Dispersion Compensation and Soliton Transmission in Optical

Fibers

André Toscano Estriga Chibeles

Dissertation submitted for obtaining the degree of Master in Electrical and Computer Engineering

Jury

President: Prof. Doutor José Bioucas Dias

Supervisor: Prof. Doutor António Luís Campos da Silva Topa

Co-Supervisor: Prof. Doutor Carlos Manuel dos Reis Paiva

Members: Profª Doutora Maria Hermínia da Costa Marçal

Abril 2011

ii

iii

Acknowledgements First I would like to thank to my supervisors Prof. Carlos Paiva and Prof. António Topa, to help

me in the process of completing my master, by giving me knowledge and guidance while I was making

this dissertation.

I would like to thank to all my friends, but specially

To my IST friends:

Nuno Couto, Rui Trindade, Filipa Henriques, André Neves, Gonçalo Carmo, João Cabrita, André

Esteves, Rafael Ferreira, Luís Pragosa, Francisco Pinto and Vera Silva.

To my friends from Évora:

Luís Almeida, Duarte Abêbora, Carlos Freixa, Tiago Toscano, Filipa Ribeiro, Daniel Engeitado,

João Rosa, Joaquim Faneca, Carlos Rosa, Tânia Pegacho, Luís Pegacho, João Roque, Luís Roque and

Filipe Louro.

To my 4th floor friends:

José Alves, Bruno Baleizão, João Vicente and António Eira.

To my friends of PIO XII university college:

André Patrão, Pedro Barata, João Caldinhas, João Fialho, Luis Rodrigues and Miguel Duarte.

And last but not least, to my family:

To my mum and dad, to my aunt Crisália, to my aunt Maria Antónia, to my uncle Zé, to my

cousin Manel and to my both grandmas, and my grandpa.

Wherever you are you will always be in my heart.

iv

Abstract In this dissertation, the propagation of pulses in a linear and non-linear regime is studied. The

topics of the temporal dispersion and laser chirp are taken into account for both the linear and non-linear

regimes. The pulse dispersion causes that the communication systems are not perfect.

To study the pulse evolution, the pulse propagation equations have to be determined, one equation

for the linear regime and another equation for the non-linear regime. The propagation equation for the

non-linear regime is the non linear Schrodinger equation (NLS). Only with this equation, the non-linear

effects can be taken into account.

In the linear regime, the techniques for dispersion compensation are addressed, the use of

dispersion compensating fibers (DCF). In this dissertation is mentioned other ways to compensate the

dispersion, they are not tested though.

Keywords:

Propagation, linear regime, non-linear regime, temporal dispersion, chirp, non-linear Schrodinger

equation, dispersion compensating fibers, soliton.

v

Resumo Nesta dissertação, foi estudada a propagação de impulsos em regime linear e não linear. Quer a

dispersão temporal quer o “chirp”, são tidos em conta no regime linear e no regime não linear. A

dispersão dos impulsos faz com que os sistemas de comunicação não sejam perfeitos.

Para se estudar a propagação de impulsos, que a equação de propagação. Uma equação tem de ser

determinada para o meio linear, outra equação tem de ser determinada para o meio não linear.

Em regime linear analisa-se as técnicas de compensação da dispersão, nomeadamente, o uso de

fibras compensadoras de dispersão.

A equação de propagação que é usada para o meio não linear é a equação não linear de

Schrodinger. Com esta equação, os efeitos não lineares podem ser considerados.

Palavras chave:

Propagação de impulsos, meio linear, meio não linear, dispersão temporal, chirp, equação não

linear de Schrodinger, fibras compensadoras de dispersão, solitões.

vi

Table of contents Acknowledgements ......................................................................................................................... iii

Abstract ............................................................................................................................................ iv

Resumo ............................................................................................................................................. v

List of figures ................................................................................................................................. viii

List of tables ..................................................................................................................................... x

List ................................................................................................................................................... xi

List of acronyms ............................................................................................................................ xiii

1. Introduction ............................................................................................................................ 1

1.1. Historical overview ............................................................................................................ 1

1.1.1. Optical fibers ............................................................................................................... 1

1.1.2. Evolution of lightwave systems ................................................................................... 2

1.1.3. Evolution of the Non-linear communication systems .................................................. 4

1.2. Motivation and objectives .................................................................................................. 5

1.2.1. Dispersion limitation in optical communication systems ............................................ 5

1.2.2. Dispersion compensation schemes .............................................................................. 6

1.3. Structure ............................................................................................................................. 7

1.4. Main contributions ............................................................................................................. 8

2. Optical fibers ......................................................................................................................... 9

2.1. Basic structure of an optical fiber ...................................................................................... 9

2.2. Normalized frequency and wavenumber ......................................................................... 10

2.3. Hybrid modes modal equation ......................................................................................... 12

2.4. Low contrast fibers .......................................................................................................... 13

2.5. LP modes ......................................................................................................................... 15

2.6. Conclusions ...................................................................................................................... 16

3. Pulse propagation in the linear regime................................................................................. 19

3.1. Propagation Equation in the linear regime...................................................................... 19

3.2. Analytical approach for pulse broadening ....................................................................... 25

3.2.1. Second and third-order moments ............................................................................... 25

3.2.2. RMS broadening of a function of the group delay .................................................... 27

3.2.3. RMS broadening of a chirped pulse .......................................................................... 28

3.3. Pulse propagation ............................................................................................................. 32

3.3.1. Bell shaped pulses ...................................................................................................... 33

3.3.2. Super-Gaussian pulse with chirp ............................................................................... 34

vii

3.4. Higher order dispersion influence effects ........................................................................ 38

3.4.1. LD=L’D ........................................................................................................................ 38

3.4.2. λ0=λD .......................................................................................................................... 40

3.5. Dispersion compensation ................................................................................................. 42

3.6. Conclusions ...................................................................................................................... 46

4. Pulse propagation in the non-linear regime ......................................................................... 47

4.1. Non-linear Kerr effect ...................................................................................................... 47

4.2. Pulse propagation equation for the a non-linear regime .................................................. 48

4.3. Solitons in optical fibers .................................................................................................. 52

4.3.1. First-order soliton ...................................................................................................... 52

4.3.2. Second-order soliton .................................................................................................. 53

4.3.3. Third-order soliton ..................................................................................................... 54

4.3.4. Gaussian pulse ........................................................................................................... 54

4.4. Interaction between solitons ............................................................................................ 55

4.5. Conclusions ...................................................................................................................... 57

5. Conclusions .......................................................................................................................... 59

5.1. Main conclusions ............................................................................................................. 59

5.2. Future work ...................................................................................................................... 59

Bibliography ................................................................................................................................... 61

Annex A .......................................................................................................................................... 63

A1. Modal equation of the hybrid modes ................................................................................... 63

Annex B .......................................................................................................................................... 71

B1. Numerical simulation of linear pulse propagation ............................................................... 71

Annex C .......................................................................................................................................... 74

C1. Numerical simulation of the NLS function: Split step Fourier method ............................... 74

viii

List of figures Figure 1.1: Increase in the bit-rate distance product during the period 1850 and 2000 [2]. ............. 2

Figure 1.2: Increase of the BL product over the period of 1975 and 2000. ..................................... 3

Figure 1.3: International undersea network of fiber-optic communication system [2]. ................... 4

Figure 1.4: Temporal dispersion that a pulse suffers after is propagated through a fiber. ............... 5

Figure 1.5: Train of pulses at the fiber input. ................................................................................... 5

Figure 1.6: Train of pulses at the receiver. ....................................................................................... 6

Figure 1.7: ISI of a train Train of pulses at the receiver input. ......................................................... 6

Figure 2.1: Waveguide structure of optical fiber. ............................................................................. 9

Figure 2.2: Normalized propagation constant as a function of the normalized frequency for the

fundamental mode LP01 for several values of ∆ . ........................................................................................ 15

Figure 2.3: Normalized propagation constant as a function of the normalized frequency, for the

first six LP modes. ....................................................................................................................................... 16

Figure 3.1: Absolute value of the pulse. The pulse at the input of the fiber is represented by a

dashed line. The impulse at the output of the fiber is represented by a full line. ........................................ 33

Figure 3.2: Absolute value of the pulse along the fiber, observed by two different angles. ........... 34

Figure 3.3: Pulse broadening along a fiber section, for different values of C. ............................... 35

Figure 3.4: Pulse amplitude at the input and output of the fiber for C=−2. ................................... 35

Figure 3.5: Evolution of the absolute value of the super Gaussian pulse for C=−2. ..................... 36

Figure 3.6: Pulse amplitude value at the input and output of the fiber, for C=0. ........................... 36

Figure 3.7: Evolution of the absolute value of the super Gaussian pulse, for C=0........................ 36

Figure 3.8: Pulse value at the input and output of the fiber for C=2. ............................................. 37

Figure 3.9: Evolution of the absolute value of the super Gaussian pulse for C=2......................... 37

Figure 3.10: Pulse amplitude for the entrance pulse and exit pulse, when the broadening is

minimum. ..................................................................................................................................................... 38

Figure 3.11: Input and output pulses for the case of third order dispersion, when D DL L′= , and

2C = − ......................................................................................................................................................... 39

Figure 3.12: Input and output pulse for the case of third order dispersion, when D DL L′= , and

0C = . .......................................................................................................................................................... 39

Figure 3.13: Input and output pulses for the case of third order dispersion, when D DL L′= , and

2C = . .......................................................................................................................................................... 40

ix

Figure 3.14: Input and output pulses for the case of third order dispersion, when 0 Dλ λ= , and

2C = − . ........................................................................................................................................................ 41

Figure 3.15: Input and output pulses for the case of third order dispersion, when 0 Dλ λ= , and

0C = . .......................................................................................................................................................... 41

Figure 3.16: Input and output pulses for the case of third order dispersion, when 0 Dλ λ= , and

2C = − . ........................................................................................................................................................ 42

Figure 3.17: Transmission system with a dispersion compensating fiber. ..................................... 42

Figure 3.18: Input and output of the DCF fiber with a chirp factor 2C = − . .................................. 43

Figure 3.19: Pulse evolution along the DCF for a 2C = − , viewed from two perspectives. .......... 44

Figure 3.20: Input and output of the DCF fiber with a chirp factor 0C = . ................................... 44

Figure 3.21: Pulse evolution along the DCF for a 0C = , viewed from two perspectives. ............ 45

Figure 3.22: Input and output of the DCF fiber with a chirp factor 2C = . ................................... 45

Figure 3.23: Pulse evolution along the DCF for a 2C = , viewed from two perspectives. ............ 46

Figure 4.1: Propagation of the first-order soliton along a fiber link. .............................................. 53

Figure 4.2: Propagation of the second-order soliton along a fiber link. ......................................... 53

Figure 4.3: Propagation of the third-order soliton along a fiber link. ............................................. 54

Figure 4.4: Propagation of a Gaussian pulse soliton along a communication link. ........................ 55

Figure 4.5: Interaction between two solitons, when θ=0 and r=1.................................................. 56

Figure 4.6: Interaction between two solitons with θ=π/2 and r=1. ................................................ 56

Figure 4.7: Interaction between two solitons when θ=0 and r=1.1................................................ 57

x

List of tables Table 3.1: Parameters of the pulse propagation with third-order dispersion, when D DL L′= . ........ 38

Table 3.2: Parameters of the pulse propagation with third-order dispersion, when 0 Dλ λ= . ........ 40

Table 3.3: Parameters used in the propagation and dispersion compensation numerical simulation.

..................................................................................................................................................................... 43

xi

List ∆ Fiber contrast a Radius of the core h Refractive index of the core α Refractive index of the cladding β Propagation constant

0k Propagation constant in vacuum u Normalized refractive index of the core w Normalized refractive index of the cladding n Modal refractive index v Normalized frequency

( )mJ u Bessel function of the first kind

( )mK u Bessel function of the second kind

modesN Number of modes

( ), , ,E x y z t Electric field

0E Amplitude of the electric field

( ),F x y Spatial distribution

( )0,B t Longitudinal variation

( )0,A t Pulse amplitude

Lβ Linear part of the propagation constant

NLβ Non-linear part of the propagation constant α Attenuation constant

gv Group velocity

DL Dispersion length ,τ ζ Normalized variables

2β Second-order dispersion

0τ Pulse initial width

3β Third-order dispersion 2σ Root-mean square t First order moment 2t Second order moment

gτ Group delay

NLφ Non-linear phase

inP Input power

0µ Magnetic permeability

0ε Electric permittivity

*E Fictional electric field 2

F Moment of the modal function

effA Effective area of the core

NLL Non-linear length

0q Initial separation

xii

r Relative amplitude

*y Admittance

xiii

List of acronyms EDFA Erbium-doped fiber amplifier ISI Inter-symbolic interference FSK Frequency shift keying DCF Dispersion compensating fiber GVD Group velocity dispersion RMS Root-mean square SPM Self-phase modulation QoS Quality of service XPM Cross-phase modulation FWM Four-wave mixing

xiv

1

1. Introduction

In this section, an historical overview, the motivation and objectives, and, the structure of the

work, as well as their contributions are presented

1.1. Historical overview

1.1.1. Optical fibers

Since the beginning of human kind, there was always a need to communicate over long distances.

Nowadays, the primary ways of communication are cell-phones, Internet and television. All this forms of

communication are supported by a network infrastructure.

A communication process consists in three main parts. The transmitter, from where the data is

generated, the communication link, responsible for transmitting the data over short, medium or long haul

distance, and the receiver, where the data is received. This work is specially focused in the communication

link.

Modern communication links of the global network are mostly optical fibers. Only the access

network is not yet completely implemented with optical fibers, but in future, the telecommunication

network will be completely composed by optical fibers.

Before the appereance of fiber optics, the communications were performed through coaxial

cables. With the use of coaxial cables in place of wires pairs, system capacity was increased considerably

[2]. In 1940, the first system using coaxial cable was implemented in the a 3 MHz band. The system was

capable of transmitting 300 voice channels and a single television channel [2]. The bandwidth of such

systems was considerably affected by the loss dependence on the frequency, which increased rapidly with

frequencies beyond 10 MHz [2].

With such a limitation in coaxial cable systen, a microwave communication system was

developed [2]. Then, both coaxial and microwave systems evolved considerably and both operated in bit

rates of the order of 100 Mb/s. Both systems had limitations in the spacing between repeaters, so that they

became very expensive.

The product BL , where B is the bit rate and L is the repeater spacing, is a system merit figure.

Figure 1.1 shows that the BL product has increased through technological advances during the last

century and a half.

2

Figure 1.1: Increase in the bit-rate distance product during the period 1850 and 2000 [2].

Such an increase in the BL product, in the second half of the twentieth century was only possible,

as optical waves were used as the carrier [2]. In May of 1960, Theodore Maiman performed the first

demonstration of a working laser [1]. The propagation medium was still needed to be invented. It was only

suggested in 1966 that optical fibers were capable of guiding the light in a manner similar to the guiding

of electrons in copper wires [2].

The first semiconductor laser was introduced by four independent groups, between September and

October of 1962 [1]. But the first lasers only worked with a cooling system of nitrogen. Only in 1970, the

first semiconductor lasers operating at room temperature made their appearance [2].

1.1.2. Evolution of lightwave systems

The research on fiber optics started near 1975. Between 1975 and 2000 an enormous progress has

happened. The communication systems evolution can be grouped into four categories, as can be seen from

Figure 1.2 which shows the BL product over the time period of 1975 and 2000.

3

Figure 1.2: Increase of the BL product over the period of 1975 and 2000.

Figure 1.2 shows a straight line corresponding to a doubling of the BL product every year and the

four generations so far developed [2]. The first generation of lightwave systems operated in the first

window (0.8 µm). Those systems became available for commercialization in 1980. The bit rate was

around 45 Mb/s and allowed a space between repeaters of up to 10 km [1]. The second generation

operated in the second window (1.3 µm) [1]. The attenuation for this generation was 1 dB/km and

dispersion was minimum [1]. The bit rate was 1.7 Gb/s and the spacing between repeaters was 50 km [1].

The second generation was first implemented at a bit rate of 100 Mb/s in the beginning of the 1980s, only

later, in 1987 the bit rate of 1.7 Gb/s was achieved [2].

Since 1979, it was known that optical fibers had a minimum loss around the 1.55 µm wavelength,

where the attenuation was 0.2 dB/km [1]. However, the dispersion in the third-order generation system

was higher [2]. In 1990, a combination of dispersion-shifted fibers and monomodals semiconductor lasers

led to the implementation of the third-order communication systems. These systems have bit rates of 10

Gb/s and the spacing between amplifiers is of 100 km [2].

The main problem of the third generation systems is that they require the use of electronic

repeaters, known as regenerators [1]. When several wavelengths were transmitted through a fiber, the

regeneration had to be performed by several regenerators, one for each wavelength. From an economical

point of view this was unconceivable. To resolve the problem of the amplification of several channels

4

using only one device, the first erbium-doped fiber amplifiers (EDFAs) was developed in 1986. EDFAs

operate in the third window and use semiconductor lasers for their pumping.

The fourth communication generation was the first all-optical generation, where the use of optical

amplifiers allowed the amplification of several wavelengths and with that, the wavelength division

multiplexing (WDM) was implemented [1]. With WDM the bit rates were increased to higher values.

In order to create a world-wide network, submarine links had to be launched; Figure 1.3 shows the

international submarine network [2].

Figure 1.3: International undersea network of fiber-optic communication system [2].

The first transatlantic submarine cable was deployed in 1956 [1], and it was called TAT-1. Later,

in 1988, the first submarine cable with optical fiber was deployed; the system was called TAT-8 and had a

monomodal fiber [1]. The TAT-8 belongs to the second generation communication system.

The submarine cables TAT-9 and TAT-10/11 were deployed in 1992, and belong to the third

generation communication systems. In 1996, the submarine cables of the fourth generation began to work;

they were the TPC-5 and TAT-12/13. They used EDFAs in the repeaters. The bit rates achieved were 5.30

Gb/s. The TPC-6 was installed in 2000 with a bit rate of 100 Gb/s.

1.1.3. Evolution of the Non-linear communication systems

The availability of low loss silica fibers led to the study of non-linear fiber optics. In 1972, the

stimulated Raman scattering and the Brillouin scattering were studied. The idea that optical fibers can

support soliton like pulses as a result of interplay between the dispersive and non-linear effects appeared

in 1973. In 1980 solitons were experimentally observed. The advances in generation and control of ultra-

5

short pulses was only possible due to the discover of solitons. The field of nonlinear fiber optics has

grown considerably since 1990s and is expected to continue during the twenty-first century [3].

1.2. Motivation and objectives

1.2.1. Dispersion limitation in optical communication systems

The main objective of this work is to study the pulse propagation in linear and non-linear regimes

for several types of pulses. In addition this work will address the dispersion effect. When the pulse

propagation is performed in a linear regime, a technique of dispersion compensation will be applied. In the

non-linear regime, soliton propagation will be studied.

When a pulse is propagating in an optical fiber, it can suffer a dispersion effect. This dispersion is

usually in the time domain. Figure 1.4 shows the dispersion suffered by a pulse can suffer after being

propagated through a fiber.

Figure 1.4: Temporal dispersion that a pulse suffers after is propagated through a fiber.

In a real situation, instead of a single pulse, a train of pulses is transmitted. Figure 1.5 shows a

train of pulses at the communication link input.

Figure 1.5: Train of pulses at the fiber input.

When the train of pulses of Figure 1.5 arrives at the receiver, the pulses shapes are not the same.

Figure 1.6 shows a possible shape of the pulses at the receiver input.

6

Figure 1.6: Train of pulses at the receiver.

In case of strong dispersion the pulse invades the bit slot of another pulse. This effect is called

inter-symbolic interference (ISI).When a pulse invades the bit slot of another pulse, the receiver may not

be able to distinguish whether the pulse corresponds to a bit “1” or “0”. Figure 1.7 shows is an example of

inter-symbolic interference in the propagation of a train of pulses situation.

Figure 1.7: ISI of a train Train of pulses at the receiver input.

Reducing the bit rate, can mitigate the effect of the IIE. So, it is very important to study the impact

of dispersion over a train of pulses and over single pulse, and limitations to the bit rate.

The pulse can be influenced by dispersion not only in the time domain, but also in the spectral

domain. When direct modulation is performed the pulse suffers a chirp effect, which consists in a

broadening of the spectrum of the pulse. The effect of the chirp in the pulse propagation will be study in

this dissertation.

1.2.2. Dispersion compensation schemes

Several techniques can be used to compensate the time dispersion and the chirp effect. To

compensate the chirp, a precompensation scheme, is generally used where the input pulse is changed at

the transmitter. There are several precompensation schemes, such as:

• Prechirp technique;

• Novel coding technique;

• Nonlinear prechirp technique.

7

The prechirp technique consists in inducing a certain amount of dispersion such that, once

conjugated with chirp, it will narrow the pulse spectrum. Then, when the pulse arrives at the receiver has

its original shape. The novel coding technique consists in the use of another propagation format, called the

frequency shift keying (FSK). The nonlinear prechirp technique amplifies the transmitter output using a

semiconductor (SOA) operating in the gain saturation regime. In practice, the nonlinear prechirp technique

induces a compression in the pulse.

The postcompensation technique consists in compensating the dispersion at the receiver, just

before the signal reaches the detector. This operation is preformed electronically, since the optical signal is

converted to its electrical form, and then, it is equalized.

The technique, considered in this dissertation is the use of dispersion compensating fibers (DCF).

The DCF fiber is between the propagation fiber and the receiver. The DCF is dimensioned to have a group

velocity dispersion (GVD) opposite to the existing of the fiber, so that the dispersion may be

compensated.

The dispersion compensating fibers introduce attenuation in a practical perspective. To counter

the imposed attenuation optical filters were designed to compensate the dispersion imposed by the optical

fibers. Since the (GVD) affects the optical signal through the spectral phase, it is evident that an optical

filter, whose transfer function cancels this phase, will restore the signal. Unfortunately, no optical filter

has a transfer function suitable for compensating the GVD exactly. Nevertheless, several optical filters

have provided partial GVD compensation by mimicking the ideal transfer function [4].

1.3. Structure This work is divided into 5 chapters. In the first chapter one, a historical overview of the optical

communication systems is presented. Then, the motivation and objectives of this work are presented.

Finally, the structure and the main contributions of this dissertation are presented.

The second chapter addresses the modal theory of optical fibers. The propagation equations are

obtained. Also in chapter 2, we address the number of modes propagating in an optical fiber, and how the

dimensions of the fiber influence the number of modes that are propagated. The hybrid nature of the

modes is also studied in this chapter.

In the third chapter the propagation equation for a linear regime is derived, the pulse broadening is

determined. It also presents several simulations for the propagation of different types of pulses. The pulses

that are used are the “bell-shaped” pulses and the super Gaussian pulses. The dispersion suffered by the

pulses is also studied. The GVD ( 2β ), the third order dispersion ( 3β ) and the chirp are also studied in

this chapter. Further in this chapter, the second order dispersion is compensated using a DCF.

8

In chapter four, the propagation of pulses is performed in a non-linear regime. The solitons are

studied as well as their interaction.

In chapter five, the main conclusions of this work are drawn.

1.4. Main contributions Efficient pulse propagation in optical fibers is, nowadays, very important, because most of the

telecommunications systems use them. With this work, where pulse propagation is studied in linear and

non-linear regimes, we hope to help to further increase the understanding of the effects that pulses can

suffer along their propagation through a fiber. Another important contribution is the systematic analysis of

the techniques used to compensate the dispersion that the pulses suffer either in the linear or non-linear

regime.

9

2. Optical fibers

When communications are performed using optical fibers, one or more modes can be propagated.

In the fiber the number of modes that can be propagated depends on the design of the optical fiber. In this

section, the parameters that influence the mode propagation and the number of modes are discussed.

2.1. Basic structure of an optical fiber The basic structure of an optical fiber is presented in Figure 2.1.

Figure 2.1: Waveguide structure of optical fiber.

Figure 2.1 shows that the optical fiber is composed by two main regions, the core region and the

cladding region. The core region is where the light is mostly propagated. The role of the cladding region is

to confine the light in the core region. The geometry of an optical fiber is cylindrical and, in most cases,

both the core and the cladding consist of silica. In order to maintain the light inside the core of the optical

fiber, the refractive index of the cladding has to be slightly lower than the refractive index of the core.

The dielectric contrast is a parameter that measures the difference between the refractive index of

the core and the cladding, and is expressed by [1]

2 21 2

212

n n

n

−∆ = (2.1)

10

where 1n is the refractive index of the core and 2n is the refractive index of the cladding.

For a step-index fiber, and using cylindrical coordinates are used (r, φ , z), the refractive index

varies along the coordinate r according to

( ) 1

2

,.

,

n r an r

n r a

≤=

> (2.2)

Then, a different transverse wave have to be considered for the core and for the cladding. They are respectively [1]

2 2 2 21 0h n k β= − (2.3)

2 2 2 22 0n kα β= − (2.4)

where β is the longitudinal wave number, and 0k is the vacuum wave number, given by

0

2.k

c

ω πλ

= = (2.5)

These two transverse wavenumbers are normalized in the next section.

2.2. Normalized frequency and wavenumber In this section, the normalization of the frequencies and of the wavenumber is presented.

Equations (2.3) and (2.4) can be normalized in order to obtain a simpler modal equation to be presented

further. The normalization of the core and the cladding wavenumber is respectively

u ha= (2.6)

w aα= (2.7)

where a is the radius of the core of the fiber.

To expand the equations (2.6) and (2.7) the longitudinal wavenumber β is defined as [1]

0nkβ = (2.8)

where n is the modal refractive index, given by

( )2 2 2 22 1 2n n b n n= + − (2.9)

where b is the normalized modal refractive index, to be defined ahead.

11

To define the normalized transverse wavenumber in the core and in the cladding in terms of the

modal refractive index, the longitudinal wavenumber β is replaced in equations (2.6) and (2.7). Then, the

parameters u and w yield

( )( )22 2 21 0u n n k a= − (2.10)

( )( )22 2 22 0 .w n n k a= − (2.11)

The parameters h and α are also called the transversal propagation constants in the core and the

cladding, respectively. Using the normalization of those parameters and summing equations (2.10) and

(2.11) a new normalized parameter is defined, which is called, normalized frequency, and is given by [1]

2 2 2v u w= + (2.12)

The parameter v can be expressed in terms of 1n , 2n , 0k and a according to

2 21 2 0 .v n n k a= − (2.13)

The equation (2.13) can also be expressed in terms of ∆ , as

1 0 2 .v n k a= ∆ (2.14)

The longitudinal wavenumber can be written in terms of v in the form

2

21.

2

vu

aβ = −

∆ (2.15)

After obtaining the previous parameters, the normalized modal refractive index is defined by [1]

2 22 2

22 2 2 2

1 2

1 .n nu w

bv v n n

−= − = =

− (2.16)

Due to the fact that 2 0 1 0n k n kβ≤ < and 2 1n n n≤ < , then the normalized modal refractive index can only vary

between 0 1b≤ < .

12

2.3. Hybrid modes modal equation In an optical fiber, the surface guided waves are generally hybrid modes. In hybrid modes the

axial electromagnetic field components zE and

zH are both different from zero [6]. It can be shown [1]

that the modal equation for the hybrid modes is given by [1]

( ) ( )42

22

1 2 ,m m

u vR u S u m

v uw

= − ∆

(2.17)

where the parameters ( )mR u and ( )mS u are defined, respectively as

( ) ( )( )

( )( )

m m

m

m m

J u K wR u

uJ u wK w

′ ′= + (2.18)

( ) ( )( )

( ) ( )( )

1 2 ,m m

m

m m

J u K wS u

uJ u wK w

′ ′= + − ∆ (2.19)

where ( )mJ u is the Bessel function of the first kind and ( )mJ u′ its derivative, ( )mK w is the Bessel function of

second kind and ( )mK w′ its derivative. The derivatives ( )mJ u′ and ( )mK w′ are, respectively, given by

( ) ( ) ( )1 1

1

2m m mJ u J u J u− +′ = − (2.20)

( ) ( ) ( )1 1

1.

2m m mK w K w K w− +′ = − + (2.21)

Derivation of equation (2.17) is presented in Annex 1.

The hybrid modes are classified into two categories: the HEmn modes and de EHmn modes. The m

parameter is the variation of the azimuthal coordinate, and the parameter n is the variation of the radial

coordinate.

While the general modal equation for the hybrid modes was derived in this section. In the next

section, a particular approximate modal equation will be presented.

13

2.4. Low contrast fibers When a fiber has a low dielectric contrast, i.e., 1∆ , the low contrast fiber linearly polarized

(LP) mode come from the approximation with 1

0

1n

n≅ . This means that the light confinement is not tight to

the core. These types of fibers are called, weakly-guided fiber. In this case the expression of ∆ can be

simplified to [2]

1 2

1

.n n

n

−∆ ≈ (2.22)

With low contrast fibers, the Gloge approximation for the modal equation can be used. The Gloge

approximation permits the use of the condition

( ) ( ).m mR u S u= (2.23)

Then, the equation (2.18) becomes

( )2

2 2m

mvR u

u w= ± (2.24)

where, when the ( + ) signal is used, only the EHmn are propagated. When the (− ) signal is used only the HEmn are

propagated.

When the EHmn modes are propagated the modal equation yields [1]

( )( )

( )( )

1 1 0m m

m m

J u K w

uJ u wK w

+ ++ = (2.25)

where the parameters ( )1mJ u+ and 1mK + are obtained using the following two equations [1]

( ) ( ) ( )1m m m

mJ u J u J u

u+′ = − + (2.26)

( ) ( ) ( )1 .m m m

mK w K w K w

w+′ = − + (2.27)

When the HEmn modes are propagated the modal equation yields

( )( )

( )( )

1 1 0m m

m m

J u K w

uJ u wK w

− −− = (2.28)

14

where the parameters 1mJ − and 1mK − are obtained through [1]

( ) ( ) ( )1m m m

mJ u J u J u

u−′ = − (2.29)

( ) ( ) ( )1m m m

mK w K w K w

w−′ = − − (2.30)

respectively.

When the objective is to propagate the modes TE0n and TM0n, the modal equation of both modes

is the same. Then, for 0m = the modal equation comes [1]

( )( )

( )( )

1 1

0 0

0.J u K w

uJ u wK w+ = (2.31)

There are two conditions that are obtained when these two modes are used, they are [1]

( ) ( ) ( )1m

m mJ u J u− = − (2.32)

( ) ( ).m mK w K w− = (2.33)

The HE0n modes are equivalent to the TE0n modes. The EH0n modes are equivalent to the TM0n

modes.

When the fibers are of low contrast the modes are linearly polarized, and are called LPpn modes.

When the propagated mode is the EHmn mode the correspondent LPpn mode has a 1p m= + . When the

propagated mode is the HEmn mode, the correspondent LPpn mode has a 1p m= − . The modal equation

that corresponds to the previous can be

( )( )

( )( )

1 1 0p p

p p

J u K wu w

J u K w

− −+ = (2.34)

or can be

( )( )

( )( )

1 1 0.p p

p p

J u K wu w

J u K w

+ +− = (2.35)

The equations (2.34) and (2.35) are equivalent [1].

After obtaining the modal equation for the LP modes, it can be simplified when the propagated

mode is the fundamental mode LP01 which is going to be done in the next section.

15

2.5. LP modes The study of the fundamental mode is very important, when single-mode fibers are used. The

SMF only support the fundamental mode HE11 or LP01. The fiber is designed such that all higher-order

modes are cut off the operating wavelength [2]. The modal equation of the fundamental mode, whether it

is equation (2.34) or equation (2.35), yields [1]

( ) ( ) ( ) ( )1 0 0 1 .uJ u K w wJ u K w= (2.36)

In order to obtain a ( )b v function, the parameters w and u are written in terms of b and v , and it is given values

to v . Then, the value of b is found in a way that the equation (2.36) has a solution. Figure 2.2 shows the ( )b v

function for the fundamental mode for various values of ∆ .

Figure 2.2: Normalized propagation constant as a function of the normalized frequency for the fundamental mode LP01 for several

values of ∆ .

Figure 2.2 shows that when the contrast increases, the required v increases so that the propagation

is possible. Then, for weakly guiding fibers with lower contrast the cut off v is lower. The propagation

coefficient increases as long as the normalized frequency increases.

Figure 2.3 shows the first six LP modes of an optical fiber when the solved equation is (2.34) or

(2.35).

16

Figure 2.3: Normalized propagation constant as a function of the normalized frequency, for the first six LP modes.

Figure 2.3 shows that a fiber with a large v supports many modes. An estimate to determine the

number of modes for a multimode fiber is to perform [2]

2

modes 2

vN = (2.37)

where modesN is the number of modes. Below a certain value of v , only the LP01 mode is propagated. The cut off

frequency 2.4048cv = is the value of v that turns the propagation of the LP01 mode possible [1]. The expression

that determines the maximum core radius is [1]

max

1

.2 2

cva an

λ

π≤ =

∆ (2.38)

The determination of the radius of the core influences the number of modes that can be

propagated. The fewer modes, the lower the core radius will be. Then, a monomodal fiber will have an

effective area lower than the effective area of the multimode fiber. Thus, the non-linear effects will be

more accentuated in the monomodal fiber than in the multimodal fiber.

2.6. Conclusions From this chapter the main conclusions that can be taken are:

• The number of modes that can be propagated in a fiber depends on the radius of the core

17

of the fiber;

• The core radius of the fiber determines whether the fiber is monomodal or multimodal;

• The cut-off frequency of a multimodal fiber is 2.4048cv = ;

• For low contrast fibers, when the contrast is lower the necessary v that is required, so that

the propagation is possible, is lower.

18

19

3. Pulse propagation in the linear regime

Even when the propagation of pulses is performed in a single-mode fiber, there is still a source of

dispersion that may influence the quality, speed, and bit rate of the communication. This type of fiber also

exhibits group velocity dispersion (GVD), and also may have higher order dispersion. The transmitted

pulses experience a time broadening due to the GVD. This pulse broadening may result in an inter-

symbolic interference (ISI) [1].

3.1. Propagation Equation in the linear regime In order to determine the shape of a pulse at the output of a communication link, the pulse

propagation equation has to be derived. Assuming that the electric field, at the input of the fiber, or 0z = ,

is linearly polarized in the direction of x, the electric field is given by

( ) ( )ˆ, ,0, , ,0,E x y y xE x y t= (3.1)

This expression be written as

( ) ( )0, ,0, , (0, )E x y t E F x y B t= ⋅ ⋅ (3.2)

where 0E is the amplitude of the electric field, ( ), F x y is the transversal distribution of the field in the

fundamental fiber mode and ( )0,B t is the time distribution of the field at 0z = . The approximation to the LP01

mode is reasonable because it is assumed that the fibers have low contrast, i.e. 1∆ << . The term ( )0,B t in equation

(3.2) defined at the entrance of the fiber, can be written as

( ) ( ) ( )00, 0, expB t A t i tω= ⋅ − (3.3)

where ( )0,A t is the pulse envelope at entrance of the fiber and 0ω is the angular frequency of the carrier.

The first point, it is important to determine the relation between ( )0,B t and ( )0,A t , using the

Fourier transform. In this thesis we use the following definitions of the Fourier transform and its inverse

Fourier transform are, respectively

( ) ( ) ( ), , expX z X z t i t dtω ω+∞

−∞= ⋅∫ (3.4)

20

( ) ( ) ( )1, , exp .

2X z t X z i t dtω ω

π

+∞

−∞= ⋅ −∫ (3.5)

Replacing ( ),X z t in equation (3.4), by the Fourier transform of ( ), ,0,E x y t yields

( ) ( ) ( ) ( ) ( )( )

0 0

0,

, ,0, , exp 0, exp .

A

E x y E F x y i t A t i t dt

ω

ω ω ω= ⋅ − ∫

(3.6)

Assuming a linear and time-invariant system and applying the properties of the Fourier transform, namely, the

frequency shift property, the equation (3.6) comes

( ) ( ) ( )0 0, ,0, , 0,E x y E F x y Aω ω ω= − (3.7)

which shows that

( ) ( )00, 0, .B Aω ω ω= − (3.8)

The spectral component ( ),B z ω propagates along the fiber with an propagation constant ( )pβ ω .

This propagation constant has several components [7], according to

( ) ( ) ( ) ( )2p L NL i

α ωβ ω β ω β ω= + + (3.9)

where ( )Lβ ω is the linear part of the propagation constant, ( )NLβ ω is the non-linear part of the propagation

constant and ( )α ω is the fiber loss parameter. In linear regime, the term ( )NLβ ω is zero. Assuming no losses, then

( )α ω is also zero. Then, equation (3.9) yields

( ) ( ).p Lβ ω β ω= (3.10)

To determine ( ),B z t , the inverse Fourier transform of ( ),B z ω needs to be performed. Replacing equation (3.10)

and, then, ( ),B z t yields

( ) ( ) ( )( )( )0

1, 0, exp .

2 LB z t A i z t dω ω β ω ω ωπ

+∞

−∞= − −∫ (3.11)

A simple way to show equation (3.11) is to make a variable change in which ω is replaced by Ω ,

such that

21

0

0

1.d

d

ω ω

ω ω

ω

Ω = −

=Ω+

=Ω

(3.12)

Then, by replacing (3.12), in (3.11), it comes

( ) ( ) ( ) ( )( )0 0

1, exp 0, exp .

2 LB z t i t A i z t dω β ωπ

+∞

−∞ = − Ω Ω + −Ω Ω ∫ (3.13)

To simplify the integral of equation (3.13), the term ( )0Lβ ω +Ω is expanded in a Taylor’s series. The result of the

expansion is

( ) ( )0 0Lβ ω β+Ω = +℘ Ω (3.14)

where ( )0 0β β ω= and

( )1 !

mm

m m

β∞

=

℘ Ω = Ω∑ (3.15)

where

0

.m

m m

d

dω ω

ββ

ω=

= (3.16)

With the expansion of ( )0Lβ ω +Ω , equation (3.13) results in

( ) ( ) ( )0 0, , expB z t A z t i z tβ ω= − (3.17)

where ( ),A z t is defined by

( ) ( ) ( )( )1, 0, exp .

2A z t A i z t d

π

+∞

−∞ = Ω ℘ Ω −Ω Ω ∫ (3.18)

In order to solve equation (3.18) the coefficients in equation (3.16) ( 1,2,m = …) have to be

determined. The first one, 1β , is physically related with the inverse of the group velocity gv as

0

1

1

g

d

d vω ω

ββ

ω =

= = (3.19)

22

Coefficients 2β and 3β are known as the second-order and third order dispersion terms and are related with gv by

0

2

2 2 2

1 g

g

vd

d vω ω

ββ

ω ω=

∂= = −

∂ (3.20)

0

32

3 3

d

dω ω

β ββ

ω β=

∂= =

∂ (3.21)

These parameters are responsible for the pulse broadening in optical fibers.

After computing the coefficients mβ , it is useful to determine ( ),A z t in terms of ( )0,A t . To do

that, it is useful to define

( ) ( ) ( ) [ ]1, 0, exp exp

2m

mA z t A i z i t dπ

+∞

−∞= Ω Ω ℘ Ω − Ω Ω ∫ (3.22)

Therefore from equation (3.18), results the general equation

( )1

, .!m

m

m

Ai A z t

z m

β∞

=

∂=

∂ ∑ (3.23)

Introducing now the fiber loss factor, equation (3.23) yields

( ) ( )1

, ,! 2m

m

m

Ai A z t A z t

z m

β α∞

=

∂= −

∂ ∑ (3.24)

where α is the attenuation constant.

After obtaining the derivative of ( ),A z t in order to z , it is necessary to derive the expression of

( ),mA z t as a function of

dAdtso that equation (3.24) can be completed. The first step is to derive

equation (3.22) in order to time for 1,2,3m = and 4 . Then the first four terms come

( )1 ,dA

iA z tdt

= − (3.25)

( )2

22,

d AiA z t

dt= (3.26)

( )3

33,

d AiA z t

dt= − (3.27)

23

( )4

44, .

d AiA z t

dt= (3.28)

Then, in a general way,

( )2 ,m

m

mm

d Ai A z t

dt

−= − (3.29)

in which ( ),mA z t is

( ) 2, .m

m

m m

d AA z t i

dt

−= − (3.30)

Due to the fact that equation (3.24) depends on ( ),mA z t , one has

( )1

1

, .! 2

mmm

mm

A d Ai i A z t

z m dt

β α∞−

=

∂= − −

∂ ∑ (3.31)

Then, by simplifying equation (3.31), it comes

( )1

1

, 0! 2

mmm

mm

A d Ai A z t

z m dt

β α∞−

=

∂+ + =

∂ ∑ (3.32)

If the fourth order and greater ( 4m ≥ ) propagation terms are ignored and the attenuation constant is neglected,

equation (3.32) comes

2 3

1 2 32 3

1 10.

2 6

A A d A d Ai

z t dt dtβ β β

∂ ∂+ + − =

∂ ∂ (3.33)

Equation (3.33) can be simplified, in order to easier derive its solution. To do so, a couple of

normalized variables is defined as [1]

0

12

0

D

D

z

LL

t z

ζτ

ββτ

τ

== →

− =

(3.34)

where DL is the dispersion length, τ0 is a measure of pulse width and 2β is the absolute value of the second-order

dispersion. In a first approach the terms dA

dzand

dA

dthave to be expressed in terms of τ and ζ , respectively

according to

24

1

1

1

1

D

D

A A A

z z z

z L

z

A A

z L z

ζ τζ τζ

βττβ τ

ζ τ

∂ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ ∂∂

= ∂

∂ = − ∂ ∂ ∂ ∂ = − ∂ ∂ ∂

(3.35)

0

0

0

.1

1

A A A

t t t

t

t

A A

t

ζ τζ τζ

ττ

τ τ

∂ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ ∂∂

= ∂ ∂ =

∂ ∂ ∂ = ∂ ∂

(3.36)

Then, equation (3.33) comes

2 3

322 2 2 30 0

1 10

2 6D D

A A Ai L L

A

ββζ τ τ τ

∂ ∂ ∂+ − =

∂ ∂ ∂ (3.37)

where the following definitions have been used

( )222

0

sgnDLββ

τ= (3.38)

330

1

6DLk

βτ

= (3.39)

By replacing equations (3.38) and (3.39) into (3.37), on has

( )2 3

2 2 3

1sgn 02

A A Ai kβ

ζ τ τ∂ ∂ ∂

+ − =∂ ∂ ∂

(3.40)

Where coefficient κ can be written as

330 2

1.

6k

βτ β

= (3.41)

The also called higher order dispersion coefficient.

25

After obtaining the basic propagation equation, the pulse shape can be determined at any distance

in the fiber, using simple computer simulation. Before the pulse propagation is performed, it is useful to

determine an expression that computes the broadening of the pulse along the fiber.

3.2. Analytical approach for pulse broadening The broadening of a pulse can induce ISI that limits the bit rate of a communication link. This

effect is more accentuated as long as the fiber length increases. Then, it is useful to determine an

analytical expression to measure the pulse in terms of the so called root-mean-square (RMS) width of the

pulse.

3.2.1. Second and third-order moments

In order to determine an expression for the width of a pulse, pulses of arbitrary shape have to be

take into account [7], because most pulses are not Gaussian and the dispersion coefficient 3β also affects

Gaussian pulses. Then, using the definition of root-mean square (RMS) value, the width of a pulse is

given by

22 2t tσ = − (3.42)

where t is the first order moment and 2t is the second order moment. The moments can be obtained by using a

general expression that is given by

( )

( )

2

2

,.

,

m

mt A z t dt

tA z t dt

+∞

−∞+∞

−∞

= ∫∫

(3.43)

In order to determine the pulse width, the non-linear effects have to be negligible. This is based on

the observation that the pulse spectrum does not change in a linear dispersive regime, irrespective of what

happens to the pulse shape [7].

The objective of this section is to determine t and 2t . In order to do that we will use the

following relation

( ) ( )22 1

, , 12

A z t dt A z dπ

+∞ +∞

−∞ −∞= Ω Ω =∫ ∫ (3.44)

26

Applying definition (3.43) and the relationship (3.44), t comes

( )2

, .t t A z t dt+∞

−∞= ∫ (3.45)

which is equivalent to write

( ) ( )*, , .t tA z t A z t dt+∞

−∞= ∫ (3.46)

By applying the Fourier transform to ( )* ,A z t , equation (3.46) can be written as

( ) ( ) ( )*1, , exp ,

2t tA z t A z i t d dt

π

+∞ +∞

−∞ −∞

= Ω Ω Ω ∫ ∫ (3.47)

And, changing the order of the integration, equation (3.47) finally yields

( ) ( ) ( )( )*1, , exp .

2t A z tA z t i t dt d

π

+∞ +∞

−∞ −∞= Ω Ω Ω∫ ∫ (3.48)

Because the system is assumed to be linear and time invariant, the derivative property can be

applied as

( ) ( ).

dX jtx t i

d

ωω

↔ (3.49)

Then, equation (3.48) becomes

( ) ( )*1, ,

2t i A z A z d

π

+∞

Ω−∞= − Ω Ω Ω∫ (3.50)

where

( ) ( ),, .

dA zA z

dΩ

ΩΩ =

Ω

(3.51)

After obtaining the first order moment, the second order moment can be easily determined.

Applying the definition of (3.43) and relation (3.44), 2t comes

( )22 2 ,t t A z t dt

+∞

−∞= ∫ (3.52)

Separating the terms inside the integral

27

( ) ( )2 , , .t tA z t tA z t dt+∞

−∞= ∫ (3.53)

After some algebra manipulation and using the Fourier transform, and its properties, equation (3.53) finally comes

( )22 , .t A z d

+∞

Ω−∞= Ω Ω∫ (3.54)

Expression for the first and second order moments were obtained in this section. In the next

sections the RMS width will be determined using these definitions and other parameters, such as, the

group delay.

3.2.2. RMS broadening of a function of the group delay

Although in the current approach the propagation constant does not include non-linear effects, it

includes dispersive effects of all orders. Considering the pulse at the input of the fiber

( ) ( ) ( )( )0, expA S iθΩ = Ω Ω (3.55)

where the spectral phase ( )θ Ω plays an important role as it is related to the frequency chirp of the pulse. In order to

take into consideration the chirp effects, the parameter ( )θ Ω and the group delay gτ have to be included in the

RMS expression of the broadening. To do that, the definitions derived in the previous sections will be used.

According to the definition of the group delay, one has

1L

g

g

d LL L

d v

βτ β

ω= = = (3.56)

where L is the fiber length and vg is the group velocity. Let’s define the derivative of ( )θ ω as

.d

d

θθΩ =

Ω (3.57)

To use the definitions (3.50) and (3.54), function ( ),A zΩ Ω has to be determined in terms of

( )S Ω and ( )θ Ω . Hence, using definition (3.51), ( ),A zΩ Ω comes

( ) ( ) ( )( ) ( ) ( )( )0 0, 0, exp 0, expd

A z A i z A iz i zd

ββ β β βΩ Ω ΩΩ = Ω − + Ω −

Ω (3.58)

where ( )0,AΩ Ω is

28

( ) ( )( ) ( ) ( )( )0, exp exp .A S i iS iθ θ θΩ Ω ΩΩ = Ω + Ω Ω (3.59)

Now, substituting equations (3.55), (3.58) and (3.59) into the definitions (3.50) and (3.54), the

moments 2t and t yield, in terms of ( )S Ω and ( )θ Ω , respectively

( ) ( ) ( )2 2 22 2 2 21 1 1 12

2 2 2 2g gt S d S d S d S dθ θ τ τπ π π π

+∞ +∞ +∞ +∞

Ω Ω Ω−∞ −∞ −∞ −∞= Ω + Ω Ω + Ω Ω + Ω Ω∫ ∫ ∫ ∫ (3.60)

( ) ( )2 2

1

1.

2 2

zt i S d S dθ β

π π

+∞ +∞

Ω−∞ −∞= Ω Ω+ Ω Ω∫ ∫ (3.61)

Introducing the following definition

( ) ( ) 21

2f f S d

π

+∞

−∞= Ω Ω Ω∫ (3.62)

the moments 2t and t can be written, respectively as

22 2 21

22 g gt S d θ τ θ τπ

+∞

Ω Ω Ω−∞= Ω + + +∫ (3.63)

.gt θ τΩ= + (3.64)

Then, replacing equations (3.63) and (3.64) in equation (3.42), the broadening parameter 2σ

finally comes

22 2 2

0 2 .g g g g

σ σ τ τ θ τ θ τΩ Ω = + − + −

(3.65)

This expression depends on the average of the group delay and of θΩ . The parameter θΩ is

responsible to the chirp effect in the broadening of the pulse. The chirp parameter will be taken into

account in the next section.

3.2.3. RMS broadening of a chirped pulse

In a single-mode fiber communication link, there are the time dispersion effects, but also chirp

effects. While the time dispersion is a broadening in the time domain, the chirp induces a broadening in

the pulse spectrum. The broadening of the pulse spectrum causes an increase in the number of frequency

components of the pulse. In this section the chirp parameter is taken into account in the expression of

equation (3.65).

29

Considering that the transmitted pulse is a super Gaussian pulse with chirp, we may write

( )2

0

0

1, exp

4

iC tA z t A

σ

+ = −

(3.66)

where 0A is the amplitude of the pulse, C is the chirp parameter, t is the time variable and 0σ is pulse width at

the input of the fiber. The group delay can be expressed in the form

( ) 21 2 3

1.

2g Lτ β β β Ω = + Ω + Ω

(3.67)

In a first stage, it is important to determine ( )0,A Ω , using the definition of the Fourier transform

( ) [ ]2

0

0

10, exp exp

4

iC tA A i t dt

σ

+∞

−∞

+ Ω = − Ω

∫ (3.68)

Using the well-known integral

( )2

2exp exp ,4

bax bx dx

a a

π+∞

−∞

− + = ∫ (3.69)

( )0,A Ω yields

( ) ( )2 2 2 2 2

2 10 0 00 2

4 10, 1 exp exp tan .

1 1 2 1A A C i C C

C C C

σ π σ σ− Ω Ω Ω = + − − +

+ + + (3.70)

In the equation (3.70), 0A has still to be determined. This parameter can be determined using the relation

( )2

, 1A z t dt+∞

−∞=∫ (3.71)

which it results in

0 240

1.

2A

πσ= (3.72)

Substituting equation (3.72) into equation (3.70), ( )0,A Ω becomes

30

( ) ( )2 2 2 2 2

10 0 042

8 10, exp exp tan .

1 1 2 1A i C C

C C C

πσ σ σ− Ω Ω Ω = − − +

+ + + (3.73)

Due to the fact that ( )0,A Ω is equivalent to

( ) ( ) ( ) 0, exp .A S θΩ = Ω Ω (3.74)

then,

( )2 2 20 042

8exp

1 1S

C C

πσ σ ΩΩ = −

+ + (3.75)

( ) ( )2 2

1 01tan .

2 1C C

C

σθ − Ω

Ω = − ++

(3.76)

To determine the expression 2σ in terms of the chirp parameter, the moments gτ , 2

gτ and

gτ θΩ have to be determined. The definitions of these moments are

( ) 21

2g g S dτ τπ

+∞

−∞= Ω Ω∫ (3.77)

( ) 22 21

2g g S dτ τπ

+∞

−∞= Ω Ω∫ (3.78)

( ) 21

2g g S dτ θ τ θπ

+∞

Ω Ω−∞= Ω Ω∫ (3.79)

where the term θΩ is

( ) 2

02

2 .1

C

C

θ σθΩ

∂ Ω Ω= =

∂Ω + (3.80)

Replacing equation (3.67) into equations (3.77), (3.78), and (3.80), the moments gτ , 2

gτ and gτ θΩ finally

comes

21 2 3

1

2g L L Lτ β β β= + Ω + Ω (3.81)

2 2 2 2 2 2 2 2 4 2 2 2 31 2 3 1 2 1 3 2 3

1 1 12

4 2 2g L L L L L Lτ β β β β β β β β β = + Ω + Ω + Ω + Ω + Ω (3.82)

31

2 2 2

2 31 0 2 0 3 02 2 2

2 2.

1 1 1g

LC LC LC

C C C

β σ β σ β στ θΩ = Ω + Ω + Ω

+ + + (3.83)

In addition to the moments gτ , 2

gτ and gτ θΩ , moments Ω , 2Ω , 3Ω and 4Ω have

also to be determined. Using the following definition of Ω

( ) 2

2 2 20 02 2

1

2

21exp 2 .

1 1

S d

dC C

π

σ σπ

+∞

−∞

+∞

−∞

Ω = Ω Ω Ω

Ω= Ω − Ω

+ +

∫

∫ (3.84)

and performing the variable change

2

20

1,

4

Cα

σ+

= (3.85)

moment Ω comes as

( )

2

2

1 1exp

2 22f

dπ αα π

+∞

−∞

Ω

ΩΩ = Ω − Ω

∫

(3.86)

where, f(Ω) is a Gaussian probability density function with a null median and variance α2. Then, the moments Ω ,

2Ω , 3Ω e 4Ω result in

0Ω = (3.87)

2

2 220

1

4

Cα

σ+

Ω = = (3.88)

3 0Ω = (3.89)

224

20

3 1.

16

C

σ +

Ω =

(3.90)

Now the moments gτ , 2

gτ and gτ θΩ can be completed by incorporating the equations (3.87)

, (3.88), (3.89) and (3.90) in their expressions as

32

2

1 3 20

1 1

3 4g

CL Lτ β β

σ +

= +

(3.91)

2 2 2

2 2 2 2 2 2 2 21 2 3 1 32 2 2

0 0 0

1 1 3 1 1

4 4 16 4g

C C CL L L Lτ β β β β β

σ σ σ + + +

= + + +

(3.92)

2 .2g LCβ

τ θΩ = (3.93)

Replacing the moments gτ , 2

gτ and gτ θΩ into equation (3.65), the parameter σ2 comes

( )22 2

22 2 2 2 2 32 20 0 02 2 3

0 0 0

1 1 .2 2 4 2

LC L LC

ββ βσ σ σ σ

σ σ σ

= + + + +

(3.94)

which it can be simplified by

( )22 22

22 32 22 2 2 30 0 0 0

1 1 .2 2 4 2

LC L LC

ββ βσσ σ σ σ

= + + + +

(3.95)

The expression of equation (3.95) gives the ratio between the width of the pulse at some point z in

the fiber and the width of the pulse at the input of the fiber. The last term of the right-hand side is the

contribution of the third-order dispersion. This equation allows the understanding of the pulse broadening

through a fiber section, for a given values of C, whether the pulse is unchirped or chirped.

3.3. Pulse propagation This section focuses on the simulation of the propagation of pulses through an optical fiber. The

“sech” shape pulse and the super Gaussian pulse are used. In the super Gaussian pulse case, the chirp

effect are also studied.

The propagation equation can be compactly written in order to simplify its simulation. The

propagation equation can be given by

( )2

2 2

1sgn 02

u ui β

ζ τ∂ ∂

− =∂ ∂

(3.96)

where u is the pulse, and ζ and τ are the normalized variables. In what follows, the third-order dispersion parameter

β3, is neglected. Assuming anomalous dispersion, one has ( )2sgn 1β = − , the propagation equation yields

33

2

2

1.

2

u ui

τ τ∂ ∂

= −∂ ∂

(3.97)

The numerical simulation is explained in the annex B1.

The following simulations were performed by solving the propagation equation (3.97). To solve

the propagation equation, it is easier to perform some operations in the frequency domain.

3.3.1. Bell shaped pulses

The pulse used in this simulation is a “sech” pulse, which expression given by

( ) ( )0, sechu τ τ= (3.98)

where τ is the normalized time. In this simulation τ goes from −20 to 20 ps, and ζ goes from zero

to 5. The pulse shape at the input and the output of the fiber are given by Figure 3.1.

Figure 3.1: Absolute value of the pulse. The pulse at the input of the fiber is represented by a dashed line. The impulse at the

output of the fiber is represented by a full line.

Figure 3.2 shows the variation of the absolute value of the pulse along the fiber link.

34

Figure 3.2: Absolute value of the pulse along the fiber, observed by two different angles.

Figure 3.1 can be taken from Figure 3.2, by making 0ζ = and 5ζ = . As both figures show, the

absolute value of the pulse decreases along the fiber, due to the dispersion, although there is no

attenuation. However, the energy of the pulse at the input is the same as at the output of the fiber. As can

be seen, the pulse at the output of the fiber is broader than the pulse at the input of the fiber. This also

happens due to the time dispersion.

3.3.2. Super-Gaussian pulse with chirp

Before presenting the simulations of the propagation of the pulses, it is very important to

understand how the width of a pulse changes along the fiber. To do that, a simple expression can be

determined based on equation (3.95). First, by the third-order dispersion factor is neglected, 3 0β = .

Secondly, considering that

20

2

2DL

σβ

= (3.99)

( )2 2 2 2sgnβ β β β= = − (3.100)

equation (3.95) comes

( )2 2

0

1 .Cσ

ζ ζσ

= + + (3.101)

The simulation of equation (3.101) is shown in Figure 3.3.

35

Figure 3.3: Pulse broadening along a fiber section, for different values of C.

Figure 3.3 shows the variation of the pulse broadening along a certain fiber section. In order to

better understand Figure 3.3, a super Gaussian pulse will be simulated. Its expression is given by

( ) 210, exp

2miC

u τ τ+ = −

(3.102)

where C is the chirp parameter and 3m = . The pulse shapes at the input of the fiber and at the output of the fiber

shown in Figure 3.4.

Figure 3.4: Pulse amplitude at the input and output of the fiber for C=−2.

Figure 3.5 shows the evolution of the pulse along a fiber section, 2C = − .

36

Figure 3.5: Evolution of the absolute value of the super Gaussian pulse for C=−2.

Figure 3.6 presents the pulse amplitude at the input and output of the fiber, for 0C = .

Figure 3.6: Pulse amplitude value at the input and output of the fiber, for C=0.

Figure 3.7 shows the evolution of the pulse along the fiber sector for 0C = .

Figure 3.7: Evolution of the absolute value of the super Gaussian pulse, for C=0.

37

Figure 3.8 presents the pulse amplitude at the input and output of the fiber with 2C = .

Figure 3.8: Pulse value at the input and output of the fiber for C=2.

Figure 3.9 shows the evolution of the pulse along the communication link with 2C = .

Figure 3.9: Evolution of the absolute value of the super Gaussian pulse for C=2.

Figure 3.4 to Figure 3.9, include all the previous effects, such as, temporal dispersion. But due to

the incorporation of the chirp effect there are more effects that have to be considered. As it was mentioned

before, the chirp parameter causes a broadening in the pulse spectrum, which means that the pulse

acquires additional frequency components. Comparing Figure 3.6 with Figure 3.8, the output pulse of

Figure 3.8 has more fluctuations than the output pulse of Figure 3.6. Then, a positive chirp value increases

the broadening of a pulse spectrum.

Figure 3.3 shows that a negative chirp value leads to greater broadening than positive values of

the chirp. When the chirp is positive the broadening curve has a minimum. The distance for which this

minimum occurs can be to determined by [7]

38

( )2

.1

C

Cζ =

+ (3.103)

The output pulse corresponding to this minimum is depicted in Figure 3.10.

Figure 3.10: Pulse amplitude for the entrance pulse and exit pulse, when the broadening is minimum.

Figure 3.10 shows that the width of the exit pulse is lower than the width of the entrance pulse.

This means that for positive chirp values the pulse is unchirped for a range of length.

3.4. Higher order dispersion influence effects In this section, the third-order dispersion influence on the pulse propagation is study. The first

example to be considered is when D DL L′= , where DL is the dispersion length due to 2β and DL′ is the

dispersion length due to 3β . The second example is when 0 Dλ λ= . The pulse used is the super Gaussian

pulse of equation (3.102) is considered.

3.4.1. LD=L’D

In this section, it is considered that the dispersion length due to 2β is equal to the dispersion

length due to 3β . The propagation method is explained in the Annex B1. Table 3.1 shows the parameters

used to simulate the super-Gaussian pulse propagation.

Table 3.1: Parameters of the pulse propagation with third-order dispersion, when D DL L′= .

Parameters Values

21β 20− ps2/km

0τ 50ps

39

1DL 125 km

1L 100 km

Figure 3.11 shows the input and output of the pulse when the chirp parameter is 2C = − .

Figure 3.11: Input and output pulses for the case of third order dispersion, when D DL L′= , and 2C = − .

Figure 3.12 shows the input and output pulse when the chirp parameter is 0C = .

Figure 3.12: Input and output pulse for the case of third order dispersion, when D DL L′= , and 0C = .

Figure 3.13 shows the input and output pulses when the chirp parameter is 2C = .

40

Figure 3.13: Input and output pulses for the case of third order dispersion, when D DL L′= , and 2C = .

The third-order dispersion causes an oscillatory behavior in one of the sides of the pulse. When

the chirp parameter is zero the oscillation is less accentuated, because the spectrum of the pulse is not so

broad as when the chirp is different from zero. In these simulation, since 2β is zero, the pulse suffers the

usual time broadening.

3.4.2. λ0=λD

When the wavelength 0λ is equal to the wavelength Dλ , the dispersion factor 2β is negligible.

Then, only 3β should be considered. The used simulation parameters are shown in Table 3.2.

Table 3.2: Parameters of the pulse propagation with third-order dispersion, when 0 Dλ λ= .

Parameters Values

3β 1 ps3/km

0τ 20 ps

DL′ 80000 km

L 4000 km

The explanation of how to simulate this type of phenomenon is presented in Annex B1. For this type of simulation, the

distances are normalized not to DL but to DL′ which is defined by [9]

30

3

.DLτβ

′ = (3.104)

Figure 3.14 shows the propagation of the super-Gaussian pulse under the effect of third order

dispersion, when 0 Dλ λ= , while the chirp parameter is 2C = − .

41

Figure 3.14: Input and output pulses for the case of third order dispersion, when 0 Dλ λ= , and 2C = − .

Figure 3.15 shows the propagation of a super-Gaussian pulse considering the third order

dispersion, when 0 Dλ λ= , while the chirp parameter is 0C = .

Figure 3.15: Input and output pulses for the case of third order dispersion, when 0 Dλ λ= , and 0C = .

Figure 3.16 shows the propagation of a super-Gaussian pulse considering the third order

dispersion, when 0 Dλ λ= , while the chirp parameter is 2C = .

42

Figure 3.16: Input and output pulses for the case of third order dispersion, when 0 Dλ λ= , and 2C = − .

The oscillating behavior is still verified when 0 Dλ λ= , but is less accentuated when the chirp is

zero. As 2β is neglected, there is no time dispersion on the pulse.

3.5. Dispersion compensation When the dispersion caused by de GVD is high, inter-symbolic interference can occur, imposing a

lower bit rate, so that high quality of service (QoS) cannot be achieved. There are several techniques to

compensate the dispersion. The use of dispersion compensating fibers (DCF) is one of the most used. A

system with a DCF fiber is presented in Figure 3.17.

Figure 3.17: Transmission system with a dispersion compensating fiber.

In Figure 3.17, the single-mode fiber (SMF) has a dispersion factor 21 0β < , while the DCF has a

dispersion factor 22 0β > . The SMF has a length 1L while the DCF has a length 2L . Considering a pulse at

the input of the receiver, whose inverse Fourier transform is [9]

( ) ( ) ( )221 1 22 2

1, 0, exp

2 2

iA L t A L L i t dω ω β β ω ω

π

+∞

−∞

= + − ∫ (3.105)

43

the condition of perfect compensation comes [9]

21 1 22 2 0.L Lβ β+ = (3.106)

Then, the parameter 22β has to be determined by solving equation (3.106), according to

21 122

2

.L

L

ββ = − (3.107)

The parameters used in this simulation are shown in the Table 3.3.

Table 3.3: Parameters used in the propagation and dispersion compensation numerical simulation.

Parameters Values

21β 20− ps2/km

0τ 50ps

1DL 125 km

1L 250 km

22β 400ps2/km

2DL 6.25 km

2L 12.5 km

The output shape of Gaussian pulse at the end of the DCF for a chirp 2C = − is shown in Figure

3.18, compared with the input pulse.

Figure 3.18: Input and output of the DCF fiber with a chirp factor 2C = − .

Figure 3.19 shows the evolution of the pulse through the DCF with 2C = − .

44

Figure 3.19: Pulse evolution along the DCF for a 2C = − , viewed from two perspectives.

To illustrate the influence of the chirp in the DCF, the input and output pulses of the DCF are

shown in Figure 3.20 with 0C = .

Figure 3.20: Input and output of the DCF fiber with a chirp factor 0C = .

Figure 3.21 shows the evolution of the pulse through the DCF with 0C = .

45

Figure 3.21: Pulse evolution along the DCF for a 0C = , viewed from two perspectives.

Dispersion compensation using a DCF with a positive chirp was also simulated. The result input

and output pulses are shown in Figure 3.22, fora 2C = .

Figure 3.22: Input and output of the DCF fiber with a chirp factor 2C = .

Figure shows the evolution of the pulse through the DCF with 2C = .

46

Figure 3.23: Pulse evolution along the DCF for a 2C = , viewed from two perspectives.

From the Figure 3.18 to Figure 3.23, the initial pulse shapes are fully recovered. The dispersion of

all the pulses is fully compensated despite the chirp value. The amplitude of the pulses at the output of the

DCF is the same as at the input, because the attenuation is negligible. There is no residual dispersion

because the high-order dispersion is also negligible. The DCF cannot compensate the high-order

dispersion.

3.6. Conclusions The conclusions from this chapter are the following:

• The pulse width increases, due to temporal dispersion;

• The amplitude of a pulse decreases along the fiber due to pulse dispersion. Although there

are no losses, the energy of the pulse must be maintained, so its amplitude decreases;

• Both positive and negative chirp values cause a pulse broadening;

• Third-order dispersion, causes an oscillatory behavior in one of the sides of the pulse, and

it is more accentuated when the chirp parameter is different from zero;

• When 0 Dλ λ= there is no temporal dispersion, while for D DL L′= there is temporal

dispersion;

• Due to the fact that there are no losses, nor fiber non-linearties, nor third-order dispersion

term, the pulse can be fully recovered through dispersion compensation;

47

4. Pulse propagation in the non-linear regime

The numerical simulation of the propagation of solitons in optical fibers is addressed in this

chapter. It is shown that the non-linear Kerr effect is the fundamental cause for the propagation of solitons

in an optical fiber. The propagation equations in the non-linear regime are determined, so that, the

numerical simulations of soliton propagation may be performed.

4.1. Non-linear Kerr effect When a pulse propagates in the linear regime, the frequency dispersion affects the pulse shape

arriving at the output of the communication link. The non-linear regime can mitigate the dispersion effect

on the pulse propagation. One of the most extraordinary effects of the non-linearity is the propagation of

solitons. The propagation of solitons is only possible due to the effect of the self-phase modulation (SPM).

The SPM is a consequence of the non-linear Kerr effect.

The non-linear phase generated by the Kerr effect is given by [11]

( ) ( )NL int P tφ γ= Μ (4.1)

where

220

n

wγ

λ′

= (4.2)

( )11 exp .Lα

αΜ = − − (4.3)

The parameter ( )inP t is the input power, related with the carried power according to

( ) ( ) ( ), exp .inP z t P t zα= − (4.4)

The parameter γ is a normalized variable, λ is the wavelength, 20w is the spot size in the Gaussian shape and 2n′ is

given by [11]

48

02 2

0

n nµε

′ = (4.5)

where µ0 and ε0 are magnetic and electric constants of the regime, respectively.

The parameter Μ is the effective length, where the parameter α of equations (4.3) and (4.4) is the

attenuation coefficient and L is the physical length of the link.

A bright pulse suffers a local instantaneous frequency shift due to the SPM, which in the front of

the pulse has a shift for the red zone and the tail of the pulse is a shift for the blue zone. Then, in the

anomalous dispersion zone ( Dλ λ< ), where the dispersion coefficient of the group velocity is 2 0β < ,

there is an antagonic action between the SPM and the GVD dispersion. Then, in this region the

propagation of bright solitons is possible, i.e. pulses that conserve their shape along the propagation.

When 2 0β < the solitons are called, bright solitons, when 2 0β > the solitons are called, dark solitons

[11].

In practice, the SPM is sort of chirp, but this chirp increases in magnitude with the travelled

distance, i.e. new frequency components are generated continuously as the optical signal propagates along

the fiber. The magnitude of the SPM-induced chirp depends on the pulse shape. The increase of the

number of frequency components leads to a spectral broadening of the pulse, and this consequence is

undesirable, because it not only increases the signal bandwidth but also distorts the pulse shape when

dispersive effects are not included [13].

To simulate the propagation of solitons, the non-linear pulse propagation equation has to be

derived. This equation will be presented in the next section.

4.2. Pulse propagation equation for the a non-linear regime

In the previous section, the basic conditions for the propagation of solitons were addressed. In this

section, the corresponding propagation equation will be derived. The anomalous dispersion zone is

considered, 2 0β < in which the propagation of bright solitons is only possible.

The nonlinear propagation is generrally performed in single-mode fibers. Each frequency

component propagates in the fiber with a slightly different propagation constant. So it is useful to work in

the spectral domain. Then, considering the pulse in the frequency domain we may write along the fiber,

( ) ( ) ( ), 0, ,A z A f zΩ = Ω Ω (4.6)

49

Where = − , is the carrier frequency, and we include the dependence of ( ),f z Ω has the propagation

constant in terms of Ω. The phase constant βp depends on ω according to

( ) ( ) ( ) ( )00 2p L NL i

α ωβ ω β ω β ω= + + (4.7)

where the term ( )Lβ ω is the linear part, the term ( )0NLβ ω is the non-linear term which only depends on the

angular frequency of the carrier, the term ( )0α ω is the attenuation constant which only depends on the angular

frequency of the carrier. Then, the factor ( ),f z ω yields

( ) ( ) ( )( )( ) ( )00, exp exp .

2L NLf z i z tα ω

ω β ω β ω ω

= + − −

(4.8)

To transform the term ( ),f z ω to the term ( ),f z Ω , the following variable change has to be

performed

0

0

1.d

d

ω ω

ω ω

ω

Ω = −

= Ω +

=Ω

(4.9)

Then, ( ),f z Ω yields

( ) ( )( ) ( ) ( )00 0, exp exp exp

2L NLf z i z t iα ω

β ω β ω

Ω = Ω+ −Ω −

(4.10)

where the second term of the right-hand (4.10) is the non-linear part.

In order to transform equation (4.6) in a propagation equation, a new amplitude must be

introduced as described in [11]

( ) ( )2*, ,Q z t y F A z t= (4.11)

where y* is an admittance that comes from a fictional electric field *E , and 2F is the moment of the modal

function ( ),F x y .

The propagation equation has a linear part and a non-linear part as

50

( ) ( ), ,L NL

QR z t R z t

z

∂= +

∂ (4.12)

where ( ),LR z t is the linear part and is already determined in the previous section as

( )2 3

1 2 32 3

1 1,

2 6 2L

Q Q QR z t i Q

t t t

αβ β β

∂ ∂ ∂= − − + −

∂ ∂ ∂ (4.13)

and the term ( ),NLR z t in equation (4.12) is the non-linear part.

( ) 2, ,NLR z t i Q Qγ= (4.14)

where the term γ is equal to

2

0

2

eff

n

A

πγ

λ= (4.15)

where 2n is a constant parameter with values around 202.6 10−× m2/W, 0λ is the carrier wavelength and effA is the

spot size or effective core area of the fiber. The parameter γ takes into account the various nonlinear effects

occurring within the fiber. As an example, 2.1γ = W−1Km-1 for a fiber with 50effA = µm2 [14]. Finally the

propagation equation yields

2 3

2

1 2 32 3

1 1.

2 6 2

Q Q Q Qi Q i Q Q

z t t t

αβ β β γ

∂ ∂ ∂ ∂= − − + − +

∂ ∂ ∂ ∂ (4.16)

As in the previous section, to solve the propagation equation in an easier way the variables t and

z are replaced by the normalized variables τ and ζ using the following definitions

D

z

Lζ = (4.17)

1

0

t βτ

τ−

= (4.18)

with

20

2

.DLτβ

= (4.19)

Then, the propagation equation can be written as

51

( )2 3

22 2 3

1sgn2 2D

Q Q Qi i L Q Q Qβ κ γ

ζ τ τ∂ ∂ ∂ Γ

+ − − = −∂ ∂ ∂

(4.20)

where

3

2 06

βκ

β τ= (4.21)

.DLαΓ = (4.22)

To easily solve the propagation equation, a normalized amplitude needs to be introduced

( ) ( )0

,,

QU

P

ζ τζ τ = (4.23)

where P0 is the maximum power of the input pulse. The propagation equation finally comes

( )2 3

222 2 3

1sgn2 2

U U Ui i N U U i Uβ κ

ζ τ τ∂ ∂ ∂ Γ

− − + = −∂ ∂ ∂

(4.24)

where the 2N is defined as [15]

2 D

NL

LN

L= (4.25)

where NLL is the nonlinear length and is defined as

0

1.NLL

Pγ= (4.26)

If 2γ = W−1Km-1 is used as a typical value, the nonlinear length will be 100NLL ∼ km at peak power levels in the

range of 2 to 4 mW. The dispersion length DL can vary between a range from 1 to 10000 km, depending on the bit

rate of the system and the type of fibers used to build it [15].

The parameter 2N represents a dimensionless combination of the pulse and fiber parameters, this

parameter can be removed from the propagation equation introducing a renormalized amplitude

( ) ( ), , .u NUζ τ ζ τ= (4.27)

Disregarding losses and second order dispersion, the propagation equation can be written in the canonical form,

doing ( )2sgn 1β = − , 0κ = and 0 Г =

52

2

2

2

10 .

2

u ui u u

ζ τ∂ ∂

+ + =∂ ∂

(4.28)

The propagation of solitons is only possible in a non-linear non-dipersive regime, where DL L>

and NLL L< . In the normal dispersion zone ( )Dλ λ< , only dark solitons can be propagated, while in the

anomalous dispersion zone ( )Dλ λ> only bright solitons can be propagated [11], which are the only ones

more useful in long-haul communications.

After obtaining the propagation equation, several simulations are going to be performed in the

next sections, in order to evaluate how the propagation of solitons is influenced by several parameters.

4.3. Solitons in optical fibers In this section, the propagation of solitons will be simulated under several conditions. Two basic

types of pulses are used in this section, the “sech” shape pulse and the Gaussian pulse. The numerical

simulations are performed using the split-step Fourier method (SSFM). It is assumed that the dispersion is

anomalous ( )2sgn 1β = − , and there are no losses ( 0Γ = ). The numerical procedure is explained in Annex

C1.

4.3.1. First-order soliton

In next sections,“sech” shape pulses are used, i.e. having the following form

( ) ( )0, sechu Nτ τ= (4.29)

where N is the order of the pulse. In this section, we use 1N = , then the initial pulse simply yields

( ) ( )0, sech .u τ τ= (4.30)

When, the pulse has a “sech” shape, and 1N = , it is called the fundamental soliton, and its propagation along the

fiber is shown in Figure 4.1.

53

Figure 4.1: Propagation of the first-order soliton along a fiber link.

Figure 4.1 shows that the soliton does not suffer dispersion and its amplitude is the same along the

fiber. Due to its unchanged shape along the fiber, this pulse is called the fundamental soliton. The pulse

shape does not change, because the fiber non-linearity exactly compensates the GVD effect [16]. Even if

N is not exactly equal to 1, i.e. if N lies between 0.5 and 1.5, the pulse tends to the fundamental soliton

for 1ζ >> . This happens, because the optical soliton is remarkably strong and stable against perturbations

[16].

4.3.2. Second-order soliton

The pulse shape of a second order soliton (N=2) is

( ) ( )0, 2sech .u τ τ= (4.31)~

The propagation of the second order soliton along the fiber is shown in Figure 4.2.

Figure 4.2: Propagation of the second-order soliton along a fiber link.

54

Figure 4.2 shows that the second-order soliton does not tend to the fundamental soliton even for

higher values of ζ . This soliton has a periodic behavior, where the initial shape is periodically recovered

along the fiber with a specific period. This period z0 is equal to

0 .2 Dz Lπ

= (4.32)

This means that, when 2

mπζ = with m=1,2,…, the pulse shape is equal to the initial one.

4.3.3. Third-order soliton

The pulse equation of the third-order soliton (N=3) is

( ) ( )0, 3sech .u τ τ (4.33)

Figure 4.3 shows the propagation of the third-order soliton along the fiber link.

Figure 4.3: Propagation of the third-order soliton along a fiber link.

Figure 4.3 shows that the third-order soliton has a periodic behavior like the second-order soliton,

and also does not tend to the fundamental soliton, for 1ζ >> . The third-order soliton recovers its initial

shape pulse with the same period of the second-order soliton, i.e. 2

mπζ = . The second and third order

solitons prove that, for N higher than 1.5, the pulse does not tend to the shape of the fundamental soliton.

4.3.4. Gaussian pulse

To tend to the fundamental soliton, the pulse shapes need to be “sech” shape, they can also have

Gaussian shape. One example of a Gaussian pulse shape is

55

( )2

0, exp .2

uτ

τ

= −

(4.34)

Figure 4.4 shows the propagation of a Gaussian shape along a fiber link.

Figure 4.4: Propagation of a Gaussian pulse soliton along a communication link.

Figure 4.4 shows that as long as the pulse propagates through the fiber, it tends to fit its shape to

the shape of the fundamental soliton. Therefore for 1ζ >> , the Gaussian shape tends to a “sech” shape.

4.4. Interaction between solitons In the real world the communications are not perform using just a single pulse, at each time. A

train of pulses is constantly propagated through a communication link. In order to propagate a train of

solitons, they need to be well sufficiently separated. Typically spacing between solitons exceeds four

times their full width at half maximum [16]. This means that apart, each soliton occupies only a fraction of

the bit slot. However, the interaction between solitons is inevitable.

To study the interaction between solitons, a particular pulse shape that simulates the interaction is

used. Its mathematical expression is given by [16]

( ) ( ) ( ) ( )0 00, sech sech expu q r r q iτ τ τ θ= − + + (4.35)

where r is the relative amplitude between the two solitons, θ is the relative phase and 02q is the initial

(normalized) separation. In the next simulations, a 0q of 3.5 is used. Figure 4.5 shows the interaction between two

solitons, when 0θ = and 1r = .

56

Figure 4.5: Interaction between two solitons, when θ=0 and r=1.

Figure 4.5 shows that the two solitons attract each other, and collide periodically, because the in-

phase is zero.

Figure 4.6 shows the interaction between two solitons when θ=π/2 and r=1.

Figure 4.6: Interaction between two solitons with θ=π/2 and r=1.

Figure 4.6 shows that the two solitons repel each other, and their spacing slowly increases with

the traveled distance.

Figure 4.7 shows the interaction between two solitons when 0θ = and 1.1r = .

57

Figure 4.7: Interaction between two solitons when θ=0 and r=1.1.

Figure 4.7 shows that the two solitons do not collide neither diverge from each other. This shows

that one way to avoid soliton collision is to change the relative amplitude, i.e. one soliton has a different

amplitude from the other.

When a soliton collides with another soliton, time jitter effects can happen, and that is not

tolerable. Another way to avoid soliton interaction is to increase their initial separation, i.e. the factor 0q .

How even, the disadvantage of increasing 0q is that the solitons need more space within the bit slot,

which decreases the bit rate. This means, that the soliton spacing limits the bit rate, because a lower

number of solitons are sent through the communication link.

4.5. Conclusions The main conclusions of this chapter are:

• The fundamental soliton keeps its shape along the fiber;

• The second and third order solitons recover their initial shapes after a certain period. That

period depends on the order of the soliton;

• The Gaussian pulse tends to the fundamental soliton when the traveled length is very

large;

• Two solitons can collide or attract each other periodically or not depending on the

parameters θ and r .

58

59

5. Conclusions

In this section, the main conclusions and future work are presented.

5.1. Main conclusions The objective of this work was to study the influence of the dispersion in a pulse propagation

situation, for a linear regime and a non-linear regime.

Regarding the fiber optics structure, the number of modes that can be propagated in a fiber

depends on the radius of the core of the fiber. The core radius of the fiber determines whether the fiber is

monomodal or multimodal. For low contrast fibers, when the contrast is lower the necessary v that is

required, so that the propagation is possible, is lower.

When pulses are propagated in a linear regime, the pulse width increases, due to temporal

dispersion. The amplitude of a pulse decreases along the fiber due to pulse dispersion. Although there are

no losses, the energy of the pulse must be maintained, so its amplitude decreases. Both positive and

negative chirp values cause a pulse broadening. Third-order dispersion, causes an oscillatory behavior in

one of the sides of the pulse, and it is more accentuated when the chirp parameter is different from zero.

When 0 Dλ λ= there is no temporal dispersion, while for D DL L′= there is temporal dispersion. Due to the

fact that there are no losses, nor fiber non-linearties, nor third-order dispersion term, the pulse can be fully

recovered through dispersion compensation.

When pulses are propagated in non-linear regimes, The fundamental soliton keeps its shape along

the fiber. The second and third order solitons recover their initial shapes after a certain period. That period

depends on the order of the soliton. The Gaussian pulse tends to the fundamental soliton when the traveled

length is very large. Two solitons can collide or attract each other periodically or not depending on the

parameters θ and r .

5.2. Future work There still remains to address a lot of aspects related to the simulations that were performed in this

work. One of them is to account with the attenuation due to losses along the pulse propagation. Another is

that the DCF technique can also be used to compensate the third-order dispersion.

60

When the propagation is performed in a non-linear regime and single-mode fibers are used, the

only linear impairment that exists is the self-phase modulation. One of the aspects that can be study is

when several modes propagate through the fiber. Then, the cross-phase modulation (XPM) and the four-

wave mixing (FWM) can be considered and studied.

Another aspect that can be further explored is the dispersion compensation in the non-linear

regime. There are also several ways to compensate the dispersion, besides the usage of DCFs. The use of

optical filters is another method to compensate the dispersion, as well as The use of other fibers such as,

fiber bragg gratings, with uniform-period gratings, and chirped fiber gratings. Another device that can be

used is the chirped mode coupler.

61

Bibliography [1] C. Paiva, “Fibras ópticas”, Departamento de engenharia electrotécnica e de computadores,

Instituto Superior Técnico, April of 2010, pp 3-5.

[2] G. Agrawal, “Fiber optic communication system: Chapter 1 – Introduction”, published by John

Wiley and Sons, USA 2005, pp. 1-19.

[3] G. Agrawal, “Nonlinear fiber optics: Chapter 1 – Introduction”, published by Academic Press,

USA 2001, pp. 1-24.

[4] G. Agrawal, “Fiber optic communication system: Chapter 7 – Dispersion management”,

published by John Wiley and Sons, USA 2005, pp. 279-320.

[5] G. Agrawal, “Fiber-optic communications systems: Chapter 2 – Optical fibers”, Published by

Jonh Wiley & Sons, 2002, pp. 23-72.

[6] K. Okamoto, “Fundamentals of optical waveguides: Chapter 3 – Optical fibers”, Published by

Elsevier 2006, pp. 57-155.

[7] G. Agrawal, “Lightwave technology telecommunication systems: Chapter 3 – Signal propagation

in fibers”, published by John Wiley and Sons, USA 2005, pp. 63-103.

[8] G. Agrawal, “Nonlinear fiber optics: Chapter 3 – Group velocity dispersion”, published by

Academic Press, USA 2001, pp. 63-93.

[9] G. Agrawal, “Lightwave technology telecommunication systems: Chapter 7- Dispersion

management “, published by John Wiley and Sons, USA 2005, pp. 279-320.

[10] D. Estrada, “Propagação de feixes ópticos em meios não-lieares: Chapter 4 – Solitões

espaciais”, Msc. Instituto Superior Técnico, June 2008, pp. 85.

[11] C. Paiva, “Solitões em fibras ópticas”, Departamento de Engenharia Electrotécnica e de

Computadores, Instituto Superior Técnico, April 2008, pp. 1-66.

[12] C. Paiva, Solitões em fibras ópticas: 2. Efeito não-linear de Kerr numa fibra óptica,

Departamento de Engenharia Electrotécnica e de Computadores, Instituto Superior Técnico, April

2008, pp. 10-17.

[13] G. Agrawal, “Lightwave technology telecommunication systems: Chapter 4 – Nonlienar

impairment”, published by John Wiley and Sons, USA 2005, pp. 126-145.

[14] G. Agrawal, “Lightwave technology telecommunication systems: Chapter 3 – Signal propagation

in fibers”, published by John Wiley and Sons, USA 2005, pp. 63-103.

[15] G. Agrawal, “Lightwave technology telecommunication systems: Chapter 8 –Nonlinearity

management”, published by John Wiley and Sons, USA 2005, pp. 284-338.

[16] G. Agrawal, “Fiber optic communication system: Chapter 9 – Soliton systems”, published by

62

John Wiley and Sons, USA 2005, pp. 404-468.

63

Annex A

A1. Modal equation of the hybrid modes

In order to determine the characteristic equation for the modes in an optical fiber, it is useful to

work in a cylindrical coordinate system ( r , φ , z ). In this discussion, the step-index fiber is considered

and for this system of coordinates, the electric and magnetic field are respectively [1]

ˆ ˆ ˆr zE E Eφ+ +E = r zϕϕϕϕ (A1.1)

ˆ ˆ ˆr zH H Hφ= + +H r zϕϕϕϕ (A1.2)

where rE , Eφ and zE and, rH , Hφ and zH are the three field components along r , ϕϕϕϕ , z are the unity vectors

of each coordinate.

Let us consider that the electric and magnetic field can be represented by A . defined as a

function of ( r , φ , z ) according to [1]

( ) ( ) [ ] ( )0, , , exp expr z t r im i z tφ φ β ω= − A A (A1.3)

where m is an integer, β is the propagation constant and ω is the angular frequency.

Considering the two following derivatives

imφ∂

=∂

(A1.4)

iz

β∂

=∂

(A1.5)

and using the definition

ˆ ˆ ˆ

1

r z

r

r r z

A rA Aφ

φ∂ ∂ ∂

∇× =∂ ∂ ∂

r z

A

ϕϕϕϕ

(A1.6)

Then, ∇×A results in

64

( )1ˆ ˆ ˆz

z r r

Ami A A i A rA imA

r r r rφ φβ β

∂ ∂ ∇× = − + − + − ∂ ∂ A r z.ϕϕϕϕ (A1.7)

The next step is to determine the r components of the electric and magnetic field. The first two

Maxwell’s equations need to be take into account

0iωµ∇× =E H (A1.8)

( )20i n rωε∇× = −H E (A1.9)

where 0µ is the vacuum magnetic permeability, 0ε is the electric permittivity in the vacuum and ( )n r is the

refractive index which is a function of r . Then, from equation (A1.8) one obtains

0z r

mi E E i H

rφβ ωµ − =

(A1.10)

0r

r

Ei E i H

rφβ ωµ

∂− =

∂ (A1.11)

( ) 0

1r zrE imE i H

r rφ ωµ

∂ − = ∂ (A1.12)

and from equation (A1.9),

( )20z r

mi H H i n r E

rφβ ωε − = −

(A1.13)

( )20

zr

Hi H i n r E

rφβ ωε

∂− = −

∂ (A1.14)

( ) ( )20

1.r zrH imH i n r E

r rφ ωε

∂ − = − ∂ (A1.15)

Considering equation (A1.11), the term rE yields

0 01 zr

k ZEE i H

rφβ β

∂= − +

∂ (A1.16)

where 0k is the propagation constant in the vacuum, and 0Z is the vacuum impedance. Considering the equation

(A1.14) field component rH comes

65

( )2

0 01 zr

k Y n rHH i E

rφβ β

∂= − −

∂ (A1.17)

where 0Y is the vacuum admittance. Replacing equations (A1.16) and (A1.17) into equations (A1.10) and (A1.13),

the electric field component Eφ and the magnetic field component Hφ yield

( )2

fE

rφ κ= (A1.18)

( )2

gH

rφ κ= (A1.19)

where ( )rκ is the generic propagation constant, defined as

( ) ( )2 2 2 20r n r kκ β= − (A1.20)

and function f and g are defined by [1]

0 0z

z

Hf m E ik Z

r r

β ∂= − −

∂ (A1.21)

( )20 0 .z

z

Eg m H ik Y n r

r r

β ∂= − +

∂ (A1.22)

Replacing now equations (A1.18) and (A1.19) into equations (A1.16) and (A1.17), the

components rE and rH finally yield

( )

0 02

zr

k ZEiE g

r rβ βκ∂

= − +∂

(A1.23)

( )

( )

20 0

2.z

r

k Y n rHiH f

r rβ βκ∂

= − −∂

(A1.24)

Next objective is to obtain the z components of the electric and magnetic field. Thus, replacing

equations (A1.18) and (A1.23) into (A1.12), zH is obtained. Replacing equations (A1.19) and (A1.24)

into equation (A1.15), zE is obtained. In order to solve the previous equations, an approximation to the

Helmholtz equation is performed [1], so that the solution is going to be a function of the Bessel functions.

Then, considering the generic form of the z components of the electric and magnetic field

66

( ) ( ) ( ) ( ), , , exp expzE r z t F r im i z tφ φ β ω= − (A1.25)

( ) ( ) ( ) ( ), , , exp exp .zH r z t G r im i z tφ φ β ω= − (A1.26)

then, from the solutions of the Helmholtz function, ( )F r and ( )G r yield

( ) ( ) ( )( ) ( )

,

,m m

m m

AJ hr A Y hr r aF r

B I r BK r r aα α

′+ ≤= ′ + >

(A1.27)

( ) ( ) ( )( ) ( )

,

,m m

m m

CJ hr C Y hr r aG r

D I r DK r r aα α

′+ ≤= ′ + >

(A1.28)

where a is the core radius, and h and α are defined by

2 2 2 2

1 0h n k β= − (A1.29)

2 2 2 2

2 0 .n kα β= − (A1.30)

( )mJ hr is the Bessel function of first kind, ( )mY hr is the Bessel function of second kind, ( )mI rα is the

modified Bessel function of first kind and the ( )mK rα is the modified Bessel function of second kind. All four

functions are of the same order m . Functions ( )F r and ( )G r can be reduced to

( ) ( )( )

,

,m

m

AJ hr r aF r

BK r r aα≤

= >

(A1.31)

( ) ( )( )

,.

,m

m

CJ hr r aG r

DK r r aα≤

= >

(A1.32)

Since ( )mY hr → −∞ when 0r → , then 0A C′ ′= = . Since ( )mI rα → ∞ when r → ∞ , then 0B D′ ′= = .

So, the equations (A1.27) and (A1.28) can be reduced into the equations (A1.31) and (A1.32), respectively.

The φ components of the electric and magnetic field are respectively

( ) ( )

0 02 2

zz

k Z HE m E i r

r r r r rφ

βκ κ

∂ = − − ∂ (A1.33)

67

( )

( )( )

20 0

2 2.z

z

k Y n r EH m H i r

r r r r rφ

βκ κ

∂ = − + ∂ (A1.34)

The only way to obtain the modal equation, it is to impose the boundary conditions r a= . The

electric field and magnetic field components need to satisfy to

( ) ( )z zE r a E r a− += = = (A1.35)

( ) ( )z zH r a H r a− += = = (A1.36)

( ) ( )E r a E r aφ φ− += = = (A1.37)

( ) ( ).H r a H r aφ φ− += = = (A1.38)

The following condition [1]

( ) ( )0 0L L

C DG r y Y F r y Y

D B= ⇒ = = (A1.39)

is imposed by the equations (A1.35) and (A1.36), which leads to

( )( )

m

m

J uB AQ Q

K w= → = (A1.40)

where u and w are the normalized transverse wavenumbers in the core and in the cladding [6], respectively,

defined as [1]

u ha= (A1.41)

.w aα= (A1.42)

Using equations (A1.37) and (A1.38), the following matrix from can be obtained

( )

( )1 1 2 2

2 21 2 2 2 1 1

1 0

0L

a bQ i a b QA

yi n a n b Q a bQ

+ + = − + +

(A1.43)

where the coefficients 1a , 2a , 1b and 2b are defined as

( ) ( )1 2

mJ ua m a

uβ= (A1.44)

68

( ) ( )2 0

mJ ua k a

u

′= (A1.45)

( ) ( )1 2

mK wb m a

wβ= (A1.46)

( ) ( )2 0 .mK w

b k aw

′= (A1.47)

The modal equation is obtained by taking the determinant of the matrix of equation (A1.43) equal

to zero. Therefore, one obtains

( ) ( )( )2 2 21 1 2 2 1 2 2 2 0.a bQ a b Q n a n b Q+ − + + = (A1.48)

The modal equation can be written the form [1]

( ) ( ) 2 2m mR u S u m= Ω (A1.49)

where Ω is defined as

22

21 2 .

u v

v uw

Ω = − ∆

(A1.50)

To determine funtions ( )mR u and ( )mS u , the two following equations are considered

2221

1 2n

n= − ∆ (A1.51)

( )( )

2 2

2 221 0

1 2 .a u

vn k a

β= − ∆ (A1.52)

Then, functions ( )mR u and ( )mS u yield,

( ) ( )( )

( )( )

m m

m

m m

J u K wR u

uJ u wK w

′ ′= + (A1.53)

( ) ( )( )

( ) ( )( )

1 2m m

m

m m

J u K wS u

uJ u wK w

′ ′= + − ∆ (A1.54)

69

and our objective is finally fulfilled.

70

71

Annex B

B1. Numerical simulation of linear pulse propagation

To solve the pulse propagation equation, we use the FFT (fast Fourier transform) and the IFFT

(inverse fast Fourier transform). Considering that the input pulse has the following form in the frequency

domain

( ) ( ) ( )1, 0, expA z A i zω ω ωβ= (B1.1)

and considering the propagation equation

( )2 3

2 2 3

1sgn 02

A A Ai β κ

ζ τ τ∂ ∂ ∂

+ − =∂ ∂ ∂

(B1.2)

where

( )2 2 2sgnβ β β= (B1.3)

and

3

2 06

βκ

β τ= (B1.4)

where the variable κ is the third-order dispersion term on the relation between the coefficients 3β and 2β , and the

width of the pulse 0τ the pulse A has to be normalized into the variables ζ and τ , according to ( ),A ζ τ .

To solve the propagation equation (B1.2) it is easier to work in the time domain, using the direct

and the inverse Fourier transform, respectively

( ) ( ) ( ), , expA A i dζ ξ ζ τ ξτ τ+∞

−∞= ∫ (B1.5)

( ) ( ) ( )1, , exp

2A A i dζ τ ζ ξ ξτ ξ

π

+∞

−∞= −∫ (B1.6)

where the term ξ is the normalized frequency given by

( )0 0 .ξ ω ω τ= − (B1.7)

If the propagation equation only includes the second order dispersion term, it becomes

72

2

2

1.

2

A Ai

ζ τ∂ ∂

= −∂ ∂

(B1.8)

Then, the solution ( ),A ζ ξ for

( ) ( ) 21, 0, exp .

2A A iζ ξ ξ ξ ζ =

(B1.9)

For the complete propagation equation, as (B1.2), then the impulse ( ),A ζ ξ is

( ) ( ) ( ) 2 32

1, 0, exp sgn .

2A A iζ ξ ξ β ξ κξ ζ

= + (B1.10)

The basic numerical simulation that has to be performed, includes three main steps:

(1) ( ) ( )0, FFT 0,A Aξ τ= is calculated;

(2) ( ),A ζ ξ is calculated using equation (B1.9) or (B1.10);

(3) ( ) ( ), IFFT ,A Aζ τ ζ ξ = is determined.

When 0 Dλ λ= , the normalized variables and the propagation equation are different. The

normalized variables become

D

z

Lζ ′ =

′ (B1.11)

30

3DL

τβ

′ = (B1.12)

where DL′ is the dispersion length associated with the third-order dispersion. Then, the propagation equation

becomes

( )3

3 3

1sgn 0 .6

A Aβ

ζ τ∂ ∂

− =′∂ ∂

(B1.13)

and the pulse is

( ) ( ) ( ) 33

1, 0, exp sgn .

6A A iζ ξ ξ β ξ ζ ′ ′= (B1.14)

73

74

Annex C

C1. Numerical simulation of the NLS function: Split step Fourier method

The split-step Fourier method (SSFM) is the most frequently used method to solve the non-linear

equations describing the pulse propagation in optical fibers. Considering that the propagation equation is

given by equation (4.28), then the same equation can be written in the form

( ) ( ),uD N uτ τ ζ τ

ζ∂

= +∂

(C1.1)

where the variables Dτ and Nτ are defined in the time domain τ , and are defined as

( )2 3

2 2 3

1sgn2

D iτ β κτ τ∂ ∂

= − +∂ ∂

(C1.2)

2.

2N i uτ

Γ= − + (C1.3)

Writing as ( )0u τ the input pulse one gets [1]

( ) ( ) ( )0, exp .u D N uτ τζ τ ζ τ= + (C1.4)

To travel along the fiber, an iterative scheme with a longitudinal pace h is considered. Then the input pulse becomes

[1]

( ) ( ) ( ), exp , .u h h D N uτ τζ τ ζ τ+ = + (C1.5)

Higher number of iterations will lead to a lower value of the pace.

The SSFM consists in two consecutive procedures, they are given by [1]

( ) ( ) ( ), exp ,v hN uτζ τ ζ τ= (C1.6)

( ) ( ) ( ), exp , .u h hD vτζ τ ζ τ+ = (C1.7)

From the equation (C1.3) and equation (C1.6), the function ( ),v ζ τ comes

( ) ( ) ( )2, exp exp , , .

2

hv ih u uζ τ ζ τ ζ τ = − Γ

(C1.8)

75

Using the Fourier transform

( ) ( ) ( ), , exp .v v i dζ ξ ζ τ ξτ τ+∞

−∞= ∫ (C1.9)

the operator Dτ becomes an operator Dξ , and is given by [1]

( ) 2 32

1sgn .2

D i iξ β ξ κξ= + (C1.10)

So, function ( ),u hζ ξ+ is given by

( ) ( ) ( ), exp ,u h hD vξζ ξ ζ ξ+ = (C1.11)

which is the same as

( ) ( ) ( ) ( )2 32, exp sgn exp , .

2

hu h i ih vζ ξ β ξ κξ ζ ξ + = (C1.12)

Finally, each iteration is performed using

( ) ( ) ( )1, , exp .

2u h u h i dζ τ ζ ξ ξτ ξ

π

+∞

−∞+ = + −∫ (C1.13)