Design and Analysis of Digital Direct-Detection Fiber-Optic CommunicationSystems Using Volterra Series Approach

A Dissertation

Presented to

the Faculty of the School of Engineering and Applied Science

University of Virginia

In Partial Fulfillment

of the requirements for the Degree of

Doctor of Philosophy

(Electrical Engineering)

by

Kumar V. Peddanarappagari

October, 1997

Abstract

Optical fiber communication systems are the most efficient means of handling the heavy data

traffic in this information age. Efforts are being made to increase the already phenomenal capacity

of these high bandwidth communication systems. Highly coherent optical sources that can generate

high power narrow pulses are being developed and fiber amplifiers are making repeaterless transmis-

sion over long distances possible. Considerable attention is being paid to the limitations placed on

these systems by linear dispersion, fiber nonlinearities, and amplified spontaneous emission (ASE)

noise from fiber amplifiers. However, most of the analysis is based on single pulse-propagation

experiments and simplified analytical expressions; pulse-to-pulse interactions are generally ignored.

The design of these systems is still empirical due to lack of analytical expressions for the output field

in terms of input, fiber, and amplifier parameters. This study develops analytical tools to analyze

and design systems to reduce the influence of linear dispersion, fiber nonlinearities, and ASE noise

in single-user and multi-user systems.

The generalized nonlinear Schroedinger (NLS) wave equation may be used to explain the effects

of linear dispersion and fiber nonlinearities on the evolution of the complex envelope of the optical

field in an optical fiber. The NLS equation is typically solved using numerical (recursive) methods.

In this work, a Volterra series based nonlinear transfer function of an optical fiber is derived based

on solving the NLS equation in the frequency-domain and retaining only the most significant terms

(Volterra kernels) in the resulting transfer function. Single pulse-propagation in single-mode optical

fibers is studied and the results are compared to available literature which uses numerical solutions.

The linear portion of the above model is then used for the theoretical study of the effects of

phase noise on linear dispersion in single-mode fiber-optic communication systems using a direct-

detection receiver. The effect of coherence time on dispersion, nonlinear interference (pulse-to-

pulse interactions), and intensity noise on the performance of a single-user communication system

is studied. A study of the effects of phase uncertainty in the received pulses due to timing jitter in

ii

modern lasers is also presented.

Using the Volterra series transfer function (VSTF), the effects of linear dispersion and fiber

nonlinearities on the performance of single-user fiber-optic communication systems are studied;

the criterion used is the signal-to-interference ratio (SIR) of intensity of the optical field and an

upper bound on the probability of error at the receiver. The model helps us choose input pulse

parameters to maximize the SIR (minimize probability of error) and design the complete optical

link or design lumped nonlinear equalizers at the receiver to compensate for the effects of linear

dispersion and fiber nonlinearities. Using the analytical expressions provided by the VSTF and the

upper bound on the probability of error, optimal dispersion parameters for the fiber segments in a

fiber amplifier based optical communication system are obtained to minimize the linear dispersion,

fiber nonlinearities and ASE noise. The effect of input power levels and amplifier gains on the system

performance in several different possible configurations of a point-to-point optical communication

system is studied. Finally, useful mathematical expressions for studying the nonlinear effects in

WDM systems are derived and possible ways of optimally extracting the information transmitted by

different users are discussed.

iii

Contents

1 Introduction 1

1.1 Fiber-optic Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Optical Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Linear Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Linear (Chromatic) Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.3 Fiber Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Optical Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Optical Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 System Analysis and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Signal Degradations within Fiber-optic Systems 22

2.1 Degradations Caused by Laser Source Imperfections . . . . . . . . . . . . . . . . 22

2.2 Degradations Caused by the Optical Fiber . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Nonlinear Schroedinger Wave Equation . . . . . . . . . . . . . . . . . . . 26

2.2.2 Linear Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.3 Nonlinear Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.4 Optical Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Photo-detector and Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iv

2.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Volterra Series Transfer Function (VSTF) 40

3.1 Derivation of Volterra Series Transfer Function . . . . . . . . . . . . . . . . . . . 43

3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Potential Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.1 Two-signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.2 Nonlinear Equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.3 Optimal Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 Summarized Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Linear Dispersion in Fiber-optic Communications 68

4.1 Derivation of SIR for Arbitrary Light Source . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Completely Coherent Light . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.2 Completely Incoherent Light . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.1 Communication System Performance . . . . . . . . . . . . . . . . . . . . 78

4.3 Summarized Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 System Design 89

5.1 Transfer Function of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Receiver Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3 Modified Chernoff Bound (MCB) . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Conclusions and Future Work 110

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

v

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A Analysis of WDM Systems Using VSTF method 116

vi

List of Figures

1.1 A block diagram of a typical fiber-optic communication system. . . . . . . . . . . 3

3.1 Normalized square deviation of the output field for no Raman effect and no dis-

persion for different input peak powers. The SSF method from the true solution is

shown with lines, the third-order VSTF method from the true solution is shown with

’ � ’, and the fifth-order VSTF method from the true solution is shown with ’+’. . . 50

3.2 Normalized square deviation of the output field for no Raman effect for different

input peak powers the third-order VSTF method from the SSF method, shown with

lines, and the fifth-order VSTF method from the SSF method, shown with ’ � ’. . . 52

3.3 Magnitude squared of the Fourier transform of output field. . . . . . . . . . . . . . 53

3.4 Normalized square deviation of the output field of third- (shown with lines) and fifth-

order VSTF method (shown with ’ � ’) from SSF method, showing the dependence

of NSD on input RMS pulse-width for a length of ��������

km. . . . . . . . . . . . 54

3.5 Interference-to-signal ratio due to the presence of a pump pulse at different frequencies. 58

3.6 Plots of output intensity of completely coherent light for different power levels. . . 62

3.7 Plot of RMS widths of output pulses, showing the effect of peak input power on

dispersion of the input pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.8 Output SIR as a function of symbol period for different power levels. . . . . . . . . 65

3.9 Output SIR as a function of peak pulse power for different symbol periods. . . . . 66

vii

4.1 Output pulse shapes for different levels of coherence of light for input pulses of�

psec RMS width: (a) GVD dominant case (b) operation at zero-dispersion wave-

length, ��� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Plot of RMS pulse-widths of output pulses, showing the effect of coherence time on

dispersion: (a) GVD dominant case (b) operation at zero-dispersion wavelength, ��� . 774.3 Plots of different waveforms of expected received signal (signal and mean nonlinear

interference) and �����

������������� showing the effect of pulse separation on the nonlin-ear interference: (a) GVD dominant case (b) operation at zero-dispersion wavelength. 79

4.4 Plot of SIR � , SIR ��� , SIR �� , SIR � , and SIR ��� showing the effect of coherence timeand timing jitter for a fixed symbol period of � � psec and a pulse-width of � psec,�

’s indicate the asymptotic values, �� � � and �! �#" , calculated from formulae inSections 4.1.1 and 4.1.2: (a) GVD dominant case, (b) operation at zero-dispersion

wavelength, ��� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5 Plot of SIR � (shown with ’o’s), and SIR ��� , showing the effect of coherence time

for an input RMS pulse-width of�

psec and different symbol periods for (a) GVD

dominant case (b) operation at zero-dispersion wavelength. . . . . . . . . . . . . . 83

4.6 Plot of the (a) SIR � (shown with ‘o’s) and SIR �� , and (b) SIR $ showing the effect ofcoherence time for a symbol period of � � psec for different pulse-widths for a GVDdominant case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7 Plot of the (a) optimal input pulse-widths and (b) SIR $ ’s for optimal pulse-width fordifferent symbol periods for a GVD dominant case. . . . . . . . . . . . . . . . . . 86

5.1 Typical communication systems used to demonstrate the design procedure. . . . . . 92

5.2 Block Diagram of the cascade of % th fiber amplifier and % th fiber segment. . . . . . 935.3 Input pulse shape used. The input pulse corresponding to &'� is shown with a solid

line to stress that this is the bit of interest. . . . . . . . . . . . . . . . . . . . . . . 102

viii

5.4 Upper bound on the probability of error using the optimal dispersion map for differ-

ent configurations shown in Figure 5.1 as a function of amplifier gain for different

input powers � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.5 The optimal total accumulated dispersion parameter determined by minimizing the

MCB for different configurations shown in Figure 5.1 and for different input powers.

The lines indicate dispersion parameters for amplifier gains close to the optimal

amplifier gains and the ’ � ’ indicate dispersion parameters for amplifier gains higher

by��� �

dB per amplifier than the optimal amplifier gains. . . . . . . . . . . . . . . . 107

ix

List of Tables

3.1 Various components of the output signal due to fiber nonlinearities and the presence

of pump pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

x

Chapter 1

Introduction

The main objectives of this study are to (i) derive analytical expressions for the nonlinear (Volterra

series) transfer function of an optical fiber, (ii) analyze the effect of phase noise (or source coherence)

on linear dispersion in direct-detection fiber-optic communication systems, (iii) analyze the effects

of linear dispersion and fiber nonlinearities in fiber-optic communication systems that use direct-

detection receiver, (iv) develop a mechanism to choose input pulse parameters to maximize the

signal-to-interference ratio (SIR) of light incident on the photo-detector, and (v) design the entire

fiber-optic communication link to minimize the effects of linear dispersion, fiber nonlinearities, and

amplified spontaneous emission (ASE) noise from the fiber amplifiers. This chapter briefly describes

how an optical fiber communication system works and how linear dispersion, fiber nonlinearities,

and ASE noise affect its performance.

1.1 Fiber-optic Communication Systems

The last decade has witnessed an unprecedented increase in data traffic and one main strategy has

emerged to handle this increase. Fiber-optic networks are used to transfer data between remote but

fixed switches at very high data rates. Data is then transferred to the nearby destinations by ei-

ther wireless (possibly mobile) or by wired copper links. The huge bandwidth of the optical fiber

is exploited to obtain high data rates between these switches, which are separated by large dis-

1

tances. Optical communication systems are becoming more and more complex due to attempts to

utilize the enormous bandwidth provided by the optical fiber; optimal system design is of great

interest. In spite of the possibility of communicating at rates of Terabits per second (Tbps) using

optical fibers, the linear dispersion in the fibers, fiber nonlinearities, ASE noise from optical am-

plifiers, and the low-speed electronic interfaces at the destinations limit data rates to Gigabits per

seconds (Gbps). To exploit this large bandwidth, a large number of channels are multiplexed with

an electronic interface for each channel on the same fiber. These channels are either multiplexed in

time (time-division multiplexed, TDM), wavelength (wavelength-division multiplexed, WDM), or

in signature-sequence (code-division multiplexed, CDM). Since optical links use guided medium,

they are expensive to install and maintain; multiplexing numerous channels is an effective approach

to minimize the cost.

Figure 1.1 shows a block diagram of a typical fiber-optic communication system using erbium

doped fiber amplifiers (EDFAs). A (possible) booster amplifier used to increase power transmitted

by a low power laser is shown, a possible receiver pre-amplifier used to increase the sensitivity of

the receiver, and a few in-line amplifiers used to compensate for the power losses in the optical fiber

are also shown.

Current commercial fiber-optic communication systems (e.g., transpacific fiber-cable system

(TPC-5), fiber-optic loop around the globe (FLAG) [1] system) use semiconductor lasers operat-

ing at a wavelength of � � � � � � � m, that can generate a series of ��� - � � � psec pulses of maximuminstantaneous power of � � mW. This stream of pulses is on-off modulated by the data to be trans-

mitted (i.e., each bit is transmitted by either sending or not sending a pulse). In WDM system, each

user’s pulse stream is modulated on a carrier with a different wavelength, and these pulse streams

are multiplexed to increase the overall data transfer possible on a single fiber. At the receiver, the

pulse stream from each user are separated by optical filtering and individual stream of pulses is

photo-detected, (sometimes integrated over the symbol period) and compared with a threshold to

determine whether a pulse is present in a given slot or not.

2

� � �� � � �

������ � �

� � � ������

�� � � �

������

Laser Receiver

BoosterAmplifier

ReceiverPre-amplifier

EDFA EDFA EDFA

��� ��� �. . .

Figure 1.1: A block diagram of a typical fiber-optic communication system.

Long-distance fiber-optic systems can broadly be categorized into terrestrial long-haul systems

and undersea systems. Networking is an important issue for terrestrial systems whereas undersea

systems are currently point-to-point links. Terrestrial systems are typically of��� �

km in length,

operating with optical amplifiers spaced at��� �

km. The undersea systems are typically of � ��� � km

in length, with the amplifier spacing of about � � km. The amplifier spacing is small to keep the

accumulated ASE noise from the large number of amplifiers to a minimum.

Unfortunately, the information transfer is not perfect. Laser imperfections like phase noise,

timing jitter, modal distribution of light from a laser, and nonlinearities introduced by the modulation

process affect the performance of a digital communication system. Linear attenuation, dispersion,

nonlinearities from the fiber, and ASE noise from optical amplifiers limit the data rate that can be

achieved and the number of users that can be placed on a given optical fiber. In addition, detector

response time, the statistical nature of the photo-detection process, which introduces shot noise, and

thermal noise from the electronic circuits increase the probability of error considerably.

1.2 Optical Transmitter

Most of the study of lasers in the recent past has concentrated on the development of highly coherent

lasers that can generate pulses as short as�

psec with coherence times as large as� � sec. It is hoped

that because of the availability of such short pulses, very high data rates can be achieved and using

low attenuation and low dispersion fibers (or using fiber amplifiers with dispersion compensation),

the performance of the optical fiber communication systems can be increased enormously. However,

such short pulses suffer more from linear dispersion and fiber nonlinearities, even when we use these

3

low attenuation and low dispersion fibers. For a given fiber length, it has been shown that because

of chromatic (linear) dispersion, the received pulses are wider if a short pulse is sent than if a wider

pulse is transmitted (as long as the input RMS pulse-width is smaller than� �

psec) [2, 3].

The stream of pulses from the laser is modulated with the data stream to be transmitted. This

modulation can be done either internally or externally. An internal modulator is driven by the elec-

tric current representing the data stream and the output of the laser is the modulated output. External

modulation involves modulating the stream of pulses from the laser with the data stream. Either

method of modulation introduces mild nonlinearities (produces chirp), which affects the perfor-

mance of the system [4, 5]. Typically the laser and the modulator are packaged together to reduce

coupling losses and other stray effects.

When a laser is on-off modulated with electric pulses representing the data stream, there is a

delay between the actual generation of the optical pulse from the time the electric pulse is applied.

This delay depends on the optical power build-up in the laser, which depends upon the bias level

of the semiconductor laser diode. When the electrical pulse exceeds the bias level of the laser,

the optical build-up starts [6]. The quantum component of the intensity noise (is an important

consideration in analog communication systems, where it is called relative intensity noise (RIN) [7])

adds to this signal making the actual pulse generation time random, giving rise to random timing

jitter. In mode-locked lasers, the timing jitter has a periodic autocorrelation properties owing to the

saturation of the laser gain medium. Timing jitter is typically on the order of�

psec for pulse-widths

of about � � psec [8].

In most studies of optical communications, it is assumed that the lasers generate monochromatic

light; however, because of finite cavity length in the laser oscillator, the light produced by a laser con-

sists of numerous distinct modes, each having very small line-width. These spectral lines (modes)

are broadened by various mechanisms in the laser as well as the fiber. Homogeneous broadening

is caused by the linear dispersion of the laser gain medium. Inhomogeneous broadening is mainly

due to (i) Doppler broadening, due to different atoms in the laser medium having different kinetic

4

energies and hence having different apparent resonance frequencies as seen by the applied signal,

and (ii) lattice broadening, in which various atoms see different frequencies due to their positions in

the lattice (local surrounding). These broadening mechanisms increase the line-width of each mode

generated, producing phase noise. Modulation with the pulse shape and fiber nonlinearities also

broaden the already broadened spectrum, increasing the severity of the linear dispersion and fiber

nonlinearities. It is well understood in the literature that the broader the line-width (i.e., the more

the phase noise), the worse the dispersion in direct-detection receivers [9, 10].

Alternatively, the phase noise at the laser can be mathematically modeled as randomly occurring

spontaneous emission events [6], which cause random changes (in magnitude and sign) in the phase

of the electro-magnetic field generated by the laser. Therefore, as time evolves, the phase executes

a random walk away from the value it would have had in the absence of spontaneous emission.

The phase noise spectrum has two components, one low frequency component that has�����

or�������

characteristic up to around�

MHz, and a white component (quantum noise) that is associated with

quantum fluctuations and is the principal cause of line broadening. Analysis of the much smaller

low-frequency component is quite tedious and attention can be restricted to quantum noise only.

It has been recognized that phase noise can severely affect the performance of coherent detection

of the information bearing signal [11, 12, 13, 14, 15, 16, 17, 18]. Considerable attention has been

paid to characterize phase noise in different kinds of lasers [8] and attempts have been made to

reduce phase noise in lasers [19]. The effect of coherence time on nonlinear processes like self-phase

modulation has been studied in [20]. Saleh [21] considered the effect of phase noise on completely

coherent and completely incoherent light when direct-detection receivers are used. Incoherent light

(intensity) gets filtered linearly [9], yet with an impulse response that is magnitude squared of the

impulse response for the coherent light (field). When coherent light is incident on the photo-detector,

the electric current is the sum of the intensity of different pulses taken individually and other cross-

terms due to pulse-to-pulse interactions. The cross-terms introduce a nonlinear interference term

which needs to be included in the error analysis at the receiver.

5

1.3 Optical Fiber

There are three major challenges faced in long-distance optical communication systems: attenuation,

linear dispersion, and fiber nonlinearities. Linear attenuation reduces the power level of the signal

below the thermal noise threshold at the receiver increasing the probability of error. Attenuation

is typically on the order of� � � dB/km, and limits the transmission distance to about ��� � km. Fiber

amplifiers are an attractive means of compensating for attenuation, since they typically have low

noise, high bandwidth, and low cost. Amplifiers gains are in the range of � � dB, so for a transmission

distance of��� ���

km, we require about 10 amplifiers. However, amplified spontaneous emission

(ASE) noise (proportional to the amplifier gain) adds to the amplified signal, which accumulates

with each amplification stage in the link, thereby degrading the signal-to-noise ratio and increasing

the probability of error at the receiver.

Linear dispersion spreads the pulses, while fiber nonlinearities introduce phase effects that accu-

mulate along the fiber due to the increased power levels (due to amplification) and large transmission

distances. Both these phenomena severely affect the performance of the communication system by

introducing inter-symbol interference (ISI) and inter-channel interference (ICI). The typical limits

on transmission distances set by linear dispersion and fiber nonlinearities are � ��� km and � � � km,respectively. These two deleterious effects can be reduced by three methods: input pulse shaping

(chirping, phase encoding, polarization scrambling, etc.), dispersion management (using dispersion

compensating fiber), or using fiber gratings.

The optical fiber supports various number of modes of the optical wave. These different modes

travel at different speeds owing to the difference in their path-lengths in the fiber. This gives rise to

what is known as modal dispersion. By adjusting the fiber core radius and core-cladding index differ-

ence, we can allow only one mode to propagate. For high-speed long-haul communication systems,

consideration can be restricted to these single-mode fibers (SMF), and the modal dispersion can

assumed to be zero. There are other random nonlinear problems like polarization mode dispersion

6

(PMD) [22, 23] that affect these long-distance transmission systems. Physically, PMD has its origin

in the birefringence that is present in any optical fiber. While this birefringence is small in absolute

terms in fibers, the corresponding beat length is only about���

m, far smaller than the dispersive or

nonlinear scale lengths which are typically hundreds of kilometers. This large birefringence would

be devastating in communication systems but for the fact that the orientation of the birefringence is

randomly varying on a length scale that is on the order of��� �

m. The rapid variation of the bire-

fringence orientation tends to make the effect of the birefringence average out to zero. The residual

effect leads to pulse spreading, referred to as polarization mode dispersion (PMD). PMD is signif-

icant only in very long-haul systems (lengths �� � � �

km); therefore, we ignore this effect in this

work, since we concentrate on links varying from��� � � � � � km. The remaining signal degradations

caused by the fiber are attenuation, chromatic (linear) dispersion and deterministic nonlinearities.

1.3.1 Linear Attenuation

Linear attenuation and dispersion are widely studied in fiber-optic systems, as they limit the rate

and reliability of information transfer [21, 24, 25, 26]. Attenuation reduces the power levels, thus

scaling the signal as the exponent of the length of the fiber. Absorption of an optical wave occur

mainly because of material absorption and Rayleigh scattering. Rayleigh scattering is generally

described by the absorption coefficient, which is known to obey the relation � ����� [20], where

��

� ���-� �

dB � m � /km for silica (depending on the constituents of the fiber core) and � is theoptical wavelength. Modern optical systems are expected to operate in the low-loss region of the

spectrum,��� �� � m- � ��� � � m, where Rayleigh scattering is less than � � � dB/km for silica. Absorption

reduces the intensity of light in the fiber, which in turn reduces the magnitude of fiber nonlinearities

thus limiting the length over which nonlinear interactions are effective. Fiber amplifiers can be

used to restore the signal to its original power levels; however, that increases the effect of fiber

nonlinearities considerably. We introduce optical amplifiers in Section 1.4.

7

1.3.2 Linear (Chromatic) Dispersion

Linear dispersion has been recognized as the primary limiting factor on the maximum data rate for

single-user optical fiber communication systems [24, 25, 26, 27] that use using optical amplifiers to

remove the effects of attenuation. The linear dispersion introduces linear inter-symbol interference

(ISI) in single-user and multi-user systems, and fiber nonlinearities introduce nonlinear ISI and

inter-channel interference (ICI) in multi-user systems.

The second-order dispersion, commonly known as group velocity dispersion (GVD), widens

the pulses, thus introducing inter-symbol interference (ISI). GVD is the primary limiting factor on

the maximum data rate achievable, as it makes the information retrieval at the receiver difficult for

long lengths of fibers. Thus, recent fiber-optic systems operate at the zero-dispersion wavelength � �(��� � � m for silica), at which GVD is zero, thus avoiding its effects. GVD consists of two distinct

components, one due to material dispersion (depends on the material used to fabricate the fiber) and

the other due to waveguide dispersion (depends upon core radius and the core-cladding index differ-

ence). By controlling the waveguide dispersion, ��� can be shifted to the vicinity of ��� � � � m wherethe fiber loss is minimum, giving us what are commonly known as dispersion-shifted fibers [20].

Using multiple claddings, waveguide dispersion and material dispersion can be carefully controlled

to give dispersion-flattened fibers; these fibers have low dispersion (�

ps/km � nm) over a relatively

large wavelength range��� � � - ����� � � m for use in WDM or CDM application.

For wavelengths � � � � , GVD is positive and the fiber is said to exhibit normal dispersion,i.e., higher frequency components of an optical pulse travels slower than the lower frequency terms.

For � � ��� , GVD is negative and the fiber is said to exhibit anomalous-dispersion. The residualdispersion in long haul applications is typically compensated by inserting a fiber having GVD with

an opposite sign that of the original fiber, so that the phase changes induced by the original fiber

are cancelled by the phase changes introduced by the inserted fiber. This methodology is generally

called dispersion mapping.

8

We have to include higher-order dispersion terms for wavelengths closer than a nanometer to � � .The third-order dispersion, which is much smaller than GVD, is the primary concern for the present

fiber-optic links operating at � � . The third-order dispersion introduces an oscillatory structure nearone of the pulse edges (intensity), i.e., the pulse shape is asymmetric [20, 28]. For positive (negative)

third-order dispersion, oscillations appear near the trailing (leading) edge of the pulse. For polariza-

tion maintaining single-mode fibers operating at � � , polarization mode dispersion, modal dispersion,and group velocity dispersion (GVD) can be ignored; third-order dispersion is the major limitation

on the data rate. Using single pulse propagation results, Marcuse [28] evaluated the effect of input

spectral width (line-width) and the input pulse-width on the output pulse-width (approximating the

output pulse shape with a Gaussian shaped pulse). It has been shown that data rates of � � � Gbps canbe achieved over fiber lengths of � ��� km when limited only by third-order dispersion [20].

Research has been conducted into choosing system parameters like input pulse-width, input

chirp, etc. to minimize output RMS pulse-width [3, 28], inter-symbol interference in the electronic

domain [24], and probability of error in the electronic domain [29, 30]. All these studies model dis-

persion properly but does not take into account the effect of laser phase noise or fiber nonlinearities

on the performance of the communication system.

The electronics and the impulse response of the photo-detector at the receiver limit the data rate

at each link (on each channel in the fiber) to about� �

Gbps (pulse separations of��� �

psec), and

typically� ��� ���

voice channels are multiplexed on such a link. These voice channels are typically

multiplexed in time, i.e., TDM. Such wide pulses do not suffer much from dispersion and together

with the use of fiber amplifiers, allow us to have long spans of the fiber without any significant

dispersion or attenuation. Future systems are expected to use � � - ��� psec input pulses, generatedby highly coherent semiconductor lasers for single-user systems and even smaller pulse-widths for

multi-user systems like TDM and CDM. For CDM systems, optical preprocessing is required before

photo-detection to separate the multiple signals so that the total throughput is not limited by the

photo-detector response. Assuming a minimum signature-sequence length of��� �

, pulse-widths of

9

�psec are required to obtain the same data rate in CDM systems as the single-user system; disper-

sion effects are more severe, which combined with fiber nonlinearities make long fibers impractical.

Thus, a realistic study of the effect of linear dispersion and fiber nonlinearities on the system perfor-

mance is required to harness the potential of these high data rate fiber-optic systems.

1.3.3 Fiber Nonlinearities

It is well recognized that fiber nonlinearities limit the performance of current optical communication

systems [20, 31, 32, 33, 34]. Various nonlinear effects such as self-phase modulation (SPM) [35],

cross-phase modulation (CPM), stimulated Raman scattering (SRS), stimulated Brillouin scattering

(SBS), and four-wave mixing (FWM) can cause significant cross-talk in multi-user systems such as

WDM [31, 33] and CDM [34] systems. Nonlinear effects increase with increase in the power level

of the laser and the length of the fiber. In previous single-user optical fiber communication systems

using power levels below�

mW (in a typical single-mode fiber of cross-sectional area��� � m � , this

power gives an intensity of � � � MW/m � , which increases the refractive index of the fiber core

by �� � ��������� ) and link lengths of � � km, fiber nonlinearities did not play a significant role in

degrading the system performance.

Current systems use peak input power levels of less than�

mW over link lengths exceeding� � � �

km. WDM systems using solitons as basic pulse shapes anticipate peak power levels of���

mW

per user at each frequency and CDM systems are expected to transmit � � - � � times that power at asingle carrier frequency. Due to the presence of multiple users in the system, power levels are much

higher in multi-user systems; fiber nonlinearities introduce inter-symbol interference (ISI) and inter-

channel interference (ICI) [36], leading to deterioration in the performance of these systems. The

availability of low attenuation (using fiber amplifiers) and low dispersion fibers makes it possible

to transmit data over long lengths without requiring reconstruction of the signal. These increased

lengths together with higher power levels make the study of fiber nonlinearities useful at this juncture

of time, especially for future WDM and CDM systems [33, 34, 37].

10

Fiber nonlinearities, especially self-phase modulation spreads the spectrum of the signals by

creating new frequencies. The frequency spread of the signal depends on the rate at which the

signal changes; the narrower the pulses, the steeper the edges and the more the time spreading due

to higher-order dispersion and higher-order fiber nonlinearities. Higher-order dispersion and fiber

nonlinearities have to be included in an accurate model when pulse-widths are smaller than� �

psec

in long-distance systems.

Current research in optical communications uses root mean-square (RMS) pulse-width, optical

and electrical signal-to-interference (SIR), power loss and crosstalk from other users (estimated from

single pulse propagation experiments) [31, 33] as measures of degradation. Other effects on pulse-

shape (e.g., spreading and steepening of the pulse) and the resulting effect on the probability of error

of a communication system have not been studied. Due to the high coherence of light from modern

lasers, the nonlinear interaction between pulses is significant. Fiber nonlinearities introduce addi-

tional pulse-to-pulse interactions that are generally ignored in the current design process. Analytical

expressions provide a powerful tool for the study and accurate modeling of these nonlinear (indi-

vidually and cumulatively) effects. In this dissertation, we present a closed-form nonlinear transfer

function that can describe linear dispersion, fiber nonlinearities and the pulse-to-pulse interactions

in single-user and multi-user fiber-optic communication system.

The generalized nonlinear Schroedinger (NLS) wave equation is commonly used to describe the

slowly varying complex envelope of the optical field (valid for pulses with widths as short as � � -

� � fsec) in the fiber. The NLS equation is derived from Maxwell’s equations either completely in thetime-domain [20, 34] or completely in the frequency-domain [38, 39]. NLS equation has also been

derived including the waveguide properties of the fiber [40]. It can explain most of the linear and

nonlinear phenomena in an optical fiber. Previous methods of solving the NLS equation, either in

the time domain or in the frequency domain, have been recursive and numerical. It is not practical

to optimize the system performance using such methods when many variables are considered since

the optimization must be performed numerically. The availability of an analytical model of the NLS

11

equation allows the design of high-performance optical amplifier based long-haul systems, which

requires the minimization of linear dispersion, fiber nonlinearities, and ASE noise from amplifiers

simultaneously.

The NLS equation is generally solved using (recursive) numerical methods such as the Split-step

Fourier (SSF) method, finite-difference methods [20] or the Runge-Kutta method [38]. The split-

step Fourier method divides the fiber into small segments and the output of each segment is found

numerically using the output of the previous segment as the input. Very small segment lengths are

required to get accurate results; therefore, the computational cost becomes prohibitively high for

long lengths of fibers and short pulse-widths, which are important for future systems. If we increase

the segment lengths to reduce these effects, the accuracy in representing the fiber nonlinearities

decreases. Furthermore, the discretization errors can accumulate along the length of the fiber, thus

generating erroneous results.

Taha et. al. [41] have investigated various recursive solutions for the time-domain NLS equation,

and have shown that the SSF method is the most accurate and computationally cheap algorithm

available (when the nonlinear portion of the NLS equation is implemented in the time domain).

Several variations of the SSF method have been proposed recently, one based on an orthonormal

expansion of the output field [42] and another based on a wavelet expansion [43]; however, both

of these methods are still recursive, and may suffer from the same problems as the original SSF

method.

These recursive methods of solving the NLS equation do not give any indication of how to re-

move the nonlinear effects, especially when dealing with communication applications, where a series

of pulses are usually transmitted and nonlinear interaction between the pulses should be considered.

A closed-form analytical description of the linear and nonlinear effects is required to optimize com-

munication system performance. An analytical method based on a Volterra series transfer function

(VSTF) is presented in this dissertation.

12

1.4 Optical Amplifiers

Linear attenuation reduces the signal levels as an exponential function of the length of the fiber.

Previous optical systems had repeaters every���

km, with the optical signals photo-detected, con-

verted to electronic signals, and then converted to optical signals using optoelectronic components.

With the advent of all optical fiber amplifiers, the need for the slow interface between optics and

electronics is bypassed and higher data rates are now possible. The fiber amplifiers have a typical

spacing of about � � - � � � km that restore the signal powers by optical means to the original values.Using fiber amplifiers, link lengths of the order of

��� � �km are in operation without ever requiring

reconstruction of the signal [27].

The invention of the erbium-doped fiber amplifier (EDFA) paved way for the development of

high bit rate, all-optical ultra long-distance communication systems [27, 44, 45, 46]. There are

three major types of optical amplifiers available for use in optical communication systems: fiber

amplifiers (mostly erbium doped, EDFA), semiconductor optical amplifiers (SOA) [47], and Raman

amplifiers. Raman amplifiers require pump powers of the order of� � �

-�

W which can not be obtained

from current semiconductor lasers and therefore these are not used. SOAs are polarization sensitive,

suffer more coupling losses, and introduce more inter-channel interference than the fiber amplifiers.

Fiber amplifiers have low insertion loss, high gain, large bandwidth, low noise, and low crosstalk;

therefore, they are most commonly used for long-haul applications. We consider only the fiber

amplifiers in this work.

Although fiber amplifiers provide a good means of compensating for attenuation, they add con-

siderable amount of ASE noise, that accumulates as the number of amplifiers increases. The ASE

noise together with the interaction of linear dispersion and fiber nonlinearities (modulation instabil-

ity) gets amplified thus affecting the receiver statistics. So while reducing the problems introduced

by linear attenuation, optical amplifiers introduce (of course less significant) ASE noise problems.

Fiber amplifiers use the energy provided by the laser pump to amplify the optical signals pro-

13

portional to the pump power and length of the doped fiber used in the amplifier. Power conversion

is most efficient for EDFAs when the pump wavelength is close to � �

nm, although recently pro-

posed distributed amplifiers use pump wavelength of� � � � nm to keep the pump power loss due to

fiber attenuation to a minimum. Amplification efficiencies of� ���

dB/mW of pump power have been

achieved at a pump of power of about���

mW. The pump power is fed to the amplifiers either in

the same direction as the signal (forward-pumping), or opposite direction to the signal (backward-

pumping), when higher gains are desired, pump power is fed from both directions using two pump

lasers. This has created considerable interest in bi-directional systems, where the huge bandwidth

of the fiber can exploited to a fuller extent. The intrinsic gain spectrum (the wavelength range

over which the signal experiences significant gain) of pure silica is about���

nm (full-width at half

maximum, FWHM). The gain of alumino-silicate glasses with homogeneous and inhomogeneous

broadening increases the amplifier bandwidth to about � � nm [20].After the pump and signal are coupled into the erbium doped fiber, there is power conversion

from pump to signal. The electrons in the doped fiber transition to a higher energy level after stim-

ulated absorption of pump energy, which return to their original energy states by either stimulated

emission or spontaneous emission. Stimulated emission due to the signal power provides the basic

amplification mechanism in the EDFA. The spontaneous emission causes the electrons to return to

the ground state randomly, thereby adding (incoherent) spontaneous noise to the amplified signal,

which gets amplified further by the amplifier gain mechanism, producing what is called amplified

spontaneous emission (ASE) noise. Therefore, both the noise power and the amplifier gain are pro-

portional to the pump power and doped fiber length, i.e., the larger the gain, the more noise power

is added to the amplified signal. Optimum pump powers and doped fiber lengths can be found to

improve the performance, which is quantified by the noise figure that is defined as the the ratio of

the signal-to-noise ratios before and after the amplifier. Typical noise figures are on the order of�

dB, and typical doped fiber lengths are���

m. For input power levels below � � dBm, the amplifieracts like a linear amplifier; however, for higher signal powers the excited carrier concentration de-

14

creases, deteriorating the gain mechanism, thereby reducing the gain. The fiber amplifier is said to

be in saturation and acts like a nonlinear device for higher signal powers.

1.5 Optical Receiver

The photo-detector is a square-law device that detects the magnitude-squared of the real envelope

of the optical field. The photo-detector produces a number of electrons following a Poisson process

with rate proportional to the intensity of the incident optical field. These electrons thus generate

an electric current proportional to the number of photons incident on the photo-detector. In a more

accurate model the impulse response of the photo-detector (due to diode capacitance and carrier

transit times) can also be included in the Poisson model, giving what is known as a filtered Poisson

model [48]. The current from ASE noise component and the beat terms between the signal and

the ASE noise add to the electronic/thermal noise from the electronic circuitry, increasing the noise

content of the output current from the photo-detector.

There are two methods of extracting the information bearing signal from the received optical

field, coherent detection and direct-detection. Coherent detection involves translating the incoming

optical wave to an intermediate frequency (IF) by mixing with an optical wave from a local oscil-

lator. The local oscillator output field is added to the incoming signal and then the photo-detector

detects the magnitude-squared of the real envelope of the sum signal and then low-pass filters it, thus

providing a scaled version of the incoming optical field at the IF (i.e., phase information is retained).

Due to the availability of highly coherent short pulse-width sources, considerable attention is being

paid to the possibility of coherent receivers [11, 12, 49]. For multi-user systems (called optical fre-

quency division multiplexed (OFDM) or dense WDM) with coherent receivers, the channel spacing

can be small of the order of��� �

GHz, as isolating the channels at the receiver is relatively simple

using a local oscillator. However, the transmitting lasers and local oscillator are required to have

a wavelength stability of the order of� � � � � �

nm, which although has been achieved in commercial

15

systems, is still not very popular. Systems using coherent detection are likely to be more sensitive

to the effects of fiber nonlinearities and laser phase noise than the present direct-detection systems.

This has created considerable interest in finding ways of reducing the line-width of lasers [19].

Modern lasers generate highly coherent light (i.e., very little phase noise), thus reducing the effect

of phase noise on the received signal. Although coherent systems offer many interesting challenges

in the area of nonlinear modeling and system design, we have focused our work on direct detection.

Direct-detection uses the intensity of the real envelope of the incoming signal to make decisions

about the transmitted bits, i.e., the phase information in the optical field is not utilized. Direct-

detection receivers are very popular because of their low cost and their insensitivity to the state of

polarization of the received signal. Since the intensity of the information bearing signal only can

be observed and the phase is lost at the detector, it is generally assumed that the phase noise does

not affect the performance of a communication system using direct-detection [3, 24, 25, 29, 30].

We show that phase noise does affect the performance of a direct-detection receiver. For multi-user

systems using direct-detection systems (called WDM systems), the channel spacing is required to

be larger than � � � GHz, so large bandwidth optical amplifiers are required.

1.6 System Analysis and Design

Several phenomena limit the transmission performance of long-haul optical transmission systems in-

cluding noise, dispersion and nonlinearities [50]. There are various methods of reducing the effects

of these fiber imperfections. Given the statistical properties of the input signals and the channel

description, an optimal receiver can be designed to minimize the deleterious effects of the fiber;

however, such a method is not popular or practical for optical communication systems due to lim-

ited optical processing capabilities. Unlike the wireless or satellite communication channels, the

fiber-optic communication channel itself can be tailored to provide the required performance; the

fiber parameters can be determined that provide the optimal performance (of course, within few

16

practical limitations). Therefore, the problem reduces to either choosing the input parameters to

suit a given channel or choose the channel to suit given input parameters. In this work, we design

the system by either choosing the input peak power and input pulse-width to maximize the optical

signal-to-interference ratio (SIR) at the receiver or choosing the channel parameters for a given in-

put parameters to minimize the probability of error. We do not consider the optimization of receiver

processing and leave that for future work.

In the initial long-haul systems using optical amplifiers, it was believed that if we maintain the

total losses to be zero and use dispersion shifted fiber (DSF), we can achieve the best performance.

However, when the system is operated at the fiber’s zero dispersion wavelength, the signals and the

amplifier noise (with the wavelengths close to the signal) travel at same velocities. Under these con-

ditions, the signal and the noise waves have long interaction lengths and can mix together. Linear

dispersion causes different wavelengths to travel at different group velocities in the SMF. Linear dis-

persion thus reduces phase matching, or the propagation distance over which closely spaced wave-

lengths interact. Therefore, we can use dispersion parameters intelligently to reduce the amount of

nonlinear interaction in the fiber. Thus, in systems operating over long distances, the nonlinear in-

teraction can be reduced by tailoring the accumulated dispersion so that the phase-matching lengths

are short, and the end-to-end dispersion is small. This technique of dispersion mapping has been

used in both single channel as well as WDM systems to reduce the nonlinear interaction between

signal and noise and different frequencies in WDM systems. Current WDM systems use non dis-

persion shifted fiber (NDSF) for most of the length, and rely on using short lengths of dispersion

compensating fiber (DCF) to get the total dispersion in the link close to zero. In the equi-modular

dispersion compensation scheme, there is a segment each of NDSF and DCF between the pairs of

amplifiers, thus keeping the total dispersion of each fiber segment between the amplifiers close to

zero.

When dispersion compensating fiber segments with negative dispersion are used in the optical

link, the interaction between dispersion and fiber nonlinearities introduces modulation instability

17

(MI), which is a parametric gain process. This parametric process amplifies the ASE noise over a

major portion of the spectrum, increasing the already accumulated noise at the receiver. Therefore,

the choice of dispersion parameters play a significant role in determining the performance of an

optical communication system. Even though there has been increasing interest in developing better

dispersion management schemes that minimize modulation instability [51, 52], the optimal disper-

sion management scheme has not been found, and there are no measures of optimality available.

Analytical results for describing MI are available only for a single fiber segment [20], and the effect

of having fiber amplifiers on modulation instability is not clear. Volterra series model includes the

effect of fiber amplifier parameters on modulation instability and the effects of modulation instabil-

ity on ASE noise, with statistical description of the output current at the receiver. This is an example

of a situation where the analysis is not possible with the SSF method due to lack of analytical ex-

pressions for the output field including all these effects; the VSTF method can excel as an excellent

design tool in this case.

For systems that have already been installed, fiber gratings are a very efficient means of com-

pensating for dispersion [53]. They are compact, passive and relatively simple to fabricate. With

the commercially available���

cm long phase masks, the bandwidth over which these gratings can

compensate for dispersion is increasing rapidly. Although it is possible to model and design the

optimal fiber gratings using the VSTF approach, we do not investigate this topic in this dissertation.

The most logical performance measure in the design of digital optical communication systems

is the probability of error. Unfortunately, analytical expressions for the probability of error are in-

tractable. Most of the performance evaluation methods in optical communication systems rely on

simulations, eye-diagrams and receiver�

. A majority of researchers rely on the receiver�

(of-

ten determined from eye-diagrams), which is proportional to the signal-to-noise ratio of the photo-

detector output current when a Gaussian distribution is used for the Poisson counting process (using

the central-limit theorem), i.e., it depends only upon the first two moments of output current of the

photo-detector. The probability of error predicted using receiver�

is very conservative [54], and

18

for the low probability of errors encountered in optical communication systems, we require tighter

bounds such as Chernoff bounds, saddle-point approximation, or approximations based on the the-

ory of large deviations [55].

The common performance measures used in other communication areas like error bounds on

the probability of error (e.g., Chernoff bounds) are often not employed in analyzing amplifier based

optical communication systems. The design of current amplifier based optical fiber systems thus is

empirical, based on simplified models, and systems parameters are varied in a heuristic fashion to get

the best performance from the system [51, 52, 56]. To design better systems, analytical methods for

studying the combined effects of dispersion, fiber nonlinearities, MI, ASE noise, and the detector

(square-law) nonlinearities are required. Ribeiro et. al. [54] have advocated the use of tighter

bounds on the probability of error. The moment generating function (MGF) for the output current

was derived and used to evaluate the performance of EDFA pre-amplified receiver. The effect of

thermal noise, photo-detector response, inter-symbol interference was included in the description of

the MCB. Their model makes some unwarranted use of the central limit theorem and assumes an

infinite optical bandwidth, which is not very practical. In this work, we derive a more accurate MGF

for the output current at the photo-detector, including the spectral distribution of the ASE noise,

which provides a more realistic bound on the probability of error.

To design better systems, we require analytical expressions for the performance criterion in

terms of the important system parameters. We derive such analytical expressions for the Chernoff

bound on the probability of error in terms of laser, fiber and amplifier parameters, including the

detector characteristics more realistically than those available in the literature. In this study, we lay

a foundation for further analysis of these effects on present and future fiber-optic communication

systems.

19

1.7 Dissertation Organization

Chapter 2 elaborates on various linear and nonlinear effects in a fiber, laser and photo-detector,

introducing mathematical notation to be used throughout this study. Two forms of the NLS equation

are presented, one derived from Maxwell’s equations in the time-domain and the other derived in

the frequency-domain.

Chapter 3 describes a general method of deriving a Volterra series transfer function (VSTF) to

model a single-mode fiber (SMF). We concentrate on showing the accuracy, advantages and limi-

tations of using the Volterra series model. A third- or fifth-order approximation to such a VSTF is

shown to provide an excellent match to the recursive methods such as the SSF method. Analysis of

the interference caused by a pump pulse (present at another frequency) on the signal pulse is given to

show the effectiveness of the VSTF method in modeling fiber nonlinearities in multi-frequency sys-

tems. We study the effectiveness of a nonlinear equalizer in restoring the original pulse shape. The

effect of fiber nonlinearities on the shape of a single pulse is studied, providing a way of choosing

the optimal input parameters (pulse-width and peak pulse power) required to get minimum output

RMS pulse-width. The effect of linear dispersion and fiber nonlinearities on the optical signal-to-

interference ratio (SIR) at the detector is presented, providing estimates of the input parameters

required to optimize the system performance.

We use the linear portion of the Volterra series model to study the effect of phase noise on

linear dispersion in direct-detection systems in Chapter 4. We derive the first two moments of the

intensity at the input of the photo-detector, and show that the moments for completely coherent

and completely incoherent light are special cases of those derived by Saleh [21]. We study the

effect of phase noise and dispersion on the pulse shapes and the determine the output RMS pulse-

width; we find optimal source parameters taking coherence time into account. We show that as the

source becomes more and more coherent to make pulse-widths smaller, dispersion effects (because

of decreased pulse-widths) and nonlinear interference or nonlinear noise due to the photo-detector

20

increases.

Using the overall nonlinear VSTF developed in Chapter 3, we derive an analytical expression

for the output field of the overall system in terms of transmitter, fiber and amplifier parameters. A

modified Chernoff bound (MCB) on the probability of error at the receiver is derived and used to

design a simple optical communication systems. As a part of the derivation, the moment generating

function (MGF) of the decision variable for an integrate-and-threshold detector is derived including

the effects of linear dispersion, fiber nonlinearities, and ASE noise from the optical amplifiers. To

show the power of the approach, optimal dispersion parameters were determined while varying the

power distribution along the fiber by varying the input peak power levels and the amplifier gains.

Four different possible amplifier chains are studied and it is shown that the configuration of the

amplifier chain is very important in determining the performance of the optical communication sys-

tems. We compare the performance of optimal dispersion parameters with equi-modular dispersion

compensating fiber to check in what conditions the approximations made in this work are valid, and

when more accurate analysis is required to get optimal dispersion parameters.

In Chapter 6, we conclude the dissertation with a few suggestions for other future work using

the analytical expressions provided by Volterra series model. An appendix is included to illustrate

the most important potential application of the VSTF method, namely, multi-user communication

systems, especially wavelength division multiplexed (WDM) systems. We show that VSTF can be

used to obtain more accurate expressions for how signals at different frequencies interact with each

other and how it can be used to design such complex systems.

21

Chapter 2

Signal Degradations within Fiber-opticSystems

This chapter introduces mathematical details of optical fiber communication systems, concentrating

on phase noise in lasers, the nonlinear Schroedinger (NLS) wave equation (derived from Maxwell’s

equations) to describe the behavior of the fiber and optical amplifiers, and the receiver statistics at

the photo-detector. We model the behavior of the system with phase noise and timing jitter at the

laser source, and linear dispersion and nonlinearities in the fiber, the gain and noise introduced by the

optical amplifier, and the square-law (nonlinear) nature of the photo-detector and shot and thermal

noise with photo-detection process.

2.1 Degradations Caused by Laser Source Imperfections

For intensity-modulated binary communication systems, the complex envelope of the optical field

(assuming monochromatic light) ��

��� � � from a laser ( � � � ) can be written as�

��� � � ���� ����� ��� $�� �

���� �� & ��� ��� ��� � � ��� � �����

������ $�� $! � �#"%$&� � ��� $�� (2.1)

where ' & ��( ��� �� are the information bits, which can take values in ' � � � ( , )

��� is the phase noise,��� is the symbol period, � is the peak power of the input pulse, and � ��� is the pulse shape. Thepulse generation timing jitter, ��� , for our purposes manifests itself as a random phase � ���* �+� �,��� $!

that is constant for each pulse. Since the carrier frequency - is generally very large, the phase

22

term )'� � - �� � can be modeled as independent from pulse to pulse and as uniformly distributedin

� ��� ����� . We can assume that )

��� and ' ) � ( ��� �� are statistically independent. Without loss ofgenerality, external modulation can be assumed to be used to keep the mathematics simple. We

introduce the random signal ��

��� so that we can study the effect of phase noise ) ��� only assuming�

��� is given, i.e., timing jitter information and data bits are given; we then consider the effectof phase noise and timing jitter together, where �� ��� becomes random because of the timing jitter' )��( ��� �� and unknown information bits ' & � ( ��� �� .

In digital communication systems using pulsed transmission, the wavelength of the carrier varies

across the pulse; there is a shift to the low frequencies on the leading edge of the pulse and a shift to

the higher frequencies on the trailing edge. This type of effect if usually called (linear) chirp. Typi-

cally, coupled with dispersion of a SMF, the chirp causes pulse broadening for wavelengths shorter

than the zero dispersion wavelength (ZDW) and for wavelengths longer than ZDW, it provides pulse

compression (used to compensate for dispersion). Chirp can also be accounted for by finding the

full-width at half-maximum (FWHM) line-width enhancement caused by chirp. Although it is pos-

sible to easily include chirp, we don’t do so in this work.

We can describe phase noise )

��� as a Wiener-Levy process

) ��� � ���� $� � ����� (2.2)where � ��� is a zero-mean Gaussian white noise process with spectral density � � . The first twomoments of )

��� can be easily seen as � � ) ����� � � and � � ) �

����� � ����� ��� , with ) � � � � . � � isequal to the laser line-width or the Lorentzian bandwidth. Phase noise can also be quantified by the

coherence time, defined as �� � �������� .Light from a laser is generally coherent only over a short period of time because of this phase

noise. The coherence of the source can be described by considering the autocorrelation properties

of� ��� � � ,

� � � ��� � � ��� � �� � � ��� ���� � ���! #"$� ���� �����

������� � � � � $�� � � $ �&% � � � � �� �����

���� ��� �&��� �(' % ' (2.3)23

Depending on the coherence time and the observation interval (interval over which intensity of

light is integrated to make the decisions about bits received, which is generally the symbol period),

the transmitted light can be assumed to follow one of three models.

Completely coherent light : The coherence time of the light is much larger than the observation

interval, which requires a highly coherent laser source like a mode-locked laser. When light

is completely coherent, there is no phase noise, i.e., � � � � , and � � ��� ����� � � " , and� � � ��� � � � � ��� � � ��� � �

���� �

��� � . Recently, research is being conducted into developinglasers that can generate highly coherent light in very short bursts, so that these lasers can be used

to send information bits at very high data rates. Mode-locked lasers can generate pulse-widths as

small as� � � �

psec and as large as��� �

psec. The coherence time can be as high as � � � sec [8, 57],thus having more than

� � � � pulses within the coherence time.Partially coherent light : The coherence time of light and the observation interval are comparable,

which is a more practical assumption for modern systems. Commercial semiconductor laser diodes

(LD) can generate pulses as small as���

psec, with coherence times of about�

psec (thus, the light is

almost incoherent). Distributed feedback (DFB) lasers can provide pulse-widths as small as � � psec

with an increase in coherence times to� �

psec [26], giving us partially coherent light.

Completely incoherent light : The coherence time is much smaller than the observation interval (bit

duration), which is true for LEDs and semiconductor laser diodes. For completely incoherent light,

� � � " and �! � � , giving � � � ��� � � � � � � � � ��� � � ��

��� � � � �� . LEDs generate true incoherentlight, with pulse-widths of about � nsec [26].

Most of the ultra-short pulse sources considered for future fiber-optic communication systems

are mode-locked lasers [20, 26]. The power spectrum of a mode-locked laser consists of � equallyspaced longitudinal modes which are locked in phase [8, 57]. The modes are only locked in phase

relative to each other; they can still share a common random phase (phase noise) which determines

the width of each mode. The mode spacing, ��� � , determines the repetition rate, � � � ������� , ofthe pulse train. The total bandwidth, � - , that is locked determines the pulse-width, � � �� � ,

24

whereas the full-width at half maximum line-width of the individual modes of the mode-locked

power spectrum, � ��� , determines the coherence time of the pulse train, � � ���� �� . For a typicalmode-locked laser [8, 57], �� �� � � sec and � � � � � � psec giving us approximately 1300 pulseswithin the coherence time. The RMS value of the timing jitter for these lasers is on the order of�

psec, and RMS amplitude jitter of ��

. Colliding-pulse mode-locked ring dye lasers [19] can

generate pulses as short as� �

fsec with a repetition time of���

nsec with a power of � W.To achieve light amplification in a laser, the amplifier gain must be high and in this high gain

region, the laser behaves in a nonlinear fashion [4, 5]; the output optical field is a nonlinear function

of the input electric current. Such laser nonlinearities are generally ignored in the analysis of com-

munication systems. Laser nonlinearities can be easily included in the Volterra series model to be

presented in Chapter 3. The only degradations due to the laser source considered in this study are

phase noise, timing jitter, and chirp.

2.2 Degradations Caused by the Optical Fiber

The polarization induced in an optical fiber is dependent upon the intensity of the light passing

through it. At low intensities, the polarization is a linear function of the applied field; however,

at high intensities this simple description is no longer valid. Polarization at a distance � from the

transmitter and time � , ��

��� � � , induced in an optical fiber by the field ��

��� � � (with a complexenvelope

�

��� � � ) with a central frequency of - � , is given by [20]

�� ��� � � ��������� � � � �� �� � � � �� �� �� � � � �� �� �� � � � �� � (2.4)� �� is a tensor of rank � � � that describes the first order susceptibility function of the material usedin the fiber, � � is the vacuum permittivity, and � �� denotes a tensor product between � and .

The linear susceptibility tensor � - � � � � � - � represents the dominant contribution to�� ��� � � � ���� ��� � ��� �� � � ��� � � , which is the linear polarization ���� ��� � � (ignoring waveguide prop-

25

erties of the fiber) given by

�� � ��� � � � ��� � �� � - � �� - � � ��� � �,� $ � - � (2.5)The nonlinear polarization �� � � ��� � � is mainly due to the third-order susceptibility, � � � - � ��- � ��- � �[20, 34], i.e.,

�� � � ��� � � � ��� � � � � � � - � ��- � ��- � � �� - � � � � �� - � � � � �� - � � � ��� � � � � �����&�����,� $ � - � � - � � - � � (2.6)We can show that �� � �

��� � � consists of a signal at - � and another term at � - � . The latter term isgenerally ignored in optical fiber communications as it lies outside the frequencies of interest.

� � � is responsible for second-harmonic generation and sum-frequency generation. � � � is zerofor � ��� � since the silica molecule is symmetric. However, dopants inside the fiber core can con-tribute to � � � under certain conditions.

The analysis of the effects of fiber nonlinearities in multiuser systems requires knowledge of the

dispersion (frequency-dependence) of the third-order susceptibility. Unfortunately, the dispersion

(frequency-dependence) of the third-order susceptibility in silica fibers is generally not known [20,

58, 59]; therefore, the third-order susceptibility is expanded in a Taylor series, and the coefficients in

the Taylor series are calculated from experimental results. Depending on the bandwidth of interest,

the higher-order terms in the Taylor expansion are ignored, and the analysis is performed with a few

of the coefficients. Whatever information various experimentally observable nonlinear effects do not

provide is lost in the process. If we expand both the linear polarization and nonlinear polarization

in a Taylor series, we obtain what is commonly known as the generalized nonlinear Schroedinger

(NLS) wave equation.

2.2.1 Nonlinear Schroedinger Wave Equation

The propagation of light in a guided medium is generally described by Maxwell’s equations. For

long lengths of fiber, the Nonlinear Schroedinger (NLS) wave equation is typically derived under a

few approximations on the waveguide properties of the guiding medium:

26

1. Slowly-varying complex envelope approximation, which means that the variation of the com-

plex envelope� ��� � � is sufficiently slow with distance � and time � , respectively:����� � � � ��� � �� � � ����� � ����� � � � �

��� � �� � ����� (2.7)����� � � � ��� � �� � � ����� � ����� - � � � ��� � �� � ����� (2.8)

where� � � � - ��� � � � � � , where � � is the linear refractive index of the fiber. This approxi-

mation is valid for pulses with widths as small as � � - � � fsec.

2. Plane-wave approximation, where the propagation constant is assumed to be given by� - � � - � � - � � � � � - � - � � � � � ��

- � - ��� � � � � �� - � - ��� � � � � � � � (2.9)

The first-order dispersion parameter is described by�� � � Re

� �������� � ����� � � - ��� ��- � � � - � �� - ��� � (2.10)

where ����� is the effective refractive index of the fiber, and � ��� is the effective cross-sectionalarea of the fiber at the frequency/wavelength of operation. The second-order dispersion (GVD)

parameter is described by�� � � Re

� ��������� � ����� - � � � � - ���� - � � � � � - � �� - ��� � (2.11)

The third-order dispersion parameter is given by�� � � Re

� - �������� � ���� � �� � � � � - ���� - � � ���- � � � � - ���� - � � �� - �� � � - � �� - ��� � (2.12)

3. The fiber acts as a single-mode fiber for the largest wavelength expected to be transmitted over

the fiber.

The NLS equation for monochromatic light in single-mode fibers can be derived completely in the

time-domain as [20, 34]� ���� � ���

�� � � � � �� � � � � �� � � �� � � � � �� � � �� � � � ��� � � � � � � ��� � � (� � � � � �� � ��� � � �� � � � � �

��� � � � � � �� � � � � ��� � � �� � � � � � � � � � � � � � � � � � � � (2.13)27

where��� ��� � � is the slowly varying complex envelope of the optical field at time � and distance

� from the transmitter. The linear attenuation coefficient of the fiber as a function of frequency is

given by � - � � Im � � ��������������� � � � - � with � � � � - ��� . The Kerr coefficient is given by� � � - �� � ��� � � � - �� � ����� � � ��� � � � � � - ���- �'��- ��� � � � � - ��'� - �'��- � � � � � � - � ��- ���'� - � � (2.14)where � � is the effective nonlinear refractive index that describes the self-phase modulation (SPM)in single-user systems and SPM, cross-phase modulation (CPM), and four-wave mixing (FWM) in

multi-user (WDM) systems.

The second nonlinear coefficient � � � �� � � � describes the self-steepening of pulses in the fiber.The third and fourth nonlinear coefficients� � � - �� � ����� � ��� � �� - � � �� - � � �� - � � � � � � � - � ��- � ��- � � � � � � - � �'� - � ��- � ��� � � � - � ��- � �'� - � � ����� � ��� � ��� � � �

(2.15)

and� � � - �� � ����� � ��� � � � � � � - � ��- � ��- � �� - � � � � � � - � �'� - � ��- � �� - � � � � � � - � ��- � �'� - � �� - � � ����� � ��� � ��� � � �(2.16)

explain the self-frequency shift and stimulated Raman scattering (SRS). The nonlinear constants � �and � � are real quantities, whereas � � and � � are complex. The imaginary part of � � � is negligiblefor most practical fibers so the losses introduced by the third-order susceptibility (real parts of � � and� � ) are negligible. The Raman coefficient is � � � � � ��������� � , where � � is the Raman gain coefficientfactor [38], and � � ��� is the inverse-Fourier transform of the Raman gain spectrum, � � - � .

Pask and Vaterescu [39] and Francois [38] have proposed to solve Maxwell’s equations com-

pletely in the frequency-domain under a weakly guiding fiber approximation. Considering the

macroscopic fiber nonlinearities like the Kerr effects (SPM, CPM, FWM, and self-steepening) and

Raman effect (self-frequency shift), the following form of the NLS equation is obtained� � - � � ���

� � � - � � - � � � � � � - � � - � � �28

� � � � -- � � � � � � � - � � � � � � - � � � � � - � - � � - � � � �� - � � - �� � � � -- � �

��� � � � � - � � - � � � - � � � � � � - � � � � � - � - � � - � � � �� - � � - � (2.17)where

� - � � � is the Fourier transform of the complex envelope � ��� � � . The linear dispersion isdescribed by � - � ��� � � � � � � � � � ������ . The shock-term due to � � in (2.13) is taken into accountthrough � � � - � . The higher-order nonlinear effects due to � � and � � in (2.13) are absorbed into themore accurate model given by � � - � .

Since the frequency-domain approach does not assume monochromatic light, the following are

the advantages of this particular approach [38]:

1. Higher-order linear dispersion and higher-order fiber nonlinearities are difficult to handle with

time-domain techniques because of time-derivatives, which get transformed into multiplica-

tions in the frequency-domain. Proper analysis of these effects for ultra-short pulses being

considered for future high-speed, high-capacity fiber-optic communication systems is better

accomplished in the frequency-domain.

2. Raman gain curves are obtained experimentally in the frequency-domain, which makes the

frequency-domain the natural domain for treating such nonlinear effects. Ultra-short pulse

measurements in applied physical research are also made in the frequency-domain. Therefore,

most of the results available in the literature are available in the frequency domain.

3. Modal distribution of light from the laser is available only in the frequency-domain, making

frequency-domain methods advantageous for such studies.

4. For WDM applications, different users are assumed to be transmitting around distinct frequen-

cies with non-overlapping spectra in the time-domain NLS equation. This is not necessarily

accurate in most systems. The frequency domain method does not impose this constraint.

The Volterra series approach can be used to evaluate the system performance with both the time

domain and frequency domain NLS equations.

29

2.2.2 Linear Dispersion

Due to the frequency dependence of the refractive index � - � , components with different frequen-cies travel with different speeds, giving rise to what is known as chromatic dispersion. Chromatic

dispersion is related to the characteristic resonance frequencies at which the medium absorbs the

electro-magnetic radiation through oscillations of bound electrons [20].

The first-order dispersion�� , called the propagation constant does not affect the performance

of the receiver for single-frequency operation except for delaying the reception of the signal by an

easily predictable amount, thus we ignore its effect on single-user communication systems. How-

ever, for WDM systems, the delay experienced by different users is different; therefore, each pulse

“walks through” thousands of pulses from other users while in transmission, and nonlinear interac-

tion between those pulses could be quite significant.

Higher-order dispersion causes pulses to broaden, thereby introducing inter-symbol interference,

which makes it difficult for the receiver to decode the information. In practice, only the second and

third-order dispersion terms�� � � � ���� �� , and � � , respectively, contribute to pulse broadening,

where � is the usual ”dispersion” used in current optical communication literature.

From (2.13), ignoring fiber nonlinearities and ignoring dispersion terms of order higher than � ,yields � � ��� � ��

�� � ���

�� ��� � � � � � � � ��� � �� � � � � � � � � ��� � �� � � � � � � � � ��� � �� � � (2.18)

where�

��� � � again is the complex envelope of the optical field at times � and at a distance � in the

fiber. The linear transfer function describing the dispersion of the complex envelope of the field in a

fiber of length � can be written as [20]

�� - � � � � � ���

�� � ��� � � � ��� � � � � ��� � � � �� (2.19)

30

2.2.3 Nonlinear Phenomenon

Now we introduce some quantitative measures for the Kerr effects and linear dispersion, and their

relation to the length of the fiber in single pulse propagation. Ignoring the higher-order derivatives

from the nonlinear portion, ignoring linear dispersion terms of order higher than 3 in (2.13), and

writing the complex envelope of the optical field�

��� � � for single pulse propagation, � ��� � � �

� � � ����� ��� ���

��� � � yields [20]� ���� � � ��� �

�� �

���

� � �� � � � � �� ��� � � �� � � � � ����� �� � � � � � � � (2.20)where

� ��� � � is the pulse shape at a distance � from the transmitter at time � , the dispersion length� � " �' � � ' , the third-order dispersion length � � � " �' � � ' , the nonlinear length � � � � ����� , and

� $ � �������" , where � is the pulse-width, � is the peak pulse power, and ��� � �� � denotes sign of� . Therefore, if the length of the fiber � � � �