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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),
and (d) Pa(z)/ Pb(z)
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)
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,
Chapter 1: Introduction
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.
Chapter 1: Introduction
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.
Chapter 1: Introduction
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.
Chapter 1: Introduction
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
Chapter 1: Introduction
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.
Chapter 1: Introduction
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].
Chapter 1: Introduction
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],
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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.
Chapter 1: Introduction
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.
Chapter 1: Introduction
13
Figure 1-1 Illustration of walk-off distance
Signal Pulse Interfering Pulse
z = zo
z = zo + Lw
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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.
Chapter 1: Introduction
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])
Chapter 1: Introduction
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.
Chapter 1: Introduction
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
( )( ) ),(ˆˆexp),( tzANLztzzA +∆=∆+ (1.17)
In the split-step Fourier method, it is assumed that the two operators commute with each
other. That is,
( ) ( ) ),(ˆexpˆexp),( 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.
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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.
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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.
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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)
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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),
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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.
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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)
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 2: Analysis of fiber nonlinearities by perturbation method
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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.
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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.
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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)
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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.
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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)
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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 === ,
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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)
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
=
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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).
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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.
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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
Chapter 3: Modeling and optimization of RMS pulse and spectrum widths
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.
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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.
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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)
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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.
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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)
Chapter 4: Performance measurements using sinusoidally modulated signal
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.
Chapter 4: Performance measurements using sinusoidally modulated signal
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|
Chapter 4: Performance measurements using sinusoidally modulated signal
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|
Chapter 4: Performance measurements using sinusoidally modulated signal
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)
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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)
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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)
Chapter 4: Performance measurements using sinusoidally modulated signal
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.
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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)
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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).
Chapter 4: Performance measurements using sinusoidally modulated signal
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].
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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,
Chapter 4: Performance measurements using sinusoidally modulated signal
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.
Chapter 4: Performance measurements using sinusoidally modulated signal
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)]
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Channel Spacing [GHz]
E.O
.P. [
dB]
Conventional Fiber, D = +17 ps/(nm km)
Equally Spaced Unequally SpacedSingle Channel
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Channel Spacing [GHz]
E.O
.P. [
dB]
Dispersion Shifted Fiber, D = -2 ps/(nm km)
Equally Spaced Unequally SpacedSingle Channel
(d)
(c)
(b)
Chapter 4: Performance measurements using sinusoidally modulated signal
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]
Chapter 4: Performance measurements using sinusoidally modulated signal
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
Channel Spacing [GHz]
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.
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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]
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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.
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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]
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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.
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
]
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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)
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
]
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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)
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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)
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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)
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 5: Noise loading analysis to characterize fiber nonlinearities
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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.
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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.
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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]
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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.
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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.
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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.
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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)
Chapter 6: Nonlinear bandwidth expansion receiver in spectrum-sliced WDM systems
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
Chapter7: Conclusions and future work
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
Chapter7: Conclusions and future work
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,
Chapter7: Conclusions and future work
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
Chapter7: Conclusions and future work
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
Chapter7: Conclusions and future work
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
Appendix A: MATLAB Programs
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’)
Appendix A: MATLAB Programs
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);
Appendix A: MATLAB Programs
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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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.
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