A multiwavelength EDF ring laser configuration using a comb filter in the cavity [6].

The first multiwavelength EDFL by liquid nitrogen cooling (77 K) was presented in 1996[6]. Figure 2 shows the configuration. 11 stable laser lines were demonstrated with 0.65 nmchannel spacing around 1535 nm. Two types of comb filters were used in the experiment.Those were a chirped fiber Bragg grating (CFBG) Fabry-Perot filter and a sampled grating.

2.3. Four-Wave Mixing

The self-stabilizing effect of four-wave mixing (FWM) can be used for a multiwavelengthEDFL. The powers of lasing wavelengths are automatically balanced by several degeneratedFWMs, 1 2 32ω ω ω= + , or nondegenerated FWMs, 1 2 3 4ω ω ω ω+ = + . For a phasematching condition, dispersion-shifted fiber (DSF) or photonic crystal fiber (PCF) arerequired. The self-stabilizing effect may be described as a “photonic Robin Hood.” Thismeans that FWM takes the energy of a rich wavelength and gives it to a poor wavelength.Because power is transferred between wavelengths by the nonlinear process, this scheme canbe thought of as automatic net gain equalization.


PC1

PC2

VOA1 VOA3

VOA4VOA2

FBG1 FBG3

FBG2 FBG4

3-dB coupler

3-dB

cou

pler

HN-PCFPC3

Output

980nm Pump laser diode

WDM coupler

EDFOutput

Fused coupler1Fu

sed

coup

ler 2

Figure 3. An experimental setup for four-wavelength EDFL (HN-PCF: highly nonlinear photoniccrystal fiber, VOA: variable optical attenuator, FBG: fiber Bragg grating, PC: polarization controller)[7].

Liu and Lu demonstrated a four-wavelength EDFL using a highly nonlinear PCF tosuppress the mode competition at room temperature as shown in Fig. 3 [7]. Experimentalresults showed lasing wavelengths of 1540.28, 1543.58, 1546.79 and 1550.08 nm.

EDFA

Output10 90

AWG(1xn)

PC

FBG1 FBGn

λ1 λ2 λn

Isolator

DSF

PC

Figure 4. A schematic of the multiwavelength EDFL based on degenerate four-wave mixing in the DSF(EDFA: erbium-doped fiber amplifier, FBG: fiber Bragg grating, AWG: arrayed waveguide grating,PC: polarization controller) [8].


Han et al. also presented similar multiwavelength EDFLs by using a DSF [8, 9]. Figure 4shows a schematic diagram of the multiwavelength EDFL employing multiple fiber Bragggratings (FBGs) and a 1 km DSF for 10 channels’ lasing with 0.8 nm channel spacing. Inaddition, they showed channel spacing tunability through the elimination of the effects ofseveral FBGs.

Output10 90

Isolator

DSF

PC1

980nm pump laser diode

EDF

PC3 PC4

PC2 PMF1L1

PMF2L2

Lyot-Sagnac filter

Figure 5. Schematics of the multiwavelength EDFL using DSF and Lyot-Sagnac filter (PMF:polarization-maintaining fiber, EDF: erbium-doped fiber, PC: polarization controller) [9].

A tunable multiwavelength EDFL was demonstrated in 2005 [9]. The tunabilityoriginated from a tunable Lyot-Sagnac filter as shown in Fig. 5. The wavelength spacing ofthe two-segment Lyot-Sagnac filter was [ ]2

1 2/ ( )n L Lλ λΔ = Δ ⋅ ± , where nΔ was the

effective birefringence between two orthogonal polarization modes and 1 2,L L were thelengths of the two polarization-maintaining fibers (PMFs) shown in Fig. 5. Thus, the channelspacing was switchable by polarization control. In the Lyot-Sagnac filter, clockwise andcounterclockwise lights experienced optical path difference due to PMF segments. Therefore,the optical path difference between two lights led to comb-like filter characteristics. Thebirefringence of the two PMF segments may be summed or subtracted depending on the stateof the polarization controllers (PCs).

Experimental results showed 11 laser lines with 1 nm spacing and 17 laser lines with 0.8nm spacing. Stable lasing characteristics and tuning capability were obtained due to FWM.


2.4. Frequency Shifting Technique

Another scheme for a multiwavelength EDFL is a frequency shifted feedback scheme [10].Figure 6 is a schematic diagram of the multiwavelength EDFL. In the scheme, an acousto-optic modulator (frequency shifter) shifts the frequency of light by 100 MHz for each roundtrip. This prevents single frequency lasing. Experimental results showed stable ~13 laser lineswith 0.8 nm spacing. A Fabry-Perot etalon with a CFBG or sampled grating was used for theperiodic filter. The experimental results showed good agreement with the simulation results.

EDFA1

EDFA2

3-dB coupler

IsolatorOutput

Frequencyshifter

Periodic filter

Figure 6. A schematic diagram of the multiwavelength EDFL employing a frequency shifted feedbackscheme (EDFA: erbium-doped fiber amplifier) [10].

Table 1. Multiwavelength EDFLs

Year First author[Reference] Comments Channel

number

Channelspacing

(nm)Scheme

1992 N. Park [3] The first multiwavelength EDFL 6 4.8 Cavity lossbalancing

1996 J. Chow [6] 11 0.65 Liquid nitrogencooling

2000 A. Bellemare [10] ~13 0.8 Frequency shifting

2002 R. Slavik [11] High uniformity 18 0.8 Frequency shifting

2005 X. Liu [7] 4 3.3 Four-wave mixing

2005 Y.-G. Han [9] Channel spacing switching 17, 11 0.8, 1 Four-wave mixing

2006 Y.-G. Han [8] Channel spacing switching 10 0.8, 1.6, 2.4 Four-wave mixing


For the frequency shifting technique, a more in-depth study was published in 2002 [11].These researchers improved the uniformity of the lasing wavelengths in the EDFL. Uniform18 laser lines with 0.8 nm channel spacing were obtained in the experiments.

Similar to the frequency shifted feedback scheme, the phase-modulation feedbackscheme was also presented [12, 13]. A LiNbO3 phase modulator was used for phasemodulation. In Ref. [12], the sawtoothed and sinusoidal phase modulation of a few tens ofkHz generated a multiwavelength operation. In Ref. [13], the authors reported that sinusoidal,sawtoothed, triangular and square waveforms are all suitable for multiwavelength lasing.They also indicated that the phase modulation of 500 Hz to a few tens of kHz is good.

Important features of the above multiwavelength EDFLs are shown in Table 1 as asummary.

3. Multiwavelength Fiber-SOA and Fiber-Raman Lasers

In a SOA, the gain medium is a semiconductor and not a single atom or ion. Therecombination of electron-hole pairs makes spontaneous or stimulated emission. The SOA iselectrically pumped. More electrons in the conduction band and more holes in the valanceband lead to higher gain. The gain spectrum of the SOA depends on materials and structure.The intrinsic inhomogeneous broadening is an advantage of the SOA in its application tomultiwavelength lasers. High gain per unit length and compact size are other advantages. Onthe other hand, the rectangular structure leads to a coupling loss for optical fiber andpolarization-dependent gain. The fast carrier lifetime (~200 ps) leads to cross saturation andstronger nonlinear processes.

SRS is an interaction between photon energy and molecular vibrational energy (opticalphonon). The amplification is performed by the energy transfer from a pump beam to thesignal beam (or light to lase). Unlike that of the EDFA and the SOA, Raman scattering doesnot require a population inversion for amplification. Very broad gain bandwidth is the maincharacteristic of the Raman amplification process. In the SRS, specific resonant frequencydoes not exist in contrast to the EDFA and the SOA. The wavelength of the pump beamdetermines the location of the gain spectrum which has a peak at 13.2 THz off the pumpwavelength. It is a main advantage of a Raman amplifier that a specific gain medium is notrequired, i.e., amplification occurs in a common optical fiber. Therefore, lumped ordistributed schemes are all possible. If several pump wavelengths are used properly, a flatgain over a wide bandwidth can be obtained [14]. Because Raman scattering is a weak effect,the SRS requires very high pump power (typically a few Watts) and a long length of fiber.Intrinsic inhomogeneous linewidth broadening is very attractive for a multiwavelength laser.

In the SOA and Raman fiber lasers, multiwavelength generation is relatively easybecause of their inhomogeneous broadening. Therefore, tunable capability has been a mainsubject of research involving these multiwavelength SOA and Raman fiber lasers. Tunabilitywas even considered in the demonstration of the first multiwavelength Raman fiber laser [15,16]. Thus, in this section, we focus on the tunable capability of SOAs and Raman fiber lasers.To avoid confusion, we use two different terminologies: switchability and tunability.Switchability and tunability denote discrete tuning and continuous tuning, respectively. Thusa wavelength switchable laser means a laser which can shift the spectral position of lasingwavelengths by some discrete steps.


3.1. Wavelength or Channel Spacing Switchability

By using a sampled high-birefringence (Hi-Bi) fiber grating as a switchable comb filter, thewavelength switchable laser was demonstrated by Yu et al. as shown in Fig. 7 [17]. We canthink of this as if two different sampled FBGs (SFBGs) are used due to the difference of therefractive indexes in the fast and slow axes of the Hi-Bi fiber. The control of a rotatablepolarizer is equivalent to the selection of one of two SFBGs. In a SFBG, the center Braggwavelength is 2B effnλ = Λ and the wavelength separation is 2 / 2B effn pλ λΔ = , where effnis the effective refractive index of the fiber core, Λ the individual grating pitch, and p is thesampling period. Therefore, while λΔ is maintained at nearly the same value, it is possibleto move only the center Bragg wavelength. Because the birefringence nΔ is of the order of10-4, the channel spacing is hardly influenced by the choice of polarization axis. However, ifwe control the polarization of light incident on the sampled grating by using the rotatablepolarizer, transmission peaks can be shifted by 2 nΔ Λ . The experimental result showed aninterleaving characteristic. The laser output was shifted by 0.4 nm with the 0.8 nm channelspacing fixed. It had a disadvantage in that the number of switchable wavelength set wasintrinsically limited to two. The amount of switchable wavelengths was determined by thechoice of PMF. Thus, the maximum switchable range was limited by the birefringence of thePMF.

PC

SOAIsolator

Isolator

Polarizer

Output

Variable coupler

SMF

Sampled Hi-Bi fiber grating

Figure 7. A schematic diagram of a wavelength switchable SOA-fiber ring laser employing sampled Hi-Bi FBG (SMF: single mode fiber) [17]

Lee et al. presented wavelength switchable SOA fiber lasers employing two SFBGs [18]and a reflection type interleaver [19]. The former used two SFBGs connected to a polarizationbeam splitter (PBS) for waveband switching as shown in Fig. 8. A rotatable linear polarizerselected one of the two SFBGs. Contrary to the work by Yu et al. [17], the amount ofwaveband switching depended on the design of the SFBGs. In other words, the amount ofwavelength shift was not limited by the choice of a PMF. The experimental result in Fig. 9showed that 5 laser lines with 0.8 nm spacing could be switched by a spectral displacement of10 nm.


SOAIsolator PC

75: 25coupler

Output

Rotatable liner polarizer

Light absorber

Sampled fiber Bragg gratting1

Sampled fiber Bragg gratting2

PBS

Figure 8. A schematic diagram of the waveband-switchable SOA-fiber laser using two SFBGs (PBS:polarization beam splitter, PC: polarization controller) [18].

1535 1540 1545 1550 1555 1560 1565

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0

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Opt

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pow

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Bm]

Figure 9. Output optical spectra showing the waveband switching operation.


SOAIsolator

Isolator

75: 25coupler

Output

50:50coupler

PC1

PC2PMF

12

Figure 10. A schematic diagram of a SOA-fiber laser employing a reflective type interleaver (PMF:polarization maintaining fiber, PC: polarization controller) [19].

L1 L2

50:50 couplerλ/4λ/4

λ/2 λ/2 λ/2 λ/2a b c

Tunable filter

WDM2WDM1 Raman fiber

Pump laser Output

Figure 11. An experimental setup for a tunable Raman fiber ring laser (WDM: wavelength divisionmultiplexer) [21].

Another scheme using a reflective interleaver is shown in Fig. 10 [19]. The interleaver iscomposed of a PBS and a PMF loop mirror. A PC in the PMF loop mirror consists of twoquarter-wave plates. The control of waveplates leads to an interleaving characteristic. Thefeature of this filter is that the transmission and reflection characteristics show interleavedsets of multiple wavelength peaks. Theoretically, for transmission, the filter has infinite


channel isolation and 3 dB insertion loss. On the other hand, for reflection, the filter shows 3dB channel isolation and 0 insertion loss. 17 wavelengths were generated with 0.8 nm channelspacing. The laser lines could be shifted by 0.4 nm with channel spacing fixed.

The PMF Lyot-Sagnac filter was also used for a multiwavelength SOA-fiber laser [20].With the PMF Lyot-Sagnac filter, the SOA-fiber laser could have channel spacingswitchability. In addition, the rotation of a quarter-wave plate made the lasing wavelengthshift. Channel spacing switchability from 0.8 nm to 4.1 nm was demonstrated. Laser linesfrom 5 to 20 were observed. Continuous wavelength tuning was also shown.

There have also been intensive research efforts for tunable multiwavelength Raman fiberlasers. Kim et al. demonstrated a multiwavelength Raman fiber ring laser with switchablechannel spacing and a tunable lasing wavelength [21]. The multiwavelength source wascomposed of a Raman fiber and a Lyot-Sagnac filter as shown in Fig. 11. The experimentalresults showed a multiwavelength generation of up to 20 laser lines with 0.43 nm spacing.

PMF2PMF1

Lyot-Sagnac filter

PC2(λ/2)

PC1(λ/2)

Fiber grating 97%

Raman gain fiber

Fiber grating 90%

Pump combiner

OutputWDM coupler

Pump

Pump laser

Figure 12. An experimental setup for a tunable multiwavelength Raman laser based on an FBG cavityincorporating PMF Lyot-Sagnac filter (PC: polarization controller, WDM: wavelength divisionmultiplexer) [22].

Han et al. demonstrated a multiwavelength Raman laser with a similar filter as shown inFig. 12 [22]. Although a similar PMF Lyot-Sagnac filter was used, there were twodifferences: a linear cavity structure employing FBGs and the use of PMFs with differentbirefringences. In the experimental result, the multiwavelength laser generated 7 channelswith 0.6 nm spacing and 5 channels with 0.8 nm spacing.

A phase modulator loop mirror filter (PM-LMF) could be used for wavelengthswitchability [23]. The PM-LMF is a sort of PMF loop mirror where a phase modulator isinserted. Because DC bias, RF power, or modulation frequency changes the birefringence ofthe phase modulator slightly, the spectral comb position can be shifted while the channelspacing is fixed. Experimentally, 21 laser lines with a 0.8 nm channel spacing were obtained.


Optical amplifier

10% Output

Coupler

Programmable Hi-Bi FLM OutputInput

Residual pump power

DCF 6.9 km

Pump laser

WSC Isolator

InOut

Combiner

3 dB coupler

PC1 PC2 PCn

Hi-Bi, L1 Hi-Bi, L2 Hi-Bi, Ln

2×2 switch

Figure 13. A schematic of a multiwavelength laser and a programmable Hi-Bi fiber loop mirror (FLM)(DCF: dispersion compensation fiber, WSC: wavelength selection coupler) [24].

Chen demonstrated channel spacing switchable fiber lasers by using a programmable Hi-Bi fiber loop mirror as shown in Fig. 13 [24]. He employed a SOA or a Raman amplifier asan optical amplifier. Although many switchable sections are theoretically possible in Fig. 13,the two sections of the PMF were demonstrated. The use of 2×2 switches changes thecombination of the PMF section more flexibly. The channel spacing expression is essentiallyequivalent to that of the PMF Lyot-Sagnac filter except that it has a more flexiblebirefringence combination. The experimental result showed 3.2 nm and 1.6 nm channelspacing switching.

A tunable Raman fiber laser employing an electro-optical tuning scheme was presented in2004 [25]. The comb filter uses an electro-optic polarization controller (EOPC) inserted in thePMF Sagnac loop filter. The PMF Sagnac loop filter is sometimes called a Lyot-Sagnac filter.Due to the PMF Sagnac loop filter, the channel spacing was switchable. When a drivingvoltage was applied to the EOPC, the additional birefringence was induced. This effect isequivalent to changing the length of the PMF slightly. Therefore, lasing wavelengths can beshifted as channel spacing is nearly fixed. This Raman laser had both channel spacingswitchability and wavelength tunability. Experimental results indicated channel spacingswitchability between 0.95 and 2.95 nm. Interleaved switching operation of 11 laser lines wasalso demonstrated with 0.88 nm channel spacing.

3.2. Wavelength Tunability

In a SOA fiber laser, wavelength tuning possibility was presented in 2001 [26]. In fact, thework in Ref. [26] was about a multiplexed sensor. The sensor was a SOA ring laser using atransmission-type filter consisting of a circulator and multiple FBGs. Eight laser lines were


observed when 8 FBGs were used. The center wavelengths of the FBGs were 1534.44,1543.68, 1546.32, 1549.38, 1552.5, 1554.06, 1556.28 and 1558.92 nm. The strain on eachFBG changed each lasing wavelength. Therefore, the sensor was indeed a sort of a tunablelaser although it was not clarified in the reference.

Isolator

Isolator PC

GC-SOA

PBS

HWP2

HWP1 QWP

PMFPort 2

Port 1

Output

75:25 coupler

CW

CCW

Figure 14. The schematic diagram of a multiwavelength SOA-fiber ring laser employing a PDLC (GC-SOA: gain-clamped semiconductor optical amplifier, HWP: half-wave plate, QWP: quarter-wave plate,PMF: polarization-maintaining fiber, PBS: polarization beam splitter, PC: polarization controller, CW:clockwise, CCW: counterclockwise) [27].

1545 1550 1555 1560 1565 1570 1575-50

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pow

er [d

Bm

]

Wavelength [nm]1557 1558 1559 1560 1561 1562 1563

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Bm

]

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0°

The

angl

e of

HW

P 1

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-50

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40°

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-10

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-30

-10

80°

(a) (b)

Figure 15. (a) The output spectrum of a multiwavelength laser (b) The tuning characteristic.


Yoon et al. published their research on the wavelength tunable SOA fiber ring laser [27].They employed a polarization-diversity loop configuration (PDLC) comb filter as shown inFig. 14. The PDLC comb filter consists of two half-wave plates, a quarter-wave plate, apolarization beam splitter, and a PMF. The PDLC filter can tune lasing wavelengths with thecontrol of a half-wave plate alone. A single polarization which comes out of the PBS entersthe PMF after passing through wave plates. The light experiences polarization changeaccording to its wavelength by the PMF. When the light meets the PBS again, thetransmittivity is determined by its polarization. The role of the wave plates is to make thepolarization-change by the rotation of the first half-wave plate reproduce the polarization-change by the PMF. When we represent polarization change in the Poincare sphere, thetrajectory of polarization change becomes a circle. Because the rotation of the first half-waveplate makes the trajectory rotate, the rotation of the first half-wave plate can shift the positionof the filter comb. In the PDLC filter, clockwise and counterclockwise lights exist. Twocounter propagating lights experience the same transmittivity and reflectivity. A 90° rotationof the angle of the first HWP corresponds to the sweep of the entire channel spacing. Thechannel spacing is determined by the length and birefringence of the PMF. In the experiment,18 laser lines were observed with 0.8 nm channel spacing. Figs. 15 (a) and (b) show theoutput spectrum and tuning characteristics. The rotation of a half-wave plate can shift theposition of lasing wavelengths linearly.

Few-mode Bragg grating

PC

Raman fiber (SMF 50 km)

Tunable chirpedFBG

Output

Pump combiner

1425nm 14351455 1465Pump laser

Pump

(a)

d

Flexible metal plate

Compression(Negative bending)

Tension(Positive bending)

(b)

Figure 16. (a) An experimental setup for a multiwavelength Raman fiber laser based on few-modeFBGs (b) A tuning method based on the symmetrical bending of a flexible metal plate (FBG: fiberBragg grating, PC: polarization controller) [28].


For a wavelength-tunable Raman fiber laser, Han et al. demonstrated the few-modeFBG scheme as shown in Fig. 16 [28]. Because the few-mode FBG has multipleresonant wavelengths, a multiwavelength Raman fiber laser could be obtainedwithout additional multichannel filters. The CFBG is used to form a linear cavity becauseof the broad reflection spectrum. Tuning of the CFBG is needed to match thereflection spectrum to that of a few-mode FBG. The lasing wavelength shift ( λΔ )can be defined as (1 ) pλ ρ ελΔ = − , where pλ is the lasing wavelength of the Raman

fiber laser, ρ is the photo-elastic coefficient, and ε is the strain induced by the bendingof the fiber. The experimental results showed 3 laser lines with 3.5 nm spacing andthe wavelength tuning characteristics. The number of laser lines was limited by thefew-mode FBG.

Han et al. also demonstrated temperature tuning of the lasing wavelength of amultiwavelength Raman laser using the few-mode FBG [29]. Three laser lineswere also obtained and the temperature sensitivity was measured as 10.5 pm/°C. Theyalso applied a similar structure to a temperature and strain sensor using amultiwavelength Raman laser with a phase-shifted FBG [30]. The experimental resultshowed that two lasing wavelengths could be shifted by strain and temperature with a fixedspacing.

3.3. Channel Spacing Tunability

Dong et al. presented a multiwavelength SOA-fiber laser and Raman fiber laseremploying a fiber Fabry-Perot filter based on a superimposed CFBG [31, 32]. Twosuper imposed CFBGs form the Fabry-Perot filter when the writing positions of thetwo CFBGs are slightly different. The tuning of the chirp rate changes the channelspacing. The SOA-fiber laser and Raman fiber laser could be implemented by using thesame filter. The SOA-fiber laser generated 10~13 laser lines with 0.3~0.6 nm channelspacing and the Raman fiber laser generated 2~10 laser lines with 0.3~0.6 nm channelspacing.

3.4. Both Wavelength and Channel Spacing Tunability

Roh et al. demonstrated a SOA-fiber laser with both wavelength and channelspacing tunability [33]. They employed a PDLC comb filter with a differential delayline (DDL) as shown in Fig. 17. The use of the DDL instead of a PMF could lead tospacing tunability as well as wavelength tunability. Both the channel spacing andlasing wavelength are continuously tunable. Channel spacing is tuned electrically andwavelength is tuned by the rotation of a half-wave plate. Because this laser adopted aPDLC comb filter, the wavelength tuning characteristics are totally equivalent to Ref. [27].Experimental results showed channel spacing tunability of 0.4 ~ 1.6 nm with up to 23laser lines.


PC

IsolatorSOAIsolator

OutputCW

DDL

QWP HWP1HWP2

CCW1

2 75:25 coupler

SF

θh1

SF

θq

SF

θD

SF

θh2

PBS

Figure 17. The schematic diagram of the channel spacing and wavelength tunable SOA-fiber ring laser(DDL: differential delay line, QWP: quarter-wave plate, HWP: half-wave plate, PBS: polarization beamsplitter, PC: polarization controller, CW: clockwise, CCW: counterclockwise) [33]

The features of the demonstrated SOA-fiber lasers and Raman fiber lasers aresummarized in Tables 2 and 3.

Table 2. Multiwavelength SOA-fiber lasers

Year First author[Reference] Advantage Channel

number

Channelspacing

(nm)Scheme

2001 S. Kim [26] 8 Fiber Bragggrating

2003 B.-A. Yu [17] Wavelength switching 4 0.8 Sampled fiberBragg grating

2004 L. R. Chen [24] Spacing tuning 11, 6 1.6, 3.2 Programmable Hi-Bi FLM

2004 Y. W. Lee [19] Waveband tuning 17 0.8 Hi-Bi FLM2005 M. P. Fok [23] Wavelength tuning 21 0.8 PM-LMF

2005 Y.-G. Han [20] Spacing and waveband switching 5~20 0.8~4.1 PMF Lyot-Sagnacfilter

2005 X. Dong [31] Spacing tuning 13 0.4 Fabry-Perot2006 I. Yoon [27] Wavelength tuning 18 0.8 PDLC

2006 S. Roh [33] Wavelength and channel spacingtuning 23 0.8 PDLC with DDL


Table 3. Multiwavelength Raman fiber lasers

Year First author[Reference] Advantage Channel

number

Channelspacing

(nm)Scheme

2001 F. Koch [15] Potential angle tuning 24 0.8 Fabry-Perot

2001 C. J. S. de Matos[16] Potential individual tuning 4 ~4.5 FBG

2003 C.-S. Kim [21] Channel spacing switching andwavelength tuning 20 0.4~3 Sagnac

2004 C.-S. Kim [25] Channel spacing switching andwavelength tuning 11 0.95, 0.88,

2.95

PMF Sagnac loop filterwith electro-opticpolarization controller

2004 Y.-G. Han [22] Channel spacing switching 7, 5 0.6, 0.8 PMF Lyot-Sagnac

2005 Y.-G. Han [28] Wavelength tuning 3 3.5 Few-mode fiber

2005 Y.-G. Han [30] Sensing (wavelength tuning) 2 1.4 Phase-shifted fiber

2006 X. Dong [32] Channel spacing tuning 2~10 0.3~0.6 Sample fiber Bragggrating

4. Conclusion

As multiwavelength light sources become more important in WDM optical communication,there is an increasing amount of research on the multiwavelength fiber lasers. In this chapterwe reviewed various schemes for a multiwavelength fiber laser to date. Feasible gain mediaare the EDFA, the SOA and the SRS. Each of these schemes has different challengingdifficulties for multiwavelength generation.

For the EDFA, the most difficult problem is its homogeneous broadening. The unstablelasing characteristic of the EDFA results from homogeneous broadening. As possibleschemes to overcome homogeneous broadening, we reviewed the techniques of cavity lossbalancing among wavelengths, self stabilization from FWM, liquid nitrogen cooling andfrequency shifted feedback.

On the other hand, inhomogeneous broadening of a SOA and Raman amplifier makesmultiwavelength generation easy. The tunability in lasing wavelength and channel spacingbecomes more important as optical communication systems become more flexible andefficient. We focused on the challenging issues of tuning multiwavelength SOAs and Ramanfiber lasers.

Practically, tunable channel spacing and tunable lasing wavelengths are required featuresin WDM optical communication systems. Tuning capabilities can be classified intoswitchability and tunability. Today’s tunable SOA-fiber and Raman fiber lasers wereclassified and reviewed. Because the tunable characteristic originates from characteristics of afilter, various filters for tunable lasers were introduced.


References

[1] Hamakawa. A.; Kato, T.; Sasaki, G.; Shigehara, M. Conf. Opt. Fiber Comm. 1997,297-298.

[2] Takahashi, H.; Toba, H.; Inoue, Y. Electron. Lett. 1994, 30, 44-45.[3] Park, N.; Dawson, J. W.; Vahala, K. J. IEEE Photon. Technol. Lett. 1992, 4, 540-541.[4] Desurvire, E.; Zyskind, J. L.; Simpson, J. R. IEEE Photon. Technol. Lett. 1990, 2,

246 – 248.[5] Goldstein, E. L.; Eskildsen, L.; da Silva, K.; Andrejco, M.; Silberberg, Y. IEEE Photon.

Technol. Lett. 1993, 5, 937 – 939.[6] Chow, J.; Town, G.; Eggleton, B.; Ibsen, M.; Sugden, K.; Bennion, I. IEEE Photon.

Technol. Lett. 1996, 8, 60-62.[7] Liu, X.; Lu, C. IEEE Photon. Technol. Lett. 2005, 17, 2541-2543.[8] Han, Y.-G.; Tran, T. V. A.; Lee, S. B. Opt. Lett. 2006, 31, 697-699.[9] Han, Y.-G.; Lee, S. B. Opt. Express 2005, 13, 10134-10139.[10] Bellemare, A.; Karasek, M.; Rochette, M.; LaRochelle, S.; Têtu, M. J. Lightwave

Technol. 2000, 18, 825-831.[11] Slavik, R.; LaRochelle, S.; Karasek, M. Opt. Commun. 2002, 206, 365-371.[12] Zhou, K.; Zhou, D.; Fengzhong, F.; Ngo, Q. N. Opt. Lett. 2003, 28, 893-895.[13] Ahmed, F.; Kishi, N.; Miki, T. IEEE Photon. Technol. Lett. 2005, 17, 753-755.[14] Liu, X.; Lee, B. IEEE Photon. Technol. Lett. 2004, 16, 428-430.[15] Koch, F.; Reeves-Hall, P. C.; Chernikov, S. V.; Taylor, J. R. Conf. Opt. Fiber Comm.

2001, 54, WDD7/1-WDD7/3.[16] De Matos, C. J. S.; Chestnut, D. A.; Reeves-Hall, P. C.; Koch, F.; Taylor, J. R. Electron.

Lett. 2001, 37, 825-826.[17] Yu, B.-A.; Kwon, J.; Chung, S.; Seo, S.-W.; Lee, B. Electron. Lett. 2003, 39, 649-650.[18] Lee, Y. W.; Yu, B.-A.; Lee, B.; Jung, J. Opt. Eng., 2003, 42, 2786-2787.[19] Lee, Y. W.; Jung, J.; Lee, B. IEEE Photon. Technol. Lett. 2004, 16, 54-56.[20] Han, Y.-G.; Kim, G.; Lee, J. H.; Kim, S. H.; Lee, S. B. IEEE Photon. Technol. Lett.

2005, 17, 989-991.[21] Kim, C.-S.; Sova, R. M.; Kang, J. U. Opt. Commun. 2003, 218, 291-295.[22] Han, Y.-G.; Lee, J. H.; Kim, S. H.; Lee, S. B. Electron. Lett. 2004, 40, 1475-1476.[23] Fok, M. P.; Lee, K. L.; Shu, C. IEEE Photon. Technol. Lett. 2005, 17, 1393-1395.[24] Chen, L. R. IEEE Photon. Technol. Lett. 2004, 16, 410-412.[25] Kim, C.-S.; Kang, J. U. Appl. Opt. 2004, 43, 3151-3157.[26] Kim, S.; Kwon, J.; Kim, S.; Lee, B. IEEE Photon. Technol. Lett. 2001, 13, 350-351.[27] Yoon, I.; Lee, Y. W.; Jung, J.; Lee, B. J. Lightwave Technl. 2006, 24, 1805-1811.[28] Han, Y.-G.; Moon, D. S.; Chung, Y.; Lee, S. B. Opt. Express 2005, 13, 6330-6335.[29] Han, Y.-G.; Lee, S. B.; Moon, D. S.; Chung, Y. Opt. Lett. 2005, 30, 2200-2202.[30] Han, Y.-G.; Tran, T. V. A.; Kim, S.-H.; Lee, S. B. Opt. Lett. 2005, 30, 1114-1116.[31] Dong, X.; Shum, P.; Xu, Z.; Lu, C. IEEE LEOS Ann. Meeting 2005, 814 – 815.[32] Dong, X.; Shum, P.; Ngo, N. Q.; Chan, C. C. Opt. Express 2006, 14, 3288-3293.[33] Roh, S.; Chung, S.; Lee, Y. W.; Yoon, I.; Lee, B. IEEE Photon. Technol. Lett. 2006, 18,

2302-2304.


Chapter 14

AGING AND RELIABILITY OF SINGLE-MODE SILICAOPTICAL FIBERS

M. Poulain1, R. El Abdi2 and I. Severin3

1UMR 6226, Université de Rennes1, F-35042 Rennes, France2LARMAUR, Fre-Cnrs 2717, Université de Rennes1, F-35042 Rennes, France

3Universita Politechnica, Splaiul Independentei, IMST, 06042 Bucarest, Romania

Abstract

The optical fiber reliability in telecommunication networks has been still an issue, that’swhy the question of how long an optical fibers might been used without a significantprobability of failure isn’t out of interest. Much work was developed around this issue, butthe optical fiber fatigue and aging process has not been yet fully understood.

The reliability of the optical fibers depends on various parameters that have beenidentified: time, temperature, applied stress, initial fiber strength and environmentalcorrosion. The major and usually unique corrosion reagent is water, either in the liquid stateor as atmospheric moisture. Glass surface contains numerous defects, either intrinsic, the so-called “Griffith’s flaws and extrinsic, in relation to fabrication process. Under permanent ortransient stress, microcracks grow from these defects, and growth kinetics depend ontemperature and humidity. Although polymeric coating efficiently protects glass surface fromscratches, it does not prevent water to reach glass fiber.

The work carried out during the last years made possible to apprehend in a more coherentway the problems of failure and rupture of fibers subjected to severe aging conditions.

In the proposed chapter, some informations on the used characterization methodology forthe silica optical fibers are given. In addition, Optical fibers analysis advantages, expectedpercussions and theoretical background are given to enlighten the potential concernedpersons. The principal optical fiber test benches are described and some results arecommented. Finally, final remarks are noted.

M. Poulain, R. El Abdi and I. Severin356

1. Introduction

Terrestrial and submarine telecommunication networks depend critically on optical fibers.While main emphasis is put on transmission and signal characteristics [1], more basic featuressuch as reliability and expected lifetime has appeared also as major concerns [2, 3]. However,these concerns become less popular with the deep crisis that occurred in the telecom marketand the emergence of more advanced fibers: one may think that new fibers should replace theexisting ones earlier than expected. In addition, until now, operators did not face seriousproblems in relation to fiber failure.

Nevertheless the reliability issue remains more than ever a topical question for severalreasons. Firstly, the impressive increase of the bit rate is accompanied by a power increasewhich is supported by the fiber core and can generate catastrophic failure phenomena andgenerate damage of the fiber ends or losses in the connectors. Secondly, current modelsinclude humidity, applied stress and temperature as major aging factors, but their accuracy forlifetime prediction is questionable.

Aging of silica fibers is now rather well understood as numerous studies have beenimplemented in this area [2-17]. One must separate the case of the fatigue static behaviorwhere fibers are subjected to a permanent strain, e.g. bended fibers, and the dynamic fatiguecorresponding to an unexpected tensile stress arising from environmental changes. Failuremechanism involves surface phenomena, which raise fundamental questions.

Surface defects, initiator for cracks grow have not yet been identified neither by ScanningElectron Microscopy (SEM) nor by Atomic Force Microscopy (AFM). The current randomnetwork model used actually to describe glass structure gives no explanation for the so-calledGriffith’s flaws [18] and does not account for density fluctuations and inhomogeneities inglass. While other models, such as the vacancy model [14, 19], may provide a physicalpicture of these defects, they are still in an emerging state that limits their application.

Water is also critical in fiber failure: fiber strength may increase by 100 % if water ismissing, for example under vacuum, in a dry box or at liquid nitrogen temperature [3, 4, 16].It is assumed that water molecules break the Si-O-Si chemical bonds of the vitreous network.This simple and logical model may be incomplete and ignore some aspects of the wholephenomenon.

Polymeric coatings are largely used to inhibit surface flaws and proved to be efficient.However the reinforcement mechanism is not well understood.

The general use of Weibull’s statistics in data processing may be inappropriate in somecases: it is widely observed that fiber strength value calculated from Weibull plots decreasesas sample length increases, but Weibull formula is precisely expressed to be independent onfiber length.

These questions, and others, have not only a fundamental interest, but still could havenotable economic implications.

The technology evolution and the research for low cost solutions lead to use new fibersand new components. Thus, polymeric fibers are being considered for the local distribution,while Bragg grating fiber components are now largely used in optical amplifiers. However,the reliability of these new components has still to be evaluated.

A traditional stake is referred to the future fibers for the local distribution networks (FiberTo The Home, FTTH). These fibers will be submitted to notable permanent stresses, for

Aging and Reliability of Single-Mode Silica Optical Fibers 357

example at door corners, thus exposure to temperature, humidity and sudden stress may belarger than in classical cables.

The better understanding of the factors ruling aging and reliability of optical fibers shouldlead not only to scientific advances, but also to economical spin-offs. The telecom market willrequire light cables for local area networks, and the design of mini cables can be optimized onthis basis. In addition various markets are likely to open in other fields where optical fibercomponents are key elements. This concerns optical fiber sensors, laser power delivery, fiberlasers, monitoring and control, remote spectroscopy, in line imaging, etc… Of particularinterest is automotive industry in those case reliability and cost make essential points.

On the fundamental level, the principal goal is to collect new information elements thatcould help answering recurrent questions.

2. Background

Failure of fibers is rather well understood as it might be considered as a particular case offragile material fracture [2-18]. While such materials exhibit a large resistance to compressivestress, they are much more sensitive to traction. Failure originates from surface flaws thatmay be described as microcracks. Their cracks growth under tensile stress as effective stressis amplified at the bottom of the crack. This growth is enhanced by water activity in mostmaterials, including oxide glasses. It is generally assumed that water acts by breaking thechemical bonds between oxygen and silicium or other cations. For this reason the intrinsicstrength K1C of the material can be observed only in extremely dry conditions, e.g. vacuum orliquid nitrogen.

In practice, optical fiber aging depends on various factors that may decrease effectivefiber strength: residual applied stress, temperature and water. It is assumed that surface flawsare enlarged, consequently crack growth promotes. Maximum water activity is in aqueoussolutions and it is expressed by the relative humidity (RH) in current atmosphere.

Figure 1. Fracture morphology of silica optical fibre (see silica core – typical fragile surface fracturesurrounded by the two layer epoxi-acrylate polymer coating).


As fiber surface has determined fracture to a large extent, external coating appearscritical. This coating is polymeric in most cases, and modern optical fibers are coated by twodifferent layers, a soft coating at glass surface and a hard coating at external surface (Fig. 1).The coating first makes a protection against scratches that occur in normal handling; it alsofills the surface flaws gluing in some a way the two sides of the micro cracks and finally, itreduces water activity at glass surface. Ideally, coating should prevent any water molecule toreach glass surface. Unfortunately, polymeric coatings, including hydrophobic coatings, arevery permeable to water. Only inorganic hermetic coating could make an efficient barrieragainst water. Polymeric coatings (e.g. epoxyacrylates) are preferred in practice because theyare more efficient to inhibit surface defects [13, 20-24].

Various theoretical models are applied for mechanical characterization of optical fibers[25- 26], but the most common one is based on Weibull's statistics.

The Weibull law expresses the failure probability F of a fiber with a length L subjected toan applied stress σ :

[ ])()(]1

1[1oLnLnm

FLn

LLn σσ −=⎥

⎦

⎤⎢⎣

⎡⎭⎬⎫

⎩⎨⎧

−(1)

where m is a size parameter and σο is a scale parameter.

The evolution of ⎥⎦

⎤⎢⎣

⎡⎭⎬⎫

⎩⎨⎧

−]

11[1

FLn

LLn in function of Ln (σ) is known as the Weibull

plot.The values of m and σο are calculated from the slope of the curve and the intersection

with the stress axis. The m parameter characterizes the defect size dispersion [26]. A high mvalue indicates that the distribution of the defect size is homogeneous while a low m valuemeans that surface defects are varying in size. When the curve appears as a broken line withtwo distinct slopes – one small for low stress and the second one large, respectively – one hasassumed two different families of defects, the first one corresponding to large extrinsicdefects, and the second one relating to intrinsic flaws. Other plots encompass several straightlines relating to different groups of defects. The failure probability F is calculated from therelation:

NiFi

5.0−= (2)

where i represents the rank of the measurement and N the total number of values. The σο

parameter represents the stress corresponding to the fiber cumulative fracture probability F ofis 50%.

In the static fatigue measurements, the fiber is subject to a constant stress and onemeasures the time to failure. This time tf is ruled by the following relation:


na

ni

fS

Btσ

2−= (3)

where B is a constant that depends on environment – typically water, S the initial inertstrength of the fiber and n the stress corrosion parameter, and σa the failure stress. The failureprobability of F can be written as:

F t L Lt

B Sff a

n

on

mn

( , ) exp.

.= − −

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−

−

1 2

2σ(4)

Or, in the logarithmic form:

( ) ( )Ln t nLn LnB Ln Sf a in= − + + −σ 2 (5)

The plot Ln t f( ) as a function of Ln a( )σ describes the static fatigue behavior and gives

access to the fatigue parameters n and BSin− 2 .

3. Experimental: Mechanical Measurements

The mechanical strength of the fibers may be measured in different ways corresponding tostatic fatigue and dynamic fatigue [27, 28]. In the static fatigue tests, fibers are subject to apermanent stress, and the time to failure is recorded for a set a fibers. Then a statisticalanalysis gives values for the mean failure strength and the mean lifetime in the testingconditions: type of fiber, temperature, applied stress and water activity. The dynamic fatiguetest consists in applying an increasing tensile strength until fiber breaks. From a convenientdata processing one finds the mean fiber strength. Special equipments are used for thesemeasurements, presented as follows.

3.1. Vertical Bench

The static fatigue under axial tensile loading consists in subjecting a fiber sample to a uniformload as a suspending weight of known value. The two fiber ends are rolled up on a pulleyprovided with a system allowing to block the fiber sample ends and to avoid any slip (Fig.2a). The higher pulley (noted 1) is fixed on a support, while the lower pulley (noted 2) ismobile and interdependent of a plate on which a chosen mass is applied.

This set up leads to carry out static tensile tests on high length fiber samples (usually 4 min length) for applied loads ranging between 5 and 50 N. A number of forty samples cansimultaneously be tested (Fig. 2b). The measurement of the fiber fracture time for different


loads allows determining the static stress corrosion parameter. Thus, the fibers are submittedto different aging conditions and subsequently to mechanical tensile testing.

Silica fiber

Mass

Beam of lightOpticalsensor

Pulley2

Pulley1

Plate

(a) (b)

Figure 2. Vertical static tensile-test bench: (a) diagram of sample fiber under mass loading, (b) generalview.

3.2. Static Fatigue under Permanent Curvature

Another testing bench can be used for static testing [29]. Optical fibers, one meter in length,are subjected to bending stresses by winding around alumina mandrel with calibrateddiameter sizes (Fig. 3a). The constant level of applied stress is adjusted by the proper choiceof the mandrel size. The time to failure is measured, and this corresponds to the time requiredfor the fiber strength to degrade until it equals the stress applied through winding round themandrel. The time to failure is measured by optical detection when the ceramic mandrelmoves out of the special holder. When fiber breaks, the mandrel rocks from its vertical staticposition and the time to failure is directly recorded with an accuracy of ±1 s. The testing setupconsists of a large number of vats containing 16 holders each (Fig. 3b).

The applied stress on the fiber depends on the mandrel diameter according to theMallinder and Proctor relation [30] as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

21

'0

εαεσ E ; αα 43' = ;

dd

fiber

glass

+=

φε (6)


where σ is the applied stress (in GPa), E0 is Young modulus (equal to 72 GPa for thesilica), ε is the relative fiber deformation, α is the constant of the elastic nonlinearity (equal to6), φ is the mandrel diameter (in μm), dglass is the glass fiber diameter and dfiber is the fiberdiameter including the polymer coating. For example, for standard silica optical fibers usedfor telecommunication networks, dglass is equal to 125 μm and dfiber is equal to 250 μm; thisleads to the corresponding stress of 3.92, 3.76, 3.34 and 3.22 GPa for the calibrated diametermandrel of 2.3, 2.4, 2.7 and 2.8 mm respectively. The testing environmental conditionsduring static fatigue measurements (temperature and relative humidity) should be also takeninto account.

E R

Clamping rings

Wound fibre oncalibratedmandrel

Light beam

(a) (b)

Figure 3. Static bending test.

3.3. Vertical Dynamic Tensile Test

Movablepulley

Dynamometriccell

Engine

Devicespeedcontrol

Fixedpulley

500

mm

Fiber

Higherplate

(a) (b)

Figure 4. Schematic description of the dynamic tensile-test bench.


During a dynamic tensile test, the fiber is subjected to a deformation under a constant speeduntil the rupture. The two fiber ends are rolled up on pulleys, having 65 mm in diameter andcovered with a powerful adhesive so as to prevent any fiber slip during the test (Fig. 4). Thelower pulley is fixed while the higher pulley is mobile and its displacement velocity (v,mm/min) corresponds to the chosen deformation speed to carry out the test. Typical fiberlength is 500 mm.

During the test, the deformation and the tensile load are measured using a dynamometriccell while the fiber deformation is deduced from the displacement between the fixed lowerpulley and the mobile higher plate.

The test velocity has an important influence for the failure stresses as this might be seenin Fig. 5. High speeds lead to failure cracks with the same geometry (not curve slope variationfor v=500 mm/min), while the low speeds lead to various crack forms.

-4

-3

-2

-1

0

1

2

0,60 0,80 1,00 1,20 1,40

Failure stress (GPa)

Ln (-

ln(1

-F))

v- 50mm/minv- 150mm/minv-300mm/minv - 500mm/min

(F represents the cumulative fracture probability)

Figure 5. Evolution of failure stresses for different tensile test velocities v (mm/min).

3.4. Long Length Dynamic Tensile Bench

This mechanical bench (Fig. 6) allows to carry out tensile tests on fibers with high lengths(from 0.5 m to 18 m) with broad speeds (ranging between 30 mm/min to 30 m/min with anaccuracy of less than 2 per 1000) and under very diverse environmental conditions(temperature, aqueous solution...). Using the set up, one can obtain information on the defectsize dispersion onto the fiber surface and can determine the dynamic stress corrosionparameter n. Indeed, this parameter is related to the velocity by the following relation:

KA nIV = (7)

where A is a parameter environment dependent, KI is the stress intensity factor and n is aparameter characterizing the material capacity to resist to a stress.


3.5. Two Point Bending Bench

Fibers can also be characterized by using a two point bending testing device (Fig. 7). Samplesof 10 cm in length are bent and placed between the grooved faceplates of the testingapparatus, in order to avoid the fiber slipping during the faceplate displacements and tomaintain the fiber ends in the same vertical plane.

Figure 6. Thirty meters long dynamic tensile bench.

Generally, a series of 30 samples are tested for different faceplate velocities (for example,100, 200, 400 and 800 μm/s, respectively). The failure stress is calculated from the distanceseparating the faceplates, using the Proctor and Mallinder relation, improved by Griffioen [8].Subsequently failure stress is obtained for each tested sample and tracing the classicalWeibull plots one might calculate the statistical parameters. Due to a very short fiber samplepart subjected under stress, this testing method is preferentially used to study the intrinsicdefects or selected flaws.

Fiber

Fiber

Faceplates

ControlBlock

Stepper motor

Faceplates

Computer

Piezo Electric Sensor

Fiber

Figure 7. Dynamic two point bending bench.


4. Results

The work carried out during the last years made possible to apprehend in a more coherentway the failure of the fibers subjected to severe aging conditions. An empirical relationdefining the fiber lifetime tf according to the temperature, residual stress and water contentwas established. For this purpose, silica optical fibers were either immersed in hot water orheated in wet atmosphere with a controlled relative humidity (RH).

The two relations derived from this set of measurements are the followings [14]:

( )⎟⎠⎞

⎜⎝⎛ ⋅−⋅+

=RTTEAt a

fσβφ0

0 exp (8)

( )⎟⎠⎞

⎜⎝⎛ ⋅−⋅+

= −

RTTEZAt a

fσβφδ 0

0 exp (9)

The first relation (8) applies to fibers aged in liquid water, while the second one (9)concerns fibers exposed to humid atmosphere with variable RH (RH = Z), temperature T andapplied stress σ. It is worth noting that the factor Φ corresponds to some kind of relaxationthat decreases the effect of the applied stress. Its magnitude grows with temperature andapplied stress.

In a second set of measurements, fibers were immersed in hot water at 65°C or at 85°Cfor a long time (up to 24 and 27 months). Then they were characterized in static and dynamicfatigue. As one could expect, fiber strength in dynamic fatigue decreases as aging timeincreases. More surprisingly, the time to failure of aged fibers subjected to static fatigueincreased enormously by comparison to non-aged fibers [31]. However this unexpected effectdid not follow a regular evolution versus time, but a rather cyclic one (Fig. 8).

(In legend calibrated mandrel diameter, in mm, in the case of the static fatigue testing set-up – Fig.3)

Figure 8. Evolution of the fiber failure time in function of different aging conditions.


The explanation for this lifetime increase can be found in the structural relaxationidentified in the previous relations. The failure mechanism of the aged fibres involves surfacephenomena, in relation to water activity. A layer of hydrated silica is likely to be formed atfibre surface [15]. This vitreous hydrated phase may relax under stress at room temperature,which partly compensates the external applied stress in static fatigue. The change of the glasssurface was exemplified by the indentation behaviour that is different from that of normalsilica [32].

New experiments are carried out as well on standard silica fibers as on new fibers [33].Fibers with a hermetic coating, fibers before and after photo-printing, fibers of polymer, onaverage have diameters between 85 µm and 125 µm. The influence of temperature, water andvarious corrosive agents on the mechanical fiber strength is determined. The coating aging isalso taken into account. Characterizations are also carried out on fibers belonging to differentvitreous systems (fluorides, oxides, sulphides) to detect and analyze less visible phenomenawhen silica fibers are studied.

For several silica fibers subjected to vertical static tensile testing (see Fig. 2) undervarious loadings, one can notice that more the suspended mass value is high; more the time ofrupture is large (Fig. 9). For weak loads (15 N), two families of cracks exist (a slope breakindicates the dispersion of the microcrack shapes).

-5

-4

-3

-2

-1

0

1

2

0 2 4 6 8ln (time to rupture) (h)

ln(-l

n(1-

F))

20N15N25N30N

Figure 9. Time to rupture evolution for different loadings (F represents the cumulative fractureprobability).

5. Final Remarks

The huge development of the telecommunication networks has been made possible by theavailability of low cost and high quality silica optical fibers. As industrial production reachesmillions of km, research rather focuses on networks and advanced components. Fiberreliability is not a critical issue at this time because few problems were encountered, most ofthem being accidental. However future fiber local loops will put fibers under large and


permanent stress. In addition, lighter and less expansive cables could be manufactured iftransient or permanent stresses have no significant influence on fiber lifetime.

Fiber aging has been the subject of numerous studies leading to theoretical models forlifetime assessment. While ground observations do not contradict these predictions, theaccuracy of the models is questionable due to the complexity of the aging mechanism. In thisrespect, experiments implemented on a long time scale are likely to bring new information.

There are some questions underlying the reliability studies. Aging parameters encompasstime, temperature, applied stress and water activity. While the critical part of water in failuremechanism is well known, its real impact varies according to the physical state - liquid orvapor - and partial vapor pressure. The stress applied to the fiber may be temporary, forexample during the proof test or network installation, or permanent when the fiber is bent incable and connecting areas. The aging mechanism is assumed to enlarge or to extend the"Griffith flaws" which are spread at the fiber surface. These defects may be described asmicro-cracks which grow under applied stress in wet environment. Although this mechanismis believed to be irreversible, water may also induce some curing effect which couldcorrespond to the geometrical smoothing of the crack tip [34, 35].

There is a practical interest in collecting quantitative information on the aging of thecommercial optical fibers over a long period of time. Such observations should allow a moreaccurate comparison between experimental and calculated strengths and make lifetimeassessments more realistic as testing periods (> 2 years) become closer to the lifetimerequired by network users that is at least 20 years.

Acknowledgments

Authors express their gratitude to France Telecom for technical assistance and equipmentsupply and to Region Bretagne for financial support.

References

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[3] Sakaguchi S., Kimura T., (1981). Influence of temperature and humidity on dynamicfatigue of optical fibers. J. Amer. Ceram. Soc. 64 [5], 259-262.

[4] Duncan W. J., France P. W. and Craig S. P., The effect of environment on the strengthof optical fiber. Pp. 309-328 in Strength of Inorganic Glass, Edited by C.R. Kurkjian,Plenum press, New York, 1985.

[5] Matthewson M. J. and Kurkjian C. R., (1988). Environmental effects of the static fatigueof silica optical fiber. J. Amer. Ceram. Soc. 71 [3], 177-183.

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[7] Michalske T., Smith W., Bunker B., (1991). Fatigue mechanisms in high-strength silica-glass fibers. J. Am. Ceram. Soc. 74, [8], 1993-1996.


[8] Griffioen, W. Optical fiber reliability. Thesis edited by Royal PTT, The NetherlandsNV, PPT Research, Leidschendam, 1994.

[9] Glaseman G. S., (1994). Assessing the long term reliability of optical fibers. Proc.National Fiber Optics Engineers Conference, 297.

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[11] Volotinen T. T. – Water tests on optical fibers – Proc. SPIE 3848, 134-143, (1999).[12] Semjonov S. L., Kurkjian C. R., (2001). Strength of silica optical fibres with micron size

flaws. J. Non-Cryst. Solids, 283, 220-224.[13] Armstrong J. M. and Matthewson M. J., (2000). Humidity dependence of fatigue of

high-strength fused silica optical fibers. J. Am. Ceram. Soc. 83, [12], 3100-3108.[14] Poulain M., Evanno N., Gouronnec A. – Static fatigue of silica fibers – Optical fiber and

fiber component mechanical reliability and testing II, M. J. Matthewson, C. R. Kurkjian,Editors, Proc. SPIE 4639, 64-74, (2002).

[15] Berger S., Tomozawa M., (2003). Water diffusion into silica optical fiber. J. Non-Cryst.Solids, 324, 256-263.

[16] Gougeon N., El Abdi R and Poulain M., (2004). Evolution of strength of silica fibersunder various moisture conditions. Optical Materials, 27, 75-79.

[17] Severin I., El Abdi R. and Poulain M., (2007). Strength measurements of silica opticalfibers under severe environment. Optics & Laser Techn.. 39, [2], 435-441.

[18] Griffith A. A., Phil. Trans. 221A, 163 (1920).[19] Poulain M., Vacancy model of ionic glasses. Proc Int. symp. Non Oxide Glasses, Part B,

pp 22-26, Corning USA and “What is glass?”(briton langage) ΣKIANT, 1, 13-26, (1996).[20] Wei T., Skutnik J., (1988). Effect of coating on fatigue behavior of optical fiber. J. Non-

Cryst. Solids, 102, 100-105.[21] Kurkjian C. R., Simpkins P. G., Inniss D., (1993). Strength, degradation and coating of

silica lightguides. J. Am. Ceram. Soc. 76, [5], 1106-1112.[22] Shiue S. T., Ouyang H., (2001). Effect of polymeric coating on the static fatigue of

double-coated optical fibers. J. App. Phy. 90, [11], 5759-5762.[23] Mrotek J. L., Matthewson M. J., Kurkjian C. R., (2001). Diffusion of Moisture through

optical fiber coatings. Journal Light-wave Technol. 19, [7], 988-993.[24] Mrotek J. L., Matthewson M. J., Kurkjian C. R., (2003). Diffusion of Moisture through

fatigue and aging-resistant polymer coatings on lightguide fibers. Journal Light-waveTechnol. 21, [8], 1775-1778.

[25] Schmitz G. K. and Metcalfe A. G., (1967). Testing of fibers. Mat. Res. Stand., 7 [4],862-865.

[26] Matthewson M. J., (1994). Optical fiber reliability models. Proc. SPIE, Critical Reviews,CR 50, 3-31.

[27] Matthewson M. J., (1994). Optical fiber mechanical testing techniques. Proc. SPIE,Critical Reviews, CR 50, 31-59.

[28] Severin I., El Abdi R., Poulain M. and Amza G., (2005). Fatigue testing of silica opticalfibres. Journal of Optoelectronics and Advanced Materials, 7 [3], 1581-1588 .

[29] International standard IEC 793-1-3, First Edition 1995-10 .[30] Mallinder, F.P., Proctor, B.A., (1964) Phys. Chem. Glass, 5, 91.[31] Gougeon N., El Abdi R. and Poulain M. (2003). Mechanical reliability of silica optical

fibers. J. Non-Cryst. Solids, 316, 125-130.


[32] Gougeon N., Sangleboeuf J. C., El Abdi R., Poulain M. and Borda C. T., (2005).Indentation Behavior of Silica Optical Fibers Aged in Hot Water. Fiber and IntegratedOptics. 24, [5], 491-500.

[33] Severin I., Poulain M., ElAbdi R. (2005). Phenomena associated to aging of silicaoptical fibers. Photonic Applications in Devices & Communication Systems, P.Mascher, A. P. Knights, eds., Proc. SPIE 5970.

[34] Hirao K., Tomozawa M. (1987). Kinetics of crack tip blunting of glasses. J. Am. Ceram.Soc. 70, [1], 43-48.

[35] Hirao K., Tomozawa M., (1987). Dymanic fatigue of treated high-silica glass:Explanation by crack tip blunting. J. Am. Ceram. Soc. 70 [6], 377-382.

INDEX

A

absorption spectra, 272 access, 53, 75, 112, 216, 359 accounting, 254 accuracy, 120, 149, 201, 215, 356, 360, 362, 366 acetone, 33 acetylene, 332 achievement, 206 acid, 31, 33, 36, 44, 104, 105 acrylate, 357 adaptability, 5 adjustment, 59, 215 adriamycin, 31, 48 adsorption, 40 aerospace, 5, 260 AFM, 356 agent, 31 aging, xii, 260, 355, 356, 357, 360, 364, 365, 366,

367, 368 aging process, xii, 355 albumin, 45 algorithm, 59, 60, 65, 68, 74, 77, 243, 246, 247, 320 alternative(s), x, 54, 65, 146, 231, 233, 238, 249,

255, 261 aluminum, 144 amplitude, x, 19, 78, 84, 85, 128, 131, 166, 171, 172,

177, 209, 279, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 293, 297, 304, 324

AN, 175, 231 annealing, 65 antibody, viii, 15, 28, 29, 30, 31, 36, 37, 38, 39, 40,

42, 43, 45, 46 antigen, 31, 38, 39, 40, 43 antimony, 188 APC, 220 apoptosis, 31 argon, 31, 260

arsenic, 273 assessment, 366 assignment, 251 assumptions, viii, 43, 52 asymmetry, 105, 152 atoms, ix, 119, 120, 122, 124, 128, 130, 131, 132,

133, 134, 136, 137, 138, 139, 140, 142, 143, 146, 147, 148, 149, 150, 151, 152, 153, 154, 336

attachment, 21, 38, 39, 40, 41, 42, 43, 44 attention, vii, ix, 3, 4, 5, 161, 163, 164, 176, 217,

258, 302 attenuated total reflectance, 261 Australia, 315 automobiles, 260 availability, 365 averaging, xi, 301, 303 avoidance, 108

B

Bacillus, 22, 28, 29, 47 Bacillus subtilis, 22 backscattering, ix, 54, 70, 71, 72, 73, 161, 162, 164,

170, 171, 172, 176, 177, 179, 180, 181, 182, 185 bacteria, 4, 28, 41, 48 bandgap, xi, 258, 260, 262, 265, 273, 315, 316, 317,

318, 319, 320, 321, 322, 332, 333 bandwidth, viii, x, 51, 54, 62, 63, 65, 68, 69, 72, 74,

76, 77, 84, 114, 115, 168, 170, 188, 206, 207, 216, 218, 221, 222, 232, 233, 241, 247, 257, 258, 259, 269, 270, 275, 342

beams, 120, 132, 140, 146, 150, 151, 154, 178, 179, 180, 181, 217, 280

beef, 22, 28, 46 behavior, viii, 15, 18, 35, 40, 72, 78, 97, 102, 115,

324, 356, 359, 367 Beijing, 3 bending, 177, 259, 262, 349, 350, 360, 361, 363

Index 370

bias, 346 binary decision, 252 binding, 29, 31, 39, 40, 41, 42, 47 biomarkers, 30 biomechanics, 84 biomolecule(s), viii, 15, 46 biosensors, 4, 15, 35 biotin, 29 birefringence, ix, 83, 84, 87, 88, 89, 90, 91, 92, 94,

98, 101, 102, 104, 106, 107, 109, 110, 170, 184, 225, 274, 340, 343, 346, 347, 349

bismuth, 184 blackbody radiation, 272 blocks, 297 blood, 26, 31, 275 BN, 175 BNP, 25, 26 Boltzmann constant, 58 bonding, 36, 44, 259 bonds, 356, 357 boundary value problem, 59 Bragg grating, viii, xii, 4, 51, 54, 83, 84, 85, 92, 97,

98, 101, 104, 106, 115, 161, 169, 170, 182, 184, 213, 227, 228, 315, 331, 332, 335, 338, 339, 349, 351, 356

branching, 210 brass, 271 breast carcinoma, 31 breathing, 280 broadband, viii, x, 51, 53, 63, 64, 79, 85, 113, 182,

187, 205, 208, 227, 261, 271 buffer, 22, 23, 24, 25, 26, 27, 29, 30, 38, 39, 177 building blocks, 297 burning, 169, 338

C

cabinets, 53 cables, vii, 53, 357, 366 calibration, 33, 94, 96, 98, 107, 192 Canada, 205 cancer screening, 31 candidates, 182, 244, 302 capillary, 120, 121, 127, 128, 131, 132, 133, 134,

135, 260, 262 carbohydrate, 37 carbon, 6, 8, 259 carboxylic groups, 36 carcinogenicity, 31 carcinogens, 31 carcinoma, 31, 48 cardiovascular disease, 30, 49

carrier, 4, 5, 114, 115, 165, 167, 206, 233, 234, 236, 317, 342

cDNA, 32 cell, 21, 26, 27, 28, 31, 35, 40, 44, 48, 148, 262, 361,

362 cell culture, 27, 31 cell growth, 28 ceramic, 360 CFBG, 338, 341, 350 CGLE, x, 279, 280, 281, 285, 286, 287, 288, 289,

290, 291, 293, 297 channels, 53, 54, 57, 63, 75, 76, 77, 84, 115, 169,

206, 213, 215, 216, 218, 219, 227, 236, 305, 338, 346

chemical bonds, 356, 357 chemical etching, 33, 98, 105, 106 chemical properties, 15 China, 3, 257 Chinese, 24 cladding, x, 6, 16, 17, 18, 19, 31, 88, 105, 106, 121,

122, 143, 144, 146, 167, 168, 177, 190, 209, 210, 215, 218, 232, 257, 258, 260, 261, 262, 263, 264, 265, 266, 269, 270, 272, 273, 274, 275, 316, 332

cladding layer, 177 classes, xi, 263, 273, 301, 303 cleaning, 36, 42 clustering, 201 CO2, 4, 207, 208, 261, 272, 274, 275 coagulation, 30, 48 coatings, 190, 258, 261, 271, 273, 356, 358, 367 codes, 112, 113, 114, 247 coding, 114, 244, 246, 247, 249 coherence, 89, 109, 151, 224, 225, 226 collaboration, 134, 309 collisions, xi, 149, 151, 279, 302, 315, 317, 324, 325,

326, 327, 328, 329, 330, 333 combined effect, 39, 312 communication, vii, viii, x, xi, xii, 51, 52, 55, 56, 79,

83, 84, 108, 169, 187, 206, 215, 231, 232, 234, 236, 238, 239, 242, 244, 246, 268, 269, 270, 275, 299, 302, 312, 335, 336, 352

communication systems, viii, x, xi, xii, 51, 52, 55, 56, 79, 108, 231, 232, 234, 238, 239, 244, 246, 275, 299, 335, 336, 352

community, 302 compatibility, 84, 205 compensation, x, 205, 206, 213, 218, 220, 221, 302,

311, 347 competition, 111, 169, 174, 179, 180, 291, 336, 339 competitiveness, 336 complementary DNA, 32 complexity, 165, 366 complications, 30

Index 371

components, ix, xi, 7, 18, 54, 57, 58, 68, 84, 87, 89, 90, 91, 93, 98, 104, 111, 122, 164, 166, 173, 176, 188, 205, 206, 207, 208, 212, 218, 220, 224, 225, 226, 233, 234, 236, 238, 250, 301, 303, 312, 356, 357, 365

composites, 295 composition, 190, 215 compounds, 4 computation, viii, 52, 61, 67 computing, 206, 253 concentration, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40,

41, 42, 43, 44, 104, 105, 106, 107, 164, 168, 200, 215

condensation, 315 conduction, 173, 342 conductivity, 8, 172, 221 configuration, vii, 3, 4, 49, 57, 65, 72, 75, 77, 78,

135, 142, 164, 165, 166, 169, 178, 189, 207, 216, 218, 220, 222, 226, 227, 303, 337, 338, 349

confinement, vii, 16, 52, 177, 264, 269, 271, 316 confusion, 342 Congress, 202 conjecture, 318, 321 consolidation, viii, 51 constituent materials, 258, 263, 272 constraints, 303, 307 contaminant, 47 contamination, 7, 32, 48, 143 continuity, 122, 173 control, 5, 68, 71, 72, 81, 154, 166, 170, 202, 212,

213, 215, 219, 220, 232, 244, 247, 302, 337, 340, 343, 345, 349, 357, 361

convergence, 8, 59, 65, 73 conversion, ix, 4, 108, 164, 187, 188, 202, 220, 233,

252, 260, 275, 332 cooling, xi, 68, 97, 111, 148, 154, 293, 335, 336,

338, 341, 352 Copenhagen, 255 corn, 26 coronary heart disease, 30 correlation(s), 176, 179, 180, 182, 193, 194, 201 correlation function, 176, 180 corrosion, xii, 84, 355, 359, 360, 362 costs, 68, 84 couples, 55 coupling, xi, 7, 16, 18, 58, 85, 87, 114, 128, 134,

136, 146, 147, 177, 178, 179, 182, 185, 187, 209, 216, 218, 264, 274, 301, 303, 315, 316, 317, 333, 342

covalent bonding, 36, 44 coverage, 40, 41, 43 crack, 357, 362, 366, 367, 368 C-reactive protein, 25, 30

critical value, 170, 179 cross-phase modulation, xi, 166, 301, 312 CRP, 25 crystal growth, 154 crystalline, 59, 273 culture, 27, 28, 31 curing, 366 CVD, 30 cytochrome, 30, 31, 48 cytokines, 30, 31, 47 cytoplasm, 31

D

damping, 298 data processing, 356, 359 data transfer, 269 decay, ix, 56, 120, 187, 188, 189, 196, 197, 199, 201,

202, 336 decibel, 62 decision making, 253 decisions, 246, 252 decoding, 114, 238, 246, 247, 252 decoupling, 108 defects, xii, 7, 164, 355, 356, 358, 363, 366 defense, 15, 272 deficiency, 30 definition, 78, 146, 240, 304, 317, 321 deformation, 84, 96, 99, 103, 207, 361, 362 degenerate, 71, 141, 274, 305, 339 degradation, 52, 54, 62, 220, 367 delivery, x, 6, 257, 258, 261, 275, 316, 333, 357 demand, 7, 52, 53, 260, 269, 270 Denmark, 255 density, vii, 3, 5, 6, 7, 9, 53, 56, 73, 154, 162, 163,

164, 172, 173, 175, 176, 181, 198, 199, 200, 202, 240, 271, 356

density fluctuations, 356 dependent variable, 59 depolarization, 222, 225 deposition, 45, 139, 154 derivatives, 199, 304, 309 destruction, 53 detection, viii, x, 4, 15, 21, 28, 29, 30, 37, 38, 40, 42,

44, 45, 46, 47, 48, 49, 131, 134, 217, 225, 231, 232, 233, 234, 235, 236, 237, 238, 239, 241, 249, 250, 255, 271, 275, 360

detection techniques, 271 detonation, 4 deviation, 68, 76, 77, 176, 179, 180, 182, 221 dielectric constant, 315 dielectric permittivity, 175 dielectrics, 271

Index 372

differential equations, 162, 165, 171, 175, 198 differentiation, 304, 309 diffraction, ix, 119, 120, 124, 125, 126, 143, 144,

152, 153, 154, 280 diffusion, 367 digital communication, 242 diode laser, 29, 133, 134, 165 diodes, 64, 65, 162, 165, 170 dipole, ix, 119, 120, 121, 128, 130, 134, 136, 146,

150, 151, 152, 154, 155, 171, 173 dipole moment, 173 dispersion, x, xi, 53, 58, 68, 71, 72, 110, 122, 123,

163, 165, 183, 206, 216, 218, 220, 226, 227, 228, 231, 232, 238, 241, 259, 261, 264, 274, 279, 280, 281, 285, 293, 301, 302, 303, 305, 307, 308, 309, 311, 312, 315, 316, 317, 318, 320, 321, 322, 324, 325, 326, 327, 328, 329, 330, 331, 333, 338, 347, 358, 362, 365

displacement, 297, 343, 362 distortions, 244 distribution, 5, 59, 69, 70, 120, 124, 126, 139, 140,

141, 142, 143, 144, 146, 150, 152, 153, 154, 173, 176, 192, 193, 198, 206, 246, 250, 356, 358

distribution function, 173 divergence, 206 diversity, 349 division, xi, 53, 112, 169, 182, 183, 187, 217, 227,

228, 270, 311, 335, 336, 337, 345, 346 DNA, viii, 4, 15, 21, 22, 27, 31, 32, 44, 48, 49 DOP, 221, 226 dopants, 215, 336 Doppler, 148 dream, 302 drinking water, 31 DRS, 72 DSC, 221 duration, 207, 244 dyes, 30 dynamic control, 220 dynamical systems, 299

E

E. coli, viii, 15,23, 28, 29, 32, 35, 37, 38, 40, 41, 44 earth, 183, 202 EEA, 79 eigenvalue, 240, 321 Einstein, Albert, 154, 306, 312, 315 elaboration, 208 elasticity, 257 electric current, 34 electric energy, 4, 5

electric field, xi, 19, 121, 124, 127, 142, 171, 176, 177, 281, 301, 303

electrical power, 221 electrodes, 34 electromagnetic, viii, ix, 83, 84, 119, 120, 121, 124,

154, 171, 302 electromagnetic fields, ix, 120, 121, 154 electromagnetic waves, 302 electromagnetism, 5 electron(s), 55, 128, 131, 134, 144, 162, 173, 175,

280, 298, 342 ELISA, 31 elongation, 208, 209 emission, ix, 28, 30, 56, 57, 72, 94, 111, 165, 187,

189, 191, 192, 193, 194, 195, 196, 197, 198, 199, 201, 202, 219, 342

encoding, 114, 251 endotoxins, 29 endurance, vii, 3, 9 energetic materials, 4 energy, vii, xi, 3, 4, 5, 6, 7, 8, 55, 62, 77, 87, 134,

139, 151, 162, 164, 166, 173, 176, 178, 189, 190, 197, 198, 199, 200, 216, 257, 269, 280, 291, 301, 303, 316, 318, 320, 321, 325, 338, 342

energy density, 5 energy transfer, 342 enlargement, 54 environment, 5, 16, 30, 44, 272, 280, 359, 362, 366,

367 environmental change, 356 environmental conditions, 212, 337, 361, 362 enzyme(s), 4, 48 epoxy, 37 equilibrium, 39, 56, 280, 297 equipment, vii, 99, 100, 101, 104, 108, 366 erbium, ix, xi, 94, 111, 162, 164, 165, 166, 167, 168,

169, 182, 183, 184, 187, 188, 201, 219, 335, 336, 339, 340, 341

Escherichia coli, 22, 28, 45, 46 estimating, 251, 252 etching, 28, 33, 44, 98, 104, 105, 106, 107, 181 ethylene glycol, 38, 39 Euro, 276 European Union, 79 evanescent waves, 132, 133, 134, 151 evaporation, 143, 144 evidence, 31, 182 evolution, 4, 52, 54, 57, 59, 61, 62, 67, 68, 73, 74,

84, 89, 94, 96, 97, 105, 106, 110, 151, 162, 165, 171, 183, 281, 290, 291, 294, 296, 313, 356, 358, 364, 365

excitation, 29, 30, 32, 46, 127, 130, 131, 132, 134, 144, 196, 210

Index 373

exercise, 254 exploitation, 258 exponential functions, 197, 309 exposure, viii, 83, 85, 104, 106, 213, 214, 357 extinction, 169, 222, 223, 224 extrusion, 273

F

fabrication, viii, ix, x, xii, 15, 33, 44, 181, 196, 205, 206, 213, 214, 216, 219, 226, 227, 229, 257, 272, 273, 355

Fabry-Perot filters, xii, 335 failure, xii, 84, 222, 355, 356, 358, 359, 360, 362,

363, 364, 365, 366 family, xi, 303, 315, 316, 317, 318, 320, 321, 328 fatigue, xii, 355, 356, 358, 359, 361, 364, 365, 366,

367, 368 feedback, xi, 54, 56, 161, 162, 163, 170, 182, 331,

335, 336, 341, 342, 352 FFT, 241 fiber aging, 357 fiber optics, 101, 104, 272, 366 fibers, vii, viii, ix, x, xii, 3, 4, 5, 6, 7, 8, 9, 15, 16, 17,

18, 21, 28, 29, 32, 34, 35, 37, 41, 42, 45, 51, 53, 55, 58, 59, 68, 72, 83, 84, 86, 87, 88, 89, 91, 92, 96, 97, 98, 104, 105, 106, 107, 108, 111, 119, 120, 121, 123, 124, 132, 133, 139, 154, 163, 167, 187, 188, 190, 191, 227, 228, 232, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 269, 270, 271, 272, 273, 274, 275, 298, 299, 302, 311, 312, 316, 332, 333, 340, 355, 356, 357, 358, 359, 360, 361, 362, 364, 365, 366, 367, 368

fibre laser, 206 fibrinogen, 27, 30 film(s), 144, 215, 261, 273 filters, xii, 4, 109, 165, 166, 206, 209, 218, 219, 227,

228, 239, 335, 338, 350, 352 financial support, 309, 366 first generation, x, 257, 258 flame, 33, 34, 44, 208 flatness, 54, 69, 219, 220 flexibility, 15, 84, 110, 212, 257, 258, 274, 275 flight, 148 fluctuations, vii, 15, 55, 166, 215, 227, 286, 356 fluid, 280, 281 fluorescence, 26, 28, 29, 30, 31, 44, 48, 49, 56, 148,

150, 197 fluorine, 259 fluorophores, 48 focusing, 5, 119 food, 29, 44, 47 food poisoning, 29

Fourier, 125, 157, 219, 227, 239, 304, 310, 320 Fourier transformation, 125 four-wave mixing, xi, 169, 170, 311, 335, 336, 338,

339 France, 157, 161, 162, 164, 166, 168, 170, 172, 174,

176, 178, 180, 182, 184, 276, 355, 366 freedom, xi, 240, 293, 301, 303 FTTH, 269, 275, 356 function values, 65 functionalization, 37 fusion, 19, 33, 34, 35, 38, 44, 190, 206, 207

G

gases, 4, 316 Gaussian, 127, 143, 144, 146, 147, 149, 153, 176,

180, 181, 239, 240, 241, 246, 250, 302, 312 gel, 45, 259 generalization, 165, 170 generation, ix, x, xi, xii, 54, 57, 108, 111, 113, 119,

120, 140, 141, 143, 152, 154, 257, 258, 260, 271, 275, 306, 312, 316, 324, 335, 336, 342, 346, 352

geometrical parameters, 219 germanium, 93, 122, 261 Germany, 81 Ginzburg-Landau equation, x, 279, 280, 297, 298,

299 Ginzburg-Landau equation (CGLE), x, 279, 297 glass(es), vii, ix, xii, 4, 5, 7, 8, 30, 88, 91, 120, 121,

122, 127, 131, 132, 133, 142, 188, 190, 192, 196, 197, 198, 200, 205, 227, 257, 258, 259, 260, 261, 262, 270, 273, 336, 355, 356, 357, 358, 360, 361, 365, 366, 367, 368

glass transition temperature, 259 global communications, 83 glucose, 35, 44, 275 glycine, 38 glycol, 38, 39 gold, 26, 37, 44, 48 graph, 41, 77, 96, 127, 181 gratings, viii, xii, 4, 54, 72, 83, 84, 91, 92, 98, 101,

102, 104, 108, 109, 114, 115, 161, 170, 184, 213, 218, 227, 232, 331, 332, 335, 340

gravity, 147, 151 grazing, 120, 132, 260 Green’s function, 176 Griffith’s flaws, xii, 355, 356 groups, 30, 36, 37, 55, 132, 163, 166, 196, 358 growth, xii, 28, 44, 46, 154, 232, 258, 269, 298, 318,

321, 323, 355, 357, 367 growth rate, 318, 321, 323 guidance, 17, 120, 127, 132, 133, 134, 136, 139, 146,

154, 155, 273, 316, 332

Index 374

H

halogen, 192 Hamiltonian, 293 HD, 35 HDPE, 271, 272 HE, 122 healing, 31, 47 heart disease, 30 heat(ing), vii, x, 15, 17, 19, 33, 35, 38, 44, 97, 130,

136, 150, 151, 207, 208, 209, 213, 214, 257, 258, 275

height, 152, 177 helium, 133 hemoglobin, 30 high power density, 164 hip, 24, 26, 113 homeland security, 44 homogeneity, 111, 112 Hong Kong, 301 host, 264, 271, 273, 274, 336 house(ing), 33, 115, 202 humidity, xii, 4, 45, 280, 355, 356, 357, 361, 364,

366, 367 hybrid, 53, 65, 66, 68, 77 hybridization, 32, 44, 48 hydrocarbons, 45 hydrochloric acid, 36 hydrofluoric acid, 33, 44, 104 hydrogen, 45, 213, 259, 316, 332 hydrogenation, 213 hydrolysis, 36 hydroxide, 36 hydroxyl groups, 36, 196 hypothesis test, 249, 250

I

identification, 233, 238 ignition energy, 8 IL-6, 27, 31 illumination, 260, 275 images, 4, 124, 142 imaging, 31, 120, 126, 143, 144, 146, 147, 150, 155,

271, 357 immobilization, viii, 15, 28, 29, 36, 37, 40, 43, 47 immunity, viii, 83, 84 impairments, 238 implementation, xii, 55, 57, 62, 66, 68, 112, 113,

115, 228, 335, 336 impurities, 55 in situ, 164

in vivo, 31, 48 incidence, 16, 120, 132, 260 inclusion, 254, 282 India, 55 indication, 31 indices, 6, 121, 122, 212, 213, 214, 263, 264, 269 indirect measure, 84 industrial production, 365 industry, 357 inelastic, 55, 293 infarction, 30 infinite, 121, 127, 207, 212, 345 information processing, 206 information technology, 83 inhibition, 48 initial state, 246, 297 inoculation, 23 insertion, viii, 83, 84, 108, 114, 165, 188, 190, 200,

201, 206, 215, 216, 218, 220, 221, 223, 336, 346 insight, 18 instability, xi, 111, 285, 289, 298, 302, 315, 316,

320, 321, 323, 328 instruments, 191 integrated optics, 178 integration, 59, 60, 218, 225, 226, 238, 240, 254 intensity, 4, 28, 30, 44, 53, 56, 102, 120, 122, 123,

124, 126, 127, 128, 130, 131, 132, 133, 134, 138, 139, 141, 142, 144, 145, 146, 147, 150, 152, 153, 154, 165, 180, 181, 188, 211, 212, 217, 232, 305, 331, 362

interaction(s), viii, xi, 17, 29, 43, 52, 53, 54, 55, 57, 64, 76, 77, 133, 136, 152, 166, 171, 279, 293, 294, 298, 299, 302, 315, 316, 317, 324, 325, 326, 332, 336, 342

interface, 16, 53, 124, 132, 142, 209, 210, 258, 260 interference, 5, 84, 108, 113, 114, 115, 152, 174,

176, 185, 217, 232, 241 internet, 80, 187, 255, 269 interpretation, 21 interval, 232, 233, 236, 237, 240, 243, 246, 247, 249 inversion, 63, 183, 233, 342 investment, 52 ionization, 134, 139 ions, 4, 128, 131, 189, 190, 198, 200, 201, 219, 261,

336 IR, 35, 42, 80, 272, 273, 274 Islam, 81 isolation, 212, 216, 218, 220, 223, 224, 346 isotope(s), 136, 137 Italy, 51, 81, 161, 227 iterative solution, 165

Index 375

J

Japan, 33, 134, 187, 276, 298, 312

K

Karhunen-Loeve Series Expansion (KLSE), x, 231, 238, 239, 240, 241, 243, 253, 254

kernel, 240, 241, 253 kinetics, xii, 32, 40, 48, 355 Korea, 119, 134, 231, 276, 335

L

laser(s), vii, viii, ix, x, xi, xii, 3, 4, 5, 6, 7, 8, 9, 29, 31, 42, 51, 53, 54, 56, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 81, 108, 111, 112, 115, 119, 120, 121, 123, 124, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 187, 189, 191, 196, 201, 202, 206, 207, 208, 215, 221, 222, 226, 233, 234, 257, 258, 260, 261, 269, 272, 274, 275, 280, 281, 287, 297, 298, 299, 300, 316, 332, 333, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 357

laser radiation, x, 257, 258, 272, 275 lasing effect, 72, 73 lattices, 313 leakage, 53, 143, 172, 175, 177, 208, 258, 262, 264,

266, 316 lectin, 29 LED, 206 lens, 7, 120, 126, 147, 150, 155 leprosy, 31 lifetime, 84, 152, 196, 197, 198, 342, 356, 359, 364,

365, 366 light beam, 136, 146 light scattering, 136 light transmission, 18, 28, 39 likelihood, 238, 246, 249 limitation, 164, 227, 259 linear function, 109 linkage, 29, 38 links, vii, 53, 275, 311 liquid nitrogen, 336, 338, 352, 356, 357 liquid phase, 45 Listeria monocytogenes, 23, 28, 46, 47 literature, 16, 98, 164, 165, 166, 170, 200, 260, 274,

303, 305, 307

local area networks, 357 location, 18, 228, 304, 305, 342 long distance, 120, 221, 232, 320, 323 LPS, 29 luminescence, 189, 196, 197, 201 lysine, 28

M

magnetic field, 150 malaria, 31 management, 52, 302, 303, 311 Manakov model, xi, 301, 303 manipulation, 119, 154, 206 manufacturer, 37 manufacturing, 207, 257, 258 mapping, 176, 236 market(s), 84, 258, 260, 275, 356, 357 Marx, 48 masking, 144 Massachusetts, 81 matrix, 59, 73, 81, 99, 100, 103, 108, 162, 173, 210,

211, 223 Maxwell's equations, 162 measurement, viii, 15, 32, 46, 47, 84, 98, 99, 100,

104, 108, 126, 148, 162, 169, 201, 202, 358, 359 measures, 60, 303, 305, 358 mechanical properties, vii, 5, 263 mechanical stress, 94 mechanical testing, 367 media, viii, xi, xii, 35, 47, 51, 169, 306, 313, 315,

331, 335, 336, 352 medical diagnostics, 44, 275 medicine, 84 melting, vii, 190 memory, 184 metals, 271 methanol, 26 microarray, 48 microcavity, 149 micrometer, 96 microorganisms, 28 microscope, 7, 31, 33, 105, 144, 146, 208 microscopy, 272 microspheres, 31 microwave, 108, 264, 271 military, 272 miniaturization, 5 mitochondria, 31 mixing, xi, 54, 169, 170, 183, 232, 302, 311, 335,

336, 338, 339, 341 model system, 305 modeling, 46, 59, 63, 68, 166, 183, 273, 308

Index 376

models, x, 162, 164, 165, 170, 231, 305, 316, 356, 358, 366, 367

modules, 52, 206, 212, 218, 222, 227 modulus, 261, 305, 310, 361 moisture, xii, 355, 367 molecular weight, 269 molecules, 15, 32, 43, 44, 49, 55, 259, 273, 332, 356 moon, 353 Morocco, 205 morphology, 357 motion, 131, 294 motivation, 305 multidimensional, 313 multiples, 54, 237 multiplicity, 280 multiplier, 128, 131, 134 muscles, 30 mutation, 65, 68 myocardial infarction, 30 myoglobin, 30, 47

N

NADH, 24 nanometers, 222 National Science Foundation, 45 national security, 29 Nd, 4, 6, 164, 196 necrosis, 27, 31 neodymium, 164 Netherlands, 255, 367 network(ing), ix, x, 52, 114, 188, 205, 207, 216, 218,

226, 227, 269, 356, 366 neural networks, 65 New York, 79, 115, 155, 156, 157, 158, 255, 297,

299, 311, 313, 366 next generation, 108 NIR, 192 nitrogen, xi, 133, 335, 336, 338, 341, 352, 356, 357 nodes, 124, 146, 154 noise, viii, 51, 52, 54, 56, 57, 62, 63, 64, 75, 162,

165, 166, 167, 168, 169, 182, 184, 189, 191, 206, 218, 220, 221, 228, 238, 239, 240, 241, 246, 247, 250, 302

nonequilibrium, 299 nonlinear dynamics, 154 nonlinear optical response, 302, 331 nonlinear optics, 4, 279, 280, 281, 302 nonlinear Schrödinger model, xi nonlinear systems, 297 normalization constant, 127 numerical analysis, 123, 262 numerical aperture, 6, 7, 168, 190, 198, 259

numerical computations, 65

O

observations, 244, 366 OFS, 188 oligomers, 48 one dimension, 233, 234 On-Off Keying (OOK), x, 231, 232, 233, 238 operator, 65, 171, 173, 304, 309 optical communications, ix, 4, 6, 76, 83, 91, 115,

120, 164 Optical Differential Phase Shift Keying (oDPSK), x,

231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 245, 246, 247, 248, 249, 251, 253, 254, 255

optical fiber, vii, viii, ix, x, xi, xii, 4, 5, 6, 7, 15, 17, 30, 31, 33, 34, 45, 46, 48, 51, 53, 54, 55, 57, 72, 76, 84, 85, 87, 102, 119, 120, 121, 123, 124, 127, 132, 136, 137, 138, 139, 146, 151, 154, 163, 164, 165, 183, 187, 188, 227, 232, 234, 257, 258, 259, 268, 269, 271, 272, 275, 279, 297, 302, 311, 312, 315, 316, 333, 335, 342, 355, 356, 357, 358, 361, 364, 365, 366, 367, 368

optical gain, 56, 221 optical polarization, 222 optical properties, 55, 56, 84 optical pulses, 163, 233, 281, 305, 311 optical solitons, 280, 302, 332 optical systems, 215, 308 optics, vii, ix, 4, 83, 101, 104, 109, 115, 119, 120,

126, 178, 207, 272, 279, 280, 281, 302, 315, 366 optimization, viii, ix, 52, 60, 64, 65, 68, 69, 76, 77,

83, 115, 225, 244, 273 optimization method, 65 ordinary differential equations, 198 orientation, 98, 236 orthogonality, 18, 19, 233 OSA, 69, 94, 167, 192, 193, 201, 208, 228 oscillation, 153, 162, 165, 166, 170, 209 oscillograph, 8 oxides, 365 oxygen, 357

P

palladium, 45, 46 PAN, 192 parameter, xi, 6, 17, 41, 71, 177, 199, 219, 266, 280,

281, 287, 289, 290, 291, 294, 295, 301, 303, 305, 307, 309, 310, 316, 358, 359, 360, 362

Paris, 276

Index 377

particles, 154, 275 particulate matter, 21 partition, 246 passive, viii, ix, x, 54, 83, 84, 165, 166, 205, 224,

269, 271, 272, 275 pathogens, 28, 44, 48 PBC, 113, 222, 223, 224 PCF, 169, 170, 262, 316, 332, 338, 339 PDEs, 281 penalties, x, 70, 231, 233 performance, x, 7, 8, 62, 63, 65, 113, 115, 162, 165,

166, 169, 171, 182, 189, 218, 219, 220, 228, 231, 232, 233, 236, 238, 239, 240, 241, 242, 243, 244, 247, 248, 254, 255, 258, 259, 260, 275

periodicity, viii, 83, 89, 166, 265 permit, 290, 304, 305 permittivity, 172, 175, 177 pH, 38, 39, 42 phase shifts, 234, 236 phase transitions, 280 phonons, 55, 56, 63 phosphate, 28 photoelastic effect, 87, 110 photographs, 91, 105 photonic crystal fiber, vii, 258, 262, 273, 316, 332,

333, 338 photonic crystal fiber (PCF), 338 photons, 55, 56, 151, 189, 194, 196, 197, 198 photosensitivity, 84, 213, 227 physics, viii, 15, 44, 183, 279, 281, 297, 305, 309,

315 pitch, 343 plane of polarization, 141 plasma, 23, 26, 30, 279, 297 plasminogen, 30 plastics, 271 PM, 69, 92, 223, 226, 346, 351 PMMA, x, 257, 258, 259, 260, 262, 263, 264, 266,

268, 269, 270, 275 POFs, x, 257, 258, 259, 260, 275 polarity, 238 polarization, vii, ix, x, xi, xii, 57, 58, 64, 83, 84, 87,

88, 89, 90, 94, 96, 97, 98, 99, 100, 102, 103, 106, 108, 109, 110, 111, 112, 113, 114, 115, 140, 141, 148, 165, 166, 167, 169, 170, 171, 172, 173, 175, 176, 183, 206, 216, 218, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 231, 264, 274, 275, 301, 303, 335, 337, 339, 340, 342, 343, 344, 345, 346, 347, 348, 349, 351, 352

polarized light, 87, 88 polycarbonate, 260, 261, 271 polyethylene, 271, 273 polyimide, 260

polymer(s), x, 257, 258, 259, 260, 261, 262, 263, 268, 269, 272, 273, 274, 275, 357, 361, 365, 367

polymer molecule, 259 polymeric materials, 257 polymerization, 259, 269 polymethylmethacrylate, x, 257 polystyrene, 22, 23, 29 poor, 35, 264, 338 population, 63, 65, 66, 68, 165, 198, 342 population size, 66 ports, 207, 223, 224 Portugal, 48, 51, 81, 83, 279 positive correlation, 194, 201 positive feedback, 56 power, vii, ix, x, 3, 4, 5, 6, 7, 8, 9, 19, 28, 30, 52, 53,

54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 88, 111, 114, 115, 132, 134, 135, 138, 139, 142, 146, 147, 149, 150, 152, 153, 162, 163, 164, 165, 167, 169, 170, 176, 177, 179, 182, 187, 188, 189, 191, 192, 193, 194, 195, 196, 197, 199, 200, 201, 202, 206, 207, 208, 209, 210, 211, 216, 218, 219, 220, 221, 222, 225, 226, 227, 234, 254, 257, 258, 260, 261, 263, 265, 275, 302, 311, 316, 338, 342, 344, 346, 347, 348, 356, 357

prediction, 290, 321, 356 pressure, viii, 4, 15, 45, 68, 131, 149, 213, 227, 280,

366 prices, 68 probability, xii, 7, 173, 197, 198, 238, 240, 247, 250,

253, 355, 358, 359, 362, 365 probability density function, 240 probe, 27, 29, 32, 57, 61, 63, 64, 67, 68, 69, 70, 72,

75, 77, 148, 150 production, viii, 83, 84, 91, 108, 268, 365 production costs, 84 production technology, viii, 83, 108 progesterone, 31, 47 prognosis, 49 program(ming), 19, 35, 65, 79, 302 propagation, ix, 18, 19, 52, 61, 65, 74, 75, 83, 84, 85,

87, 89, 108, 109, 122, 127, 144, 146, 147, 165, 171, 172, 177, 189, 191, 206, 208, 210, 211, 225, 226, 257, 258, 260, 262, 271, 281, 283, 285, 287, 288, 289, 291, 292, 294, 295, 296, 298, 299, 302, 305, 311, 312, 316, 317, 321, 322, 331

propane, 33 protein(s), viii, 15, 24, 25, 26, 28, 29, 30, 31, 42, 44,

46, 47 protocol, 32, 37, 39 prototype, 54 PTT, 367

Index 378

pulse(s), x, xi, 6, 108, 128, 131, 163, 164, 165, 166, 169, 183, 196, 232, 233, 238, 241, 243, 244, 279, 280, 281, 283, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 308, 311, 312, 316, 324, 331, 333

pumps, viii, 51, 53, 54, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 74, 76, 77, 78, 79, 108, 112, 221, 222, 226

purification, 273 pyrene, 31

Q

QED, 136, 139, 154 quantum phenomena, 171 quantum well, 161 quartz, 26

R

radial distribution, 153 radiation, x, 4, 7, 80, 149, 166, 171, 172, 176, 179,

198, 213, 214, 257, 258, 260, 271, 272, 275, 302, 320, 326

radio, 114 radius, 6, 17, 18, 19, 33, 121, 127, 144, 145, 146,

147, 162, 171, 172, 176, 177, 181, 182, 260, 264, 265, 269

Raman, viii, xi, xii, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 163, 164, 165, 182, 206, 207, 218, 219, 221, 222, 227, 228, 229, 232, 275, 299, 305, 312, 316, 332, 335, 336, 342, 345, 346, 347, 349, 350, 351, 352

Raman and Brillouin scattering, 55 Raman spectroscopy, 55 Raman-scattering, 312 range, vii, x, 7, 28, 29, 30, 35, 41, 56, 58, 64, 65, 76,

84, 102, 108, 110, 132, 133, 162, 163, 166, 167, 168, 180, 182, 184, 191, 193, 207, 208, 212, 214, 215, 221, 222, 224, 226, 228, 234, 249, 257, 258, 261, 265, 266, 267, 268, 269, 271, 272, 274, 280, 285, 289, 293, 317, 319, 323, 343

reactant, 41 reactive sites, 41 reality, 56, 280, 293 reasoning, 305, 307, 310 reception, 76, 77 receptors, 48 recognition, 29, 32, 44 recombination, 175, 217, 342

recovery, 167 recurrence, 297 redistribution, 176 reduction, 40, 53, 54, 61, 131, 232, 244, 259, 262,

281, 293 redundancy, 238 reflection, vii, viii, ix, x, 7, 16, 70, 72, 83, 85, 90, 91,

93, 94, 96, 97, 98, 101, 102, 103, 105, 106, 108, 113, 136, 142, 168, 182, 189, 191, 216, 227, 257, 258, 260, 266, 316, 343, 345, 346, 350

reflectivity, 71, 85, 86, 90, 114, 172, 177, 258, 260, 261, 273, 349

refraction index, 85, 88, 89 refractive index(ices), viii, ix, 5, 6, 8, 15, 16, 17, 21,

35, 40, 42, 43, 44, 83, 84, 85, 87, 89, 121, 122, 132, 146, 168, 175, 190, 212, 213, 214, 215, 218, 220, 232, 258, 259, 260, 261, 264, 265, 266, 268, 271, 281, 343

regenerate, 36, 218 regeneration, 52 rehydration, 39 reinforcement, 356 relationship, 8, 9, 19, 146, 147, 150, 172, 173, 177,

178, 180, 199, 267, 268 relaxation, 59, 63, 175, 196, 317, 320, 364, 365 reliability, xii, 5, 355, 356, 357, 365, 366, 367 remote sensing, 44 reparation, viii, 15 resistance, 84, 258, 259, 275, 357 resolution, 61, 84, 119, 144, 162, 167, 192, 252, 272 resonator, 162, 163, 166, 172, 176, 177, 178, 179,

183 response time, 221 RF, 346 rings, viii, 51, 166, 262, 263, 264, 273, 361 RNA, 27, 32 robustness, 44, 249 Romania, 355 room temperature, 8, 36, 111, 338, 339, 365 root-mean-square, 139 roughness, ix, 161, 162, 171, 172, 176, 177, 178,

179, 180, 181, 182, 262 routines, 304 routing, 108, 205 Royal Society, 309 rubidium, 122, 134 Russia, 55

S

SA, 43, 44, 51 safety, 5, 53 Salmonella, 22, 28, 46

Index 379

sample(ing), viii, 15, 16, 17, 23, 36, 37, 39, 40, 42, 43, 44, 94, 102, 105, 154, 196, 197, 233, 237, 239, 252, 343, 356, 359, 360, 363

sapphire, 8, 131, 147, 261, 271, 333 saturation, ix, 43, 44, 54, 120, 161, 162, 168, 174,

176, 179, 221, 342 scaling, 164 scattered light, 55, 70, 132, 143 scattering, viii, ix, 28, 39, 51, 53, 54, 55, 56, 57, 62,

79, 136, 143, 151, 153, 162, 164, 174, 176, 177, 187, 189, 191, 192, 193, 194, 197, 200, 201, 219, 221, 232, 264, 275, 299, 305, 312, 316, 332, 336, 342

schema, x, 231, 233, 234, 235, 236, 237, 241, 247, 249, 254

scholarship, 79 Schrödinger equation, 279, 281, 298 science, vii, 271, 302 scientific community, 302 search, 65, 166 security, 29, 44 seed(ing), 54, 74, 165, 320 selecting, 7, 63, 235 selectivity, 32, 40, 44 self-phase modulation, 183 semiconductor, ix, xi, 4, 54, 161, 162, 168, 169, 170,

171, 172, 173, 178, 180, 182, 184, 185, 335, 336, 342, 348

semiconductor lasers, 173 sensing, viii, ix, 15, 16, 21, 30, 31, 44, 45, 46, 83, 84,

98, 104, 115, 169, 182, 205, 272, 274, 275 sensitivity, 28, 29, 35, 41, 44, 94, 96, 97, 98, 114,

162, 170, 178, 182, 217, 227, 244, 249, 350 sensors, vii, viii, ix, xi, 4, 15, 16, 21, 41, 44, 45, 46,

47, 83, 84, 93, 98, 108, 111, 115, 260, 272, 275, 335, 357

separation, 93, 98, 136, 137, 213, 293, 294, 306, 317, 324, 329, 343

sepsis, 29 series, 134, 219, 240, 268, 304, 310, 363 serum, 23, 26, 31, 45 shape(ing), 17, 57, 88, 108, 127, 146, 151, 180, 208,

219, 280 sharing, 167 shock, 275 sign(s), 57, 120, 122, 237, 238, 250, 251, 252, 253,

288, 294 signaling, 49, 232, 233, 236, 237, 243, 246, 247, 249 signals, 5, 28, 53, 54, 57, 59, 61, 62, 63, 64, 67, 68,

69, 70, 73, 77, 98, 114, 130, 134, 148, 150, 164, 187, 188, 189, 206, 207, 216, 218, 221, 226, 236, 250, 302

signal-to-noise ratio, 54, 165

silane, 32 silica, xii, 5, 6, 16, 23, 24, 25, 26, 31, 32, 48, 87, 93,

122, 182, 183, 188, 191, 199, 202, 206, 213, 215, 221, 226, 227, 257, 258, 260, 261, 262, 270, 271, 273, 316, 332, 333, 336, 355, 356, 357, 361, 364, 365, 366, 367, 368

silicon, 154, 271 silver, 6, 183, 273 simulation, viii, 15, 18, 19, 20, 21, 64, 66, 67, 68, 69,

73, 74, 77, 81, 86, 90, 92, 110, 145, 149, 152, 162, 171, 178, 294, 341

SiO2, 190 sites, 7, 29, 41, 52, 71, 216 smoothing, 366 sodium, 36 sodium hydroxide, 36 software, 33, 34 solitons, x, xi, 183, 279, 280, 281, 286, 287, 288,

289, 290, 293, 294, 297, 298, 299, 302, 303, 305, 306, 307, 311, 312, 313, 315, 316, 317, 318, 319, 320, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333

solvent, 37 species, 28, 29, 196, 201, 280, 293 specificity, 40 spectral component, 58 spectroscopy, xi, 30, 55, 162, 271, 335, 357 spectrum, ix, 35, 56, 58, 65, 69, 70, 71, 72, 74, 76,

77, 83, 91, 92, 93, 94, 98, 101, 102, 103, 106, 112, 131, 132, 136, 137, 140, 163, 166, 167, 188, 192, 201, 216, 218, 219, 226, 270, 291, 302, 315, 316, 317, 336, 342, 348, 349, 350

speed, 18, 65, 162, 187, 188, 205, 208, 217, 227, 255, 302, 316, 361, 362

speed of light, 18, 316 spin, 357 sprouting, 22, 46 stability, x, xi, 44, 59, 68, 73, 149, 165, 167, 205,

285, 288, 291, 293, 298, 299, 309, 315, 316, 318, 320, 321

stabilization, 72, 73, 74, 75, 178, 352 stages, 247 standard deviation, 179, 180, 182 standardization, 259 standards, 68, 149 staphylococcal, 47 Staphylococcus, 22, 29, 47 statistics, 356, 358 steel, 271 sterile, 37 stimulant, 29 stock, 39 storage, 272, 316

Index 380

strain, 40, 45, 84, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 107, 108, 115, 348, 350, 356

strength, xii, 7, 17, 32, 34, 58, 132, 275, 286, 355, 356, 357, 359, 360, 364, 365, 366, 367

stress, xii, 7, 87, 88, 91, 93, 94, 104, 106, 107, 109, 110, 115, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366

stress intensity factor, 362 stretching, 259 strong interaction, 29 structural relaxation, 365 subtraction, 310 sulphur, 273 sun, 116, 183, 202, 277 supply, 44, 192, 280, 366 suppression, 162, 167, 170, 185, 338 surface layer, 43 susceptibility, 56 suspensions, 35 switching, 108, 170, 331, 341, 343, 344, 347, 351,

352 symbols, 217, 236, 237, 238, 244, 246, 247, 249,

250, 251, 253 symmetry, 211 synchronization, 163 synthesis, 32 systems, viii, ix, x, xi, xii, 51, 52, 53, 54, 55, 56, 63,

69, 70, 75, 76, 77, 78, 79, 83, 84, 108, 114, 162, 164, 165, 169, 171, 182, 187, 188, 205, 206, 207, 215, 221, 222, 231, 232, 234, 236, 238, 239, 241, 243, 244, 246, 247, 254, 255, 271, 275, 279, 280, 281, 293, 297, 298, 299, 301, 303, 305, 306, 308, 311, 312, 316, 332, 335, 336, 352, 365, 366

T

technical assistance, 366 technology, vii, viii, 3, 7, 51, 52, 55, 68, 83, 84, 108,

187, 213, 222, 232, 259, 271, 302, 356 teflon, 273 telecommunication networks, xii, 206, 218, 355, 356,

361, 365 telecommunications, x, 53, 56, 111, 162, 166, 205 temperature, viii, x, xii, 5, 7, 8, 15, 33, 36, 45, 57,

58, 84, 92, 93, 97, 98, 99, 100, 101, 102, 103, 104, 107, 108, 110, 111, 115, 140, 148, 150, 152, 166, 169, 184, 191, 206, 208, 212, 213, 214, 215, 219, 220, 226, 257, 258, 259, 260, 261, 272, 280, 338, 339, 350, 355, 356, 357, 359, 361, 362, 364, 365, 366

temperature dependence, 57, 97, 102 temperature gradient, 110 tensile strength, 359

tensile stress, 356, 357 tension, 33, 34 theory, vii, x, 3, 18, 85, 86, 121, 124, 125, 127, 144,

154, 162, 171, 249, 279, 281, 286, 291, 297, 299, 302

therapy, 31, 275 thermal expansion, 213 threat, 29, 44 threshold(s), vii, 3, 6, 8, 9, 53, 56, 72, 74, 132, 138,

139, 163, 164, 168, 170, 178, 179, 180, 181, 233 threshold level, 163 thrombin, 30, 31, 48 tics, x, 231, 233, 234, 235, 236, 237, 241, 247, 249,

254 time(ing), viii, xi, xii, 5, 8, 30, 32, 33, 37, 41, 42, 43,

45, 46, 47, 52, 57, 59, 61, 64, 65, 66, 67, 72, 84, 104, 105, 106, 108, 112, 113, 114, 115, 148, 150, 151, 152, 162, 163, 165, 170, 171, 175, 187, 188, 192, 197, 198, 199, 213, 214, 217, 221, 228, 233, 239, 244, 254, 258, 262, 280, 281, 293, 299, 301, 303, 309, 311, 317, 324, 355, 358, 359, 360, 364, 365, 366

tin, 175 TIR, 16 TNF-alpha, 27, 31 Tokyo, 158, 203 topology, 68 total energy, 320, 321 total internal reflection, vii, x, 16, 257, 258, 316 toxin, 23, 29 tracking, 183 trade, 258 traffic, 53, 187, 269 trajectory, 294, 349 transference, 81 transformation, 125, 288 transition(s), 130, 134, 138, 148, 150, 151, 164, 173,

189, 190, 197, 198, 199, 208, 245, 246, 259, 280 transition rate, 198 transition temperature, 259 translation, 93, 96 transmission, vii, viii, x, 6, 7, 16, 18, 19, 20, 21, 28,

33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55, 57, 58, 63, 70, 71, 75, 76, 77, 78, 84, 114, 115, 136, 137, 138, 139, 182, 187, 188, 205, 206, 207, 208, 211, 212, 213, 215, 217, 220, 221, 223, 226, 228, 231, 232, 244, 246, 247, 255, 257, 258, 259, 260, 261, 262, 263, 264, 268, 269, 270, 271, 272, 273, 274, 275, 280, 293, 297, 298, 300, 302, 311, 343, 345, 347, 356

transmission path, 78 transmits, x, 8, 21, 232, 257 transparency, 316, 331, 332

Index 381

transport, 30, 48 transverse section, 91, 94, 105 trend, 43, 106, 130 triggers, 56 tuberculosis, 31 tumor necrosis factor, 27, 31 tumors, 275 tunneling, 176 typhoid, 31

U

UK, 81, 301 ultraviolet light, viii, 83, 85 uniform, 16, 17, 18, 21, 22, 23, 68, 85, 86, 87, 110,

152, 218, 331, 359 urine, 23 users, 112, 113, 114, 366 UV, 213, 214, 227 UV radiation, 213, 214

V

vacuum, 6, 36, 124, 128, 131, 132, 134, 142, 175, 356, 357

Vakhitov-Kolokolov criterion, 318 valence, 173 values, 19, 20, 21, 59, 60, 65, 67, 68, 70, 76, 87, 92,

96, 97, 99, 100, 101, 104, 108, 126, 139, 147, 162, 176, 178, 179, 180, 181, 200, 237, 244, 247, 248, 249, 252, 267, 273, 282, 283, 290, 291, 293, 294, 295, 308, 358, 359

vapor, 45, 131, 148, 366 variability, 214 variable(s), xi, 4, 59, 174, 220, 240, 253, 301, 302,

303, 306, 309, 312, 339, 364 variance, 240, 254 variation, xi, 92, 93, 94, 98, 142, 167, 200, 219, 226,

229, 252, 269, 287, 301, 303, 315, 362

vector, 127, 172, 173, 175, 176, 183, 210, 211, 250, 313

vehicles, 84 velocity, xi, 57, 87, 105, 133, 134, 139, 150, 163,

172, 212, 281, 291, 294, 295, 296, 297, 301, 302, 305, 316, 317, 318, 321, 322, 324, 325, 327, 329, 330, 362

vibration, 259 viscosity, 4 voice, 4, 302

W

water absorption, 224 wave number, 6, 122, 132, 304, 305, 309 wave propagation, 257, 305 wavelengths, viii, xii, 42, 51, 53, 54, 63, 64, 68, 71,

72, 75, 76, 77, 87, 89, 97, 98, 106, 107, 109, 112, 113, 114, 163, 170, 188, 193, 208, 213, 216, 220, 221, 222, 260, 261, 262, 270, 272, 335, 336, 337, 338, 339, 342, 343, 346, 347, 348, 349, 350, 352

welding, 272 wells, 177 windows, 7, 8, 53 workers, 164, 259 wound healing, 31 writing, 350

X

X-axis, 95, 97

Y

Y-axis, 95, 97, 102 yield, 304, 305 ytterbium, 164, 333

Fo Research Advances

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