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Doctoral Dissertation
Study on Nonlinear Effects in Optical
Fiber Communication Systems with
Phase Modulated Formats
MOHAMMAD FAISAL
Department of Electrical, Electronic and
Information EngineeringGraduate School of Engineering
OSAKA UNIVERSITY
January 2010
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This thesis is dedicated to my parents, Khodeza Begum and
Mohammad Solaiman, my wife Naima and daughter Faiza
for their eternal love, steady support and continuous
encouragements.
Faisal
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Preface
This thesis presents a theoretical study on effects of fiber nonlinearity in single channel and
multi-channel dispersion-managed (DM) optical fiber communication systems for phase
modulation schemes and their mitigation techniques. The content of the dissertation is based on the
research which was carried out during my Ph. D. course at the Department of Electrical, Electronic
and Information Engineering, Graduate School of Engineering, Osaka University. The dissertation
is organized as follows:Chapter 1 is a general introduction which gives the background, the purpose of the study and
overview of the dissertation. It briefly states the researches on advanced modulation format
particularly phase modulation formats that are promising for high speed long-haul lightwave
communications. Then nonlinear effects are asserted and DM transmission systems have been
discussed for soliton and quasi-linear pulse which show a considerable research attention to
achieve ultra high speed optical networks. The recent researches on self-phase modulation (SPM)
and cross-phase modulation (XPM) induced phase fluctuations have been summarized and then
the motivation of this study is stated.
Chapter 2 presents the basics of optical fiber communications along with a brief discussion on
the modulation formats for ultra-high speed long-haul transmission systems. Phase modulated
formats have been discussed addressing the background of this study. Next the basic theories for
the analyses employed in this thesis for DM transmission is presented after making a brief
discussion on fiber nonlinearities. First fundamental equations of optical pulse propagation in a
fiber have been studied. Then variational method is described and coupled ordinary differential
equations have been deduced assuming a suitable solution for the Nonlinear Schrdinger (NLS)
equation. The pulse dynamics in optical fiber with periodic dispersion compensation and
amplification is investigated considering a Gaussian-shape ansatz.
Chapter 3 describes the phase jitter mechanism followed by theoretical study of phase jitter in
constant dispersion soliton, DM soliton and quasi-linear pulse transmission systems. After
introducing ASE noise by periodically located optical amplifiers into the system, the ordinary
differential equations derived in chapter 2 are linearized. Due to noise, the pulse parameters
(amplitude, width, chirp, frequency, center pulse position and phase of pulse) get affected
randomly. The noise power is much weaker than the signal power but it is accumulated along the
transmission line. The dynamics of noise-perturbed pulse parameters have been derived. Therefore,
the variances and cross-correlations of these parameters have been evaluated. The phase jitter
effect in DM soliton systems is examined with physical interpretation. Various DM models have
been assumed and the impact of dispersion management on phase jitter has been investigated. The
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results obtained for DM models are compared to that of a constant dispersion soliton system. The
variational results are verified by numerically solving the NLS equation using split-step Fourier
method and carrying out Monte Carlo simulations.
Next the quasi-linear pulse propagation in DM transmission systems has been investigated.Using the same analytical calculations, phase jitter for different quasi-linear DM models has been
explored. The analytical results obtained by variational method are supported by numerical
simulations. Phase jitter effect is further studied taking into account the variation in fiber length
constituting the DM period for a strong DM system utilizing standard telecommunication fibers.
By altering the fiber dispersion, phase jitter is calculated for a particular DM map. Upgradation of
dispersion maps have been studied by achieving lower phase noise. Impact of amplifier spacing
and different periodic dispersion configurations using high dispersion fibers is also investigated.
Chapter 4 explains the fundamental mechanism of collision-induced phase fluctuations in a
periodically dispersion compensated two-channel WDM transmission system. Dynamicalequations for pulse propagation in WDM system has been deduced using variational analysis
assuming XPM as a perturbation source. Phases shift due to XPM has been estimated for 50 GHz
channel spacing considering two different bit rate systems. Impact initial pulse spacing between
inter-channel pulses on phase shift is investigated for different dispersion models. Furthermore,
influence of channel spacing and residual dispersion on phase fluctuation has been explored.
Chapter 5 concludes the thesis by summarizing the results stating the significance of this study
concerning the high speed long-haul optical fiber communication systems based on phase
modulation data formats.
All the results described in this dissertation were published in Optics Communications,
Proceedings of 13th Optoelectronics and Communications Conference (OECC 2008), Proceedings
of 7th International Conference on the Optical Internet (COIN 2008), Proceedings of 8th
International Conference on Optical Communications and Networks (ICOCN 2009), Technical
Report of IEICE, and in the proceedings of International Symposium (EDIS 2009) and Conference
(SCIENT 2008) organized by Global COE CEDI of Osaka University.
Mohammad Faisal
Osaka, Japan
January 2010
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Acknowledgements
This research has been carried out during my tenure of doctoral course at the Department of
Electrical, Electronic and Information Engineering, Graduate School of Engineering, Osaka
University. First of all, I would like to express my deep sense of appreciation and gratitude to
Professor Ken-ichi Kitayama for giving me the opportunity to study in his laboratory and
providing me the support and encouragement as a Guardian during the academic years I have been
living in Japan. I am much thankful to him for his kind recommendation, and for the review anddiscreet suggestions to this dissertation.
I would like to give my sincere thanks to Professor Shozo Komaki for his careful review and
constructive suggestions which have improved this thesis.
I would like to express my great thanks and gratefulness to Associate Professor Akihiro Maruta
for his instructions, continuous encouragement, valuable discussions, and careful review during
the period of this research. His keen sight and a wealth of farsighted advice and supervision have
always provided me the precise guiding frameworks of this research. I have learned many valuable
lessons and information from him through my study in Osaka University, which I have utilized to
develop my abilities to work innovatively and to boost my knowledge. I am profoundly indebted
to Associate Professor Masayuki Matsumoto for his invaluable informative discussions and useful
suggestions.
I express my thanks to all the past and present colleagues in the Photonic Network Laboratory
(Kitayama Lab.), Department of Electrical, Electronic and Information Engineering, Graduate
School of Engineering, Osaka University. They have always provided me encouragement and
friendship. I thank Assistant Professor Yuki Yoshida and specially Dr. Yuji Miyoshi for various
helpful discussions and support. Cordial thanks go to Dr. Giampiero Contestabile, Mr. Takahiro
Kodama, Mr. Shougo Tomioka, Mr. Shinji Tomofuji, Mr. Seiki Takagi, Mr. Iori Takamatsu, Mr.
Yousuke Katsukawa, and Mr. Nozomi Hasimoto for generous support and hearty friendship. I also
appreciate the other students and staff of this laboratory for their continuous cooperation and
encouragement.
I wish to acknowledge the Ministry of Education, Culture, Sports, Science and Technology of
Japan for granting me the (MEXT) scholarship during my three and half years study in Japan. I
express my thanks and gratitude to Global COE program Center for Electronics Devices
Innovations under the Ministry of Education, Culture, Sports, Science and Technology of Japan
for the financial support as RA in the last (4th
) year of my doctoral course. I also express my
gratefulness to ICOM foundation for granting me a small scholarship during the last year which
has been helpful to me to continue my study without economic apprehension.
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I feel thankful to my friends for their brotherly support and encouragement which provide me a
great help during my study and stay in Japan. Their cooperation was helpful to face the challenges
and stress, and eventually it gave me confidence and stamina to accomplish the doctoral program.
Finally I would like to express my heartfelt thanks and deepest gratitude to my family and myparents, brothers and sister for their deep understanding, endless devotion and love, unwavering
patience, and steady support during the period of my education.
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Contents
Preface
Acknowledgements
Chapter 1 Introduction 1
Chapter 2 Fundamentals of Optical Fiber Transmissions 7
2.1 Introduction 7
2.2 Modulation Formats for Optical Fiber Transmissions 8
2.3 Fiber Nonlinearities 10
2.4 Fundamental Theories of Dispersion-Managed Pulse 12
2.4.1 Elementary Equation of Lightwave Propagation 12
2.4.2 Variational Analysis of Optical Pulse 14
2.4.3 Dispersion-Managed Soliton 16
2.5 Conclusion 22
Chapter 3 Theoretical Analysis of Phase Jitter in Dispersion-Managed Systems 23
3.1 Introduction 23
3.2 Mechanism of Phase Jitter 24
3.3 Analytical Calculation of Phase Jitter 26
3.4 Analytical and Numerical Simulation for DM Soliton 29
3.5 Quasi-Linear Pulse Transmission 32
3.6 Analytical and Numerical Simulation for Quasi-linear Systems 36
3.7 Upgradation of Dispersion Map for Quasi-Linear Pulse Transmission 39
3.8 Effect of Amplifier Spacing 43
3.9 Effect of Dispersion Compensation Configuration 45
3.10 Conclusion 47
Chapter 4 XPM Effects in Dispersion-Managed Transmission Line 49
4.1 Introduction 49
4.2 Analytical Calculation of XPM Induced Phase Shift 50
4.3 System Description 52
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4.4 Basic Mechanism of Collision-Induced Phase Shift 53
4.5 Effect of Initial Pulse Spacing Between Channels 57
4.6 Effect of Channel Spacing and Residual Dispersion 58
4.7 Conclusion 59
Chapter 5 Conclusions 61
Appendix A 63
Appendix B 73
Bibliography 91
List of Publications 101
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Chapter 1
Introduction
On-off keying (OOK)-based wavelength-division multiplexed (WDM) transmission systems
with erbium-doped fiber amplifiers (EDFA) are the current state-of-the art technology for
lightwave communications. Almost all commercially available optical fiber transmission systems
employ OOK format for coding the information. Due to increased demand of global broadband
data services and advanced Internet applications including text, audio and video,
telecommunication networks based on fiber-optics are getting huge popularity and facing more
and more pressure to cope up with the demand. The next generation lightwave transmission
systems should provide this high capacity and at the same time, at a lower cost. This shifts the
research trend from OOK-based system to the advanced modulation formats such as differentialphase shift keying (DPSK), differential phase amplitude shift keying (DPASK), amplitude phase
shift keying (APSK), and multilevel PSK/DPSK etc. to enhance the per-fiber transmission
capacity. Enhancing the spectral efficiency of a WDM network is considered as an economical
way to expand the system capacity. For these reasons, in recent years, the differential phase
modulation schemes, particularly DPSK and differential quadrature phase shift keying (DQPSK),
draw huge research attention and are becoming the promising transmission formats for next
generation spectrally efficient high speed long-haul optical transmission networks [1-6]. In this
section, some features of phase modulated formats are briefly described referring some recent
researches and technological developments, and dispersion-managed (DM) optical transmission
with periodic amplification is discussed to clarify the background of this thesis.
Phase modulated data formats like PSK and differential PSK have compact spectrum with
constant envelope which yield some advantages over other data formats. They are, particularly
differential PSK is robust to fiber dispersion and nonlinearity and have low intrachannel effects at
high bit rate ( 40 Gb/s) [6, 7]. Early works on phase modulated optical communications were
based on coherent detection process to improve the receiver sensitivity. But coherent detection is
much complex and costly. With deployment of fiber amplifiers like EDFA, direct detection for
differential phase modulation schemes are becoming popular because of simpler receiver structure
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Chapter 1. Introduction2
with the merits of phase modulation and low-cost implementation. Recently phase-modulated
transmission based on direct detection of DPSK becomes the promising data format for future
lightwave communications, which was rediscovered in 1999 by Atia et al. [8]. In PSK format,
message lies in phase, whereas in DPSK transmissions, information is coded into the phasedifference rather than phase so that direct-detection receiver can be used. Using balanced receiver,
DPSK requires around 3 dB lower receiver sensitivity than OOK for a given bit-error rate (BER)
and it is shown that DPSK is better than OOK in terms of receiver sensitivity and resilience against
fiber impairments at about 40 Gb/s [7-9]. It is also experimentally proved that DPSK provides
better performance at 40 Gb/s [10] and even 10 Gb/s [11-12] considering dense WDM systems. In
2006, Pinceman et al. [13] has shown that properly optimized carrier suppressed return-to-zero
DPSK (CSRZ-DSPK) format might double the error-free transmission distance with respect to
amplitude shift keying (ASK). At 160 Gbps, DPSK gives 4 dB optical signal to noise ratio
(OSNR) improvements over OOK [14]. This remarkable performance improvement is important to
increase the system margin i.e. to extend the transmission distance along with achievement of
much higher speed.
Optical M-ary PSK and quadrature amplitude modulation (QAM) offer higher spectral
efficiency at higher bit rates and these are quite competent for dense WDM systems [15-19].
Among these, quadrature phase shift keying (QPSK) is becoming the most promising because of
its superior transmission characteristics. These schemes require coherent detection and even after
advancement in EDFA, it can provide better receiver sensitivity than OOK (IM/DD), but at the
cost of receiver complexity. Direct detection gives one degree of freedom per polarization,
whereas, coherent detection permits the use of two degrees of freedom per polarization increasing
the spectral efficiency. Coherent detection increases receiver sensitivity compared to direct
detection while experiences the drawbacks of local oscillator laser synchronization and
polarization control. In 2005, Gagnon etal. [20]has reported a QPSK transmission with coherent
detection and digital signal processing (DSP) which can provide higher SNR (i.e., higher bit error
rate (BER) performance) over the conventional differential detection without phase locking of the
local oscillator to the carrier phase. In 2006, Koc et al. [21] has proposed another novel approach
for realization of coherent QPSK without need of synchronization. They designed an algorithm
and showed error-free transmission by simulation and experimentally as well. 8-PSK, 8- and 16-
QAM offer much higher spectral and SNR efficiencies but again at the sacrifice of more
complexity and cost. They face limitations due to laser linewidth requirements, which may be
resolved by devising new laser or other ways and the advantages could overcome the complexity
and cost. Furthermore, research has been going on to reduce these drawbacks. However, we can
strongly predict that the phase modulation formats with high spectral efficiency are attractive
alternatives to upgrade the capacity of currently deployed fiber-optic transmission systems.
In long distance optical fiber transmission, dispersion management is employed which is one
of the key techniques to handle the dispersion problem. DM transmission line consists of
alternating fiber segments with anomalous and normal dispersion, can be used to maintain a
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Chapter 1. Introduction 3
desirable path-average dispersion and mitigate inter-channel cross talks due to four-wave mixing
(FWM), and cross-phase modulation (XPM) [22, 23]. Fiber nonlinearity could be the essential
limiting concern for long-haul fiber-optic transmission. However, fiber nonlinearity, particularly
self-phase modulation (SPM), can be positively and effectively utilized to form soliton pulsewhere SPM and dispersion balance each other to sustain the pulse shape in fiber [24, 25]. In 1973
Hasegawa and Tappert first proposed the existence of soliton pulse in optical fiber and deduce an
equation governing the attributes of a slowly varying complex envelope of an electric field
propagating in a fiber [26]. In this pioneering work they showed NLS equation has a stationary
solution which indicates a stationary pulse can propagate in a fiber with dispersion and Kerr
nonlinearity for a long distance without any distortion. Forysiak et al. [27] incorporated the
dispersion compensation in soliton system to reduce the timing jitter effect in 1993. Periodic
dispersion management has been introduced into optical soliton by Suzuki et al. [28] in 1995.
Hasegawa et al. [29] has studied the dispersion-managed (DM) soliton and quasi-soliton
transmission in WDM as well as TDM and described their feasibility for high speed long-haul
systems in 1997. Since then, many simulation and experimental works on DM soliton have been
carried out by various researchers. DM soliton system offers extra benefit of reduction of timing
jitter [28, 30], reduce modulational instability [31], robustness to inter-channel collisions [32] and
improved signal to noise ratio at the receiver. Thats why, DM soliton shows a great prospect for
WDM systems with very high bit rate (as high as 160 Gbps) and system capacity [32-35].
Recently quasi-linear systems with periodic dispersion management attract considerable
research attention in fiber-optic communications, where fiber nonlinearity can be managed
successfully to develop stationary like pulse comparable to soliton [36-38]. The DM quasi-linear
transmission system is robust to collision-induced timing jitter, inter-channel crosstalk and stable
pulse evolution can be achieved with lower energy compared to DM soliton [39].
A lot of researches and development efforts have been done on advanced optical modulation
schemes, both theoretically and experimentally, to address different aspects of those formats, and
consequently to implement phase-modulated signals on currently deployed Metro or backbone
networks. For phase modulation formats, SPM-induced nonlinear phase noise is observed as a
major limiting factor to materialize the long distance transmission. In case of multi-channel DM
transmission system, XPM-induced phase fluctuations are also deleterious for phase modulated
formats, particularly with lower channel spacing. These key issues should be addressed for DM
soliton and quasi-linear pulse which are prospective candidates for desirable high speed long-haul
lightwave transmission systems.
The physical impairments of optical fiber transmission can be categorized into two main parts
irrespective of modulation/detection schemes: linear and nonlinear. Linear barriers include fiber
loss and dispersion, and nonlinear part comprises SPM, XPM, and FWM etc. In previous
paragraphs, management of chromatic dispersion and SPM has been discussed in relation to DM
soliton and quasi-linear pulse. The remaining impairment is the signal attenuation in fiber. Periodic
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Chapter 1. Introduction4
amplification, either lumped or distributed, is usually employed to compensate for fiber loss along
the transmission line. The periodically installed optical amplifiers amply the signal and produce
amplified spontaneous emission (ASE) noise inherently, which causes phase noise by interacting
with signal. This phase noise is realized to be the major performance limiting factor for phasemodulated lightwave communication systems [12, 40-43]. This noise is composed of two different
parts. The first, termed as linear phase noise, is due to the accumulation of the additive white
Gaussian noise results from amplified spontaneous emission (ASE) noise of amplifiers. The
second one is referred to as nonlinear phase noise, in which ASE noise causes amplitude
fluctuations of signal which are converted to nonlinear phase noise by fiber nonlinearity, mainly
SPM and this phenomenon is widely recognized as Gordon-Mollenauer effect [42]. The phase
noise impedes the phase modulated lightwave system by corrupting the phase of the signal which
conveys the information. On the other hand, phase fluctuation induced by XPM is also a great
concern for WDM/dense WDM systems with phase modulation formats.
Kikuchi [43] has studied the amplifier noise considering the dispersion and nonlinearity in
optical fibers. He has included the dispersion effect which was not assumed in the pioneer work of
Gordon-Mollenauer. After calculating the spectral density of ASE noise he has pointed out the
effect of dispersion on phase noise in multi-span long transmission system. Similar study has been
made by Green et al. [44] considering those issues. They have investigated the effect of chromatic
dispersion on phase noise and shown that it can either enhance or suppress the nonlinear noise
amplification. Nonlinear phase noise in single channel DPSK systems has been analyzed by
Zhang etal. [45] taking into account the intrachannel effect in a highly dispersive system. Demir
[46] has studied the nonlinear phase noise in multi-channel multi-span optically amplified dense
WDM systems considering DPSK and DQPSK signal formats. Cartaxo etal. [47] has described
the contribution of fiber nonlinearity to the relative intensity noise spectra. This intensity noise
results from the phase modulation to intensity modulation conversion of laser phase noise which is
a major impairment of direct detection systems. Phase noise in soliton systems have been
investigated by McKinstrie et al. [48]. Periodic dispersion compensation can affect the phase jitter
of soliton and quasi-linear systems and these are the discussed in this thesis.
Concerning phase modulation formats, collision-induced phase fluctuations in DM multi-
channel systems are also important to be noticed. XPM effect is more dominant than self-phase
modulation (SPM) induced distortion in WDM system with narrow channel spacing [49]. Though
FWM is a limiting nonlinearity for WDM system, its impact is much low in highly dispersive
fibers and it can be reduced by unequal channel spacing and dispersion management. Recently
XPM has drawn considerable research attention since phase modulated signals are going to be
introduced in WDM networks [50-53]. Malach et al. [54] has studied the effect of residual
dispersion on XPM and SPM theoretically for 10 Gb/s WDM NRZ system. Jansen et al. [55] has
experimentally studied the effect of XPM in two different dispersion maps for 10 Gb/s NRZ
system and shown that both maps are impaired by XPM at 50 GHz channel spacing. XPM-induced
distortions in DPSK system has been numerically studied with a particular dispersion management
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Chapter 1. Introduction 5
scheme [56]. Narrow filtering has been suggested for hybrid system of DPSK and OOK to
suppress XPM but at the cost of complexity [57]. XPM effect on single and dual-polarization RZ-
DQPSK signals have been investigated reporting larger tolerance achievement by single-
polarization format over non-zero dispersion shifted fiber (NZDSF) [58]. In this dissertation, wealso investigate XPM effects, discuss its behavior on different aspects, and study its mitigation in
periodically DM WDM transmission systems consists of highly dispersive fibers.
Phase fluctuations caused by nonlinear effects are the key limiting factors to achieve the
maximum transmission distance by phase modulated systems. The other limitations of such
modulation formats arise from the stringent requirement of laser linewidth, laser phase noise,
additive white Gaussian noise in coherent detectors etc. The complete performance analyses of this
sort of transmission systems considering phase jitter and other linear/nonlinear noise are still under
research. Getting motivations from these facts, we examine the phase jitter effect in DM soliton
and quasi-linear transmission systems and evaluate the impact of periodic dispersion management
on phase jitter taking into account the fiber loss, dispersion, SPM and amplifier ASE noise. We
further study XPM-induced phase fluctuations in DM transmission line. The aim is to realize ultra-
high speed long distance/transoceanic dense WDM networks.
In the chapters of this dissertation, firstly, the fundamentals of optical fiber communications
will be outlined emphasizing on modulation formats followed by the brief description on fiber
nonlinearities and variational method. Secondly, phase jitter in constant dispersion soliton, DM
soliton and in DM quasi-linear transmission systems will be discussed. Upgradation oftransmission maps will be proposed by obtaining reduced phase jitter. After that, phase shift
induced by XPM in periodically DM line will be explained. The contents of each chapter are
summarized as follows:
In Chapter 2, the basics of optical fiber communications will be introduced highlighting the
different modulation formats for ultra-high speed long-haul transmission systems. Phase
modulated formats will be discussed addressing the background of this study. Next the basic
theories for the analyses employed in this thesis for DM transmission will be presented.
Fundamental equations of optical pulse propagation in a fiber have been studied. Variational
method will be described and coupled ordinary differential equations will be deduced assuming a
suitable solution for the nonlinear Schrdinger (NLS) equation. The pulse dynamics in optical
fiber with periodic dispersion compensation and amplification is investigated considering a
Gaussian-shape ansatz.
In Chapter 3, after introducing ASE noise by periodically located optical amplifiers into the
system, the ordinary differential equations derived in Chapter 2 are linearized considering that
noise as a perturbation. Due to noise, the pulse parameters (amplitude, width, chirp, frequency,
center pulse position and phase of pulse) get affected randomly. The noise power is much weaker
than the signal power but it is accumulated along the transmission line. The dynamics of noise-
perturbed pulse parameters have been derived. Therefore, the auto-correlations (variances) and
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Chapter 1. Introduction6
cross-correlations of these parameters have been evaluated. The phase jitter effect in DM soliton
systems is explained with physical interpretation. Various DM models have been assumed and the
impact of dispersion management on phase jitter has been investigated. The results obtained for
DM models are compared to that of a constant dispersion soliton-based system. The variationalresults are verified by numerically solving the NLS equation using split-step Fourier method and
carrying out Monte Carlo simulations [59-63]. Chapter 3 is also devoted for the quasi-linear pulse
propagation in DM transmission systems. Utilizing the same variational analysis, phase jitter for
different quasi-linear DM models has been explored. For quasi-linear transmission, linear phase
noise is included with nonlinear part to find the total phase jitter. Phase jitter effect is further
studied taking into account the variation in fiber length constituting the DM period for a stronger
DM system. Phase jitter is also calculated for different dispersion map strength [61, 62]. Next, this
chapter proposes upgraded dispersion maps to achieve longer transmission length by mitigating
phase noise [64]. Effect of amplifier spacing and dispersion map configuration on phase jitter is
also investigated. In all cases, analytical results are supported by numerical simulations.
Chapter 4 explains the fundamental mechanism of collision-induced phase shift in a two-
channel WDM system with periodic dispersion management for RZ pulse with 40% duty cycle.
This chapter shows the phase shift due to XPM for different bit rate systems and checks different
transmission models with highly dispersive fibers. It presents the analytical calculation obtained by
variational method for a two-channel system, and the result is verified by numerical simulation [63,
65]. Impact of initial pulse spacing, channel spacing and residual dispersion on phase fluctuations
caused by XPM are also studied [65, 66].
Chapter 5 provides the summary of the results with stating the significance of this study
concerning the long-haul high-speed optical fiber communication networks.
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Chapter 2
Fundamentals of Optical Fiber
Transmission
2.1 Introduction
The introduction of WDM with optical amplifiers has revolutionized the optical fiber
transmission system by increasing the system capacity both by the number of channels and
distance. Transmission capacity can be further enhanced by increasing the channel bit rate. The
channel bit rate is upgraded to 10 Gb/s from 2.5 Gb/s and it is predicting that the next generation
lightwave communications will be based on 40 Gb/s rate. However, this high bit rate systems willface many problems due to fiber dispersion and nonlinearity which are interrelated with
transmitting power, number of channels, channel spacing and transmission length etc. To increase
the system capacity overcoming these difficulties while maintaining a low system cost, phase
modulation formats have been proposed which are spectrally efficient and has tolerance to fiber
nonlinearities.
PSK formats have been reported with enhanced OSNR compared to currently deployed OOK
based transmission networks. In this chapter, we will discuss the basic modulation formats with
distinctive stress on phase modulated schemes followed by detailed description on the basic
theories and necessary equations for optical pulse propagation in fiber. Section 2.2 will define the
modulation formats for optical communications. Section 2.3 presents a brief discussion on fiber
nonlinearities. Afterward, section 2.4 introduces the fundamental equation that describes the
propagation of optical pulse in a fiber with dispersion management, which can be derived from
Maxwells equation and can be transformed into nonlinear Schrdinger (NLS) equation. We may
find an analytical solution of NLS equation, which is called soliton solution when the coefficients
of that equation are constant. In the sub-section 2.4.2, variational method is explained, which is the
main tool for theoretical analyses of DM soliton and quasi-linear pulse transmission. Assuming a
proper solution of NLS equation with varying coefficients, the pulse dynamics can be ascertained
by evaluating the variational equations for the pulse parameters, such as, amplitude, inverse of
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Chapter 2. Fundamentals of Optical Fiber Communications8
pulse width, chirp, center frequency, center position and phase. Sub-section 2.4.4 presents the
attributes of soliton pulse in DM line considering a Gaussian ansatz. We also check their
dependence on the variation of dispersion management and find the evolution of pulse in
periodically compensated fiber with and without periodic amplification.
2.2 Modulation Formats for Optical Fiber Transmissions
Modulation format is a critical issue in the design and development of optical network. In
order to transmit data, the system should have a modulator which will convert the electrical data
signals to optical pulses. There are several choices to transport optical pulses in the communication
networks. We can classify them in several ways. But there are three basic types of digital
modulation formats.
1.
ASK2. FSK3. PSK
OOK is one version of ASK, wavelength-shift keying (WSK) and minimum shift keying (MSK)
are the versions of FSK, PSK has a lot of versions, like DPSK, DPASK, QPSK, DQPSK, 8-PSK
etc. QAM is a combination of ASK and PSK. Again, we can regroup them according to binary,
e.g., binary ASK, binary FSK, binary PSK, and multi-level coding, like, M-ary ASK, M-ary FSK,
and M-ary PSK, e.g., QPSK, DQPSK, 8-PSK, QAM, 16-QAM etc. All these can be categorized
into two groups according to duty cycle or line coding.
1.Non-return-to-zero (NRZ)2. Return-to-zero (RZ)
Now we are going to define the basic modulation formats:
OOK is a simple format in which information lies in the amplitude, and its transmitter and
receiver configurations are straight forward. But the receiver sensitivity is low. Furthermore,
OOK-based transmission system is vulnerable to dispersion and nonlinearity, and at high bit rate
like 40 Gb/s or more, intrachannel nonlinearities cause severe performance degradation. That's
why, OOK data format is not suitable for high speed optical transmission.
FSK modulation technique has relatively higher receiver sensitivity but at the expense of
complex transceiver configurations. Moreover, bandwidth expands drastically with the increase of
number of channels. For these reasons, FSK data format may not be so popular for high speed
fiber-optic WDM and dense WDM networks.
In PSK format, information is coded into the phase of the carrier signal. It has a constant
envelope with compact spectrum. Furthermore, it is more robust to dispersion and nonlinearity.
PSK with differential scheme enables simple direct detection with increased OSNR. However, it
has some drawbacks, like precise alignment of transmitter and receiver which is complex, stringent
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2.3 Fundamental Theories of Dispersion-Managed Pulse 9
requirement of laser linewidth etc. In the subsequent section, we are going to discuss about the
phase modulated schemes which are under investigation of our current study.
2.2.1 PSK Format for Fiber-Optic Network
The phase of the optical carrier signal generated by laser diode is modulated by the digital data
information. In binary PSK, when data changes from 1 to 0 or vice versa, the phase of the
carrier is altered to 180 degree and thereby information is encoded into the phase. This modulated
signal is transmitted through fiber and received at the receiver. For PSK, either homodyne or
heterodyne coherent detection is used, which is complex and costly. Thats why, differential
encoding of the phase modulated signals are performed and preferred as they enable simple direct
detection with enhanced OSNR.
Tx Rx
PhaseMod.LD
Data
Amplifier
PSK
Fiber
Tx Rx
PhaseMod.LD
Data
Amplifier
PSK
Fiber
Tx Rx
PhaseMod.LD
Data
Amplifier
PSK
Fiber
Figure 2.1: Typical schematic diagram of PSK scheme for single user.
In differential encoding of PSK, i.e., DPSK format, information lies in phase transition rather
than in phase like PSK. Differentially encoded data is phase modulated by a modulator at the
transmitter. The receiver is composed of a delay interferometer and a balanced receiver. DPSK
with direct detection balanced receiver requires almost 3 dB lower OSNR compared to OOK to
achieve a given BER.
LD
Phase Mod.
Differentially encoded
NRZ Data
Pre-coder
NRZ Data
Balanced ReceiverOne-bitDelay
Interferometer
Error
Detector
LD
Phase Mod.
Differentially encoded
NRZ Data
Pre-coder
NRZ Data
Balanced ReceiverOne-bitDelay
Interferometer
Error
Detector
Figure 2.2: Typical DPSK transmission and reception.
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Chapter 2. Fundamentals of Optical Fiber Communications10
Multi-level phase modulated signal formats, like QPSK, DQPSK, 8-PSK, 16-QAM etc. can
offer the following extra advantages:
More information transmission per unit bandwidth Saving modulation and detection bandwidth More efficient use of amplifier bandwidth Increase tolerance to chromatic dispersion and polarization mode dispersion (PMD)
etc.
2.3 Fiber Nonlinearities
Fiber nonlinearities arise from the two basic mechanisms. Firstly, most of the nonlinear effects
in optical fibers originate from nonlinear refraction, a phenomenon that refers to the intensitydependence of refractive index of silica resulting from the contribution of )3( . The refractive
index of fiber core can be expressed either as
2
20
2)(),(~ EnnEn += (2.1)
or as
0 2
eff
Pn n n= + (2.2)
where n0 is the linear part and n2 is the nonlinear-index coefficient related to)3( by the relation
(3)
2 (3 / 8 ) Re( )n n =
.Pis the power of the light wave inside the fiber and Aeff is the effective areaof fiber core over which power is distributed. The intensity dependence of refractive index of silica
leads to a large number of nonlinear effects, such as, SPM, XPM and FWM.Fiber Nonlinearities
Kerr
effects
Stimulated
scattering effects
XPMSPM SRSFWM SBS
Fiber Nonlinearities
Kerr
effects
Stimulated
scattering effects
XPMSPM SRSFWM SBS
Figure 2.3: Nonlinear effects in fibers
The second mechanism for generating nonlinearities in fiber is the stimulated scattering
phenomena. These mechanisms give rise to stimulated Brillouin scattering (SBS) and stimulated
Raman scattering (SRS). Fiber nonlinearities that now must be considered in designing state-of-
the-art fiber optic systems may be categorized as Kerr effects, which include SPM, XPM, and
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2.3 Fundamental Theories of Dispersion-Managed Pulse 11
FWM, and scattering effects that include SBS and SRS. Different fiber nonlinear effects are
briefly narrated below.
2.3.1 Self-Phase Modulation (SPM)
Self-phase modulation (SPM) is due to the power dependence of the refractive index of the
fiber core. SPM refers the self-induced phase shift experienced by an optical field during its
propagation through the optical fiber; change of phase shift of an optical field is given by
( )2
2 0 L NLn n E k L = + = + (2.3)
where 0 2k = and L is fiber length. L is the linear part and NL is the nonlinear part that
depends on intensity. NLis the change of phase of the optical pulse due to the nonlinear refractive
index and is responsible for spectral broadening of the pulse. Thus different parts of the pulse
undergo different phase shifts, which gives rise to chirping of the pulses. The SPM-induced chirp
affects the pulse broadening effects of dispersion.
SPM interacts with the chromatic dispersion in the fiber to change the rate at which the pulse
broadens as it travels down the fiber. Whereas increasing the dispersion will reduce the impact of
FWM, it will increase the impact of SPM. As an optical pulse travels down the fiber, the leading
edge of the pulse causes the refractive index of the fiber to rise causing a blue shift. The falling
edge of the pulse decreases the refractive index of the fiber causing a red shift. These red and blue
shifts introduce a frequency chirp on each edge, which interacts with the fibers dispersion tobroaden the pulse.
2.3.2 Cross Phase Modulation (XPM)
Cross phase modulation (XPM) is very similar to SPM except that it involves two pulses of light,
whereas SPM needs only one pulse. In Multi-channel WDM systems, all the other interferingchannels also modulate the refractive index of the channel under consideration, and therefore its
phase. This effect is called Cross Phase Modulation (XPM).
XPM refers the nonlinear phase shift of an optical field induced by copropagating channels atdifferent wavelengths; the nonlinear phase shift be given as
( )2 2
2 0 1 22
NLn k L E E = + (2.4)
where E1 and E2 are the electric fields of two optical waves propagating through the same fiber
with two different frequencies.
XPMSPM
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2.3 Fundamental Theories of Dispersion-Managed Pulse 13
(GVD). (z) presents the Kerr nonlinear coefficient, which is related to nonlinear refractive index
of fibern2 and its effective core areaAeff by effAn 22= , where is the wavelength of light.
g(z) represents the fiber loss ifg(z) < 0 or gain ifg(z) > 0. For fiber with loss [dB/km],g(z) is
expressed as ( ) 2010logeZg = . Using chain rule and introducing new time coordinate
( ) dtz
= 0 1 moving at group velocity, we can derive the following equation
( ) ( )EzigEEzE
z
Ei =+
22
22
2
. (2.6)
Next we introduce new non-dimensional variables as follows:
0tT = ,
0zzZ= ,
0
EuP
= ,
where t0, z0 and P0 are the arbitrary constants in real units for normalizing the quantities that
describe the time, distance and electric field for optical signal, respectively. Now we can achieve
the normalized form of Eq. (2.5) as follows:
( )( ) ( )
22
2
2
b Zu ui s z u u i Z u
Z T
+ =
, (2.7)
where b(Z), s(Z) and (Z) indicate the dispersion profile, the fiber nonlinearity and loss in
normalized form, respectively and these normalized quantities are denoted as
( )20
02
t
zZb
= , ( ) ( ) 00PzZZs = , ( ) ( ) 0zZgZ = . (2.8)
In actual optical fiber communication systems, fiber amplifiers, either lumped or distributed, are
periodically installed along the transmission line to compensate for the loss between two
successive amplifiers. Pulse envelope will change periodically due to this periodic amplification.
We use the transformation ( ) ( ) ( ), ,u Z T a Z U Z T =
, where U(Z, T) is a slowly varying amplitude ofpulse envelope and a(Z) is a rapidly varying term, which is a periodic real function with period of
amplifier spacing can be given by
( ) ( ) =Z
ZdZaZa00
exp , (2.9)
where a0 is a constant determined by the gain of amplifier and calculated as
( )a
a
Z
Za
=
2exp1
20
, (2.10)
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Chapter 2. Fundamentals of Optical Fiber Communications14
where is fiber loss and Za is amplifier spacing, both are in normalized units. We obtain the
following equation after the transformation
( )( ) 02
2
2
2
=+
UUZST
UZb
Z
U
i , (2.11)
where ( ) ( ) ( )ZsZaZS 2= and represents the normalized effective fiber nonlinearity, which
includes the effect of fiber nonlinearity, fiber loss and periodic gain. We can apply this equation in
a system with periodic dispersion compensation while retaining periodic or constant nonlinearity.
If fiber dispersion and nonlinearity are kept constant and fiber is lossless, b(Z) and S(Z) can be
normalized to unity, then Eq. (2.11) can be written as
0
2
1 22
2
=+
UU
T
U
Z
Ui . (2.12)
This is the well known nonlinear Schrdinger (NLS) equation with constant coefficients, the basic
equation for optical soliton, which is integrable and can be solved analytically. The fundamental
stationary solution of Eq. (2.12) is known as solitary wave (soliton) solution and is given as
( ) ( ){ } ( )2 20 0sech exp2
iU Z ,T T Z T Z i T Z i
= + + +
, (2.13)
where represents the amplitude as well as pulse width of soliton, represents its speed which
indicates the deviation from the group velocity as well as the frequency. T0 and 0 represent the
initial center position of soliton pulse in time and initial phase, respectively.
When b(Z) and/orS(Z) is not constant with respect to Z, Eq. (2.11) is termed as NLS equation
with varying coefficients, and it is no longer integrable. It implies that we can not find any exact
solution, but we can obtain an approximate analytical solution.
2.4.2 Variational Analysis of Optical Pulse
As we have mentioned in previous sub-section, Eq. (2.11) can not be solved analytically.Several methods have been developed to explain and to study the propagation of nonlinear return-
to-zero stationary pulse under those conditions: the perturbation theory [67], the guiding center
theory [68, 69], the variational method [70], and the numerical averaging method [71, 72] etc. The
fundamental nature of these methods is to reduce the original perturbed system with dispersion
management into a simpler model or an approximate equation is assumed which can be easily
solved. The methods provide us a way to explore the characteristics of DM soliton or quasi-linear
pulse overlooking the details of the solution and even considering some perturbations.
In this thesis, considering a known function for the solution of pulse waveform, we study the
variational method to examine the attributes and evolution of pulse in fiber-optic transmission line
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2.3 Fundamental Theories of Dispersion-Managed Pulse 15
with periodic dispersion compensation and/or amplification. We also accomplish direct numerical
calculations of the perturbed NLS equation to analyze the evolutionary properties and verify our
variational results.
Optical pulse propagation in fiber described by the NLS equation with periodically varyingeffects and small perturbations can be written in Langevin form as
( )( )
( )
( )
22
2
,
2
b Z R Z T U Ui S Z U U
Z T a Z
+ =
, (2.14)
where R(T, Z) represents the perturbation term and R
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Chapter 2. Fundamentals of Optical Fiber Communications16
( ) ( )( )2 2 212
C
S ZdCb Z p C A R
dZ= + + , (2.22)
dR
dZ
= , (2.23)
( )0
0TdT b Z R
dZ= + , (2.24)
( )( )
( )2 2 25
2 4 2
b Z S Z dp A R
dZ
= + + , (2.25)
where
( )( )
221
Im 3 2 exp22
i
AR R e da Z
=
, (2.26)
( )( )
22Im 1 2 exp
2
ip
pR R e d
Aa Z
=
, (2.27)
( ){ }( )
222
Re Im 1 2 exp
2
i iCR R e C R e d
Aa Z
=
, (2.28)
( ){ }
22Re Im exp
2
i ipR R e C R e dAa Z
=
, (2.29)
( )0
22
Im R exp2
iTR e d
Apa Z
=
, (2.30)
( )( ){ }
221 Re 3 2 4 Im exp
22
i iR p R e R e dApa Z
= +
. (2.31)
Here, Re iR e and Im iR e
represent the real and imaginary parts of iR e , respectively. Eqs.
(2.20) - (2.25) are the equation of motion for each parameter under the perturbation, and describe
the pulse dynamics in DM transmission system.
2.4.3 Dispersion-Managed Soliton
Dispersion management scheme has become a necessary technology for long-haul and ultra-
high speed lightwave transmission systems. In this thesis, we theoretically analyze the pulse
behaviour along the DM transmission fiber for both soliton and quasi-linear pulse separately. We
assume a two-step periodic dispersion map as shown in Fig. 2.4 for DM transmission line and
optical amplifiers are positioned at the middle of anomalous dispersion fibers regularly. We
consider it as a general model for both soliton and quasi-linear systems. Each amplifier adds ASE
noise to the signal when it restores the pulse energy to its original value. The noise is considered as
the perturbation and the pulse properties have been altered randomly during propagation. For DM
soliton, we consider an average dispersion within a period, whereas for quasi-linear system,
dispersion is fully compensated at the end of each period.
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2.3 Fundamental Theories of Dispersion-Managed Pulse 17
Fig. 2.4: Schematic diagram of a periodic two-step dispersion map.
b(Z) is a periodic function ofZwith periodZb. When b(Z) > 0, it means the normal dispersion
fiber is used, when b(Z) < 0, anomalous dispersion fiber is deployed, and the difference between
two dispersion is denoted as21 bbb = . The amplifier spacing Za is assumed to be equal to
dispersion map period Zb and the average dispersion bav is taken as 1 for DM soliton, and 0 for
quasi-linear system.
The variational method already discussed in previous section will be utilized here to investigate
the pulse behavior in DM line. Considering Gaussian assumption for the solution of Eq. (2.14) and
in absence of perturbation (R = 0), the pulse properties in a dispersion-managed system with fiber
loss and gain can be determined by the following the equations,
( ) 03
00 CpZb
dZ
dp= , (2.32)
( ) ( ) ( ) 0020
20
0
21 pE
ZSCpZb
dZ
dC
+= . (2.33)
Here, we set the initial values as ( ) 000 ==Z , ( )00 0 0T Z= = , whereA0(Z),p0(Z) and C0(Z) are the
pulse parameters in absence of perturbation,0
200 pAE = is a constant for any Zand represents
the pulse energy.
Now we are going to describe the pulse dynamics in dispersion-managed line using the systemparameters mentioned in Table 1. Considering a Gaussian pulse, one can find stable pulse
propagation for an appropriate energy in which inverse of pulse width and chirp are periodically
varying with distance with periodZb as shown in Figs. 2.5, 2.6, 2.7 and 2.8. Eqs. (2.32) and (2.33)
are used to analytically evaluatep(Z) and C(Z) respectively. We directly numerically solve the Eq.
(2.14), plot them, and find that the variational results are in good agreement with numerical values.
This confirms the validity of the assumption of Gaussian-type pulse shape function and the
validity of variational analysis.
b2
b1
3
4bZ1
4bZ
b0
bav
Za
Zb
b(Z)
Z
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Chapter 2. Fundamentals of Optical Fiber Communications18
Table 1: Fiber and system parameters used in the analysis
Parameter Real unit Normalized unit
Wavelength m55.1 =
Fiber loss dB/km0.2= 256.23=
Effective area of fiber core 2m50 =effA
Spontaneous emission factor 5.1=spn
Minimum pulse width, FWHM 10s ps = 763.1Ts =
Nonlinear coefficient of fiber /Wm100.3220
2
=n
Average dispersion ps/nm/km1.0=avd 0.1=avD
DM period 40 0 kmbz .= 0 1414bZ .=
Amplifier spacing km0.40=az 1414.0=aZ
We can explain the pulse dynamics more in details using the illustrations. Figures 3.2 and 3.3
show the periodic evolutions ofp(Z) and C(Z) for b = 70 (7.0 ps/nm/km, where b1 = 3.6
ps/nm/km and b2 = 3.4 ps/nm/km) and pulse energy E0 = 5.65 (0.0493 pJ) for loss free fiber. We
observe the minima of absolute value of pulse chirp at the ends and mid-point of DM period and
minima of pulse width at the ends of DM period. The maxima of absolute value of pulse chirp and
pulse width occur at the junctions of two different fiber segments. Figure 2.7 shows the closed
orbit in p-Cplane which proves the periodicity in DM line. Figure 2.8 gives the smaller closed
orbit for
b = 42 (4.2 ps/nm/km, where b1 = 2.2 ps/nm/km and b2 =
2.0 ps/nm/km) and pulseenergy E0 = 3.57 (0.0311 pJ) in a loss less line.
Fig. 2.5: Chirp, C forb = 70. Fig. 2.6: Inverse of pulse width, p forb = 70.
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2.3 Fundamental Theories of Dispersion-Managed Pulse 19
Fig. 2.7:p-Cplane forb = 70, E0 = 5.65. Fig. 2.8:p-Cplane forb = 42, E0 = 3.57.
We next discuss the DM soliton with fiber loss and periodic amplification for the same systemsdescribed above, of course, in absence of perturbation. We obtain the periodic solutions ofp(Z)
and C(Z) in Figs. 2.9 and 2.10, respectively, like before except that the curves become asymmetric.
For loss less case, we consider initial chirp C0 = 0 for both models, now for fiber with loss 0.2
dB/km, C0 = 0.4279 forb = 70 and C0 = 0.174 forb = 42, but the initial value of inverse of
pulse widthp0 = 1 for all cases. Figures 2.11 and 2.12 demonstrate the closed orbit inp-Cplane for
these systems with pulse energy E0 = 12.41 (0.1082 pJ) and 7.47 (0.0652 pJ), respectively, which
again prove the periodic nature of soliton pulse in DM line with fiber loss and lumped gain. For
both loss less and lossy systems, we find that stronger DM line (largerb) possesses bigger closed
orbit and require larger energy for evolution of soliton pulse along the line [73].
Fig. 2.9: Chirp, C forb = 70. Fig 2.10: Inverse of pulse width, p forb = 70
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Chapter 2. Fundamentals of Optical Fiber Communications20
Fig 2.11:p-Cplane forb = 70, E0 = 12.41. Fig 2.12:p-Cplane forb = 42, E0 = 7.47.
Figures 2.13 and 2.14 show the pulse evolution within one DM period for loss less and lossy
lines respectively. The waveform of soliton exhibits the characteristic breathing shape in both
cases. In loss less case, soliton pulse regains its original value and shape at the end of period.
However, in lossy case, we have to use amplifier to compensate for the loss and restore the pulse
to its initial value at a regular interval. Figures 2.15 and 2.16 display the pulse propagation for
multi-period (5 periods) along with and without fiber loss and periodic gain by amplifier,
respectively. We consider the same DM period and amplifier spacing. At the starting of each
period, soliton pulse retains its initial value and shape as shown in Fig. 2.16.
One major objective of this thesis is to examine the phase behavior of soliton and quasi-linear
pulse in a periodically dispersion compensated lightwave transmission line. For that purpose, we
also evaluate the phase shift change and explore the trend of variation using the variational method.
Assuming the initial conditions as follows: ( ) ( )0 00 0Z T Z = = and ( ) 000 ==Z , we assess the
following expression for phase shift of DM soliton in absence of perturbation
( ) ( ) ( ) ( ) ( )
dpSE
dpbZZZ
+=
00
0
0
200
24
5
2
1 . (2.34)
The phase variation of DM soliton with periodic amplification is shown in Fig. 2.17 and derived
by Eq. (2.34) along with Eqs. (2.32) and (2.33). Numerical results are obtained by directly solving
Eq. (2.14) in absence of perturbation using split-step Fourier method. There is a little difference
between analytical result and numerical calculation. There will be slight error due to this
difference which may be ignored, particularly in case of comparisons presuming the same trend for
other models.
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2.3 Fundamental Theories of Dispersion-Managed Pulse 21
Fig. 2.13: Propagation of a DM soliton in one
period without loss and amplification.Fig. 2.14: Propagation of a DM soliton in one
period with loss and amplification.
Fig. 2.15: Propagation of a DM soliton in multi-
period transmission line without loss. Fig. 2.16: Propagation of a DM soliton in multi-period multi-span transmission line with loss andperiodic amplification.
From Eq. (2.34), it is evident that the pulse phase depends on pulse energy, width, fiber
dispersion and nonlinearity. Pulse phase shift increases linearly with transmission distance as
shown in Fig. 2.17 and predicted in the above equation.
Fig. 2.17: Phase variation of DM soliton pulse.
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Chapter 2. Fundamentals of Optical Fiber Communications22
2.5 Conclusion
This chapter introduces the phase modulation formats for fiber-optic network and hence discuss
briefly about fiber nonlinear effects. This chapter has explained the essential analytical theories for
this thesis and discussed the characteristics of dispersion-managed soliton elaborately. The
fundamental equation for optical pulse propagation in a fiber has been introduced and NLS
equation with constant and varying coefficients is deduced. The dispersion-management scheme
has been described for soliton and quasi-linear systems and pulse evolution along the periodic
dispersion compensated line considering loss less fiber has been evaluated. The transmission line
with fiber loss and periodic gain by amplifiers has also been enlightened. Assuming Gaussian
ansatzfor the NLS equation, we have derived the coupled ordinary differential equations for the
pulse parameters using the variational method. The pulse dynamics in DM line is evaluated by that
set of equations. The pulse energy has increased due to higher DM map strength, which is a
prospective feature and can accomplish some significant role in case of Gordon-Haus timing jitter
and pulse-pulse interaction within the channel. We have also shown the phase behavior of DM
soliton with periodic amplification but in absence of perturbation.
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Chapter 3
Theoretical Analysis of Phase Jitter in
Dispersion-Managed Systems
3.1 Introduction
Optical amplifiers are periodically installed in long distance transmission line in order to
compensate for the fiber loss. The amplifiers restore the signal power and at the same time produce
ASE noise inherently. This noise perturbs the pulse parameters and may degrade the performance
of transmission systems. Dispersion management can improve the system performance by
introducing some extra advantages and mitigating some detrimental effects as we discussed in the
previous chapters. However, amplifier noise could affect the transmission particularly the systems
with phase modulation schemes. Fiber dispersion along with nonlinearity might complicate the
situation. In this chapter, first we define the phase jitter and explain the mechanism of how ASE
noise involves in forming the phase jitter. Then, we carry out theoretical analysis to model the
amplifier noise effect in DM line taking into account the fiber Kerr nonlinearity, particularly SPM.
Applying the variational method and linearization scheme, we develop ordinary differential
equations for the variances and cross-correlations of the six pulse parameters perturbed by
amplifier noise in section 3.3. In section 3.4, we evaluate the phase jitter for DM soliton
analytically employing those equations and then verify the results by directly solving the NLS
equation using split-step Fourier method and conducting Monte Carlo simulations.
Research and development have been going on to enhance the bit rate to 40 Gbps or beyond in
currently deployed standard telecommunication fiber with periodically installed fiber amplifiers
like EDFAs [74-77]. Return-to-zero (RZ) pulses with short pulse width have to be launched into
the transmission line and be recovered at the end of fiber span or dispersion-managed period or
transmission line using proper dispersion compensation. Due to use of strong dispersion-managed
line, conventional single-mode fiber (SMF) of 17 ps/nm/km followed by dispersion shifted fiber
(DSF) or dispersion compensating fiber (DCF), the pulses get rapidly dispersed and be reproduced
23
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3.3 Analytical Calculation of Phase Jitter24
with minor impairments of fiber nonlinearity. This fiber nonlinearity could be minimized by
appropriate choice of dispersion compensation technique and pulse width [77]. This leads to the
quasi-linear propagation of signal in fiber-optic transmission line.
In this chapter, we also deal with the quasi-linear pulse transmission in periodically dispersioncompensated lightwave systems. Section 3.5 enlightens the features of quasi-linear pulse in DM
line. In section 3.6, we evaluate the phase jitter for quasi-linear systems employing the same
analytical method as described in the previous section and carry out numerical simulations to
validate the analytical results. Section 3.7 proposes upgraded dispersion maps to achieve lower
phase noise and higher Q-factor. In section 3.8, effect of amplifier spacing on phase noise is
investigated. Finally section 3.9 shows the effect of dispersion map configuration on phase noise
and recommends a particular map suitable for long-haul DM transmission system.
3.2 Mechanism of Phase Jitter
J. P. Gordon and L. F. Mollenauer [42] have first noticed the phase jitter impairment in
lightwave transmission systems with linear amplifiers in 1990. In this pioneering work, they have
intuitively analysed the effect of amplifier noise on phase of the transmitted signal ignoring the
fiber dispersion. The ASE noise introduced by periodically located optical amplifiers along the
transmission line perturbs the pulse amplitude, width, chirp, frequency, center position and phase.
Due to the stochastic nature of the phenomena, we have to determine the correlations of these
pulse parameters influenced by noise to explore their behaviour analytically.
Phase fluctuation caused by ASE noise is termed as phase jitter. Phase jitter can be categorised
into two parts, linear phase noise and nonlinear phase noise. Linear phase noise results from the
accumulation of additive white Gaussian noise generated by ASE. Nonlinear phase jitter is
occurred as follows: signal amplitude varies due to ASE noise, these amplitude variations are
transformed into phase fluctuations by fiber Kerr effects, mainly SPM. This nonlinear phase jitter
mechanism is demonstrated in Fig. 3.1. For single channel transmission system, nonlinear phase
noise induced by SPM is the major nonlinear impairment to be addressed.
Both linear and nonlinear phase noise accumulate span after span. Linear phase noise is
considerable if signal power is small. In case of long-haul communication, large signal power is
required to maintain the desired receiver sensitivity, so nonlinear phase noise becomes significant.
The mechanism of linear and nonlinear phase noise is illustrated in vector representation in Fig.
3.2. For soliton or DM soliton, the system is nonlinear, in such cases with long-haul transmission
line the linear phase noise remains very small compared to nonlinear part and can be neglected.
The nonlinear phase noise is induced mainly by the beating of the signal and ASE noise from the
same polarization as the signal and within an optical bandwidth matched to the signal. It results
from the interaction of fiber Kerr effects and ASE noise produced by optical amplifiers. The
effects of amplifier noise outside the signal bandwidth and amplifier noise from orthogonal
polarization are all ignored for simplicity.
3.2 Mechanism of Phase Jitter
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3.2 Mechanism of Phase Jitter 25
Signal Signal + ASEAmplifier
By Kerr effectRandom shiftof phase
1. SPM (mainly)2. XPM
Nonlinearphase jitterSignal Signal + ASEAmplifier
By Kerr effectRandom shiftof phase
1. SPM (mainly)2. XPM
Nonlinearphase jitter
Fig. 3.1: Nonlinear phase noise mechanism in fiber-optic transmission system.
Refractive index of silica based fiber at high power can be expressed as
+=+=
effA
PnnEnnn 20
2
20, (3.1)
where, n0 is the linear refractive index, E is field intensity, n2 is the non-linear refractive index
depending on optical powerP, andAeff is effective core area. This intensity dependence refractive
index leads to a large number of nonlinear effects, such as, SPM, XPM and four-wave mixing
(FWM), which are commonly denoted as Kerr effects. Since our concern is a single channel, we
focus on SPM only. SPM refers to the self-induced phase shift experienced by an optical field
during its propagation through fiber. Phase shift change of an optical field can be given as
zkEnn 02
20 += ,
L NL = + , (3.2)
where, 20 =k is propagation constant andzis the fiber length. L is the linear phase shift and
NL is the nonlinear part which depends on signal power. NL is responsible for spectral broadening
of the pulse and noise could affect it because of direct relation to signal intensity. Within one
amplifier spacing, the overall nonlinear phase shift is
( )eff
L
eff
NL PLdzA
zPkn == 0 02
, (3.3)
where,effAn 22= is known as the fiber nonlinear coefficient,Pis assumed to be the launched
power of ( )0PP= and with fiber loss coefficient , ( ) zPezP = . L is span length
and ( ) Leff eL = 1 is the effective span length.
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3.3 Analytical Calculation of Phase Jitter26
Signal
Noise
Signal
Linear phase
fluctuations
Signal
N
Amplitude
fluctuations
N
Nonlinear phase
fluctuations
Sign
al
Nonlinear
phase fluctuation
due to noise (NL)
AB
CNN
Signal
Noise
Signal
Linear phase
fluctuations
Signal
N
Amplitude
fluctuations
N
Nonlinear phase
fluctuations
Sign
al
Nonlinear
phase fluctuation
due to noise (NL)
AB
CNN
Fig. 3.2: Phase jitter mechanism in vector representation. Signal moves from A to B when there is no
noise and moves from A to C if there is noise. The difference gives the phase jitter due to ASE noise.
If amplifier noise is denoted by Nand the electric field of optical signal E, both are complex
quantities with proper unit, considering the noise effect on signal intensity, the nonlinear phase
shift within a fiber span can be written as
+=L
NL dzNEkn02
02
2NELA effeff += .
2 2* *
eff eff A L E E.N E .N N = + + + (3.4)
For ASE noise within the bandwidth of the signal, we find the mean nonlinear phase shift is
2ELA effeffNL = and the rest of the phase variation is occurred due to noise as implied in Eq. (3.4).
ForMnumber of spans, the overall phase shift with accumulated ASE noise is
{ }2 2 2 2
1 1 2 1 2 3 1NL eff eff MA L E N E N N E N N N E N N = + + + + + + + + + + + +" " , (3.5)
whereN1,N2,N3, . . . ,NM represent the white random noise with Gaussian distribution generated
by 1st, 2
nd, 3
rd, . . . , M
thamplifiers located along the transmission line and assuming all are
independent with identical distribution.
3.3 Analytical Calculation of Phase Jitter
The ASE noise added by each periodically located amplifier along the transmission line
perturbs the pulse parameters randomly. The noise interacts with the pulse and causes phase noise.
The noise having the same phase and frequency like the signal affects the pulse parameters. We
assume the following perturbation term which models the amplifier noise effect added at the m-th
amplifier located ataZ mZ=
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3.2 Mechanism of Phase Jitter 27
( ) ( ){ , , imR mI R n Z T in Z T e
= + , (3.6)
where nmR and nmIare real random functions which satisfy the following correlations,
( ) ( ) ( ) ( ) ( ) ( ) ( ), , , ,2
mmR mR mI mI
N Zn Z T n Z T n Z T n Z T Z Z T T = = , (3.7)
( ) ( ), , 0mR mI n Z T n Z T = . (3.8)
Here,Nm(Z) is the spectral density of the m-th amplifier noise and is given by
( ) ( )0m aN Z N Z mZ= . (3.9)
Here, ( )0 1spN n h G= , nsp is spontaneous emission factor, h is the photon energy and
( )aZG = 2exp is amplifier gain, where is fiber loss and Za is amplifier spacing, both are in
normalized units. For variational analysis,Nm is calculated in terms of soliton unit as
( )( )
( )3 3 3
2 0
26
8 1.
sp
m a
av eff a
c hn n t GN Z Z mZ
b A Z
=
(3.10)
Here, n2 is nonlinear coefficient of fiber, c is the speed of light, h is the Planck constant, Aeff is
fiber effective core area, is the fiber loss, and t0 is the normalization factor for time which is
obtained dividing time by 1.665 for Gaussian pulse and 1.763 for soliton.
To simulate the effect of noise in pulse parameters, we make linearization by using
)()()( 0 ZxZxZx += , where x is a small noise contribution and x0 indicates the noise-free pulse
parameter. The linearization is valid as noise power is much weaker than the signal power.
Spontaneous-spontaneous beat noise is assumed to be small compared to signal-spontaneous beat
noise and hence it is ignored. For the noise-induced part of pulse parameters the ordinary
differential Eqs. (2.20) - (2.25) of Chapter 2 can be re-written in linearized form as
( ) ( )( )0 0 0 0 0 0 02
2mA
m
d A b Z p p C A A C p A p C R
dZ
= + + + , (3.11)
( ) ( ) ( )20 0 03 mpm
d pb Z p C p p C R
dZ
= + + , (3.12)
( )( ) ( ){ } ( )20 0 0 0 02 1 2 mC
m
d Cb Z p C p p C C S Z A A R
dZ
= + + + , (3.13)
( )m
m
dR
dZ
= , (3.14)
( )( )
0
0mT
m
d Tb Z R
dZ
= + , (3.15)
( )( ) ( )0 0
5 2
4m
m
db Z p p S Z A A R
dZ
= + + , (3.16)
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3.3 Analytical Calculation of Phase Jitter28
where
( )( )
221
3 2 exp22
mA mI R n da Z
=
, (3.17)
( ) ( )
220
01 2 exp 2mp mI
p
R n dA a Z
= , (3.18)
( )( )( )
22
0
0
21 2 exp
2mC mR mI R n C n d
A a Z
=
, (3.19)
( )( )
2
00
0
2exp
2m mR mI
pR n C n d
A a Z
=
, (3.20)
( )0
2
0 0
2exp
2T mIR n d
A p a Z
=
, (3.21)
( )( )
22
0
13 2 exp
22m mRR n d
A a Z
=
, (3.22)
Using the above expressions from Eqs. (3.11) to (3.16), the auto-correlations (variances) and the
cross-correlations of the noise-perturbed part of pulse parameters can be deduced in the form of
ordinary differential equations (ODE) as (see Appendix B)
( ) ( )( )
( )
2 22 0
0 0 0 0 0 0 0 20
32
4
m
m
d A A N Zb Z p p C A A C A p A p A C
dZ E a Z
= + + + , (3.23)
( )( )2 20 0 0 0 0 0 0 07 2 2
2
b Zd A pp p C A p p A C A C p A p p C
dZ
= + + +
( )( )0 0
20
1 ,2
m
m
A p N Z
E a Z+ (3.24)
( )( ){ }2 20 0 0 0 0 0 0 04 1 3 2
2
b Zd A Cp C A p p C A C A C p C A p C
dZ
= + +
( )( )
( )2 0 0
0 20
12 m
m
C N ZS Z A A
E a Z , (3.25)
( )( )0 0 0 0 0 0 02
2
b Zd Ap p C A A C p A p C
dZ
= + + , (3.26)
( )( )0 2 20 0 0 0 0 0 0 0 0 02 2
2
d A T b ZA p C A T A p C p T A p C T
dZ
= + + + , (3.27)
( ) ( ) ( ) 20 0 0 0 0 0 0 05 2
2 2 ,2 4
b Zd Ap A p p C A A C p A p C S Z A A
dZ
= + + + +
(3.28)
( ) ( )( )
( )
2 2
2 2 0
0 0 0 2
0
12 3 m
m
d p p N Zb Z p C p p p C
dZ E a Z
= + + , (3.29)
( ) ( ){ } ( )2 2 2 20 0 0 0 0 02 1 2d p C
b Z p C p p C p C p C S Z A A pdZ
= +
( )
( )0 0
20
2,m
m
p C N Z
E a Z (3.30)
( ) ( )2
0 0 03d p
b Z p C p p C dZ
= + , (3.31)
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3.2 Mechanism of Phase Jitter 29
( )( )0 2 30 0 0 0 03d p T
b Z p p C p T p C T dZ
= + + , (3.32)
( ) ( ) ( )2 20 0 0 0 05 2
34
d pb Z p p p C p p C S Z A A p
dZ
= + + + , (3.33)
( ) ( ){ } ( ) ( )( )
( )
2202 2
0 0 0 0 0 20
144 1 2 2 ,
m
m
C N Zd Cb Z p C p C p C C S Z A A C
dZ E a Z
+= + + +
(3.34)
( ) ( ){ } ( )20 0 0 0 02 1 2d C
b Z p C p p C C S Z A AdZ
= + + , (3.35)
( ) ( ){ } ( )0 2 20 0 0 0 0 0 0 02 1 2 2d C T
b Z p C p T C p C C T S Z A A T dZ
= + + , (3.36)
( ) ( ){ } ( ) ( )20 0 0 0 02
2 1 2 5 44
d Cb Z p p C C p p C C S Z A A C A
dZ
= + +
( )
( )20
1,m
m
N Z
E a Z (3.37)
( ) ( )( )
2 220 0
20
11 m
m
p C N Zd
dZ E a Z
+= , (3.38)
( )( )
( )0 2 0
20
1 m
m
d T C N Zb Z
dZ E a Z
= , (3.39)
( ) ( )0 05 2
4
db Z p p S Z A A
dZ
= + , (3.40)
( )( )
( )
20
0 2 20 0
12 m
m
d T N Zb Z T
dZ E p a Z
= + , (3.41)
( )( ) ( )0 0 0 0 05 24
d T b Z p p T S Z A A T dZ
= + + , (3.42)
( ) ( )( )
( )
2
0 0 20
5 2 32
2 4
m
m
d N Zb Z p p S Z A A
dZ E a Z
= + + . (3.43)
The analytical result for the phase variance is obtained from Eq. (3.43) by solving the above
correlated ODEs from Eqs. (3.23) to (3.43). The coupled ODEs are numerically solved using
Runge-Kutta method.
3.4 Analytical and Numerical Simulations for Dispersion-
Managed Soliton
We numerically simulate the soliton pulse evolution in constant dispersion and DM lines with
the same path-averaged dispersion of 0.1 ps/nm/km and for the same total transmission length of
9000 km. For soliton, we consider hyperbolic secant-shaped pulse for conventional soliton and
Gaussian-shaped pulse for DM soliton. We show pulse evolution along the periodic DM fiber with
period Zb in absence and presence of noise in Figs. 3.3 and 3.4, respectively. We observe
stationary pulse propagation in both cases.
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3.3 Analytical Calculation of Phase Jitter30
Fig. 3.3: DM soliton pulse evolution in absence of noise along the transmission line.
Fig. 3.4: DM soliton pulse evolution in presence of noise along the transmission line.
Next we consider two different DM soliton models consisting of 2.2 and 2.0 ps/nm/km fibres
with equal length concatenated alternately for model (a) and 3.6 and
3.4 ps/nm/km fibres formodel (b). We follow the two-step dispersion map as shown in Fig. 3.4 for DM line. The system
parameters used in the analysis are: DM period 40 km, amplifier spacing 40 km, optical carrier
wavelength 1.55 m, nonlinear coefficient 2.434 W-1
km-1
, fiber loss 0.2 dB/km, spontaneous
emission factor 1.5 and pulse width (FWHM) 10 ps. The dimensionless dispersion map strength S
which implies the degree of DM effects is calculated as 1.07 and 1.79 for model (a) and (b),
respectively. The definition ofSis given as [78]
1 1 2 2 ,F
b z b z S
+= (3.44)
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3.2 Mechanism of Phase Jitter 31
where, b1,z1 and b2,z2 are the dispersion coefficients and lengths of two fiber sections constituting
the DM map period (z1+z2 =zb) and Fis the minimum pulse width (FWHM).A single pre-chirped Gaussian pulse is launched into periodically DM line of model (a) and (b)
with a pulse energy of 0.065 pJ and 0.108 pJ, respectively. White Gaussian noise is adjoined topulse at every amplifier position. We add random noise with zero mean and variance of 2mN
separately to the real and imaginary part of signal in frequency domain. Monte Carlo simulations
have been carried out by directly solving Eq. (2.14) of Chapter 2 based on split-step Fourier
method for 1000 realizations and the variance of phase at pulse peak is calculated along the
transmission line.
Fig. 3.5: Variance of phase noise vs. transmission distance for soliton and DM soliton. The solid and
dashed curves show the analytical results obtained by the variational method and the circles represent
the results by numerical simulation.
Fig. 3.5 is the plot of the variance of the phase noise as a function of transmission distance.
The agreement between analytical and numerical simulation results is fairly satisfactory. We find
that the model (a) yields the lowest phase noise compared to soliton and model (b). Due to pulse
broadening, the degree of SPM is reduced in DM case compared to that of constant dispersion
soliton, which causes lower nonlinear phase noise. However, periodic dispersion management
enhances pulse energy, which further increases with the increase of dispersion difference between
two fibers and/or due to elongated DM period [73]. The enhanced energy in model (b) increases
the fiber nonlinear phase shift which consequently enhances the nonlinear phase variances as
compared to model (a) as predicted in Ref. [42].
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3.3 Analytical Calculation of Phase Jitter32
Fig. 3.6: Phase variance versus dispersion map strength Sfor DM soliton for transmission distance of
9000 km.
Fig 3.6 shows the variation of phase jitter against different dispersion map strength after
transmission of 9000 km. The analytical predictions supported by numerical calculation
recommend that weaker (S < 1) and moderately strong dispersion maps (S 1) are suitable to
achieve lower phase noise. The results also suggest that weaker dispersion management may allow
lower phase noise compared to constant dispersion soliton and stronger DM maps step-up the
phase noise and deteriorate the performance. However, weaker DM maps might enhance timing
jitter and other inter-channel effects which should be considered to attain more practical optimized
value.
3.5 Quasi-Linear Pulse Transmission
In any case of constant dispersion soliton or DM soliton, both dispersion and nonlinearity are
indispensable to preserve the pulse in fiber. Quasi-linear system is a different case, which assumes
Gaussian-shaped pulses that propagate along the transmission line having zero or very low path-
averaged dispersion. In a DM quasi-linear system, local dispersion is utilized to mitigate the
impairments caused by the fiber nonlinearity, i.e., nonlinearity is technically controlled while
maintaining almost zero path-averaged dispersion. Here interaction between fiber dispersion and
nonlinearity adjusts the amount of energy to be launched into the fiber links [68]. Smaller power
for the quasi-linear pulse can be chosen to transmit through the fiber compared to soliton or DM
soliton and this transmitting power is largely limited by the effects of nonlinearity.
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3.2 Mechanism of Phase Jitter 33
Fig. 3.7: Phase noise against pulse peak power for a particular distance. The solid and dashed lines are
obtained by variational method and the plus signs indicate the numerical simulation results.
The pulse evolution along the transmission line depends on the peak power, initial chirp and
the relative position of amplifier within a dispersion map period [77]. We study such
communication links to address the phase jitter and find its dependence on dispersion