study on nonlinear effects in optical fiber communication systems with phase modulated formats

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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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i

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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iii

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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iv

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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v

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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vi

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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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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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

7


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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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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.


20/112


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


21/112


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


22/112


23/112


(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)


24/112


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


25/112


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


26/112


( ) ( )( )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.


27/112


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


28/112


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.


29/112


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


30/112


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.


31/112


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.


32/112


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.


33/112

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


34/112

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.



35/112


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.



36/112


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=



37/112


( ) ( ){ , , 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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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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( )( )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.

3.4 Analytical and Numerical Simulations for Dispersion-Managed Soliton


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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)

3.4 Analytical and Numerical Simulations for Dispersion- anaged Soliton


41/112


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].

3.4 Analytical and Numerical Simulations for Dispersion-Managed Soliton


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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.

3.5 Quasi-Linear Pulse Transmission


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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

study on nonlinear effects in optical fiber communication systems with phase modulated formats

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