Analysis and Characterization of Fiber Nonlinearities.pdf

Analysis and Characterization of Fiber Nonlinearities with Deterministic and Stochastic Signal Sources

By

Jong-Hyung Lee

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Electrical Engineering

Ira Jacobs, Chair

Ioannis M. Besieris

John K. Shaw

Brian D. Woerner

Maïté Brandt-Pearce (University of Virginia)

February 10, 2000

Blacksburg, Virginia

Keywords: Fiber Nonlinearity, Optical Communication, Fiber Optics, WDM

Copyright 2000, Jong-Hyung Lee

Analysis and Characterization of Fiber Nonlinearities with Deterministic and Stochastic Signal Sources

Jong-Hyung Lee

(ABSTRACT)

In this dissertation, various analytical models to characterize fiber nonlinearities have

been applied, and the ranges of validity of the models are determined by comparing with

numerical results.

First, the perturbation approach is used to solve the nonlinear Schrödinger equation, and

its range of validity is determined by comparing to the split-step Fourier method. In addition, it is

shown mathematically that the perturbation approach is equivalent to the Volterra series

approach. Secondly, root-mean-square (RMS) widths both in the time domain and in the

frequency domain are modeled. It is shown that there exists an optimal input pulse width to

minimize output pulse width based on the derived RMS models, and the functional form of the

minimum output pulse width is derived. The response of a fiber to a sinusoidally modulated input

which models an alternating bit sequence is studied to see its utility in measuring system

performance in the presence of the fiber nonlinearities. In a single channel system, the sinusoidal

response shows a strong correlation with eye-opening penalty in the normal dispersion region

over a wide range of parameters, but over a more limited range in the anomalous dispersion

region. The cross-phase modulation (CPM) penalty in a multi-channel system is also studied

using the sinusoidally modulated input signal. The derived expression shows good agreement

with numerical results in conventional fiber systems over a wide range of channel spacing, Df,

and in dispersion-shifted fiber systems when Df > 100GHz. It is also shown that the effect of

fiber nonlinearities may be characterized with stochastic input signals using noise-loading

analysis. In a dense wavelength division multiplexed (DWDM) system where channels are spaced

very closely, the broadened spectrum due to various nonlinear effects like SPM (self-phase

modulation), CPM, and FWM (four-wave mixing) is in practice indistinguishable. In such a

system, the noise-loading analysis could be useful in assessing the effects of broadened spectrum

due to fiber nonlinearities on system performance. Finally, it is shown numerically how fiber

nonlinearities can be utilized to improve system performance of a spectrum-sliced WDM system.

The major limiting factors of utilizing fiber nonlinearities are also discussed.

iii

Acknowledgements

It has been more than 5 years since I came to Virginia Tech, but I still remember

the excitement I felt when I saw the Drillfield for the first time. Dr. Jacobs, my advisor,

helped me keep that feeling alive through many long nights of research. I wish I could

adequately express my deep appreciation for his consistent guidance and encouragement

throughout my doctoral program.

I also wish to express my most sincere appreciation to the other members of my

committee, Drs. Besieris, Shaw, Woerner, and Brandt-Pearce. Their insight, expertise,

and wealth of knowledge were invaluable to me. Without their help, I could not have

completed this dissertation.

Finally, my deepest appreciation goes to my mother, brother, and sister. Their

understanding, encouragement and love have made this work both possible and

meaningful.

iv

Table of Contents

Title page

Abstract

Acknowledgements

Table of Contents

List of Tables and Illustrations

List of Acronyms

List of Symbols

Chapter 1. Introduction 1

Chapter 2. Analysis of Fiber Nonlinearities by Perturbation Method 23

1-1. Historical Perspective of Fiber Optic Communication Systems

1-2. Nonlinear Effects in Optical Fibers

1-2-1. Stimulated Scattering

1-2-2. Optical Kerr Effects

1-3. Nonlinear Schrödinger Equation and Split-Step Fourier Method

1-4. Motivation and Outline of the Dissertation

3

6

7

8

15

20

2-1. Introduction

2-2. Normalized Nonlinear Schrödinger Equation

2-3. Perturbation Solution of the normalized NLSE

2-4. Perturbation Approach and Volterra Series Transfer Function

2-5. Comparison of Perturbation Solution with Split-Step Fourier

Method

2-6. Summary

23

24

28

33

34

37

i

ii

iii

iv

viii

xiii

xiv

v

Chapter 3. Modeling and Optimization of RMS Pulse and Spectrum Widths 43

Chapter 4. Performance Measurements Using Sinusoidally Modulated Signal 66

3-1. Introduction

3-2. RMS Width Variation in a Dispersive Nonlinear Fiber

3-2-1. RMS Pulse Width with a Gaussian Input Pulse

3-2-2. RMS Spectrum Width with a Gaussian Input Pulse

3-3. Optimization of RMS Widths

3-3-1. Optimum Input Pulse Width to Minimize st(z)

3-3-2. Optimum Input Pulse Width to Minimize sw(z)

3-3-3. Optimum Input Pulse Width to Minimize the Product

of st(z) and sw(z)

3-4. Summary

66

68

68

71

75

76

87

89

95

98

4-1. Introduction

4-2. Self-Phase Modulation Analysis using Sinusoidally Modulated

Signal

4-2-1. Theoretical Background

4-2-2. Sinusoidal Response of NLSE

4-2-3. Eye-Opening Penalty and Sinusoidal Response

4-2-4. Sinusoidal Response using Perturbation Analysis

4-3. Cross-Phase Modulation Analysis using Sinusoidally Modulated

Signal

4-3-1. Pump-Probe Analysis with Sinusoidally Modulated

Pump Signal

4-3-2. Eye-Opening Penalties of 3-Channel WDM systems

4-4. Summary

43

45

46

49

53

53

61

62

64

vi

Chapter 5. Noise Loading Analysis to Characterize Fiber Nonlinearities 106

Chapter 6. Nonlinear Bandwidth Expansion Receiver in Spectrum-Sliced WDM Systems 130

Chapter 7. Conclusions and Future Work 149

5-1. Introduction

5-2. Noise Loading Analysis using Split-Step Fourier Method

5-2-1. Evolution of Spectral Density

5-2-2. Evaluation of Pa, Pb, and NPR

5-2-3. Noise Loading Analysis with Different Dispersion

Maps

5-3. Evaluation of Pb using the third-order Volterra Series Model of

Single-Mode Fiber

5-4. Summary

106

109

110

110

112

123

129

6-1. Introduction

6-2. Auto-covariance of Photo-Detected Signals with Nonlinear

Bandwidth Expansion Receiver (NBER)

6-3. Optimum Optical Filter Bandwidth and Q-factor of Nonlinear

Bandwidth Expansion Receiver (NBER)

6-4. Limitations of Nonlinear Bandwidth Expansion Receiver

6-4-1. Effects of Non-ideal Optical Amplifier

6-4-2. Effects of Spectrum-Slicing Filter Shape

6-5. Summary

130

132

137

143

143

144

146

7-1. Summary of Major Contributions

7-2. Suggestions for Future Research

149

153

vii

Appendix A. MATLAB Programs 155

References 159 Vita 167

viii

List of Tables and Illustrations

Chapter 1

Chapter 2

Table 1-1 Major Progress of Optical Communications

Table 1-2 Major Progress in Dense WDM system in recent years

Figure 1-1 Illustration of walk-off distance

Figure 1-2 Illustration of side-bands generation due to FWM in two-channel

system

5

6

13

14

Table 2-1 Comparison of Output RMS pulse width (st) with To = 70ps

Figure 2-1 (a) Power fluctuation in an optically amplified system (Eq.(2.3)),

(b) its equivalent model (Eq.(2.5))

Figure 2-2 Comparison of fundamental soliton output by the split-step Fourier

method with theoretical prediction

Figure 2-3 NSD evolutions of soliton transmission by the split-step Fourier

method

26

29

38

39

Figure 2-4 NSD by perturbation method with N=1 (solid with * = 1st order

and b2 > 0, dash dot with * = 1st order and b2 < 0, solid with o = 2nd order and

b2 > 0, dash dot with o = 2nd order and b2 < 0)

Figure 2-5 Comparison of pulse shapes by the first order perturbation method

and split-step method (a) z/LD =0.2, b2 > 0, (b) z/LD =0.2, b2 < 0, (c) z/LD =0.5,

b2 > 0, (d) z/LD =0.5, b2 < 0

Figure 2-6 Normalized critical distances at NSD = 10-3

(a) b2 > 0 (*:1st order, o: 2nd order), (b) b2 < 0 (*:1st order, o: 2nd order)

40

41

42

ix

Chapter 3

Chapter 4

Table 3-1 Summary of the optimum input pulse widths and the minimum

output pulse widths in the normal dispersion region by the various methods.

(NL

z=ζ ø1)

Figure 3-1 Comparison of RMS pulse width models with the simulated one.

Input pulse is a Gaussian shape and normal dispersion region is assumed.

(x = z/LD)

Figure 3-2 Comparison of RMS spectrum width model by the variational

method (-.) with the simulated one (solid) in the normal dispersion region.

Figure 3-3 Normalized output widths as a function of normalized input width

(so) at three distances ζ=z/LN= 0.2, 10, and 20. Blue solid curves by split-step

Fourier method and red dotted curves by variational method

Figure 3-4 Comparison of simulated so,opt with curve-fitting and square-root of

z

Figure 3-5 T(z) as a function of so2

in the normal dispersion region with a

Gaussian input.

58

51

52

59

60

65

Figure 4-1 Frequency response of fiber. As dispersion increases, the

bandwidth of Hp(w) decreases [37].

Figure 4-2 Fourier series coefficients evolution with dispersion (normal)

alone. (a) at three different distances, (b) |C1| and |C2| as a function of

transmission distance.

Figure 4-3 Fourier series Coefficients evolution with nonlinearity (N=2). (a) at

three different distances, (b) |C1|,|C2|, and |C3| as a function of distance.

70

80

81

x

82

83

84

85

86

88

100

101

Figure 4-4 Evolution of the fundamental Fourier series coefficient magnitude

(|C1|) as a function of transmission distance. (a) Normal dispersion region

(b2 > 0), (b) Anomalous dispersion region (b2 < 0)

Figure 4-5 Eye-opening penalties in the normal dispersion region.

Figure 4-6 Eye patterns in the normal dispersion region;

(a) Back-to-back, (b) Dispersion alone at z/LD = 0.0556,

(c) N =3 at z/LD = 0.0556, and (d) N=6 at z/LD = 0.0556

Figure 4-7 1dB power penalty distances as a function of N2; (a) in the normal

dispersion region (b2 > 0), (b) in the anomalous dispersion region (b2 < 0)

Figure 4-8 (a) Comparison of the critical distance at NSD = 10-3 using up to

the first order perturbation solution and the simulated 1dB penalty distance of

sinusoidal response in the normal dispersion region. (b) Comparison of the

fundamental Fourier series coefficient, |C1| when N2 = 3

Figure 4-9 Pump-probe set-up for CPM effect study

Figure 4-10 The probe signal’s intensity fluctuations after z = 100km.

Df =100GHz, Rb=10Gb/s, a=0.2dB/km, g=2�10-3[1/(km¼mW)], P1 = 0.2mW,

and P2 = 20mW; (a) D = +17 [ps/(nm¼km)], (b) D = -2 [ps/(nm¼km)]

Figure 4-11 The normalized intensity interferences, M(%), after z = 100km.

Rb=10Gb/s, a=0.2dB/km, g=2�10-3[1/(km¼mW)], P1 = 0.2mW, and P2 =

20mW; (a) D = +17 [ps/(nm¼km)], (b) D = -2 [ps/(nm¼km)]

Figure 4-12 Eye-opening penalties as a function of Df after z =100km.

(Conventional fiber system)

Figure 4-13 Eye-opening penalties as a function of Df after z =100km. (DSF

system)

Figure 4-14 Eye-patterns of DSF system after z =100km; (a) Back-to-back

case, (b) Single channel case, (c) Center channel of Df=75GHz case (unequally

spaced), and (d) Center channel of Df=75GHz case (equally spaced)

Figure 4-15 CPM penalty (a) conventional fiber system, and (b) DSF system

102

103

104

105

xi

Chapter 5

Table 5-1 Simulation Parameters for Noise Loading Analysis

Table 5-2 Parameters of Three Different Dispersion Maps (L= L1+L2=150km)

Table 5-3 Comparison of Pb at z = 50km

Figure 5-1 Noise loading test set-up

Figure 5-2 Normalized spectral densities of noise source (a) without notch

filter, (b) with notch filter

Figure 5-3 Normalized spectral densities with notch filter;

(a) b2 = 0.1 [ps2/km], z=50km, (b) b2 = 0.1 [ps2/km], z=200km

(c) b2 = -0.1 [ps2/km], z=50km, and (d) b2 = -0.1 [ps2/km], z=200km

Figure 5-4 Normalized spectral densities with notch filter;

(a) b2 = 10 [ps2/km], z=50km, (b) b2 = 10 [ps2/km], z=200km

(c) b2 = -10 [ps2/km], z=50km, and (d) b2 = -10 [ps2/km], z=200km

Figure 5-5 Spectral growth within the notch filter bandwidth

when b2 = 3 [ps2/km]; (a) with the notch filter, and (b) without the notch filter

Figure 5-6 NPR simulation results as a function of transmission distance.

Simulation parameter is b2.

109

113

126

108

115

116

117

118

119

Figure 5-7 b2 = +0.1 ps2/km; (a) Pa(z), (b) Pa(z)/ Pa(0), (c) Pb(z),

and (d) Pa(z)/ Pb(z)

Figure 5-8 b2 = -0.1 ps2/km; (a) Pa(z), (b) Pa(z)/ Pa(0), (c) Pb(z),


Figure 5-9 b2 = +10 ps2/km; (a) Pa(z), (b) Pa(z)/ Pa(0), (c) Pb(z),


Figure 5-10 b2 = -10 ps2/km; (a) Pa(z), (b) Pa(z)/ Pa(0), (c) Pb(z),


Figure 5-11 Noise loading analysis with different dispersion maps;

(a) Pa(z)/ Pa(0), (b) Pb(z), and (c) NPR [dB]

120

120

121

121

122

xii

Figure 5-12 Modeling of single-mode fibers with Volterra series

Figure 5-13 Spectral densities at z = 50km by the Volterra series approach

Figure 5-14 Spectral densities at z = 50km by the split-step Fourier method;

(a) b2 = +3ps2/km, (b) b2 = -3ps2/km, (c) b2 = +10ps2/km,

and (d) -10ps2/km

Chapter 6

Figure 6-1 Spectrum-Sliced WDM system

Figure 6-2 Nonlinear Bandwidth Expansion Receiver

Figure 6-3 Comparison of Eye-Diagrams; (a) without bandwidth expansion (b)

with bandwidth expansion

Figure 6-4 Normalized auto-covariance (correlation coefficient) curves of the

photo-detected signal when m=5

Figure 6-5 Modified Correlation Time (a) m = 5, (b) m = 2.5

Figure 6-6 Block Diagram of the Q-factor Simulation with NBER

Figure 6-7 Q-factor as a function of bandwidth of the optical filter in NBER

(Pv=30mW)

Figure 6-8 Q-factor of NBER with non-ideal EDFA (a) EDFA noise effects on

the sensitivity of NBER (Pv= 40mW), (b) NBER sensitivity with gain

modeling (Ps =15dBm)

Figure 6-9 NBER performance with a rectangular transmitting filter

(bandwidth = 19.6GHz) (a) Modified correlation time as a function of the

optical filter bandwidth after the nonlinear fiber, (b) Q-factor vs. received

signal power

131

132

135

136

140

141

142

147

148

123

127

128

xiii

List of Acronyms

ASE

BER

BPF

CPM

DSF

DWDM

EOP

FWM

GVD

ICI

ISI

LD

LED

NBER

NLSE

NPR

NRZ

NSD

PRBS

RF-FDM

RMS

RZ

SBS

SNR

SPM

SRP

SRS

SS-WDM

WDM

Amplified Spontaneous Emission

Bit Error Rate

Band-Pass Filter

Cross-Phase Modulation

Dispersion-Shifted Fiber

Dense-WDM

Eye-Opening Penalty

Four-Wave Mixing

Group Velocity Dispersion

Inter-Channel Interference

Inter-Symbol Interference

Laser Diode

Light Emitting Diode

Nonlinear Bandwidth Expansion Receiver

Non-Linear Schrödinger Equation

Noise Power Ratio

Non-Return to Zero

Normalized Square Deviation

Pseudo-Random Bit Sequence

Radio-Frequency Frequency Division Multiplexing

Root Mean Square

Return to Zero

Stimulated Brillouin Scattering

Signal to Noise Ratio

Self-Phase Modulation

Sinusoidal Response Penalty

Stimulated Raman Scattering

Spectrum-Sliced WDM

Wavelength Division Multiplexing

xiv

List of Symbols

D Dispersion parameter, 22

21 βλπ

λc

vd

dD

g

−=

=

G Optical amplifier gain

LD Dispersion distance, 2

2

βo

D

TL =

LN Nonlinear distance, o

N PL

γ1=

Lw Walk-off distance, λλλ ∆

≈−

=−− D

T

vv

TL o

gg

ow

)()( 21

11

M(%) Normalized intensity interference

N Nonlinearity parameter, N

D

L

LN =2

Po Peak power of optical signal

Pavg Path-averaged power of optical signal, dzePz

Paz z

oa

avg ∫ −=0

1 α

Pa Output power of the BPF without the notch filter at the input

Pb Output power of the BPF with the notch filter at the input

Q Q-factor, 01

01

σσµµ

+−

=Q where m1,0 and s1,0 are the mean and

standard deviation of the marks and spaces at the decision circuit

Rb Bit rate

Tb Bit period

A(z,t)

Aeff

Bo

Bt

CT

C(t)

Slowly varying envelope of optical field

Effective core area of fiber

Optical filter bandwidth

Channel bandwidth

Total transmission capacity

Auto-covariance

xv

Tc Full-width of auto-covariance at 1/e maximum value

To An arbitrary temporal characteristic value of the initial pulse

Tr Rise time of the pulse

U(z,t) Normalized slowly varying envelope, ),(1

),( tzAP

tzUo

=

d Walk-off parameter, λλλλλλ

λ

∆=−≈= ∫ DDdDd 12

2

1

)(

m Transmission parameter of a SS-WDM system, m = Bt/Rb

t Local time, t = t′ -z/vg ( t′= physical time)

to Initial pulse width (half-width at 1/e intensity)

vg Group velocity

z Fiber length, Propagation distance

za Amplifier spacing

zc Critical distance (the distance at which NSD =10-3)

Df Channel spacing in frequency

Dl Channel spacing in wavelength

a Fiber loss

b2 Second order group-velocity dispersion parameter

b3 Third order group-velocity dispersion parameter

g Fiber nonlinear coefficient

l Wavelength

lo Center wavelength

lZD Zero dispersion wavelength

smin Minimum output RMS pulse width

so RMS pulse width of input signal

so,opt Optimum input RMS pulse width

xvi

st RMS pulse width of output signal

sw RMS spectrum width

t Normalized time variable, oT

t=τ

t Time difference of a wide-sense stationary process, t=|t1-t2|.

(Chapters 5 and 6 only.)

tc Correlation time, ∫∞

ττ=τ0

c d)(C)0(C

1

cτ~ Modified correlation time, ∫−

=2

cT

2cT

)(C(0)

1~ τττ dCc

w Angular frequency

wp Fundamental angular frequency of a periodic signal

x Normalized distance by the dispersion distance, x = z/LD

z Normalized distance by the nonlinear distance, z= z/LN

Chapter 1: Introduction

1

Chapter 1

Introduction We are now faced with the arrival of a multi-media society built around the

sharing of voice, text, and video data. It is predicted that over 20 million computers will

be interconnected by the year 2000. One of the key foundations of this information

society is high capacity optical fiber communications, which has been one of the fastest

growing industries since the 1980s.

In recent years, the advent of erbium-doped fiber amplifiers (EDFAs) is one of

the most notable breakthroughs in fiber optic communication technology. Before the

emergence of EDFAs, the standard method of compensating fiber loss was to space

electronic regenerators periodically along the transmission link. A regenerator consists of

a photo-detector, electronic processing and amplification block, and a transmitter.

Functionally, it performs optical-to-electrical conversion, electronic processing and

electrical-to-optical conversion, and retransmission of the regenerated signal. The

advantage of regenerative systems is that transmission impairments such as noise,

dispersion, and nonlinearities do not accumulate, which makes it easy to design

transmission links. However, electronic blocks in regenerators prevent exploitation of the

huge bandwidth of the fiber. Furthermore, since the electronics are normally designed for

a specific bit rate and modulation format, it is necessary to replace all the regenerative

repeaters along the link when the system capacity must be increased. On the other hand,


2

optical amplifiers like EDFAs simply amplify the optical signal by several orders of

magnitude without being limited by electronic speed. In addition, optical amplification is

bit-rate and modulation format independent, which implies that optically amplified links

can be upgraded by replacing terminal equipments alone. The optically amplified

transmission lines can be considered as a transmission pipe which is transparent to data

rate and signal modulation format.

However, transmission impairments, which are in general not significant in a

regenerative system, accumulate along the transmission link when linear amplifiers

(analog repeaters) are used, so that they can not be simply ignored, and this puts a new

challenge to transmission design engineers. Among those impairments, dispersion, fiber

nonlinearities, and noise accumulation from optical amplifiers are the key limiting

factors. Dispersion, a linear phenomenon, is relatively well understood, and various

effective dispersion compensation techniques have been devised to cope with dispersion

induced performance degradation. Fiber nonlinearities, on the other hand, have not been

fully analyzed and understood especially when other impairments like dispersion are also

present. Their effects on the system performance are usually estimated by numerical

simulations or by experiments. Therefore, it is of interest to have analytical tools for

estimation of fiber nonlinearity induced performance degradation which might give us

better physical insight in designing and analyzing optical transmission systems. This is

the main subject of this dissertation.

The contents of the remaining sections of this chapter are as follows. First, a brief

historical perspective (Section 1-1) of modern fiber optic communications is provided to

have a better understanding of where we are, and where we are headed to in the next

century. Next, various nonlinear effects of fiber are reviewed and their relative

importance in communication systems is discussed (Section 1-2). Thirdly, the nonlinear

Schrödinger equation, which is the crucial equation in a fiber transmission system, is

introduced and the so-called symmetrized split-step Fourier method is described (Section

1-3). The split-step numerical method is used throughout the dissertation as a reference to

evaluate the accuracy of the new analytical models developed. Finally, Section 1-4 gives

the motivation and the organization of this dissertation.


3

1-1. Historical Perspective of Fiber Optic Communication Systems

Even though an optical communication system had been conceived in the late

18th century by a French Engineer Claude Chappe who constructed an optical telegraph,

electrical communication systems became the first dominant modern communication

method since the advent of telegraphy in the 1830s. Until the early 1980s, most of fixed

(non-radio) signal transmission was carried by metallic cable (twisted wire pairs and

coaxial cable) systems. However, large attenuation and limited bandwidth of coaxial

cable limited its capacity upgrade. The bit rate of the most advanced coaxial system

which was put into service in the United States in 1975 was 274 Mb/s. At around the

same time, there was a need of conversion from analogue to digital transmission to

improve transmission quality, which requires further increase of transmission bandwidth.

Many efforts were made to overcome the drawbacks of coaxial cable during the 1960s

and 1970s. In 1966, Kao and Hockham proposed the use of optical fiber as a guiding

medium for the optical signal [1]. Four years later, a major breakthrough occurred when

the fiber loss was reduced to about 20dB/km from previous values of more than

1000dB/km. Since that time, optical communication technology has developed rapidly to

achieve larger transmission capacity and longer transmission distance. The capacity of

transmission has been increased about 100 fold in every 10 years. There were several

major technological breakthroughs during the past two decades to achieve such a rapid

development, and their major characteristics are summarized in Table 1-1 with the first

year when the major breakthroughs were commercially available.

The first generation of optical communication was designed with multi-mode

fibers and direct bandgap GaAs light emitting diodes (LEDs) which operate at the 0.8µm-

0.9µm wavelength range. Compared to the typical repeater spacing of coaxial system

(~1km), the longer repeater spacing (~10km) was a major motivation. Large modal

dispersion of multi-mode fibers and high fiber loss at 0.8µm (> 5dB/km) limited both the

transmission distance and bit rate. In the second generation, multi-mode fibers were

replaced by single-mode fibers, and the center wavelength of light sources was shifted to

1.3µm, where optical fibers have minimum dispersion and lower loss of about 0.5 dB/km.


4

However, there was still a strong demand to increase repeater spacing further, which

could be achieved by operating at 1.55µm where optical fibers have an intrinsic

minimum loss around 0.2dB/km. Larger dispersion in the 1.55µm window delayed

moving to a new generation until dispersion shifted fiber became available. Dispersion

shifted fibers reduce the large amount of dispersion in the1.55µm window by modifying

the index profile of the fibers while keeping the benefit of low loss at the 1.55µm

window. However, growing communication traffic and demand for larger bandwidth per

user revealed a significant drawback of electronic regenerator systems, namely

inflexibility to upgrade. Because all the regenerators are designed to operate at a specific

data rate and modulation format, all of them needed to be replaced to convert to a higher

data rate. The difficulty of upgradability has finally been removed by optical amplifiers,

which led to a completely new generation of optical communication. An important

advance was that an erbium-doped single mode fiber amplifier (EDFA) at 1.55µm was

found to be ideally suited as an amplifying medium for modern fiber optic

communication systems. Invention of the EDFA had a profound impact especially on the

design of long-haul undersea systems. Trans-oceanic systems installed recently like TAT

(Transatlantic Telephone)-12/13 [2] and TPC (Transpacific Crossings)-5 [3] were

designed with EDFAs, and the transmission distance reaches over 8000km without

electronic repeaters between terminals. The broad gain spectrum (3~4THz) of an EDFA

also makes it practical to implement wavelength-division-multiplexing (WDM) systems.

It is highly likely that WDM systems will bring another big leap of transmission

capacity of optical communication systems. Some research groups have already

demonstrated that it is possible to transmit almost a Tbits/s of total bit rate over thousands

of kilometers. Some of the important experimental results of dense WDM systems are

summarized in Table 1-2 [4-9]. In 1999, for example, N. Bergano et al. successfully

demonstrated transmission of 640 Gb/s over 7200km using a re-circulating loop [8] while

G. Vareille et al. demonstrated the transmission capacity of 340Gb/s over 6380km on a

straight-line test bed [9]. These results indeed show that remarkable achievements have

been made in recent years, and let us forecast that optical communication systems in the

next generation will have a transmission capacity of a few hundreds of Gb/s.


5

While high capacity dense WDM systems keep heading to closer channel spacing

and broader bandwidth of optical amplifiers to fully exploit the fiber bandwidth, on the

other hand, upgrading embedded systems remains as another challenge. As of the end of

1997, about 171 million km of fiber have been deployed world wide, of which 69 million

km is deployed in North America [10]. Unfortunately, most of the embedded fibers are

conventional single-mode fibers which have a large dispersion at the 1.55µm window.

Upgrading these systems will require various dispersion combating techniques which are

highly tuned at a specific system to optimize system performance.

Table 1-1 Major Progress of Optical Communications

Year Bit Rate Repeater Spacing Major Technologies

1980 45Mb/s 10km -. λ = 0.8 µm -. Multi-mode fiber -. GaAs LED

1987 1.7Gb/s 50km -. λ = 1.3 µm -. Single-mode fiber -. InGaAsP Laser Diode

1990 2.5Gb/s 60~70km -. λ = 1.55 µm -. Dispersion shifted fiber

1996 5Gb/s Optical Amplifier Spacing

33~82km

-. λ = 1.55 µm -. Optical Amplifier -. WDM


6

Table 1-2 Major Progress in Dense WDM system in recent years

Year Bit Rate/ch.

Channel Number

Transmission Distance

Amplifier Spacing

Signal Format

Ref.

1996 20 Gb/s 3 10,000 [km] 50 [km] Soliton [4]

1996 5 Gb/s 20 9,000 [km] 45 [km] NRZ [5]

1997 5 Gb/s 32 9,300 [km] 45 [km] NRZ & Soliton

[6]

1998 10 Gb/s 64 500 [km] 100 [km] NRZ [7]

1999 10 Gb/s 64 7,200 [km] 50 [km] Chirped RZ

[8]

1999 10 Gb/s 34 6380 [km] 50 [km] RZ [9]

1-2. Nonlinear Effects in Optical Fibers

The nonlinearities in optical fibers fall into two categories. One is stimulated

scattering (Raman and Brillouin), and the other is the optical Kerr effect due to changes

in the refractive index with optical power. While stimulated scatterings are responsible

for intensity dependent gain or loss, the nonlinear refractive index is responsible for

intensity dependent phase shift of the optical signal. One major difference between

scattering effects and the Kerr effect is that stimulated scatterings have threshold power

levels at which the nonlinear effects manifest themselves while the Kerr effect doesn’t

have such a threshold.


7

1-2-1. Stimulated Scattering

Stimulated Brillouin Scattering (SBS)

Optical waves and acoustic waves in a fiber can interact to cause stimulated

Brillouin scattering. In stimulated Brillouin scattering, a strong optical wave traveling in

one direction (forward) provides narrow band gain for light propagating in the opposite

direction (backward). Some of the forward-propagating signal is redirected to backward,

resulting in power loss at the receiver. If the SBS threshold is defined as the input power

at which the scattered power increases as large as the input power in the undepleted pump

approximation, the SBS threshold power is proportional to[14,26],

11

~

∆∆+

B

s

B

thB g

Pνν

(1.1)

where gB is the Brillouin gain coefficient, Dns is the linewidth of the source, and DnB is

the Brillouin linewidth.

Eq. (1.1) indicates that the threshold power will be increased as the linewidth of

the source increases. For optical fibers at 1550nm, the Brillouin linewidth is about

20MHz, so optical signals modulated at higher bit rates will experience lesser effects of

SBS. From a system point of view, the relatively narrow gain spectrum of SBS prevents

interactions among channels in a WDM system, which makes SBS independent of

channel number. Only each individual channel signal needs to be below the threshold

power. Another characteristics of SBS which make it less troublesome compared to other

nonlinear effects is that the threshold of SBS does not decrease in a long amplified

system because practical optical amplifiers have one or more optical isolators. The

optical isolators prevent accumulations of the backscattered light from SBS.

Therefore, although SBS could be a detrimental nonlinear effect in an optical

communication system, system limitations are usually set by other nonlinear effects [15].


8

Stimulated Raman Scattering (SRS)

SRS is due to the interaction of photons with a fiber’s molecular vibrations.

Unlike SBS, SRS scatters light waves in both directions, forward and backward.

However, the backward-propagating light can be eliminated by using optical isolators.

Therefore, the forward scattered light is of more concern. The Raman gain coefficient is

about three orders of magnitude smaller than the Brillouin gain coefficient, and the SRS

threshold is known to be around 1W for a single-channel system [13]. In a single-channel

system, the large threshold power makes SRS a negligible effect. However, the gain

bandwidth of SRS is of the order of 12THz, which is about 6 orders of magnitude greater

than that of SBS. The large gain bandwidth of SRS enables it to couple different channels

in a WDM system, which can cause performance degradation through cross talk.

Chraplyvy and Tkach estimated the worst case of signal-to-noise ratio (SNR) degradation

in an amplified system due to SRS [16]. According to the estimate, the requirement to

ensure a SNR degradation of less than 0.5 dB in the worst channel is that the product of

total power, total bandwidth, and the total effective length of the system should be less

than 10 THz-mW-Mm. Although it was assumed in their estimate that all the channels

are transmitting mark states simultaneously, the probability of which is very low in a

multi-channel system, it indicates that SRS may impose a fundamental limit on the

capacity of future optical communication systems. However, the SRS threshold is high

enough such that other nonlinear effects caused by nonlinear refractive index are more

limiting factors in contemporary communication networks.

Various effects caused by nonlinear refractive index are discussed in the

following section.

1-2-2. Optical Kerr Effect

The refractive index of silica fiber for communication is weakly dependent on

optical intensity, and is given by [17],


9

)(2 tInnn o += (1.2)

where no 1.5, n2 2.6 � 10-20 m2/W, and I(t) = optical intensity.

Although the refractive index is a very weak function of signal power, the higher power

from optical amplifiers and long transmission distances make it no longer negligible in

modern optical communication systems. In fact, phase modulation due to intensity

dependent refractive index induces various nonlinear effects, namely, self-phase

modulation (SPM), cross-phase modulation (CPM), and four-wave mixing (FWM).

Self-Phase Modulation (SPM)

The dependence of the refractive index on optical intensity causes a nonlinear

phase shift while propagating through an optical fiber. The nonlinear phase shift is given

by

ztInNL )(2

2λπφ = (1.3)

where l is the wavelength of the optical wave, and z is the propagation distance.

Since the nonlinear phase shift is dependent on its own pulse shape, it is called self-phase

modulation (SPM). When the optical signal is time varying, such as an intensity

modulated signal, the time-varying nonlinear phase shift results in a broadened spectrum

of the optical signal. If the spectrum broadening is significant, it may cause cross talk

between neighboring channels in a dense wavelength division multiplexing (DWDM)

system. Even in a single channel system, the broadened spectrum could cause a

significant temporal broadening of optical pulses in the presence of chromatic dispersion.

However, under some circumstances SPM and chromatic dispersion can be beneficial.

One extreme example is the soliton [18], which is known to be stable and dispersion-free.

Even with non-return-to-zero (NRZ) pulses, it is known that pulse compression could be


10

achieved partially in the anomalous dispersion region1where the linear chirp induced by

chromatic dispersion and the nonlinear one due to SPM have opposite signs. When a

transmission system is designed to achieve the optimum compensation of the linear chirp

and the nonlinear chirp, it is often called a nonlinear assisted transmission system [12].

Cross-Phase Modulation (CPM)

Another nonlinear phase shift originating from the Kerr effect is cross-phase

modulation (CPM). While SPM is the effect of a pulse on it own phase, CPM is a

nonlinear phase effect due to optical pulses in other channels. Therefore, CPM occurs

only in multi-channel systems. In a multi-channel system, the nonlinear phase shift of the

signal at the center wavelength li is described by [12],

+= ∑

≠ jiji

iNL tItIzn )(2)(

22λ

πφ (1.4)

The first term is responsible for SPM, and the second term is for CPM. Eq. (1.4) might

lead to a speculation that the effect of CPM could be at least twice as significant as that of

SPM. However, CPM is effective only when pulses in the other channels are

synchronized with the signal of interest. When pulses in each channel travel at different

group velocities due to dispersion, the pulses slide past each other while propagating.

Figure 1-1 illustrates how two isolated pulses in different channels collide with each

other. When the faster traveling pulse has completely walked through the slower

traveling pulse, the CPM effect becomes negligible. The relative transmission distance

for two pulses in different channels to collide with each other is called the walk-off

distance, Lw [11].

λλλ ∆≈

−=

−− D

T

vv

TL o

gg

ow

)()( 21

11

(1.5)

1 The anomalous dispersion region has a negative sign of b2, the second order propagation constant. b2 is


11

where To is the pulse width, vg is the group velocity, and l1, l2 are the center wavelength

of the two channels. D is the dispersion coefficient, and Dl = |l1-l2|.

When dispersion is significant, the walk-off distance is relatively short, and the

interaction between the pulses will not be significant, which leads to a reduced effect of

CPM. However, the spectrum broadened due to CPM will induce more significant

distortion of temporal shape of the pulse when large dispersion is present, which makes

the effect of dispersion on CPM complicated.

Four-Wave Mixing (FWM)

Four-wave mixing (FWM), also known as four-photon mixing, is a parametric

interaction among optical waves, which is analogous to intermodulation distortion in

electrical systems. In a multi-channel system, the beating between two or more channels

causes generation of one or more new frequencies at the expense of power depletion of

the original channels. When three waves at frequencies fi, fj, and fk are put into a fiber,

new frequency components are generated at fFWM=fi+fj-fk [19]. In a simpler case where

two continuous waves (cw) at the frequencies f1 and f2 are put into the fiber, the

generation of side bands due to FWM is illustrated in Figure 1-2.

The number of side bands due to FWM increases geometrically, and is given by [20],

)(2

1 23chch NNM −= (1.6)

where Nch is the number of channels, and M is the number of newly generated sidebands.

For example, eight channels can produce 224 side bands. Since these mixing products

can fall directly on signal channels, proper FWM suppression is required to avoid

significant interference between signal channels and FWM frequency components.

also called as the group-velocity dispersion parameter and will be defined in Section 1-3.


12

When all channels have the same input power, the FWM efficiency, h, can be

expressed as the ratio of the FWM power to the output power per channel, and is

proportional to [21],

( )

2

22

∆∝

λη

DA

n

eff

(1.7)

where Aeff is the effective area of fiber.

Eq.(1.7) indicates that FWM of a fiber can be suppressed either by increasing channel

spacing or by increasing dispersion. Large dispersion can cause unacceptable power

penalties especially in high bit rate systems. However, careful design of the dispersion

map (often called dispersion management) which allows large local dispersion but limits

the total average dispersion to be below a certain level is found to be very effective to

combat both dispersion and FWM induced degradations. There is a rich collection of

literature on dispersion management, and a few examples can be found in [22-24].

Three different effects from the nonlinear refractive index, namely, SPM, CPM,

and FWM have been discussed. However, in a real system, especially in a DWDM

system where channels are packed very closely to each other, the broadened spectrum

due to the three nonlinear effects is usually indistinguishable. The system performance

degradations by fiber nonlinearities are, in general, assessable by solving the nonlinear

Schrödinger equation (NLSE). The NLSE and a numerical algorithm to solve the NLSE

– the split-step Fourier method - will be introduced in the following section.


13

Figure 1-1 Illustration of walk-off distance

Signal Pulse Interfering Pulse

z = zo

z = zo + Lw


14

Figure 1-2 Illustration of side-bands generation due to FWM in two-channel system

f1

z = 0 km

z = zo km

Original Frequencies

New Frequencies

f

f

f2

f1 f2 2f1-f2 2f2-f1


15

1-3. Nonlinear Schrödinger Equation and Split-Step Fourier Method

The propagation of optical waves in a single mode fiber is governed by

Maxwell’s equations which lead to the wave equation

ttc o 2

2

2

2

22 )(1

∂∂−=

∂∂−∇ EPE

E µ (1.8)

where E is the electric vector, mo is the vacuum permeability, c is the speed of light, and P

is the polarization density field.

At very weak optical powers, the induced polarization has a linear relationship with E

such that

tdtttt o ′′⋅′−= ∫∞

∞−),()(),(

)1(

L rErP χε (1.9)

where εo is the vacuum permittivity, and χ(1) is the first order susceptibility.

To account for fiber nonlinearities, the polarization can be written in two parts.

),(),(),( NLL ttt rPrPrP += (1.10)

where ),(NL trP is the nonlinear part of the polarization.

In silica fiber, the nonlinear part of the polarization usually comes from the third order

susceptibility [11]. That is,

∫ ∫ ∫∞

∞−

∞

∞−

∞

∞−−−−= 321321321

)3(

NL ),(),(),(),,(),( dtdtdttttttttttt o rErErErP Mχε

¡(1.11)

The third order susceptibility, c(3), is a fourth rank tensor, and could have 81 different

terms. However, in isotropic media like a single mode fiber operating far from any


16

resonance, the number of independent terms in the third order susceptibility is reduced to

one [25]. Eq.(1.9) to (1.11) can be used in Eq.(1.8) to derive the propagation equation in

nonlinear dispersive fibers. However, a few simplifying assumptions are generally made

to solve Eq.(1.8). First, NLP is treated as a small perturbation of LP , and the field

polarization is maintained along the fiber. Another assumption is that the index difference

between core and cladding is very small (so called weakly guiding approximation), and

the center frequency of the wave is assumed to be much greater than the spectral width of

the wave (so called quasi-monochromatic assumption). The quasi-monochromatic

assumption is analogous to low-pass equivalent modeling of bandpass electrical systems,

and is equivalent to the slowly varying envelope approximation in the time domain.

Finally, the propagation constant, b(w), is approximated by a few first terms of a Taylor

series expansion about the carrier frequency, wo, that is,

( ) ( ) ( ) ( ) L+−+−+−+= 33

22

1 6

1

2

1 βωωβωωβωωβωβ oooo (1.12)

where

o

n

n

n d

d

ωωωββ

=

=

The second order propagation constant, b2 [ps2/km], accounts for the dispersion effects in

fiber-optic communication systems. Depending on the sign of b2, the dispersion region

can be classified into two regions, normal (b2 > 0) and anomalous (b2 < 0). Qualitatively,

in the normal-dispersion region, the higher frequency components of an optical signal

travel slower than the lower frequency components. In the anomalous dispersion region,

the opposite occurs. Fiber dispersion is often expressed by another parameter, D

[ps/(nm¼km)], which is called the dispersion parameter2. D is defined as

=

gvd

dD

1

λ, and

the relationship between b2 and D is given by [30]

2 The parameter, D, has the opposite sign of b2. That is, in the normal dispersion region (b2 > 0), D < 0 and in the anomalous dispersion region (b2 < 0), D > 0.


17

Dcπ

λβ2

2

2 −= (1.13)

where l is the wavelength and vg is the group velocity.

The cubic and higher-order terms in Eq.(1.12) are generally negligible as long as the

quasi-monochromatic assumption holds. However, when the center wavelength of an

optical signal is near the zero-dispersion wavelength (that is, b2 0), the b3 term should be

included.

If the input electric field is assumed to propagate in the +z direction and is

polarized in the x direction, Eq.(1.8) becomes

),(2

),( tzAtzAz

α−=∂∂

(linear attenuation)

),(2 2

22 tzA

tj

∂∂+

β (second order dispersion)

),(6 3

33 tzA

t∂∂+

β (third order dispersion)

),(),(2

tzAtzAjγ− (Kerr effect)

),(),(2

tzAtzAt

Tj R ∂∂+ γ (SRS)

),(),(2

tzAtzAto ∂

∂−ωγ

(self-steepening effect)

�(1.14)

where A(z,t) = the slowly varying envelope of the electric field

z = propagation distance

t = t′ -z/vg ( t′= physical time, vg = the group velocity at the center wavelength)

α = the fiber loss coefficient ([1/km])

β2 = the second order propagation constant ([ps2/km])

β3 = the third order propagation constant ([ps3/km])


18

γ = the nonlinear coefficient =2pn2/(loAeff)

n2 = the nonlinear index coefficient

Aeff = the effective core area of fiber

lo = the center wavelength

wo = the center angular frequency

TR = the slope of the Raman gain ( ~5fs)

Eq.(1.14) is often called the generalized nonlinear Schrödinger equation, and is known

to be applicable for propagation of pulses as short as ~50fs. This corresponds to a spectral

width of ~20THz. When the pulse width is greater than 1ps, Eq.(1.14) can be

considerably simplified (as indicated below) because the Raman effect term and the self-

steepening effect term are negligible compared to the Kerr effect term [11].

AAiAt

Ai

z

A 2

2

2

2 22γα

∂∂β

∂∂ +−−= (1.15)

In Eq.(1.15), the third order dispersion term is also ignored, because this is negligible

compared to the second order dispersion term unless operation is near the zero-dispersion

wavelength. Considering that the bit period of a 10Gb/s non-return-to-zero (NRZ) system

is 100ps ( > 1ps), Eq.(1-15) can serve as a propagation equation in contemporary optical

communication systems with a fairly good accuracy.

Split-Step Fourier Method

It is required to solve the nonlinear Schrödinger equation to understand various

impairments occurring during signal transmission. However, it is not possible to solve it

analytically when both the nonlinearity and the dispersion effect are present, except in the

very special case of soliton transmission. Therefore, numerous numerical algorithms have

been developed to solve Eq.(1.14) or Eq.(1.15). The split-step Fourier method is one of

these, and is the most popular algorithm because of its good accuracy and relatively

modest computing cost.


19

The algorithm is briefly discussed in the following.

Eq.(1.15) can be expressed as

( ) ),(ˆˆ),(tzANL

z

tzA +=∂

∂ (1.16)

where the linear operator,2

2

222ˆ

t

jL

∂∂−−= βα

, and the nonlinear operator,2

),(ˆ tzAjN γ= .

When the electric field envelope, A(z,t), has propagated from z to z+Dz, the analytical

solution of Eq.(1.16) will have a form of

( )( ) ),(ˆêxp),( tzANLztzzA +∆=∆+ (1.17)

In the split-step Fourier method, it is assumed that the two operators commute with each

other. That is,

( ) ( ) ),(êxpêxp),( tzANzLztzzA ∆∆≈∆+ (1.18)

Eq.(1.18) suggests that A(z+Dz,t) can be estimated by applying the two operators

independently. If Dz is sufficiently small, Eq.(1.18) can give a fairly good result. Dz is

usually chosen such that the maximum phase shift ( zAp ∆=Φ2

max γ , Ap=peak value of

A(z,t)) due to the nonlinear operator is below a certain value. It has been reported that

when maxΦ is below 0.05 rad, the split-step Fourier method gives a good result for

simulation of most contemporary optical communication systems [12]. The simulation

time of Eq.(1.18) will greatly depend on the size of Dz. To reduce simulation time, a

more refined algorithm, the so called symmetrized split-step Fourier method, was

devised3 [11,82], and that method is used throughout this dissertation.

Mathematically, the symmetrized split-step Fourier method can be expressed as follows.

3 The symmetrized split-step Fourier method was apparently first applied in fiber propagation in [11], but was initially used in [82] for study of the interaction of intense electromagnetic beams with the atmosphere.


20

),(ˆ2

exp)(ˆexpˆ2

exp),( tzALz

zdzNLz

tzzAzz

z

∆

′′

∆≈∆+ ∫

∆+

(1.19)

While Eq.(1.18) assumes that nonlinearities are lumped at every Dz, Eq.(1.19) assumes

the nonlinearities are distributed through Dz, which is more realistic. When Dz is

sufficiently small, the evaluation of the nonlinear operator is approximated as

[ ]∫∆+

∆++∆≈′′zz

z

zzNzNz

zdzN )(ˆ)(ˆ2

)(ˆ (1.20)

However, Eq.(1.20) requires iterative evaluation because )(ˆ zzN ∆+ is not known at

z+Dz/2. Initially, )(ˆ zzN ∆+ will be assumed to be the same as )(ˆ zN . Although the

iterative evaluation is time-consuming, the improved numerical algorithm allows us to

use larger Dz than that of Eq.(1.18), which will result in saving overall computational

time.

1-4. Motivation and Outline of the Dissertation

One of the most important changes in fiber-optic communication systems brought

about by EDFAs is the expansion of regenerator spacing up to transoceanic distances.

However, a new problem has arisen, that is, the accumulation of fiber nonlinearities along

the links. The high optical power levels available from EDFAs makes system

performance more vulnerable to various nonlinear effects. In a multi-channel system, the

effect of fiber nonlinearities should be addressed more properly to understand inter-

channel effects in addition to intra-channel effects. While the other two conventional

limiting factors in designing optical communication systems, namely, fiber loss and

dispersion, are relatively well understood, and can be easily overcome by optical

amplifiers and dispersion compensation, fiber nonlinearities have not been fully analyzed


21

and understood despite a rich collection of literature dealing with fiber nonlinearities (For

example, [11-13, 26]). Therefore, it is crucial to understand fiber nonlinearities and their

effects on fiber-optic communication systems.

In optical communication systems, the input signal to the fiber is usually a

composite optical signal modulated with information bit streams. When all the input

signal frequencies interact due to fiber nonlinearities, the output bit stream may behave in

a complicated way giving adverse effects on system performance. The output waveform

can be obtained by solving the nonlinear Schrödinger equation (NLSE). In general, it is

not possible to solve the equation analytically. Conventional ways of analyzing fiber

nonlinearities either rely on pure numerical methods such as the split-step Fourier method

or rely on analytical solutions with over simplifications such as the assumption of

nonlinearity alone.

The key objective of this dissertation is to develop analytical models to

characterize fiber nonlinearities. First, the perturbation approach will be used to solve

the NLSE. Secondly, an alternate characterization technique, the root-mean-square

(RMS) width, will be studied. The response of fiber to sinusoidally modulated input will

also be studied to see its utility in measuring system performance in the presence of the

fiber nonlinearity both in a single channel system and in a multi-channel system. Finally,

the combined effect of fiber nonlinearity and the stochastic nature of the input signal on

the system performance will be studied.

The organization of the dissertation in the remaining chapters is as follows. In

Chapter 2, it is shown that the perturbation approach predicts there is no even order

nonlinearity in pulse distortion. Furthermore, we can separate the perturbation solution of

the NLSE into two parts, one resulting from dispersion alone, and the other from the

interaction of dispersion and nonlinearity. Numerical results by the perturbation analysis

and by the split-step Fourier method are compared for a broad range of parameter values

to determine a valid range for the perturbation analysis. The advantages and

disadvantages of the perturbation method are discussed, in addition to a mathematical

derivation showing the equivalence of the perturbation method to the Volterra transfer

function approach. Chapter 3 starts with the motivation of modeling the RMS width


22

followed by the derivations of the analytical modeling of the RMS widths both in the

time and spectral domains. With the developed RMS models, the optimum pulse width to

minimize the output pulse width is found and compared with the simulation results. The

optimum pulse width to minimize the product of the output RMS pulse width and the

output RMS spectrum width is also studied. In Chapter 4, first the theoretical

background will be discussed on how the sinusoidal response can be used to assess the

worst case system performance of an optical fiber communication system. In a single

channel system, eye opening penalties are compared with the magnitude reduction of

fundamental Fourier series coefficient of the output field in both dispersion regions,

normal and anomalous. Numerical results indicate that the sinusoidal analysis can be a

useful metric in assessing worst-case performance in the presence of fiber nonlinearities.

In a multi channel system, an analytical expression is derived to estimate intensity

interferences due to cross-phase modulation when the interfering channel is sinusoidally

modulated. The valid range of the expression is compared with three-channel system

simulations by the split-step Fourier method. In Chapter 5, noise loading analysis is

studied to assess transmission effects of the stochastic nature of input signal in the

presence of fiber nonlinearities. Furthermore, in Chapter 6, a study is made of how fiber

nonlinearities can be utilized to improve the performance of a spectrum-sliced WDM

system in which each channel signal is noise-like. Numerical simulations show that

bandwidth expansion obtained by fiber nonlinearities can reduce the correlation time of

the signal process when combined with a passive optical filter. The reduced correlation

time will contribute to reduced excess noise of the photo-detected signal. Optimum

bandwidth of the optical filter after bandwidth expansion has also been determined

through simulation of correlation time and Q factor. The chapter closes by discussion of

limitations of the bandwidth expansion technique. Finally, a summary of the primary

contributions of the dissertation is given in Chapter 7. Future research direction is also

suggested.

Chapter 2: Analysis of fiber nonlinearities by perturbation method

23

Chapter 2

Analysis of Fiber Nonlinearities by Perturbation Method

2-1. Introduction

Recently, K. V. Peddanarappagari and M. Brandt-Pearce solved the nonlinear

Schrödinger equation by the Volterra series transfer function approach [27,28]. Because

this approach gives a closed-form solution, it can be a useful design tool for a nonlinear

equalizer at the output of the fiber. However, its complicated analytical form not only

makes it hard to get physical insight, but also in many cases makes it less attractive in

computational time compared to the split-step Fourier method. Additionally, its range of

validity, that is, the valid range of the various physical parameters involved to assure

accuracy within an allowable tolerance, has not been fully studied.

In this chapter, firstly the normalized NLSE is derived. The normalized NLSE

will make it more convenient to treat various physical parameters in a unified way. Next,

the perturbation approach is applied to solve the normalized NLSE, and it is shown

mathematically that the approach is equivalent to the Volterra series transfer function


24

method. Finally, numerical results by the perturbation method will be compared with the

result of the split-step Fourier method to determine its valid range of parameters.

2-2. Normalized Nonlinear Schrödinger Equation

For pulse width To �1ps, Eq.(1.15) can be used as a propagation equation instead

of the generalized NLSE (Eq.(1.14)) because of negligible effects of higher-order

nonlinearities - the stimulated Raman scattering (SRS) and the self-steepening. Eq.(1.15)

describes the propagation of an optical pulse in single-mode fibers under the effects of

loss, group velocity dispersion (GVD), and the nonlinear Kerr effect which are the most

important transmission effects in contemporary optical communication systems. Since

Eq.(1.15) involves various physical parameters, it is often convenient to convert to

normalized units by defining two length scales LD (dispersion distance) and LN (nonlinear

distance). These two distances are defined as

2

2

βo

D

TL = (2.1)

o

N PL

γ1= (2.2)

where b2 is the second order propagation constant, g is the nonlinear coefficient, and Po is

the peak power of the slowly varying envelope, A(z,t). The parameter To is an arbitrary

temporal characteristic value of the initial pulse. To is often defined as either full width

half maximum (the pulse 3dB width) or the rise time of the pulse, Tr [12]. In NRZ

system, typically, Tr is about 25% of the bit duration, Tb.


25

When the slowly varying envelope, A(z,t), is normalized by its peak value such that

),(),( tzUPtzA o= , Eq.(1.15) can be expressed in terms of LD and LN as below.

UUL

zU

Lz

Ui

ND

2

2

22 )exp(

2

)sgn( ατ

β −−∂∂=

∂∂

(2.3)

where sgn(b2) = +1 when b2 > 0, and sgn(b2) = -1 when b2 < 0. t represents a

normalized time unit such that t = t/To.

In an amplified optical transmission system, signal power fluctuates periodically along

the link, and Eq.(2.3) can be further simplified by defining average power along the link.

When optical amplifiers are placed uniformly along the transmission link with amplifier

spacing za, the average power is expressed by

)1(1

0

a

a

z

a

oz

zo

aavg e

z

PdzeP

zP αα

α−− −== ∫ (2.4)

In the evaluation of Eq.(2.4), it is assumed the physical length of the amplifier is

negligible compared to transmission distance, which is a good approximation in real

systems. Now Eq.(2.3) can be expressed without a loss term by approximating the power

fluctuation as a constant value of Pavg.

UUL

U

Lz

Ui

ND

2

2

22 1

2

)sgn(−

∂∂=

∂∂

τβ

(2.5)

and avg

N PL

γ1= (2.6)

Eq.(2.5) is equivalent to Eq.(2.3) without a loss term, but with a constant optical power,

Pavg, and its validity will be justified subsequently. Figure 2-1 illustrates an optically


26

amplified system and its equivalent lossless system. To check the validity of Eq.(2.5),

Eq.(2.3) and Eq.(2.5) are compared by the split-step Fourier method with a Gaussian

input pulse of 70ps initial half-width at 1/e-intensity point. To exaggerate power

fluctuation, fiber loss is assumed to be 0.25 [dB/km], which is somewhat larger than the

typical value of 0.2 [dB/km], and, in addition, amplifier spacing is assumed to be 80km,

which is also somewhat larger than typical value of ~ 50km. Amplifier gain (G) is set to

compensate fiber loss exactly such that G = exp(+aza). In this case, the average power

can be obtained either by Eq.(2.4) or by the relationship GlnG

GPP oavg

1−= . Simulation

parameters are summarized below and the results are in Table 2-1.

Physical Parameters in Simulation

lo = 1.55 [mm] n2 = 6�10-13 [1/mW] g = 2.43�10-3 [1/(km¼mW)] b2 = 3 [ps2/km] Po = 2 [mW] ; initial input peak power a = 0.25 [dB/km] ; power loss G = 20 [dB] ; optical amplifier gain za = 80 [km] ; amplifier spacing

Simulation Results

Table 2-1 Comparison of Output RMS pulse width (st) with To = 70ps

Output Pulse Width

(st) by Eq.(2.3)

Output Pulse Width

(st) by Eq.(2.5)

z =2,400km 121.04ps 121.36ps

z =9,600km 438.56ps 442.93ps


27

To compare the two equations, the output root-mean-square (RMS) pulse widths are

calculated at two transmission distances, z =2,400km and z =9,600km. Simulation results

(Table 2-1) show that the output pulse widths obtained by the two equations are very

close to each other even with 20dB power fluctuation and with 120 amplifiers or a total

propagation distance of 9,600km. Therefore, a lossless system modeling with average

power is a good approximation, and from now on, it is assumed that signal power is its

averaged value along the transmission link and NL will be denoted as NL unless it is

necessary to distinguish these.

If we normalize distance by the dispersion distance, LD, such that x = z/LD,

Eq.(2.5) can be further simplified as below.

UUiNU

iU 22

2

2

2 )sgn(2

1 +∂∂⋅−=

∂∂

τβ

ξ (2.7)

where 2

22

βγ oavg

N

DTP

L

LN == .

The resulting Eq. (2.7) is called the normalized NLSE. It has some advantages over

Eq.(1.15) since it involves only a single dimensionless parameter N, which makes the

equation easier to deal with and might give better physical insight. Additionally, the

normalized units allow us to use the perturbation approach [29]. Since the perturbation

approach is based on the approximation by mathematical modeling, it is necessary to

determine the order of magnitude of the physical parameters involved. However it is hard

to use Eq.(1.15) because the various physical parameters involved make it difficult to

determine their relative strength. For example, if the length of a certain parameter is 1m,

that is a very small number compared to the propagation distance of a fiber optic

communication system while it’s a very large number compared to the wavelength of the

light source.

The range of N values can be estimated using the typical values of fiber

parameters. The typical values of dispersion coefficient, b2, and the nonlinear coefficient,

g, of conventional single mode fiber at the 1.55mm window are –20 [ps2/km] and


28

2 [km-1W-1], respectively. When we assume the average power of optical signal, Pavg, is

in the range of 0.1mw to 10mW, the nonlinear distance, LN, ranges from 50 km to

5,000km. Similarly, bit rates of 2.5Gb/s to 10Gb/s result in 31.25 km to 500 km for the

dispersion distance, LD. In the calculation of LD, the parameter To is assumed as the pulse

rise time which is set to 25% of bit period which is a typical value in NRZ systems. The

calculated LD and LN give 0625.02min =N , and 102

max =N .

In the case of dispersion-shifted fiber (DSF), which has typical values of |b2| = 3

[ps2/km] and g = 2.7 [km-1W-1], the range of N2 is from 0135.02min =N to 1352

max =N .

Again the bit rate is assumed in the range of 2.5Gb/s to 10Gb/s, and the average power

from 0.1mW to 10mW.

In the next section, perturbation solution of the normalized NLSE will be derived,

and the result will be compared with the Volterra series transfer function.

2-3. Perturbation Solution of the normalized NLSE

In general, the solution of the NLSE can not be found in analytical form.

However, we can find the solution of the NLSE in two extreme cases. In the limit of LD

÷ LN (dispersion is dominant), Eq.(2.7) degenerates into the linear partial differential

equation as below.

2

2

2 )sgn(2

1

∂τ∂β

∂ξ∂ U

iU ⋅−= (2.8)

By taking the Fourier transform, the equation can be expressed as an ordinary differential

equation.

uu

i 22 )sgn(

2

1 ωβ∂ξ∂ −= (2.9)

where )],([ τξUu ℑ= . [ ]⋅ℑ is the Fourier transform operator.


29

(a) Power fluctuation in an optically amplified system

(b) its equivalent model

Figure 2-1 (a) Power fluctuation in an optically amplified system (Eq.(2.3)) (b) its

equivalent model (Eq.(2.5))

z [km] za

P [mW]

Po

z [km]

P [mW] Pave

2za 3za

za 2za 3za


30

The solution has the form

= ξωβωωξ 2

2 )sgn(2

exp),0(),(i

uu (2.10)

where ),0( ωu is the Fourier transform of the incident field at ξ = 0, U(0, τ).

In the other extreme case where LD ø LN (nonlinearity is dominant), we can again find

the analytical solution of Eq.(2.7). In this limit, the NLSE becomes

UUL

i

z

U

N

2=∂∂

(2.11)

and its solution is of the form

[ ]),(exp),0(),( τττ ziUzU NΦ= (2.12)

where N

N L

zUz

2),0(),( ττ =Φ .

Even though we can find the analytical solutions of the NLSE in these two extreme cases,

typical optical communication systems usually do not fall into either of the extreme

cases. In that case, numerical approaches are usually required to solve the NLSE. While a

numerical approach like the split-step Fourier method is known to be accurate, it is time

consuming and doesn’t provide any physical insight.

In this section, we will attempt to find the perturbation solution of the normalized

NLSE. While the perturbation approach may not give a solution as accurate as numerical

approaches, the approach can provide better physical insight to understand how

dispersion and nonlinearity interact.

To find the perturbation solution of the normalized NLSE, let

L+++= ),(),(),(),( )2(2)1()0( τξετξετξτξ VVUU (2.13)


31

where ε = N2 and put it in Eq.(2.7).

( )LL

L

L

++++++−

+++=

+++

),(),(),(),(),(),(

)),(),(),((2

1)sgn(

)),(),(),((

)2(2)1()0(2)2(2)1()0(

)2(2)1()0(2

2

2

)2(2)1()0(

τξετξετξτξετξετξε

τξετξετξ∂τ∂β

τξετξετξ∂ξ∂

VVUVVU

VVU

VVUi

�(2.14)

By equating the terms proportional to εn separately for each value of n,

2

)0(2

2

)0(

2

1)sgn(

∂τ∂β

∂ξ∂ UU

i = (2.15)

)0(2)0(2

)1(2

2

)1(

2

1)sgn( UU

VVi −=

∂τ∂β

∂ξ∂

(2.16)

{ } )(22

1)sgn( )0(*)1()0()1(2)0(

2

)2(2

2

)2(

UVUVUVV

i +−∂

∂=∂

∂τ

βξ

(2.17)

M

and so forth. Here, * denotes the complex conjugate.

Eq.(2.13) can also be expressed in the frequency domain as follows.

L+++≈ ),(),(),(),( )2()1()0( ωξωξωξωξ uuuu (2.18)

where u(0)(x,w ) is the solution of the dispersion alone case which is given by Eq.(2.10).

Higher order terms in the frequency domain are defined to include N parameter for


32

simplicity such that u(1)(x,w) = [ ]),()1( τξεVℑ , u(2)(x,w) = [ ]),()2(2 τξε Vℑ , and so forth. In

the frequency domain, higher order terms are solutions of coupled ordinary differential

equations. From Eqs.(2.16) and (2.17),

{ } ),(),( ),()sgn(2

),( )0(2)0(2)1(22

)1(

ξτξτξωωβξ

ξωUUjNu

ju ℑ+=∂

∂ (2.19)

( ){ } ),(),(),(),(),(2),()sgn(2

),( )0(*)1()0()1(2)0(2)2(22

)2(

ξτξτξτξτξτξωωβξ

ξωUUUUUjNu

ju +ℑ+=∂

∂

�(2.20)

M

where ),(),( )1()1( τξεξτ VU = , and ℑ{�} denotes Fourier transform with respect to τ.

We may expect that including higher order terms will improve the accuracy of the

perturbed solution of Eq.(2.18). However, calculations of higher order terms will increase

the numerical load tremendously, which makes the perturbation approach less attractive.

Therefore it will be interesting to find the valid range of N and propagation distance for

which the perturbation solution, up to the first or the second order terms, is valid within a

given tolerance. The valid range of parameter values will be discussed in Section 2.5.

2-4. Perturbation Approach and Volterra Series Transfer Function

In the Volterra series approach [27], the linear transfer function, H1(x,w), and the

third-order nonlinear transfer function (third-order Volterra kernel), H3(x,w1,w2,w3), can

be expressed as below in the normalized units (H2(x,w1,w2) = 0),


33

= ξωβωξ 2

21 )sgn(2

exp),(i

H (2.21)

))(sgn(

2))(sgn(

2

))(sgn(2

exp))(sgn(2

exp

),,,(

23212

23

22

212

23212

23

22

212

2

3213

ωωωβωωωβ

ξωωωβξωωωβ

ξωωω

+−−+−

+−−

+−

=jj

jj

jN

H

(2.22)

Therefore, the normalized field spectrum is approximated as below by ignoring higher

order terms.

21212*

12121321

)3()1(

)()()(),,,()2(

1)(),(

),(),(),(

ωωωωωωωξωωωωωπ

ωξω

ξωξωξω

dduuuHuH

uuu VV

+−+−+=

+≈

∫∫�(2.23)

where u(ω)=u(ω,0).

In Eq.(2.23), ),()1( ξωVu is equivalent to the unperturbed solution, ),()0( ξωu , because

both are the linear solution of the NLSE. In addition, we can show that the third-order

Volterra kernel output, ),()3( ξωVu , is equivalent to the first-order perturbed solution,

),()1( ξωu in Eq.(2.18).

By differentiating ),()3( ξωVu with respect to ξ,

21212*

121213

2

)3(

)()()(),,,(

)2(

1),( ωωωωωωωξ

ξωωωωωπξ

ξωdduuu

HuV +−∂

+−∂=

∂∂

∫∫

�(2.24)

where


34

),(),(),(),,,())(sgn(2

1),,,(312

*111

23213

23212

21213 ξωξωξωξωωωωωωβξ

ξωωωωωHHHjNHj

H ++−=∂

+−∂

Then

[ ]

{ } ),(),( ),()sgn(2

),(),(),()2(

1),()sgn(

2

)()()(),(),(),()2(

),()sgn(2

)()()(),(),(),()2(

1

)()()(),,,()sgn(2

1

)2(

1),(

)0(2)0(2)3(22

2121)1(

2)1(

1)1(

2

2)3(2

2

21212*

12112*1112

2)3(

22

21212*

12112*111

2

2

21212*

1212132

22

)3(

ξτξτξωωβ

ωωξωωωξωξωπ

ξωβω

ωωωωωωωξωωωξωξωπ

ξωβω

ωωωωωωωξωωωξωξωπ

ωωωωωωωξωωωωωωβπξ

ξω

UUjNuj

dduuujNuj

dduuuHHHjN

uj

dduuuHHHjN

dduuuHju

V

VVVV

V

V

ℑ+=

+−+=

+−+−+=

+−+−+

+−+−=∂

∂

∫∫

∫∫

∫∫

∫∫

∗

�(2.25)

where { }),(),( )0()1( ξτξω UuV ℑ= .

The resulting equation has the same form of Eq.(2.19) which comes from the perturbation

approach. Therefore we can conclude that both of these approaches are equivalent at least

up to the third-order of the Volterra approach.

2-5. Comparison of Perturbation Solution with Split-Step Fourier Method In this section, the valid range of the perturbation solution developed in Section 2-

3 will be determined. The split-step Fourier method will be used as a reference, and its

accuracy will be addressed first by comparing with known theoretical predictions. It is

known that Eq.(2.7) leads to soliton solutions by applying the inverse scattering method

[11]. To support solitons, the dispersion should be in the anomalous region (b2 < 0, D

>0), and the input pulse should have a hyperbolic secant shape. One of the solutions is


35

the fundamental soliton which propagates without change of pulse shape for arbitrarily

long distance in an ideal case. When the input pulse is )(sech),0( ττ =U , the analytical

solution to Eq.(2.7) in the anomalous dispersion region with N=1 gives [11]

( )2exp)(sech),( ξττξ jU −= (2.26)

where x = z/LD.

Eq.(2.26) is ideally suited to see the accuracy of the split-step Fourier method because the

solution is the result of interplay between dispersion and nonlinearity, and it has a simple

form. Figure 2-2 compares Eq.(2.26) with the simulation result by the split-step Fourier

method at x = z/LD = 15. From the figure, we can observe that the difference between the

two curves is negligible. (Note the magnitude scale is logarithmic.) Since typical values

of LD are in the range of a few hundreds km to thousands km, the simulation distance

x=15 could be over transoceanic distances. In the simulation, the step size Dx = 0.01 is

used, which will result in 0.01 rad of the maximum phase shift by the nonlinear operator.

To compare two curves generated by two different methods, say, ‘A’ method and

‘B’ method, the normalized square deviation (NSD) is defined as [27],

∫

∫∞

∞−

∞

∞−

−=

ττ

ττξτξξ

dU

dUU

NSDBA

2

2

),0(

),(),(

)( (2.27)

where UA(x,t) = output field envelop by method ‘A’, and UB(x,t) = output field envelop

by method ‘B’.

Figure 2-3 shows the calculated NSDs as a function of propagation distance resulting

from the split-step method compared to the analytical solution, Eq.(2.26). We observe

that the NSD is greatly affected by the simulation step size Dx as expected. However,

NSDs remain almost constant at very small values up to the transmission distance x = 15.

For example, when Dx = 0.01, the NSD remains below 10-11 up to x = 15. These results

indicate that the split-step Fourier method is very reliable, and can serve as a reference to


36

measure the valid range of perturbation method if Dx is small enough. For the remainder

of this chapter, Dx = 0.01 will be used for the split-step method

Normalized square deviation (NSD) between split-step method and perturbation

solution

Now the perturbation solution of the normalized NLSE is compared with

simulation results. Input pulse shape is assumed to be a Gaussian such that

−= 2

2

1exp),0( ττU . Figure 2-4 shows the NSD between perturbation solutions and

numerical simulations when N =1. As expected, the perturbation solutions up to the

second order give about one order smaller NSD’s compared to the results from the first

order solutions. However, as the propagation distance increases, the perturbation solution

results in larger errors (larger NSD values) regardless of dispersion region. These results

suggest that the perturbation solution is limited in its numerical accuracy compared to the

split-step method. Figure 2-5 compares output pulse shapes by the first order perturbation

solution with split-step simulation results at x = 0.2 ((a) and (b)) and at x = 0.5 ((c) and

(d)). When x = 0.5, the differences between the two curves become noticeable while the

differences are negligible when x = 0.2. To decide the valid range of parameters for use

of the perturbation solution, we need to determine an allowable tolerance level. From

Figure 2-4 and Figure 2-5, the maximum allowable NSD value is chosen to be 10-3,

which occurs between (a) and (c) in Figure 2-4.

Figure 2-6 shows the critical distance (= xc), which is defined as the distance at

which NSD =10-3, as a function of N. The curves of xc in logarithmic plots are almost

straight lines for a broad range of N values. The good linearity between the calculated xc

and N2 indicates a near constant value for their product. That is, the first order

perturbation solution gives 3.022 ≈==N

c

D

cc L

z

L

zNN ξ in both the normal and anomalous

dispersion regions. When the second order is included, the product is approximately 0.7.

Since N2 = LD/LN, and LN = 1/gPavg, we can estimate the critical distance zc. With


37

112 −−= mWkmγ , the critical distance by the first order perturbation is estimated as

][150

mWkmP

zavg

c ⋅≈ . If we take the numerical example in Section 2-2, Pavg = 0.43mW

(Po=2mW and G = 20dB), this results in zc 350km. This means that we can get less

than 10-3 of NSD using the first order perturbation solution up to z = 350km. When we

include the second order term, the critical distance extends to nearly 800km. However,

Figure 2-4 indicates that the critical distances can be substantially shorter if a smaller

value of NSD is required to have more accurate results.

2-6. Summary

Applying the perturbation method to the nonlinear Schrödinger equation results in

a coupled set of first order differential equations in the frequency domain. We have also

shown that the perturbation approach is equivalent to the Volterra series method at least

up to the third-order of the Volterra approach.

The normalized square deviations (NSD) are evaluated for a broad range of

parameters using the split-step method as a reference. When we use the first-order

perturbation solution, the critical distance at which NSD reaches its maximum allowable

value (10-3, in this work) is inversely proportional to the average pulse power, Pavg. The

proportionality constant is evaluated to be around 150 [km¼mW]. The second-order

solution will increase the critical distance more than a factor of two, but the increased

computation load makes it less attractive. Finally, there are little differences in the

critical distances between the normal and anomalous dispersion regions. This is because

the critical values are relatively small; therefore, pulse shapes depending on the

dispersion region are not changed greatly.


38

Figure 2-2 Comparison of fundamental soliton output by the split-step Fourier method

with theoretical prediction

-20 -15 -10 -5 0 5 10 15 2010

-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

τ = t/To

Mag

nitu

de

Soliton Output at z/LD

= 15 with ∆ξ = 0.01

Input = sech(τ) Output at z/LD = 15


39

Figure 2-3 NSD evolutions of soliton transmission by the split-step Fourier method

0 5 10 15

10-14

10-12

10-10

10-8

10-6

10-4

z/LD

NS

D

Fundamental Soliton Propagation by Split-Step Fourier Method

∆ξ=0.05 ∆ξ=0.01 ∆ξ=0.002


40

Figure 2-4 NSD by perturbation method with N=1 (solid with * = 1st order and b2 > 0,

dash dot with * = 1st order and b2 < 0, solid with o = 2nd order and b2 > 0, dash dot with

o = 2nd order and b2 < 0)

10 -2

10 -1

10 0

10 1

10 -7

10 -6

10 -5

10 -4

10 -3

10 -2

10 -1

10 0

z/L D

NS

D

NSD by Perturbation Method with N = 1

(a)

(b)

(d)

(c)

1st order ( β 2 > 0) 2nd order ( β 2 > 0) 1st order ( β 2 < 0) 2nd order ( β 2 < 0)


41

Figure 2-5 Comparison of pulse shapes by the first order perturbation method and split-

step method (a) z/LD =0.2, b2 > 0, (b) z/LD =0.2, b2 < 0, (c) z/LD =0.5, b2 > 0, (d) z/LD

=0.5, b2 < 0

-10 -5 0 5 10 0

0.2

0.4

0.6

0.8

1

Mag

nitu

de

(a) z/L D = 0.2, β

2 > 0

ssf 1st

-10 -5 0 5 10 0

0.2

0.4

0.6

0.8

1

Mag

nitu

de

(b) z/L D = 0.2, β

2 < 0

ssf 1st

-10 -5 0 5 10 0

0.2

0.4

0.6

0.8

1

Mag

nitu

de

(c) z/L D = 0.5, β

2 > 0

τ

ssf 1st

-10 -5 0 5 10 0

0.2

0.4

0.6

0.8

1

Mag

nitu

de

(d) z/L D = 0.5, β

2 < 0

τ

ssf 1st


42

Figure 2-6 Normalized critical distances at NSD = 10-3 (a) b2 > 0 (*: 1st order, o: 2nd

order) (b) b2 < 0 (*: 1st order, o: 2nd order)

10-1

100

101

102

10-4

10-2

100

102

z c/LD

β2 > 0

1st2nd

10-1

100

101

102

10-4

10-2

100

102

N2 = LD

/LN

z c/LD

β2 < 0

1st2nd

Chapter 3: Modeling and optimization of RMS pulse and spectrum widths

43

Chapter 3

Modeling and Optimization of RMS Pulse and Spectrum Widths

3-1. Introduction

Root-mean-square (RMS) pulse width is of interest since it provides a useful

metric for assessing performance limitations in fiber-optic communication systems. The

RMS pulse width is directly related to the maximum data rate through the commonly

used design criterion

4

1<bt Rσ (3.1)

where st is the RMS pulse width at the output of the fiber, and Rb is the bit rate.

Also RMS spectral width (sw) determines basic design parameters of a wavelength

division multiplexed (WDM) system such as channel spacing and bandwidth of optical

filters. In a WDM system, the total transmission capacity (CT) is defined by CT = Nch⋅Rb,

where Nch = the number of channels and Rb = bit rate per channel. To maximize CT, it is

required to have the largest possible Nch, which can be achieved by having the smallest


44

RMS spectrum width at a given distance. Additionally, it is also required to have the

largest bit rate, but bit rate and RMS pulse width at the output of fiber should satisfy

some condition like Eq.(3.1), which says that the output RMS pulse width should be

decreased to increase bit rate, Rb. Then, the question, how can we maximize the total

transmission capacity, CT, is equivalent to the question, how can we minimize the

product, swst at a given distance? Therefore, the product, st(z)sw(z), should be inversely

proportional to the capacity of WDM systems.

In a fiber transmission system where dispersion is dominant (negligible fiber

nonlinearities), it is known that there exists an optimum input RMS pulse width, so, to

minimize the output width, st(z), when a transform-limited pulse is transmitted. The

optimum input RMS pulse width, so,opt, and the resulting minimum output pulse width,

smin, is given as a function of transmission distance, z, by [30]

22, zopto βσ = (3.2)

z2min βσ = (3.3)

In the case of dispersion alone, the magnitude of the pulse spectrum is invariant, and

consequently sw(z) remains constant at its initial value, swo. Therefore, the product

st(z)sw(z) will have the same functional form as st(z). In this case, the optimum input

pulse width given by Eq.(3.2) will also minimize the product, st(z)sw(z).

However, there appears to be no published results on maximizing CT in terms of

RMS widths as well as on minimizing st(z) when fiber nonlinearities are no longer

negligible. The main objective of this chapter is to study the possible existence and the

functional form of the optimum input pulse width to minimize the RMS quantities, st(z),

sw(z), and the product of the two, sw(z)st(z) in the presence of fiber nonlinearities. Even

though the RMS quantities are strictly applicable only for the case of transmission of an

isolated pulse, it is of interest to see how their functional forms compare to Eqs.(3.2)

and.(3.3) in the presence of nonlinearities. The derived results can provide basic design

parameters for optimum performance of WDM systems.


45

In this chapter, more accurate modeling than prior treatments [31,32] will be

attempted first. The functional forms of the optimum input pulse width based on the

developed RMS models follow, and the results will be compared with numerical results

obtained by the split-step Fourier method.

3-2. RMS Width Variation in a Dispersive Nonlinear Fiber

In a dispersive nonlinear fiber, the pulse shape at the fiber output can deviate

considerably from the input pulse shape and can have much more complicated forms. In

such a case, the FWHM (full width at half maximum) is not a good measure of pulse

width, and the root-mean-square (RMS) width, s, is often used to describe pulse width

more accurately. The RMS pulse width is defined as below [11].

[ ] 2/122 tt −=σ (3.4)

where

dttzU

dttzUt

t

n

n

∫

∫∞

∞−

∞

∞−=2

2

),(

),(

(3.5)

and U(z,t) = optical field envelope.

When the higher order dispersion coefficients can be ignored compared to the b2 term,

the first moment term in Eq.(3.4) is always zero if the input pulse shape is symmetrical

about its center. In that case, the RMS pulse width in the normalized units (t = t/To, x =

z/LD, and ),(),( tzUPtzA o= ) defined in section 2-2 can be expressed as follows.


46

2/1

2

22

),(

),(

=

∫

∫∞

∞−

∞

∞−

ττξ

ττξτσ

dU

dU

t (3.6)

and the RMS spectrum width is

2/1

2

22

),(

),(

=

∫

∫∞

∞−

∞

∞−

ωωξ

ωωξωσ ω

du

du

(3.7)

where ),( ωξu is the Fourier transform of U(x, τ).

3-2-1. RMS Pulse Width with a Gaussian Input Pulse

If the pulse shape at the input of the fiber has a Gaussian form, the pulse shape at

the output of the fiber can be easily found in analytical form in the case of dispersion

alone. In that case, the RMS widths given by Eq.(3.6) and Eq.(3.7) can also be found in

analytical form. However, it is impossible, in general, to get analytical forms of output

pulse shape and the RMS widths considering fiber nonlinearity. M. J. Potasek et al.[31]

derived the RMS pulse width by approximating the nonlinearity as a lumped effect at the

input to the fiber. In the normalized units, the square of the broadening factor in terms of

the RMS width is given by

22

4

2

22

2

33

41)sgn(21

)0(

)(

++

+=

DDDt

t

L

z

L

zN

L

zN

z βσσ

(3.8)


47

where LD is the dispersion length, and N is the normalized nonlinearity parameter as

defined in section 2-2.

The above expression can give a rough idea of how the interaction of dispersion

and nonlinearity can increase the output pulse width, but it is found that the expression

seriously overestimates the output pulse width. For a better result, we can model the

nonlinearity as lumped at the center of the propagation distance. In this model, it is

assumed that the nonlinearity has no effects on the pulse phase until the middle of the

propagation distance. However, at the middle of the propagation distance, the

nonlinearity changes the pulse phase suddenly by an amount given for the case of

nonlinearity alone occurring over the total propagation distance, z. The resulting pulse

propagates the remaining half of the distance again neglecting nonlinearity. If the

normalized input pulse, U(0,t), has a Gaussian shape such that

−=

2

2

2exp),0(

ot

ttU , the

pulse shape at the middle of fiber can be expressed by Eq.(2-10) and Eq.(2-12).

=

2

,2

exp,2

,2

tz

UL

zit

zUt

zU D

ND (3.9)

where ( ) ( )

−−

−=

22exp

2

,2

22

2

21

22 zit

t

zit

tt

zU

oo

oD ββ

and to is the initial pulse width

(half-width at 1/e intensity).

Now the output pulse shape will be determined by propagating

tz

U ,2

in the remaining

half of the distance by assuming dispersion alone.

The numerator of the RMS pulse expression in Eq.(3.6) can now be calculated

using Parseval’s theorem and the property of Fourier transform,

[ ] ( )n

nnn

df

fWdjtwt

)(2)( −−=ℑ π . The resulting RMS expression in normalized units is

given by


48

( )4

22

42

2

22

2

)(41133)(4112

)sgn(11

)0(

)(

++

+++=

DDDDt

t

L

z

Lz

N

L

z

Lz

Nz βσσ

�(3.10)

Since the method to derive Eq.(3.8) and Eq.(3.10) is analogous to the numerical

algorithm of the split-step Fourier method with the step size, z and z/2, respectively, we

may call them the one-step method and the two-step method, respectively.

Recently, a more elegant mathematical way, namely the variational method, has

been reported to model the RMS pulse width more accurately [33,34]. The method

assumes a given functional form for the pulse and allows the width, chirp, and height to

vary with propagation distance. When the input pulse is a Gaussian shape, the output

pulse profile is also assumed to have the Gaussian form,

+−=Ψ )(

)(2exp)(),(

2

2

zjbzt

tzatz

o

(3.11)

where a(z) is the pulse center height, to(z) is the pulse width (the half-width at 1/e-

intensity point) and b(z) is the chirp parameter. This Gaussian ansatz is substituted in the

NLSE to get the relation of the output RMS pulse width to the input RMS pulse width.

For a Gaussian pulse, to2(z)=2s2(z), and the square of the broadening factor is found to be

[33]

4

22

4

222

2

24

2)sgn(

24

1

2

)sgn(11

)0(

)(

++

++=

DDt

t

L

zNN

L

zNz ββσσ

(3.12)

Figure 3-1 compares Eq.(3.8), Eq.(3.10) and Eq.(3-12) with the simulated results by the

split-step Fourier method when b2 > 0 (normal dispersion). While the one-step method

(Eq.(3.8)) overestimates the RMS pulse width significantly as the propagation distance is

increased even with a modest nonlinearity (N=2), the two-step method (Eq.(3.10)) and


49

the variational method (Eq.(3.12)) follows the simulated result quite closely. As

expected, the more sophisticated the method is, the closer is the fit.

3-2-2. RMS Spectrum Width with a Gaussian Input Pulse

The magnitude of the pulse spectrum will not change at all in the case of

dispersion alone as predicted by Eq.(2.10). But fiber nonlinearity can cause a significant

distortion of spectrum shape at the fiber output. By assuming nonlinearity is dominant

(negligible dispersion), the RMS spectrum width, as defined by Eq.(3.7), can be found in

analytical form for a Gaussian input shape, as reported in [31]. Using the normalized

units, the expression is repeated in Eq.(3.13).

2/12

4

33

41

)0(

)(

+=

DL

zN

z

ω

ω

σσ

(3.13)

This equation predicts that the RMS spectrum width keeps increasing as a function of

distance. However, we expect this expression may be grossly inaccurate as the distance

becomes comparable to or greater than the dispersion distance, LD. This is because the

dispersion effect tends to make the spectrum magnitude become invariant.

Recently, it was reported that the spectrum width could be modeled more

accurately by the variational method, which gives the following result [35]

( ) ( )

2/1

2

0

1121

)0(

)(

=

−+=ξξσ

ξσω

ω

oo ttN (3.14)

where x = z/LD. In the variational method, the pulse shape is assumed to remain a

Gaussian as given in Eq.(3.11). Therefore, the pulse broadening factor in terms of the

half-width at 1/e-intensity, ( ) ( )0oo tt ξ , is the same as the pulse broadening factor in


50

terms of the RMS pulse width, )0()( tt σξσ . It is interesting to note that the variational

method (Eq.(3.14)) predicts the output RMS spectral width asymptotes to

( ) ( )021212

ωσN+ as the pulse broadening factor increases while Eq.(3.13) doesn’t

predict such an asymptotic behavior. When N is greater than 1, the asymptote is

approximately proportional to the nonlinear parameter N since

( ) ( ) ( )02021 41212ωω σσ NN ≈+ . Therefore, in physical units, the asymptote is

approximately given by ( )018.12

ωσβ

γ oavg

tP where g is the nonlinearity constant, Pavg

is the average signal power, and b2 is the second order propagation constant.

Figure 3-2 compares the normalized RMS spectrum width predicted by Eq.(3.14)

with the simulation results by the split-step Fourier method. In calculation of Eq.(3.14),

the simulated pulse width is used for ( ) ( )0oo tt ξ . In Figure 3-2, we observe both curves

agree very well each other for all values of N, but the deviation increases as N increases.

For comparison purposes, if Pavg = 1mW, g =2�10-3[km-1¼mW-1], to=100ps, and b2 = 5

[ps2/km], these parameter values result in N = 2, and the variational method predicts the

asymptote to be 2.58sw(0) which is very close to the simulated result (~2.53sw(0)) as

observed in Figure 3-2. However, Eq.(3.13) with the same parameter values gives

sw(x=5) 17.6sw(0) which is a gross overestimate.


51

Figure 3-1 Comparison of RMS pulse width models with the simulated one. Input pulse

is a Gaussian shape and normal dispersion region is assumed. x = z/LD.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

z/LD

RMS Pulse Width Evolution in Normal Dispersion Region (N=2)

Simulation One-step Two-step Variational

st(x

)�s

t(0)


52

Figure 3-2 Comparison of RMS spectrum width model by the variational method (-.)

with the simulated one (solid) in the normal dispersion region.

0 1 2 3 4 5 6 7 8 9 10 1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6 Normalized RMS Spectrum

N=5

N=4

N=3

N=2

N=1

sw(x

)�s

w(0

)

x=z/LD


53

3-3. Optimization of RMS Widths

3-3-1. Optimum Input Pulse Width to Minimize st(z)

In the normalized NLSE, transmission distance is often normalized by the

dispersion distance, LD. However, LD is defined in terms of the input pulse width as in

Eq.(2-1). Since we want to optimize the RMS widths with respect to the input pulse

width, it is more convenient to normalize distance by the nonlinear distance, LN, which is

independent of the initial pulse width. Additionally, it is convenient to define a new

normalized quantity, s, such that

2

)()(

2 N

t

Ls

βζσζ = (3.15)

where z=z�LN.

Notice that N

D

N

oo L

L

Lss ===

2)0(

2βσ

is the same as the nonlinear parameter, N, as

defined in Eq.(2-7). Now the broadening factor, st(z)�so, is the same as the ratio, s(z)�so,

and the optimization of st(z) with respect to so is the same as the optimization of s(z)

with respect to so. With fixed physical parameters, the optimum input pulse will indicate

the optimum nonlinearity constant, so,opt (=Nopt) in the system.

In the following, the functional forms of the optimum so will be derived based on

the one-step method (Eq.(3.8)) and the two-step method (Eq.(3.10)).

One-Step Method

Eq.(3.8) can be expressed in terms of z=z�LN using the relationship,

22oD

N

ND sNL

L

L

z

L

z ζζ === ,


54

4

4

2

2

222

22

33

412)sgn(1

)0(

)(

oooot

t

ssss

sz ζζβσσ

+

++==

(3.16)

By differentiating s2 with respect to so2, then setting equal to zero, we get the optimum so

value such that

424,

33

4 ζζ +=optos (3.17)

In the extreme case of z÷1,

ζζζζ ≈≈+= optoopto ss ,2424

, ,33

4 (3.18)

which gives 22

22,opto,

zLs N

opto

ββσ =≈ . With this optimum value and using the

condition (z÷1) in Eq.(3.8) gives

z2min βσ ≈ (3.19)

In this extreme case, the optimum input pulse width and the minimum output pulse width

are the same as the case of dispersion alone.

In the other extreme case of zø1,

ζζζ 94.033

4 ,

33

4 41

opto,44

, =

≈≈ ss opto (3.20)

which gives the optimum input pulse width

N

N

Nopto L

zL

L

z 22, 662.0

294.0

ββσ =≈ (3.21)


55

Substituting Eq.(3.20) or Eq.(3.21) into Eq.(3.16) gives

Nopto L

zs 2,minmin 26.19.1or 786.1

βσσζ =≈≈ (3.22)

Two-Step Method

Eq.(3.10) can be rewritten in terms of z.

4

2

2

4

2

2

2

2

4

2

22

22

4

1133

1

4

112

)sgn(1

)0(

)(

o

o

o

o

ot

t

s

s

s

s

s

sz ζ

ζ

ζζζ

βσσ

+

++

+

+==

(3.23)

Again, by differentiating s2 with respect to so2 and setting to zero, we get the quartic

equation below. Here x=so2.

0

4

11

1

33

4

11

33

4

11

24

)sgn(3

2

2

6

2

2

2

24

2/3

2

2

42224 =

+

+

+

−

+

+−

xx

x

x

xxx

ζ

ζ

ζ

ζ

ζ

ζβζ

�(3.24)

In the extreme case of z÷1, the above equation is simplified greatly such that

224 xx ζ≈ . From the simplified relation,

22 , 22

,opto,,

zLss N

optoopto

ββσζ =≈≈ (3.25)

which gives


56

zs 2minmin ,2 βσζ ≈≈ (3.26)

in agreement with the one-step result.

In the other extreme case of zø1, Eq.(3.24) reduces to

03324

)sgn(

33

6422

44 =++− ζζβζ

xxx (3.27)

In Eq.(3.27), it is assumed 4

1

4

1 2

2

2

osx

ζζ = ÷1, the validity of which will be checked later.

In principle, the resulting quartic equation can be solved analytically with the help of

standard mathematical software. However, the solutions have a very long and

complicated form, which makes them hardly useful. To attempt further simplification of

Eq.(3.27), we compare the magnitude of each term in Eq.(3.27) with the help of

simulated data at z=25 (so,opt 8).

1st term ~ 1.68û107

2nd term ~ 3.08û108

3rd term ~ 4.48û106

4th term ~ 4.7û107

It’s a rough approximation, but to get an analytical solution, the third term is ignored.

Then the solution is

−±== 6

844,

2

27

4

27272

1 ζζζoptosx (3.28)

Since we used the assumption 2

2

4

1

x

ζ÷ 1, the positive sign (+) is appropriate in

Eq.(3.28). Then


57

N

Noptooptoopto L

zL

ss 22,,8/1, 468.0

2 ,66.0

27

ββσζζ =≈=≈ (3.29)

From Eq.(3.29), 24

,

2 27

4

1

4

1

ζζ =

optos÷ 1 when z ø1, which validates the assumption made

in Eq.(3.27).

With Eq.(3.29) and (3.23),

+≈++=

2

)sgn(

27

227

2

)sgn(

27

2 24/1

24/12224/1

2min

βζζβζs

Although Eq.(3.29) predicts that the optimum pulse width is independent of the sign of

b2, the RMS pulse width is more meaningful for estimating distortion effects in fiber

transmission in the case of normal dispersion. In this case, sgn(b2) = +1. Then

Noptoopto L

zss 2,min,min 89.09.1 ,9.1259.1

βσσζ =≈=≈ (3.30)

In the limit of z÷1, both methods (one-step and two-step methods) lead to the

case of dispersion alone. This is not a surprising result since the condition of z÷1

indicates the propagation distance is much smaller than the nonlinear distance, LN, which

means the nonlinearity has little effect on the transmission of the pulse.

In the other limit, zø1, which is the more interesting case, the analytical results

are summarized in Table 3-1. For comparison purposes, the analytical result by the

variational method and the simulation result by the slit-step Fourier method are also

included. (Figure 3-3 compares the two methods, the variational and the split-step Fourier

methods, which indeed demonstrates the existence of the optimum input pulse width to

minimize the output pulse width.) Normal dispersion is assumed in all the cases. It is

interesting to observe that all the analytical methods predict that so,opt and smin are

linearly proportional to the propagation distance, z unlike the case of dispersion alone

where so,opt and smin are proportional to the square root of the propagation distance, z


58

(Eq.(3.2) and Eq.(3.3)). As the analytical methodology gets more sophisticated, the

proportionality constants get smaller and closer to the simulated values. This is because,

in simpler models, the interaction of nonlinearity and dispersion is underestimated such

that a larger nonlinearity (larger so= N) gives a narrower output pulse width (nonlinearity

alone makes the output pulse width invariant.).

Since we obtained analytical expressions in two extreme cases, z÷1 and zø1, it

is of interest to find the critical distance, zc, which divides the two regions. Figure 3-4

compares the simulated so,opt with the dispersion dominant case, so,opt = ζ . When z is

relatively small, the dependence of so,opt on z is pretty well predicted by the square root of

z. As a rule of thumb, when z < 3, so,opt ζ . Otherwise, so,opt can be more accurately

predicted by the curve fitting result. Therefore the critical distance, zc 3.

Table 3-1 Summary of the optimum input pulse widths and the minimum output pulse

widths in the normal dispersion region by the various methods. (NL

z=ζ ø1)

One-Step Method

Two-Step Method

Variational Method [33,34]

Split-step Fourier Simulation

and Curve Fitting

so,opt (=Nopt)

ζζ 94.027

28/1

=

ζζ 66.027

18/1

=

ζζ 452.024

14/1

=

9056.02897.0 +ζ

smin 1.786ûz 1.254ûz 1.056ûz 6427.09751.0 +ζ

opto,σ NL

z 2662.0β

NL

z 2468.0β

NL

z 232.0β

opto,

min

,

min

s

s =

optoσσ

1.9

1.9

2.34

3.35


59

Figure 3-3 Normalized output widths as a function of normalized input width (so) at three

distances ζ=z/LN= 0.2, 10, and 20. Solid curves by split-step Fourier method and dotted

curves by variational method

0 2 4 6 8 10 120

5

10

15

20

25

Normalized input width, so

Nor

ma

lized

out

put w

idth

, s(s o)

ζ = 20

ζ = 10

ζ = 0.2


60

0 5 10 15 20 25 0

1

2

3

4

5

6

7

8

9

z=z/LN

Figure 3-4 Comparison of simulated so,opt with curve-fitting and square-root of z

So,opt

Simulated data(*) & Curve fitted

NLz=ζ


61

3-3-2. Optimum Input Pulse Width to Minimize sw(z)

For a transform-limited Gaussian input pulse, Eq.(3.14) from the variational

method can be rewritten using the relation, 41)0()0( 2222 ==== zztoo ωω σσσσ . In terms

of the normalized distance z (=z/LN),

−+=

2

22

22

)(

11

2

1

4

1)(

o

tNo L

σζσβσ

ζσ ω (3.31)

In Eq.(3.31), the relationship, N

o

LN

2

22 2

βσ

= , is used as defined in Eq.(3.15). From

Eq.(3.12), the square of the pulse broadening factor can be rewritten in terms of z as

below.

6342212

2

2

2 1111

)()(

ooooo

t

sC

sC

sC

s

s +++== ζσ

ζσ (3.32)

where 43

422

21 24

2 and ,

24

1,

2

1 ζζζζ =+== CCC .

Eq.(3.32) shows that 2

2 )(

o

t

σζσ

is a monotonically decreasing function of so. Therefore, we

observe that the RMS spectrum width expressed in Eq.(3.31) is a monotonically

decreasing function of so as well. This is because

−

2

2 )(11

o

t

σζσ

term in Eq.(3.31) is

monotonically decreasing with so. That is, the variational method predicts that there is no

optimum input pulse width to minimize the output spectrum width when the input pulse

is a transform-limited Gaussian. As input pulse width is increased, N

o

LN

2

22

βσ

=


62

becomes larger, and then the spectral broadening factor,)0(

)(

ω

ω

σξσ

, also increases as

observed in Figure 3-2. However, the initial spectral width ( ) 12)0(

−== oo σσσ ωω

decreases as input pulse width increases. Apparently, the increased spectral width

resulting from the nonlinearity is less than the reduction in the initial spectral width due

to the increase initial pulse width.

3-3-3. Optimum Input Pulse Width to Minimize the Product of st(z) and sw(z)

Now consider the product of output pulse width and output spectrum width.

Using Eq.(3.31) and (3.32),

−+=

o

oo

t

ss

ss

s)(

1121

)(

4

1 22

222

ζζσσ ω (3.33)

If we define ( ) ( )ζσζσζ ω224)( tT = ,

−+=

ooo

o s

s

s

ss

s

sT

)()(2

)()(

2

22

2

2 ζζζζ (3.34)

For simplicity of notation, define . and 222oo ssysx == Then Eq.(3.34) can be

expressed in terms of x and y as below.

( ) ( ) ( )yyxyT t −+== 24)( 22 ζσζσζ ω (3.35)

To have an optimum input pulse width, the derivative of T(z) with respect to x should

have zero value(s).


63

( )yyx

y

yx

x

T −+∂∂

−+=

∂∂

22

121 (3.36)

where

( )1 1 and 0) ( 32

33

221

43

32

21 >+++=<−−−=

∂∂

x

C

x

C

x

Cy

x

C

x

C

x

C

x

y

We can rearrange Eq.(3.36) such that

( )

+−−+

∂∂

−=

−−

−+

∂∂

−=

−++++−−−+

∂∂

−=

−+

∂∂+

∂∂

−=

∂∂

33

22

43

32

43

32

21

43

32

21

2212

2

11

212

2

11

1322

2

11

22

11

x

C

x

Cy

x

y

y

x

C

x

C

x

yx

x

y

y

x

y

x

C

x

C

x

C

xx

C

x

C

x

Cx

x

y

y

x

y

x

y

x

yx

x

y

yx

T

Since y = 22oss > 1 and

x

y

∂∂

< 0, the first and the second terms are always negative.

Furthermore, because x, C2 and C3 are all positive quantities, the third term is also

negative, which means that the derivative of T(z) with respect to x is always negative

regardless of the initial pulse width. This result leads to the conclusion that the variational

method predicts there is no optimum input pulse width which minimizes the product of

st(z) and sw(z) when the input pulse is a transform-limited Gaussian. This is mainly

because sw(z) is a monotonically decreasing function of the initial pulse width, so (or the

normalized initial pulse width, so).

In Figure 3-5, T(z), as calculated by the split-step Fourier method at a few fixed

distances, is plotted as a function of input pulse width. It is seen from Figure 3-5 that T(z)

monotonically decreases as the input pulse width (so) increases.


64

3-4. Summary

Output RMS pulse width is modeled by lumping the fiber nonlinearity at the

middle of the propagation distance. The methodology is fairly simple and the resulting

two-step model predicts the output RMS pulse width much closer to the simulated one

compared to the previous one-step model, in which the fiber nonlinearity is lumped at the

input of the fiber. The two-step model is also used to derive the optimum input pulse

width to minimize the output pulse width at a given distance. While the two-step model is

not as good as the variational model (it gives a larger deviation from the simulation

results), it is interesting to see that all of the analytical models including the one-step

model predict the same functional form of so,opt, which is linearly proportional to the

propagation distance, z. If the maximum bit rate is taken to be 1/(4st) (see Eq.(3.1)), all

the analytical models predict the maximum bit rate-transmission distance product has a

functional form of (Table 3-1)

22

1~

βγβ avg

Nb P

LzR = (3.37)

if zø1(zøLN).

Eq.(3.37) predicts that the maximum bit rate-transmission distance product is inversely

proportional to the square roots of both the average power of the signal and the fiber

dispersion coefficient.

When z÷1 (z ÷ LN), so,opt degenerates to the case of dispersion alone, where so,opt is

proportional to the square root of z. In this case, 2~ βzzRb . The simulation results

(Figure 3-4) shows that the boundary between the two extreme cases is near z=3 (The

transmission distance is 3 times the nonlinear distance LN).

Unlike the output pulse width, there is no optimum input pulse width to minimize

the output spectrum width because the RMS spectrum width, sw(z) is a monotonically

decreasing function of the input pulse width. When we desire to optimize the product of

st(z) and sw(z) in the case of dispersion alone, the initial pulse width which minimizes

the output pulse width will also be the optimum value to minimize st(z)�sw(z) because


65

sw(z) is invariant. However, with fiber nonlinearity, it is shown mathematically that there

is not an optimum input pulse width regardless of the propagation distance. The reason is

that the output spectrum is a monotonically decreasing function of input pulse width so

and the optimum pulse width is not a strong function of so as observed in Figure 3-3.

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600

700

800

900

1000

Figure 3-5 T(z) as a function of so

2 in the normal dispersion region with a Gaussian input.

T(z)

z=5

z=20

z=10

x=so2

Chapter 4: Performance measurements using sinusoidally modulated signal

66

Chapter 4

Performance Measurements Using Sinusoidally Modulated Signal

4-1. Introduction

In a digital communication link, bit error rate (BER) is the most important

parameter to measure the performance of the communication link between a transmitter

and a receiver. In an optical fiber communication system, BER may often be measured

only experimentally. This is because the high quality performance of a conventional

optical fiber communication link (BER =10-9) requires an extremely large number of bits

to evaluate BER, which makes numerical simulation of BER generally impractical. BER

is often evaluated indirectly using Q-factor1, which is commonly used to measure system

performance. Another simpler way of estimating performance is to observe eye-opening.

The eye-opening is quantified by measuring the minimum value between the sampled

values of marks (ones) and spaces (zeros) in the received bit sequence, r(t). The eye-

1Q-factor is essentially the signal-to-noise ratio at the decision circuit. It will be defined more precisely in Chapter 6.


67

opening is a useful system performance metric when signal distortion is a more limiting

factor than noise. Mathematically, it is defined as below [12].

( ) ( )o

jjjj

P

btrbtr )0,(max)1,(min opening-Eye

=−== (4.1)

where tj represents the sampling instant of the jth-bit interval and Po is the peak power of

r(t). The first term in the numerator represents the minimum value at the sampling instant

when a mark is transmitted, and the second term is the maximum value when a space is

transmitted. In this chapter, the sampling instant of the received signal is assumed to be at

the center of each bit period. To assess system performance degradation due to signal

transmission through the fiber, eye-opening penalty (EOP) is often used. EOP is the

measure of the relative eye opening after transmission compared to eye opening in the

back-to-back case (no transmission effect). That is,

[ ]

−=

back)-to-(backfiber without opening-eye

nsmission)(after trafiber with opening-eyelog10dB EOP (4.2)

The RMS pulse width models developed in Chapter 3 may give a good estimation of

EOP because the pulse spreading at a given pulse energy often indicates the decrease of

the peak value of the pulse. However, since the RMS pulse width models are based on

the transmission of a single pulse, they may not be a good indicator of EOP when

intersymbol interference(ISI) due to dispersion and/or nonlinearities is not negligible.

In this chapter, the input optical signal is assumed to have a raised-cosine form

which models an alternating bit sequence of ones and zeros. The sinusoidally modulated

signal enables us to analyze the optical transmission impairments due to fiber

nonlinearities, including the effects of intersymbol interference. Indeed, when ISI is

predominantly caused by the neighboring pulses, it may be argued that the alternating

pattern is the worst-case pattern. In the following section, the self-phase modulation

effect will be studied using a sinusoidally modulated signal (section 4-2). Next, a


68

sinusoidally modulated signal will also be used to study system performance degradation

due to cross-phase modulation in a multi-channel system (section 4-3). The sinusoidal

analyses in both sections will be compared to more realistic cases by simulations in

which a pseudo-random bit sequence (PRBS) is used for the input bit sequence.

4-2. Self-Phase Modulation Analysis using Sinusoidally Modulated Signal

4-2-1. Theoretical Background

Dispersion is an undesirable characteristic of the fiber and its effects on the

performance of optical communication system may be expressed in both the time and

frequency domains. In the time domain, the dispersion effect is often expressed in terms

of RMS pulse width (st) which provides an estimate of the maximum allowable bit rate

by a simple relationship like Eq.(3.1), 4

1<bt Rσ where st is the RMS pulse width at the

output of the fiber, and Rb is the bit rate.

The effect of dispersion on the system performance may also be approximated in

the frequency domain by defining the transfer function of the fiber in power (Hp(w)).

Under some conditions, the fiber can be modeled as a pseudo-linear system in power for

digital communication purposes [36], in which case Hp(w) is related to the optical input

and output power by

Pout(w) = Hp(w)Pin(w) (4.3)

The above relationship can be used in determining the bandwidth of the fiber. If we

consider an optical input signal intensity-modulated by a constant amplitude sinusoidal

wave, the amplitude of the output optical signal decreases as the modulation frequency

increases, resulting in a low-pass system response (Figure 4-1) because the dispersion


69

becomes more significant as the modulation frequency (or bit rate) increases [37].

Therefore the effect of finite bandwidth on system performance is to limit the bit rate that

can be transmitted over a given distance, and it is often quantified by the dispersion

power penalty, which is the required input power increase to compensate for the decrease

of output peak power (that is, decrease of signal to noise ratio) caused by dispersion.

Similarly, the amplitude of the output optical signal will decrease at a given modulation

frequency as the transmission distance increases because of fiber dispersion.

One way of estimating the dispersion power penalty at a given bit rate is to

consider the alternating sequence, i.e., ¡,1,0,1,0,1,0,¡, because that sequence has the

highest possible freqency component w/2p = Rb/2. If we assume that Hp(w) has a

Gaussian shape such that ( ) ( )2exp)0( 22ωσω tpp HH −= [61],

Optical Power Penalty (dB) = -10log10Hp(p Rb) =21.4(stRb)2 (4.4)

For example, if we allow the power penalty to be 1dB, the pulse spreading satisfies the

relationship

st � 0.216/Rb (4.5)

which results in a slightly more stringent condition than Eq.(3.1). The alternating

sequence has also been considered in [83] to study how coding may be used to counter

the effect of dispersion.

In practice, the situation is much more complicated when nonlinearities are

present, but measuring the sinusoidal response at w/2p = Rb/2 may still give an

indication of the system performance degradation. We wish to test this supposition, and

to determine the extent to which sinusoidal response may be used to measure

performance. The measurement of sinusoidal response may give a better system

performance estimate than the measurement of the rms width of an isolated pulse. Also,

it should be a much simpler test scheme compared to a BER measurement which requires

a very long pseudo-random bit sequence.


70

Figure 4-1 Frequency response of fiber. As dispersion increases, the bandwidth of Hp(w)

decreases [37].

Transmitter Receiver Fiber

Frequency

Hp(w)

Fiber Response


71

In modern optical communication systems, fiber nonlinearities can not be simply

ignored because the input peak power is increasing and the nonlinearities are

accumulating with the advent of optical amplifiers. Therefore it is of interest to study

how the fiber nonlinearities affect the sinusoidal response, and to see whether the

sinusoidal response is still valid to measure the worst case ISI effect on the system

performance when the fiber nonlinearities are not negligible. In the following, the

sinusoidal response of dispersion alone case will be studied first, and the result will then

be extended to include the effect of fiber nonlinearity.

4-2-2. Sinusoidal Response of NLSE

First consider the dispersion alone case with a periodic input signal, for which we

can solve the NLSE analytically. The resulting analytical expression can be used to

estimate the sinusoidal response in a dispersion dominant system, and it will also be

useful to check numerical simulation results.

The normalized NLSE without nonlinear terms is given in Eq.(2-8) and repeated

below.

2

2

2 )sgn(2

1

τβ

ξ ∂∂⋅−=

∂∂ U

iU

(4.6)

If the input pulse sequence is periodic, we can express it in a Fourier series as below.

)0(),0( ∑∞

−∞=

=n

jnn

peCU τωτ (4.7)

where wp is the fundamental angular frequency of ),0( τU . Since we are interested in a

periodic input signal in this chapter, it is convenient to normalize the time variable t by

the bit period Tb such that t=t/Tb. Therefore, the dispersion distance, LD, and the


72

nonlinear constant, N, are also defined in terms of bit period Tb. That is, 2

2

βb

D

TL = and

2

22

βγ bavg

N

DTP

L

LN == . These conventions will be used throughout this chapter.

Since the linear response of a periodic signal will also result in a periodic signal, the

output signal can also be expressed in a Fourier series.

∑∞

−∞=

=n

jnn

peCUτωξτξ )(),( (4.8)

Now the Fourier series coefficients, which are functions of transmission distance, can be

derived by substituting Eq.(4.8) into Eq.(4.6), and the solution can be easily found.

)()sgn(2

)(

)(n)sgn(2

)(

222

-n

222

ξωβξ

ξ

ξωβξ

ξ τωτω

npn

jnnp

jn

n

n

Cnj

d

dC

eCj

eC

pp

=

=∂

∂ ∑∑∞

∞=

∞

−∞=

=∴ ξωβξ 22

2 )sgn(2

exp)0()( nj

CC pnn (4.9)

Finally the output may be written as

∑∞

−∞=

=

n

jnpn

penj

CU τωξωβτξ )sgn(2

exp)0(),( 222 (4.10)

For example, if we model the alternating bit sequence as a raised-cosine wave with its

period Tp = 2�Tb (Tb = bit period), the normalized input signal can be written as below.

)(4

1

2

1cos

2

1

2

1),0( pp fU ωτωτ +=+= (4.11)


73

where .2

1

2 and)( p =+= −

πω

ω τωτω pp jjp eef

The Fourier series coefficients of the input signal are C0(0) = 1/2, C1(0) = C-1(0) = 1/4,

and Cn(0) = 0 (for all n ≠0,±1). The output signal by assuming β2 > 0 is then

τωωτξξωξω

p

j

p

jpp

efeU cos2

1

2

1)(

4

1

2

1),(

22

22 +=+= (4.12)

Therefore when the input field signal is given by Eq.(4.11), the input and output optical

power signals are,

τωτωττ ppUP cos2

12cos

8

1

8

3),0(),0(

2 ++== (4.13)

τωξω

τωτξτξ pp

pUP cos2

cos2

12cos

8

1

8

3),(),(

22 ++== (4.14)

From the output power signal, P(x,t), we can see that the fundamental frequency

component (wp), C1, is periodic as a function of distance while the DC and the second

harmonic components remain constant. Because of its periodicity with respect to distance

parameter, x, the magnitude of C1 will have its first null at xo = zo�LD = p�wp2 = 1�p =

0.3183, where the fundamental frequency component will die out completely. Since we

may not get any further information from the magnitude of C1 after the first null, the

magnitude of C1 should be measured before the first null occurs at xo = 0.3183 in the case

of dispersion alone. In physical units, xo corresponds to 0.3183�LD = 0.3183�Tb2/|b2|.

For example, in 10Gb/s systems, the first null distance (zo) will occur around 160km for a

typical dispersion coefficient of conventional single mode fiber, |b2| = 20 [ps2/km]. At the

same bit rate, the first null distance (zo) is around 1060km for a typical dispersion

coefficient of dispersion-shifted fiber, |b2| = 3 [ps2/km].

Figure 4-2(a) shows the magnitude of the Fourier series coefficients of the optical

power signal with dispersion alone at a few fixed distances, while Figure 4-2(b) shows


74

their evolution as a function of distance. Both figures are generated numerically by the

split-step Fourier method with the input optical field given by Eq.(4.11), and the results

agree well with the analytical expression of Eq.(4.14).

Figure 4-3(a) and (b) show the evolution of the Fourier series coefficients in the

normal dispersion region (β2 > 0) when fiber nonlinearity is non-negligible, specifically

N = 2. Unlike the dispersion alone case, it is observed that new frequency components,

mainly at w=3wp, are generated, which indeed shows that the fiber acts as a nonlinear

system. Figures 4-2(b) and 4-3(b) show that the magnitude of the fundamental frequency

component behaves like a low-pass filter as a function of propagation distance before the

first null. Since the difference between the two figures is whether or not fiber nonlinearity

is present, the curves may reveal how the nonlinearity affects the system performance.

Figure 4-4 ((a) normal dispersion, (b) anomalous dispersion) shows that the

evolution of the magnitude of the fundamental frequency component, |C1|, at a few

different N values. While |C1| decreases as N increases in the normal dispersion region at

a given normalized distance, z/LD, before the first null, the opposite occurs in the

anomalous region. This is because the anomalous dispersion region supports solitons.

The input pulse will evolve into a fundamental soliton if (1/2) < N < (3/2), and a second-

order soliton if (3/2) < N < (5/2), and so forth. Second and higher order solitons break up

into spiked pulses and reassemble periodically while they propagate. Therefore, in the

anomalous region, the sinusoidal test to see the worst case ISI effect may not be

appropriate because other effects like modulation instability2 or optical amplifier noise

can be more limiting factors on system performance [11,12]. However, when the N value

is sufficiently small (N < 1/2) such that the input pulse does not evolve into a

fundamental soliton, the sinusoidal method may still be useful to assess the worst case

system performance even in the anomalous dispersion region. In the following section,

the sinusoidal analysis will be compared with EOP to determine the extent to which these

are correlated.

2 Modulation instability is known to be observable in the anomalous region only. It is often interpreted in terms of a four-wave-mixing process phase-matched by SPM.


75

4-2-3. Eye-Opening Penalty and Sinusoidal Response

In the previous section, it is observed that the magnitude of the fundamental

Fourier series coefficient, |C1(x)|, may be a good indicator of system performance

degradation even in the presence of fiber nonlinearity. In this section, |C1(x)| will be

compared with a more general measure of system performance, namely the eye-opening

penalty (EOP).

Figure 4-5 illustrates how EOP increases with transmission distance in the normal

dispersion region. Figure 4-6 shows eye-diagrams of the received signals for the

conditions indicated in Figure 4-5. In the calculation of EOP, a 32 bit pseudo-random

sequence, ∑=

−bN

kbk kUb

1

)~~( ττ , is used as the input. Nb = 32 and bk = the information bit

sequence (‘01011000101111011010100000101110’), and a Gaussian pulse shape

−

−== 2

2

~2

1exp

2

1exp)~( ττ

oo t

t

t

tU is assumed. The bit period in normalized unit,

bτ~ , is taken to be 3, which corresponds to Tb = 3�to in physical units where to is the initial

half width at half maximum of the Gaussian pulse.

To compare the EOP using pseudo-random bit sequence (PRBS) with the

sinusoidal response, sinusoidal response penalty (SRP) is defined as below.

−=

)0(

)(log10[dB] SRP

1

1

C

C ξ (4.15)

where |C1(x)| is the magnitude of the fundamental Fourier series coefficient of the

received signal at x. For example, 1dB penalty of SRP corresponds to |C1(x)| = 0.1986

since |C1(0)| = 0.25 in Eq.(4.11).

Figure 4-7 compares the critical transmission distances, Dcc Lz=ξ , when EOP

and SRP reach 1dB respectively as a function of N2. In the normal dispersion region, the

two curves agree very well over a wide range of N values. This result strongly indicates

that the sinusoidal analysis can be used either experimentally or computationally as an

alternate way of EOP measurement. Figure 4-7 shows that the transmission distance for a


76

1dB penalty remains almost constant when the N value is less than 1. This suggests that

the fiber can be considered as a linear device as long as N < 1. However, when N > 1, the

1dB penalty distance decreases as N increases. Physically, this can be interpreted as the

maximum transmission distance for an allowable 1dB penalty decreases as the signal

power increases when other fiber parameter values are fixed. If we use g =

2.43�10-3[1/(km¼mW)], and b2 = 3 [ps2/km], and bit rate Rb =10Gb/s (Tb = 100ps), N =1

corresponds to a signal power 0.12 mW (path averaged). The 1dB penalty distance with

N = 1 is approximately 0.1 in normalized units, and it corresponds to zc 0.1LD =

0.1Tb2/|b2| = 333 km in physical units.

Figure 4-7 (b) compares the 1dB penalty distances of EOP and SRP in the

anomalous dispersion region. Unlike the normal dispersion case, the 1dB penalty distance

increases as N increases. As we observed in Figure 4-4(b), the sinusoidal analysis may

not be appropriate to assess system performance because |C1(x)| behaves irregularly and

doesn’t drop below its initial value when the N value is greater than 3. However, within a

limited range of parameter values, the sinusoidal analysis could serve as an easy alternate

way of estimating EOP even in the anomalous dispersion region.

4-2-4. Sinusoidal Response using Perturbation Analysis

In the previous section, it was demonstrated that the sinusoidal analysis could be

an alternate much simpler way of measuring EOP. In this section, the sinusoidal response

of NLSE is solved by the perturbation method developed in Chapter 2.

From Eq.(2.16), the first-order perturbed output, ),()1( τζU , is related to the linear

output, ),()0( τζU , as below.

),(),(),(

2

),( )0(2)0(22

)1(2)1(

τξτξτ

τξξ

τζUUjN

UjU +∂

∂−=∂

∂ (4.16)

In Eq.(4.16), the fiber is assumed to be in the normal dispersion region where the

sinusoidal analysis has a broader range of agreement with EOP than in the anomalous


77

region. When the input sinusoid is given as the raised-cosine form (Eq.(4.11)), ),()0( τζU

is the same as Eq.(4.12), which will make the nonlinear term in Eq.(4.16) periodic.

Therefore the first-order perturbed output ),()1( τζU will also be periodic, and we can

express the first-order output in terms of a Fourier series.

∑∞

−∞=

=n

jnn

peCU τωξτξ )(),( )1()1( (4.17)

with initial condition Cn(1)(0) = 0 for all n.

By putting Eq.(4.17) into Eq.(4.16), we get

),(),()(2

)( )0(2)0(2)1(22)1(

τξτξξωξ

ξ τωτω UUjNeCnj

eC

pp jn

no

jnn ∑∑ +=∂

∂ (4.18)

where

)3(64

1)2(

2cos

16

1

32

1

)(64

7

2cos

8

1

2cos

8

1

16

3),(),(

22

22

2

2

2

2

22

2)0(2)0(

p

j

pp

j

p

jpp

j

fefe

feeUU

pp

pp

ωωξω

ωξωξω

τξτξ

ξωξω

ξωξω

+

++

+++=

The Fourier series coefficients, Cn(1), can be evaluated by equating terms of the same

frequency. Since the highest frequency component in ),(),( )0(2)0( τξτξ UU is 3ωp, we

need to set up differential equations up to n = 3. The resulting Fourier series coefficients

for each n value up to n = 3 are

n = 0, ( )

++−= ξω

ωξξω

ωξ 2

2

222

2

2)1(

0 sin164

11cos

16)( p

pp

p

NNj

NC

n = 1,

+

+

−

−

=

2cos

64

11

2sin

8

1

2sin

64

11

2cos

2

3cos

32)(

22

2

22

22

2

2

2)1(

1

ξωξ

ξωω

ξωξ

ξωξω

ωξ pp

p

pp

p

p

jNNN

C


78

n = 2,

( )[ ]ξωξωξωξωω

ξ 22222

2)1(

2 sin()2sin(21)cos()2cos(232

)( ppppp

jN

C −+−−=

n = 3,

−+−−= ξωξωξωξω

ωξ 2222

2

2)1(

3 sin()2

9sin(1)cos()

2

9cos(

256)( pppp

p

jN

C

Now the output field is approximated as the sum of the linear solution and the first-order

perturbation solution such that

[ ] [ ]( ) ( )

)3()2()(

)3()2()(

)3()2()()(

),(),(),(

321

)1(3

)1(2

)1(1

)0(1

)1()0(

)1(3

)1(2

)1(1

)1()0(1

)0(

)1()0(

pppo

pppoo

pppopo

fCfCfCC

fCfCfCCCC

fCfCfCCfCC

UUU

ωωωωωω

ωωωωτξτξτξ

+++=

+++++=

+++++=

+≈

�(4.19)

where

( )

)1(33

)1(22

22

22

222

2

22

2

2

)1(1

)0(11

22

222

2

2)1()0(

2cos

64

11

2sin

8

1

4

1

2sin

64

11

2

3cos

322cos

324

1

sin164

1cos1

162

1

CC

CC

jN

NNN

CCC

NNj

NCCC

pp

p

pp

p

p

p

pp

pp

ooo

=

=

+

++

−

+

−=

+=

+−

−−=+=

ξωξ

ξωω

ξωξξω

ωξω

ω

ξωω

ξξωω

Finally, the output optical power signal is obtained as

2

321

2)1()0(2)3()2()(),(),(),(),( pppo fCfCfCCUUUP ωωωτξτξτξτξ +++=+≈=

�(4.20)


79

Eq.(4.20) has various frequency components. However, we are interested only in the ωp

component (the fundamental Fourier series component), which is expressed as

( ) ( ) ( ){ }*32

*21

*1 ReReRe2|),( CCCCCCP op

++=ωτξ (4.21)

where * denotes the complex conjugate.

It is worth remembering that the perturbation analysis has a limited range of applicability

because of accuracy as discussed in Chapter 2. To estimate the valid range of Eq.(4.21),

the critical distance at NSD (normalized square deviation) = 10-3 using the first-order

perturbation solution evaluated in Chapter 2 is plotted again in Figure 4-8 (a). NSD is

defined in Eq.(2.27), and is expressed as

∫

∫∞

∞−

∞

∞−

−=

ττ

ττξτξξ

dU

dUU

NSDBA

2

2

),0(

),(),(

)( where

UA(x,t) = output field envelop by method ‘A’, and UB(x,t) = output field envelop by

method ‘B’.

The NSD curve compares the distance of 1dB SRP (sinusoidal response penalty,

defined in Eq.(4-15)) resulting from simulations as a function of N2. The normal

dispersion region is assumed in both cases. Figure 4-8 (a) indicates that Eq.(4.21) can

give a large error when N2 is greater than 3. In Figure 4-8 (b), the |C1(x)| by simulation is

compared with Eq.(4.21) when N2 = 3. Even with a modest value of N parameter, the

two curves show a significant discrepancy. For example, the normalized distance

corresponding to 1 dB SRP is around 0.07 from Eq.(4.21), but the simulation result gives

approximately 0.11. This result suggests that the perturbation method (Eq.(4.21)) should

not be used to get numerical results (except for very small N), although the expression

can give some physical insight. Even though Eq.(4.21) can be accurate when the N value

is small (dispersion dominant case), Eq.(4.12) (dispersion alone case) may serve better in

that case because of its simplicity.


80

(a)

(a)

(b)

Figure 4-2 Fourier series coefficients evolution with dispersion (normal) alone. (a) at

three different distances (b) |C1| and |C2| as a function of transmission distance.

00.05

0.10.15

0.20.25

-4

-2

0

2

40

0.1

0.2

0.3

0.4

z/LD

Harmonics

Mag

nitu

de

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

z/LD

Mag

nitu

de

Evolution of Fourier Coefficients (Dispersion alone)

|C1||C2|


81

(a)

(b)

Figure 4-3 Fourier series Coefficients evolution with nonlinearity (N=2). (a) at three

different distances (b) |C1|,|C2|, and |C3| as a function of distance.

00.05

0.10.15

0.20.25

-4

-2

0

2

40

0.1

0.2

0.3

0.4

z/LD

Harmonics

Mag

nitu

de

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

z/LD

Ma

gnitu

de

Evolution of Fourier Coefficients (N=2)

|C1||C2||C3|


82

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

z/LD

Mag

nitu

de

(a) Normal Dispersion (β2 > 0)

N=4N=2

N=1

Dispersion alone

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

z/LD

Mag

nitu

de

(b) Anomalous Dispersion (β2 < 0)

N=2N=1

N=4

N=3

Figure 4-4 Evolution of the fundamental Fourier series coefficient magnitude (|C1|) as a

function of transmission distance. (a) Normal dispersion region (b2 > 0) (b) Anomalous

dispersion region (b2 < 0)


83

Figure 4-5 Eye-opening penalties in the normal dispersion region.

(Eye patterns, corresponding to conditions (a), (b), (c), (d), are contained in Figure 4-6.)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.5

1

1.5

2

2.5

3

z/LD

EO

P(d

B)

Eye Opening Penalty in Normal Dispersion Region

(a) (b)

(c)

(d)

N=0N=3N=6


84

Figure 4-6 Eye patterns in the normal dispersion region; (a) Back-to-back, (b) Dispersion

alone at z/LD = 0.0556, (c) N =3 at z/LD = 0.0556, and (d) N=6 at z/LD = 0.0556

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1a) Back-to-Back

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

b) Dispersion alone (z/LD

=0.0556)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

τ = T/Tb

c) N=3 (z/LD

=0.0556)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

τ = T/Tb

d) N=6 (z/LD

=0.0556)


85

Figure 4-7 1dB power penalty distances as a function of N2; (a) in the normal dispersion

region (b2 > 0), (b) in the anomalous dispersion region (b2 < 0)

10-2

10-1

100

101

102

103

10-2

10-1

100

N2=LD/LNL

z c/LD

(a) β2 > 0Sinusoidal AnalysisPRBS

10-2

10-1

100

101

10-1

100

N2=LD/LNL

z c/LD

(b) β2 < 0

Sinusoidal AnalysisPRBS


86

Figure 4-8 (a) Comparison of the critical distance at NSD = 10-3 using up to the first

order perturbation solution and the simulated 1dB penalty distance of sinusoidal response

in the normal dispersion region. (b) Comparison of the fundamental Fourier series

coefficient, |C1| when N2 = 3

10-2

10-1

100

101

102

103

10-4

10-3

10-2

10-1

100

N2=LD/LNL

z c/L D

(a)

N2 ≈ 3

Sinusoidal Analysis

NSD=10-3

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

z/LD

|C1|

(b)

N2 = 3

≈0.07 ≈0.11

SimulationEq.(4-21)


87

4-3. Cross-Phase Modulation Analysis using Sinusoidally Modulated Signal

So far, the nonlinear effects in a single channel system have been discussed.

However, in a WDM system where multiple-channel signals are transmitted

simultaneously, system performance is further limited by inter-channel nonlinear effects

such as cross-phase modulation (CPM) and four-wave mixing (FWM). It is known that

FWM effect can be suppressed effectively by a proper design of the dispersion map [38-

46]. However, the effects of dispersion on CPM have not been fully understood even

though extensive studies have been reported recently [47-57]. A larger local dispersion

will always mitigate FWM effects, but will not necessarily reduce CPM because of the

improved conversion efficiency of phase modulation to intensity modulation in the

presence of dispersion. Therefore the effect of dispersion on CPM will likely be more

complicated than on FWM, and the understanding of CPM on the performance of modern

WDM systems is important to optimize system performance.

CPM is often studied by a simple two-channel system where one channel signal

(called pump signal) is a strong intensity-modulated signal and the other (called probe

signal) is a weak continuous wave (unmodulated) signal [51-56]. Figure 4-9 shows the

pump-probe scheme. The power level of the probe signal is usually set to be very low

such that the intensity fluctuation due to the SPM in the presence of dispersion is

negligible. Therefore, the intensity fluctuation of the probe signal at the receiver will

reveal the amount of CPM distortion due to its neighboring pump signal. In this section,

the intensity-fluctuation of the probe signal at the receiver will be analytically derived

when the pump signal is sinusoidally modulated. The derived expression will be

compared with simulation results. In addition, the validity of the probe signal

measurements to estimate system performance degradation in a real WDM system will be

examined. For that purpose, the eye opening degradations in 3-channel systems will be

compared with the intensity fluctuations of the probe signal to see the correlations

between them.


88

Figure 4-9 Pump-probe set-up for CPM effect study

l1

LD

External Modulator

PRBS Generator

l2

LD

1:1

LD = Laser Diode

l1 Fiber

Optical Filter

Electrical Filter


89

4-3-1. Pump-Probe Analysis with Sinusoidally Modulated Pump signal

It has been reported that the CPM induced performance degradation is not a

strong function of channel number in WDM systems [50]. Consequently, CPM is often

studied using the simple two-channel scheme, namely the pump-probe set-up. Recently,

M. Shtaif and M. Eiselt derived an analytical method for determining the intensity

fluctuation of the received probe signal using the so-called first walk-off approximation

[52,53]. In the first walk-off approximation, it is assumed the dispersion effect is

negligible during the first walk-off distance, Lw, but the different group velocities of the

pump and probe signals are considered. After Lw, fiber nonlinearities are then neglected.

In the pump-probe scheme, the amount of the intensity fluctuation (peak-to-peak

value or RMS value) is of more interest than the exact shape of the intensity fluctuation.

Therefore, if the pump signal is modulated sinusoidally in the form of Eq.(4.9), the first

walk-off approximation may lead to a more convenient expression than the result of

references [52] and [53]. This is the main objective of this section; that is, derivation of

an analytic expression for the probe signal when the pump signal is a raised-cosine

function which models an alternating bit sequence (�010101�).

Field propagation of a two-channel system can be expressed as the coupled NLSE

[11],

( ) 1

2

2

2

11121

2

211 2

22AAAiA

t

Ai

z

A+=++ γα

∂∂β

∂∂

(4.22a)

( ) 2

2

1

2

22222

2

2222 2

22AAAiA

t

Ai

t

Ad

z

A+=++

∂∂

+ γα∂

∂β∂

∂ (4.22b)

where b2j (j =1 or 2) is the j-th channel dispersion parameter (second derivative of the

phase constant with respect to frequency), a is the fiber loss, and gj is the nonlinearity

coefficient. The walk-off parameter 11

12

−− −= gg vvd where vgj is the group velocity of

channel j, and the time scale is normalized by vg1 such that 1gv

ztt −′= ( t ′ is physical


90

time). The walk-off parameter can also be expressed in terms of the dispersion coefficient

D since λλλλλλ

λ

∆=−≈= ∫ DDdDd 12

2

1

)( where lj (j=1 or 2) is the center wavelength of

channel j. Aj(j=1,2) is the field envelope of channel j. In the pump-probe set-up, the input

probe signal placed at the wavelength l1 is modeled as a constant field with a weak

power level, P1. The pump signal at l2 has a much larger power than the probe signal,

and is assumed to be sinusoidally modulated such that

+= tPtA pωcos

2

1

2

1),0( 22 (4.23)

where P2 (ø P1) is the peak power of the pump signal and wp = p/Tb (Tb=bit period). In

the context of the first walk-off approximation, the fiber dispersion term in Eq.(4.22a)

can be neglected for 0 < z < Lw, and then the probe signal will experience phase

distortion alone as expressed below [52,53].

( ) ( )),(expexp2

exp),( 1111 tzjjz

PtzA ϕφα

−= , z < Lw (4.24)

The constant phase term, α

γα

γφαα zz e

PtAe −− −=−= 1

),0(1

11

2

111 , arises from self-phase

modulation, and the nonlinear phase shift due to cross-phase modulation, j1(z,t), is

expressed as [52,53]

( ) ( ) ςςλ

αςλ

γϕλ

dtD

AD

tzt

zDt∫∆−

−

∆−

∆= exp),0(

2,

2

21

1 (4.25)

The sinusoidally modulated pump signal, Eq.(4.23), allows the integral to be evaluated

analytically. The result is


91

( ) ( )( )

( )

+

∆∆++

+

∆∆++∆

∆=≈

ttDD

ttDD

DP

Dttz

ppp

p

ppp

p

ωωωλ

αλαω

ωωωλ

αλαωα

λλ

γϕϕ

2sin22cos4

1

8

1

sincos1

2

1

8

32,

22

2221

11

�(4.26)

In the derivation of Eq.(4.26), e-az ÷ 1 is assumed. The first term (constant phase shift

with respect to the time variable) will not induce intensity fluctuation at all. However, the

second and the third term can cause intensity fluctuation in the remaining section of fiber

in the presence of dispersion. Eq.(4.26) can be further simplified if we ignore the third

term compared to the second term. The magnitude of the third term is at most 1/4 of the

second term’s magnitude, and could be around 1/16 of the second term’s when wp ø

a/|DDl|. This condition is satisfied when dispersion and/or the channel spacing is large.

For example, wp of 5 Gb/s NRZ systems is p/Tb = p/200ps = 0.016[1/ps]. With a =

0.2dB/km, Dl = 1.5nm, and D = 17 [ps/(nm¼km)], a/|DDl| results in 0.0018 [1/ps] which

is around 1/10 of wp. This will make the magnitude of the third term around 1/16 of the

second term’s. If Dl = 0.3nm with other parameters fixed, a/|DDl| results in a larger

number, 0.009 [1/ps], but the magnitude of the third term (1/13 of the second term’s) is

still small compared to the second term’s. Therefore the analytical expression of the

probe signal within the first walk-off distance is approximately given by

( ) ( ))(expexp2

exp),( 1111 tjjz

PtzA tc ϕφα

−≈ (4.27)

where f1c is the constant phase shift term and j1t(t) is the second term of Eq.(4.26); that

is,

( ) ( )

( )( )

( )ψωβ

ψωλωα

γ

ωλωα

λωω

λωααγϕ

+=

+∆+

=

∆+

∆+

∆+=

t

tD

P

tD

Dt

DPt

pw

p

p

p

p

pp

p

t

sin

sin

sincos)(

22

21

2222211

(4.28)


92

where pD λω

αψ∆

= −1tan and ( )22

21

p

w

D

P

λωα

γβ∆+

=

In the first walk-off approximation, when z > Lw, it is assumed that the dispersion effect

is dominant over the nonlinear effects. For z > Lw, the wave propagation is then modeled

by ignoring the nonlinear terms in Eq.(4.22a). Then, the propagation equation after z >

Lw is given by

022 12

12

211 =++ A

t

Ai

z

A α∂

∂β∂∂

, z > Lw (4.29)

Eq.(4.27) with z = Lw is the initial probe signal to Eq.(4.29). In NRZ systems, Lw is often

defined by the pulse rise time, Tr, such that λ∆

=D

TL r

w because only the time-varying

part of the pump signal will cause the intensity fluctuation of the probe signal. Therefore,

the first walk-off distance can be interpreted as the fiber transmission distance required

for the rising part of the pump signal to move completely away from its original position

relative to the probe signal. The rise time (10% to 90% of its peak intensity) of the

sinusoidally modulated pump signal is 0.474�Tb, and this will be used for calculating the

first walk-off distance, Lw. For example, if Tb=100ps (10Gb/s NRZ system), a =

0.2dB/km, D = 17 [ps/(nm¼km)], and Dl = 1nm, then the walk-off distance Lw = 2.8km.

In DSF systems (D = -2 [ps/(nm¼km)]), the same parameter values give Lw = 23.7km.

Since Eq.(4.27) is periodic and the probe signal propagation can be considered

linear after Lw, the probe signal will also remain as a periodic signal after Lw. Therefore

the probe signal after z = Lw can be expressed as a Fourier series.

∑=n

tjnn

pezCtzAω)(),(1 (n = integer) (4.30)

The Fourier series coefficient )(zCn can be obtained in a similar way as in Section 4-2-1,

and the result is


93

( ) ( ) ( )wnwo

Lz

n LCLznj

ezCw

−=

−−22

212

1

2exp)( ωβ

α, z > Lw (4.31)

where ( )wn LC is the Fourier series coefficient at z = Lw. From the definition of Fourier

series,

( ) ( )

( ) ( )dtee

Tj

zP

dteeT

jz

PLC

T

T

tjntjc

T

T

tjntjcwn

ppw

pt

∫

∫

−

−+

−

−

−=

−=

2

2

sin11

2

2

)(11

1exp

2exp

1exp

2exp 1

ωψωβ

ωϕ

φα

φα

(4.32)

In the above expression, T is the period of j1t(t).

By letting ψωθ += tp ,

( ) ( ) θπ

ψπ

ψπ

θθβψωψωβdeedtee

Tnjjn

T

T

tjntjwppw ∫∫

+

+−

−

−

−+ = sin2

2

sin

2

11 (4.33)

If y is very small compared to p, ( )wn LC can be expressed in terms of the Bessel function

of the first kind of the nth order because ( ) )(2

1 sinω

π

π

θθβ βθπ n

nj Jde w =∫−

− . In that case, the

intensity fluctuation of the probe signal may be approximated by the Bessel function

expansion as below.

( ) ( )2

22211

2

1 2exp),( tjn

wpn

wnz peLzn

jJePtzA ωα ωββ

−≈ ∑− (4.34)

At a given bit rate, a larger dispersion and/or a larger channel spacing can make Eq.(4.34)

a better approximation because pD λω

αψ∆

= −1tan is smaller when |DDl| is larger.

Eq.(4.34) may be readily evaluated using conventional engineering software (MATLAB

in this dissertation).


94

Figure 4-10 compares Eq.(4.34) with the simulated results using the split-step

Fourier method in two different systems, one for conventional single-mode fiber (D = 17

[ps/(nm¼km)]) and the other for dispersion shifted fiber (D = -2[ps/(nm¼km)]. The channel

spacing (Df) is assumed to be 100GHz (Dl = 0.8nm in 1.55mm window), and Rb=10Gb/s,

a=0.2dB/km, g=2�10-3[1/(km¼mW)], P1 = 0.2mW, and P2 = 20mW are used. Fiber loss is

assumed exactly compensated by a preamplifier at the receiver. While the derived

expression agrees very closely with the simulated probe channel’s intensity for the

conventional fiber system, it does not agree very well for the dispersion-shifted fiber

(DSF) system. This is mainly because the Bessel function approximation of Eq.(4.33)

causes a larger error in the DSF system than in the conventional fiber system. With the

given parameter values, the conventional fiber system will result in y = 0.101 (÷ p)

while the DSF system gives y = 0.548.

In the pump-probe scheme, the minimum value of the interfered intensity of the

probe signal may be of more interest than the exact shape of the interfered intensity

because the minimum value may directly indicate the amount of eye-closing of the

received signal in WDM systems. Figure 4-11 compares the normalized intensity

interferences, M(%), from the derived analytical expression with the simulated ones as a

function of the channel spacing Df. M(%) is defined as

100))(Probe(mean

))min(Probe())(Probe(mean(%) ×−=

t

ttM (4.35)

where Probe(t) = intensity of the probe signal, and the physical parameter values are the

same as used in Figure 4-10. The derived analytical expression, Eq.(4.34), predicts M(%)

of the conventional fiber system very closely over a wide range of channel spacings. On

the other hand, as expected in the DSF system, there are significant discrepancies

between the analytical results and the simulated results as seen in Figure 4-10. However,

even in this case the analytical results show the same qualitative tendency as the

simulated ones. In both systems, M(%) is inversely proportional to the channel spacing,

Df, except when Df < 75GHz in the DSF system. These results agree well with reported

experimental results [51,53,55].


95

When Df < 75GHz in the DSF system, M(%) has a tendency to decrease with

decreasing Df. Physically, this can be explained by the decrease of conversion efficiency

of phase distortions to intensity fluctuations. When the first walk-off distance is

increased, the phase distortions due to nonlinear interactions between pump and probe

channels become significant, but the remaining length of dispersion fiber which is

responsible for conversion of the phase distortions to intensity fluctuations decreases at a

given transmission distance, z (Eq.(4.34)). Furthermore, the argument of the Bessel

function in Eq.(4.34), bw, becomes independent of the channel spacing in the limit of Leff

÷ Lw (Leff = effective fiber length = zdez z ′∫ ′−

0

α 1/a) because

( )

( ) N

eff

weffN

eff

p

w

L

L

LLL

L

D

P

≈+

=

∆+=

2

22

2

5.11

1

λωα

γβ

(4.36)

which is independent of Dl. In the other limit of Leff ø Lw, λ

β∆

∝ 1 ~

N

ww L

L. These

explain the qualitative agreements between the simulated and analytical results in Figure

4-11 (b).

When the channel spacing becomes smaller, the other nonlinear effect, FWM, can

be significant in real WDM systems where each channel signal with equal power level is

modulated by a random bit sequence. Therefore, the intensity interference of the probe

signal may not measure the degradation of system performance because the pump-probe

scheme is specifically designed to see the CPM effect alone. The correlation between the

intensity interference and the system performance degradation will be studied in the next

section.

4-3-2. Eye-Opening Penalties of 3-Channel WDM systems

In the previous section, the intensity fluctuation of the probe signal has been

derived. The derived analytical expression shows very good agreement with the

simulated results in the conventional fiber system, and predicts the qualitative tendency


96

in the DSF system even though quantitative discrepancies are not negligible. However,

another important question still remains to be answered. What is the correlation between

the intensity fluctuations of the weak probe signal (originally a continuous wave) and the

performance degradation in a real WDM system? To examine the correlation of these

quantities, 3-channel WDM systems are considered. Each channel is modulated at

Rb=10Gb/s with a 32bit-long random sequence. The peak power of each channel is

assumed to be 20mW and the pulse shape is assumed Gaussian. Fiber loss and the

nonlinearity constant are a=0.2dB/km, g=2�10-3[1/(km¼mW)], respectively. The system

performance degradation is measured using the eye-opening penalty (EOP) defined in

Eq.(4.2). To see the effect of FWM on EOP, two cases are simulated, one for equally-

spaced channels and the other for unequally-spaced channels. The use of unequal

spacing is a technique that has received considerable recent investigation to reduce FWM

effects on WDM systems [20,26,59,60]. In this work, the unequally spaced system has a

10% offset, that is, the left channel is spaced 10% closer to the center channel while the

right channel is spaced 10% further away from the center channel compared to the

equally spaced system. For example, the left, and the right channel spacing are 90GHz

and 110GHZ, respectively in the unequally spaced system, corresponding to a 100GHz

equally spaced system.

Figure 4-12 shows the calculated EOP of the conventional fiber system (D = 17

[ps/(nm¼km)]) as a function of Df after z = 100km. The differences between the EOPs of

the equally spaced case and the unequally spaced case are negligible, which suggests that

CPM is the dominant multi-channel nonlinearity (negligible FWM) in the conventional

fiber system. Therefore, the pump-probe measurements can be very useful to estimate

system performance degradation in the conventional fiber system. (Direct correlation

between EOP penalties and the probe signal fluctuation will be compared at the end of

this section.) In Figure 4-12, the EOP of the single-channel case is also plotted as a

reference. The large dispersion coefficient results in around 2.1dB of EOP in the single-

channel case.

Figure 4-13 shows the calculated EOP of the DSF system (D = -2 [ps/(nm¼km)]).

Unlike the conventional fiber system, the unequal spacing results in a significant

improvement of EOP especially when Df is small. For example, the unequal spacing


97

improves the EOP by more than 1.5dB compared to the equally spaced case when Df =

50GHz. These results indicate that FWM is also a significant effect in degrading

performance in the DSF system because the difference between the equal and the unequal

spaced channels is a result of whether the FWM effect is suppressed or not. Therefore,

the pump-probe measurements may not serve to directly indicate the performance

degradation in a real WDM system when DSF and equal channel spacings are used.

Figure 4-14 shows the eye-patterns of the center channel of the DSF system when Df =

75GHz. Figure 4-14(b) shows the eye-pattern of the single channel case. The EOP of

Figure 4-14(b) (~0.5dB) can be considered as a power penalty due to the combined effect

of dispersion and SPM. The EOP of Figure 4-14(c) (unequally spaced) can be considered

as the added penalty due to CPM in addition to the penalty of dispersion and SPM, while

the EOP of Figure 4-14 (d) (equally spaced) is the result of the further addition of FWM

penalty.

Finally, Figure 4-15 compares the CPM penalties resulting from the sinusoidal

pump-probe measurements and the EOP simulations in the 3-channel WDM systems. In

the 3-channel systems, the CPM penalty is defined as the difference of the EOP of the

unequally spaced case and the EOP of the single channel case. For example, the CPM

penalty of the DSF system is the difference of curve (c) and curve (b) in Figure 4-13. In

the pump-probe measurements, the CPM penalty is defined as below.

( )( ))(~

mean

)(~minlog10- ]Penalty[dB CPM

ti

ti

p

p= (4.37)

where )(~

tip is the photo-detected probe-signal after the electrical filter at the receiver. A

third-order Butterworth filter with bandwidth = 0.8�Rb is used for the electrical filter. In

the conventional fiber system (Figure 4-15(a)), the CPM penalty of the 3-channel WDM

system agrees well with the CPM penalty from the sinusoidal pump-probe measurements

over a wide range of Df. Therefore, the derived analytical expression can be very useful

to estimate the performance degradation in a conventional fiber WDM system.

Unlike the conventional fiber system, Figure 4-15(b) shows that the CPM

penalties of the pump-probe measurements, whether from simulations or from the derived

analytical expression, do not agree well with the 3-channel system’s penalty especially


98

when Df < 100GHz. This is mainly because FWM is also a significant contribution to the

performance degradation in the DSF system. However, when Df > 100GHz, the derived

analytical result gives a good estimation of CPM penalty even in the DSF system.

4-4. Summary

In this chapter, the transmission impairments due to fiber nonlinearities have been

analyzed using a sinusoidally modulated signal which models an alternating bit sequence

of ones and zeros in on-off keying. In the analysis of SPM in single channel transmission,

the sinusoidal response of nonlinear fiber shows a strong correlation with EOP in the

normal dispersion region over a wide range of values of the normalized nonlinearity

parameter N (0.1 < N2 <100). This result strongly indicates that the measurement of the

sinusoidal response can be an alternate way of measuring EOP without having a long

sequence of randomly modulated input bits. However, in the anomalous dispersion region

where soliton formation is possible, the sinusoidal response has a much more limited

range of application to estimate system performance.

The sinusoidal response has also been derived analytically based on the

perturbation analysis developed in chapter 2. Since the perturbation analysis has a limited

range of validity, the derived analytical expression also has a limited range of

applicability. Comparison with numerical results reveals that the derived expression may

result in a significant error when N2 > 3.

The sinusoidal analysis has also been applied in a multi-channel system to

estimate CPM-induced performance degradation. The intensity fluctuation of the probe

signal has been derived in the context of the first-walk-off approximation. The derived

expression shows good agreement with numerical results in conventional single-mode

fiber systems. The derived expression also shows qualitative agreement with numerical

results in DSF systems even though it results in larger errors than in the conventional

fiber case especially when the channel spacing is small. When Df > 100GHz, however,


99

the derived expression also gives a good estimate of the CPM induced power penalty

even in the DSF system.

In addition, the correlation of the probe signal’s intensity fluctuation and the EOP

in a real WDM system has been examined. Numerical studies show that FWM induced

power penalty becomes comparable to the CPM induced power penalty in the DSF

system when the channel spacing is less than 100GHz. Therefore, the derived analytical

solution could find most application in a WDM system where the performance

degradation is mostly from CPM rather than FWM.


100

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 10-9

0.16

0.18

0.2

0.22

0.24

Pro

be In

tens

ity [m

W] Simulation

Theory

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 10-9

0.16

0.18

0.2

0.22

0.24

Pro

be In

tens

ity [m

W]

time [sec]

SimulationTheory

(a) D = +17 ps/(nm km), Df = 100GHz

(b) D = -2 ps/(nm km), Df = 100GHz

Figure 4-10 The probe signal’s intensity fluctuations after z = 100km. Df =100GHz,

Rb=10Gb/s, a=0.2dB/km, g=2�10-3[1/(km¼mW)], P1 = 0.2mW, and P2 = 20mW (a) D =

+17 [ps/(nm¼km)] (b) D = -2 [ps/(nm¼km)]


101

Figure 4-11 The normalized intensity interferences, M(%), after z = 100km. Rb=10Gb/s,

a=0.2dB/km, g=2�10-3[1/(km¼mW)], P1 = 0.2mW, and P2 = 20mW (a) D = +17

[ps/(nm¼km)] (b) D = -2 [ps/(nm¼km)]

50 100 150 2005

10

15

20

25

30

35

Pro

be M

odul

atio

n [%

]

(a)Conventional Fiber

SimulationTheory

50 100 150 2005

10

15

20(b)Dispersion Shifted Fiber

Channel Spacing [GHz]

Pro

be M

odul

atio

n [%

]

SimulationTheory


102

Figure 4-12 Eye-opening penalties as a function of Df after z =100km.

(Conventional fiber system)

40 50 60 70 80 90 100 110 120 130 140 1501.5

2

2.5

3

3.5

4

4.5


E.O

.P. [

dB]

Conventional Fiber, D = +17 ps/(nm km)

Equally Spaced Unequally SpacedSingle Channel


103

Figure 4-13 Eye-opening penalties as a function of Df after z =100km. (DSF system)

(Eye patterns, corresponding to conditions (b), (c), (d), are contained in Figure 4-14.)

40 50 60 70 80 90 100 110 120 130 140 150-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


E.O

.P. [

dB]

Dispersion Shifted Fiber, D = -2 ps/(nm km)

Equally Spaced Unequally SpacedSingle Channel

(d)

(c)

(b)


104

Figure 4-14 Eye-patterns of DSF system after z =100km (a) Back-to-back case, (b)

Single channel case, (c) Center channel of Df =75GHz case (unequally spaced), and (d)

Center channel of Df =75GHz case (equally spaced)

0 50 100 150 2000

5

10

15

20(a) Back-to-Back

0 50 100 150 2000

5

10

15

20(b) Single Channel

0 50 100 150 2000

5

10

15

20

[ps]

(c) Unequally Spaced

0 50 100 150 2000

5

10

15

20(d) Equally Spaced

[ps]


105

Figure 4-15 CPM penalty (a) conventional fiber system, and (b) DSF system

50 60 70 80 90 100 110 120 130 140 1500

0.5

1

1.5

2

CP

M P

enal

ty [d

B]

(a) Conventional Fiber

PRBS Sinusoidal(sim.) Sinusoidal(theory)

50 60 70 80 90 100 110 120 130 140 1500

0.5

1

1.5

2


CP

M P

enal

ty [d

B]

(b) Dispersion Shifted Fiber

PRBS Sinusoidal(sim.) Sinusoidal(theory)

Chapter 5: Noise loading analysis to characterize fiber nonlinearities

106

Chapter 5

Noise Loading Analysis to Characterize Fiber Nonlinearities

5-1. Introduction

One of the major concerns in multi-channel communication systems is the system

impairment due to cross-talk between channels. Cross-talk is often induced by system

nonlinearities because nonlinearities can generate new spectral components which may

fall into the neighboring channels. Therefore, it has been of interest to develop analysis

and characterization methods to evaluate system impairments due to cross-talk. Among

the various techniques developed are the two-tone test, three-tone test, and noise loading

analysis using a Gaussian noise source [62]. When there are a significant number of

channels, noise loading is preferred because Gaussian noise having a broad spectrum is a

good approximation of multi-channel signal loading in a broadband system, whereas

sinusoidal loading is not representative of multi-channel loading. In the noise loading

analysis, a sharp notch filter is used to remove a part of the noise spectrum before input to

the system. At the output of the system, a bandpass filter (BPF) tuned to the notch filter

will indicate the spectral shape due to nonlinearities within the filter bandwidth.

Therefore, the output of the BPF is the quantity of interest in the noise loading analysis.


107

Conventionally, noise loading is used in the distortion analysis of RF FDM

(radio-frequency frequency division multiplexing) systems where the high power

amplifier at the transmitter is the major source of nonlinearities. Some papers have

discussed the noise loading analysis when the nonlinearities are modeled as memoryless

[63-65]. Maqusi applied the Volterra series representation of nonlinearities in the noise

loading analysis to model memory in nonlinear systems [66].

However, the noise loading analysis, to our knowledge, has not been applied to

assess fiber nonlinearities in optical communication systems. (Unfortunately, ‘noise

loading’ has been used for a different meaning in optical communication systems. In the

references [67,68], noise loading means adding noise at the receiver in a controlled way

to estimate BER margin.) The reason that the noise loading analysis has not been applied

in fiber optic communication systems might be because the major nonlinearities in fiber

optic communication systems come from the fiber, which is distributed throughout the

transmission path unlike in RF FDM systems where the major nonlinearity is lumped at

the transmitter. In a DWDM system where channels are spaced very closely, the

broadened spectrum due to various nonlinear effects like SPM, CPM, and FWM is in

practice indistinguishable. In such a system, the noise loading analysis could be useful in

assessing the effects of broadened spectrum due to fiber nonlinearities on system

performance. In addition, in a spectrum-sliced system where the transmission signal is

modulated noise, the noise loading analysis could be more appropriate to assess the

effects of nonlinearities rather than analyses using deterministic signals.

Figure 5-1 shows the noise loading test setup for fiber optic communication

systems. The test consists of a broadband noise source followed by a notch filter. The

bandwidth (Bo) rejected by the notch filter is designed to be much narrower than the

bandwidth of the flat source. At the output of the fiber, an optical BPF, the bandwidth

and center frequency of which is tuned to the notch filter, is inserted. Ideally, the output

of the BPF corresponds to intermodulation noise due to fiber nonlinearities that falls into

the bandwidth defined by the notch filter. In practice, following detection thermal noise

from electronic circuits of the detector may also fall into the same bandwidth, but this

noise could be easily calibrated out. In this dissertation, it is assumed that the power


108

spectrum that falls into the bandwidth of the BPF consists of only the intermodulation

noise due to fiber nonlinearities.

In the noise loading analysis, noise power ratio (NPR) is often used as a figure of

merit rather than absolute power. NPR is defined by

=

β

α

P

P10log10NPR (5.1)

where Pa = average output power of the BPF without the notch filter at the input to the

fiber, and Pb = average output power of the BPF with the notch filter at the input to the

fiber. NPR should be a good estimate of signal to noise ratio degradation due to fiber

nonlinearities within the bandwidth of interest as long as Pa ø Pb. This is because Pb is

the noise power due to fiber nonlinearities, and Pa is approximately the sum of Pb and the

signal power within the bandwidth of the notch filter. That is, the signal to noise ratio

within the bandwidth may be approximated as ( )( )ββα PPPSNR −≈ log10

( )1log10 −= βα PP NPR≈ .

In this chapter, the noise loading analysis has been simulated using the split-step

Fourier method to see the possible applications of the noise loading analysis in fiber optic

communication systems. In addition, the Volterra series representation of the fiber is used

to calculate Pb analytically, and the analytic results are compared with the split-step

Fourier numerical results.

Figure 5-1 Noise loading test set-up

PowerMeter

Notch Filter(Bo) BPF(Bo)

Broad Band Noise Source


109

5-2. Noise Loading Analysis using Split-Step Fourier Method

In this section, the noise loading analysis is numerically studied using the split-

step Fourier method. The simulation block diagram closely follows Figure 5-1. The

broadband noise source is modeled by a complex Gaussian noise source with average

power of 10mW and 180GHz bandwidth. The bandwidth of the notch filter and the BPF

at the output is 30GHz. All the filters are assumed to be ideal (rectangular) shape, and the

transmission distance is 200km. The major simulation variable is b2, the second order

group velocity dispersion (GVD) parameter. Numerical values of the major parameters

used in the simulations are summarized in Table 5-1.

Table 5-1 Simulation Parameters for Noise Loading Analysis

Parameters Symbols Values Note

Sampling frequency Fs 1.28 [THz]

Data size ND 213=8192

Step distance Dz 0.2 [km] Simulation Step

Noise source

bandwidth

f2 – f1 180 [GHz] Dl 1.5nm

Notch filter bandwidth Bo 30 [GHz] Dl 0.24nm

Average signal power

to the fiber

Pavg 10 [mW] Fixed regardless of the

notch filter

Fiber length z 200 [km]

Fiber loss a 0 Lossless case

Fiber nonlinear coeff. g 2�10-3 [mW-1km-1]

The second order

GVD parameter

b2

�0.1 [ps2/km]

�3 [ps2/km]

�10 [ps2/km]

Major simulation variables

(Typical DSF b2: -3 to +3 [ps2/km])

The third order GVD

parameter

b3 0.063 [ps3/km]


110

5-2-1. Evolution of Spectral Density

In the noise loading analysis, it may not be important to know the exact spectral

shape at the output of the fiber because NPR is defined by powers at the output of the

BPF filter the bandwidth of which is much narrower than the noise source bandwidth.

However, it is of interest to see how the spectral density shape evolves depending on

fiber parameters. Since we can observe a random signal only over a finite time interval in

practice, it is required to estimate the power spectral density of a random process using

the observed finite length of data. One of the most popular methods to estimate the power

spectral density is the Bartlett method – averaging periodograms [69]. The Bartlett

method allows to trade-off between resolution and variance of the estimator.

Figure 5-2 shows the estimated spectral density of the Gaussian noise source with

the total data size, 8192, before and after the notch filter. The Bartlett method with data

segment size, 512, is used to reduce the variance of the estimator. Figure 5-3 (a) and (b)

show how the spectral shape changed after transmission of 50km and 200km respectively

when b2 = 0.1 [ps2/km]. Figure 5-3 (c) and (d) are the corresponding results when b2 =

-0.1 [ps2/km]. With relatively small values of dispersion parameters, the notched out

bandwidth is quickly filled in, and the overall bandwidth is increased significantly even at

the relatively short transmission distance of 50km. It is interesting to observe the

bandwidth expansion is more significant in the anomalous dispersion region (β2 < 0).

Note that the frequency range in Figure 5-3 is increased to 1000GHz from 400GHz in

Figure 5-2. In Figure 5-4 the corresponding results are shown for the case where the

dispersion is much larger. Now, with |b2| = 10 [ps2/km], the spectral density shapes

shown in Figure 5-4 have not been changed drastically regardless of the dispersion

region. Unlike the |b2| = 0.1 [ps2/km] case, the notched bandwidth is clearly observable.

This is not a surprising result because the relative strength of fiber nonlinearities is

decreased at a given signal power when the fiber dispersion is increased (larger

magnitude of b2).

5-2-2. Evaluation of Pa, Pb, and NPR


111

As defined in Eq.(5.1), to determine NPR it is required to evaluate Pa and Pb first.

The integration of the estimated spectral density over the bandwidth of the BPF will give

the estimate of Pa or Pb depending on whether the notch filter at the input is inserted or

not. Figure 5-5 (a) and (b) show the growth with distance of the spectral densities which

fall within the bandwidth of the output BPF with and without the notch filter at the input

of the fiber, respectively. b2 = 3 [ps2/km] is used in Figure 5-5, and the spectral densities

are estimated by the Bartlett method again, but without dividing data within the

bandwidth into segments due to the relatively small number of the data size within the

bandwidth (= 192). However, the segment size will not affect the evaluation of Pa or Pb

because the Bartlett method is an unbiased estimator.

Figure 5-6 shows the resulting NPR with various b2 values as a function of

propagation distance. The simulation results clearly show that NPR is a strong function of

the magnitude of dispersion parameter, b2. When |b2| = 0.1 ps2/km, NPR approaches 0 dB,

which means that Pa and Pb become comparable to each other as transmission distance

increases. However, when dispersion is significant such as |b2| = 10 ps2/km, NPR remains

about 15dB regardless of the sign of b2. In the other two cases, |b2| = 0.1 ps2/km and |b2| =

3ps2/km, transmission in the normal dispersion region gives about 2 dB advantage of

NPR over transmission in the anomalous dispersion region. These results suggest that we

may obtain better performance in the normal dispersion region due to a reduced cross-

talk. It is also interesting to observe that NPRs asymptote after around z = 50km which

corresponds to the nonlinear length, LN = 1/(gPavg) = 50 km, regardless of the dispersion

parameter values.

Figures 5-7 to 5-10 show the numerical results for Pa, Pb and their ratio for b2 =

0.1 ps2/km, b2 = -0.1 ps2/km, b2 = 10 ps2/km, and b2 = -10 ps2/km, respectively. In each

figure, Pa(z) is plotted in (a), Pa(z)/Pa(0) in (b), Pb(z) in (c), and Pa(z)/Pb(z) is plotted in

(d). When the magnitude of b2 is small (|b2| = 0.1 ps2/km) (Figure 5-7 and 5-8), Pa

reduces to about half of its initial value at the propagation distance around 100km, but

remains almost constant after that, which suggests that the bandwidth expansion after

100km may not be significant. This is because the increased spectrum width makes the

dispersion effect more significant, and therefore the nonlinearity less significant, in which


112

case the magnitude of signal spectrum becomes invariant. Pb starts to level off around

50km. The earlier saturation occurrence of Pb than Pa may be interpreted as a balance

between energy increase within the notched bandwidth due to the intermodulation

components from outside of the bandwidth and energy loss due to the intermodulation to

outside of the bandwidth.

When the magnitude of b2 becomes significant (|b2| = 10 ps2/km) (Figure 5-9 and

5-10), Pa behaves oppositely depending on the dispersion region. In the normal

dispersion region (Figure 5-9), Pa tends to grow as distance increases (Pa(z)/Pa(0) > 1),

but in the anomalous region (Figure 5-10), it decreases as distance increases

(Pa(z)/Pa(0) < 1). This is related to the spectral shape as we observed in Figure 5-4 (b)

(convex in the normal dispersion) and Figure 5-4 (d) (concave in the anomalous

dispersion). Pb in the normal dispersion region is larger than Pb in the anomalous

dispersion region. Therefore, NPR which is given by the ratio of Pa and Pb remains at

almost the same level in both dispersion regions. The different behaviors of Pa and Pb

depending on the dispersion region may imply different performance for the b2 = +10

ps2/km and b2 = -10 ps2/km cases even though NPRs are almost the same. It is interesting

to note, however, that Pa and Pb behave in a similar way regardless of the dispersion

region when |b2| = 0.1 ps2/km as observed in Figure 5-7 and 5-8.

In a real system, a fiber span is often designed using a proper dispersion map [22-

24] to combat nonlinearity and dispersion effects simultaneously. The noise loading

analysis of different dispersion maps will be discussed in the following section.

5-2-3. Noise Loading Analysis with Different Dispersion Maps

The principle of a dispersion map is to allow large local dispersion to reduce the

effects of nonlinearity, but to limit the average dispersion to be below a certain level by

alternately placing fibers with opposite sign of dispersion. In an installed system with

conventional single-mode fibers (CF) operating in the 1.55 µm window, the dispersion is

large and anomalous. Consequently, the compensating fibers (dispersion compensating


113

fibers) after the conventional fibers should have normal dispersion in the 1.55mm window

with a relatively large dispersion parameter because the conventional fibers have a typical

value of about b2 = -20 ps2/km. However, various dispersion maps could be considered to

optimize performance when a new system is designed with dispersion shifted fibers

(DSF, typically b2 = -3 to +3 ps2/km in the 1.55mm window). In this section, one span of

fiber links with three different dispersion maps is considered, and the performance of

each dispersion map is compared using the noise loading analysis. Table 5-2 shows the

fiber parameters of each map. Total fiber length is 150km and the total average

dispersion is designed to be zero, that is, 21

222

121

2 LL

LL

++

=βββ = 0, where Li (i=1,2) is the

fiber length, and i2β is the dispersion coefficient of i-th segment fiber, respectively. Map 1

and Map 2 are designed with DSFs having the same magnitude of the dispersion

coefficients but with opposite signs. In Map 3, a conventional fiber is used as a second

segment fiber to compensate the DSF of the first section. In each map, input is Gaussian

noise with 0.5mW average power and 200GHz bandwidth. The bandwidth of the notch

filter is 20GHz, and all the filters have the ideal (rectangular) spectral response.

Table 5-2 Parameters of Three Different Dispersion Maps (L = L1+L2=150km)

First Section 12β [ps2/km] g1[mW-1km-1] 1L [km]

Second Section 22β [ps2/km] g2[mW-1km-1] 2L [km]

2β

Map 1 +0.64 2�10-3 75 -0.64 2�10-3 75 0

Map 2 -0.64 2�10-3 75 +0.64 2�10-3 75 0

Map 3 +0.64 2�10-3 146 -23.32 3�10-3 4 0


114

Figure 5-11 shows the noise loading analysis with the three different dispersion

maps. From Figure 5-11 (c), we can observe that Map 2 (DSF(anomalous) +

DSF(normal)) gives the poorest performance among the three maps. Since Map 1

(DSF(normal) + DSF(anomalous)) gives about a 2.5dB advantage of NPR over Map 2,

we may conclude that it is advantageous to place normal dispersion fiber first. This is

because the spectral broadening in the normal dispersion region is less significant than in

the anomalous region. The performance of Map 3 (DSF(normal) + CF(anomalous)) is

similar to that of Map 1. This is because the first section of the fiber link is the normal

dispersion with the same magnitude in both of Map 1 and Map 3. Most of spectrum

broadening will occur in the first section of the link because the broadened spectrum in

the first section makes the dispersion effect comparable to or dominant over the nonlinear

effects in the second section. In the second fiber, propagation is essentially linear and

performance depends on only the total dispersion of this fiber ( 22β � 2L ). Therefore,

conventional fibers (CF) may be used instead of anomalous dispersion-shifted fibers

(DSF) when the period of the dispersion map is large enough such that most spectrum

broadening occurs in the first section of the fiber link.


115

-200 -150 -100 -50 0 50 100 150 200-60

-40

-20

0

dB

(a) Input Spectral Density w/o Notch

-200 -150 -100 -50 0 50 100 150 200-60

-40

-20

0

[GHz]

dB

(b) Input Spectral Density with Notch

Figure 5-2 Normalized spectral densities of noise source (a) without notch filter (b) with

notch filter


116

-500 0 500-60

-50

-40

-30

-20

-10

0

dB

(a) β2 = 0.1ps2/km, z = 50km

-500 0 500-60

-50

-40

-30

-20

-10

0

dB

(b) β2 = 0.1ps2/km, z = 200km

-500 0 500-60

-50

-40

-30

-20

-10

0

[GHz]

dB

(c) β2 = -0.1ps2/km, z = 50km

-500 0 500-60

-50

-40

-30

-20

-10

0

[GHz]

dB

(d) β2 = -0.1ps2/km, z = 200km

Figure 5-3 Normalized spectral densities with notch filter (a) b2 = 0.1 [ps2/km], z=50km

(b) b2 = 0.1 [ps2/km], z=200km (c) b2 = -0.1 [ps2/km], z=50km (d) b2 = -0.1 [ps2/km],

z=200km


117

-200 -100 0 100 200-60

-40

-20

0

dB

(a) β2 = 10ps2/km, z = 50km

-200 -100 0 100 200-60

-40

-20

0

dB

(b) β2 = 10ps2/km, z = 200km

-200 -100 0 100 200-60

-40

-20

0

[GHz]

dB

(c) β2 = -10ps2/km, z = 50km

-200 -100 0 100 200-60

-40

-20

0

[GHz]

dB

(d) β2 = -10ps2/km, z = 200km

Figure 5-4 Normalized spectral densities with notch filter (a) b2 = 10 [ps2/km], z=50km

(b) b2 = 10 [ps2/km], z=200km (c) b2 = -10 [ps2/km], z=50km (d) b2 = -10 [ps2/km],

z=200km


118

Figure 5-5 Spectral growth within the notch filter bandwidth when b2 = 3 [ps2/km]

(a) with the notch filter (b) without the notch filter

050

100150

200

-4

-2

0

2

4

x 1010

0

0.2

0.4

0.6

0.8

1

1.2

x 10-10

Distance [km]

(a) with Notch Filter (β2=3ps2/km)

Frequency [Hz]

[mW

/Hz]

050

100150

200

-4

-2

0

2

4

x 1010

0

2

4

x 10-10

Distance [km]

(b) without Notch Filter (β2=3ps2/km)

Frequency [Hz]

[mW

/Hz]


119

0 20 40 60 80 100 120 140 160 180 200 220

0

5

10

15

20

25

Distance [km]

NP

R [d

B]

β2=10ps2/km

β2=-10ps2/km

β2=3ps2/km

β2=-3ps2/km

β2=0.1ps2/km

β2=-0.1ps2/km

Figure 5-6 NPR simulation results as a function of transmission distance. Simulation

parameter is b2.


120

0 50 100 150 2000.8

1

1.2

1.4

1.6(a)

0 50 100 150 2000.5

0.6

0.7

0.8

0.9

1(b)

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

z [km]

(c)

0 50 100 150 2000

5

10

15

20

25

z [km]

(d)

Figure 5-7 b2 = +0.1 ps2/km (a) Pa(z), (b) Pa(z)/ Pa(0), (c) Pb(z), and (d) Pa(z)/ Pb(z)

Figure 5-8 b2 = -0.1 ps2/km (a) Pa(z), (b) Pa(z)/ Pa(0), (c) Pb(z), and (d) Pa(z)/ Pb(z)

P

a

(z)

[mW

] P

b

(z)

[m

W]

P

a

(z)/

P

a

(0)

P

a

(z)/

P

b

(z)

0 50 100 150 2000.6

0.8

1

1.2

1.4

1.6(a)

0 50 100 150 2000.4

0.6

0.8

1(b)

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

z [km]

(c)

0 50 100 150 2000

5

10

15

20

z [km]

(d)

P

b

(z)

[m

W]

P

a

(z)/

P

b

(z)

P

a

(z)/

P

a

(0)

P

a

(z)

[mW

]


121

0 50 100 150 2001.2

1.3

1.4

1.5

1.6(a)

0 50 100 150 2000.8

0.85

0.9

0.95

1(b)

0 50 100 150 2000

0.01

0.02

0.03

0.04

0.05

z [km]

(c)

0 50 100 150 20020

40

60

80

100

z [km]

(d)

Figure 5-9 b2 = +10 ps2/km (a) Pa(z), (b) Pa(z)/ Pa(0), (c) Pb(z), and (d) Pa(z)/ Pb(z)

Figure 5-10 b2 = -10 ps2/km (a) Pa(z), (b) Pa(z)/ Pa(0), (c) Pb(z), and (d) Pa(z)/ Pb(z)

0 50 100 150 2001.4

1.5

1.6

1.7

1.8

1.9(a)

0 50 100 150 2001

1.05

1.1

1.15

1.2

1.25(b)

0 50 100 150 2000

0.02

0.04

0.06

0.08

z [km]

(c)

0 50 100 150 20020

40

60

80

100

120

z [km]

(d)

P

b

(z)

[m

W]

P

a

(z)/

P

b

(z)

P

a

(z)/

P

a

(0)

P

a

(z)

[mW

] P

a

(z)

[mW

] P

b

(z)

[m

W]

P

a

(z)/

P

a

(0)

P

a

(z)/

P

b

(z)


122

Figure 5-11 Noise loading analysis with different dispersion maps (a) Pa(z)/ Pa(0),

(b) Pb(z), and (c) NPR [dB]

0 50 100 1500.98

0.985

0.99

0.995

1

1.005(a)

z [km]

Map1Map2Map3

0 50 100 1500

2

4

6

8x 10

-4 (b)

z [km]

Map1Map2Map3

0 50 100 15015

20

25

30

35

40(c)

z [km]

Map1Map2Map3

P

a

(z)/

P

a

(0)

P

b

(z)

[m

W]

NPR

[dB

]


123

5-3. Evaluation of Pb using the third-order Volterra Series Model of Single-Mode Fiber

When fiber nonlinearities are weak, we may treat the solution of the nonlinear

Schrödinger equation as a perturbation of the linear solution (Chapter 2) or equivalently

we may express the output of the fiber with Volterra series transfer functions [27,28]. If

we take only the first higher order Volterra series, the output of a single-mode fiber may

be modeled as below [27].

Figure 5-12 Modeling of single-mode fibers with Volterra series

The linear transfer function, )2

exp()2

exp(),( 221 zjzzH ωβαω −= (a = the fiber loss

coefficient, b2 = the second order propagation constant, and z = transmission distance), is

derived from the nonlinear Schrödinger equation by assuming dispersion alone, and the

third-order Volterra transfer function, H3, may be expressed as below [27].

))((

1),,,(

23212

))(()(22

3213

2321223212

ωωωωβαγωωω

ωωωωβαωωωβα

−−+−−=

−−+−+−+−

j

eejzH

zjzzj

z (5.2)

where g is the nonlinearity coefficient.

),(1 zH ω

),,,( 3213 zH ωωω

y1(t)

y3(t)

y(t)= y1(t)+ y3(t)

x(t)


124

In Eq.(5.2), the b3 term is omitted for simplicity. The effect of the b3 term is usually

negligible except when the signal wavelength is close to the zero dispersion wavelength,

lZD.

From Figure 5-12, the output auto-correlation function, )(τyyR , is obtained as

below. x(t) and y(t) denote the input process to the fiber and the output process of the

fiber, respectively.

{ } ( )( ){ }

)()()()(

)()()()(

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

33*

313111

33133111

31*3

*1

*

ττττ

ττττττττ

yyyyyyyy

yyyyyyyy

yy

RRRR

RRRR

tytytytyEtytyER

+++=

+++=

++++=+=

(5.3)

where * denotes the complex conjugate and E{�} is the ensemble average operator. The

output spectral density function, Syy(f), which is the Fourier transform of )(τyyR , is then

)()()()()( 31313311 fSfSfSfSfS yyyyyyyyyy∗+++= (5.4)

where { })()( 1111 τyyyy RfS ℑ= , { })()( 3333 τyyyy RfS ℑ= , { })()( 3131 τyyyy RfS ℑ= , and

{ })()( *3131 τyyyy RfS ℑ=∗ . {}⋅ℑ is the Fourier transform operator.

The spectral density functions for the third-order nonlinearity are derived in the

literature [70], and the results are

)(),(),(2

111 fSzfHzfS xxyy = (5.5)

∫ −= duuSzfuuHfSzfHzfS xxxxyy )(),,,()(),(3),( 3*131 (5.6)

dudvufSuvSuSzvfuvuH

duuSzfuuHfSzfS

xxxxxx

xxxxyy

)()()(),,,(6

)(),,,()(9),(

2

3

2

333

−−−−+

−=

∫∫∫

(5.7)


125

where )( fS xx is the input spectral density function.

In the evaluation of Pb (= average output power of the BPF at the fiber output with the

notch filter at the input to the fiber), the input spectral density Sxx(f) has a sharply

notched-out spectral shape from –Bo/2 to +Bo/2. Therefore, from Eq.(5.5) to (5.7), we

observe that the second term of Eq.(5.7) is the only term that can contribute to Pb.

If we denote the second term of Eq.(5.7) as ),( zfΦ ,

dudvufSuvSuSzvfuvuHzf xxxxxx )()()(),,,(6),(2

3 −−−−=Φ ∫∫ , Pb can be evaluated

by integrating ),( zfΦ from –Bo/2 to +Bo/2.

∫∫−−

Φ==2

2filternotch with

2

2

),(),(

o

o

o

o

B

B

B

Byy dfzfdfzfSPβ (5.8)

When the fiber loss term is ignored, the third order Volterra series transfer function may

be expressed as ( ) 2

222

22

2

3 )2)(2()2)(2(2sin

2),,,(

+−−

+−−

=−−

uvfvu

zuvfvuzvfuvuH

βπβπ

γ.

Therefore,

�(5.9)

We can observe that ),( zfΦ is an even function of b2, which means the Volterra series

approach does not predict the dependence of Pb on the dispersion region.

Figure 5-13 shows the numerical evaluation result of Eq.(5.9) at z = 50km with

|b2| = 3ps2/km and |b2| = 10ps2/km. Fiber loss and b3 terms are ignored, and the

nonlinearity coefficient, g = 2�10-3 mW-1km-1, is used. Signal power is assumed to be

10mW, and the spectral shape input to the fiber is flat ranging from –90GHz to +90GHz

( )dudvvfSuvSuS

uvfvu

zuvfvuzf

zfS

xxxxxx

yy

)()()()2)(2(

)2)(2(2sin

2

3),(

),(

2

222

22

filternotch with

−−

+−−

+−−

=Φ= ∫∫

βπβπ

γ


126

with 30GHz notched-out. The same parameter values will be used in the split-step

Fourier method for comparison, and the results are shown in Figure 5-14. From the

evaluated spectral densities, we can obtain Pb by integrating them over Bo (notch-

bandwidth). The numerical results at z = 50km are compared in Table 5-3. From Table 5-

3, it is observed that Volterra series approach using up to the third order transfer function

of the fiber considerably overestimates the noise power output, Pb, compared to the split-

step method. For example, when |b2| = 3ps2/km, the Volterra series approach gives Pb =

0.6878 mW regardless of the dispersion region while the split-step method gives Pb =

0.1369 mW (normal) and Pb = 0.1975 mW (anomalous). Furthermore, numerical

evaluation of ),( zfΦ (Eq.(5.9)) often requires more computational resources than the

split-step method due to its double convolution. Therefore, it may not be appropriate to

apply the Volterra approach to obtain quantitative results in noise loading analysis.

Table 5-3 Comparison of Pb at z = 50km

Dispersion Parameter Volterra Series Approach Split-Step Fourier Method

|b2| = 3ps2/km 0.6878 mW 0.1369 mW (normal)

0.1975 mW (anomalous)

|b2| = 10ps2/km 0.2330 mW 0.0317 mW (normal)

0.0350 mW (anomalous)


127

-20 -15 -10 -5 0 5 10 15 20

0.5

1

1.5

2

2.5

3x 10

-11

Frequency [GHz]

Syy

[mW

/Hz]

Figure 5-13 Spectral densities at z = 50km by the Volterra series approach

|b2| = 3ps2/km

|b2| = 10ps2/km


128

Figure 5-14 Spectral densities at z = 50km by the split-step Fourier method;

(a) b2 = +3ps2/km, (b) b2 = -3ps2/km, (c) b2 = +10ps2/km, and (d) b2 = -10ps2/km

-4 -2 0 2 4

x 1010

0

1

2

3

4x 10

-11

Syy

[mW

/Hz]

(a) β2=3ps2/km

-4 -2 0 2 4

x 1010

0

1

2

3

4x 10

-11

Syy

[mW

/Hz]

(b) β2=-3ps2/km

-4 -2 0 2 4

x 1010

0

0.5

1

1.5x 10

-11

Frequency [Hz]

Syy

[mW

/Hz]

(c) β2=+10ps2/km

-4 -2 0 2 4

x 1010

0

0.5

1

1.5x 10

-11

Frequency [Hz]

Syy

[mW

/Hz]

(d) β2=-10ps2/km


129

5-4. Summary

In this chapter, the noise loading technique is applied to fiber optic transmission

systems to characterize fiber nonlinearities. In the noise loading analysis, NPR (noise

power ratio) is the critical parameter defined as the ratio of Pa (average output power of

the BPF without the notch filter at the input to the fiber) and Pb (average output power of

the BPF with the notch filter at the input to the fiber). Simulation results using the split-

step method show that NPR is a strong function of the magnitude of the dispersion

parameter, b2. NPR is larger when the magnitude of b2 is larger, which suggests that

larger dispersion is always beneficial to reduce nonlinear cross-talk. In addition,

simulation results indicate that it is advantageous to propagate in normal dispersion

region when the magnitude of b2 is around 3 ps2/km or less, which is the typical range of

b2 in dispersion-shifted fibers. The noise loading analysis is also applied to fiber links

with different dispersion maps. Numerical study shows that there is about a 2.5dB

advantage in NPR in using normal dispersion fiber first in alternating dispersion maps

even though the total average dispersion is equal to zero in both cases. This may be

because there is less spectral broadening in the normal dispersion regime as observed in

Figure 5-3.

The Volterra series approach is also applied to the noise loading analysis.

However, compared to the split-step method, the Volterra method using up to the third

order transfer function overestimates noise power at the output even at the relatively short

transmission distance (z = 50km). Furthermore, numerical load to evaluate noise power

using Volterra method is relatively heavy. Therefore, it may not be appropriate to use the

Volterra series approach to obtain quantitative results in noise loading analysis when

nonlinearities are significant.

Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems

130

Chapter 6

Nonlinear Bandwidth Expansion Receiver in Spectrum-Sliced WDM Systems

6-1. Introduction

In the previous chapter, the transmission of stochastic signals in nonlinear fibers

using noise loading analysis was discussed. In this chapter, we will investigate how fiber

nonlinearities can be utilized to improve the performance of spectrum-sliced WDM (SS-

WDM) systems (Figure 6-1). In a spectrum-sliced system an incoherent (stochastic)

broadband source is used to generate the carrier signals of each channel by slicing the

source spectrum with passive optical filters. Spectrum-sliced systems are known to be

applicable for local area networks because although performance is limited it is

potentially much lower in cost compared to conventional WDM systems. Recently,

however, it has been demonstrated that spectrum slicing can be used for much larger

scale networks with over Gb/s rate and hundreds of kilometers of transmission distance

[71]. One of the key factors in the design of Gb/s spectrum-sliced WDM systems is the

trade-off in optical bandwidth between signal-to-excess optical noise ratio and dispersion

penalty. A small optical bandwidth of the spectrum-sliced signal will cause a significant

intensity noise (called excess noise), but a larger optical bandwidth will induce significant


131

dispersion. In recent experiments, it was reported that good performance might be

achieved with a narrow transmitted bandwidth if the bandwidth is expanded at the

receiver using a short section of nonlinear fiber [72-74]. The proposed method implies we

could also take advantages of narrower transmitted bandwidth to maximize transmission

capacity for a given total bandwidth. However, the theoretical bases of this technique,

and the factors limiting the performance improvement, have not previously been

reported.

In this chapter, the performance improvement obtained by using a nonlinear fiber

at the receiver will be explained by observing the auto-covariance curves of the photo-

detected signal (section 6-2). In section 6-3, it will be shown that there exists an optimum

filter bandwidth to maximize system performance. In addition the ‘modified correlation

time’ will be introduced to design the optimum filter bandwidth. The limiting factors of

the technique will be discussed in section 6-4, and finally a summary of this chapter will

be presented in Section 6-5.

Figure 6-1 Spectrum-Sliced WDM system

l1

l2

l1

Noisy Source

Signal 1

i1(t)

ln

ln

in(t)

Signal n

Optical Filter

Photo-Detector

MUX

Electrical Filter

Fiber


132

6-2. Auto-covariance of Photo-Detected Signals with Nonlinear Bandwidth Expansion Receiver (NBER)

High bit rate transmission using spectrum-slices in a WDM system becomes

possible as a result of the relatively high powers of broadband amplified spontaneous

emission (ASE) noise that can be obtained from EDFAs (erbium-doped fiber amplifiers).

However, because of its noise nature, there is an intrinsic intensity noise in the source.

This noise is called excess noise, and is known to be inversely proportional to optical

bandwidth. Therefore, it is desirable to have a large optical bandwidth of the spectrum-

sliced signal at a given bit rate. However, a larger bandwidth will not only induce

significant dispersion, but also limits the total transmission capacity of the system.

Recently, J. H. Han et al. demonstrated experimentally that the performance of SS-WDM

systems could be improved significantly by expanding the bandwidth of the signal at the

receiver utilizing fiber nonlinearities [72-74]. Figure 6-2 shows the structure of the so-

called nonlinear bandwidth expansion receiver (NBER).

Figure 6-2 Nonlinear Bandwidth Expansion Receiver

li

EDFA

G

Nonlinear Fiber Optical Filter

Optical Filters

w(t) )(tip


133

In NBER, a channel is selected first by an optical filter. The channel-selected signal is

then amplified by a following EDFA such that the signal power is large enough to induce

significant nonlinearities in a short segment of fiber which follows the EDFA. The

bandwidth expansion caused by the fiber could be significant depending on the power

level of the signal and the physical parameters of the fiber. However, the bandwidth

expansion alone, resulting from fiber nonlinear index of refraction, does not improve

system performance because phase nonlinearities do not affect the signal intensity.

Statistically, the new frequency components generated by the fiber nonlinearities are not

independent. However, optical filtering following the nonlinearity can reduce this

dependency. Numerical determination of eye-diagrams shows that NBER can indeed

significantly reduce excess noise in the mark state (Figure 6-3) consistent with

experimental results in [72,73]. Figure 6-3 is for the case of m = Bt/Rb = 5, where Bt is

the channel bandwidth and Rb is the bit rate. For m = 5, Rb=2.5Gb/s corresponds to an

optical bandwidth of the signal around 0.1nm in the 1.55µm window. The optical filter

following the nonlinear fiber is modeled by a first-order Butterworth filter with 334GHz

bandwidth, and the electrical filter after photo-detection is assumed to be a third-order

Butterworth filter with bandwidth of 0.7�Rb. Input power to the fiber is 40mW, and the

fiber is 20km long, and has the nonlinearity coefficient g = 2.4�10-3 mW-1km-1. In Figure

6-3, it is assumed that the fiber is nonlinear alone; i.e., dispersion is negligible.

The assumption of nonlinearity alone can be justified when the power level of the

input signal to the nonlinear fiber is significant due to the amplification of EDFA, and the

fiber length is relatively short. Similar to the case of deterministic signals (Eq.(2.12), for

nonlinearity alone we can obtain the output random process of the nonlinear fiber, w(t),

in analytical form.

))(exp()()(2

tzjtt vvw γ= (6.1)

where v(t) = the input random process to the fiber, g = nonlinearity coefficient, and z =

the fiber length.

For example, if the average power of v(t), Pv, is 40mW, g = 2�10-3 mW-1km-1, |b2| = 3

ps2/km, the dispersion distance (LD = To2�|b2|) is 3,333km for a 100ps initial pulse width


134

while the nonlinear distance (LN = (gPv)-1) is 12.5km. Therefore, Eq.(6.1) is justified if

the fiber length, z, is comparable to or smaller than LN when LD ø LN [11].

The bandwidth of w(t) could be significantly larger than v(t), but the photo-

detected signal of w(t) will be exactly the same as that of v(t) because the photo-detector

is insensitive to signal phase. However, optical filtering before photo-detection could

modify the statistical properties of w(t) to reduce the excess noise in the photo-detected

signal as observed in Figure 6-3. The photo-detected signal, )(ti p , with the nonlinear

bandwidth expansion receiver (Figure 6-2) may be expressed as

2)()()( tthkti oppp w∗= where kp [mA/mW] is the gain of the photo-detector, hop(t) is the

impulse response the optical filter after the nonlinear fiber, and * denotes convolution. To

determine the effect on system performance of the broadened spectrum combined with

the following optical filter it is necessary to study the statistical properties of )(ti p . Since

the reduced fluctuation of )(ti p is dependent on having many uncorrelated samples of the

received signal in the bit period, we expect that the auto-covariance of the photo-detected

current should give insight into the performance. The degree of statistical correlation of a

process can be observed by the correlation coefficient which is defined by the normalized

auto-covariance, )0(

)(

C

C τ [75].

Figure 6-4 shows the normalized auto-covariance of the photo-detected signal in

the mark state without bandwidth expansion, and with bandwidth expansion followed by

optical filters of different bandwidths. The bit rate is 2.5Gbits/s and the transmitted

bandwidth is 12.5 GHz (0.1nm) which corresponds to m = 5. The photo-detector gain (kp)

is assumed to be 1, and the optical filter is assumed a first-order Butterworth. Without

bandwidth expansion, the auto-covariance has a broad peak corresponding to the narrow

transmitted bandwidth. With bandwidth expansion the auto-covariance peak narrows.

However, as the optical filter bandwidth is increased, the width of the auto-covariance

asymptotes (dependent on the expansion bandwidth), but the height of the tails of the

auto-covariance increases (increased correlation of spectral components). This suggests

that there is an optimum filter bandwidth to achieve maximum performance improvement

by NBER. This aspect will be further discussed in the next section.


135

Figure 6-3 Comparison of Eye-Diagrams (a) without bandwidth expansion (b) with

bandwidth expansion

0 2 4 6 8 0

50

100

150

200

250

300

350 A

.U.

[sec]

0 2 4 6 8 0

10

20

30

40

50

60

A.U

.

[sec]

(b) with bandwidth expansion

(a) without bandwidth expansion

�10-10

�10-10


136

Figure 6-4 Normalized auto-covariance (correlation coefficient) curves of the photo-

detected signal when m=5

-30 -20 -10 0 10 20 30

0

0.2

0.4

0.6

0.8

1 Bo 2 =30GHz

Bo 2 =100GHz

Bo 2 =250GHz

Without BW Expansion

Time [ps]

)0(

)(

C

C τ


137

6-3. Optimum Optical Filter Bandwidth and Q-factor of Nonlinear Bandwidth Expansion Receiver (NBER)

The normalized auto-covariance curves imply that there might be an optimum

optical filter bandwidth to achieve maximum performance improvement by NBER as

observed in Figure 6-4. It will be shown in this section that there is indeed an optimum

bandwidth to minimize an effective correlation time. The Q-factor, a more general system

performance measurement metric, will also be used to verify the existence of an optimum

bandwidth. Q-factor will be defined subsequently.

To characterize the auto-covariance we consider the correlation time defined as

[75]

∫∞

ττ=τ0

c d)(C)0(C

1 (6.2)

where C(τ) is the auto-covariance function.

However, since we are interested in auto-covariance within one bit period, it is not

appropriate to integrate to infinity in the calculation of τc. Instead, we define a modified

correlation time, cτ~ in terms of Tc.

∫−

=2

cT

2cT

)(C(0)

1~ τττ dCc (6.3)

where Tc is a reference time interval defined as the range of the auto-covariance without

bandwidth expansion to drop to 1/e of its peak value.

Figure 6-5 shows the calculated cτ~ as a function of optical filter bandwidth, Bo, for

various input powers to the nonlinear fiber. (The greater the input power the greater the

bandwidth expansion.) Figure 6-5 (a) is the case of m = 5 and Figure 6-5 (b) is for m =

2.5 which corresponds to transmitted bandwidths of 12.5 GHz (Dl = 0.1nm) and 6.25

GHz (Dl=0.05 nm), respectively when the bit rate is 2.5Gbits/s. The solid line is the cτ~

for the reference case without the nonlinear bandwidth expansion. It is interesting to

observe that there is an optimum bandwidth of the optical filter to minimize cτ~ , which


138

suggests the existence of an optimum bandwidth to minimize bit error rate (BER) at a

given m.

BER can be estimated by the Q-factor which is defined as [30]

01

01

σσµµ

+−

=Q (6.4)

where m1, m0 are the mean values, and s1, s0 are the standard deviations of the mark

and space states at the receiver, respectively. If it is assumed that the mark and space

states are Gaussian distributed then it is easily shown [30] that the Q-factor and BER are

related by

=

22

1 QerfcBER (6.5)

where dyexerfcx

y∫∞ −=

22)(

π.

It has been reported that BER estimation using Eq.(6.5) may be significantly inaccurate

because the assumption of Gaussian statistics may not be valid in describing the photo-

detected signal in direct-detection receivers [76-79]. Nevertheless, the Q-factor can give a

good estimate of relative system performance with a reasonable amount of computational

load. Figure 6-6 shows the block diagram to simulate the Q-factor with the nonlinear

bandwidth expansion receiver. A total 1240 bits are used to calculate the Q-factor.

Figure 6-7 shows the simulated Q-factor as a function of the bandwidth of the

optical filter in NBER when input power to the nonlinear fiber is fixed at 30mW. It is

significant that large Q-factors can be achieved with relatively small values of m. It has

been shown that in the absence of bandwidth expansion the Q-factor, assuming a

polarized source, asymptotically approaches m as signal-to-noise ratio increases [80].

Therefore, m should be at least 36 to achieve Q = 6 (BER = 10-9) without the bandwidth

expansion block. However, Figure 6-7 shows that Q = 6 can be achieved with NBER

even when m is 2.5 if the optical filter bandwidth is set to between 50 to 120GHz. In

addition, by comparing Figures 6-5 and 6-7 it is seen indeed that the maximum Q-factor

occurs when cτ~ is minimized. In Figure 6-7, the optimum bandwidths to maximize Q for


139

m =2.5 and for m = 5 occur near 70GHz and 150GHz, respectively. They correspond to

the optimum bandwidth to minimize cτ~ when Pv = 30mW in Figure 6-5. Thus, the

modified correlation time, which is much easier to compute than the Q-factor, appears to

be a very useful means for determining optimum filter bandwidth.

Physically, the existence of the optimum filter bandwidth can be explained by

considering two extreme cases. When the filter bandwidth is too small, the signal will

suffer too much energy loss and there is inadequate bandwidth expansion to achieve

performance improvement. At the other extreme where the filter bandwidth is very large,

then the bandwidth expansion consists solely of phase modulation which does not change

the signal intensity.


140

Figure 6-5 Modified Correlation Time (a) m = 5, (b) m = 2.5

0 50 100 150 200 2503.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5x 10

-12

Bo [GHz]

Cor

rela

tion

time

(a)m=5

Without ExpansionPv=30mW

Pv=50mW

0 50 100 150 200 2500.5

1

1.5

2x 10

-11

Bo [GHz]

Cor

rela

tion

tim

e

(b)m=2.5

Without ExpansionPv=30mW

Pv=50mW

Pv=70mW


141

Optical Filter1

Optical Filter2 Ideal EDFA

w(t) v(t)

Nonlinear Bandwidth Expansion Block

)(~ tw Ideal Photo-Detector Electrical Filter

ip(t) (•)2

)(~

tpi

01

01

σσµµ

+−

=Q

Figure 6-6 Block Diagram of the Q-factor Simulation with NBER

Broad-Band Complex Gaussian Random Process

( )dTT

bT

b∫ •0

1

PRBS

Calculation of m1, s1 & m0, s0

G

20km Pv


142

0 50 100 150 200 250 300 5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

Figure 6-7 Q-factor as a function of bandwidth of the optical filter in NBER (Pv=30mW)

01

01

σσµµ

+−

=Q

m=2.5

m=5

Bo [GHz]


143

6-4. Limitations of Nonlinear Bandwidth Expansion Receiver

Nonlinear bandwidth expansion at the receiver could be a very useful technique to

improve performance of spectrum-sliced WDM systems. In this section, the limiting

factors of the technique will be discussed. In the previous section, the optical amplifier in

the NBER is assumed to be ideal, that is; ASE (amplified spontaneous emission) noise

and gain saturation of the optical amplifier in NBER is ignored. The performance of

NBER including those non-ideal effects will be discussed, and the effect of the optical

filter shape will also be discussed.

6-4-1. Effects of Non-ideal Optical Amplifier

The system performance of NBER has been studied so far by assuming an ideal

EDFA in NBER. However, to investigate the potential limitations of NBER, it is

necessary to model the EDFA more realistically by including ASE noise and finite

amplifier gain with the effect of gain compression. To find the dominant non-ideal

effects, first, Q-factor is simulated by fixing the amplifier output power at 40mW

(Pv=40mW) to ensure enough bandwidth expansion at the output of the following fiber,

but including the effects of ASE noise of the EDFA (Figure 6-8 (a)). The spectral density

of the ASE noise is given by [30]

Ssp(f) = (G-1)nsphf (6.6)

where G = amplifier gain, nsp = population-inversion factor, h = Planck’s Constant

(6.62617û10-34 J sec) and f is the optical frequency. In Figure 6-8 (a), the Q-factor is

plotted as a function of input power to the amplifier. As the input power is reduced, the

gain of the amplifier is increased (Pv is fixed at 40mW), and consequently the ASE noise

of the amplifier is increased according to Eq.(6.6). From Figure 6-8 (a), we can observe

that the sensitivity of NBER is indeed degraded by the ASE noise when the input power

to the EDFA decreases. It is necessary for the input power to the EDFA to be more than

–34dBm to achieve Q = 6.


144

Now instead of fixing the amplifier output power, the gain model of the EDFA is

included. The large signal gain of an EDFA can be modeled as below [30].

−−=s

out

P

P

G

GGG

1exp0

(6.7)

where G0 is the unsaturated small signal gain, Ps is the saturation power, and Pout is the

EDFA output power.

Figure 6-8 (b) shows that when the input power is low the sensitivity of the NBER is

severely limited by including EDFA gain model for both Go=30dB and 35dB. m = 5 and

Ps = 15dBm is used for both cases. To achieve Q = 6, Pin should be around –15dBm for

Go=30dB and around –20dBm for Go=35dB. These simulation results indicate that the

finite gain of the EDFA in the NBER is the more limiting factor on sensitivity than is

ASE noise. This result is not surprising because NBER requires a large EDFA output

power so that it can create enough nonlinearities in the following fiber.

6-4-2. Effects of Spectrum-Slicing Filter Shape

So far, analysis has been performed assuming the spectrum-slicing filter in the

transmitter has a shape of the first-order Butterworth filter to model a fiber Fabry-Perot

filter. However, it’s more advantageous to have a filter with sharper spectral response,

which will make it possible to put more channels without deteriorating system

performance by inter-channel interference. In the extreme case, a rectangular filter to

slice the spectrum is ideal to achieve maximum transmission capacity at a given ISI

penalty. A sharper spectrum of transmitting signal, however, may give a poorer

performance due to less efficient bandwidth expansion even if the signal power remains

the same. In this section, an ideal (rectangular) spectrum-slicing filter is considered to see

the effect of the filter shape on the performance of NBER.

In the previous section, a first-order Butterworth filter with a 3dB bandwidth of

12.5GHz is considered corresponding to m = 5 for Rb=2.5Gb/s. The same filter has


145

19.6GHz of equivalent noise bandwidth1, Beq. Equivalent noise bandwidth is the width

of a fictitious rectangular spectrum such that the power in that rectangular band is equal

to the power associated with the actual spectrum [81]. Therefore, a rectangular filter

having a 19.6GHz bandwidth will make the output power of the spectrum-sliced signal

equal to that of the first order Butterworth filter having a 3dB bandwidth of 12.5 GHz.

Figure 6-9 shows the modified correlation time and Q-factor with the rectangular slicing

filter. From the calculated cτ~ (Figure 6-9 (a)), we expect that the optimum filter

bandwidth following the nonlinear bandwidth expansion will occur around 30GHz with

Pv=30mW, and around 50GHz with Pv=50 and 70mW. These optimum bandwidths occur

approximately at one third of the first-order Butterworth filter case (Figure 6-5 (a)). For

example, the optimum bandwidth occurred around 150GHz with Pv=50mW when the

12.5GHz 3dB bandwidth of the first order Butterworth filter is used. This result suggests

that the bandwidth expansion when the input spectrum is very sharp is much less

effective than when input spectrum has long tails. Figure 6-9 (b) shows the Q-factor

with the optimum bandwidth of 50GHz of the optical filter in NBER. The optical

amplifier is modeled to include ASE noise and the gain saturation effect with Go=30dB as

described in Eq.(6-6) and Eq.(6-7). There are performance improvements when the

received signal power is large. However, it is observed that the improvement achieved is

much less (smaller values of Q) compared to the case where a first order Butterworth

filter is used for the spectrum slicing (Figure 6-8 (b)). The Q factor with the rectangular

filter is less than 5 while it was more than 8 for the case of a first order Butterworth filter

when the received signal power is –10dBm. This is because the optimum bandwidth is

smaller as indicated by the minimum point of the modified correlation time (Figure 6-9

(a)). Therefore, we may conclude that the nonlinear bandwidth expansion technique

requires a broad spectral shape of the transmitted signal to achieve a significant

performance improvement.

1The equivalent noise bandwidth is defined as ∫∞

∞−

= dffHH

Beq

2

2)(

)0(2

1.


146

6-5. Summary

Simulation studies show that in spectrum-sliced WDM systems nonlinear

bandwidth expansion at the receiver may be used to reduce excess noise while keeping

the transmitted optical bandwidth small. This is important because small transmitted

bandwidths are crucial to minimize dispersion effects, and to maximize transmission

capacity for a given total bandwidth. The performance improvement can be explained by

observing the auto-covariance curves of the photo-detected signal. The optical filter after

the nonlinear fiber in the NBER makes the covariance curve narrower than that of the

input signal, but can cause a rise in the tails if the filter bandwidth is too large.

Simulations of auto-covariance and Q-factor indicate that to maximize system

performance there is an optimum bandwidth of the optical filter following the nonlinear

bandwidth expansion.

To have optimum performance improvement, the input spectrum slice should

have fairly long tails to make the bandwidth expansion more efficient. Even when the

spectrum has a broad shape, the sensitivity of NBER is severely limited by the finite gain

of the optical amplifier. To increase the receiver sensitivity, it is desirable to have large

fiber nonlinearities, and an optical amplifier with larger gain.


147

Figure 6-8 Q-factor of NBER with non-ideal EDFA (a) EDFA noise effects on the

sensitivity of NBER (Pv= 40mW), (b) NBER sensitivity with gain modeling (Ps =15dBm)

-40 -35 -30 -25 -20 -15 -10 1

2

3

4

5

6

7

8

9

10

11

B o 2 =50GHz B o 2 =200GHz B o 2 =500GHz

-40 -35 -30 -25 -20 -15 -10 3

4

5

6

7

8

9

10

11

Pin [dBm]

Pin [dBm]

Q

Q

Go =35dB

Go =30dB

(a)

(b)


148

Figure 6-9 NBER performance with a rectangular transmitting filter (bandwidth =

19.6GHz) (a) Modified correlation time as a function of the optical filter bandwidth after

the nonlinear fiber (b) Q-factor vs. received signal power

0 50 100 150 200 250 1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9 x 10 -11

Without Expansion P v =30mW P v =50mW P v =70mW

-40 -35 -30 -25 -20 -15 -10 2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

B o 2 =30GHz

B o 2 =50GHz

B o 2 =100GHz

Pin [dBm]

Q

Cor

rela

tion

Tim

e

Bo [GHz]

(a)

(b)

Chapter7: Conclusions and future work

149

Chapter 7

Conclusions and Future Work In this final chapter, we summarize the conclusions that can be drawn from the

research performed for this dissertation, and then provide suggestions for future research.

7-1. Summary of Major Contributions

The main motivation of this work was to obtain and compare analytical models to

characterize the effects of fiber nonlinearities on fiber optic communication systems.

Fiber nonlinearities have become one of most significant limiting factors of system

performance since the advent of erbium-doped fiber amplifiers (EDFAs) because input

power is increasing and the effects of fiber nonlinearities are accumulating with the use

of EDFAs. In wavelength-division-multiplexing (WDM) systems, inter-channel

interference due to fiber nonlinearities may limit the system performance significantly.

Therefore, understanding of fiber nonlinearities is crucial to optimize system

performance of optical fiber transmission links. However, very few analytical models

exist to analyze the effects of fiber nonlinearities except in very special cases such as

solitons. Conventionally, pure numerical methods such as the split-step Fourier method

have been used to analyze fiber nonlinearities. However, relying on pure numerical

methods is not desirable particularly in the design stage of a new system. In this


150

dissertation, several analytical models have been presented to give better physical insight

of the effect of fiber nonlinearities on fiber optic communication systems. The major

results obtained from each approach are summarized as follows:

1. The perturbation approach is applied to solve the nonlinear Schrödinger

equation, and its valid range has been determined by comparing with the results

of the split-step Fourier method over a wide range of parameter values. With

11mWkm2 −−=γ , the critical distance for the first order perturbation approach is

estimated to be ]mWkm[150 ⋅≈

avgc P

z . The critical distance, zc, is defined as the

distance at which the normalized square deviation compared to the split-step

Fourier method reaches 10-3. Including the second order perturbation will

increase zc more than a factor of two, but the increased computation load makes

the perturbation approach less attractive. In addition, it is shown mathematically

that the perturbation approach is equivalent to the Volterra series approach.

2. Output root-mean-square (RMS) pulse width is modeled by lumping the fiber

nonlinearity at the middle of the propagation distance. The resulting two-step

model predicts the output RMS pulse width much closer to the simulated one

compared to the existing one-step model. We show that there exists an optimal

input pulse width to minimize output pulse width based on the derived RMS

models. The derived analytical model predicts that the maximum bit rate-

transmission distance product has a functional form of 22

1~

βγβ avg

Nb P

LzR =

if zø1(zøLN), and 2~ βzzRb when z÷1 (z ÷ LN) [33,34]. It is also shown

that there is no optimum input pulse width to minimize the output spectrum

width or to minimize product of st(z) and sw(z).

3. The response of a fiber to a sinusoidally modulated input has been studied to see

its utility in measuring system performance in the presence of fiber


151

nonlinearities both in a single channel system and in a multi-channel system.

The sinusoidally modulated signal models an alternating bit sequence of ones

and zeros in on-off keying. In single channel transmission, the sinusoidal

response of normally dispersive fiber shows a strong correlation with eye-

opening penalty over a wide range of the nonlinearity parameter N (0.1 < N2

<100). This result implies that the measurement of the sinusoidal response can

be an alternate way of measuring eye-opening penalty (EOP) without having a

long sequence of randomly modulated input bits. But in the anomalous

dispersion region, the sinusoidal response has a much more limited range of

application to estimate system performance. The sinusoidal response has also

been derived analytically based on the perturbation analysis. Since the

perturbation analysis has a limited range of validity, the derived analytical

expression also has a limited range of applicability. Comparison with numerical

results reveals that the derived expression may result in a significant error when

N2 > 3.

4. The sinusoidal analysis has also been applied in a multi-channel system to

estimate CPM (cross-phase modulation)-induced performance degradation using

the pump-probe scheme. An analytical form of the intensity fluctuation of the

probe signal has been derived, which shows good agreement with numerical

results in conventional single-mode fiber systems over a wide range of channel

spacing Df, and in dispersion-shifted fiber systems when Df > 100GHz. The

pump-probe measurement or calculation is a useful measure of system

performance when CPM is the dominant degradation, but is less useful when

four-wave mixing (FWM) is significant.

5. It is shown that the effect of fiber nonlinearities can also be characterized with a

stochastic signal using the noise loading technique. Numerical results show that

NPR is a strong function of the magnitude of the dispersion parameter, b2. NPR

is larger when the magnitude of b2 is larger, which suggests that larger

dispersion is always beneficial to reduce nonlinear cross-talk. In addition,


152

simulation results indicate that it is advantageous to propagate in the normal

dispersion region when the magnitude of b2 is around 3 ps2/km or less, which is

the typical range of b2 in dispersion-shifted fibers. It is also shown that there is

about a 2.5dB advantage in NPR in using normal dispersion fiber first in

alternating dispersion maps even though the total average dispersion is equal.

The Volterra series approach is also applied to the noise loading analysis.

However, compared to the split-step method, the Volterra method using up to the

third order transfer function overestimates noise power at the output even at

relatively short transmission distances (z = 50km).

6. Finally, it is shown numerically how fiber nonlinearities can be utilized to

improve system performance. Simulation studies show that in spectrum-sliced

WDM systems nonlinear bandwidth expansion at the receiver may be used to

reduce excess noise while keeping the transmitted optical bandwidth small. The

performance improvement is explained by observing the auto-covariance curves

of the photo-detected signal. Simulations of auto-covariance and Q-factor

indicate that to maximize system performance there is an optimum bandwidth of

the optical filter following the nonlinear bandwidth expansion. The proposed

modified auto correlation time to characterize the auto-covariance curves is

shown to be a simple means for estimating the optimum bandwidth of the optical

filter.

The limitations of the nonlinear bandwidth expansion technique have also been

studied. The sensitivity of the nonlinear bandwidth expansion receiver is

severely limited by the finite gain of the optical amplifier. To increase the

receiver sensitivity, it is desirable to have large fiber nonlinearities, and an

optical amplifier with larger gain.

It is worth recalling that each developed analytical model has its own valid range of

parameters or valid systems. For example, RMS models developed in Chapter 3 may not

be applicable where ISI (inter-symbol interference) and/or ICI (inter-channel


153

interference) are not negligible since the models are developed assuming a single pulse

transmission in a single channel system. In a single channel system where ISI is

significant, the sinusoidal response developed in Chapter 4 may be more suitable because

it includes the ISI effect. In a multi-channel system where CPM is the dominant nonlinear

effect to cause ICI, the derived intensity fluctuation of the probe signal in Chapter 4 can

be used to estimate the performance degradation due to CPM. When FWM is also

significant, the noise loading analysis could be used.

7-2. Suggestions for Future Research

In this dissertation, the symmetrized split-step Fourier method is used as a

reference to evaluate the accuracy of the new analytical models developed. The nonlinear

operator is distributed within the step size Dz in the symmetrized split-step method,

whereas it is lumped at the center of Dz in the conventional split-step method. A larger

Dz requires less computational time, but results in poorer accuracy. Therefore, a trade-off

is required between accuracy and computational time. At a given Dz, the symmetrized

method will give a more accurate result, but with longer computational time due to the

numerical iteration described in Chapter 1. Therefore, it is of interest to see which

method allows faster computation at a given tolerance. Since in the case of solitons an

analytical solution is available this could be used to measure the accuracy of the two

split-step methods.

We have neglected the polarization of the optical field (single polarization

assumption) so far. However, it is known that polarization effects become significant in

installed conventional fiber systems when upgrading to bit rates of 10Gb/s or higher.

Therefore, it is of interest to see how the analytical models developed in this dissertation

can be modified in the presence of polarization effects. The first step might be to find the

valid range of the single polarization assumption. When both polarization modes are

considered, the combined effect of the polarization and the fiber nonlinearities may be

treated in analytical forms by considering the worst case. Another assumption made so

far is that the chirp in light sources is negligible. In the linear regime, it is well known


154

that that the chirp may broaden or compress the output pulse width depending on the

dispersion region. Therefore, in the presence of the fiber nonlinearities, it is of interest to

see how the results in Chapter 3 change with the chirp parameter. Since the chirp can be

controlled by an optical device (e.g. fiber Bragg grating), the chirp parameter could give

more design freedom to optimize system performance.

As a continuation of Chapter 5, the NPR’s resulting from the noise loading

analysis may be compared with a more general system performance metric, the Q-factor

to see the correlation between them. Another interesting problem is to find alternate way

of bandwidth expansion in NBER discussed in Chapter 6. The bandwidth expansion

technique requires a fairly long fiber (a few tens of km) for each channel at the receiver.

The fiber might need to be customized to have its zero-dispersion wavelength to be at the

center wavelength of the selected channel to induce enough bandwidth expansion when

there are a large number of channels. The fiber also acts as a power loss device as does

the following optical filter, and a single optical amplifier may not be sufficient to amplify

multiple channels at the receiver. Therefore, even though the technique using nonlinear

fiber has potential applications in improving system performance of spectrum-sliced

WDM systems, it is desirable to find altenate ways to expand signal bandwidth.

Appendix A: MATLAB Programs

155

Appendix A. MATLAB Programs 1. fiber_run.m % This program simulates a single-channel fiber transmission link % using the symmetrized split-step Fourier algorithm. % % written by Jong-Hyung Lee clear all %============================================= % Define Time Window and Frequency Window %============================================= taum = 2000; dtau = 2*taum/2^11; tunit= 1e-12; % make time unit in psec tau = (-taum:dtau:(taum-dtau))*tunit; fs = 1/(dtau*tunit); tl = length(tau)/2; w = 2*pi*fs*(-tl:(tl-1))/length(tau); % w=angular freq. wst = w(2)-w(1); %============================================= % Define Physical Parameters %============================================= c = 3e5; %[km/sec] speed of light ram0 = 1.55e-9; %[km] center wavelength k0 = 2*pi/ram0; n2 = 6e-13 ; %[1/mW] gamm = k0*n2 ; %[1/(km*mW)] alphaDB = 0.2 ; % [dB/km] Power Loss alpha = alphaDB/(10*log10(exp(1))); %[1/km] Power Loss in linear scale % Dispersion parameters (beta3 term ignored) Dp = -2; % [ps/nm.km] beta2 = -(ram0)^2*Dp/(2*pi*c); % [sec^2/km] %============================================= % Define Input Signal %============================================= % A single Gaussian pulse is assumed. Po = 2; % [mW] initial peak power of signal source C = 0; % Chirping Parameter m = 1; % Super Gaussian parameter (m=1 ==> Gaussian) t0 = 50e-12; %[sec] initial pulse width


156

at = sqrt(Po)*exp(-0.5*(1+i*C)*(tau./t0).^(2*m)); % Input field in the time domain a0 = fft(at(1,:)); af = fftshift(a0); % Input field in the frequency domain %============================================= % Define Simulation Distance and Step Size %============================================= zfinal = 100; %[km] propagation distance pha_max = 0.01; %[rad] maximum allowable phase shift due to the nonlinear operator % pha_max = h*gamma*Po (h = simulation step length) h = fix(pha_max/(gamm*Po)); % [km] simulation step length M = zfinal/h; % Partition Number % Define Dispersion Exp. operator % Dh = exp((h/2)*D^), D^=-(1/2)*i*sgnb2*P, P=>(-i*w)^2 Dh = exp((h/2)*(-alpha/2+(i/2)*beta2*w.^2)); % %================================================% % Propagation Through Fiber % %================================================% % Call the subroutine, sym_ssf.m for the symmetrized split-step Fourier method [bt,bf] = sym_ssf(M,h,gamm,Dh,af); % Preamplifier at the receiver % Optical amplifier is assumed ideal (flat frequency response and no noise) GdB = 20; % [dB] optical amplifier power gain gainA = sqrt(10^(GdB/10)); % field gain in linear scale rt = gainA*bt; % plot the received power signal figure(1) plot(tau,abs(rt).^2,’r’)


157

2. sym_ssf.m

function [to,fo] = sym_ssf(M,h,gamma,Dh,uf0) % Symmetrized Split-Step Fourier Algorithm % % ==Inputs== % M = Simulation step number ( M*h = simulation distance ) % h = Simulation step % gamma = Nonlinearity coefficient % Dh = Dispersion operator in frequency domain % uf0 = Input field in the frequency domain % % ==Outputs== % to = Output field in the time domain % fo = Output field in the frequency domain % % written by Jong-Hyung Lee for k = 1:M %============================================================= % Propagation in the first half dispersion region, z to z+h/2 %============================================================= Hf = Dh.*uf0; %========================================================== % Initial estimate of the nonlinear phase shift at z+(h/2) %========================================================== % Initial estimate value ht = ifft(Hf); % time signal after h/2 dispersion region pq = ht.*conj(ht); % intensity in time u2e = ht.*exp(h*i*gamma*pq); %Time signal %============================================================= % Propagation in the second Dispersion Region, z+(h/2) to z+h %============================================================= u2ef = fft(u2e); u3ef = u2ef.*Dh; u3e = ifft(u3ef); u3ei = u3e.*conj(u3e); %======================================================== % Iteration for the nonlinear phase shift(two iterations) %======================================================== u2 = ht.*exp((h/2)*i*gamma*(pq+u3ei)); u2f = fft(u2) ; u3f = u2f.* Dh; u4 = ifft(u3f);


158

u4i = u4.*conj(u4); u5 = ht.*exp((h/2)*i*gamma*(pq+u4i)); u5f = fft(u5); uf0 = u5f.*Dh; u6 = ifft(uf0); u6i = u6.*conj(u6); %============================================================= % Maximum allowable tolerance after the two iterations etol = 1e-5; if abs(max(abs(u6i))-max(abs(u4i)))/max(abs(u6i)) > etol disp(’Peak value is not converging! Reduce Step Size’),break end %============================================================= end to = u6; fo = uf0;

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167

Vita

Jong-Hyung Lee was born in Korea on August 2, 1964. He received his Bachelor

of Science in Electronic Engineering from Yonsei University, Seoul, Korea in 1987. He

received his Master of Science in Electronic Engineering from the same school in 1990.

From January 1990 to June 1994, he worked with Daewoo Telecommunications

in Seoul, Korea. He was involved in the development of commercial integrated circuits.

His main responsibility was to design analog integrated circuits with bipolar processes.

He joined the graduate program at Virginia Tech in August 1994. His research

efforts have been focused on analysis and characterization of fiber nonlinearities. His

research interests include the modeling, simulation, and design of high data rate fiber

optic communication systems. Working under Dr. Jacobs, he received his Ph.D. in

February 2000 from Virginia Tech.

Analysis and Characterization of Fiber Nonlinearities.pdf

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