Study On Nonlinear Effects In Optical Fiber Communication Systems With Phase Modulated Formats

Doctoral Dissertation

Study on Nonlinear Effects in Optical Fiber Communication Systems with Phase Modulated Formats

MOHAMMAD FAISAL

Department of Electrical, Electronic and Information Engineering Graduate School of Engineering

OSAKA UNIVERSITY January 2010

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

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 Schrödinger (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. Dynamical equations 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 and discreet 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 my parents, 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････････････････････････････････････････････････････････････････ⅰ Acknowledgements･･････････････････････････････････････････････････････ⅲ 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

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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

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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 differential phase 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

1

2

Chapter 1. Introduction

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 phase difference 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 et al. [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 16QAM 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

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 pulse where 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. That’s 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

4

Chapter 1. Introduction

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 phase modulated 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 et al. [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 et al. [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 multichannel 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

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 RZDQPSK signals have been investigated reporting larger tolerance achievement by singlepolarization format over non-zero dispersion shifted fiber (NZDSF) [58]. In this dissertation, we also 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 ultrahigh 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 of transmission 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 Schrödinger (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 noiseperturbed pulse parameters have been derived. Therefore, the auto-correlations (variances) and

6

Chapter 1. Introduction

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 variational results 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 twochannel 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.

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 will face 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 Maxwell’s equation and can be transformed into nonlinear Schrödinger (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

8

Chapter 2. Fundamentals of Optical Fiber Communications

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. ASK 2. FSK 3. 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

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. That’s why, differential encoding of the phase modulated signals are performed and preferred as they enable simple direct detection with enhanced OSNR.

LD

Phase Mod.

Fiber Amplifier

Rx

Tx Data

PSK

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.

NRZ Data Pre-coder LD

Differentially encoded NRZ Data One-bit Balanced Receiver Delay Interferometer

Phase Mod.

Error Detector

Figure 2.2: Typical DPSK transmission and reception.

10

Chapter 2. Fundamentals of Optical Fiber Communications

Multi-level phase modulated signal formats, like QPSK, DQPSK, 8-PSK, 16-QAM etc. can offer the following extra advantages: ƒ

More information transmission per unit bandwidth

ƒ

Saving modulation and detection bandwidth

ƒ

More efficient use of amplifier bandwidth

ƒ

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 intensity dependence of refractive index of silica resulting from the contribution of χ (3) . The refractive index of fiber core can be expressed either as 2 2 n~ (ω , E ) = n0 (ω ) + n2 E

(2.1)

or as n = n0 + n2

P Aeff

(2.2)

where n0 is the linear part and n2 is the nonlinear-index coefficient related to χ (3) by the relation n2 = (3 / 8n ) Re( χ (3) ) . P is the power of the light wave inside the fiber and Aeff is the effective area of 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

Stimulated scattering effects

Kerr effects

SPM

XPM

FWM

SBS

SRS

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-ofthe-art fiber optic systems may be categorized as Kerr effects, which include SPM, XPM, and

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

(

ϕ = n + n2 E

2

)k L = ϕ 0

+ ϕ NL

L

(2.3)

where k 0 = 2π λ and L is fiber length. φL is the linear part and φNL is the nonlinear part that depends on intensity. φNL is 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 fiber’s dispersion to broaden 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 interfering channels 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 at different wavelengths; the nonlinear phase shift be given as

(

2

ϕ NL = n2 k0 L E1 + 2 E2

SPM

2

)

(2.4)

XPM

where E1 and E2 are the electric fields of two optical waves propagating through the same fiber with two different frequencies.

12

Chapter 2. Fundamentals of Optical Fiber Communications

In XPM, two pulses travel down the fiber, each changing the refractive index as the optical power varies. If these two pulses happen to overlap, they will introduce distortion into the other pulses through XPM. Unlike, SPM, fiber dispersion has little impact on XPM. Increasing the fiber effective area will improve XPM and all other fiber nonlinearities.

2.3.3 Four-Wave Mixing (FWM) In the stimulated scattering processes, the optical fiber acts as a nonlinear medium and plays an active role through the participation of molecular vibrations. In many nonlinear phenomena the fiber plays a passive role except for mediating the interaction among several optical waves through a nonlinear response of bound electrons. Such processes are referred to as the parametric processes as they originate from light-induced modulation of a medium parameter such as refractive index. Nonlinear phenomena like harmonic generation, four-wave mixing and parametric amplification fall into this category. FWM is caused by the nonlinear nature of the refractive index of optical fiber itself. And nonlinear refractive index of fiber depends on the intensity of propagating light. FWM is a thirdorder parametric process in which three waves of frequencies fi, fj and fk interact through thirdorder susceptibility χ ( 3) of fiber material and generate a fourth wave of frequency fijk = fi + fj − fk. FWM effect is usually only observed in fiber optic communication systems with multiple channels and it depends on signal power, number of channels, channel spacing, and fiber dispersion etc. FWM effect is harmful for WDM system with low dispersive fibers and smaller channel spacing. Moreover, it could be substantially mitigated by dispersion management and/or unequal or repeated unequal channel spacing. In this work, our main concern is SPM and XPM as they cause phase fluctuations which are detrimental for phase modulated signals.

2.4 Fundamental Theories of Dispersion-Managed Pulse 2.4.1 Elementary Equation of Lightwave Propagation The complex envelope of a slowly varying optical field propagating in a fiber taking into account the effects of fiber dispersion, Kerr nonlinearity and loss, can be described by the following equation, which is derived from Maxwell’s equation by using reductive perturbation method and making some assumptions as ∂E ⎫ β ( z ) ∂ 2 E 2 ⎧ ∂E i ⎨ + β1 ( z ) ⎬ − 2 + γ ( z ) E E = −ig ( z )E , 2 2 ∂t ∂t ⎭ ⎩ ∂z

(2.5)

where E(z, t), z and t are the complex electric field, propagation distance and time, in real world units, respectively. β1(z) is the inverse of group velocity and β2(z) is the group velocity dispersion

2.3 Fundamental Theories of Dispersion-Managed Pulse

13

(GVD). γ(z) presents the Kerr nonlinear coefficient, which is related to nonlinear refractive index of fiber n2 and its effective core area Aeff by γ = 2π n2 λ Aeff , where λ is the wavelength of light.

g(z) represents the fiber loss if g(z) < 0 or gain if g(z) > 0. For fiber with loss α [dB/km], g(z) is expressed as g (Z ) = α × loge 10 20 . Using chain rule and introducing new time coordinate z

τ = t − ∫ 0 β1 (ζ ) dζ moving at group velocity, we can derive the following equation

i

∂E β 2 ∂ 2 E 2 − + γ ( z ) E E = −ig ( z )E . 2 ∂z 2 ∂τ

(2.6)

Next we introduce new non-dimensional variables as follows: T=

τ , z Z= ,

t0

u=

z0

E , P0

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: i

∂u b ( Z ) ∂ 2 u 2 − + s ( z ) u u = −iΓ ( Z ) u , ∂Z 2 ∂T 2

(2.7)

where b(Z), s(Z) and Γ(Z) indicate the dispersion profile, the fiber nonlinearity and loss in normalized form, respectively and these normalized quantities are denoted as b (Z ) =

β 2 z0 ,

s(Z ) = γ (Z )z0 P0 ,

t02

Γ(Z ) = g (Z )z0 .

(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 of pulse envelope and a(Z) is a rapidly varying term, which is a periodic real function with period of amplifier spacing can be given by a (Z ) = a0 exp

{∫

Z 0

}

Γ(Z ′)dZ ′ ,

(2.9)

where a0 is a constant determined by the gain of amplifier and calculated as a0 =

2ΓZ a , 1 − exp(− 2ΓZ a )

(2.10)

14

Chapter 2. Fundamentals of Optical Fiber Communications

where Г is fiber loss and Za is amplifier spacing, both are in normalized units. We obtain the following equation after the transformation i

∂U b (Z ) ∂ 2U 2 − + S (Z ) U U = 0 , 2 ∂Z 2 ∂T

(2.11)

where S (Z ) = a 2 (Z ) s (Z ) 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 i

∂U 1 ∂ 2U 2 − +U U = 0. 2 ∂Z 2 ∂T

(2.12)

This is the well known nonlinear Schrödinger (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 i ⎡ ⎤ U ( Z ,T ) = η sech {T + κ Z − T0 ( Z )} exp ⎢ −iκ T + (η 2 − κ 2 ) Z + iθ 0 ⎥ , 2 ⎣ ⎦

(2.13)

where η represents the amplitude as well as pulse width of soliton, κ represents its speed which indicates the deviation from the group velocity as well as the frequency. T0 and θ0 represent the initial center position of soliton pulse in time and initial phase, respectively. When b(Z) and/or S(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 returnto-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

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 varying effects and small perturbations can be written in Langevin form as i

R ( Z ,T ) , ∂U b ( Z ) ∂ 2U 2 − + S (Z ) U U = 2 ∂Z a (Z ) 2 ∂T

(2.14)

where R(T, Z) represents the perturbation term and R<<1. The Lagrangian for Eq. (2.14) with perturbation can be written as ⎡ i ⎛ ∂ U * ∂ U * ⎞ b ( Z ) ∂U 2 S ( Z ) 4 i L=∫ ⎢ ⎜ U − U⎟+ U − U *R − UR* + − ∞ 2 ∂Z a (Z ) 2 ∂T 2 ∂Z ⎢⎣ ⎝ ⎠

(

∞

⎤

)⎥ dT .

(2.15)

⎥⎦

As pulse is impaired by the perturbation R(Z, T), it is essential to examine the dynamics of pulse amplitude, width, chirp, frequency, position and phase under the influence of perturbation. We use the variational method to investigate the pulse dynamics with the perturbation. We assume the following function for the solution of Eq. (2.14) without perturbation, U ( Z , T ) = A ( Z ) f (τ ) exp ( iϕ ) , with τ (Z , T ) = p (Z ){T − T0 (Z )}, C (Z ) 2 κ (Z ) ϕ ( Z ,T ) = τ − τ +θ (Z ), 2 p(Z )

(2.16) (2.17) (2.18)

where A(Z), p(Z), C(Z), κ(Z), T0(Z) and θ(Z) are the six pulse parameters representing the amplitude, the inverse of pulse width, the linear chirp, the central frequency, the central time position and the phase of the pulse, respectively. The pulse propagating in a dispersion-managed system has almost Gaussian shape and we can then assume the Gaussian pulse,

(

)

i.e., f (τ ) = exp − τ 2 2 , the approximate solution becomes ⎡ 1 ⎤ κ U ( Z , T ) = A exp ⎢ − (1 − iC )τ 2 − i τ + iθ ⎥ . p ⎣ 2 ⎦

(2.19)

By applying the variational method with the pulse assumed in Eq. (2.19), the differential equations for these parameters with respect to Z can be written as evaluated in Appendix A, dA b ( Z ) = A p 2 C + RA , dZ 2 dp = b( Z ) p 3C + R p , dZ

(2.20) (2.21)

16

Chapter 2. Fundamentals of Optical Fiber Communications S (Z ) 2 dC = −b( Z ) p 2 1 + C 2 − A + RC , dZ 2 dκ = Rκ , dZ dT0 = b( Z ) κ + RT0 , dZ b( Z ) 2 5S (Z ) 2 dθ κ − p2 + A + Rθ , =− dZ 2 4 2

(

)

(

(2.22) (2.23) (2.24)

)

(2.25)

where ∞ ⎛ τ2 ⎞ , 1 2 − iϕ ⎡ ⎤ − Im 3 2 τ exp R e ⎜ − ⎟ dτ ∫ ⎦ 2 π a ( Z ) −∞ ⎣ ⎝ 2⎠ ∞ ⎛ τ2 ⎞ , p 2 − iϕ ⎡ ⎤ Im 1 2 exp − Rp = R e τ ⎜ − ⎟ dτ ⎦ π Aa ( Z ) ∫ −∞ ⎣ ⎝ 2⎠

RA =

(

)

(2.26)

(

)

(2.27)

∞ ⎛ τ2 ⎞ 2 Re ⎡⎣ R e−iϕ ⎤⎦ − C Im ⎡⎣ R e−iϕ ⎤⎦ 1 − 2τ 2 exp ⎜ − ⎟ dτ , ∫ π Aa ( Z ) −∞ ⎝ 2⎠ ∞ ⎛ τ2 ⎞ 2p Rκ = Re ⎡⎣ R e−iϕ ⎤⎦ − C Im ⎡⎣ R e−iϕ ⎤⎦ τ exp ⎜ − ⎟ dτ , ∫ π Aa ( Z ) −∞ ⎝ 2⎠

RC =

{

}(

{

}

)

RT0 =

∞ ⎛ τ2 ⎞ , 2 − iϕ ⎡ ⎤ Im R τ exp e ⎜ − ⎟ dτ ⎦ π Apa ( Z ) ∫ −∞ ⎣ ⎝ 2⎠

Rθ = −

∞ ⎛ τ2 ⎞ . 1 2 − iϕ − iϕ ⎡ ⎤ ⎡ ⎤ − + Re 3 2 τ 4 κ Im τ exp p R e R e ⎜ − ⎟ dτ ∫ ⎣ ⎦ ⎣ ⎦ 2 π Apa ( Z ) −∞ ⎝ 2⎠

{

(

)

(2.28) (2.29) (2.30)

}

(2.31)

Here, Re ⎡ R e−iϕ ⎤ and Im ⎡ R e−iϕ ⎤ represent the real and imaginary parts of R e−iϕ , 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 ultrahigh 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.

2.3 Fundamental Theories of Dispersion-Managed Pulse

17

b(Z) bav

Zb b1

∆b

0 3 Zb 4

1 Zb 4

b2

Z

Za Fig. 2.4: Schematic diagram of a periodic two-step dispersion map.

b(Z) is a periodic function of Z with period Zb. 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 as ∆b = b1 − b2 . 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, dp0 = b(Z ) p03C0 , dZ dC0 S (Z ) = −b(Z ) p02 1 + C02 − E 0 p0 . dZ 2π

(

)

(2.32) (2.33)

Here, we set the initial values as κ 0 (Z = 0 ) = 0 , T00 ( Z = 0 ) = 0 , where A0(Z), p0(Z) and C0(Z) are the pulse parameters in absence of perturbation, E0 = π A02 p0 is a constant for any Z and represents the pulse energy. Now we are going to describe the pulse dynamics in dispersion-managed line using the system parameters 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 period Zb as shown in Figs. 2.5, 2.6, 2.7 and 2.8. Eqs. (2.32) and (2.33) are used to analytically evaluate p(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.

18

Chapter 2. Fundamentals of Optical Fiber Communications Table 1: Fiber and system parameters used in the analysis

Parameter Wavelength Fiber loss Effective area of fiber core

Real unit

Normalized unit

λ = 1.55 µ m α = 0.2 dB/km Aeff = 50 µ m 2

Γ = 23.256

Spontaneous emission factor

nsp = 1.5

Minimum pulse width, FWHM

τ s = 10 ps

Ts = 1.763

Nonlinear coefficient of fiber

n2 = 3.0 × 10 −20 m 2 /W d av = 0.1 ps/nm/km zb = 40.0 km z a = 40.0 km

Dav = 1.0 Zb = 0.1414 Z a = 0.1414

Average dispersion DM period Amplifier spacing

We can explain the pulse dynamics more in details using the illustrations. Figures 3.2 and 3.3 show the periodic evolutions of p(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-C plane 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 pulse energy E0 = 3.57 (0.0311 pJ) in a loss less line.

Fig. 2.5: Chirp, C for ∆b = 70.

Fig. 2.6: Inverse of pulse width, p for ∆b = 70.

2.3 Fundamental Theories of Dispersion-Managed Pulse

Fig. 2.7: p-C plane for ∆b = 70, E0 = 5.65.

19

Fig. 2.8: p-C plane for ∆b = 42, E0 = 3.57.

We next discuss the DM soliton with fiber loss and periodic amplification for the same systems described above, of course, in absence of perturbation. We obtain the periodic solutions of p(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 for ∆b = 70 and C0 = −0.174 for ∆b = 42, but the initial value of inverse of pulse width p0 = 1 for all cases. Figures 2.11 and 2.12 demonstrate the closed orbit in p-C plane 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 (larger ∆b) possesses bigger closed orbit and require larger energy for evolution of soliton pulse along the line [73].

Fig. 2.9: Chirp, C for ∆b = 70.

Fig 2.10: Inverse of pulse width, p for ∆b = 70

20

Chapter 2. Fundamentals of Optical Fiber Communications

Fig 2.11: p-C plane for ∆b = 70, E0 = 12.41.

Fig 2.12: p-C plane for ∆b = 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: κ 0 ( Z ) = T00 ( Z ) = 0 and θ 0 (Z = 0 ) = 0 , we assess the following expression for phase shift of DM soliton in absence of perturbation

θ 0 (Z ) =

5E0 1 Z b(ζ ) p02 (ζ )dζ + ∫ 0 2 4 2π

Z

∫ 0 S (ζ ) p0 (ζ ) dζ .

(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.

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 multiperiod transmission line without loss.

Fig. 2.16: Propagation of a DM soliton in multiperiod multi-span transmission line with loss and periodic 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.

22

Chapter 2. Fundamentals of Optical Fiber Communications

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 ansatz for 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.

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

24

3.3 Analytical3.2 Calculation Jitter Mechanismof of Phase Phase Jitter

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 dispersion compensated 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.

Mechanism of Phase 3.2 3.2 Mechanism of Phase Jitter Jitter

25 By Kerr effect

Random shift of phase

∆φ Amplifier

Signal

Signal + ASE

1. SPM (mainly) 2. XPM

Nonlinear phase 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

⎛ P 2 n = n0 + n2 E = n0 + n2 ⎜⎜ ⎝ Aeff

⎞ ⎟, ⎟ ⎠

(3.1)

where, n0 is the linear refractive index, E is field intensity, n2 is the non-linear refractive index depending on optical power P, and Aeff 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

(

φ = n0 + n2 E = ϕ L + ϕ NL ,

2

)k z , 0

(3.2)

where, k0 = 2π λ is propagation constant and z is the fiber length. φL is the linear phase shift and

φNL is the nonlinear part which depends on signal power. φNL 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 L

φ NL = ∫ 0 n2 k0

P(z ) dz = γ PLeff , Aeff

(3.3)

where, γ = 2π n2 λ Aeff is known as the fiber nonlinear coefficient, P is assumed to be the launched

power of P = P(0) and with fiber loss coefficient α, P ( z ) = Pe −α z . L is span length

(

)

and Leff = 1 − e −αL α is the effective span length.

26

3.33.3 Analytical Jitter AnalyticalCalculation Calculationof of Phase Phase Jitter

Nonlinear phase fluctuation due to noise (∆φNL)

Noise N

N

N

Signal

Signal

Signal

N

C

Linear phase fluctuations

B

Amplitude fluctuations

al gn Si

A

Nonlinear phase fluctuations

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 N and 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

2

φNL = ∫ 0 n2 k0 E + N dz 2 = γAeff Leff E + N .

= γ Aeff Leff ⎡⎣ E + E.N * + E * .N + N ⎤⎦ 2

(3.4)

2

For ASE noise within the bandwidth of the signal, we find the mean nonlinear phase shift is 2 φ NL = γAeff Leff E and the rest of the phase variation is occurred due to noise as implied in Eq. (3.4).

For M number of spans, the overall phase shift with accumulated ASE noise is

{

2

2

2

ϕ NL = γ Aeff Leff E + N1 + E + N1 + N 2 + E + N1 + N 2 + N 3 " + E + N1 + " + N M

2

},

(3.5)

where N1, N2, N3, . . . , NM represent the white random noise with Gaussian distribution generated by 1st, 2nd, 3rd, . . . , Mth amplifiers 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 at Z = mZ a

3.2 Mechanism of Phase JitterJitter 3.3 Analytical Calculation of Phase

27

R = {nmR ( Z , T ) + inmI ( Z , T )} eiϕ ,

(3.6)

where nmR and nmI are real random functions which satisfy the following correlations, nmR ( Z , T ) nmR ( Z ′, T ′) = nmI ( Z , T ) nmI ( Z ′, T ′) =

nmR ( Z , T ) nmI ( Z ′, T ′) = 0 .

Nm ( Z ) δ ( Z − Z ′) δ (T − T ′) , 2

(3.7) (3.8)

Here, Nm(Z) is the spectral density of the m-th amplifier noise and is given by N m ( Z ) = N 0δ ( Z − mZ a ) .

(3.9)

Here, N 0 = nsp hν ( G − 1) , nsp is spontaneous emission factor, hν is the photon energy and G = exp(2 ΓZ a ) is amplifier gain, where Г 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 Nm ( Z ) =

8π 3c 3hn2 nspt03 ( G − 1) 2

λ 6 bav Aeff Z a

(3.10)

δ ( Z − mZ a ) .

Here, n2 is nonlinear coefficient of fiber, c is the speed of light, h is the Planck constant, Aeff is fiber effective core area, α 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 x(Z ) = x0 (Z ) + δ x(Z ) , where δx 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 d (δ A ) b ( Z ) = p0 ( p0C0δ A + 2 A0C0 δ p + A0 p0δ C ) + ∑ δ RmA , dZ 2 m d (δ p ) 2 = b ( Z ) p0 ( 3C0 δ p + p0δ C ) + ∑ δ Rmp , dZ m d (δ C ) = −2b ( Z ) p0 1 + C02 δ p + p0C0 δ C − 2 S ( Z ) A0δ A + ∑ δ RmC , dZ m d (δκ ) = ∑ δ Rmκ , dZ m d (δ T0 ) = b ( Z ) δκ + ∑ δ RmT0 , dZ m d (δθ ) 5 2 S ( Z ) A0δ A + ∑ δ Rmθ , = b ( Z ) p0 δ p + dZ 4 m

{(

)

}

(3.11) (3.12) (3.13) (3.14) (3.15) (3.16)

28

3.3 Analytical Calculation of Phase Jitter

where ∞ ⎛ τ2 ⎞ , 1 2 n 3 2 τ exp − ⎜ − ⎟dτ mI ∫ 2 π a ( Z ) −∞ ⎝ 2⎠ ∞ ⎛ τ2 ⎞ p0 nmI 1 − 2τ 2 exp ⎜ − ⎟dτ , = ∫ π A0 a ( Z ) −∞ ⎝ 2⎠

(

δ RmA =

δ Rmp

(

(3.17)

)

(3.18)

∞ ⎛ τ2 ⎞ , 2 2 1 2 τ exp − − n C n ( ) ⎜ − ⎟dτ 0 mI mR π A0 a ( Z ) ∫ −∞ ⎝ 2⎠ 2 ∞ ⎛ τ ⎞ 2 p0 = ( nmR − C0 nmI )τ exp ⎜ − ⎟dτ , ∫ −∞ π A0 a ( Z ) ⎝ 2⎠

(

δ RmC = δ Rmκ

)

δ RT0 =

)

(3.19) (3.20)

∞ ⎛ τ2 ⎞ , 2 τ exp n ⎜ − ⎟dτ mI π A0 p0 a ( Z ) ∫ −∞ ⎝ 2⎠

δ Rmθ = −

(3.21)

∞ ⎛ τ2 ⎞ 1 nmR 3 − 2τ 2 exp ⎜ − ⎟dτ , ∫ 2 π A0 a ( Z ) −∞ ⎝ 2⎠

(

)

(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) d δ A2 dZ

(

)

= b ( Z ) p0 p0C0 δ A2 + 2 A0C0 δ Aδ p + A0 p0 δ Aδ C +

3 4 E0

A02 N m ( Z ) , ∑ a2 ( Z ) m

d δ Aδ p b (Z ) = p0 7 p0C0 δ Aδ p + 2 p02 δ Aδ C + 2 A0C0 δ p 2 + A0 p0 δ pδ C dZ 2 A0 p0 N m ( Z ) 1 + , ∑ a2 ( Z ) 2 E0 m

(

)

b(Z ) d δ Aδ C =− p0 4 1 + C02 δ Aδ p + 3 p0C0 δ Aδ C − 2 A0C0 δ pδ C − A0 p0 δ C 2 2 dZ A C N (Z ) , 1 − 2 S ( Z ) A0 δ A2 − ∑ 0 02 m E0 m a (Z )

{ (

)

(3.23)

(3.24)

} (3.25)

b(Z ) d δ Aδκ (3.26) = p0 ( p0C0 δ Aδκ + 2 A0C0 δ pδκ + A0 p0 δ Cδκ ) , dZ 2 d δ Aδ T0 b(Z ) (3.27) = 2 δ Aδκ + p02C0 δ Aδ T0 + 2 A0 p0C0 δ pδ T0 + A0 p02 δ Cδ T0 , dZ 2 b(Z ) d δ Aδθ 5 2 = p0 ( 2 δ Aδ p + p0C0 δ Aδθ + 2 A0C0 δ pδθ + A0 p0 δ Cδθ ) + S ( Z ) A0 δ A2 , dZ 2 4

(

)

(3.28) d δp dZ

2

(

)

= 2b ( Z ) p02 3 C0 δ p 2 + p0 δ pδ C +

1 E0

∑ m

p Nm ( Z ) , a2 ( Z ) 2 0

d δ pδ C = −b ( Z ) p0 2 1 + C02 δ p 2 − p0C0 δ pδ C − p02 δ C 2 dZ p C N (Z ) 2 − ∑ 0 02 m , E0 m a (Z )

{ (

)

d δ pδκ = b ( Z ) p02 ( 3C0 δ pδκ + p0 δ Cδκ dZ

),

(3.29)

}−

2 S ( Z ) A0 δ Aδ p

(3.30) (3.31)

3.2 Mechanism of PhaseSimulations Jitter 3.4 Analytical and Numerical for Dispersion-Managed Soliton

29

d δ pδ T0 (3.32) = b ( Z ) δ pδκ + 3 p02C0 δ pδ T0 + p03 δ Cδ T0 , dZ d δ pδθ 5 2 (3.33) S ( Z ) A0 δ Aδ p , = b ( Z ) p0 δ p 2 + 3 p0C0 δ pδθ + p02 δ Cδθ + dZ 4 1 + C02 N m ( Z ) d δ C2 4 2 2 = − 4b ( Z ) p0 1 + C0 δ pδ C + p0C0 δ C − 2 2 S ( Z ) A0 δ Aδ C + ∑ , dZ E0 m a2 ( Z )

(

)

(

)

{(

(

}

)

)

(3.34) d δ Cδκ (3.35) = −2b ( Z ) p0 1 + C02 δ pδκ + p0C0 δ Cδκ − 2 S ( Z ) A0 δ Aδκ , dZ d δ Cδ T0 = −b ( Z ) 2 p0 1 + C02 δ pδ T0 − δ Cδκ + 2 p02C0 δ Cδ T0 − 2 S ( Z ) A0 δ Aδ T0 , (3.36) dZ d δ Cδθ 2 = b ( Z ) p0 δ pδ C − 2 (1 + C02 ) δ pδθ − 2 p0C0 δ Cδθ + S ( Z ) A0 ( 5 δ Aδ C − 4 δ Aδθ ) dZ

d δκ 2 dZ

{(

)

{ (

)

}

}

{

}

N (Z ) 1 − ∑ 2m , E0 m a ( Z ) 1 = E0

∑

(

(3.37)

)

p02 1 + C02 N m ( Z )

m

a (Z ) 2

d δκδ T0 1 = b( Z ) δκ 2 − dZ E0

4

∑ m

,

C0 N m ( Z ) , a2 ( Z )

d δκδθ 5 2 = b( Z ) p0 δ pδκ + S ( Z ) A0 δ Aδκ , dZ 4 d δ T02 N (Z ) , 1 = 2b ( Z ) δκδ T0 + ∑ m dZ E0 m p02 a 2 ( Z ) d δ T0δθ 5 2 S ( Z ) A0 δ Aδ T0 , = b ( Z ) ( p0 δ pδ T0 + δκδθ ) + dZ 4 d δθ 2 Nm ( Z ) . 5 2 3 = 2b ( Z ) p0 δ pδθ + S ( Z ) A0 δ Aδθ + ∑ 2 4 E0 m a 2 ( Z ) dZ

(3.38) (3.39) (3.40) (3.41) (3.42) (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 DispersionManaged 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.

30

3.4 Analytical and Numerical Simulations for Dispersion-Managed Soliton 3.3 Analytical Calculation of Phase Jitter

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 for model (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 µm, nonlinear coefficient 2.434 W-1km-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 of S is given as [78] S=

−b1 z1 + b2 z2

τF

,

(3.44)

3.2 Mechanism of PhaseSimulations Jitter 3.4 Analytical and Numerical for Dispersion-Managed Soliton

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 τF is 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 to pulse at every amplifier position. We add random noise with zero mean and variance of N m 2 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].

32

3.3 Analytical Calculation Phase Jitter 3.5 Quasi-Linear PulseofTransmission

Fig. 3.6: Phase variance versus dispersion map strength S for 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 pathaveraged 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.

3.2 Mechanism of Transmission Phase Jitter 3.5 Quasi-Linear Pulse

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 management. To investigate the phase jitter effect for 1000-km long quasi-linear transmission line with strong dispersion management of 17/−17 ps/nm/km fibers, we choose the peak power such that we may obtain lower phase noise. Fig. 3.7 shows the variation of phase noise against the pulse peak power for two different distances. Two curves are plotted for 600 km and 1000 km respectively, which show a bit different variation with respect to peak power because of nonlinearity. For our calculation, we assume the periodic dispersion map as shown in Fig. 1 with zero path-averaged dispersion and select the optimal peak power as 4.5 mW. We follow the same analytical process as described in section 3.3 and numerically simulate using split-step Fourier method to validate the variational results.

34

3.3 Analytical Calculation Phase Jitter 3.5 Quasi-Linear PulseofTransmission

Fig. 3.8: (a) Pulse chirp and (b) inverse of pulse width as a function of distance for DM quasi-linear pulse. The solid lines are obtained by numerical simulations.

Fig. 3.8 illustrates the variation of pulse chirp and inverse of pulse width as a function of distance in a quasi-linear system for 17/−17 ps/nm/km dispersion management with fiber loss of 0.2 dB/km. The analytical results are confirmed by the numerical simulations. The results indicate that pulse chirp and width are periodic in nature with respect to distance.

Fig. 3.9: Pulse shape after transmission of 1000 km (solid line) and initial pulse (dotted line), (a) time domain and (b) frequency domain.

3.2 Mechanism of Phase Jitter 3.5 Quasi-Linear Pulse Transmission

35

Fig. 3.9 shows the shape of quasi-linear pulse with fiber loss and amplification, before and after propagation of 1000 km DM line, and in absence of ASE noise. The pulse has no initial chirp. In temporal domain, a considerable pulse broadening is noted, which is due to dispersion effect. Small spectral broadening is the result of nonlinear chirp.

Fig. 3.10: Pulse shape with initial chirp after propagation of 1000 km (solid line) and initial pulse (dotted line), (a) time domain and (b) frequency domain.

In Fig. 3.10, the pulse shape is plotted again incorporating an appropriate initial chirp. We vary the initial value of chirp to achieve less distorted pulse at the receiving end. Figure shows that the pulse broadening is much reduced in temporal domain and spectral broadening is also decreased. We may adjust the initial chirp to control the pulse shape.

36

3.3 Analytical Calculation of Phase Jitter 3.6 Analytical and Numerical Simulations for Phase Jitter in Quasi-Linear Systems

Fig. 3.11: DM quasi-linear pulse evolution in absence of noise along the transmission line.

Fig. 3.12: DM quasi-linear pulse evolution in presence of noise along the transmission line.

Figs. 3.11 and 3.12 demonstrate the quasi-linear pulse evolution over 1000 km fiber in absence and presence of amplifier noise, respectively. We follow the transmission model as mentioned above. Stable pulse evolution is observed in quasi-linear regime by properly adjusting the peak power and chirp.

3.6 Analytical and Numerical Simulations for Phase Jitter in Quasi-Linear Systems We conduct numerical simulations to evaluate the variance of phase fluctuations for three different quasi-linear DM models consisting of 7.0/-7.0 ps/nm/km fibres with equal length of 20 km concatenated alternately for model (x), 12.0/-12.0 ps/nm/km fibers for model (y) and 17.0/17.0 ps/nm/km fibers for model (z) following the two-step dispersion map as shown in Fig. 3.4. The system parameters used in the analysis are same as for DM soliton except the pulse width

Mechanism of Phase Simulations Jitter 3.63.2 Analytical and Numerical for Phase Jitter in Quasi-Linear Systems

37

(FWHM) 20 ps, nonlinear coefficient 1.853 W-1km-1 and average dispersion is assumed to be zero. A single pre-chirped Gaussian pulse is launched into each periodically DM line with a peak power of 4.5 mW. White Gaussian noise having the spectral strength given by Eq. (4.9) is added to pulse at every amplifier location. Monte Carlo simulations have been carried out and the variance of phase at pulse peak is estimated along the transmission line. In quasi-linear system, linear phase noise is considerable and we calculate it by using the following expression as given in Ref. [42]

θ2

lin

=

M N0 , 2 Ea

(3.45)

where M is the total number of amplifiers and Ea is the signal energy just after every amplifier.

Fig. 3.13. Phase variance vs. transmission distance for quasi-linear systems. The solid and dashed curves show the analytical results. The circles represent the numerical simulation results.

The agreement between analytical calculation and numerical simulation results is quite satisfactory as shown in Fig. 3.13. We find that the model (z) offers the lowest phase noise among the three models. We can infer that the phase noise decreases for stronger DM line. For a constant energy, when dispersion increases, pulse width broadens, and the peak power is suppressed locally which consequently reduces the effect of SPM. So, SPM induced phase noise becomes smaller for stronger DM line. The total phase noise is calculated by adding linear and nonlinear phase noise. Similar results have been obtained by S. Kumar, who has used different perturbation approach [79].

b1(Z) Zb b2(Z)

Z-

Z+

Z− Lr = + Z

Z-

Fig. 3.14: Length ratio in a DM map period.

38

3.3 Analytical Calculation of Phase Jitter 3.6 Analytical and Numerical Simulations for Phase Jitter in Quasi-Linear Systems

Fig. 3.15: Phase noise against the ratio of anomalous to normal dispersion fiber length constituting a DM period. The total transmission length is 1000 km with periodic DM map.

We further explore the effect of phase noise on quasi-linear DM lines by varying the fiber length of both type of dispersion. Within a DM map period of 40 km, we reduce the fiber length of negative dispersion and adjust the amount of negative dispersion while keeping the positive dispersion fixed at 17 ps/nm/km retaining the zero-path averaged dispersion. Actually we reduce the length of dispersion compensating fiber by increasing its value to find its effect on phase noise. Fig. 3.14 shows the definition of length ratio, Lr, which is the ratio of anomalous to normal dispersion fiber length and the results are shown in Fig. 3.15 where the phase noise is plotted against the length ratio. We find that the phase noise reduces with the increase length ratio. By optimizing the length ratio, we can design a transmission link with lower phase noise.

Fig. 3.16: Phase noise is plotted against dispersion map strength S for unity length ratio.

3.2 Mechanism of Phase Map Jitterfor Quasi-Linear Pulse Transmission 3.7 Upgradation of Dispersion

39

Fig. 3.16 depicts the phase jitter behavior with fiber dispersion map strength after transmission of 1000 km for unity Lr. We change the dispersion of both type of fibers of equal length of 20 km forming the DM period, monitor the phase jitter, and plot against DM map strength. We observe that higher dispersive fibers offer lower phase noise, i.e, phase jitter decreases with larger values of S. It could be suggested that S > 2 regime is suitable for quasi-linear pulse transmission to maintain lower phase noise. However, further upgradation of quasi-linear systems might be achieved by modifying the DM map and/or amplifier spacing etc. which will be discussed in the next sections.

3.7 Upgradation of Dispersion Map for Quasi-Linear Pulse Transmission We discuss the upgradation of dispersion map by means of achieving lower phase noise. First of all we consider two different periodic dispersion maps with lumped amplifiers as shown in Fig. 3.17 (Map (a) and Map (b)). Map (a) consists of non-zero dispersion shifted fiber (NZDSF) followed by dispersion compensating fiber (DCF) and Map (b) comprises of standard single mode fiber (SMF) known as standard telecommunication fiber (STF) followed by DCF in each period of Zd. We assume that dispersion is fully compensated at the end of each period and dispersion map

period Zd is equal to amplifier spacing Za. For Map (a), dispersion of NZDSF is taken as 4.0 ps/nm/km with fiber length of 48 km and DCF is −96.0 ps/nm/km with length of 2.0 km. For Map (b), SMF has a dispersion of 17.0 ps/nm/km with length of 42.5 km and DCF has −96.33 ps/nm/km dispersion with length of 7.5 km. The nonlinear coefficient and fiber loss of SMF and NZDSF are taken as 1.52 W−1km−1 and 0.2 dB/km, respectively. For DCF, nonlinear coefficient is 5.06 W−1km−1 and loss is 0.5 dB/km. The system parameters used in the analysis are: DM period 50 km, optical carrier wavelength 1.55 µm, and spontaneous emission factor 1.5. The minimum pulse width (FWHM) is taken as 20 ps. The dimensionless dispersion map strength S is calculated as 1.22 and 4.61 for Map (a) and (b), respectively.

40

Analytical Calculation Phase Jitter 3.7 Upgradation of Dispersion3.3 Map for Quasi-Linear Pulseof Transmission

Fig. 3.17: Different dispersion maps for upgradation.

A single unchirped Gaussian pulse is launched into each periodic dispersion compensated line with a peak power of 2 mW over a 3000 km long fiber link. White Gaussian noise having the power spectral density given by Eq. (3.9) is added to pulse at every amplifier position. Monte Carlo simulations have been carried out for 1000 realizations by directly solving Eq. (2.14) based on split-step Fourier method and the variance of phase at pulse peak is calculated numerically along the transmission line.

Fig. 3.18: Phase of transmitted pulse is plotted against distance in absence of perturbation. The solid curves are obtained by variational equations and the dotted curves show the solutions by direct PDE simulations.

3.2 Mechanism of Phase Map Jitterfor Quasi-Linear Pulse Transmission 3.7 Upgradation of Dispersion

41

Fig. 3.18 shows the pulse phase as a function of transmission distance in absence of perturbation, i.e., without ASE noise for Map (a) and (b). Dynamical equation Eq. (2.25) is utilized in absence of perturbation to ascertain the analytical results including Eqs. (2.32) and (2.33). A fair agreement is achieved between variational analysis and direct PDE simulations for phase. In stronger dispersive fibers pulse width broadens and phase decreases more rapidly compared to weaker dispersive fibers. After adding the amplifier noise, the variance of phase noise is plotted against transmission distance for the maps under discussion in Fig. 3.19. The oscillations occur due to variations of local dispersion along the periodic dispersion compensated line. In Fig. 3.19, numerical simulations show that phase fluctuations increase highly after 2500 km for Map (a). This is because pulse distortion increases much at the end of this moderate dispersion model due to accumulated SPM effects and ASE noise. The analytical values do not give good prediction in that region. However, the agreement between analytical estimation and numerical simulation results is satisfactory except the above mentioned region. The figure reveals reasonably that Map (b) offers lower phase noise than Map (a). Though the analytical results include the linear part of variance of phase noise, we show it separately to indicate that it is same for both maps and increases linearly as it is related to the number of amplifiers. On the other hand, the nonlinear part increases nonlinearly along transmission line as it is related as the cube of the distance. It is quite evident that nonlinear phase noise is larger in Map (a). From the results we can infer that stronger dispersion managed maps are suitable to mitigate phase noise. The physical reason is that in strongly dispersion compensated line pulse width expands much more for a constant energy compared to a weaker one and the peak power is suppressed locally, as a result the effect of SPM is diluted. Therefore fiber nonlinearity induced phase noise becomes smaller for stronger dispersion compensated line.

Fig. 3.19: Variance of phase fluctuations versus transmission distance. The solid and dashed curves represent the analytical results. The triangles and circles indicate the solutions obtained by numerical simulations. The linear phase noise is shown by dash-dotted line.

42

3.3Map Analytical Calculation Phase Jitter 3.7 Upgradation of Dispersion for Quasi-Linear PulseofTransmission

Fig. 3.20: Variance of phase noise is plotted against transmission distance for Map (b), (c) and (d). The solid, dotted and dashed curves display the analytical solutions. Circles, cross and plus signs present the values acquired by numerical calculations.

Secondly, we compare three different map configurations, Map (b), (c), and (d) as shown in Fig. 3.17. For all the three configurations we assume the same fiber and system parameters of Map (b) except the dispersion parameter of DCF, length of SMF and the map period. For Map (c), Zd is 100 km, SMF length is 92.5 km and DCF length is 7.5 km with dispersion of −209.66 ps/nm/km. Map (d) has 200 km map period with 192.5 km SMF and 7.5 km DCF having dispersion of −436.33 ps/nm/km. The results are plotted in Fig. 3.20. We find that Map (d) offers the minimum phase noise. Map (b), (c) and (d) have dispersion map strengths of 4.61, 9.21 and 20.03 respectively, which imply that Map (d) is the strongest one. Hence, the upgraded Map (d) is the best in reducing the phase noise. It is noted that oscillations are different for Map (b) and (c) as amplifier/s along with the DCF are placed at the middle of two SMF sections within one map period. Linear phase noise will remain as usual as the number of amplifiers is same for all the maps. To perceive the affect of data transmission on those models, eye-pattern is evaluated at the receiver end for each map numerically for 20Gsymbol/s RZ-DQPSK signal with a PRBS of length 27−1. We choose the PRBS length by making a trade off between simulation time and intersymbol interference level. Therefore, we asses the system quality of those models by calculating the Qfactor considering a typical direct detection DQPSK receiver [4]. The Q-factor is defined as Q=

m1 − m0 , σ1 + σ 0

(34)

where m1 and m0 are average values of signals and σ1 and σ0 are the standard deviation of noise at two signal levels. We assume 40% duty cycle for RZ pulses. A Gaussian band-pass filter (2 nm) is used after every amplifier to remove the out-of-band amplifier noise. Fig. 3.21 shows the eyepattern of different dispersion maps for a transmission distance of 2000 km. Q-factor is calculated and plotted against transmission distance as shown in Fig. 3.22. We find that Q-factor is above the

3.2 Mechanism of Phase Map Jitterfor Quasi-Linear Pulse Transmission 3.7 Upgradation of Dispersion

43

standard value (Q = 6) required for error-free data transmission for all the maps which implies that further longer transmission distance could be achieved. However, it is clear that upgraded maps offer higher values of Q-factor.

Fig. 3.21: Eye-pattern at the receiver end for different dispersion maps after transmission of 2000 km.

Fig. 3.22. Q-factor vs. transmission distance for different dispersion maps.

3.8 Effect of Amplifier Spacing In this section we explore the effect of amplifier spacing in dispersion compensated line and show the trend of phase noise with spacing or number of amplifiers within a specific transmission

Effect of Amplifier Spacing 3.3 Analytical3.8Calculation of Phase Jitter

44

distance. We take the Map (b) as the basic model and modify its amplifier spacing to 75 km and 100 km maintaining equal dispersion map period, i.e., Zd is kept equal to Za. However, the ratio of SMF to DCF in a period of Zd is retained same. Therefore, we compare the phase noise effect in three models of different amplifier spacing for the total transmission distance of 3000 km.

Fig. 3.23: Variance of phase noise with different amplifier spacing. The solid, dash-dotted and dashed curves present analytical predictions and triangles display the numerical values.

Fig. 3.23 shows the results with acceptable agreement between calculations by variational method and numerical simulations. The results reveal that increase in amplifier spacing causes enhancement of phase noise. If amplifier spacing increases, amplifier gain G increases which ultimately boosts the ASE noise as it is evident from Eq. (3.9). Linear phase noise will be reduced as the number of amplifiers decreases with longer spacing for a given transmission distance. Dispersion map strength S for the 50 km, 75 km and 100 km spacing is 4.61, 6.91 and 9.21, and they require 60, 40 and 30 repeaters, respectively. Although map strength increases as we increase the period Zd and linear phase noise decreases, the enhancement in ASE noise offsets these advantages.

Fig. 3.24 Eye-pattern at the receiver end for different amplifier spacing after transmission of 2000 km.

Fig. 3.24 shows the eye-pattern after transmission distance of 2000 km for the three cases of

3.2 Mechanism of Phase Jitter Configuration 3.9 Effect of Dispersion Compensation

45

amplifier spacing. Like previous section, we assume 20 Gsymbol/s RZ-DQPSK data transmission with 40% duty cycle RZ pulse. We find that Q-factor decreases with the increase of spacing and in case of 100 km spacing it goes much below the minimum value required for error-free data transmission. Consequently shorter amplifier spacing can be suggested for mitigation of phase noise. By using the upgraded dispersion maps with shorter amplifier spacing, further reduction of phase noise might be possible. However, we can not reduce amplifier spacing to as short as we desire in order to mitigate phase noise, because it is not practicable and also it will incur high cost. So, one should be prudent about designing transmission system with periodic dispersion map and amplification with regard to phase noise.

3.9 Effect of Dispersion Compensation Configuration In conventional dispersion compensated system, the high value of anomalous GVD of SMF is periodically compensated by normal GVD of DCF by adjusting its appropriate length placed at the end of map period. This is the post-compensation configuration for periodic dispersion map that has been considered so far. Now we consider other two, pre- and bi-end compensation configurations for periodic dispersion map with in-line amplifiers as depicted in Fig. 3.25. We compare these three maps assuming the system parameters of Map (b) in Fig. 1 with modifications in periodic dispersion map and check the phase noise for each case. We assume quasi-linear pulse transmission along the line. Since the interaction between dispersion with fiber nonlinearity will be different depending on positive or negative GVD value and the relative position of DCF in a period with respect to amplifier, all these maps may give different results.

Fig. 3.25: Pre-, post- and bi-end dispersion compensation configurations for periodic dispersion map.

In pre-compensation configuration DCF is followed by SMF. The length of DCF is chosen such that it satisfies the equation L1D1 + L2 D2 = 0 for zero residual dispersion at the end of each

3.9 Effect3.3 of Dispersion Analytical Compensation Calculation ofConfiguration Phase Jitter

46

period, where L1, D1 and L2, D2 are the length and dispersion parameter of SMF and DCF, respectively, and L1 and L2 constitute a period or span (L1 + L2 = Za = Zd). In our model of precompensation, 7.5 km (L2) DCF precedes 42.5 km (L1) SMF with amplifier spacing of 50 km. This map period is repeated to reach 3000 km of a single channel lightwave transmission. The postcompensation configuration is opposite of pre-compensation. In case of bi-end compensation configuration, in every period, two equal length (L2/2) DCFs are added at the both ends of SMF. In fact, it is a combination of pre- and post-compensation configurations. In that model, first we connect 3.75 km DCF on either side of 42.5 km SMF and follow this map for the total transmission line. The results obtained from variational method are shown in Fig. 3.26 where we plot smooth curves avoiding oscillations to display the results clearly. The results imply that neither pre-compensation nor bi-end compensation configuration could be suitable to achieve lower phase noise in long transmission line.

Fig. 3.26: Effects of pre-, post- and bi-end compensation configurations for periodic dispersion map on phase noise. The solid, dotted and dashed curves represent estimations of variational calculation. The triangles, circles and squares show the numerical solutions.

Fig. 3.27 shows the eye-pattern after transmission distance of 2000 km for the three dispersion compensation configurations assuming 20 Gsymbol/s RZ-DQPSK data transmission with 40% duty cycle RZ pulse. The Q-factor calculated from the eye-patterns indicates that postcompensation configuration offers the maximum value among the three models.

3.9 3.2 Effect of Dispersion Compensation Mechanism of Phase Jitter Configuration

47

Fig. 3.27: Eye-pattern at the receiver end for different periodic compensation configuration after transmission of 2000 km.

In case of post-compensation configuration, the pulse is compressed first by SMF and then it is allowed to expand by DCF. DCF has higher nonlinearity and higher loss which is placed at the end of period prior to amplifier where power is low. So the degree of SPM effect is low in such configuration. In pre-compensation configuration map optical pulse is launched into DCF and it gets chirped and broadened much and then the pulse is allowed to recompress during its propagation through SMF. In this configuration DCF is placed just after amplifier, so SPM effect is higher which gets accumulated span after span and deteriorates the pulse. In bi-end configuration, in each period, one DCF is placed just after amplifier and another is placed at the end of period before another amplifier. Here SPM effect is much higher and phase noise increases more. Finally we can infer that the post-compensation configuration excels other configurations for long-haul transmission line. In this section, we consider full compensation of dispersion for all the three configurations, i.e., there is no residual dispersion at the end of transmission line. Actually there is always some residual dispersion in practical system. The future work might include the effect of residual dispersion along with further study on optimization on each compensation method.

3.10 Conclusion This chapter has presented the mechanism of phase jitter and described the theoretical analysis of phase jitter in periodically DM line. The impact of periodic dispersion management on phase jitter has been analyzed using variational method assuming necessary linearization. Variational equations have been derived to model the amplifier noise effect on pulse parameters and the phase jitter is evaluated for long-haul DM soliton and quasi-linear transmission systems, respectively, for different dispersion management schemes. Nonlinear phase noise is ascertained for two different DM models and they are compared to a conventional constant dispersion soliton-based system. The analytical results are supported by numerical simulations. The results show that the reduction of phase noise is possible by appropriately choosing the dispersion management. The results recommend that weaker dispersion compensation is useful to mitigate the phase jitter and may allow lower phase noise compared to

48

3.3 Analytical Calculation3.10 of Phase Jitter Conclusion

constant dispersion soliton. On the other hand, stronger DM systems enhance phase noise and deteriorate the performance. Quasi-linear pulse behavior and phase jitter in DM quasi-linear systems have been investigated and depicted in the last few sections. We estimate the phase jitter for quasi-linear systems with different DM models. The results suggest that the stronger dispersion management can be more efficient to suppress phase jitter. We also evaluate the effect of phase jitter on DM quasi-linear systems by altering the fiber dispersion and the length ratio of fibers composing the DM period. Upgraded dispersion maps are proposed by achieving lower phase noise and higher Q-factor. The impact of amplifier spacing and dispersion map configuration on phase noise has also been studied, and it has been found that amplifier spacing should be chosen discreetly. Post-compensation map is observed as an appropriate map configuration in order to manage dispersion periodically for long-haul multi-span DM transmission lines.

Chapter 4 XPM Effects in Transmission Line

Dispersion-Managed

4.1 Introduction XPM is one of the crucial nonlinear impairments in multi-channel lightwave systems, which causes phase fluctuations of copropagating channels, and it is detrimental for phase sensitive modulation formats. SPM-induced phase fluctuations are deleterious for single channel transmission, however, XPM-induced phase fluctuations are much concern for WDM transmission [80]. In WDM system, FWM is also a significant nonlinear impairment. Since we are considering dispersion management with highly dispersive fibers like standard single mode fiber (SMF) and dispersion compensating fiber (DCF), the effect of FWM is negligible. In this Chapter, we study the dynamics of inter-pulse collisions in a two-channel WDM system and explore the influence of XPM on phase of RZ pulse considering single polarization. There have been several reports on inter-pulse collisions in dispersion-managed (DM) WDM systems mainly focusing on timing shift and frequency shift [81-87]. Here, we use variational method to examine the collision-induced phase shift analytically and split-step Fourier method is utilized to prove the results numerically. Furthermore, we show the effect of frequency shift and initial pulse spacing between channels on phase fluctuations and therefore discuss its mitigation in WDM system. Section 4.2 states variational analysis for two-channel WDM transmission system with periodic dispersion management. We assume XPM as a source of perturbation and derive the dynamical equations for pulse parameters. Section 4.3 describes the transmission system with different DM models and section 4.4 explains the basic mechanism of inter-pulse collision considering a conventional map. Impact of initial pulse spacing between launched pulses of two channels on phase fluctuation is delineated in section 4.5 for

49

50

4.2 Analytical Calculation of XPM-Induced Phase Shift

different transmission models. Section 4.6 shows the influence of channel spacing and residual dispersion per span on phase fluctuation induced by XPM.

4.2 Analytical Calculation of XPM-Induced Phase Shift Optical pulse propagation in a fiber governed by the nonlinear Schrödinger equation (NLSE) is described as i

∂U b ( Z ) ∂ 2U 2 − + S ( Z ) U U = R, ∂Z 2 ∂T 2

(4.1)

where U, Z, T are normalized amplitude of pulse envelope, distance and retarded time, respectively. b(Z) stands for fiber dispersion, S(Z) represents fiber nonlinearity and R denotes perturbation. For a two-channel WDM system, we consider RZ pulse Uj(Z, T) (j = 1, 2) where two pulses U1 and U2 interact each other through the XPM effect only. Assuming XPM as a perturbation, NLSE can be separated to the equations for U1 and U2 as i

∂U j ∂Z

2

−

2 b (Z ) ∂ U j + S (Z ) U j U j = R j , 2 2 ∂T

(4.2)

2

where R j = −2S ( Z ) U 3− j U j . We assume that Uj has a Gaussian waveform with linear chirp and can be given as ⎛ τ2 ⎞ U j (Z,T ) = Aj exp⎜ − j + iφ j ⎟ , ⎜ 2 ⎟ ⎝ ⎠

(4.3)

where τ j ( Z,T ) = p j ( Z ) {T − Tj ( Z )},

(4.4)

φ j ( Z, T ) =

(4.5)

Cj ( Z ) 2

τ 2j −

κ j ( Z)

pj ( Z )

τ j +θ j ( Z ) .

Here Aj(Z), pj(Z), Cj(Z), κj(Z), Tj(Z) and θj(Z) are the pulse parameters representing the amplitude, the inverse of pulse width, the linear chirp, the central frequency, the central time position and the phase of the pulse, respectively. Applying variational method [70] to Eq. (4.2), the dynamical equations of these parameters under perturbation can be derived as (see Appendix B) dp j dZ

= b ( Z ) p 3j C j ,

(4.6)

51

4.3 Analytical Calculation of XPM-induced Phase Shift

dC j dZ dθ j

(

)

= − b ( Z ) p 2j 1 + C 2j − S ( Z )

Ej 2π 5E j

p j − 4S (Z )

E 3− j p j p33− j P5

π

{P

2

}

(4.4)

− 2( ∆ τ ) 2 F ,

E pp b( Z ) 2 2 κ j − p 2j + S ( Z ) p j + S ( Z ) 3− j 1 5 2 P 2 2 P 2 + p32− j − 2 p32− j ( ∆τ ) F , (4.8) dZ 2 4 2π π P d (∆κ ) E1 + E 2 p12 p 22 ∆ τ (4.9) = 4S (Z ) F, dZ P3 π d ( ∆τ ) (4.10) = b( Z ) p12C1 + p22C2 ∆τ + p1 p2∆κ , dZ

(

=−

)

{(

{ (

)

}

}

)

where ∆κ = κ 1 − κ 2 , ∆τ = p1 p2 (T1 − T2 ) ,

(4.11) (4.12)

p12 + p22 ,

(4.13)

⎧⎪ ⎛ ∆τ ⎞2 ⎫⎪ , F = exp ⎨ − ⎜ ⎟ ⎬ ⎩⎪ ⎝ P ⎠ ⎭⎪

(4.14)

P=

2

and E j = ∞ U j dT = π A2j p j which represents a constant pulse energy of Uj. Momentum ∫∞ conservation law represented by E1κ1 ( Z ) + E2κ 2 ( Z ) = E1κ1 (0) + E2κ 2 (0) is satisfied for any Z. Eq. (4.11) indicates the frequency separation of the two channels. Eq. (4.12) is related to pulse widths and pulse spacing ∆T(Z) = T1(Z) –T2(Z) between inter-channel pulses, and at Z = 0, ∆T(0) gives the initial pulse spacing. By integrating Eq. (4.8), the pulse phase can be written as θ j ( Z ) = θ j ( 0 ) + θ Disp ( Z ) + θSPM ( Z ) + θ XPM ( Z ) , where

(4.15)

1 Z b(ζ ) κ 2j (ζ ) − p 2j (ζ ) d ζ , 2 ∫0 5E j Z θSPM ( Z ) = ∫ S (ζ ) p j (ζ ) d ζ , 4 2π 0 E Z pp 2 θXPM ( Z ) = 3− j ∫ 0 S (ζ ) 1 5 2 P2 2P2 + p32− j − 2 p32− j ( ∆τ ) F (ζ ) dζ . P π

θ Disp ( Z ) = −

{

}

{ (

)

}

(4.16) (4.17) (4.18)

Here θj(0) indicates the initial pulse phase. θDisp, θSPM and θXPM terms imply the contributions from dispersion, SPM, and XPM, respectively. The phase shift observed at any channel is deduced as

δθ = θ 2ch − θ1ch = δθ Disp + δθSPM + δθ XPM ,

(4.19)

where θ2ch is the phase observed at one channel when both channels carry signals and θ1ch is the phase of the pulse for a singe-channel system. The total phase shift of Uj can be similarly written as

52

4.2 Analytical Calculation of XPM-Induced Shift 4.3 System Phase Description

δφ = φ2ch − φ1ch = δφC + δφκ + δφθ ,

(4.20)

where

δφC =

C2ch 2 C1ch 2 , τ 2ch − τ 1ch 2 2

(4.21)

⎛κ ⎞ κ δφκ = − ⎜ 2ch τ 2ch − 1ch τ 1ch ⎟ , p p 1ch ⎝ 2ch ⎠

(4.22)

δφθ = δθ = θ 2ch − θ1ch .

(4.23)

Here δφC , δφκ and δφθ are the phase shift components contributed by pulse chirp (C), frequency (κ) and phase (θ), respectively. Note here that both δφC and δφκ are also related to pulse width (p) and pulse spacing (∆T).

4.3 System Description Model (A)

SMF

DCF

λ1

λ1 λ2

M U X

L1

L2

D E M U X

λ2

× N spans

Model (B)

λ1

λ1 λ2

M U X

L1/2

Rx

SMF

DCF

SMF

Rx

L1/2

L2

D E M U X

λ2

Rx

Rx

× N spans

Model (C)

DCF

SMF

λ1

λ1 λ2

M U X

L2

L1 × N spans

D E M U X

λ2

Rx

Rx

Fig. 4.1: N-span WDM transmission line models consisting of different dispersion maps.

A long-haul fiber-optic transmission line is simulated following periodic dispersion compensated two-channel WDM system. Transmission line with highly dispersive fibers like standard single mode

4.3 Analytical Calculation of XPM-induced Phase 4.4 Basic Mechanism of Collision-Induced Phase Shift Shift

53

fiber (SMF) is considered and dispersion compensating fiber (DCF) is used to control the total residual dispersion. We consider three distinct periodic dispersion models, Model (A), (B) and (C) as shown in Fig. 4.1 with a period or span of L1+L2 = 50.31 km which is repeated 40 times to cover the total transmission distance of 2012.4 km. L1 = 43 km and L2 = 7.31 km, are the lengths of SMF and DCF, respectively. In Model (A), SMF is followed by DCF in each period. Configuration of Model (C) is opposite of Model (A). In Model (B), SMF length is halved ( L1 2 ) and is connected to either end of DCF in each period. The fiber parameters for SMF are taken as dispersion 17 ps/nm/km, effective core area 80 µm2 and nonlinear index coefficient 2.5×10−20 m2/W, when these are −100 ps/nm/km, 20 µm2, and 3.0×10−20 m2/W for DCF. We do not consider fiber loss for simplicity. The wavelength of channel 1 (λ1) is 1555.0 nm and channel 2 (λ2) is 1555.4 nm giving a channel spacing of 0.4 nm (50 GHz). Two different systems are envisaged. In the first system, both channels carry 40 ps pulses with a peak power of 0.673 mW (−1.7 dBm) assuming bit rate of 10 Gb/s with 40% duty cycle, and in the second system, channel 1 carries 10 ps pulse with a peak power of 2.692 mW (4.3 dBm) and channel 2 carries 40 ps pulse with a peak power of 0.673 mW simulating a combined system where 40 Gb/s signal is copropagated with 10 Gb/s signal for the same 40% duty cycle. In order to increase the capacity of currently deployed 10 Gb/s system, we have to replace gradually some 10 Gb/s channels by 40 Gb/s channels. So there is a possibility of copropagation of such different bit rate signals through the same transmission fiber. We consider this second hybrid bit rate system on this context. We evaluate the analytical results by variational analysis and numerical simulations have been carried out by directly solving Eq. (4.1) using split-step Fourier method taking into account the parameters mentioned above.

4.4 Basic Mechanism of Collision-Induced Phase Shift Several works have been done on collision-induced timing shift and frequency shift. Golovchenko et al. [81] has numerically shown reduction of timing shift in periodic DM transmission system using split-step Fourier method. Soliton collisions in strong DM WDM system has been theoretically studied showing mitigation of residual frequency shift with map strength [82]. Kaup et al. [83] has developed a perturbation theory to analyze collision-induced position shift and frequency shift. Wald et al. has [84] analyzed the complete and incomplete collisions between DM solitons and reported their impact on frequency shift with map strength in WDM system using variational approximation. Interactions between DM solitons have been numerically investigated characterizing complete and incomplete collisions under various design parameters [85]. Sugahara et al. [86] has theoretically studied collision-induced timing jitter with different dispersion management. Sinkin et al. [87] has used a time shift function to calculate timing shift and frequency shift in quasi-linear systems. In this section, we investigate the fundamental mechanism of collision-induced phase shift analytically employing

54

4.4 Basic Mechanism of of Collision-Induced 4.2 Analytical Calculation XPM-Induced Phase PhaseShift Shift

variational method and verify it numerically. We also check the effect of frequency shift and initial pulse spacing between channels on phase fluctuations in two-channel WDM system.

Fig. 4.2: Pulse locus, frequency shift and phase shift of propagating pulse experiencing collisions in transmission line following Model (A) for 2ch×10 Gb/s system. Solid, dashed and dotted curves represent the results obtained by variational analysis. Crosses are obtained by direct numerical calculation of Eq. (4.1).

55

4.3 Analytical Calculation of XPM-induced Phase 4.4 Basic Mechanism of Collision-Induced Phase Shift Shift

ch1

ch2 40ps

40ps FWHM

150ps 10Gb/s+10Gb/s ch1

ch2 10ps

40ps

150ps

Fig. 4.3. Sum of pulse widths (FWHM) of two pulses of two channels at the initial position.

In Fig. 4.2, pulse loci of both channels, sum of pulse widths, frequency shift, and phase shift as outlined in Eqs. (4.19) and (4.20) are plotted against transmission distance for two dispersion compensation periods assuming Model (A) in which both channels are operated at 10 Gb/s. Pulse widths (FWHM) of both pulses are added along the transmission line to clearly exhibit the collision state (C.S.) as shown in the top of Fig. 4.2. In this work, the collision state is defined by the pulse spacing being shorter than the sum of pulse widths. The upper figure in Fig. 4.3 shows the initial allocation of two pulses for two channels. In this case, the sum of pulse widths at the initial allocation of two inter-channel pulses is 80 ps. The initial pulse spacing (∆T) between two launched pulses into the two channels is −150 ps, i.e., channel 2 is delayed by 150 ps at the input of fiber. The total fiber dispersion is perfectly compensated at the end of each period so that there is no residual dispersion at the receiving end. Pulse locus of channel 2 is relatively zigzag to that of channel 1. Complete collisions [83-86] occur between inter-channel pulses twice in each period. The frequency shift (∆κ)

varies alternately in SMF and DCF of each period and there is no deviation of ∆κ at the end of each period. From the third and fourth figures of Fig. 4.2, it is found that phase shift takes place in every collision and it increases monotonically with transmission distance. From the third figure, we observe that the total phase shift stems from δφθ , and the other components, i.e, δφC and δφκ have negligible effect. From the fourth figure, it is also evident that XPM causes dominant phase alteration, whereas, dispersion and SPM-induced phase changes are much smaller. If phase shift occurred in one

56

4.2 Analytical Calculationofof XPM-Induced Phase Shift 4.4 Basic Mechanism Collision-Induced Phase Shift

period is ∆φ, after N spans the total phase shift could be predicted as N×∆φ. We also plot the phase shift induced by intra-channel XPM (δθIXPM) against transmission distance and find that it is much smaller compared to inter-channel XPM (δθXPM). We use the same dynamical equations to calculate δθIXPM between two-pulses in a single channel.

Fig. 4.4: Pulse locus, frequency shift and phase shift of propagating pulse experiencing collisions in transmission line following Model (A) for the combined system of 40 Gb/s and 10 Gb/s. Solid, dashed and dotted curves represent the results obtained by variational analysis. Crosses are obtained by direct numerical calculation of Eq. (4.1).

Fig. 4.4 shows the pulse loci, sum of pulse widths, frequency and phase fluctuations of the combined system of 40 Gb/s and 10 Gb/s for the same ∆T of −150 ps. The lower figure in Fig. 4.3

4.3Effect Analytical Calculation of XPM-induced Phase Shift 4.5 of Initial Pulse Spacing Between Channels

57

shows the initial allocation of two pulses for two channels. In this case, the sum of pulse widths at the initial allocation of two inter-channel pulses is 50 ps. We notice incomplete collisions between interchannel pulses [83-86], i.e., one pulse does not go through the second pulse completely like the previous case. Hence accumulated deviation of ∆κ is occurred which is increasing with transmission distance. Phase shift also increases with transmission distance in both channels and small phase fluctuation is obtained in dispersion-induced part due to accumulation of ∆κ as predicted in Eq. (4.16). The variational results for phase fluctuation are well supported by numerical calculations.

4.5 Effect of Initial Pulse Spacing Between Channels

Fig. 4.5: Phase shift after propagation of 2012.4 km versus initial pulse spacing (∆T) between two launched channels for 2ch×10 Gb/s system. Solid, dashed and dash-dotted curves represent the results obtained by variational analysis. Plus, circles and triangles display the numerical results.

In the previous section, we have taken ∆T as −150 ps, but in practice this will not remain fixed. In WDM system, the initial pulse spacing between launched pulses of different channels will vary. In this section, we investigate the effect of this initial pulse spacing on phase shift induced by XPM. We vary ∆T and plot the phase shift for all the three models considering full dispersion compensation for the total 40 spans operating at 10 Gb/s with a channel spacing of 50 GHz which is shown in Fig. 4.5. Eq. (4.18) perceptibly implies that phase shift stimulated by XPM depends on ∆T. The above figure correctly shows that phase shift is sensitive to ∆T and every model experiences higher XPM effect for a certain range of ∆T. All the three models observe almost the same amount of maximum phase shift. In all the cases, numerical results show acceptable agreement with analytical prediction obtained by variational method.

58

4.2 Analytical of XPM-Induced Shift 4.6 Effect ofCalculation Channel Spacing and ResidualPhase Dispersion

4.6 Effect of Channel Spacing and Residual Dispersion We examine the effect of channel spacing on phase fluctuation stimulated by XPM for the three models. We calculate the phase shift for different channel spacing after transmission of 40 spans considering 2ch×10 Gb/s system and plot the results in Fig. 4.6. The ∆T is chosen at which the phase shift has the maximum value for each channel spacing. All the models show almost the same phase deviation and it decreases with larger channel spacing. This is expected since larger channel spacing yields shorter collision state. We can approximately predict that maximum phase shift is inversely proportional to channel spacing.

Fig. 4.6: Maximum phase shift after propagation of 2012.4 km as a function of channel spacing for 2ch×10 Gb/s system. Solid, dashed and dotted curves represent the results obtained by variational analysis. Plus, circles and triangles show the numerical results.

So far we have considered the perfect dispersion compensation at the end of each period, i.e., there is no residual dispersion at the receiver end for all three models. Actually there is always some residual dispersion in practical system. Furthermore, in WDM system, full compensation might be achieved for a single channel. For other channels, there is some residual dispersion depending on their separation from the zero-dispersion wavelength channel. In this section, we also check the impact of residual dispersion on phase fluctuation for Model (A) which is a conventional post-compensation map. The amount of residual dispersion per span is varied up to ±30 ps/nm by changing the dispersion of DCF. The ∆T is chosen at which the phase shift has the maximum value for each residual dispersion per span. Calculations for 40 spans are carried out for both bit-rate systems for 50 GHz spacing. Phase

4.6 Effect of Channel Spacingofand Residual Dispersion 4.3 Analytical Calculation XPM-induced Phase Shift

59

shift, pulse width (FWHM), and the worst initial pulse spacing are plotted against residual dispersion per span in Fig. 4.7. The result suggests that some amount of (positive or negative) residual dispersion could be effective to suppress XPM. Maintaining residual dispersion is better than perfect dispersion compensation concerning XPM. The walk-off is an important parameter to determine XPM. The walkoff can be defined as the number of bit periods two adjacent channels are separated from each other due to accumulated dispersion. Using the expression of walk-off [54], woff = Dacc × ∆λ T0 , where Dacc is accumulated dispersion, ∆λ is channel spacing and T0 is bit period, we find that due to residual dispersion per span, walk-off increases with

accumulated dispersion and XPM is suppressed.

However, if higher amount of residual dispersion is used, pulse width broadens drastically, particularly in the high bit rate system as shown in Fig. 4.7. So optimization of residual dispersion and pulse width is necessary in order to achieve reduction of phase deviation caused by XPM.

Fig. 4.7: Effect of residual dispersion on phase shift, pulse width (FWHM) and initial pulse spacing (∆T). For both cases, Model (A) is taken into account and phase shift and pulse width are observed in channel 1 after propagation of 40 spans. The asterisks and circles display the numerical results.

4.7 Conclusion This chapter has theoretically studied the fundamental mechanism of collision-induced frequency shift and phase shift in a two-channel WDM system. Therefore, collision-induced phase fluctuation is estimated for 50 GHz channel spacing for periodic dispersion maps using highly dispersive fibers operating at 10 Gb/s. A hybrid transmission of 40 Gb/s and 10 Gb/s is also investigated. RZ pulse is

4.7 Conclusions 4.2 Analytical Calculation of XPM-Induced Phase Shift

60

considered with 40% duty cycle. XPM is the prime impairment that causes phase deviation which should be noticed carefully in devising phase modulated optical fiber multi-span dispersion-managed WDM transmission systems. XPM-induced phase fluctuation is found to be highly dependent on the initial pulse spacing between the launched signals for all the dispersion compensation models. The analytical estimation supported by numerical simulations reveals that XPM increases with transmission distance depending on initial pulse spacing and it affects every dispersion model. However, phase fluctuations can be appreciably mitigated by properly choosing channel spacing and/or residual dispersion. This chapter shows the results for a two-channel WDM transmission system. For multi-channel WDM system, it could be predicted that those channels will suffer from XPM effect much which will be positioned around zero-dispersion wavelength having very low residual dispersion. Furthermore, multi-channel system with smaller channel spacing might be affected seriously due to XPM.

Chapter 5 Conclusions This thesis has been devoted to the investigation of phase fluctuations in dispersion-managed optical fiber transmission systems in order to obtain long-haul high capacity optical communication networks employing phase modulation formats. In this study, we have focused on the impact of periodic dispersion management on the phase jitter in DM soliton and quasi-linear pulse transmission systems considering amplifier noise, fiber nonlinearity, mainly SPM, and chromatic dispersion. We have also explored the XPM-induced phase fluctuations in DM WDM systems. Here we summarize the results which we have achieved from this study and make some suggestions for future works. In Chapter 2, the fundamental equations and theories of optical pulse transmission in DM line has been introduced. Some features of DM soliton pulse have been studied considering a Gaussian solution of NLS equation. A set of ordinary differential equations for pulse dynamics in DM line have been evaluated. We have obtained a periodic variation of pulse chirp and width, both for loss less and lossy fibers, with respect to transmission distance with the period of DM map period. We have found that stronger DM map yields larger pulse energy for soliton transmission which play an important role in G-H timing jitter, pulse-to-pulse interactions etc. The phase characteristic of DM soliton has also been checked in absence of amplifier noise. In Chapter 3, Phase jitter has been estimated for long-haul optical fiber transmission systems by evaluating the analytical model based on variational method and the results are verified by numerical simulations using split-step Fourier algorithm. Firstly, nonlinear phase noise is ascertained for two different periodic DM models and those models are compared to a conventional constant dispersion soliton-based system. The results show that the reduction of phase noise is possible by appropriately choosing the dispersion management. Furthermore, it can be concluded that nonlinear phase noise increases for stronger DM lines. Secondly, applying the same procedures we estimate the phase jitter for quasi-linear pulse transmission in different DM models. In this case, the results suggest that the

61

62

Chapter 5: Conclusions

stronger dispersion management can be more efficient to suppress phase jitter. We also evaluate the effect of phase jitter in DM quasi-linear systems by altering the fiber dispersion and the length ratio of fibers composing the DM period. Furthermore, we have proposed upgraded models for quasi-linear systems which offer lower phase noise and higher Q-factor. The implications of these results in obtaining DM soliton or quasi-linear pulse based phase modulated systems are substantial. Future effort may include a detailed analysis of phase jitter on WDM systems with a single or multiple pulses taking into account intra and inter-channel effects. After evaluating the probability density function of phase noise properly, BER performance of phase-modulated DM systems can be calculated accurately considering other linear/nonlinear noise. Further study can also include experimental investigation of phase noise in those systems we have studied. In Chapter 4, we have illustrated the basic mechanism of phase fluctuation due to XPM in a periodically DM two-channel WDM system. We have envisaged two different bit rate systems: firstly both channels have been operated at 10 Gb/s and secondly the hybrid bit rate system in which one channel carries pulse corresponding to 40 Gb/s signal and other channel carries pulse corresponding to 10 Gb/s signal. XPM-induced phase shift has been found to be dependent on initial pulse spacing between channels. Different transmission models were investigated to observe the phase shift behavior. All the models have offered higher phase shift for a certain range of initial pulse spacing. Next reduction of phase shift has been achieved by adjusting the channel spacing and residual dispersion per span. Future study may include data transmission in more than two channels to attain more realistic results including intra-channel nonlinear effects. Experiments can be adopted to verify the results obtained analytically and numerically. From the results obtained, it can be concluded that phase fluctuations due to fiber nonlinearity can be a major limiting factor in the way to realize long-haul high speed phase modulated transmission systems. However, these impairments could be mitigated by properly designing DM transmission systems along with consideration of other design factors like amplifier spacing, channel spacing, and residual dispersion. In all cases, our results obtained by variational analysis are validated by numerical simulations.

Appendix A A.1 Basic Equation of Optical Pulse Propagation in Optical Fibers We derive the basic equation that governs the propagation of optical pulse in a dispersive nonlinear medium like optical fiber. Optical signal is an electromagnetic wave whose electric field vector, E and magnetic field vector, H are governed by Maxwell’s equations as ∇×E = − ∇×H =

∂B , ∂t

∂D

∂t ∇ ⋅ D = 0,

(A.1) (A.2)

,

(A.3) (A.4)

∇ ⋅ B = 0,

The medium has no free charges or electric current. Here, D and B are the electric and magnetic flux densities, respectively, which are related to E and H by the following relations (A.5) (A.6)

D = ε 0 E, B = µ 0 H,

where, ε 0 and µ 0 are the permittivity and permeability of free space, respectively. By taking curl of Eq. (A.1) and using Eqs. (A.2), (A.5) and (A.6), we obtain ∇ × ∇ × E = − µ 0ε 0

∂ 2E . ∂t 2

(A.7)

We can define the Fourier transform of the electric field as ∞ E = ∫ − ∞ E exp(iω t ) dt.

(A.8)

We transform the Eq. (A.7) in Fourier domain and find

63

64

Appendix A ~ ~ ∇ × ∇ × E = ω 2 µ 0ε 0 E.

(A.9)

~ ~ ~ Using the identity ∇ × ∇ × E = ∇(∇ ⋅ E ) − ∇ 2 E and Eq. (A.3), we can derive

~ ~ − ∇ 2 E = ω 2 µ 0 ε 0 E.

(A.10)

We now consider the fiber medium with refractive index of n. Assuming the single polarization toward x and propagation direction towards z, the above equation can be written as ~ ∂ 2E ~ + β 2 E = 0, 2 ∂z

(A.11)

where β = nω c is the mode propagation constant and c is the velocity of light in free space and is given by c = 1

ε0 µ0 . Let E be represented as the following form

Ε (z, t ) = F (z, t ) exp[i (β 0 z − ω0 t )].

(A.12)

In Fourier domain, we can write ∞ ~ E (z, ω − ω0 ) = ∫ Ε (z , t ) exp[i (ω − ω0 )t ] dt , −∞

(A.13)

which is found to satisfy the following equation ~ ∂2E 2 ~ + n 2 ω, E β 02 E = 0. ∂z 2

(

)

(A.14)

To solve this equation, we can assume the following form

~ ~ E (z , ω − ω0 ) = F (z , ω − ω0 ) exp(iβ 0 z ),

(A.15)

where F~ (z , ω − ω0 ) is a slowly varying function of z and β 0 is wave number and is given as ~ β 0 = ω c . Eq. (A.14) leads to the following equation for F (z , ω − ω0 ) : 2iβ 0

~ ∂F ~ + β 2 − β 02 F = 0. ∂z

(

)

(A.16)

Here we neglect the term ∂ 2 F~ ∂z 2 as F~ (z , ω − ω0 ) is assumed to be a slowly varying function of z. For β − β 0 << 1 , we can approximate β 2 − β 02 = ( β + β 0 ) ( β − β 0 ) ≈ 2β 0 (β − β 0 ) . The equation is transformed to i

~ ∂F ~ + (β − β 0 )F = 0. ∂z

(A.17)

65 Appendix A Since fiber refractive index n is a function of optical wave frequency ω (due to fiber dispersion) and field intensity (due to fiber nonlinearity) as mentioned in Eq. (A.14), β can be expressed as

β=

ω c

(

n ω, F

2

)

ω ⎧⎪

n F = ⎨n0 (ω ) + 2 c ⎪⎩ Aeff

2

⎫⎪ ⎬ + ig , ⎪⎭

(A.18)

where, n0(ω) is the linear refractive index of real part of fiber index and n2 is intensity dependent refractive index which is a measure of fiber nonlinearity. The imaginary part of β represents the decay of optical signal along the fiber and is related to the fiber loss coefficient α as g = α 2 . The inverse Fourier transform of Eq. (A.17) yields the propagation equation of F (z , t ) . For this reason, the linear term of real part of β can be expanded in a Taylor series about the carrier frequency ω0 like 1 2

1 6

β = β 0 + β1 (ω − ω0 ) + β 2 (ω − ω0 )2 + β 3 (ω − ω0 )3 + " + m where β = d β m dω m

ω0 n2 F

2

cAeff

(A.19)

+ ig ,

. The third and higher order terms are negligible if the spectral ω =ω0

width ∆ω << ω0 . Considering up to the second term and after substituting Eq. (A.19) in Eq. (A.17), we obtain 2 ~ ⎫⎪ ~ ω0 n2 F ∂F ⎧⎪ 1 2 i + ⎨β1 (ω − ω0 ) + β 2 (ω − ω0 ) + + ig ⎬ F = 0. cAeff ∂z ⎪⎩ 2 ⎪⎭

(A.20)

We take inverse Fourier transform of this equation by using F (z , t ) =

1 2π

~

∫ − ∞ F (z,ω − ω0 )exp[− i(ω − ω0 )t ]dω. ∞

(A.21)

In Fourier transform process, ω − ω0 is replaced by the differential operator i ∂ , and we then obtain ∂F ⎞ 1 ∂2F 2 ⎛ ∂F i⎜ + β1 − β + γ F F = −igF , ⎟ 2 2 ∂ z ∂ t 2 ∂ t ⎝ ⎠

∂t

(A.22)

where γ is termed as the nonlinearity coefficient and is defined as γ = ω0 n2 cAeff . This is the basic propagation equation of optical pulse in a dielectric medium like fiber.

66

Appendix A

A.2 Variational Analysis The detailed analysis of Lagrangian formulation and derivation of variational equations for optical pulse parameters are presented here. The basic equation which describes the optical pulse propagation with perturbation in fiber is given by i

∂U b (Z ) ∂ 2U 2 − + S (Z ) U U = R ( Z , T ) . 2 2 ∂T ∂Z

(A.23)

The description of its symbols and derivation are presented in Chapter 2 and the aforementioned section. We know that this equation is not integrable because of varying coefficients S(Z) and b(Z) and perturbation term R. That’s why, we apply Lagrangian variational method to simplify the equation and find an approximate solution assuming a known function. Applying the variational principle which is called the principle of stationary action or principle of least action to the Lagrangian, we can write for infinite dimensional case

δ S = δ ∫ −∞ ∫ −∞ L (U ,U * ,U Z ,U Z* ,U T ,U T* ) dTdZ = 0, ∞

∞

(A.24)

where δ S is the variation of action, L is the Lagrangian density which is a function of U ,U * and their derivative with respect to Z and T (the subscript denotes the partial derivative). Now we allow small arbitrary variations in the U and we obtain ∞

∞

⎛ ∂L ⎞ ∂L ∂L ∂L ∂L ∂L δU + δU * + δU Z + δU Z* + δU T + δU T* ⎟ dTdZ = 0, * * * ∂U ∂U Z ∂U Z ∂U T ∂U T ⎝ ∂U ⎠

δ S = ∫ −∞ ∫ −∞ ⎜ ∞

∞

⎛ ∂L

⎞

∞

∞

⎛ ∂L

∞

∞

∫ −∞ ∫ −∞ ⎝⎜ ∂U δ U ⎠⎟ dTdZ + ∫ −∞ ∫ −∞ ⎝⎜ ∂U * δ U +∫

∞

∞

⎛ ∂L

⎞

*

⎛ ∂L

δ U Z ⎟ dTdZ + ∫ −∞ ∫ −∞ ⎜ −∞ ∫ −∞ ⎜ ∂U * ∂U *

∞ ∞ ⎛ ∂L ⎞ ⎞ δ U Z ⎟ dTdZ ⎟ dTdZ + ∫ −∞ ∫ −∞ ⎜ ⎠ ⎝ ∂U Z ⎠

⎞

∞

∞

⎛ ∂L ⎞ δ U T* ⎟ dTdZ = 0, * ⎝ ∂U T ⎠

δ U T ⎟ dTdZ + ∫ −∞ ∫ −∞ ⎜

⎝ T ⎠ ⎝ Z ⎠ ∞ ∞ ∂L ∞ ∞ ∂L ∞ ∞ ∂L ∂ (δ U ) * ∫ −∞ ∫ −∞ ∂U δ UdTdZ + ∫ −∞ ∫ −∞ ∂U * δ U dTdZ + ∫ −∞ ∫ −∞ ∂U ∂Z dTdZ Z

(

)

(

)

* * ∞ ∞ ∂L ∂ ( δ U ) ∞ ∞ ∂L ∂ δ U ∂L ∂ δ U + ∫ −∞ ∫ −∞ dTdZ + ∫ −∞ ∫ −∞ dTdZ + ∫ −∞ ∫ −∞ dTdZ = 0, ∂U T ∂T ∂U Z* ∂Z ∂U T* ∂T (A.25) Using the rule of integration by parts, we get ∞

∞

∞

∞ ∂ ⎛ ∂L ⎡ ∂L ⎤ ∂L ∂ (δ U ) ∫ −∞ ∂U ∂Z dZ = ⎢ ∂U δU ⎥ − ∫ −∞ ∂Z ⎜ ∂U ⎣ Z ⎦ −∞ ⎝ Z Z ∞

= −∫

∞ −∞

∂ ⎛ ∂L ⎜ ∂ Z ⎝ ∂U Z

⎞ ⎟ δUdZ , ⎠

⎞ ⎟ δUdZ ⎠

(A.26)

67 Appendix A ∞

∞ ∂ ⎛ ∂L ⎡ ∂L ⎤ ∂L ∂ (δU ) ∫ −∞ ∂U ∂T dT = ⎢ ∂U δU ⎥ − ∫ − ∞ ∂T ⎜ ∂U ⎣ T ⎦ −∞ ⎝ T T ∞

= −∫

∂ −∞ ∂T ∞

⎛ ∂L ⎜ ⎝ ∂U T

⎞ ⎟ δUdT ⎠

⎞ ⎟ δ UdT . ⎠

(A.27)

Similar derivations for their conjugates can be achieved. The integrated terms for δU and

δ U * vanish since the variations of U and U * at the end-points are defined as zero. Eq. (A.25) is then transformed into ⎛ ∂L ⎞ ∂ ⎛ ∂L ⎞ ⎫ ⎜ ⎟− ⎜ ⎟ ⎬ δ UdTdZ ⎝ ∂U Z ⎠ ∂T ⎝ ∂U T ⎠ ⎭ ⎩ ⎧ ∂L ∞ ∞ ⎪ ∂ ⎛ ∂L ⎞ ∂ ⎛ ∂L ⎞ ⎪⎫ * + ∫ −∞ ∫ −∞ ⎨ * − ⎜ ⎟− ⎜ ⎟ ⎬ δ U dTdZ = 0. ∂Z ⎝ ∂U Z* ⎠ ∂T ⎝ ∂U T* ⎠ ⎪⎭ ⎪⎩ ∂U ∞

∞

⎧ ∂L

∫ −∞ ∫ −∞ ⎨ ∂U

−

∂ ∂Z

(A.28)

Since δU and δU * are arbitrary and independent from each other, the above equation becomes ∂ L ∂ ⎛ ∂ L ⎞ ∂ ⎛ ∂L ⎞ − ⎜ ⎟− ⎜ ⎟ = 0, ∂ U ∂ Z ⎝ ∂U Z ⎠ ∂T ⎝ ∂U T ⎠ ∂L ∂ ⎛ ∂L ⎞ ∂ ⎛ ∂L ⎞ − ⎜ ⎟− ⎜ ⎟ = 0. * ∂U ∂Z ⎝ ∂U Z* ⎠ ∂T ⎝ ∂U T* ⎠

(A.29) (A.30)

In Lagrangian formulation of classical mechanics, these equations are termed as Euler-Lagrange equation in case of infinite dimension. The classical equations of motion in Lagrangian dynamics are given by these equations. We can generalize these equations like the form ∂ L ∂ ⎛ ∂L − ⎜ ∂K ∂Z ⎝ ∂K Z

⎞ ∂ ⎛ ∂L ⎟− ⎜ ⎠ ∂T ⎝ ∂K T

⎞ ⎟ = 0, ⎠

(A.31)

where K is a generalized coordinate which indicates U or U * . The Lagrangian density for the basic equation of optical pulse propagation in fiber given by A.23 can be obtained by Eq. (A.31) with K = U * as L=

b(Z ) S (Z ) 4 i 2 U ZU * − U Z*U + UT + U − U * R + UR* . 2 2 2

(

)

(

)

(A.32)

Now we derive the dynamical equations of optical pulse propagation in fiber that we described in Chapter 2. We assume the following ansatz

U ( Z , T ) = A ( Z ) f (τ ) exp ( iϕ ) ,

with τ ( Z , T ) = p ( Z ) {T − T0 ( Z )} , and

68

Appendix A C (Z ) 2 κ (Z ) τ − τ +θ ( Z ). 2 p(Z )

ϕ ( Z ,T ) =

(A.33)

Employing this ansatz, we deduce the Lagrangian function L by integrating the Lagrangian density as given in Eq. (A.32) as L=∫

∞ −∞

L dT

b(Z ) S (Z ) 4 ⎧i ⎫ 2 * * UT + U − U *R + UR* ⎬ dT ⎨ U ZU − U ZU + 2 2 ⎩2 ⎭ 4 2 S (Z ) A A ⎧ ⎛ 1 dC C dp ⎞ ⎛ dT dθ ⎞ ⎫ =− + + IL ⎜κ 0 + ⎨ IC ⎜ ⎟⎬ + I N ⎟ 2p p ⎩ ⎝ 2 dZ p dZ ⎠ ⎝ dZ dZ ⎠ ⎭ =∫

+

(

∞

−∞

)

(

)

b(Z ) 2 ⎛ κ2 ⎞ pA ⎜ I D + IC C 2 + I L 2 ⎟ + N , 2 p ⎠ ⎝

(A.34)

where I C , I D , I L , and I N are real constants defined by

IC = ∫

τ 2 f 2 dτ , I D = ∫ ⎛⎜ −∞ −∞ ∞

∞

2

∞ ∞ df ⎞ 2 4 ⎟ d τ , I L = ∫ − ∞ f dτ , I N = ∫ − ∞ f d τ , ⎝ dτ ⎠

(A.35)

and N is the noise term defined by

N =−∫

∞

(U R + UR )dT . *

−∞

*

(A.36)

We find that Lagrangian L is a function of x and its derivative with respect to Z, where x is any one of the parameters defined in Eq (A.33). The variation of action δ S can be written as ∞

δ S = δ ∫ −∞ L ( x, xZ ; Z ) dZ = 0.

(A.37)

This gives ∞

⎛ ∂L ⎞ ∂L δx+ ∂xZ ⎟dZ = 0. ∂xZ ⎝ ∂x ⎠

δ S = ∫ −∞ ⎜

(A.38)

Using integration by parts, we obtain ∞

⎧ ∂L d ⎛ ∂L ⎞ ⎫ − ⎜ ⎟ ⎬δ xdZ = 0. ⎩ ∂x dZ ⎝ ∂x ⎠ ⎭

δ S = ∫ −∞ ⎨

(A.39)

As a result we find the well-known Euler-Lagrange equation ∂L d ⎛ ∂L ⎞ − ⎜ ⎟ = 0. ∂x dZ ⎝ ∂xZ ⎠

(A.40)

69 Appendix A Applying Eq. (A.34) to the above Euler-Lagrange equation, we find: I. When x = A ,

∂ L d ⎛ ∂L ⎞ − ⎜ ⎟=0 ∂A dZ ⎝ ∂AZ ⎠ gives 3 2 A ⎧ ⎛ 1 dC C dp ⎞ ⎛ dT0 dθ ⎞ ⎫ 2 I N S ( Z ) A κ I I − + + + + ⎨ C ⎟⎬ L⎜ p ⎩ ⎜⎝ 2 dZ p dZ ⎟⎠ p ⎝ dZ dZ ⎠ ⎭ ⎛ κ2 ⎞ +b ( Z ) Ap ⎜ I D + I C C 2 + I L 2 ⎟ + N A = 0, p ⎠ ⎝ b(Z ) 2 ⎛ ⎛ C dp 1 dC ⎞ κ2 ⎞ p ⎛ dT0 dθ ⎞ 2 2 κ IC ⎜ I I S Z A p I I C I N A, + + + − − + + ( ) ⎜ D C ⎟= L ⎟ L ⎜ dZ dZ ⎟ N 2 p2 ⎠ 2 A ⎝ ⎠ ⎝ p dZ 2 dZ ⎠ ⎝ (A.41) where, NA = − ∫

⎛ ∂U * ∂U * ⎞ R+ R ⎟dT . − ∞ ⎜ ∂A ∂A ⎠ ⎝ ∞

(A.42)

II. When x = p , ∂ L d ⎛ ∂L ⎞ − ⎜ ⎟=0 ∂A dZ ⎝ ∂pZ ⎠ gives S ( Z ) A2 b(Z ) 2 ⎛ κ2 ⎞ p2 ⎛ dT0 dθ ⎞ ⎛ 2C dA 3 dC ⎞ 2 IC ⎜ I I κ + + + − + + − = − p I I C I Np, ⎟ L⎜ ⎟ N ⎜ D C ⎟ L 2 2 p2 ⎠ A2 ⎝ A dZ 2 dZ ⎠ ⎝ dZ dZ ⎠ ⎝ (A.43) where, * ∞ ⎛ ∂U ∂U * ⎞ (A.44) Np = − ∫ ⎜ R+ R ⎟dT . −∞ ∂ ∂ p p ⎝ ⎠

III. When x = C , ∂L d ⎛ ∂L ⎞ − ⎜ ⎟=0 ∂C dZ ⎝ ∂CZ ⎠ gives ⎛ A dA 3 A2 dp ⎞ IC ⎜ − + − b ( Z ) A2 pC ⎟ = N C , 2 ⎝ p dZ 2 p dZ ⎠ where, * ∞ ⎛ ∂U ∂U * ⎞ NC = − ∫ ⎜ R+ R ⎟dT . −∞ ∂C ⎠ ⎝ ∂C IV. When x = κ ,

(A.45)

(A.46)

70

Appendix A

∂L d ⎛ ∂L ⎞ − ⎜ ⎟=0 ∂κ dZ ⎝ ∂κ Z ⎠ gives dT p −b ( Z ) κ + 0 = 2 N κ , dZ A I L where, * ∞ ⎛ ∂U ∂U * ⎞ Nκ = − ∫ ⎜ R+ R ⎟dT . −∞ ∂κ ⎝ ∂κ ⎠

(A.47)

(A.48)

V. When x = T0 , ∂ L d ⎛ ∂L ⎞ − ⎜ ⎟=0 ∂T0 dZ ⎝ ∂T0 Z ⎠ gives ⎛ 2 dA 1 dp ⎞ dκ p − + =− κ⎜ NT , ⎟ I L A2 0 ⎝ A dZ p dZ ⎠ dZ where, * ∞ ⎛ ∂U ∂U * ⎞ NT0 = − ∫ ⎜ R+ R ⎟dT . − ∞ ∂T ∂T0 ⎠ ⎝ 0

(A.49)

(A.50)

VI. When x = θ , ∂L d ⎛ ∂ L ⎞ − ⎜ ⎟=0 ∂θ dZ ⎝ ∂θ ⎠ gives 2 dA 1 dp p − =− Nθ , A dZ p dZ I L A2 where, * ∞ ⎛ ∂U ∂U * ⎞ Nθ = − ∫ ⎜ R+ R ⎟dT . −∞ ∂θ ⎝ ∂θ ⎠

(A.51)

(A.52)

Solving the coupled equations (A.41), (A.43), (A.45), (A.47), (A.49), and (A.51), the dynamics of the six pulse parameters under the perturbation can be determined as follows: dA b ( Z ) = A p 2 C + RA , dZ 2 dp = b( Z ) p 3C + R p , dZ ⎛I ⎞ I dC = −b( Z ) p 2 ⎜ D + C 2 ⎟ − N S ( Z ) A2 + RC , dZ ⎝ IC ⎠ 2 IC dκ = Rκ , dZ dT0 = b( Z ) κ + RT0 , dZ

(A.53) (A.54) (A.55) (A.56) (A.57)

71 Appendix A ⎛κ2 I ⎞ 5I dθ = − b( Z ) ⎜ − D p 2 ⎟ + N S ( Z ) A2 + Rθ , 2 dZ I ⎝ ⎠ 4IL L

(A.58)

where, ⎞, ⎟ ⎠ 2 N ⎞ p ⎛N Rp = 2 ⎜ C − θ ⎟ , A ⎝ IC 2 I L ⎠ p ⎛ CNθ N A pN p ⎞ , − − RC = ⎜ ⎟ A ⎝ AI L 2 I C AI C ⎠ p Rκ = 2 κ Nθ − NT0 , A IL pN RT0 = 2 κ , A IL

RA =

p ⎛ N C 3 Nθ − ⎜ 2 A ⎝ IC 2 I L

(

Rθ = −

(A.59) (A.60) (A.61)

)

(A.62) (A.63)

p ⎛ 2CN C 2κ Nκ 3N A pN p ⎞ . + − − 2 AI L ⎜⎝ A 2 A A ⎟⎠

(A.64)

Assuming the Gaussian trial function, i.e., f (τ ) = exp(− τ 2 2 ) , the real constant IC , I D , I L , and I N are calculated as

IC =

π, π , π . IL = π , IN = ID =

2

2

2

(A.65)

Applying the above real constants and using the Eqs. (A.59) - (A.64), the Eqs. (A.53) - (A.58) can be deduced as dA b ( Z ) = A p 2 C + RA , dZ 2 dp = b( Z ) p 3C + R p , dZ S (Z ) 2 dC = −b ( Z ) p 2 1 + C 2 − A + RC , dZ 2 dκ = Rκ , dZ dT0 = b( Z ) κ + RT0 , dZ b( Z ) 2 5S (Z ) 2 dθ A + Rθ , =− κ − p2 + dZ 2 4 2

(

(

)

)

(A.66) (A.67) (A.68) (A.69) (A.70) (A.71)

where, ⎛ τ2 ⎞ , −iϕ 2 ⎡ ⎤ R e Im 3 2 τ exp − ⎜ − ⎟ dτ ∫ ⎦ 2 π −∞ ⎣ ⎝ 2⎠ ∞ ⎛ τ2 ⎞ , p 2 − iϕ ⎡ ⎤ − Im 1 2 exp Rp = R e τ ⎜ − ⎟ dτ ⎦ π A ∫ −∞ ⎣ ⎝ 2⎠ RA =

1

∞

(

)

(A.72)

(

)

(A.73)

72

Appendix A ⎛ τ2 ⎞ , 2 − iϕ ⎦⎤ − C Im ⎣⎡ R e ⎦⎤ 1 − 2τ exp ⎜ − 2 ⎟ dτ ⎝ ⎠ 2 ⎛ τ ⎞ Re ⎡⎣ R e−iϕ ⎤⎦ − C Im ⎡⎣ R e−iϕ ⎤⎦ τ exp ⎜ − ⎟ dτ , ⎝ 2⎠

RC =

{Re ⎣⎡ R e π A ∫ −∞

Rκ =

2p ∞ π A ∫ −∞

2

RT0 = Rθ = −

∞

}(

− iϕ

{

)

}

∞ ⎛ τ2 ⎞ 2 Im ⎡⎣ R e−iϕ ⎤⎦ τ exp ⎜ − ⎟ dτ , ∫ π Ap −∞ ⎝ 2⎠

{ p Re ⎡⎣ R e π Ap ∫ −∞ 1

2

∞

− iϕ

)

(A.75) (A.76)

⎛ τ2 ⎞ ⎤⎦ 3 − 2τ 2 + 4κ Im ⎡⎣ R e−iϕ ⎤⎦ τ exp ⎜ − ⎟ dτ . ⎝ 2⎠

(

(A.74)

}

(A.77)

Here, Re ⎡ R e−iϕ ⎤ and Im ⎡ R e−iϕ ⎤ represent the real and imaginary parts of R e−iϕ , respectively. ⎣ ⎦ ⎣ ⎦

Appendix B B.1 Derivation of Ordinary Differential Equations of Auto and Cross-Correlations of Pulse Parameters As described in Chapter 3, the perturbation term which models the amplifier ASE noise generated at m-th amplifier can be expressed by

R = {nmR ( Z , T ) + inmI ( Z , T )} eiϕ ,

(B.1)

where nmR and nmI are real random functions. To model the effect of noise in pulse parameters, we make linearization by using x( Z ) = x0 ( Z ) + δ x(Z ) , where x0 represents any of the six pulse parameters in absence of noise and δx is a small noise induced part of the related parameter. We utilize the linearization technique to treat the ordinary differential equations derived by variational

(

)

method for the pulse parameters assuming Gaussian pulse, i.e., f (τ ) = exp − τ 2 2 . Now the approximate solution of Eq. (A.23) without the perturbation becomes

⎡ 1 ⎤ κ U ( Z , T ) = A exp ⎢ − (1 − iC )τ 2 − i τ + iθ ⎥ . p ⎣ 2 ⎦

(B.2)

Considering the above ansatz, Eq. (A.53) can be written without perturbation as

dA0 b ( Z ) = A0 p02C0 . 2 dZ

(B.3)

In presence of ASE noise which is assumed as perturbation, the same equation leads to

d ( A0 + δ A) b ( Z ) 2 = ( A0 + δ A)( p0 + δ p ) ( C0 + δ C ) + ∑ δ RmA . dZ 2 m

(B.4)

Subtracting Eq. (B.3) from Eq. (B.4) and neglecting the higher order terms, we obtain the equation of motion of pulse amplitude under amplifier ASE noise in linearized form,

73

74

Appendix B d (δ A ) b ( Z ) = p0 ( p0C0δ A + 2 A0C0 δ p + A0 p0δ C ) + ∑ δ RmA , dZ 2 m

(B.5)

where ∞ ⎛ τ2 ⎞ 1 2 − iϕ ⎡ ⎤ Im 3 2 τ exp Re − ⎜ − ⎟ dτ ∫ ⎦ 2 π a ( Z ) −∞ ⎣ ⎝ 2⎠ ∞ ⎛ τ2 ⎞ 1 2 Im 3 2 τ exp n in = + − [ ] ⎜ − ⎟ dτ mR mI ∫ 2 π a ( Z ) −∞ ⎝ 2⎠

(

δ RmA =

)

(

=

)

∞ ⎛ τ2 ⎞ . 1 2 − n 3 2 τ exp ⎜ − ⎟ dτ ∫ mI 2 π a ( Z ) −∞ ⎝ 2⎠

(

)

(B.6)

Similarly we can determine the linearized equations for other pulse parameters which are given in Chapter 3. Now we can deduce the auto-correlations (variances) and the cross-correlations of the noiseperturbed part of pulse parameters in the form of ordinary differential equations (ODE) using the Eqs. (3.11) to (3.16). Let’s start with Eq. (3.11)

d (δ A ) b ( Z ) = p0 ( p0C0δ A + 2 A0C0 δ p + A0 p0δ C ) + ∑ δ RmA . dZ 2 m

(B.7)

After integrating over short distance from Z1 to Z2, Eq. (B.7) becomes Z2

δ A ( Z 2 ) = δ A ( Z1 ) + ∫Z

1

+∫

Z2

Z1

Z2 b ( Z ′) 2 p0 C0δ A ( Z ′) dZ ′ + ∫ b ( Z ′) A0 p0C0δ p ( Z ′) dZ ′ Z 1 2

Z2 b ( Z ′) A0 p02δ C ( Z ′) dZ ′ + ∫ δ RmA ( Z ′) dZ ′ . Z 1 2

(B.8)

Since δx, i.e., δA, δp, δC, δκ, δT0 and δθ are the noise induced parts, they possess the random nature and we can derive the expression of their correlations. For the case of δA only, the autocorrelation can be calculated as b ( Z ′) 2 b ( Z ′′) 2 p0 C0 ⋅ p0 C0 δ A ( Z ′ ) δ A ( Z ′′ ) dZ ′dZ ′′ 1 1 2 2 Z2 Z2 + ∫ ∫ b ( Z ′) A0 p0C0 ⋅ b ( Z ′′) A0 p0C0 δ p ( Z ′) δ p ( Z ′′) dZ ′dZ ′′ Z2

δ A2 ( Z 2 ) = δ A2 ( Z1 ) + ∫Z Z1

Z1

+∫

Z2

+∫

Z2

Z1

Z1

Z2

∫Z

1

Z2

∫Z

1

Z2

∫Z

b ( Z ′) b ( Z ′′) A0 p02 ⋅ A0 p02 δ C ( Z ′) δ C ( Z ′′) dZ ′dZ ′′ 2 2 Z2 b ( Z ′ ) 2 δ RmA ( Z ′) δ Rm′A ( Z ′′) dZ ′dZ ′′ + 2 δ A ( Z1 ) ∫Z p0 C0δ A ( Z ′) dZ ′ 1 2

1

Z2 b ( Z ′) 2 p0 C0 δ A ( Z ′) dZ ′∫Z b ( Z ′′) A0 p0C0 δ p ( Z ′′) dZ ′′ 1 2

Z2

Z2

1

1

Z2

+2

∫Z

+2

∫Z b ( Z ′) A0 p0C0 δ p ( Z ′) dZ ′∫Z

b ( Z ′′) A0 p02δ C ( Z ′′) dZ ′′ 2

Appendix B

75

+2

Z2

∫Z

1

Z2 b ( Z ′) A0 p02δ C ( Z ′) dZ ′∫ δ Rm′A ( Z ′′) dZ ′′ Z 1 2 Z2

Z2

Z1

Z1

+2 δ A ( Z1 ) ∫ δ RmA ( Z ′) dZ ′ + 2 δ A ( Z1 ) ∫ b ( Z ′) A0 p0C0 δ p ( Z ′) dZ ′ +2 δ A ( Z1 ) ∫

Z2

Z1

Z2

b ( Z ′) A0 p02δ C ( Z ′) dZ ′ 2

Z2 b ( Z ′′ ) b ( Z ′) 2 p0 C0 δ A ( Z ′) dZ ′∫ A0 p02δ C ( Z ′′) dZ ′′ Z 1 2 2

+2

∫Z

+2

∫Z

+2

∫Z b ( Z ′) A0 p0C0 δ p ( Z ′) dZ ′∫Z

1

Z2 1

Z2 b ( Z ′) 2 p0 C0 δ A ( Z ′) dZ ′∫ δ Rm′A ( Z ′′) dZ ′′ Z1 2

Z2

Z2

1

1

δ Rm′A ( Z ′′) dZ ′′

(B.9)

Since the distance from Z1 to Z2 is short, we can take the approximation δ x ( Z ′) ≈ δ x ( Z1 ) . We also make the following assumptions: Z2

Z2

1

1

Z2

Z2

1

1

∫Z ∫Z

∫Z ∫ Z

δ x ( Z ′) δ x ( Z ′′) dZ ′dZ ′′ = 0, δ x ( Z ′) δ y ( Z ′′) dZ ′dZ ′′ = 0, Z2

δ x ( Z1 ) ∫Z δ y ( Z ′) dZ ′ = 0, 1

Z2

Z2

1

1

∫Z ∫Z

δ x ( Z ′) δ Rm′x ( Z ′′) dZ ′dZ ′′ = 0, Z2

δ x ( Z1 ) ∫Z δ Rmx ( Z ′) dZ ′ = 0,

(B.10)

1

then Eq. (B.9) is transformed into

δ A2 ( Z 2 ) = δ A2 ( Z1 ) + bp02C0 δ A2 ( Z1 ) ( Z 2 − Z1 ) + 2bA0 p0C0 δ Aδ p ( Z1 ) ( Z 2 − Z1 ) +bA0 p02 δ Aδ C ( Z1 ) ( Z 2 − Z1 ) + ∫

Z2

Z1

when m = m′

δ RmA ( Z ′) δ RmA ( Z ′′) =

Z2

∫Z

1

δ RmA ( Z ′) δ Rm′A ( Z ′′) dZ ′dZ ′′,

A0 N m ( Z ) 3 . ∑ 4 E0 m a 2 ( Z ) 2

(B.11)

(B.12)

when m ≠ m′ δ RmA ( Z ′) δ Rm′A ( Z ′′) = 0.

(B.13)

The derivation of this expression (and similar for others) are given in the next section. Now considering zero initial value and taking derivative with respect to Z, Eq. (B.11) can be written as d δ A2 dZ

2 = b ( Z ) p02C0 δ A2 + 2 b ( Z ) A0 p0C0 δ Aδ p + b ( Z ) A0 p02 δ Aδ C + ∑ δ RmA m

76

Appendix B

(

)

= b ( Z ) p0 p0C0 δ A2 + 2 A0C0 δ Aδ p + A0 p0 δ Aδ C +

A N (Z ) 3 ∑ 0 m . 4 E0 m a 2 ( Z ) 2

(B.14)

Let us consider Eqs. (3.11) and (3.12) of Chapter 3, after integrating over short distance from Z1 to Z2, the later equation becomes Z2

Z2

Z2

1

1

1

δ p ( Z 2 ) = δ p ( Z1 ) + 3∫Z b ( Z ′) p02C0δ p ( Z ′) dZ ′ + ∫Z b ( Z ′) p03δ C ( Z ′) dZ ′ + ∫Z δ Rmp ( Z ′) dZ ′ . (B.15) The cross-correlation between δA and δp is derived as Z2

δ Aδ p ( Z 2 ) = δ Aδ p ( Z1 ) + 3 δ A ( Z1 ) ∫Z b ( Z ′) p02C0δ p ( Z ′) dZ ′ 1

Z2

+ δ A ( Z1 ) ∫Z b ( Z ′) p03δ C ( Z ′) dZ ′ 1

Z2

Z2

Z1

Z1

+δ A ( Z1 ) ∫ δ R p ( Z ′) dZ ′ + δ p ( Z1 ) ∫ +3∫

Z2

Z2 b ( Z ′)

∫Z

b ( Z ′) 2 p0 C0δ A ( Z ′) dZ ′ 2

p02C0 ⋅ b ( Z ′′) p02 C0 δ A ( Z ′) δ p ( Z ′′) dZ ′dZ ′′

2 b ( Z ′) 2 p0 C0 ⋅ b ( Z ′′) p03 δ A ( Z ′) δ C ( Z ′′) dZ ′dZ ′′ +∫ ∫ Z1 Z1 2 Z 2 Z2 b ( Z ′) 2 p0 C0 δ A ( Z ′) δ Rmp ( Z ′′) dZ ′dZ ′′ +∫ ∫ Z1 Z1 2 Z1

Z2

1

Z2

Z2

+ δ p ( Z1 ) ∫Z b ( Z ′) A0 p0C0δ p ( Z ′) dZ ′ 1

+3∫

Z2

Z1

+∫

Z2

+∫

Z2

Z1

Z2

2 ∫Z b ( Z ′) p0 C0 ⋅ b ( Z ′′) A0 p0C0 1

Z2

3 ∫Z b ( Z ′) p0 ⋅ b ( Z ′) A0 p0C0 1

Z1

Z2

∫Z b ( Z ′) A0 p0C0 1

δ p ( Z ′) δ p ( Z ′′) dZ ′dZ ′′

δ p ( Z ′) δ C ( Z ′′) dZ ′dZ ′′ Z2

δ p ( Z ′) δ Rm′p ( Z ′′) dZ ′dZ ′′ + δ p ( Z1 ) ∫Z

1

b ( Z ′) A0 p02δ C ( Z ′) dZ ′ 2

b ( Z ′) A0 p02 ⋅ b ( Z ′′) p02C0 δ p ( Z ′) δ C ( Z ′′) dZ ′dZ ′′ 1 2 Z2 Z2 b ( Z ′) A0 p02 ⋅ b ( Z ′′) p03 δ C ( Z ′) δ C ( Z ′′) dZ ′dZ ′′ +∫ ∫ Z1 Z1 2 Z2 Z 2 b ( Z ′) Z2 A0 p02 δ C ( Z ′) δ Rm′p ( Z ′′) dZ ′dZ ′′ + δ p ( Z1 ) ∫ δ RmA ( Z ′) dZ ′ +∫ ∫ Z1 Z1 Z1 2 +3∫

Z2

+3∫

Z2

Z1

Z1

+∫

Z2

Z1

Z2

∫Z

Z2

2 ∫Z b ( Z ′) p0 C0 1

Z2

3 ∫Z b ( Z ′) p0 1

δ p ( Z ′) dZ ′δ Rm′A ( Z ′′) dZ ′dZ ′′ Z2

δ C ( Z ′) δ Rm′A ( Z ′′) dZ ′dZ ′′ + ∫Z

1

Z2

∫Z

1

′′

δ RmA ( Z ′) δ Rm′p ( Z ′′) dZ ′dZ .

Taking into account the assumptions made in Eq. (B.10), the above equation leads to

δ Aδ p ( Z2 ) = δ Aδ p ( Z1 ) + 3bp02C0 δ Aδ p ( Z1 ) ( Z2 − Z1 ) + bp03 δ Aδ C ( Z1 ) ( Z 2 − Z1 ) 1 + bp02C0 dZ ′ δ Aδ p ( Z1 ) ( Z 2 − Z1 ) + bA0 p0C0 δ p 2 ( Z1 ) 2

(B.16)

Appendix B

77

Z2 Z 2 ′′ 1 + bA0 p02 δ pδ C ( Z1 ) + ∫ ∫ δ RmA ( Z ′) δ Rm′p ( Z ′′) dZ ′dZ , Z Z 1 1 2

(B.17)

when m = m′

δ RmA ( Z ′) δ Rmp ( Z ′′) = when m ≠ m′

A0 p0 N m ( Z ) 1 . ∑ a2 ( Z ) 2 E0 m

(B.18)

δ RmA ( Z ′) δ Rm′p ( Z ′′) = 0.

(B.19)

With initial value as zero, the differential form of Eq. (B.16) becomes

d δ Aδ p 7 = b ( Z ) p02C0 δ Aδ p + b ( Z ) p03 δ Aδ C + b ( Z ) A0 p0C0 δ p 2 ( Z1 ) 2 dZ 1 + b ( Z ) A0 p02 δ pδ C + ∑ δ RmAδ Rmp , 2 m

b(Z ) p0 7 p0C0 δ Aδ p + 2 p02 δ Aδ C + 2 A0C0 δ p 2 + A0 p0 δ pδ C 2 A0 p0 N m ( Z ) 1 + . ∑ a2 ( Z ) 2 E0 m

=

(

) (B.19)

Similarly the rest of the auto-correlation and cross-correlation equations can be derived and expressed as given in Chapter 3.

B.2 Calculation of Auto and Cross-Correlations of Noise Terms The noise terms given in Chapter 3 can be re-written here, ∞ ⎛ τ2 ⎞ 1 nmI 3 − 2τ 2 exp ⎜ − ⎟dτ , ∫ 2 π a ( Z ) −∞ ⎝ 2⎠ ∞ ⎛ τ2 ⎞ p0 nmI 1 − 2τ 2 exp ⎜ − ⎟dτ , = ∫ π A0 a ( Z ) −∞ ⎝ 2⎠

(

δ RmA =

δ Rmp

(

(B.20)

)

(B.21)

∞ ⎛ τ2 ⎞ , 2 2 1 2 τ exp − − n C n ( ) ⎜ − ⎟dτ 0 mI mR π A0 a ( Z ) ∫ −∞ ⎝ 2⎠ 2 ∞ ⎛ τ ⎞ 2 p0 = ( nmR − C0 nmI )τ exp ⎜ − ⎟dτ , ∫ −∞ π A0 a ( Z ) ⎝ 2⎠

(

δ RmC = δ Rmκ

)

δ RT0 =

)

∞ ⎛ τ2 ⎞ , 2 τ exp n ⎜ − ⎟dτ mI π A0 p0 a ( Z ) ∫ −∞ ⎝ 2⎠

δ Rmθ = −

∞ ⎛ τ2 ⎞ 1 nmR 3 − 2τ 2 exp ⎜ − ⎟dτ , ∫ 2 π A0 a ( Z ) −∞ ⎝ 2⎠

(

)

(B.22) (B.23) (B.24) (B.25)

78

Appendix B

Their correlation equations are derived as follows: 1) ∞ ∞ 1 δ RmA ( Z , T ) δ RmA ( Z ′, T ′) = nmI ( Z , T ) nmI ( Z ′, T ′) 3 − 2τ 2 3 − 2τ ′2 ∫ ∫ 2 −∞ −∞ 4π a ( Z )

(

=

∞ ∞ N0 ∫ ∫ 2 −∞ −∞ 8π a ( Z )

=

(

m

)

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 3 − 2τ 2 3 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

)(

)

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0 N 0 3 − 2τ 2 3 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ ( Z − Z ′) dτ ′ ∫ 2 8π a ( Z ) −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)(

)

Taking T = T ′ and Z ′ = mZ a

∑

)(

δ RmA ( Z ) δ RmA ( Z ′) =

1 8π

∑ m

p0 N m ( Z ) ∞ 9 − 12τ 2 + 4τ 4 exp −τ 2 dτ ∫ 2 −∞ a (Z )

(

1 ⋅6 π 4π

=

A Nm ( Z ) 3 ∑ 4 E0 m a 2 ( Z )

m

( )

p0 N m ( Z ) a2 ( Z )

=

∑

)

2 0

2)

δ RmA ( Z , T ) δ Rmp ( Z ′, T ′) =

∞ ∞ p0 nmI ( Z , T ) nmI ( Z ′, T ′) 3 − 2τ 2 1 − 2τ ′2 ∫ ∫ 2 2π A0 a ( Z ) −∞ −∞

(

∞ ∞ p0 N 0 ∫ ∫ 2 −∞ −∞ 4π A0 a ( Z )

=

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p02 N 0 3 − 2τ 2 1 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ ( Z − Z ′) dτ ′ ∫ 2 4π A0 a ( Z ) −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)(

)(

)

)

Taking T = T ′ and Z ′ = mZ a

∑ m

)

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 3 − 2τ 2 1 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

=

(

)(

δ RmA ( Z ) δ Rmp ( Z ′) = =

=

1 4π

∑ m

p02 N m ( Z ) ∞ 3 − 8τ 2 + 4τ 4 exp −τ 2 dτ A0 a 2 ( Z ) ∫ −∞

1 ⋅2 π 4π

(

∑ m

p02 N m ( Z ) A0 a 2 ( Z )

A0 p0 N m ( Z ) 1 ∑ 2 E0 m a2 ( Z )

)

( )

Appendix B

79

3)

δ RmA ( Z , T ) δ RmC ( Z ′, T ′) =

∞ ∞ 2 nmI ( Z , T ) {nmR ( Z ′, T ′) − C0 nmI ( Z ′, T ′)} ∫ ∫ 2 2π A0 a ( Z ) −∞ −∞

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ 3 − 2τ 2 1 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 3 − 2τ 2 1 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

(

)(

)

=−

∞ ∞ C0 N 0 ∫ ∫ 2 −∞ −∞ 2π A0 a ( Z )

=−

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0C0 N 0 2 2 ′ 3 2 1 2 exp exp τ τ − − − ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ 2π A0 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)(

(

)

)(

)

Taking T = T ′ and Z ′ = mZ a

∑ m

δ RmA ( Z ) δ RmC ( Z ′) = − =−

1 2π

∑ m

p0C0 N m ( Z ) ∞ 3 − 8τ 2 + 4τ 4 exp −τ 2 dτ ∫ 2 −∞ A0 a ( Z )

(

1 ⋅2 π 2π

∑ m

)

( )

p0C0 N m ( Z ) A0 a 2 ( Z )

A C N (Z ) 1 = − ∑ 0 02 m E0 m a (Z ) 4)

δ RmA ( Z , T ) δ Rmκ ( Z ′, T ′) =

∞ ∞ 2 p0 nmI ( Z , T ) {nmR ( Z ′, T ′) − C0 nmI ( Z ′, T ′)} 3 − 2τ 2 τ ′ 2π A0 a 2 ( Z ) ∫ −∞ ∫ −∞

(

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ∞ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0 N 0 =− 3 − 2τ 2 τ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ∫ ∫ 2 2π A0 a ( Z ) −∞ −∞ ⎝ 2⎠ ⎝ 2 ⎠ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p02 N 0 2 ′ 3 2 exp exp τ τ =− − − ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ 2π A0 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

(

)

)

Taking T = T ′ and Z ′ = mZ a

∑ m

δ RmA ( Z ) δ Rmκ ( Z ′) = − =0

1 2π

∑ m

p02 N m ( Z ) ∞ 3τ − 2τ 3 exp −τ 2 dτ A0 a 2 ( Z ) ∫ −∞

(

)

( )

)

80

Appendix B

5)

δ RmA ( Z , T ) δ RmT0 ( Z ′, T ′) =

∞ ∞ 2 nmI ( Z , T ) nmI ( Z ′, T ′) 3 − 2τ 2 τ ′ ∫ ∫ 2 2π A0 p0 a ( Z ) −∞ −∞

(

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 3 − 2τ 2 τ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

=

∞ ∞ N0 ∫ ∫ 2 −∞ −∞ 2π A0 p0 a ( Z )

=

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ N0 2 ′ − − 3 2 exp exp τ τ ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ 2π A0 p0 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)

(

)

Taking T = T ′ and Z ′ = mZ a

∑ m

)

(Z )

N

1 2π =0

∑ A pma 2 ( Z ) ∫ −∞ ( 3τ − 2τ 3 ) exp ( −τ 2 ) dτ

δ RmA ( Z ) δ Rmκ ( Z ′) =

m

0

∞

0

6)

δ RmA ( Z , T ) δ Rmθ ( Z ′, T ′) = −

∞ ∞ 1 nmI ( Z , T ) nmR ( Z ′, T ′) 3 − 2τ 2 3 − 2τ ′2 ∫ ∫ 2 −∞ −∞ 4π A0 a ( Z )

(

)(

)

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ =0

Taking T = T ′ and Z ′ = mZ a ∑ δ RmA ( Z ) δ Rmθ ( Z ′) = 0 m

7)

δ Rmp ( Z , T ) δ Rmp ( Z ′, T ′) =

∞ ∞ p02 nmI ( Z , T ) nmI ( Z ′, T ′) 1 − 2τ 2 1 − 2τ ′2 ∫ ∫ 2 2 −∞ −∞ π A0 a ( Z )

(

m

)

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 1 − 2τ 2 1 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

=

∞ ∞ p02 N 0 ∫ ∫ 2 2 −∞ −∞ 2π A0 a ( Z )

=

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p03 N 0 2 2 ′ 1 2 1 2 exp exp τ τ − − − ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ 2π A02 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

(

)(

)

)(

)

Taking T = T ′ and Z ′ = mZ a

∑

)(

δ Rmp ( Z ) δ Rmp ( Z ′) =

1 2π

∑ m

p03 N m ( Z ) ∞ 1 − 4τ 2 + 4τ 4 exp −τ 2 dτ A02 a 2 ( Z ) ∫ −∞

1 = ⋅2 π 2π

(

∑ m

p03 N m ( Z ) A02 a 2 ( Z )

)

( )

Appendix B

81

=

p02 N m ( Z ) 1 ∑ E0 m a 2 ( Z )

8)

δ Rmp ( Z , T ) δ RmC ( Z ′, T ′) =

∞ ∞ 2p nmI ( Z , T ) {nmR ( Z ′, T ′) − C0 nmI ( Z ′, T ′)} ∫ ∫ −∞ −∞ π A a (Z ) 0 2 2 0

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ 1 − 2τ 2 1 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ 2 2 ⎛ τ ⎞ ⎛ τ′ ⎞ 1 − 2τ 2 1 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

(

)(

)

=−

2 p0C0 N 0 ∞ ∞ 2π A02 a 2 ( Z ) ∫ −∞ ∫ −∞

=−

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p02C0 N 0 ∞ 2 2 ′ τ τ 1 2 1 2 exp exp − − − ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ π A02 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)(

(

)

)(

)

Taking T = T ′ and Z ′ = mZ a

∑ m

1

δ Rmp ( Z ) δ RmC ( Z ′) = − =−

π 1

π

p02C0 N m ( Z ) ∞ 1 − 4τ 2 + 4τ 4 exp −τ 2 dτ ∫ 2 2 −∞ A0 a ( Z )

(

∑ m

⋅2 π

∑ m

2 =− E0

)

( )

p02C0 N m ( Z ) A02 a 2 ( Z )

p0C0 N m ( Z ) a2 ( Z )

∑ m

9)

δ Rmp ( Z , T ) δ Rmκ ( Z ′, T ′) =

∞ ∞ 2 p02 nmI ( Z , T ) {nmR ( Z ′, T ′) − C0 nmI ( Z , T )} 1 − 2τ 2 τ ′ ∫ ∫ 2 2 −∞ −∞ π A0 a ( Z )

(

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ 2 ∞ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0 N 0 2 ′ =− − − τ τ 1 2 exp exp ⎜ ⎟ ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ π A02 a 2 ( Z ) ∫ −∞ ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

=−

)

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p02 N 0 2 ′ 1 2 exp exp − − τ τ ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ π A02 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)

Taking T = T ′ and Z ′ = mZ a

∑ m

δ Rmp ( Z ) δ Rmκ ( Z ′) = − =0

1 2π

∑ m

p02 N m ( Z ) ∞ τ − 2τ 3 exp −τ 2 dτ A02 a 2 ( Z ) ∫ −∞

(

) ( )

)

82

Appendix B

10)

δ Rmp ( Z , T ) δ RmT0 ( Z ′, T ′) =

∞ ∞ 2 n ( Z , T ) nmI ( Z ′, T ′) 1 − 2τ 2 τ ′ ∫ ∫ π A a ( Z ) −∞ −∞ mI

(

2 2 0

)

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ∞ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ N0 = 1 − 2τ 2 τ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ∫ ∫ 2 2 π A0 a ( Z ) −∞ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

=

)

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0 N 0 2 ′ − − τ τ 1 2 exp exp ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ π A02 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)

Taking T = T ′ and Z ′ = mZ a

∑ m

1

∑ π

δ Rmp ( Z ) δ RmT0 ( Z ′) =

m

=0

p0 N m ( Z ) ∞ τ − 2τ 3 exp −τ 2 dτ ∫ 2 2 −∞ A0 a ( Z )

(

)

( )

11)

δ Rmp ( Z , T ) δ Rmθ ( Z ′, T ′) = −

∞ ∞ 1 nmI ( Z , T ) nmR ( Z ′, T ′) 1 − 2τ 2 3 − 2τ ′2 ∫ ∫ −∞ −∞ 2π A a ( Z )

(

2 2 0

)(

)

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

=0

Taking T = T ′ and Z ′ = mZ a

∑ m

δ Rmp ( Z ) δ Rmθ ( Z ′) = 0

12) δ RmC ( Z , T ) δ RmC ( Z ′, T ′) =

∞ ∞ 4 {nmR ( Z , T ) − C0nmI ( Z , T )}{nmR ( Z ′, T ′) − C0nmI ( Z ′, T ′)} ∫ ∫ −∞ −∞ π A a (Z ) 2 2 0

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ 1 − 2τ 2 1 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ∞ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 2N0 2 2 ′ 1 2 τ 1 2 τ exp exp = − − − ⎜ ⎟ ⎜− ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ π A02 a 2 ( Z ) ∫ −∞ ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

=

(

)

2 p0 1 + C02 N 0

(

)(

)

(

)(

)

⎛ τ2 ⎞

∞

Taking T = T ′ and Z ′ = mZ a

∑ m

⎛ τ ′2 ⎞ ⎟ δ ( Z − Z ′) dτ ′ 2 ⎠

1 − 2τ 2 )(1 − 2τ ′2 ) exp ⎜ − ⎟ exp ⎜ − ( ∫ 2 2 −∞ π A0 a ( Z ) ⎝ 2⎠ ⎝

δ RmC ( Z ) δ RmC ( Z ′) =

2

∑ π m

(

)

p0 1 + C02 N m ( Z ) A a (Z ) 2 2 0

∫ −∞ (1 − 4τ ∞

2

) ( )

+ 4τ 4 exp −τ 2 dτ

Appendix B

83

2

=

⋅2 π

π

∑

(

A a (Z ) 2 2 0

m

(

)

p0 1 + C02 N m ( Z )

)

1 + C Nm ( Z ) 4 ∑ E0 m a (Z )

=

2 0 2

13) ∞ ∞ 4p {n ( Z , T ) − C0nmI ( Z , T )}{nmR ( Z ′, T ′) − C0nmI ( Z ′, T ′)} ∫ ∫ π A a ( Z ) −∞ −∞ mR

δ RmC ( Z , T ) δ Rmκ ( Z ′, T ′) =

0 2 2 0

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ 1 − 2τ 2 τ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

(

=

=

(

)

(

)

2 p0 1 + C02 N 0

)

⎛ τ2 ⎞

2 p02 1 + C02 N 0

π A02 a 2 ( Z )

∞

∞

⎛

m

⎞

⎛

′2 ⎞

τ τ ∫ (1 − 2τ )τ ′ exp ⎜ − 2 ⎟ exp ⎜ − 2 ⎟ δ ( Z − Z ′) dτ ′ ⎝ ⎠ ⎝ ⎠ ∞

2

2

−∞

Taking T = T ′ and Z ′ = mZ a

∑

⎛ τ ′2 ⎞ ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ 2 ⎠

1 − 2τ 2 )τ ′ exp ⎜ − ⎟ exp ⎜ − ( ∫ ∫ 2 2 −∞ −∞ π A0 a ( Z ) ⎝ 2⎠ ⎝

δ RmC ( Z ) δ Rmκ ( Z ′) =

1

∑ π

(

)

2 p02 1 + C02 N m ( Z )

m

=0

A a (Z ) 2 2 0

3 2 ∫ −∞ (τ − 2τ ) exp ( −τ ) dτ ∞

14)

δ RmC ( Z , T ) δ RmT0 ( Z ′, T ′) =

∞ ∞ 4 {nmR ( Z , T ) − C0nmI ( Z , T )} nmI ( Z ′, T ′) ∫ ∫ 2 −∞ −∞ π A p0 a ( Z ) 2 0

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ 1 − 2τ 2 τ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ 2 ∞ ∞ ⎛ τ ⎞ ⎛ τ ′2 ⎞ 2C N = − 2 02 0 ∫ ∫ 1 − 2τ 2 τ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ π A0 a ( Z ) −∞ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)

(

=−

)

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 2C0 N 0 2 ′ − − 1 2 τ τ exp exp ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ π A02 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)

Taking T = T ′ and Z ′ = mZ a

∑ m

δ RmC ( Z ) δ RmT0 ( Z ′) = − =0

2

∑ π m

C0 N m ( Z ) ∞ τ − 2τ 3 exp −τ 2 dτ ∫ 2 2 −∞ A0 a ( Z )

(

)

( )

84

Appendix B

15)

δ RmC ( Z , T ) δ Rmθ ( Z ′, T ′) = −

∞ ∞ 1 {n ( Z , T ) − C0nmI ( Z , T )} nmR ( Z ′, T ′) ∫ ∫ π A a ( Z ) −∞ −∞ mR 2 2 0

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ 1 − 2τ 2 3 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ∞ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ N0 1 − 2τ 2 3 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ =− ∫ ∫ 2 2 2π A0 a ( Z ) −∞ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)(

(

=−

)

)(

)

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0 N 0 2 2 1 2 3 2 exp exp τ τ − − − ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ 2π A02 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)(

)

Taking T = T ′ and Z ′ = mZ a

∑ m

δ RmC ( Z ) δ Rmθ ( Z ′) = − =−

=−

1 2π

p0 N m ( Z ) ∞ 3 − 8τ 2 + 4τ 4 exp −τ 2 dτ A02 a 2 ( Z ) ∫ −∞

(

∑ m

1 ⋅2 π 2π

1 E0

N

)

( )

p0 N m ( Z ) A02 a 2 ( Z )

∑ m

(Z )

∑ a 2m( Z ) m

16) ∞ ∞ 4 p02 {n ( Z , T ) − C0nmI ( Z , T )}{nmR ( Z ′, T ′) − C0nmI ( Z ′, T ′)} π A02 a 2 ( Z ) ∫ −∞ ∫ −∞ mR

δ Rmκ ( Z , T ) δ Rmκ ( Z ′, T ′) =

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ τ ⋅ τ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

= =

(

)

(

)

2 p02 1 + C02 N 0

π A02 a 2 ( Z ) 2 p03 1 + C02 N 0

π A02 a 2 ( Z )

∞

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ exp ⎟ ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

∞

∫ −∞ ∫ −∞ ττ ′ exp ⎜ −

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ exp ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

∞

∫ −∞ττ ′ exp ⎜ −

Taking T = T ′ and Z ′ = mZ a

∑ δ Rmκ ( Z ) δ Rmκ ( Z ′)

=

m

2

∑ π

(

A a (Z ) 2 2 0

m

=

2

π

⋅

)

p03 1 + C02 N m ( Z )

π 2

∑ m

(

(

∞

∫ −∞τ

)

2

p03 1 + C02 N m ( Z ) A a (Z ) 2 2 0

)

p02 1 + C02 N m ( Z ) 1 = ∑ E0 m a2 ( Z )

( )

exp −τ 2 dτ

Appendix B

85

17)

δ Rmκ ( Z , T ) δ RmT0 ( Z ′, T ′) =

∞ ∞ 4 {n ( Z , T ) − C0nmI ( Z , T )} nmI ( Z ′, T ′) ∫ ∫ π A a ( Z ) −∞ −∞ mR 2 2 0

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ τ ⋅ τ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ 2 ∞ ∞ ⎛ τ ⎞ ⎛ τ ′2 ⎞ 2C N = − 2 02 0 ∫ ∫ ττ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ π A0 a ( Z ) −∞ −∞ ⎝ 2⎠ ⎝ 2 ⎠

=−

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 2 p0C0 N 0 ∞ ′ exp ⎜ − ⎟ exp ⎜ − ⎟ δ ( Z − Z ′) dτ ′ ττ ∫ 2 2 π A0 a ( Z ) −∞ ⎝ 2⎠ ⎝ 2 ⎠

Taking T = T ′ and Z ′ = mZ a

∑ m

δ Rmκ ( Z ) δ RmT0 ( Z ′) = − =−

2

∑ π m

2

π

⋅

p0C0 N m ( Z ) ∞ 2 τ exp −τ 2 dτ ∫ 2 2 −∞ A0 a ( Z )

π 2

( )

∑ m

p0C0 N m ( Z ) A02 a 2 ( Z )

C N (Z ) 1 =− ∑ 02m E0 m a ( Z ) 18)

δ Rmκ ( Z , T ) δ Rmθ ( Z ′, T ′) = −

∞ ∞ p {nmR ( Z , T ) − C0nmI ( Z , T )} nmR ( Z ′, T ′) ∫ ∫ −∞ −∞ π A a (Z ) 0 2 2 0

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ τ 3 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ∞ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0 N 0 2 ′ =− − − 3 2 exp exp τ τ ⎜ ⎟ ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ 2π A02 a 2 ( Z ) ∫ −∞ ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

)

(

=−

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0 N 0 3 − 2τ 2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ ( Z − Z ′) dτ ′ τ ∫ 2 2 2π A0 a ( Z ) −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

Taking T = T ′ and Z ′ = mZ a

∑ m

)

δ Rmκ ( Z ) δ Rmθ ( Z ′) = − =0

1 2π

∑ m

)

p0 N m ( Z ) ∞ 3τ − 2τ 3 exp −τ 2 dτ A02 a 2 ( Z ) ∫ −∞

(

)

( )

86

Appendix B

19) 4 πA p a

δ RmT0 ( Z , T ) δ RmT0 ( Z ′, T ′) =

2 0

∞

2 2 0

∞

( Z ) ∫ −∞ ∫ −∞

nmI ( Z , T ) nmI ( Z ′, T ′) τ ⋅ τ ′

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ∞ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 2 N0 ′ ττ exp exp = − ⎜ ⎟ ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ π A02 p02 a 2 ( Z ) ∫ −∞ ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

=

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ 2 N0 ′ exp ⎜ − ⎟ exp ⎜ − ⎟ δ ( Z − Z ′) dτ ′ ττ ∫ 2 π A p0 a ( Z ) −∞ ⎝ 2⎠ ⎝ 2 ⎠ 2 0

Taking T = T ′ and Z ′ = mZ a

∑ m

N

2

(Z )

∞

N

(Z )

∑ A2 pma 2 ( Z ) ∫ −∞ τ 2 exp ( −τ 2 ) dτ π

δ Rmκ ( Z ) δ Rmκ ( Z ′) =

m

=

=

2

π

⋅

0

π 2

0

∑ A2 pma 2 ( Z ) m

0

0

Nm ( Z ) 1 ∑ E0 m p02 a 2 ( Z )

20)

δ RmT0 ( Z , T ) δ Rmθ ( Z ′, T ′) = −

∞ ∞ 1 n ( Z , T ) nmR ( Z ′, T ′) ∫ ∫ 2 π A p0 a ( Z ) −∞ −∞ mI 2 0

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ τ 3 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠

(

)

=0

Taking T = T ′ and Z ′ = mZ a

∑ m

δ RmT0 ( Z ) δ Rmθ ( Z ′) = 0

21)

δ Rmθ ( Z , T ) δ Rmθ ( Z ′, T ′) =

∞ ∞ 1 nmR ( Z , T ) nmR ( Z ′, T ′) 3 − 2τ 2 3 − 2τ ′2 ∫ ∫ 4π A a ( Z ) −∞ −∞

(

2 2 0

)(

)

⎛ τ2 ⎞ ⎛ τ ′2 ⎞ ⋅ exp ⎜ − ⎟ exp ⎜ − ⎟ dτ dτ ′ ⎝ 2⎠ ⎝ 2 ⎠ ∞ ∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ N0 = 3 − 2τ 2 3 − 2τ ′2 exp ⎜ − ⎟ exp ⎜ − ⎟ δ (T − T ′) δ ( Z − Z ′) dτ dτ ′ ∫ ∫ 2 2 8π A0 a ( Z ) −∞ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

=

)(

)

∞ ⎛ τ2 ⎞ ⎛ τ ′2 ⎞ p0 N 0 2 2 ′ − − − τ τ 3 2 3 2 exp exp ⎜ ⎟ ⎜ − ⎟ δ ( Z − Z ′) dτ ′ 8π A02 a 2 ( Z ) ∫ −∞ ⎝ 2⎠ ⎝ 2 ⎠

(

Taking T = T ′ and Z ′ = mZ a

)(

)

Appendix B

∑

87

1 8π

δ Rmθ ( Z ) δ Rmθ ( Z ′) =

m

=

=

∑ m

p0 N m ( Z ) ∞ 9 − 12τ 2 + 4τ 4 exp −τ 2 dτ ∫ 2 2 −∞ A0 a ( Z )

(

1 ⋅6 π 4π

∑ m

)

( )

p0 N m ( Z ) A02 a 2 ( Z )

Nm ( Z ) 3 ∑ 4 E0 m a 2 ( Z )

B.3 Calculation of Dynamical Equations for Two-Channel WDM System In this section, we develop the dynamical equations for a two-channel WDM system with periodic dispersion management in order to estimate the phase fluctuations caused by XPM. Assuming XPM as the sole perturbation and using Eq. (4.3) of Chapter 4, Rj is can be defined as 2

R j = −2 S ( Z ) U 3− j U j

⎛ τ2 ⎞ = −2S ( Z ) A32− j A j exp −τ 32− j exp ⎜ − j ⎟ exp ( iφ j ) ⎜ 2⎟ ⎝ ⎠

(

)

(B.26)

The dynamical equations with perturbation introduced in Chapter 2 can be written for two-channel WDM (j = 1, 2) system as dp j dZ dC j

dZ dκ j

dZ dT j

= b( Z ) p 3j C j + R p j ,

(

(B.27)

)

= −b( Z ) p 2j 1 + C 2j −

S (Z ) 2

A2j + RC j ,

(B.28)

= Rκ j ,

(B.29)

= b( Z ) κ j + RT j , dZ dθ j b( Z ) 2 5S (Z ) 2 κ j − p 2j + A j + Rθ j , =− dZ 2 4 2

(

(B.30)

)

(B.31)

where Rp j =

pj

π Aj

∞

∫ −∞

Im ⎣⎡ R j e

− iϕ j

⎛ τ2 ⎞ ⎤ 1 − 2τ 2j exp ⎜ − j ⎟ dτ , ⎦ ⎜ 2⎟ ⎝ ⎠

(

)

(B.32)

⎛ τ 2j ⎞ , − iϕ j 2 ⎤ ⎡ ⎤ τ − − C R e Im 1 2 exp ⎜⎜ − ⎟⎟ dτ j j ∫ −∞ ⎦ ⎣ j ⎦ ⎝ 2⎠ ⎛ τ 2j ⎞ , 2pj ∞ − iϕ j − iϕ j ⎡ ⎤ ⎡ ⎤ Rκ j = Re R e C Im R e τ exp dτ − − ⎜ j ⎣ j ⎦ ⎣ j ⎦ j ⎜ 2 ⎟⎟ π A j ∫ −∞ ⎝ ⎠ ⎛ τ 2j ⎞ , ∞ 2 − iϕ j ⎡ ⎤ Im R j e ⎦ τ j exp ⎜ − ⎟ dτ RT j = ⎜ 2⎟ π A j p j ∫ −∞ ⎣ ⎝ ⎠ RC j =

2 π Aj

∞

{

Re ⎡⎣ R j e

{

− iϕ j

}(

}

)

(B.33) (B.34) (B.35)

88

Appendix B

Rθ j = −

1 2 π Aj p j

∞

∫ −∞

{ p Re ⎡⎣ R e ⎤⎦ (3 − 2τ ) + 4κ j

− iϕ j

j

2 j

j

Im ⎡⎣ R j e

− iϕ j

⎛ τ2 ⎞ ⎤ τ j exp ⎜ − j ⎟ dτ . ⎦ ⎜ 2⎟ ⎝ ⎠

}

(B.36)

Here Re ⎡ R j e−iϕ j ⎤ and Im ⎡ R j e−iϕ j ⎤ represent the real and imaginary parts of R j e−iϕ j , respectively, ⎣ ⎦ ⎣ ⎦ and can be given as Re ⎡⎣ R j e

− iϕ j

Im ⎡⎣ R j e

− iϕ j

⎛ τ2 ⎞ ⎤ = −2 S ( Z ) A32− j A j exp −τ 32− j exp ⎜ − j ⎟ ⎦ ⎜ 2⎟ ⎝ ⎠

(B.37)

⎤ = 0. ⎦

(B.38)

(

)

2

Now applying E j = ∫ ∞ U j dT = π A2j p j which represents a constant pulse energy of Uj, Eqs. ∞ (B.32) – (B.36) can be deduced as Rp j = 0 ,

(B.39)

⎛ τ2 ⎞ ⎛ τ2 ⎞ ∞ 4 S ( Z ) ∫ A32− j A j exp −τ 32− j exp ⎜ − j ⎟ 1 − 2τ 2j exp ⎜ − j ⎟ dτ , −∞ ⎜ 2⎟ ⎜ 2⎟ π Aj ⎝ ⎠ ⎝ ⎠ ∞ 4 2 2 2 2 2 2 = − S ( Z ) E3− j p j p3− j ∫ 1 − 2 p1 (T − T1 ) exp − p1 (T − T1 ) − p2 (T − T2 ) dT ,

(

RC j = −

π

=

4

π

−∞

S ( Z ) E3− j p j p

3 3− j

= − 4S ( Z )

E3− j p j p33− j

π

P

5

)

{

2 1

− p22 + 2 p12 p22 (T1 − T2 )

(p

2 1

2

)

} {

{− p {P

(

+ p12

)

5

}

2

} exp ⎪⎧⎨− p p (T − T ) ⎪⎫⎬ , 2 1

1

2

p12 + p22

⎪⎩

2

2

2 2

⎭⎪

}

− 2 ( ∆τ ) F , 2

(B.40)

⎛ τ2 ⎞ ⎛ τ2 ⎞ ∞ 4 S ( Z ) p j ∫ A32− j A j exp −τ 32− j exp ⎜ − j ⎟τ j exp ⎜ − j ⎟ dτ , −∞ ⎜ 2⎟ ⎜ 2⎟ π Aj ⎝ ⎠ ⎝ ⎠ ∞ 4 2 2 2 2 2 = − S ( Z ) E j p1 p2 ∫ p1 (T − T2 ) exp − p1 (T − T1 ) − p2 (T − T2 ) dT ,

(

Rκ j = −

π

=

4

π

)

{

−∞

( (p + p )

p12 p22 (T1 − T2 ) p12 + p22

S ( Z ) E j p1 p2

= ( −1)

3− j

4S ( Z )

2 1

2 1

5

2

) exp ⎧⎪− p ⎨ ⎪⎩

}

2 1

p22 (T1 − T2 ) p12 + p22

2

⎫⎪ , ⎬ ⎭⎪

E j p12 p22 ∆τ F , π P3

(B.41)

RT j = 0 , Rθ j = −

= =

1

π

(B.42)

⎛ τ2 ⎞ ⎛ τ2 ⎞ ∞ 1 S ( Z ) ∫ p j A32− j A j exp −τ 32− j exp ⎜ − j ⎟ 3 − 2τ 2j exp ⎜ − j ⎟ dτ . −∞ ⎜ 2⎟ ⎜ 2⎟ 2 π Aj p j ⎝ ⎠ ⎝ ⎠

(

π

S ( Z ) E3− j

(

)

{3 − 2 p (T − T ) }exp{− p (T − T ) − p (T − T ) } dT , ⎡ 2 ( p + 2 p p + p ) + p { p + p − 2 p p (T − T ) }⎤ ⎦ exp ⎧⎪ − p p (T − T ) ⎫⎪ , pp ⎣

S ( Z ) E3− j p2 ∫

1

)

∞

−∞

4 1

1 2

2

2 1

2 1

2 2

2

2 1

1

4 2

2 2

(p

2 1

1

2 1

+ p12

2 2

)

5

2

2

2 2

2 1

2

2 2

2

1

2

⎨ ⎪⎩

2 1

2 2

2

1

p12 + p22

2

⎬ ⎭⎪

Appendix B

89

{

}

⎡ 2 P 4 + p 2 P 2 − 2 ( ∆τ )2 ⎤ 2 3− j ⎣ ⎦ exp ⎧⎪ − ⎛ ∆τ ⎞ ⎫⎪ , p1 p2 ⎨ ⎜ ⎟ ⎬ P5 π ⎩⎪ ⎝ P ⎠ ⎭⎪

= S (Z )

E3− j

= S (Z )

E3− j p1 p2 2 2 P 2 P 2 + p32− j − 2 p32− j ( ∆τ ) F , 5 π P

{ (

}

)

where ∆τ = p1 p2 (T1 − T2 ) , P =

{

dZ dC j

dZ dκ j

dZ dT j

}

2 p12 + p22 , and F = exp − ( ∆τ P ) . Taking the above Eqs. (B.39) –

(B.43), Eqs. (B.27) – (B.31) are transformed into

dp j

(B.43)

= b( Z ) p 3j C j ,

(B.44)

(

)

= −b( Z ) p 2j 1 + C 2j −

= ( −1)

3− j

4S ( Z )

S (Z ) 2

A2j − 4 S ( Z )

E3− j p j p33− j

π

P5

{P

2

}

− 2 ( ∆τ ) F , 2

(B.42)

E j p12 p22 ∆τ F , π P3

(B.43)

= b( Z ) κ j , dZ dθ j E pp b( Z ) 2 5S (Z ) 2 2 A j + S ( Z ) 3− j 1 5 2 P 2 2 P 2 + p32− j − 2 p32− j ( ∆τ ) F . κ j − p 2j + =− dZ 2 π P 4 2

(

{ (

)

)

}

(B.44) (B.45)

The frequency separation (∆κ(Z)) and pulse spacing (∆T(Z)) between two copropagating channels can be determined as ∆κ ( Z ) = κ 1 ( Z ) − κ 2 ( Z ) ,

(B.46)

∆T ( Z ) = T1 ( Z ) − T2 ( Z ) .

(B.47)

Considering Eq. (B.44), Eq. (B.47) can be written in differential equation as d ( ∆T ) = b( Z ) ∆κ . dZ

(B.48)

Now using Eq. (B.44) and Eq. (B.48), ∆τ = p1 p2 (T1 − T2 ) can be deduced in differential equation as d ( ∆τ ) = b( Z ) p12C1 + p22C2 ∆τ + p1 p2∆κ . dZ

{(

)

}

(B.49)

And using Eq. (B.43), Eq. (B.46) can be written in differential equation as d (∆κ ) E + E 2 p12 p 22 ∆ τ . = 4S (Z ) 1 F dZ P3 π

(B.50)

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List of Publications by the Author Journal Papers [1] M. Faisal and A. Maruta, “Phase jitter in periodically dispersion-managed optical transmission systems,” Optics Communications, vol. 282, pp. 1893-1901, 2009. [2] M. Faisal and A. Maruta, “Cross-phase modulation induced phase fluctuations in optical RZ pulse propagating in dispersion compensated WDM transmission systems,” in press, Optics Communications, doi:10.1016/j.optcom.2009.12.034, 2010.

International Conferences [1] M. Faisal and A. Maruta, “Impact of dispersion management on phase jitter in long haul optical transmission line,” in Proc. of 13th Optoelectronics and Communications Conference/33rd Australian Conference on Optical Fibre Technology (OECC/ACOFT 2008), Paper P-96, in CD-ROM, Sydney, Australia, 7-10 July, 2008. [2] M. Faisal and A. Maruta, “Phase noise in dispersion managed quasi-linear optical transmission systems,” in Proc. of 14th Asia-Pacific Conference on Communications/7th International Conference on the Optical Internet (APCC/COIN 2008), in CD-ROM, Tokyo, Japan, 14-16 October, 2008. [3] M. Faisal and A. Maruta, “Effects of XPM on phase of optical RZ pulse in dispersion compensated transmission line,” in Proc. of 8th International Conference on Optical Communications and Networks (ICOCN 2009), Paper conf09b361, in CD-ROM, Beijing, China, 15-17 September, 2009.

Technical Report [1] A. Maruta, M. Faisal and T. Nakamura, “Influence of amplifier noise and cross-phase modulation on phase in optical RZ pulse,” IEICE Technical Report, pp. 85-89, OCS2008-121, (2009-1).

101

102

List of Publications by the Author

Others [1] M. Faisal and A. Maruta, “Upgradation of dispersion compensation map for photonic transmission systems,” in Proc. of 2nd Global COE International Symposium (EDIS 2009), Paper WS3-B1, pp. 111-112, Osaka, Japan, December 2, 2009 - February 2, 2010. [2] M. Faisal and A. Maruta, “Timing and phase jitter in dispersion-managed photonic networks,” in Proc. of 1st Global COE Student Conference on Innovation Electronic Topics (SCIENT 2008), Paper P-13, pp. 89, Osaka, Japan, July 31-August 1, 2008. [3] M. Faisal, M. N. Islam and S. P. Majumder, “Performance comparison of wavelength shift keying WDM system and conventional on-off WDM system in presence of four-wave mixing,” Optik-International Journal for Light and Electron Optics, vol. 117, no. 12, pp. 555-562, December, 2006. [4] M. Faisal, M. N. Islam and M. S. Alam, “Comparative performance of wavelength shift keying and repeated unequal channel spacing schemes in reducing the four-wave mixing effect in optical WDM system,” Optical Engineering, vol. 45(1), pp. 015002-1 to 015002-6, January 2006. [5] M. Faisal, A. Hasib, and A. B. M. H. Rashid, “Fault Characterization, Testability Issue and Design for Testability of Complementary Pass Transistor Logic Circuits,” IEE Proc.- Circuits, Devices and Syst., Vol. 152, No. 3, pp. 215-222, June, 2005. [6] H. Rahman, M. Faisal, and A. B. M. H. Rashid, “Fault characterization and testability issue of low capacitance full-swing BiCMOS logic circuits,” in Proc. of 3rd International Conference on Electrical & Computer Engineering (ICECE 2004), Dhaka, Bangladesh, pp. 43-48, December 28-30, 2004. [7] M. Faisal, M. N. Islam, and S. P. Majumder, “Performance of wavelength shift keying technique in reducing the four-wave mixing effect in optical WDM systems,” in Proc. of 3rd International Conference on Electrical, Electronics & Computer Engineering (ICEECE 2003), Dhaka, Bangladesh, pp. 157-160, December 22-24, 2003.

Journal Paper under Peer-Review Process [1] M. Faisal and A. Maruta, “Mitigation of phase noise in dispersion compensated quasi-linear optical transmission systems: Analysis,” under review, Optics Communications.