Timing Jitter In Long-haul WDM
Return-To-Zero Systems
vorgelegt von
Diplom-Ingenieur
André Richter
aus Berlin
von der Fakultät IV Elektrotechnik und Informatik
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
- Dr.-Ing. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr.-Ing. Noll
1. Gutachter: Prof. Dr.-Ing. Petermann
2. Gutachter: Prof. Dr.-Ing. Voges (Universität Dortmund)
Tag der wissenschaftlichen Aussprache: 19. Februar 2002
Berlin 2002
D 83
A. Richter - Timing Jitter in WDM RZ Systems i
Table of Contents
1Introduction 1
1.1 Long-haul fiber-optic transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Chapter overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Long-haul WDM transmission systems 5
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Fiber propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Group-velocity dispersion (GVD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Kerr effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3.1 Self- and cross-phase modulation . . . . . . . . . . . . . . . . . . . . 12
2.3.3.2 Four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.4 Propagation over nonlinear and dispersive fiber . . . . . . . . . . . . . . 15
2.3.4.1 Generalized nonlinear Schrödinger equation . . . . . . . . . . 15
2.3.4.2 Split-step Fourier method . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.4.3 Characteristic scale distances . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Erbium-doped fiber amplifier (EDFA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Erbium ions in glass hosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.2 Amplifier gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.3 Amplified spontaneous emission (ASE) noise . . . . . . . . . . . . . . . . . 24
2.4.4 Design aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.2 Noise contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Characteristics of RZ pulse propagation 31
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Dispersion-managed soliton (DMS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Evolution from classical soliton theory . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1.1 Lossless fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1.2 Periodically amplified fiber link . . . . . . . . . . . . . . . . . . . . . 35
3.2.1.3 Dispersion-managed, lossy fiber link . . . . . . . . . . . . . . . . . 37
ii A. Richter - Timing Jitter in WDM RZ Systems
3.2.2 Pulse dynamics of DMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Chirped return-to-zero (CRZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Convergence of DMS and CRZ schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Modeling of timing jitter 45
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Main system distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Noise from optical amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2 Intrachannel pulse-to-pulse interactions . . . . . . . . . . . . . . . . . . . . 47
4.2.3 Interchannel cross-phase modulation . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.4 Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Timing jitter due to optical inline amplification (ANTJ) . . . . . . . . . . . . . 51
4.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.2 Linearization approximation for arbitrary pulse shapes . . . . . . . 53
4.3.2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Timing jitter due to interchannel cross-phase modulation (CITJ). . . . . . 59
4.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.2 Elastic collision approximation for arbitrary pulse shapes . . . . . . 62
4.4.2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.2.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.2.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 WDM system simulations - timing jitter 77
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 ANTJ in dispersion-managed systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 CITJ in WDM transmission systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.1 Dependence of CITJ on dispersion map and amplifier positioning 82
5.3.2 Dependence of CITJ on dispersion slope . . . . . . . . . . . . . . . . . . . . 85
5.3.3 Dependence of CITJ on RZ modulation scheme . . . . . . . . . . . . . . . 86
5.3.4 Dependence of CITJ on channel spacing . . . . . . . . . . . . . . . . . . . . 89
5.3.5 Dependence of CITJ on initial pulse positioning in bit interval . . 92
A. Richter - Timing Jitter in WDM RZ Systems iii
6 Estimation of system performance 99
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2.1 Optical signal-to-noise ratio (OSNR) . . . . . . . . . . . . . . . . . . . . . . . 100
6.2.2 Eye-opening penalty (EOP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.3 Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.4 Bit error rate (BER) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.4.1 Monte Carlo (MC) experiment . . . . . . . . . . . . . . . . . . . . . 103
6.2.4.2 Gaussian approximation (GA) . . . . . . . . . . . . . . . . . . . . . 104
6.2.4.3 Deterministic noise approximation (DNA) . . . . . . . . . . . . 108
6.3 Impact of pulse timing jitter on BER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.4 WDM system simulations - BER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4.1 RZ system over dispersion-managed link with mainly SSMF . . . 113
6.4.2 Dispersion-managed soliton system . . . . . . . . . . . . . . . . . . . . . . . 115
6.4.3 Chirped RZ system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7 Summary 121
References 123
A List of Acronyms 135
B List of Symbols 137
Acknowledgements 141
iv A. Richter - Timing Jitter in WDM RZ Systems
A. Richter - Timing Jitter in WDM RZ Systems 1
Chapter 1
Introduction
1.1 Long-haul fiber-optic transmission
Long distance fiber-optic telecommunication systems carry digital infor-
mation over terrestrial distances ranging from 3,000 km to 5,000 km and
transoceanic distances ranging from 5,500 km to 12,000 km. During the last
10 years these systems have evolved significantly. In 1988, only 2% of long
distance traffic was carried by submarine cables. At this time, most traffic
was transmitted via satellite connections. However, in 2000, 80% of the traf-
fic was carried over optical fiber links already. The popularity of optical fiber
systems is mainly due to the potentially huge bandwidth, and thus channel
capacity, which was made available due to the dramatic progress in the
development of optical fibers and amplifiers, transmitters and receivers dur-
ing the past 10 years.
In the late 1980s, electro-optic repeaters and fiber optics were introduced
to long-haul transmission systems, replacing the copper cables. This first
generation of transoceanic fiber systems carried 280 Mbit/s in a single chan-
nel at 1300 nm. With the invention of single-frequency laser diodes at
around 1550 nm, and Erbium-doped fiber amplifiers (EDFA), Giga-
bit-per-second systems could be built in the 1990s. The first transoceanic
projects employing these technologies since 1996 were the TAT-12/13 project
[148] and the TPC-5 network [14], initially, both operated at 5 Gbit/s.
Today’s long-haul transmission systems represent the fourth generation
utilizing multiple carrier wavelengths, which had lead to an explosion of
channel capacity. At the same time, deregulation of telecommunication mar-
2 A. Richter - Timing Jitter in WDM RZ Systems
Introduction
kets and global success of the internet has driven the demand for higher and
higher system capacity. In 1998, existing systems were upgraded to carry up
to four coarsely spaced wavelengths. Today, new dense wavelength-division
multiplexing (DWDM) systems are under construction that will soon deliver
up to 1 Tbit/s of data per fiber over transoceanic distances.
Conventionally, non return-to-zero (NRZ) modulation format has been
used in long-haul transmission systems [148], [14]. These systems are based
on the fact that fiber dispersion and nonlinearities are detrimental effects.
NRZ is used advantageously as it provides minimum optical bandwidth, and
minimum optical peak power per bit interval for given average power.
However, with increased bitrates it has been shown that RZ modulation
formats offer certain advantages over NRZ, as they tend to be more robust
against distortions [23]. For instance, RZ modulation is more tolerant to
non-optimized dispersion maps than NRZ schemes [113]. This can be
explained by the fact that optimum balancing between fiber nonlinearities
and dispersion is dependent on the pulse shape. A RZ modulated signal
stream consists of a sequence of similar pulse shapes, whereas a NRZ modu-
lated stream does not. The dispersion tolerance of a signal stream can be
derived from the superposition of the dispersion tolerance of the individual
pulse shapes [113]. In fact, for the majority of cases, the best results of WDM
transmission experiments regarding the distance-bitrate product have been
achieved using RZ modulation formats in both terrestrial and transoceanic
systems (see Table 1-1).
When designing high capacity systems, it becomes increasingly important
to carefully model system performance before performing laboratory experi-
ments and field trials, as these experiments are costly and time consuming.
The huge design space can only be limited by analytical approximations and
computer modeling using powerful simulation tools.
This work focuses on the characteristics of RZ pulse propagation over
modern long-haul fiber-optic transmission systems. Major distortions of
those systems arise from pulse timing jitter, which are introduced by various
sources along the propagation path. It is the subject of this work to numeri-
cally investigate the timing jitter in long-haul WDM RZ systems. The follow-
ing section presents a short overview of each chapter in this thesis.
Introduction
A. Richter - Timing Jitter in WDM RZ Systems 3
.
1.2 Chapter overview
In Chapter 2, recent trends in long-haul terrestrial and submarine sys-
tems design are presented. Drastic changes have been reported over the last
decade driven by the dramatic increase of capacity demand. Design consider-
ations of transmitter, optical fiber link and receiver for typical long-haul
transmission systems are discussed.
Chapter 3 presents different RZ modulation formats, which are considered
to be of great potential for long-haul propagation. The time and frequency
dynamics of the most favorite modulation formats, namely dispersion-man-
aged soliton (DMS) and chirped return-to-zero (CRZ), are discussed in detail.
Lately, convergence between DMS- and CRZ-based transmission systems
has been subject to discussion. Overall results of these discussions are briefly
summarized.
Chapter 4 starts with a brief overview of the main distortions that occur in
long-haul WDM transmission. This is followed by a detailed discussion pre-
Table 1-1. Results of recent long-haul experiments.
Capacity
[Gbit/s]
Distance
[km]
Notes Reference
OFC 1998 320 (64x5) 7,200 CRZ (chirped return-to-zero),
LCF (large core fiber)
[19]
ECOC 1998 500 (25x10) 9,288 nonlinearity supporting RZ,
HODM (higher order dispersion management)
[123]
OFC 1999 640 (64x10)
1,020 (51x20)
7,200
1,012
CRZ, LCF, FEC (forward error correction)
DMS (dispersion-managed soliton), mainly SMF
[20]
[63]
ECOC 1999 1,100 (55x20)
1,000 (100x10)
3,020
6,200
DMS, HODM, C/L-band
chirped RZ, HODM using LCF, FEC
[47]
[149]
OFC 2000 1,800 (180x10) 7,000 CRZ, HODM, FEC [32]
ECOC 2000 1,120 (56x20)
2,110 (211x10)
6,200
7,221
CRZ, HODM, FEC
chirped RZ, HODM using LCF, Raman,
C/L-band, FEC
[24]
[147]
OFC 2001 2,400 (120x20) 6,200 CRZ, HODM using LCF, enhanced FEC [25]
4 A. Richter - Timing Jitter in WDM RZ Systems
Introduction
senting the two main sources of pulse timing jitter in long-haul fiber-optic
transmission systems. These jitter sources are, noise generated from optical
inline amplifiers, and interchannel cross-phase modulation (XPM). Firstly,
an overview of common techniques used to estimate ASE-noise induced tim-
ing jitter (ANTJ) is presented. A recently reported approach for ANTJ esti-
mation is presented, which is derived for arbitrary pulse shapes. This
approach takes into account the impact of pulse chirping on accumulated
timing jitter [60]. Secondly, an overview of common approximations to esti-
mate collision-induced timing jitter (CITJ) due to interchannel XPM is given.
The limitations of all these techniques are discussed in detail. Then, a new
approach for estimating CITJ is presented, which can be applied to RZ pulses
of arbitrary shapes, undergoing an arbitrary number of collisions with pulses
propagating in an arbitrary number of channels.
In Chapter 5, results from typical WDM system simulations are pre-
sented. Timing jitter values estimated from the two semi-analytical tech-
niques discussed in Chapter 4 are used to explore the dependence of ANTJ
and CITJ on several system design parameters such as dispersion map,
amplifier positioning, channel spacing, and initial pulse positioning.
In Chapter 6, commonly used methods for estimating system performance
are presented. The focus is on techniques used for calculating the bit error
rate (BER), and their applicability in software modeling of long-haul trans-
mission systems. The translation of pulse timing jitter to BER is outlined,
and a simple method of considering its impact is presented. This method is
then applied to different WDM system simulations.
Finally, Chapter 7 gives a summary of the work.
A. Richter - Timing Jitter in WDM RZ Systems 5
Chapter 2
Long-haul WDM transmission
systems
2.1 Overview
This chapter provides general information on components and subsystems
of a typical long-haul WDM transmission system. The scope of this chapter is
not to provide detailed knowledge, but rather to provide information, which
characterizes the most important system components, and is of need in the
following chapters. This chapter should help to motivate, why system param-
eter values are set the way they are, and why certain system impairments
are not regarded in this work.
Firstly, the externally modulated transmitter, which performs intensity
modulation (IM) of the optical carrier wave is briefly introduced. A WDM
transmitter consists typically of one externally modulated transmitter per
channel, which are connected together via filters and couplers to finally feed
the optical WDM signal into the optical fiber.
Secondly, fiber propagation is investigated in more detail. The major fiber
propagation effects are presented, which are the linear effects of fiber attenu-
ation and chromatic dispersion, and the nonlinear effects due to the Kerr
nonlinearity. Polarization-dependent propagation effects are not considered,
as this is outside the scope of this work. Afterwards, the generalized nonlin-
ear Schrödinger equation (GNLS) is introduced, which describes the propa-
gation of optical waves over the nonlinear, dispersive fiber. A numerical
6 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
method for solving the GNLS is introduced, and important scale distances
are listed.
Thirdly, the Erbium-doped fiber amplifier (EDFAs) is introduced as the
major mean for amplification in long-haul WDM transmission systems.
EDFAs are employed as booster amplifiers at the WDM transmitter, as
inline amplifiers compensating periodically for the fiber attenuation, and as
pre-amplifiers in front of the optical receivers to limit the impact of the
receiver noise. After providing some information about the amplification pro-
cess in Erbium-doped fibers, the characteristics of amplifier gain and noise
generation are discussed. Finally, design aspects of EDFAs are briefly pre-
sented.
This chapter will be finished with a section on optical direct detection (DD)
receivers. Their architecture is briefly presented, and major noise contribu-
tions are listed.
2.2 Transmitter
For modern WDM transmission systems employing channel bitrates of
10 Gbit/s and higher, external modulation is commonly applied for intensity
and phase modulation of the optical carrier. Externally modulated transmit-
ters provide a high wavelength-stability, a small amount of distortions, a
high extinction ratio, and a defined frequency chirp characteristic. Fre-
quency chirp denotes the time-dependence of the phase of the optical signal,
and can be controlled to counteract for fiber propagation degradations due to
chromatic dispersion and self-phase modulation1 [49].
Externally modulated transmitters are based on the principle that a wave-
length-stable laser is operated to emit a continuous wave (CW) into an exter-
nal modulator device, which is controlled by an electrical voltage carrying
the data and pulse shape information. Two types of modulators are com-
monly used, namely Mach-Zehnder modulators (MZM) and electro-absorp-
tion modulators (EAM) [72]. MZMs provide usually a better defined transfer
1. See Chapter 2, p. 9 and Chapter 2, p. 12 for details.
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 7
characteristic. They can be designed to have zero frequency chirp, or a chirp,
which can be controlled by the electrical drive voltage. EAMs are more easy
to integrate, however, provide an intrinsic chirp already, which is propor-
tional to the gradient of the emitted optical power.
Figure 2-1 shows the schematic of the transmitter for the case that a MZM
is used for external modulation. The electrical drive signal is created by feed-
ing the output of the bit source into a pulse generator.
Figure 2-1. Schematic of externally modulated laser.
The Mach-Zehnder modulator is based on the interference principle. That
is, the electric field of the incident optical signal is split to propagate over two
branches, over which the field experiences different amounts of phase change
due to the electro-optic effect. Then, the optical signals of the two branches
are recombined again, which results in an interference pattern that is
directly related to the phase difference between the two branches. The
amount of phase change over each branch is controlled by electrical voltages.
The transfer function of the MZM can be written as [155]
(2-1)
where is the electric field at the input and the output of the
modulator, respectively.
is the phase difference of the electric fields in the two branches
of the MZM,
is the bias point of operation,
is the so-called α-factor defining the chirping behavior of the MZM.
It is given as [96]
1011
CW
Laser
Random
Bit Source
MZM
Pulse
Generator
10111011
CW
Laser
Random
Bit Source
MZM
Pulse
Generator
Eout t()
Ein t()
------------------- cos ∆Φ t()[]j–αΦ
bias
∆()tan ∆Φ t()[]exp=
Ein out,t()
∆Φ t()
∆Φbias
α
8 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
(2-2)
where is the phase change at the output of the MZM,
is the intensity change at the output of the MZM,
with , speed of light in a vacuum, permeability.
When operating the MZM with , ideal intensity modulation is
achieved for .
2.3 Fiber propagation
With the invention of low-loss optical fibers [78], the telecommunications
industry discovered the optical fiber as medium for efficient information
transfer between locations being several kilometers apart from each other.
Today, a multitude of different fiber types are commercially available, which
offer different signal propagation characteristics. For details, see [44].
In the following section, the main characteristics of signal propagation in
a single-mode optical fiber are presented. Afterwards, an expression for the
propagation of the slowly varying amplitude of the electric field in sin-
gle-mode optical fibers is derived, and a numerical algorithm for solving it is
briefly presented.
2.3.1 Attenuation
The first investigated effect is the fiber attenuation, which describes the
fact that optical signal power decreases exponentially when propagated in
optical fibers. This can be written in logarithmic units as follows
.(2-3)
α2Itd
dϕt() td
dIt()⁄=
dϕdt⁄
dI dt⁄
IE
2cµ⁄=cµ
∆Φbias π4⁄=
α0=
PdBm 0() PdBm L()–αdB km⁄L=
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 9
where is the fiber attenuation in [dB/km],
is the fiber distance in [km],
is the optical power at the input of the fiber in [dBm],
is the optical power after the fiber distance in [dBm].
The fiber attenuation is mainly caused by absorption and scattering pro-
cesses. Absorption arises from impurities and atomic effects in the fiber
glass. Scattering is mainly due to intrinsic refractive index variations of fiber
glass with distance (Rayleigh scattering) and imperfections of the cylindrical
symmetry of the fiber. The usable bandwidth ranges from approximately
800 nm (increased Rayleigh scattering) to approximately 1600 nm (infrared
absorption due to vibrational transitions). In this region, attenuation is
mainly governed by Rayleigh scattering, which scales with λ-4. It can reach
values below 0.2 dB/km at around 1550 nm. For details on contributions to
fiber attenuation in optical fibers, see [54], [91].
Sometimes it is useful to work with attenuation values in linear units. The
following equation defines the relation between linear and logarithmic
expression
. (2-4)
2.3.2 Group-velocity dispersion (GVD)
The second considered propagation effect is the group-velocity dispersion
(GVD). It arises from the frequency dependence of the modal propagation
constant of an optical wave traveling in silica fiber. Expanding in
a Taylor series around an arbitrary frequency gives [4]
(2-5)
αdB km⁄
L
PdBm 0()
PdBm L() L
α10()ln
10
------------------αdB km⁄0.23026αdB km⁄
≈=
βω() βω()
ω0
βω() nω()
ω
c
----
=
β0β1ωω
0
–()
1
2
---β2ωω
0
–()
21
6
---β3ωω
0
–()
3
++ +=
10 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
where is the effective refractive index of the optical fiber,
is the speed of light in a vacuum,
.
The coefficients have the following physical interpretation:
• accounts for a frequency independent phase offset during propagation.
• denotes the inverse of the group velocity , which determines the
speed of energy propagated through the fiber.
• describes the frequency dependence of the inverse of . It defines the
broadening of a pulse due to the fact that its Fourier components propa-
gate with different group velocities1. This effect is known as chromatic dis-
persion or group velocity dispersion (GVD).
• is known as the slope of the GVD, or second order GVD. It accounts for
the frequency dependence of the GVD and, therefore, for different broad-
ening properties of signals or signal portions propagating at different fre-
quencies. It is important to be considered for frequency regions where
is close to zero, or for wideband transmission problems2.
It is commonly of more interest to determine the dependence of the inverse
of the group velocity on wavelength rather than on frequency. This depen-
dence is described by the dispersion parameter and its slope with respect
to wavelength, . The following relationships hold
.(2-6)
is typically measured in units ps/nm-km. It determines the broadening
for a pulse of bandwidth after propagating over a distance , or equiva-
lently, the time offset of two pulses after a distance , which are separated in
the spectral domain by .
1. Equivalently, it defines the different propagation speeds of pulses in frequency separated
channels, and hence, is basis for interchannel pulse collisions in WDM transmission sys-
tems.
2. Such as wideband dispersion compensation in DWDM systems, or estimation of crosstalk
due to stimulated Raman scattering.
nω()
c
βkωk
k
∂
∂βω()
ωω
0
=
=k0123,,,=
βkk0123,,,=,
β0
β1vg
β2vg
β3
β2
D
S
Dd
dλ
-------1
vg
----- 2πc
λ2
----------β2
–== SdD
dλ
-------- 2πc()
2
λ3
----------------- 1
λ
---β3
1
πc
------β2
+
==
β2
d
dω
--------1
vg
----- λ2
2πc
----------– D== β3
dβ2
dω
----------λ3
2πc()
2
----------------- λS2D+()==
D∆T
∆λ z
z
∆λ
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 11
(2-7)
GVD consists mainly of two additive parts, namely intrinsic material dis-
persion (frequency dependence of the refractive index) and waveguide disper-
sion (frequency dependence of the guiding properties of the fiber). Depending
on the manufacturing process and the radial structure of the fiber, fiber
types with various dispersion profiles can be designed. For further informa-
tion see [92].
2.3.3 Kerr effect
The third propagation effect, which is presented here, is the Kerr effect. It
denotes the phenomenon that actually the refractive index of optical fibers
is slightly dependent on the electric field intensity of the optical sig-
nal passing through the fiber.
(2-8)
where is the linear refractive index,
is the nonlinear refractive index.
Note that the electric field intensity varies in time with the transmitted
pulse stream. It induces intensity dependent modulation of the refractive
index, and hence modulation of the phase of the transmitted pulse stream.
This effect is called the Kerr effect.
Compared to other nonlinear media, is very small1. However, optical
fiber is quite an effective nonlinear medium as field intensities of several
milliWatts are focused in the small fiber core of 50 −75 µm2 over interaction
lengths of tens to hundreds of kilometers. A comparison with bulk media
using typical parameters shows that the enhancement factor of nonlinear
processes in single mode fibers is 107−109 [5]. Thus, effects from nonlinear
interaction between signal pulses might accumulate during transmission
and become of system limiting importance.
1. Typically of the order of 10-20 [m2/W].
∆T∆λ λd
dT
≈∆λzλd
d1
vg
-----
∆λzD==
nωt,() I
nωt,()n0ω() n2It()+=
n0
n2
n2
12 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
Kerr nonlinearity accounts for diverse intensity dependent propagation
effects. The most important ones are:
• Self-Phase Modulation (SPM)
• Cross-Phase Modulation (XPM)
• Four Wave Mixing (FWM).
The phenomenological aspects of these effects are discussed in the following
sections.
2.3.3.1 Self- and cross-phase modulation
When sending pulse streams with intensity and initial phase at
different carrier frequencies into the fiber, phase modulation of the signal in
channel m depends on the local power distribution of all channels as follows
(2-9)
where is the phase modulation of channel m,
is the initial phase of channel m,
is the linear refractive index of channel m,
is the nonlinear refractive index,
is an index denoting the neighboring channels of channel m.
The first term in the square brackets on the right-hand side of Eq. (2-9)
corresponds to the accumulated linear phase shift due to transmission.
The second term corresponds to the accumulated nonlinear phase shift
due to self-phase modulation (SPM) in channel m. The SPM-induced phase
shift is proportional to the local electric field intensity. It induces frequency
chirp and spectral broadening, so pulses behave differently in the presence of
GVD. As shown in Chapter 3, p. 32, linear chirp from GVD and nonlinear
chirp from SPM combine and can be used advantageously for pulse propaga-
tion.
The third term describes phase modulation induced by intensity fluctua-
tions in neighboring channels k. This effect is called cross-phase modulation
(XPM). XPM introduces additional nonlinear pulse chirp, which interacts
It() φ
0
φmtz,()φ
0m,
–2π
λ
------ n0m,zn
2zI⋅mt() n2z2⋅Ik
km≠
∑t()++=
φmtz,()
φ0m,
n0m,
n2
k
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 13
with the local dispersion as well. Note that it occurs only over distances,
where pulses overlap. So, XPM effects reduce in general with increased dis-
persion as pulses at different frequencies propagate faster through each
other1. More details on the impact of XPM in WDM transmission is outlined
in Chapter 4, p. 59.
2.3.3.2 Four-wave mixing
When propagating waves at different carrier frequencies over the fiber,
parametric interactions might induce the generation of intermodulation
products at new frequencies. This nonlinear effect is called four-wave mixing
(FWM). It occurs, for instance, when two photons at frequencies and
are absorbed to produce two other photons at frequencies and such
that
. (2-10)
It could also be understood as mixing of three waves producing a fourth one
(2-11)
where is the electric field of the wave
propagating at ,
is the modal propagation constant at ,
is the carrier frequency of the wave .
The energy of the wave is given as superposition of mixing products
of any three waves for which holds2. Note that the propa-
gation constant is frequency dependent. So, efficient interactions only occur
if contributions to given at different times add up over distance. This
so-called phase matching condition can be written as
(2-12)
1. The distance over which pulses propagating in different channels overlap is the so-called
collision length. See also Eq. (2-24).
2. Additionally to FWM between three waves at different frequencies, degenerate FWM
occurs when two of the waves coincide at the same frequency ( ).
ω1ω2
ω3ω4
ω1ω2
+ω3ω4
+=
E
klm EkElEm∗
=
EkElEmjωkωlωm
–+()t{}jβω
k
()βω
l
()βω
m
()–+[]z–{}expexp=
EiEωi
() Eijωitβω
i
()z–[]{}exp==
ωi
βω
i
() ω
i
ωklm ωkωlωm
–+= Eklm
Eklm
ωklm ωkωlωm
–+=
ωkωl
=
Eklm
∆β 0→with ∆β ω() βω
k
()βω
l
()βω
m
()–βω
klm
()–+=
14 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
where describes the phase mismatch between the intermodulating
electric fields. With Eq. (2-5) substituted into Eq. (2-12), is given
with respect to fiber dispersion as
(2-13)
where is the reference frequency of and .1
The power of the newly created wave is proportional to the power of the three
interacting waves2
(2-14)
where is the so-called FWM efficiency taking into account the phase
mismatch . It is given by [84]
(2-15)
where is the fiber attenuation, is the propagation distance.
Figure 2-2 shows the FWM efficiency versus channel spacing for differ-
ent dispersion values after one span of 50 km length. Only the degenerate
case is considered3.
1. The second relation in Eq. (2-13) has been derived using Eq. (2-6).
2. Assuming no pump depletion due to FWM, which is satisfied, if power transfer between
waves is small.
3. The reference frequency is set to .
∆β
∆β
∆β ω() ω
kωm
–()ω
lωm
–()β
2
–β3
ωlωk
+
2
-------------------ω–
+
=
2πcωkωm
–
ω0
---------------------
ωlωm
–
ω0
--------------------
Dωlωk
+
2ω0
-------------------ω
ω0
-------–
2πc
ω0
----------S2D+
+
=
ω0DS
Eklm z()
2ηEkz()Elz()Emz()()
2
∼
η
∆β
ηα2
α2∆β+
--------------------14αz–() ∆βz2⁄()sin 2
exp
1αz–()exp–[]
2
----------------------------------------------------------------+
=
αz
η
ωkωl
=
ω0ωl
=
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 15
Figure 2-2. FWM efficiency η versus channel spacing for different dispersion
values after propagation over one span of 50 km (S = 0.1 ps/nm2-km,
α= 0.25 dB/km); from left to right: D = 17.0, 4.4, 2.3, 0.1 ps/nm-km.
Increasing local dispersion results in an increased walk-off of Fourier compo-
nents and thus, phase mismatch after shorter propagation distances, which
results in steeper decrease of FWM efficiency with channel spacing.
Figure 2-2 shows that for 2.3 ps/nm-km, FWM efficiency is decreased
below -25 dB for 100 GHz channel spacing and still below -20 dB for 75 GHz
channel spacing.
2.3.4 Propagation over nonlinear and dispersive fiber
2.3.4.1 Generalized nonlinear Schrödinger equation
In this section, the propagation equation for the slowly varying amplitude
of the electric field in optical single-mode fibers is presented. Details of the
derivation can be found in [103], [6].
Starting from Maxwell’s equations, the optical field evolution in a dielec-
tric medium can be described by the wave equation as follows
(2-16)
D ≥
E1
c2
-----
t2
2
∂
∂E–∆µ
0t2
2
∂
∂P–0=
16 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
where is the electric field vector,
is the electric polarization vector,
is the speed of light in a vacuum, is the vacuum permeability.
For silica-based optical fibers, can be described via as follows
(2-17)
where is the first order susceptibility, defining the linear evolution
behavior,
is the third order susceptibility, responsible for the nonlinear
propagation characteristics. It is related to the ratio of nonlinear and
linear refractive index via .
Assuming that the fundamental mode of the electric field is linearly polar-
ized in the x or y direction1, its value can be approximately described using
the method of separation of variables by
(2-18)
where is the transversal field distribution,
is the complex field envelope2 describing electric field
evolution in the propagation direction z and time t, with
corresponding to the optical power.
For single-mode fiber, represents the fundamental fiber mode
HE11, approximately given by a Gaussian distribution over the fiber radius.
is determined as a solution of the generalized nonlinear Schrödinger
equation (GNLS), which is given by
(2-19)
1. With z being the propagation direction.
2. Also known as slowly varying amplitude.
E
PP
LPNL
+=
cµ0
P E
Pµ0
c2
------χ1()
Eχ3()
E3
+{}≈
χ1()
χ3()
n2n0
⁄3
8
---Re χ3()
{}=
Exyzt,,,()Re F x y,()Azt,() jω0tβ0z–()[]exp{}=
Fxy,()
Azt,()
A2
Fxy,()
Azt,()
j
z∂
∂Aβ1t∂
∂A+1
2
---+ β2t2
2
∂
∂Aj
1
6
---β3t3
3
∂
∂A–jα
2
--- A+γA2A=
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 17
where is the fiber attenuation in linear units,
are defined in Eq. (2-5),
is the nonlinear coefficient,
with nonlinear refractive index, reference frequency,
speed of light in a vacuum, effective core area of the fiber given
by .
The approximate term for is derived assuming a Gaussian distribution
of ,
(2-20)
where is the effective mode radius.
Eq. (2-19) describes the evolution of the slowly varying field amplitude
over a nonlinear, dispersive fiber. It describes the most important propaga-
tion effects for pulses of widths larger than 5 ps. It can, however, be extended
to include higher order GVD and other nonlinear effects1, which might be of
importance for ultra-short pulse propagation or wide bandwidth applica-
tions. Polarization dependent propagation effects are not considered here, as
they will not be regarded throughout this work.
2.3.4.2 Split-step Fourier method
The generalized nonlinear Schrödinger equation (GNLS), as given in
Eq. (2-19), can not be solved analytically for the general case of arbitrarily
shaped pulses launched into the fiber. However, powerful numerical proce-
dures have been developed over the years to solve it. Among them, the
split-step Fourier method has proven to be the most robust technique [83]. It
is based on the principle that linear and nonlinear propagation effects can be
considered separately from each other over short fiber distances .
1. Such as stimulated Raman scattering and pulse self-steepening.
α
βii,123,,=
γn2ω0
()cAeff
()⁄=
n2ω0
cA
eff
Aeff
Fxy,()
2xdyd
∞–
∞
∫
∞–
∞
∫
2
Fxy,()
4xdyd
∞–
∞
∫
∞–
∞
∫
----------------------------------------------------------------- πρm
2
≈=
Aeff
Fxy,()
Fxy,() x2y2
+()
ρm
2
-----------------------–exp=
ρm
∆z
18 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
(2-21)
where is the linear operator, accounting for fiber
dispersion and attenuation.
is the nonlinear operator accounting for the Kerr
nonlinearity.
If , the so-called split-step size, becomes too large the condition for separa-
ble calculation of and breaks, and the algorithm delivers wrong results.
So careful determination of the optimum split-step size is of importance in
order to use minimal computational effort for a given accuracy. Typically,
is adaptively adjusted, for instance, according to
(2-22)
where is the maximum acceptable phase shift due to the nonlinear
operator1,
is the nonlinear scale length2,
is a maximum split-step size, which is set to limit spurious
FWM tones [22].
The linear operator is most efficiently solved in the spectral domain,
while the nonlinear operator is more favorably solved in the time domain.
Assuming a discrete signal description in the time and frequency domain,
the Fast Fourier Transform (FFT) is used for converting between both [129].
As the speed of the FFT is proportional to Nlog2N, where N is the number of
signal samples in the time or frequency domain, careful determination of the
simulation bandwidth and the time window is important for minimizing
computational effort given specific accuracy constraints3.
1. Typically in the range of 0.05 −0.2rad.
2. See also Eq. (2-26).
3. See also Chapter 4, p. 61 for a numerical effort estimation.
Az ∆z+()Az() ∆zN
ˆ
()exp{}∆zD
ˆ
()exp=
D
ˆj1
2
---β2t2
2
∂
∂1
6
---β3t3
3
∂
∂α
2
---–+=
N
ˆjγ–A2
=
∆z
D
ˆ
N
ˆ
∆z
∆zmin ∆φNLLNL ∆zmax
,{}=
∆φNL
LNL 1γA2
⁄=
∆zmax
D
ˆ
N
ˆ
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 19
2.3.4.3 Characteristic scale distances
Certain characteristics of pulse evolution over optical fiber are governed
by several scale lengths, which are described in the following:
• The effective length is defined as the equivalent fiber interaction
length with respect to constant power [29]. Thus, it is derived from the ex-
ponential power decay in optical fiber as follows
. (2-23)
It is important when, for instance, rescaling the pulse evolution to account
for attenuation and periodic amplification.
• The walk-off length is defined as the distance it takes for one pulse
traveling at frequency to overtake another pulse traveling at frequency
(2-24)
where , and is the pulse duration (half-width 1/e-intensity)1.
It is also called the collision length , as it accounts for the distance
where two pulses at different frequencies collide during propagation.
Thus, it is of importance when determining XPM effects.
• The dispersion length defines the distance over which a chirp-free
Gaussian pulse broadens by a factor of due to GVD
. (2-25)
It denotes the distance where dispersive effects become important.
• The nonlinear length defines the distance over which the phase
change due to the Kerr nonlinearity becomes one rad
. (2-26)
1. Measured as the width from the pulse center to the point where the intensity level dropped
to 1/e of the maximum intensity level.
Leff
Leff αz1
–()exp z1
d
0
z
∫1αz–()exp–
α
----------------------------------==
Lw
ω1
ω2
Lw
T0
β1ω2
()β
1ω1
()–
----------------------------------------------T0
D∆λ12
--------------------
≈=
β1vg
1–
=T0
Lc
LD
2
LD
T0
2
β2
---------=
LNL
LNL
1
γA2
------------=
20 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
It denotes the distance where nonlinear effects become important.
The ratio between and describes the dominating behavior for
pulse evolution over optical fiber. Of special interest is the region
, where cancellation of nonlinear and dispersive effects can be
observed for certain parameter settings (see Chapter 3, p. 32).
Note also that solutions of the GNLS, as given in Eq. (2-19), remain
invariant from certain scale transformations. When dividing, for instance,
both sides of Eq. (2-19) by an arbitrary scaling factor , a set of new parame-
ters can be defined, for which the solution of Eq. (2-19) remains invariant.
These new parameters are related to the old ones by [106]
. (2-27)
So when, for instance, the bitrate is increased from 10 Gbit/s to 40 Gbit/s,
and thus, the pulse width is reduced to one fourth, p is calculated to 0.25.
With the decreased pulse width, sensitive scale lengths for dispersion and
nonlinearity reduce also to one fourth. This implies that dispersion map
lengths as well as average and local dispersion values need to be adjusted,
and nonlinear propagation effects become four times more important than
for 10 Gbit/s. The latter could eventually be compensated by reducing the
pulse power. This however, would also reduce the signal-to-noise ratio1, and
thus, give rise to other distortions.
2.4 Erbium-doped fiber amplifier (EDFA)
With the invention of Erbium-doped fiber amplifiers (EDFA) in the late
1980s [109], the development of fiber-optic communication systems acceler-
ated rapidly. Electro-optic repeaters could be replaced by the more robust,
flexible and cost-efficient EDFAs, allowing all-optic links over transoceanic
distances in the mid 1990s.
1. See Chapter 2, p. 24, and Chapter 6, p. 100 for details.
LDLNL
LDLNL
∼
p
t
˜pt⋅=z
˜pz⋅=β2
˜pβ2
⋅=γ
˜1
p
---γ⋅=α
˜1
p
---α⋅=
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 21
Apart from the optical fiber itself, the EDFA is the most important compo-
nent of long-haul transmission in determining system performance. Depend-
ing on the application, different design criteria are of importance. The main
amplifier characteristics are large gain, low noise figure, gain flatness, and
large output power. In the following, the main concepts of EDFAs are pre-
sented. A detailed discussion on the design of EDFAs is given in [35], [15].
2.4.1 Erbium ions in glass hosts
When doping silica fiber with Er3+ ions (or with ions of other rare-earth
elements), the fiber can be operated as an active laser medium when
pumped. Figure 2-3 shows the energy diagram of Erbium ions in glass hosts.
Figure 2-3. Energy diagram of Er3+-ions in glass hosts.
For 980 nm pumping, carriers are absorbed from ground level to the third
laser level. Because of phonon relaxation within one microsecond, they tran-
sit down to the second level almost immediately1, where they distribute due
to thermalization processes. For 1480 nm pumping, carriers are absorbed
from the ground level directly to the second laser level, where they again dis-
tribute due to thermalization processes. Thus, it is a reasonable assumption
to model gain and noise behavior in EDFAs using only a two-level laser
medium.
1. Compared to the lifetime at the second stage .
stimulated
emission
phonon
relaxation
4I11/2
4I13/2
4I15/2
energy states
τ ~ 1µs
spontaneous
emission
thermalization
τ ~ 10 ms
1
2
3’
3
equivalent
laser levels
1480-nm
pump
980-nm
pump
absorption stimulated
emission
phonon
relaxation
4I11/2
4I13/2
4I15/2
energy states
τ ~ 1µs
spontaneous
emission
thermalization
τ ~ 10 ms
1
2
3’
3
equivalent
laser levels
1480-nm
pump
980-nm
pump
absorption stimulated
emission
phonon
relaxation
4I11/2
4I13/2
4I15/2
energy states
τ ~ 1µs
spontaneous
emission
thermalization
τ ~ 10 ms
1
2
3’
3
equivalent
laser levels
1480-nm
pump
980-nm
pump
absorption
τ10 ms∼()
22 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
From the upper level, carriers can transit down to the ground level via
spontaneous emission, or via stimulated emission for the case where signal
energy is co-launched in the wavelength range 1530 nm to 1600 nm. Stimu-
lated emission provides signal gain; spontaneous emission is detected as
noise.
2.4.2 Amplifier gain
The gain in EDFAs is strongly dependent on the carrier inversion, i.e, the
amount of carrier population in the upper state compared to the total num-
ber of carriers. It is determined by the pumping scheme, and co-dopants such
as germanium and alumina. The wavelength dependence of local gain can be
written as [36]
(2-28)
where , denote the carrier populations of the upper and lower states,
respectively,
, denote the absorption and emission cross-sections,
respectively,
is the overlap factor, i.e., the area of overlap between Erbium
ions and the optical signal mode in the fiber.
Note that the population inversion shows a strong local dependence, so
gain may differ over the length of the doped fiber. Figure 2-4 shows the
wavelength dependence of the gain for the two pump wavelength regions and
the signal bandwidth around 1550 nm.
gλ() N2σeλ() N1σaλ()–[]Γλ()=
N2N1
σaλ() σ
eλ()
Γλ()
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 23
Figure 2-4. Exemplary gain/loss profile for various values of population
inversion.
The left diagram shows the gain as a function of population inversion at
980 nm (pump) and neighborhood. Perfect inversion is possible as the emis-
sion cross-section is zero at 980 nm. This results in small noise generation
for 980 nm pumped EDFA configurations. The right diagram in Figure 2-4
shows gain as a function of population inversion at 1480 nm (pump) and
around 1550 nm (signal band). For the depicted inversion profile, the trans-
parency point (g = 0 dB/m) for 1480 nm pumping is at about 75% population
inversion. So, perfect inversion is not possible, which results in increased
noise generation. The figure depicts the strong wavelength dependence of the
gain, which results from the wavelength dependence of population inversion.
For large inversion levels, a high gain peak is observed at 1530 nm; for lower
inversion levels, the gain decreases at 1530 nm, but increases at 1550 nm,
eventually delivering a flat gain over several nanometers.
Ignoring noise for the moment, the average signal power dependence on
doped fiber length is given by
. (2-29)
For small signal power values, the total amplifier gain is usually indepen-
dent of the incoming power. If the power launched into the doped fiber is
increased over a certain level, stimulated recombination starts to affect car-
rier population inversion. Every photon created by stimulated emission
transfers one ion from the upper state to the lower state. This results in gain
reduction until absorption of pump power and stimulated plus spontaneous
1.48 1.52 1.56 1.6
-4
-2
0
2
4
λ
0.8
0.6
0.4
0.2
0.0
1.0
980 nm region 1550 nm region
pump
1000
970 980 990
-4
-3
-2
-1
0
[dB/m]
N2/ (N1 + N2)
0.0
0.2
0.4
0.6
0.8
1.0
λ
g
[nm]
[dB/m]
g
[nm]
N2/ (N1 + N2)
1.48 1.52 1.56 1.6
-4
-2
0
2
4
-4
-2
0
2
4
λ
0.8
0.6
0.4
0.2
0.0
1.0
980 nm region 1550 nm region
pump
1000
970 980 990
-4
-3
-2
-1
0
[dB/m]
N2/ (N1 + N2)
0.0
0.2
0.4
0.6
0.8
1.0
λ
g
[nm]
[dB/m]
g
[nm]
N2/ (N1 + N2)
zd
dPgλP,()P=
24 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
recombinations are balancing each other. This effect is known as gain satu-
ration.
The saturation characteristic of EDFAs is very complex, as it is locally
dependent. Assuming a homogenous power distribution along the EDFA, the
gain is given by [80]
(2-30)
where is the gain of the amplifier, given by with
the length of doped fiber,
is the small-signal or linear gain of the amplifier,
is the saturation power, defined as input power for which
=0.5.
For long-haul applications, inline amplifiers are typically operated in satura-
tion to avoid gain fluctuations due to incoming power fluctuations.
2.4.3 Amplified spontaneous emission (ASE) noise
Spontaneous transitions of photons from the upper state to the ground
state add up along the doped fiber and stimulate other transitions, which
results in self-amplification. This effect is called amplified spontaneous emis-
sion (ASE). ASE-noise evolution propagates bidirectionally along the fiber.
So it builds up in the forward and backward directions. The amount of
ASE-noise created at each end of the doped fiber depends on the local popula-
tion inversion.
The ASE-noise can be approximated by a white, Gaussian random process.
The power spectral density (PSD) of the ASE-noise in the x-polarization at
the amplifier output can be written as
(2-31)
P
Psat
-----------
input
0.7213G02–
G1–
---------------- ln G0
G
-------
⋅=
GGPL
Amp
()P0()⁄=LAmp
G0
Psat input
GG
0
⁄
SASE x,
PASE x,
∆f
--------------------nsp G1–()hf==
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 25
where is the spontaneous emission factor1,
is total amplifier gain, is the photon energy,
is the ASE-noise power measured over a bandwidth .
As the ASE-noise power is proportional to the gain and , it can be limited
when operating the EDFA at high population inversion.
The noise performance of amplifiers is usually characterized by the noise
figure NF. It is defined as degradation of the electrical signal-to-noise ratio
(SNR) due to the amplifier, measured with an ideal photodetector2 [34].
(2-32)
The approximation relation in Eq. (2-32) is derived for low noise amplifiers,
neglecting ASE-ASE beat noise [93]. The NF is typically given in dB. For
amplifiers with large gain, the minimum NF is 3 dB, as . With
Eq. (2-31) and Eq. (2-32), the noise power spectral density can be written as
function of NF and gain; both values are measurable from the outside of the
amplifier
. (2-33)
2.4.4 Design aspects
Figure 2-5 shows a typical architecture for a two-stage EDFA. The first
stage operates as a low-noise pre-amplifier, preferably pumped at 980 nm to
ensure small NF3. The second stage operates as a power amplifier, prefera-
bly pumped at 1480 nm as this concept provides higher power conversion
efficiency. Generally, the doped fiber can be pumped from either side. The
isolator between both stages prevents saturation of the first stage due to
1. Also called population inversion factor, with 1.0, typically 1.4 −2.0.
2. No thermal noise, no dark current, 100% quantum efficiency.
3. According to the chain rule, the NF of an amplifier cascade is mainly determined by the NF
of the first amplifier in the chain [31].
nsp
N2σeλ()
N2σeλ() N1σaλ()–
----------------------------------------------------=
nsp ≥
Ghf
PASE x,∆f
nsp
NF SNRin
SNRout
----------------------=
2nsp
G1–()
G
------------------ 1
G
----+≈
nsp 1.0≥
SASE x,
PASE x,
∆f
--------------------1
2
---NF G⋅1–()hf==
26 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
backward propagating ASE-noise from the second stage. Also, the filters are
used to suppress ASE-noise outside the signal bandwidth and perform gain
equalization.
Figure 2-5. Two-stage EDFA design.
EDFA designs as depicted in Figure 2-5 have a NF of 4 −6dB and can
provide flat gain over a bandwidth of about 30 nm. More sophisticated
designs achieving gain flatness over 70 nm can be developed using more than
two amplifier stages or other co-dopings [158]. For transoceanic fiber links,
the requirements for gain flatness and low NF are very high as up to 200
EDFAs are cascaded over such links. Also, robustness is of great importance,
as each failure may cause severe repair costs. Thus, redundant pump config-
urations are usually used for inline EDFAs in submarine links.
2.5 Receiver
As considered in this work, the receiver performs the optical-electrical sig-
nal conversion, eventually some signal enhancement features, and finally
the decision about the transmitted bit stream.
2.5.1 Architecture
Figure 2-6 shows the schematic of an intensity-modulation direct detec-
tion (IM-DD) receiver. The received electric field of the optical signal is first
Isolator
Mux
980 nm pump 1480 nm pump
Filter
EDF1 EDF2
Isolator
Mux
980 nm pump 1480 nm pump
Filter
EDF1 EDF2
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 27
pre-amplified by an EDFA with low noise figure1 in order to reduce noise
limiting effects of the following electrical components. It is followed by an
optical filter, which rejects noise outside the signal bandwidth. In case of
WDM transmission, the filter serves as demultiplexer as well, rejecting the
signal power of the neighboring WDM channels. Then, the optical power is
detected by a photodiode. It is translated into an electrical current resem-
bling the same time-characteristics as the incident optical power. The photo-
diode is assumed to inhabit an electrical load resistor and an electrical
amplifier. It is typically followed by an electrical filter, which performs pulse
shaping and further noise reduction. Finally, the signal is passed to the deci-
sion circuitry consisting of a clock recovery and a decision gate.
Figure 2-6. Schematic IM-DD receiver.
Typically two types of photodiodes are used for detection, namely PIN and
Avalanche photodiodes. Details on both can be found in [55]. Throughout this
work, the IM-DD receiver is assumed to be based on a PIN photodiode. The
received current after the PIN photodiode can be written as
(2-34)
where is the quantum efficiency of the photodiode,
is the electron charge, is the photon energy,
is the electric field amplitude of the optical signal wave,
is an additive noise term summarizing sources of noise.
1. See Chapter 2, p. 24 for details on design issues.
Pre-
Amplifier
Optical
Filter
Electrical
Filter
Photo-
diode
Clock
Recovery
DeciderPre-
Amplifier
Optical
Filter
Electrical
Filter
Photo-
diode
Clock
Recovery
Decider
it() ηq
hf
-------Est()
2
⋅Nt()+=
η
qhf
Est()
Nt()
28 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
Ignoring the noise term first, note that the electrical power is propor-
tionally related to the square of the optical signal power. The proportion con-
stant is the so-called responsivity of the photodiode, defined as
. (2-35)
2.5.2 Noise contributions
In the following section, the noise term in Eq. (2-34) is investigated in
more detail. Details to receiver noise can be found in [7].
Shot noise
Shot noise arises from the fact that electric current is not continuous, but
consists of discrete electrons, generated randomly by the photodiode in
response to the incident optical power. Shot noise is actually described by a
Poisson process [39], but is well approximated in practice by a Gaussian
probability density function (PDF). Its variance is given by
(2-36)
where is the incident optical power,
is the dark current of the photodiode,
is the effective noise bandwidth of the receiver considering the
limited bandwidth of photodiode and the electrical filter.
Thermal noise
Thermal noise arises from random motions of electrons for non-zero tem-
peratures [56]. The induced random current fluctuations can be well approx-
imated by a Gaussian PDF with variance
(2-37)
where is the Boltzmann’s constant, is the absolute temperature,
is the electrical load resistor of the photodiode,
is the noise figure of the electrical amplifier in the photodiode.
Nt()
Rηq
hf
-------= 0R1≤≤
Nt()
σsh
22qRP
oId
+()∆fel
⋅=
Po
Id
∆fel
σth
24kBTR
l
⁄()NF ∆fel
⋅=
kBT
Rl
NF
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 29
ASE-noise beating
Additionally to electrical receiver noise sources, optical noise that falls in
the same frequency band as the optical signal passes the optical filter and is
detected by the photodiode. There are mainly two beating terms of interest,
namely beating of ASE-noise with itself and with the electric field of the opti-
cal signal. In a first approximation, these beating terms can also be assumed
to follow a Gaussian distribution. More details are given in Chapter 6,
p. 108. The variance of the signal-ASE beat noise is given by [86]
(2-38)
where is the detected electrical signal power level without considering
ASE-noise,
is the power spectral density of the ASE-noise,
is the transfer function of the electrical filter.
The variance of the ASE-ASE beat noise can be written as
(2-39)
where ⊗ denotes convolution.
σsignal ASE,
22R2PSSASE ff
s
–()SASE ff
s
+()+[]HT
2f() fd
0
∞
∫
=
PS
SASE
HT
σASE ASE,
24R2SASE f() SASE f()⊗[]HT
2f() fd
0
∞
∫
=
30 A. Richter - Timing Jitter in WDM RZ Systems
Long-haul WDM transmission systems
A. Richter - Timing Jitter in WDM RZ Systems 31
Chapter 3
Characteristics of RZ pulse
propagation
3.1 Overview
The first long-haul transmission systems employing RZ modulation
schemes were demonstrated for classical soliton systems in 1988 using
Raman amplification over 4,000 km [115], and in 1990 using EDFAs over
10,000 km [116]. The experimental proof was given: RZ modulation can be
utilized for long-haul transmission. However, further development of system
concepts and components needed to be achieved before the first massive
WDM (7x10 Gbit/s) transmission experiment over transoceanic distances
could be performed [120] in 1996. Again, solitons were used, which were at
that time the most promising RZ modulation scheme to cover long-haul dis-
tances.
However, the first commercial long-haul system utilizing a RZ modulation
scheme was built in 1998, using chirped RZ (CRZ) pulses. Contrary to classi-
cal soliton systems, which are based on balancing linear and nonlinear
effects, the CRZ modulation scheme allows propagation in a quasi-linear
regime. It turned out that dense WDM systems using classical solitons were
difficult to design. This was mainly due to nonlinear channel interactions
raised by the fact that the local dispersion values need to stay small. In the
mid 1990s, the first dispersion-managed soliton (DMS) systems were experi-
mentally demonstrated. The results showed that the soliton concept could be
applied for dispersion-managed fiber links, which in effect triggered inten-
32 A. Richter - Timing Jitter in WDM RZ Systems
Characteristics of RZ pulse propagation
sive research on how to design dense WDM systems using the newly found
DMS format.
This chapter presents, firstly, the evolution of dispersion-managed soli-
tons from classical soliton theory. DMS characteristics are analyzed. Then,
the CRZ modulation format is discussed. Finally, an outlook on the latest
trends of RZ modulation formats is presented.
3.2 Dispersion-managed soliton (DMS)
3.2.1 Evolution from classical soliton theory
3.2.1.1 Lossless fiber
Considering the general pulse evolution equation presented in Chapter 2,
p. 15, linear propagation effects and fiber attenuation are balancing the
intensity dependence of the refractive index1. Ignoring fiber loss (α=0), and
introducing normalized units, the GNLS in Eq. (2-19) can be written as
(3-1)
where is the normalized field amplitude given by , with
peak power at fiber input,
is the normalized transmission distance given by ,
is the dispersion length, and is nonlinear length2,
is the normalized, retarded time given by
is the signum function.
Using the inverse scattering method [69], Eq. (3-1) can be solved analytically
for a launched pulse shape, which satisfies Eq. (3-1) at any distance point.
1. Introduced as Kerr nonlinearity, see Chapter 2, p. 11.
2. See Chapter 2, p. 19 for definitions.
j
Z∂
∂qLD
LNL
----------- q2q1
2
---sgn β2
()
T2
2
∂
∂q–=
qqAP
0
⁄=P0
ZZzL
D
⁄=
LDLNL
TTtβ1z–()LDβ2
⁄=
sgn x()
Characteristics of RZ pulse propagation
A. Richter - Timing Jitter in WDM RZ Systems 33
Among others, so-called soliton solutions exist in the anomalous dispersion
regime ( ), which satisfy the criteria
. (3-2)
Solitons in optical fiber transmissions where first predicted theoretically
in 1973 [64] and observed experimentally in 1980 [114]. Since then, they
have been widely used to investigate and explain fundamental pulse propa-
gation characteristics. Of special interest is the fundamental soliton solution,
as its pulse shape is not altered during propagation. It is given for the case
to
. (3-3)
Only the phase of the fundamental soliton is undergoing a circular change
with period . Other, higher order soliton solutions of interest for
fiber-optic communications are those of the same initial pulse shape as the
fundamental one
. (3-4)
Note that these solutions change their shape periodically during propaga-
tion, recovering their original shape after soliton periods of length
. (3-5)
At the fiber input, the peak power and width of the soliton are related by
Eq. (3-2). Using definitions for dispersion and nonlinear lengths, as intro-
duced in Chapter 2, p. 19, one gets
(3-6)
where is the pulse duration (half-width 1/e-intensity)1
is the FWHM pulse duration2.
1. Measured as the width from the pulse center to the point where the intensity level dropped
to 1/e of the maximum intensity level.
2. Measured as the width between the two points left and right of the pulse center where the
intensity level dropped to 1/2 of the maximum intensity level (FWHM: full-width half max-
imum).
β20<
Ns
LD
LNL
----------- 12…,,==
LDLNL
=
qZT,() T() jZ 2⁄()expsech=
zπLD
=
q0T,()NsT()sech=Ns23…,,=
LSπ2⁄()LD
=
P0Ns
2β2
γT0
2
--------- Ns
23.107 β2
γTF
2
----------
≈=Ns12…,,=
T0
TF
34 A. Richter - Timing Jitter in WDM RZ Systems
Characteristics of RZ pulse propagation
Figure 3-1 shows the pulse shape evolution of the fundamental soliton in
comparison with a Gaussian pulse. For both pulses, the peak power is
7.53 mW and the FWHM duration is 20 ps. It is nice to see how the funda-
mental soliton retains its shape, while the Gaussian pulse shape is changing
rapidly as it tries to balance linear (pulse-spreading) and nonlinear
(pulse-compressing) forces.
Figure 3-1. Evolution of sech (a) and Gaussian (b) shaped pulses
of equal FWHM duration, (no fiber attenuation).
Soliton interactions
Solitons are derived from Eq. (3-1) as a stable solution for single pulse
propagation. Thus, interactions with pulses in neighboring bit intervals
might lead to perturbations, which could destroy the soliton characteristics if
their impact becomes too large. Considering two solitons of equal amplitude
and phase, they will periodically collapse at distance intervals of [52]
(3-7)
time [ps]
distance
100
0
[km]
0
10,000
distance
[km]
time [ps] 100
00
10,000
(a)
(b)
LCollapse
π
2
---LD∆T0
()exp=
Characteristics of RZ pulse propagation
A. Richter - Timing Jitter in WDM RZ Systems 35
where is the dispersion length,
is the initial pulse separation, with bit duration,
and pulse duration (half-width 1/e-intensity),
For transmission distances shorter than , no collapsing occurs. This
constraint would limit initial pulse separation to assuming trans-
oceanic transmission over up to 15,000 km. However, it can be reduced to
about for the case where the two collapsing solitons are not of equal
amplitude and phase (which is the case in most realistic scenarios) due to, for
example, random initial chirp from the transmitter, second order GVD, and
random amplifier noise [71], [85]. More information on intrachannel interac-
tions is given in Chapter 4, p. 47.
3.2.1.2 Periodically amplified fiber link
A remarkable property of solitons is their robustness against small pertur-
bations. Therefore, it is generally not of great importance to launch an
exactly sech-shaped pulse with a correct power-width ratio. Small deviations
will be repaired by re-shaping, yielding a pulse with slightly different pulse
width, and energy shedding during propagation [68]. In the very late 1980s,
the first soliton experiments with EDFAs were performed [115], [125]. They
showed that solitons can propagate stably in lossy fiber with periodically
spaced amplifiers as long as the amplifier spacing is small enough.
Considering the normalized GNLS as given in Eq. (3-1) for the anomalous
dispersion regime and , fiber loss and amplification can be incorpo-
rated as follows
(3-8)
where is the fiber loss, over one dispersion length ,
is the amplifier gain as function of distance.
The term on the right-hand side of Eq. (3-8) represents a perturbation to the
classical soliton propagation. The application of perturbation theory [70]
showed that as long as its influence is small, the soliton will still develop.
LD
∆T0Tb2T0
()⁄=Tb
T0
LCollapse
∆T010≥
∆T04≥
LDLNL
=
j–Z∂
∂q1
2
---
T2
2
∂
∂qq
2q++ jα
2
--- LDGLD
–
q=
α
2
--- LDLD
GZ()
36 A. Richter - Timing Jitter in WDM RZ Systems
Characteristics of RZ pulse propagation
However, this requirement limits lumped amplifier spacings to about
20 −30 km or the usage of distributed amplification1.
Average soliton regime
In the early 1990s, it was discovered that solitons exist even for propaga-
tion scenarios where . The pulse evolution does not follow the classical
soliton regime as pulses periodically changed their time and frequency
shape. However, the same stable pulse shape could be observed in average
over several amplifier spans. As the pulse evolution is governed by the aver-
age soliton energy, these pulses are called guiding center or average solitons.
In this propagation regime, restrictions to amplifier spacing can be
relaxed. Stable pulse propagation occurs as long as [65]. However,
the initial peak power needs to be increased with respect to Eq. (3-6) to
accommodate for the fiber loss. The so-called pre-emphasis [97] assures that
average pulse energy over one amplifier span equals the energy of the funda-
mental soliton. This is, assuming that there will be no changes of pulse width
during propagation over one span, this means that
(3-9)
where is the effective fiber length, as defined in Eq. (2-23),
is the amplifier spacing.
When the pulse power is not enhanced, the average soliton is not able to
focus itself. The resonance effect between the average soliton length and
amplifier span length is also of interest. With Eq. (3-5), resonances occur
at [67]
. (3-10)
An average soliton emitted under such conditions, will emit dispersive waves
that adjust amplitude and width until it eventually matches another soliton
solution, for which Eq. (3-10) does not hold.
1. The usage of Raman amplification was considered in the very early investigations of soli-
tons and has become of great interest during the past two years again.
Γ1»
LA
LALD
<
P
˜0
LA
Leff
----------P0
=
Leff
LA
LS
LA
LA4πkLD
=k123…,,,=
Characteristics of RZ pulse propagation
A. Richter - Timing Jitter in WDM RZ Systems 37
3.2.1.3 Dispersion-managed, lossy fiber link
Very soon after the average soliton was found for periodically amplified
systems, it was discovered that periodic dispersion compensation has a simi-
lar effect on pulse dynamics [66]. In 1995, it has been shown that the
so-called dispersion-managed soliton (DMS) can be propagated over
long-haul distances [146]. Inside each dispersion map, the characteristics of
DMS evolution is governed by local dispersion values. Thus, the DMS
changes its width inside each dispersion map in a periodic fashion; it
’breathes’ with the local dispersion. Average dispersion and fiber nonlinear-
ity support the pulse behavior on average over several dispersion maps1. Sta-
ble propagation of DMSs is possible, as long as the dispersion map length is
small compared to [137].
The advantage of DMS systems compared to classical soliton systems is
the possibility to utilize larger local dispersion values, which results in
increased robustness against disturbing fiber nonlinear effects (such as
FWM, XPM) and timing jitter due to ASE-noise2. With the invention of
DMSs, the door was open to WDM applications featuring competing channel
spacings. It allowed the usage of soliton propagation characteristics to be
applied for high-capacity long-haul system applications.
3.2.2 Pulse dynamics of DMS
Design of initial pulse power and width is critical for successful propaga-
tion of DMS. One important parameter determining the DMS behavior is the
dispersion map strength , given as [139]
(3-11)
1. See also Figure 3-2, p. 40.
2. See Chapter 4, p. 46 for details.
LNL
S
Sλ2
2πc
----------D1Dave
–()L1D2Dave
–()L2
–
T0
2
-----------------------------------------------------------------------------------
=
38 A. Richter - Timing Jitter in WDM RZ Systems
Characteristics of RZ pulse propagation
where , are the dispersion coefficients, and , are the lengths of
the anomalous and normal dispersion spans, respectively,
is the average dispersion of the dispersion map.
determines for instance the ratio of minimum and maximum pulse width
over a single dispersion map [38]
. (3-12)
Utilizing the advantages of DMS propagation in the anomalous dispersion
regime on the one hand, and avoiding interactions with neighboring pulses
on the other hand, the dispersion map strength should be in the range of
[21].
Also, the energy scaling with respect to classical soliton propagation can
be derived as function of . Several approaches have been published, for
proper energy scaling [138], [159], [162], [163]. Using so-called second order
moment analysis, the ratio between DMS energy and energy of the
classical soliton can be determined for a two-section dispersion map as
[16]
. (3-13)
The increase of power for DMS systems compared to classical soliton systems
using pulses of comparable duration results in an increased optical sig-
nal-to-noise ratio (OSNR)1 without the need to increase the average disper-
sion. This is advantageously as it allows higher robustness against
ASE-noise induced timing jitter and amplitude fluctuations2.
Dispersion-managed solitons are not sech-shaped anymore; they tend to
be more Gaussian-shaped. In general, it is difficult to determine the proper
pulse shape, width and power for a DMS, as these parameters depend
strongly on the applied dispersion map and amplifier positioning. There are
1. See Chapter 6, p. 100 for definition.
2. See Chapter 4, p. 46 for further explanations.
D1D2L1L2
Dave
D1L1D2L2
+
L1L2
+
------------------------------------=
S
TFmax,
TFmin,
--------------------1S2
2
------+=
4S10≤≤
S
EDMS
ES
EDMS
ES
---------------- 1.18 11.92S2
+
11ε
1ε–
----------- 1.92S2
++
2
---------------------------------------------------------------
≈with εDave
D1
-------------=
Characteristics of RZ pulse propagation
A. Richter - Timing Jitter in WDM RZ Systems 39
three main rules for launching the proper DMS into a system, and thus avoid
energy shedding throughout the propagation [150]. Firstly, pulses should be
launched with the proper frequency chirping at the correct points into the
dispersion map. Secondly, the energy of the launched pulse should match the
energy of the true DMS solution for the particular dispersion map. Thirdly,
the pulse shape should be close to the true DMS solution to reduce oscilla-
tions of pulse width and power. A method for finding optimal dispersion
maps including chirp-free points is presented in [160].
Using the so-called variational approach [10], pulse dynamics based on the
Gaussian ansatz can be derived, that is, assuming that the DMS shape is
Gaussian with variable amplitude, width, chirp and phase [58]. In [152], the
so-called path-average mapping method is applied to derive an analytical
expression for the transfer function1 of a single section of a periodically cas-
caded DMS system. It is based on the assumption that nonlinear effects are
small over one span length, such that the quasi-linear evolution of main
pulse parameters can be assumed. This method is of advantage when the dis-
persion span length is much larger than the amplifier spacing and a low
power signal is propagated. In the path-average model, the complex envelope
of DMS can be described using an orthogonal set of chirped Gauss-Hermite
functions [150].
However, it is not important that the launched pulse reflects exactly the
true GNLS solution of the considered dispersion-managed fiber link. When
selecting the shape of the launch pulse, it should have steeply falling tails.
The extinction ratio should be at least 15 dB for 10 Gbit/s, and 20 dB for
20 Gbit/s.2 It is not essential that the DMS is launched with the correct pulse
width [62].
Figure 3-2 shows typical DMS pulse evolution over a symmetric dispersion
map, e.g., anomalous and normal dispersion fiber spans are of equal
lengths3. Here the DMS is found to be approximately of Gaussian shape with
16.74 ps FWHM duration and 2.8 mW peak power at the middle of the nor-
mal dispersion fiber, where it is launched. Note that no chirp is added.
1. Typically a nonlinear, integral operator.
2. Measured 50 ps away of the pulse center.
3. Details of the dispersion map are listed in Table 4-1, p. 57.
40 A. Richter - Timing Jitter in WDM RZ Systems
Characteristics of RZ pulse propagation
Figure 3-2. Evolution of dispersion-managed soliton. (a) over 10,000 km with
snap shots after each dispersion map (200 km), (b) over one dispersion map of
200 km, (no fiber attenuation). Details of the dispersion map are listed in
Table 4-1, p. 57.
The upper graph in Figure 3-2 shows stable evolution over 10,000 km,
where snap shots are taken after each dispersion map. Slight modifications
of pulse peak power and width are recognizable over the distance, which
results from the fact that the launched Gaussian pulse is not the true solu-
tion of the propagation equation. The lower graph shows pulse dynamics
inside the map, where pulse ’breathing’ is recognizable. The DMS pulse
width is smallest in the middle of the spans, and widest at the edges between
anomalous and normal dispersion fiber spans, where chirp is broadening the
spectrum as well.
time [ps]
distance
100
0
[km]
0
10,000
distance
[km]
time [ps] 100
00
200
(a)
(b)
Characteristics of RZ pulse propagation
A. Richter - Timing Jitter in WDM RZ Systems 41
3.3 Chirped return-to-zero (CRZ)
The first successful RZ propagation over long-haul distances employing no
soliton technique was first proposed in [19] using the so-called chirped
return-to-zero (CRZ) modulation format. CRZ pulses are typically following a
raised-cosine shape with a superimposed sinusoidal phase modulation [51].
(3-14)
where is the maximum amplitude, is the bit duration,
is the modulation depth of phase modulation.
The FWHM width is approximately 33% of the bit duration. For comparison,
the FWHM width of a typical DMS pulse is approximately 15 −20% of the bit
duration.
The main difference of CRZ and DMS systems lies in the applied disper-
sion maps. While dispersion maps of DMS systems have originally been
designed such that pulses do not spread outside the bit duration, CRZ sys-
tems used from their first implementation dispersion maps where pulses
accumulate large values of dispersion before they are compensated again.
Strong pulse to pulse interactions are the result. However, the initial pulse
peak power and width are selected such that these interactions are of linear
nature and could easily be reversed by proper dispersion compensation.
While DMS dispersion maps have a length of 100 −200 km with
2−4 amplifiers in between, CRZ systems employ dispersion maps of length
250 −550 km with 5 −11 amplifiers in between. The CRZ pulse shape is usu-
ally totally destroyed over the dispersion map length, recovering only at its
end again. In the presence of non-negligible dispersion slope, pulses might be
detectable only when channel by channel compensation for the dispersion
slope is applied at the transmitter and receiver ends [51].
Figure 3-3 shows pulse evolution of a CRZ over a mostly anomalous dis-
persion map1.
1. Details of the dispersion map are listed in Table 5-4, p. 87.
qt() Aπ
2
---πt
Tb
-------
sincos jmpπ2πt
Tb
---------
cosexp⋅=Tb
2
-------tTb
2
-------
<<–
AT
b
mp
42 A. Richter - Timing Jitter in WDM RZ Systems
Characteristics of RZ pulse propagation
Figure 3-3. Evolution of CRZ. (a) over 9,990 km with snap shots after each
dispersion map (495 km), (b) over one dispersion map of 495 km, (no fiber
attenuation). Details of the dispersion map are listed in Table 5-4, p. 87.
The CRZ pulse is launched at about the middle of the normal dispersion
fiber. An optimum pre-chirping at the fiber input is of key importance for
successful propagation. In [161], several lists are derived for selecting the
optimum chirp required for CRZ pulses depending on the applied dispersion
map. General design goal is to produce a chirp-free pulse at the receiver,
which corresponds to a pulse with minimum time-bandwidth product, and
thus minimum crosstalk with other pulses in time and frequency. For the
evolution diagram in Figure 3-3, a depth of the phase modulation of
was used. The upper graph in Figure 3-3 shows stable evolution
over 9,990 km, where snap shots are taken after each dispersion map. The
pulse shape changes with distance, however, retains a well detectable for-
mat. The lower graph shows pulse dynamics inside the map. Strong pulse
dynamics inside the map lead to pulse power spreading over neighboring bits
on the one hand, and development of large, sharp-edged power spikes on the
other hand.
9,990
distance
[km]
time [ps]
100
00
495
(a)
(b)
distance
[km]
time [ps]
100
00
mp0.6=
Characteristics of RZ pulse propagation
A. Richter - Timing Jitter in WDM RZ Systems 43
The CRZ transmitter design is quite complex, as the combined modulation
of amplitude and phase requires a set of cascaded external modulators.
Recently, the first integrated CRZ transmitter for 10 Gbit/s has been demon-
strated, which eases application tremendously as board space and insertion
losses between the components is reduced [57]. Also, an alternative way of
generating CRZ pulse streams is published recently, which allows CRZ mod-
ulation with only a single MZM [100] or electro-absorption modulator [135].
However, more effort need to be put into the generation of the proper electri-
cal drive voltage.
3.4 Convergence of DMS and CRZ schemes
There have been debates as to whether DMS or CRZ systems will be the
best performing modulation format for the future. It began with the discus-
sion of whether soliton or NRZ propagation is the optimal choice. While soli-
ton systems evolved into DMS systems over the years, CRZ evolved from
design aspects typically applied to NRZ systems. Recent suggestions for
DMS systems apply ultra-short dispersion maps. They are especially
designed for high-speed applications (≥40 Gbit/s), while maintaining narrow
channel spacings [76], [151]. Other suggestions go in the direction of apply-
ing dispersion maps with high local dispersion values (SMF) resulting in
large map strengths [9]. When is increased, interactions between pulses
become larger. So pulse powers need to be reduced, which limits the average
dispersion that can be balanced by fiber nonlinearities. Thus, with increased
map strength DMS propagation transits toward the quasi-linear propaga-
tion regime of CRZ systems.
Remaining differences between DMS and CRZ systems are the different
dispersion map lengths and local dispersion values, as well as launch points,
which determine the initial pulse shapes and chirps. While DMS are typi-
cally launched close to a chirp-free point1 in the dispersion map, where the
1. Point in the dispersion map, where the pulse representing the optimal DMS solution is
chirp-free. This point does not need to exist.
SS
S
44 A. Richter - Timing Jitter in WDM RZ Systems
Characteristics of RZ pulse propagation
time-bandwidth product is minimum, CRZ pulses are launched at points
requiring a large initial pre-chirp. In [124] a comparison was presented
between a typical CRZ system experiment [20] and a recent experiment of a
DMS system [63]. The dispersion maps for both systems are very different,
and so is the pulse evolution. However, similar propagation behavior is
observed, when the launch power and ASE noise build-up are scaled properly
by the applied amplifier spacings and total transmission distances. Both sys-
tems support fiber nonlinearities and operate in a quasi-linear mode.
Modern systems will utilize RZ modulation formats, which incorporate the
flexibility of DMS with respect to SPM and dispersion effects, and the robust-
ness against nonlinear multichannel effects (XPM, FWM) of CRZ. One candi-
date is the modulation format carrier suppressed return-to-zero (CS-RZ). It
modulates neighboring RZ pulses with an alternating relative optical phase
difference of π. This alternating phase results in carrier suppression of the
generated RZ format. The effect is a reduced sensitivity against intrachannel
pulse distortions, but also an increased signal bandwidth.
The superior performance of CS-RZ has been demonstrated in recent
high-speed transmission experiments [74]. In [94] dense WDM applications
(50x20 Gbit/s) employing pure RZ, CS-RZ and SSB-RZ have been compared
over long-haul distances (4,000 km). CS-RZ performed best, promising to be
the ideal solution for suppressing linear and nonlinear interactions. In
another long-haul experiment [25], it has been demonstrated that CS-RZ
was outperformed by the CRZ modulation format. However, this was mainly
achieved because of received filters were optimized for the CRZ modulation
format.
A. Richter - Timing Jitter in WDM RZ Systems 45
Chapter 4
Modeling of timing jitter
4.1 Overview
In Chapter 3, p. 37, general conditions were presented for building nonlin-
earity-supporting, dispersion-managed soliton (DMS) systems operating in a
stable regime over optical fiber. This chapter investigates propagation distor-
tions of DMSs and other RZ pulse shapes.
There are actually many different sources responsible for pulse degrada-
tions. A distinction can be made between distortions causing energy fluctua-
tions and pulse position fluctuations. While mostly negligible for NRZ
systems, the impact of timing jitter on overall system performance is not neg-
ligible in WDM systems using RZ modulation formats. Initially low, timing
deviations accumulate over distance from small perturbations, to eventually
become of system limiting importance.
This chapter provides an overview of the main sources of timing jitter in
long-haul WDM RZ transmission systems, and presents modeling techniques
used to investigate them. After listing the main distortions and their typical
impact on system performance, two dominating sources of timing jitter are
investigated in more detail.
Firstly, ASE-noise added by optical amplifiers along the transmission line
influences not only the pulse energy but also the relative frequency position
of the propagated pulses. These frequency shifts are stochastic in nature as
pulses experience different amounts of random ASE-noise. Secondly, pulses
propagating in different WDM channels collide with each other due to their
different propagation velocities. These collisions introduce residual fre-
46 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
quency shifts in the presence of non-negligible interchannel XPM efficiency.
Because of the randomness of channel bit streams, each pulse accumulates
an individual frequency shift. Due to fiber dispersion, random frequency
shifts lead to a variation of channel velocities with time, and thus timing jit-
ter.
For each of the two sources of timing jitter, estimation techniques are pre-
sented, which are the basis for semi-analytical algorithms. They solve
numerically an analytically derived solution for the pulse central time vari-
ance considering local pulse shape information, which is gathered by solving
the generalized nonlinear Schrödinger equation (GNLS) using the split-step
Fourier method1. These algorithms allow a reduction in computational effort
by orders of magnitude compared to full numerical simulations. For many
long-haul WDM RZ transmission systems, these fast techniques make sys-
tem parameter optimization possible, as scanning of large parameter sets
becomes feasible.
4.2 Main system distortions
4.2.1 Noise from optical amplifiers
Long-haul transmission links experience performance degradations due to
ASE-noise from optical amplifiers2 along the line. In transoceanic systems,
over 100 amplifiers are cascaded; each of them adds noise onto the signal
stream. Besides changing the pulse energy, ASE-noise affects the pulse posi-
tion in nonlinear pulse propagation as well. In fact, it turned out that
ASE-noise induced timing jitter (ANTJ) is the ultimately limiting effect in
single channel soliton propagation. For more linear propagation scenarios, as
the case for CRZ systems, energy fluctuations especially noise accumulation
at spaces has a more severe impact on system performance.
1. See Chapter 2, p. 17 for details.
2. See Chapter 2, p. 20 for details on noise generation in optical amplifiers.
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 47
Disturbing effects of ASE-noise are closely dependent on amplifier spac-
ing, Noise Figure, and fiber dispersion. A detailed discussion will be given in
Chapter 4, p. 51. Examples of system simulations will be presented in
Chapter 5, p. 78.
4.2.2 Intrachannel pulse-to-pulse interactions
In order to increase the throughput in fiber-optic communication systems,
it is tempting to increase the bitrate per channel while maintaining a narrow
channel spacing. However, pulses in neighboring bit intervals tend to inter-
act with each other, if they are placed too close together. When estimating
the impact of intrachannel pulse-to-pulse interactions1, it is helpful to distin-
guish between nonlinearity-supporting systems, like soliton or DMS sys-
tems, and quasi-linear systems.
Soliton and DMS systems
Interactions between neighboring solitons introduce timing jitter, and
finally the collapse of the solitons, if the initial soliton separation in time is
not wide enough2. To maintain a safe pulse separation, the pulse width
needs to be reduced and peak powers increased, when the channel bitrate is
increased. This is possible to some extent but hardly efficient for bitrates of
40 Gbit/s over NZDSF, for example. However, it has been shown that pulse
separation can be halved, if neighboring pulses are launched in orthogonal
polarizations [122]. The parameters of the investigated DMS system simula-
tions in this work are selected such that pulses propagate in regions of safe
pulse separation, and hence, pulse interactions are negligible.
Quasi-linear systems
In quasi-linear systems, the dispersion length is much shorter than
the nonlinear length . Thus, pulses change rapidly their power distribu-
tion in time due to dispersion-induced pulse spreading and compression over
, which results in an averaging out of nonlinear propagation effects3
[112]. Pulse-to-pulse interactions in quasi-linear systems can be best
1. Dependence of pulse propagation from other pulses in the same channel.
2. As outlined in Chapter 3, p. 34.
3. See also Chapter 3, p. 41.
LD
LNL
LNL
48 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
described by time-equivalent terms of FWM and XPM, namely intrachannel
four-wave mixing (I-FWM) and intrachannel cross-phase modulation
(I-XPM) [40], [102]. I-FWM induces pulse echoes, which grow logarithmically
with distance and decrease with the square of the pulse separation. I-XPM
induces timing jitter, which can be described by the variance of the pulse
central time for the case of two interacting pulses of Gaussian shape as [30]
(4-1)
where is the bit duration, is the FWHM pulse duration,
is the GVD, is the nonlinear coefficient,
is the optical pulse power,
is a term considering pulse chirp and fiber loss versus distance.
Pulse-to-pulse interactions in quasi-linear systems become important for
high bitrate (≥40 Gbit/s) transmission. Also, it has been shown that for a
careful selection of the dispersion map, the timing jitter due to pulse-to-pulse
interactions can be drastically reduced [30], [107]. Thus, it will not be
regarded in the following analysis.
4.2.3 Interchannel cross-phase modulation
In WDM systems, pulses propagate with different velocities. This leads to
interchannel interferences1 caused by unbalanced collisions between pulses
of different channels [164], [118]. This effect is schematically shown in
Figure 4-1 for a collision between two pulses. When the faster traveling pulse
approaches the slower traveling pulse, both pulses are attracted to each
other, due to cross-phase modulation. As a result, their velocities modify
slightly, i.e., the faster traveling pulse becomes slower and the slower travel-
ing pulse faster. While both pulses are at the same location, no velocity mod-
ifications occur. When the faster traveling pulse finally passes the slower
traveling pulse, both pulses repel each other; again under slight modification
of their velocities. But this time, the correction of the velocities is in the oppo-
site direction. Due to fiber attenuation, the temporal velocity changes
1. Dependence of pulse propagation from pulses in neighboring channels.
σt
2CγP()
2
β2
-------------------TF
Tb
-------
3
=
TbTF
β2γ
P
C
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 49
induced by XPM do not cancel each other out completely. Thus, the pulses
shift in time.
Figure 4-1. Collision between two pulses in frequency-separated channels
results in deviation from original central time (drawn for anomalous disper-
sion regime).
The net timing shift of each pulse in a WDM transmission system depends
on the number of collisions it experiences, and the pulse power during colli-
sion. As the propagated bit sequences are usually random, each pulse experi-
ences a unique shift in time, which results in random pulse position
deviations at the detector. This effect is called collision-induced timing jitter
(CITJ), or XPM-induced timing jitter. CITJ is one of the major sources of bit
errors in WDM RZ systems. The individual time shifts are strongly depen-
dent on the relative velocity difference between the colliding pulses, and
their pulse shape and power during the collision. Hence, the applied disper-
sion map, the number of WDM channels, and their spacing, as well as the
positioning of optical amplifiers are of strongest influence for the accumula-
tion of XPM-induced timing jitter.
In Chapter 4, p. 59, techniques for estimating CITJ are presented. In
Chapter 5, p. 81, results of WDM system simulations are presented, which
help to determine the accumulation behavior of CITJ in dependence on vari-
ous system parameters.
distance
time
∆t
f1
f2
f1< f2
distance
time
∆t
f1
f2
f1< f2
50 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
4.2.4 Others
In the following, additional distortion effects are listed. They are not
regarded in detail in this work, as they are of no importance and cannot be
discussed in depth without broadening the scope of this work.
Polarization dependent propagation effects
The polarization dependence of optical fiber links might introduce severe
system degradations in optical communication systems [45], [48]. In particu-
lar, systems employing high bitrates of 40 Gbit/s or systems covering large
distances are affected. In long-haul transmission, pulses suffer mainly from
the coupled effect of polarization mode dispersion (PMD) in the fiber and
polarization dependent loss (PDL) in the optical repeater blocks. For details
on nonlinear propagation problems in the presence of PMD, see [79].
Transmitter and receiver imperfections
Beside direct fiber propagation effects, which might translate into timing
jitter of the received pulse sequence, imperfections of the optical transmitter
might add to these pulse position fluctuations. The initial timing jitter of
laser sources used as carriers in external modulation transmitters is usually
of less importance. More important is the initial phase noise of optical carri-
ers, which translates into pulse position fluctuations during propagation as a
direct result of fiber dispersion. The induced timing jitter, defined by the
variance of pulse position fluctuations, can be derived for a classical soliton
to be [156]
(4-2)
where is the carrier linewidth,
is the dispersion length, as defined in Eq. (2-25),
is the group velocity dispersion,
is the transmission distance.
Note that the timing jitter scales with distance squared. For example, the
linewidth-induced timing jitter is about = 2.04 ps2 for = 12 MHz,
D= 0.1 ps/nm-km, λ=1550nm, =20ps, and z=10,000km.
σt
2z() δf2
3
---πβ2
3
LD
------------z2
⋅≈
δf
LD
β2
z
σt
2δf
T0
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 51
Additionally to imperfection at the transmitter, the optical receiver might
add to the accumulated timing jitter. At the decision circuitry in the optical
receiver, the pulse sample circuit and the received pulse stream need to be
synchronized. This synchronization is usually performed using a phase
locked loop, which might be infected by timing fluctuations itself.
FWM and dispersive waves
Four wave mixing between WDM channels can lead to crosstalk, as dis-
cussed in Chapter 2, p. 13. For most of the scenarios discussed in this work,
FWM impact is considered to be of additive character or even negligible, as
the FWM efficiency is small due to the fast walk-off of channels1.
When launching nonlinearity supporting RZ pulses into the fiber with
pulse shape, power, width and chirp not exactly corresponding to a solution
of the nonlinear propagation equation, pulse energy will be reduced, under
emission of dispersive waves, until the transmitting pulse is transferred to a
stable solution of the propagation equation2. Throughout this work, great
care is taken to launch proper pulses into the fiber, and thus omit pulse dis-
tortions caused by energy shedding effects. However, this has not always
been achieved.
Some of the discrepancies in comparing approximation techniques for tim-
ing jitter and BER with full numerical simulations, which are performed in
Chapter 5 and Chapter 6, p. 112, arise from FWM and dispersive waves.
4.3 Timing jitter due to optical inline
amplification (ANTJ)
4.3.1 Motivation
The ASE-noise induced timing jitter was calculated for the first time by
Gordon and Haus [53]. They applied linearization approximation to the ana-
1. See Figure 2-2, p. 15 for details.
2. See also Chapter 3, p. 35.
52 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
lytically known soliton solution for the case that noise energy is much lower
than signal energy. For a classical soliton system employing equally spaced,
lumped amplifiers, the variance of the pulse central time due to ASE-noise
induced timing jitter (ANTJ) can be written as [82]
(4-3)
where is nonlinear coefficient as defined in Eq. (2-19),
is the fiber attenuation,
is the power spectral density of ASE-noise in a
single polarization as defined in Eq. (2-33),
is the effective length as defined in Eq. (2-23),
is the dispersion length as defined in Eq. (2-25),
is the amplifier spacing, is the propagation distance,
is the pulse duration (half-width 1/e-intensity).
The first term in the brackets of Eq. (4-3) denotes the effect of direct coupling
of ASE-noise power onto the soliton pulse. It increases in proportion with dis-
tance and in inverse proportion with dispersion. The second term denotes the
Gordon-Haus effect, e.g., the coupling of ASE-noise with the soliton spec-
trum, which causes a random walk of the pulse central frequency, and thus
its velocity. It increases proportionally with dispersion and with the cube of
distance, and is thus the dominating effect for soliton propagation1.
Several techniques were proposed to reduce or even limit ASE-noise
induced timing jitter. Weak spectral filtering can be applied to control the
soliton central frequency and thus its position in time [110]. It has been
shown that the growth of ANTJ with distance could be reduced from cubic to
linear. As filters reduce the soliton energy, excess gain needs to be provided
by the inline amplifiers, which will increase ASE-noise build-up just at the
main soliton frequency range. This problem has been overcome by imple-
menting sliding spectral filters along the link, which slide the solitons slowly
1. Note that Eq. (4-3) is derived for classical soliton systems, e.g., it tends to go to zero when
the dispersion goes to zero.
σt
2γαSASE
Leff
LA
----------T0
π2
12
------ LDz1
9
---1
LD
--------z3
+≈
γ
α
SASE PASE ∆f⁄=
Leff
LD
LAz
T0
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 53
into different frequency regions, leaving the ASE-noise behind as it is not
moving with the filter [119].
When ANTJ should be limited, active devices need to be put on the link,
which lock the soliton to a fixed position in time. One approach is to use
amplitude modulators, which however, destabilize the soliton propagation.
So amplitude modulation is applied together with frequency filters, which
compensate this instability again [126]. Another way to reduce ANTJ in soli-
ton systems is by using one or multiple phase modulators along the trans-
mission line [136]. The phase modulation translates into frequency
modulation, which in the presence of filters translates into amplitude modu-
lation again.
All these techniques have been proposed for classical soliton systems.
However, they are also applicable to DMS systems, eventually with less
degree of effectiveness. Additionally, it has been demonstrated experimen-
tally [146], [26] and proven theoretically [98] that ANTJ is naturally reduced
in DMS systems compared to classical soliton systems, due to the required
power enhancement.
4.3.2 Linearization approximation for arbitrary pulse
shapes
In the following, an approach is presented, which extends the idea of the
linearization approximation that it holds for arbitrarily shaped RZ pulses,
which solve the perturbed GNLS. The derivations presented below are lent
from [60].
4.3.2.1 Modeling
The approach is based on a normalized version of the GNLS, as given in
Eq. (2-19), considering small perturbations due to additive ASE-noise. It can
be written using the Langevin form1 as follows.
(4-4)
1. Based on notations first time used in [73].
j
z∂
∂q1
2
---D
t2
2
∂
∂qC–q2q+jgq N+=
54 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
where corresponds to the photon flow, with ,
with photon energy,
is the transmission distance multiplied by , with arbitrary
scaling dispersion1,
is the retarded time given by ,
is the local dispersion divided by ,
, with nonlinear coefficient,
is the local fiber loss/gain normalized to , given by
, (4-5)
with fiber attenuation, , gain per amplifier,
fiber length of amplifier, and amplifier spacing,
accounts for small perturbations due to ASE-noise. In the case
of lumped amplifiers its autocorrelation function is given by
(4-6)
with , and
spontaneous emission factor.
Figure 4-2 shows the assumed pulse power profile versus distance.
1. Typically average dispersion of the dispersion map.
qzt,()
2qzt,()Azt,()hf⁄=
hf
zβ20,β20,
ttt
˜β1z–=
Dz() β
20,
Chfγβ
20,
⁄=γ
gz() β
20,
gz()β
20,
g0 kLAzkL
ALAmp
+<<
α2⁄–otherwise
=k12…,,=
αg0G() 2LAmp
()⁄ln=G
LAmp LA
N
zt,()
Nzt,()N∗z1t1
,(),〈〉ψz()δzz
1
–()δtt
1
–()=
ψz() 2g0nsp kLAzkL
ALAmp
+<<
0otherwise
=k12…,,=
nsp
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 55
Figure 4-2. Power profile versus distance.
To derive statistics of pulse position fluctuations, it is helpful to define
central time and frequency of by its center of mass in time and frequency,
respectively, as
(4-7)
where is the pulse energy.
Ignoring pulse spreading outside the bit interval, integration in Eq. (4-7)
needs to be performed over a single bit only1. Differentiating and with
respect to and using Eq. (4-4) one obtains the following dynamic equations
for central time and frequency
(4-8)
. (4-9)
Eq. (4-8) can be solved directly with respect to yielding two contribu-
tions to the deviations of the pulse central time as follows
. (4-10)
1. Note that when ISI effects should be taken into account, integration needs to be performed
over multiple bit durations.
Power
[dB]
LALAmp z
Power
[dB]
LALAmp z
q
T1
U
---- tq2td
∞–
∞
∫
=Ω1
U
---- Im t∂
∂qq∗
td
∞–
∞
∫
=
Uq
2td
∞–
∞
∫
=
TΩ
z
zd
dTDΩ2
U
----+tT–()Im Nq∗
()td
∞–
∞
∫
=
zd
dΩ2
U
----–ΩIm Nq∗
()t2
U
---- Re Nt∂
∂q∗
td
∞–
∞
∫
+d
∞–
∞
∫
=
Ω
T∆F∆T+=
D
0
z
∫ΩN()dz 2
U
----
0
z
∫tT–()Im Nq∗
()td
∞–
∞
∫dz+=
56 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
As seen from Eq. (4-10), timing deviations are due to the superposition of two
effects, namely noise-induced frequency shifts in the presence of fiber disper-
sion ( ), and the direct overlay of noise onto the pulse ( ).
Following the derivations in [60], the variance of the pulse central time
can be calculated from Eq. (4-9) and Eq. (4-10) yielding three contributing
terms:
(4-11)
with
(4-12)
where is the field description for the retarded frequency,
meaning that the pulse is investigated during evolution over the fiber
at its central frequency defined in Eq. (4-7),
is defined in Eq. (4-6).
Term A denotes the contribution from frequency shift alone, and term C
denotes the contribution from direct time offset alone. Term B represents the
contribution caused by pulse chirp, which could be understood as interfer-
ence between time and frequency shift. Consider a chirped pulse, meaning
that its local frequency depends on time. An ASE-noise induced time offset
would lead to a frequency shift, which would translate in additional time
shift. Depending on the actual RZ system under investigation, the individual
terms vary in importance.
The result in Eq. (4-12) differs from the classical theory of ANTJ derived
by Gordon and Haus [53]. Firstly, term A is proportional to the pulse band-
width squared over the pulse energy squared. Secondly, term B is totally new
as initial derivations are derived for the fundamental soliton solution1,
which is not chirped.
1. See Chapter 3, p. 32 for details.
∆F∆T
σt
2T2
〈〉 T〈〉
2
–∆F2
〈〉2∆F∆T⋅〈〉∆T2
〈〉++==
A∆F2
〈〉2D
0
z
∫z1
() D
0
z1
∫z2
() ψ
0
z2
∫t∂
∂q
˜2td
∞–
∞
∫
U2
⁄dz3dz2
dz1
==
B2∆F∆T⋅〈〉2D
0
z
∫z1
() ψ
0
z1
∫Re q
˜∗t∂
∂q
˜
td
∞–
∞
∫
U2
⁄dz2dz1
==
C∆T2
〈〉 ψ
0
z
∫tT–()
2q
˜2td
∞–
∞
∫
U2
⁄dz1
==
q
˜qj–Ωt()exp=
ψz()
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 57
4.3.2.2 Numerical implementation
Eq. (4-11) and Eq. (4-12) completely determine the variance of the pulse
central time. The numerical implementation is straightforward.
1. Calculate the propagation of the pulse shape by numerically solving
the GNLS given in Eq. (2-19) for one signal pulse per channel.
2. Substitute the calculated pulse shape per distance in Eq. (4-12) and
calculate the variance of the pulse central time .
The model is implemented in an iterative algorithm. Calculation of ANTJ is
very sensitive to correct capturing of local propagation characteristics of sig-
nal pulses. Thus, it is important to choose the step sizes such that a certain
dispersion-length product per step is not exceeded. The main advantage of
the presented algorithm is that it dramatically reduces the computational
effort compared to full numerical simulations, as it is not necessary to simu-
late the propagation of a long stream of pseudo-random bits in order to find
numerical estimates of the central time variance.
4.3.2.3 Validation
The approach has been validated, both numerically and experimentally in
great detail [60]. One validation case is presented below, other application
examples are discussed in Chapter 5, p. 78.
A typical single channel DMS system with a bitrate of 10 Gbit/s is consid-
ered here. The initially Gaussian pulses are launched with a peak power of
2.8 mW and a FWHM duration of 16.75 ps. The dispersion map parameters
are described in Table 4-1.
Figure 4-3 illustrates graphically the meaning of parameters listed in
Table 4-1. The dispersion map consists of a span of anomalous DSF with dis-
Table 4-1. Parameters of dispersion map A.
L1100.0 km D12.34 ps/nm-km
L2100.0 km D2-2.19 ps/nm-km
Llaunch 50.0 km
LA50.0 km
q
q
σt
2
58 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
persion D1 and length L1, and a span of normal DSF with dispersion D2 and
length L2. Amplifiers are spaced in distance by LA. They are placed at the
midpoints and the edges of each span. Pulses are launched in each channel
at Llaunch, here, from the middle of the anomalous dispersion span.
Figure 4-3. Schematic of dispersion map A.
Figure 4-4 shows ASE-noise induced timing jitter versus distance for
NF = 6.3 dB of all inline amplifiers [132]. Timing jitter results are compared
with data gathered by propagating random streams of 256 bits and estimat-
ing the variance of the pulse position from the received pulse sequence
directly (full numerical simulations). Excellent agreement is observed.
Figure 4-4. ANTJ versus distance for a single channel system. Results of lin-
earization approximation (dashed line) are compared with results using full
numerical simulations (solid line). Dispersion map A was applied (see
Table 4-1).
L1-Llaunch Llaunch
D1
D2
L2
LA
L1-Llaunch Llaunch
D1
D2
L2
LA
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 59
4.4 Timing jitter due to interchannel
cross-phase modulation (CITJ)
4.4.1 Motivation
One of the major sources of bit errors in WDM systems using RZ modula-
tion formats is collision-induced timing jitter (CITJ). It arises from nonlinear
interactions of pulses in different WDM channels due to cross-phase modula-
tion1. So it is also called XPM-induced timing jitter.
This topic has been subject to extensive studies over the past years, result-
ing in different approximation techniques for XPM-induced timing jitter. The
residual wavelength shift of a two-soliton collision can be derived to [118],
[88]
(4-13)
where is the position of collision, is the amplifier spacing,
,
with FWHM duration of the soliton, dispersion length,
, amplifier spacing, collision length2.
It has been found that when the collision length is at least four times
larger then the amplifier spacing , the spectral shift resulting from a colli-
sion between two solitons can be neglected [118]. With Eq. (2-24), this
implies a maximum channel spacing of approximately
. (4-14)
1. See Chapter 2, p. 12 for details on XPM.
2. See Chapter 2, p. 19 for the definition of .
∆λ z() cm
2πmz
LA
--------------- 2πm
αLA
------------
tan 1–
–sin
m1=
∞
∑
=
zL
A
cm0.2274 λ2
cTF
---------- LA
LD
-------- m3x4
mx()sinh 2
-----------------------------12πm
αLA
------------
2
+
0.5–
≈
TFLD
x2.8LcLA
⁄≈ LALc
LD and Lc
Lc
LA
∆λ12 0.15 TF
DL
A
---------------
<
60 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
Similar to Eq. (4-13), approximations of the residual frequency shift are
found for Gaussian-shaped dispersion-managed solitons (DMS) [37], [143],
[2], [128], [145] or the induced time shift directly [90]. It was shown that
DMS suffer less from interchannel pulse collisions compared to classical soli-
tons [75]. In [144], the dispersion map is optimized to minimize the accumu-
lated CITJ for a DMS system.
All the methods mentioned so far have in common that they only consider
two-pulse collisions. In [41] three-pulse collisions were analyzed. In [27] the
solution of N coupled GNLS is derived, where each GNLS describes the prop-
agation of an individual, frequency-shifted soliton. One important result
from this work is that an approximate formula for the total collision-induced
frequency shift is gained by considering the superposition of all the individ-
ual two-soliton collisions. In [88], the randomness of the propagated bit
streams is taken into account. The CITJ of channel m was estimated for an
amplified soliton system with N channels to
(4-15)
where is the bit duration,
is the spacing between channel m and k,
is the propagation distance,
is the distance between collisions,
is the variance of wavelength shift, given by
with defined in Eq. (4-13).
All the techniques presented above lack certain features, which limit their
usability for more general applications. RZ pulse shapes applied in today’s
WDM systems may differ significantly from classical solitons or Gauss-
ian-shaped DMS. Besides, the number of channels in modern WDM systems
is typically larger than three. Their spacing is drastically reduced compared
to the time of early soliton investigations. Thus, four and higher number
pulse collisions must be considered when estimating the accumulation of
timing jitter due to XPM.
∆t2
〈〉
m Tb
∆λmk
--------------
2L
Zmk
-----------
3∆λmk
2
〈〉
6
--------------------
k
km≠
∑
≈
Tb
∆λmk
L
Zmk
∆λmk
2
〈〉 ∆λmk
2
〈〉cm
2
∑
=
cm
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 61
Beside analytical approximations, another solution for estimating
XPM-induced timing jitter would be to use full numerical simulations. How-
ever, the computational effort for such simulations is often very large, as it
dramatically increases with the number of channels in the WDM system.
This is due to two reasons. Firstly, the simulation bandwidth needs to be
increased when adding more channels. In turn, the step size in distance1
needs to be reduced to resolve the relatively fast sliding of channels due to
their large difference in group velocity. Secondly, pseudo-random bit streams
of at least the order of 100 need to be simulated per channel to reach a rea-
sonable statistical confidence for the estimated variance of the pulse central
time.
Figure 4-5 shows an example for the computational effort required to eval-
uate the timing jitter for a 10-channel WDM system with a transmission dis-
tance of 10,000 km using full numerical simulations. Note that the inverse of
the time resolution defines the simulation bandwidth and that an average
split-step size of 100 m is assumed. In practice, it would be set to an adaptive
value depending on local peak power and dispersion.
Figure 4-5. Estimation of numerical effort for calculating timing jitter using
full numerical simulations based on the split-step Fourier method.
According to benchmark tests of FFTW, a professional FFT package devel-
oped at MIT [46], approximately 90x106flops per second are performed on a
typical Pentium II with 300 MHz and 256 MByte RAM2. This leaves an
1. The differential distance step when solving the GNLS using the split-step Fourier method,
see Chapter 2, p. 17 for details.
bitrate 10 Gbit/s
# of bits 256
time resolution 0.2 ps
# of samples N 131,072
Example: 10-Channel WDM Transmission
fiber distance 10,000 km
split-step size 0.1 km
# of FFT’s 100,000
# of flops/FFT 5 N log2(N)
≈11.14 x 106
total # of flops ≈1.11 x 1012
bitrate 10 Gbit/s
# of bits 256
time resolution 0.2 ps
# of samples N 131,072
Example: 10-Channel WDM Transmission
fiber distance 10,000 km
split-step size 0.1 km
# of FFT’s 100,000
# of flops/FFT 5 N log2(N)
≈11.14 x 106
total # of flops ≈1.11 x 1012
62 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
approximate run time of 2 hours and 25 minutes. The FFT accounts in aver-
age for about 70% of the calculation effort in the split-step Fourier method1.
So all together, the time taken to estimate timing jitter for the presented sce-
nario is of the order of three hours. Note that this time is needed for the esti-
mation of only one point in an eventually multi-dimensional parameter
space.
In summary, existing analytical or semi-analytical approximations are not
applicable for general case RZ WDM systems. This leaves full numerical sim-
ulations which, however, are not at all suited for optimization tasks consider-
ing multiple system parameters. However, they serve as robust technique for
gathering reference data when deriving new estimation techniques.
4.4.2 Elastic collision approximation for arbitrary pulse
shapes
In this section, a semi-analytical approach is described, which results in
the derivation of an efficient numerical algorithm to accurately evaluate col-
lision-induced timing jitter in WDM systems with RZ modulation format.
The key factor of this approach is to find an analytical description for the
variance and the average of the pulse central time as a function of arbitrary
pulse shape evolution with distance for each channel.
As shown in Chapter 4, p. 74 and Chapter 5, p. 86, the approach is valid
for systems in which RZ signal pulses do not spread outside the bit length,
and also for systems with large pulse spreading but low local pulse energies
outside the bit interval.
4.4.2.1 Modeling
Details of the derivation below were published in [61].
Basic equations
The derivation starts with the normalized version of the GNLS given in
Eq. (2-19) for the n-th pulse in channel m of a WDM system. When fiber loss
2. See [43] for details on the benchmark test results.
1. If no special effects, like stimulated Raman scattering are included.
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 63
is periodically amplified, and FWM is neglected, the dynamic propagation
equation is given by
(4-16)
where is the normalized field amplitude of the n-th bit in channel m
given by , with actual field amplitude,
nonlinear coefficient, dispersion scale length, given by
, with FWHM pulse duration,
arbitrary scaling dispersion1,
is the transmission distance divided by ,
is the time divided by ,
is the group velocity of channel m divided by ,
is the local dispersion of channel m divided by ,
is the net gain given by ,
with fiber attenuation and amplifier gain.
The assumption that FWM can be neglected in Eq. (4-16) holds for cases of
large wave vector mismatch, e.g., due to high local fiber dispersion values
and reasonable channel separation2.
To derive statistics for pulse position fluctuations, it is helpful to define
central time and frequency of from its center of mass in time and fre-
quency, respectively, as
(4-17)
1. Typically average dispersion of the dispersion map.
2. See Chapter 2, p. 13 for details.
j
z∂
∂1
vm
------- t∂
∂
+qmn
1
2
---+ Dmt2
2
∂
∂qmn qmn
22qkl
2
l
∑
k
km≠
∑
+– qmn jgqmn
=
qmn
qmn Amn γLD
=Amn
γLD
LDTF
2β20,
⁄=TF
β20,
zL
D
tT
F
vmz() LDTF
⁄
Dmz() β
2z() β
20,
gz() gz()
LD
-----------GzkL
A
=
α2⁄()–zkL
A
≠
=k12…,,=
αG
qmn
Tmn
1
Umn
------------ tmqmn
2tm
d
∞–
∞
∫
=
64 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
(4-18)
where is the energy of the n-th pulse in channel m,
is the retarded time of channel m.
Ignoring pulse spreading outside the bit interval, integration of Eq. (4-17)
needs to be performed over only a single bit. Differentiating and
with respect to and using Eq. (4-16) one obtains the following dynamic
equations for central time and frequency
(4-19)
(4-20)
where is the accumulated time
offset between the n-th pulse in channel m and the l-th pulse in
channel k at distance point z.
denotes the time with respect to the center of the n-th
pulse in channel m.
Interferences between adjacent pulses within each channel are neglected1.
It is also assumed that all signal pulses within a channel have the same
shape, though, the shape may differ over distance z from channel to channel.
With these assumptions, one can write
(4-21)
where is the average shape of signal pulses in channel m,
accounts for the randomness of the bit stream in channel m. It
equals 1 for marks and 0 for spaces, where equal probabilities of
marks and spaces are assumed.
1. Cases for which this assumption holds are outlined in Chapter 4, p. 47.
Ωmn
1
Umn
------------ Im tm
∂
∂qmnqmn∗
tm
d
∞–
∞
∫
=
Umn qmn
2tm
d
∞–
∞
∫
=
tmtv
m
1–z1
()z1
d
0
z
∫
–=
Tmn Ωmn
z
zd
dTmn DmΩmn
=
zd
dΩmn
2
Umn
------------ qmn
∞–
∞
∫2
T∂
∂qkl zT θmn kl,
+,()
2Td
l
∑
k
km≠
∑
=
θmn kl,vm
1–z1
()vk
1–z1
()–[]z1
d
0
z
∫Tmn Tkl
–()+=
Tt
mTmn
–=
qmn αmnqm
=
qmz()
αmn
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 65
Typically, there is no significant change of pulse shape resulting from col-
lisions in dispersion-managed WDM systems as the collision length, , is
much shorter than the nonlinear length, .1 Thus, changes of pulse shape
due to interchannel pulse collisions are neglected. This approximation is
known as elastic collision approximation (ECA). Note, however, that pulse
shape changes versus distance due to dispersion and SPM effects are consid-
ered.
Regarding the points above, formal solutions of Eq. (4-19) and Eq. (4-20)
can be derived as
(4-22)
(4-23)
where is the initial central time of the n-th pulse in channel m,
describes the
frequency shift due to collision between the n-th pulse in channel m
with the l-th pulse in channel k per distance.
Substituting from Eq. (4-23) into Eq. (4-22) yields the deviation from
the initial central time at distance point of the n-th pulse in channel m,
. (4-24)
is a random variable as it depends on , which accounts for the ran-
domness of transmitted bit streams in neighboring channels.
Statistics of random pulse position fluctuations are typically described by
their first two moments only, assuming Gaussian statistics2. The average
1. See Chapter 2, p. 19 for definitions.
2. A reasonable assumption, as outlined in Chapter 6, p. 109.
Lc
LNL
Tmn z() Tmn
0Dmz1
()Ω
mn z1
()z1
d
0
z
∫
+=
Ωmn z() 2αkl
Ψmn kl,z1
()
Umz1
()
------------------------------ z1
d
0
z
∫
l
∑
k
km≠
∑
=
Tmn
0
Ψmn kl,z() qmzT,()
∞–
∞
∫2
T∂
∂qkzT θmn kl,
+,()
2Td=
Ωmn
z
∆Tmn Tmn T–mn
0
=
2Dmz1
() 1
Umz2
()
--------------------αkl
l
∑
k
km≠
∑Ψmn kl,αkl z2
,()z2
d
0
z1
∫z1
d
0
z
∫
=
∆Tmn αkl
66 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
deviation and deviation variance of the initial pulse central time in
channel m are defined as [141]
(4-25)
(4-26)
where the summation is performed over a combination of a large number of
pulses N.
In the following section, the average deviation and secondly, the vari-
ance of the pulse central time for channel m will be derived.
Average central time
After substituting Eq. (4-24) into Eq. (4-25) and taking the average, one
finds that
(4-27)
where
(4-28)
with denoting the average over all n.
In the following, it is assumed that a pulse in channel m interacts at an
arbitrary instance of time only with one pulse in channel k. That is, pulses
are separable during transmission, i.e., pulse broadening due to dispersion
does not introduce intersymbol interferences. This separation of collisions
becomes invalid when RZ pulses expand beyond the limits of the bit dura-
tion. However, it is shown in Chapter 5, p. 86 that this restriction can be
relaxed. In fact, it should be reformulated to be: Contribution to accumulated
pulse central time deviations due to the collision between a pulse in channel
m with the distant pulses in channel k can be neglected. Note, however, that
µmσtm,
2
µtm,∆Tmn
〈〉
1
N
---- ∆Tmn
n1=
N
∑
==
σtm,
2∆Tmn
2
〈〉∆Tmn
〈〉–21
N
---- ∆Tmn
()
2µtm,
2
–
n1=
N
∑
==
µtm,
σtm,
2
µtm,Dmz1
() Qmz2
()
Umz2
()
--------------------z2
d
0
z1
∫
z1
d
0
z
∫
=
Qmz() 2αklΨmn kl,αkl z2
,()〈〉
l
∑
k
km≠
∑
=
x〈〉
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 67
all possible interactions where only one pulse per channel is taking part are
considered.
Fixing k in the sum of Eq. (4-28) and selecting for each n-th bit in channel
m only the term that corresponds to the nearest bit in channel k,
Eq. (4-28) becomes
(4-29)
where means averaging over all n for which ,
is the accumulated time offset between
channel m and k due to dispersion,
is the difference of
deviations from the pulse central times between the n-th pulse in
channel m and its nearest pulse in channel k,
denotes the differential time delay between the closest bits in
channel m and k, normalized to the bit duration (defined further
down).
Expanding in Eq. (4-29) into a Taylor series with respect to
around 0, averaging, and taking into account the symmetry
consideration , one obtains that up to small corrections of the
order of , can be written as
(4-30)
(4-31)
where is the relative time delay of a pulse in
channel k compared to the closest pulse in channel m,
is the
differential time delay of channel k relative to channel m,
lnJ
mk
+=
Qmz() Ψ
mnkn,Jmk
+;,θmk ∆θmnkn,Jmk
+;,
+z,()〈〉
k
km≠
∑
=
x〈〉 αkn,Jmk
+1=
θmk vm
1–z1
()vk
1–z1
()–[]z1
d
0
z
∫
=
∆θmnkn,Jmk
+;,∆Tmn,µm
–()∆Tkn,Jmk
+µk
–()–=
Jmk
Ψmnkn,Jmk
+;,
∆θmnkn,Jmk
+;,
∆θ〈〉 ∆θ()
3
〈〉=
O∆θ〈〉
4
()Qmz()
Qmz() Ψ
mk Θmk z()z,[]
1
2
---
Θmk
2
2
∂
∂Ψmk Θmk z()z,[]σ
tm,
2σtk,
2
–()+
k
km≠
∑
=
with Ψmk Θmk z()z,[]qmzT,()
∞–
∞
∫2
T∂
∂qkzT Θmk z()+,[]
2Td⋅=
Θmk z() τ
mk z() Jmk z()Tb
–=
τmk z() vm
1–z1
()vk
1–z1
()–[]z1
d
0
z
∫Tm
0Tk
0
–()µ
tm,µtk,
–()++=
68 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
is the maximum number of completed bit
durations within ,
is the initial time delay in channel i.
The three contributions to account for the dispersion-induced timing dif-
ference, the initial pulse position difference, and the average position devia-
tion due to collisions. Figure 4-6 shows a schematic diagram of the functions
and versus distance.
Figure 4-6. Schematic plot of , dashed line, and , solid line, between
channel m and k in multiples of bit duration versus distance.
are distance intervals of single bit crossings.
Note that the contribution of the second term on the right-hand side of
Eq. (4-30) is typically small, as:
• For adjacent channels k, is close to , so that is small.
• For distant channels k, is not small. However, their impact on
CITJ can be assumed to be small as large walk-off due to dispersion im-
plies short distances of pulse interactions.
Thus, the second term on the right-hand side of Eq. (4-30) can be
neglected, and one obtains from Eq. (4-27) the following equation for the cen-
tral time average
Jmk z() floor τmk z()
Tb
------------------
=
τmk
Ti
0
τmk
τmk z() Jmk z()
Tb
0
1
2
-1
τmk Jmk
z
z11z12z13z14z15z16
z21z22
τmk Jmk
zJmk,()
lzJmk,()
l1+
,()
σtm,
2σtk,
2σtm,
2σtk,
2
–
σtm,
2σtk,
2
–
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 69
(4-32)
where is given by Eq. (4-31).
Eq. (4-32) can be understood as follows: accounts for the strength of a
collision between a pulse traveling in channel m with its closest pulse travel-
ing in channel k at a distance . After weighting by the local pulse energy
, integration over distance is performed. The accumulated frequency shift
is then multiplied with the local dispersion to give the local timing shift.
All time shifts are accumulated over distance via integration. Finally, sum-
mation over all neighboring channels k of channel m is performed.
Central time variance
In the previous paragraph, an expression was derived for the average
pulse deviation . So to determine , only needs to be
found. From Eq. (4-24) one can derive that
(4-33)
where
with denoting the average over all n.
As mentioned before, only single pulse collisions with the closest pulses in
neighboring channels are considered for each distance point. Thus, it is suffi-
cient to choose for the n-th pulse only the terms and .
Also, using the assumption that marks and spaces in different channels are
mutually independent, one can show that the term evaluates to
[61]
µtm,Dmz1
()
Ψmk Θmk z2
()z,2
[]
Umz2
()
-------------------------------------------------z2
d
0
z1
∫
z1
d
0
z
∫
k
km≠
∑
=
Ψmk
Ψmk
z2
Um
Dm
∆Tmn
〈〉 σtm,
2∆Tmn
2
〈〉
∆Tmn
2
〈〉 Dmz1
()Dmz2
()
Φmz3z,4
()
Umz3
()Umz4
()
-----------------------------------------z4
d
0
z2
∫z3
d
0
z1
∫
z2
d
0
z
∫z1
d
0
z
∫
=
Φmz1z,2
()4αklαk′l′Ψmn kl,z1
()Ψ
mn k′l′, z2
()〈〉
l'
∑
k'
k'm≠
∑
l
∑
k
km≠
∑
=
x〈〉
lnJ
mk
+= l'nJ
mk
+=
Φmz1z,2
()
70 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
(4-34)
where is defined by Eq. (4-31),
is the discrete Dirac-delta function.
Substituting Eq. (4-34) into Eq. (4-33), extracting defined by Eq. (4-27),
and integrating in Eq. (4-33) by parts one obtains
(4-35)
where is the accumulated dispersion of channel m
between distance and .
In dispersion-managed fiber links, local dispersion values are typically
much larger than the average dispersion over the dispersion map. Since the
propagation velocity is determined by the local dispersion values, a pulse in
channel m might experience multiple collisions with another pulse in chan-
nel k. It is important to collect these collisions separately for each pulse as
their occurrences are correlated. This is depicted in Figure 4-7. Assuming
low average time delay differences due to collisions, the distance at which
these collisions occur can be determined.
Φmz1z,2
() Ψ
mk Θmk z1
()z,1
[]Ψ
mk′Θmk′z2
()z,2
[] +
k'
k'm≠
∑
k
km≠
∑
=
δJmk z1
()Jmk z2
(),[]Ψ
mk Θmk z1
()z,1
[]Ψ
mk Θmk z2
()z,2
[]
k
km≠
∑
Ψmk
δxy,()
µtm,
2
σtm,
2Dmz1
()Dmz2
()
Umz1
()Umz2
()
----------------------------------------- ×
0
z
∫
0
z
∫
k
km≠
∑
=
δJmk z1
()Jmk z2
(),[]Ψ
mk Θmk z1
()z,1
[]Ψ
mk Θmk z2
()z,2
[]dz1dz2
Dmzi
() Dmz
˜
()z
˜
d
zi
z
∫
=
ziz
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 71
Figure 4-7. Two pulses can collide more than once because their relative
speeds depend on local dispersion.
Thus, the whole propagation distance can be expressed as a sum of the bit
crossing distance intervals where collisions between signal pulses in chan-
nels m and k occur. Collisions occurring between the same pair of pulses are
grouped. So, each of the integrals in Eq. (4-35) can be represented as
(4-36)
where , with and are the bit crossing
distance intervals in which the differential time delay between
channel m and k, , falls within the -th bit (see Figure 4-6).
Substituting Eq. (4-36) into Eq. (4-35) and keeping only the terms with
one finally obtains the following formula for the central
time variance,
(4-37)
where the summation with respect to is taken over with
, while the summation with respect to is taken over
.
relative time position
?
!
channel k
channel m
(fk<fm)
normal
dispersion
anomalous
dispersion
xd
0
z
∫xd
zJmk,()
l
zJmk,()
l1+
∫
l
∑
Jmk
∑
=
zJmk,()
lzJmk,()
l1+
,()l2j1+=
j
012…,,,=
τmk Jmk
Jmk z1
() Jmk z2
()=
σtm,
2Dmz1
()
Umz1
()
--------------------Ψmk Θmk z1
()z,1
[]z1
d
zJmk,()
l
zJmk,()
l1+
∫
l
∑
Jmk
∑
2
k
km≠
∑
=
ll2j1+=
j
012…,,,=Jmk
Jmk 0123…,±,±,±,=
72 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
Eq. (4-37) can be understood as follows. The strength of collision
between pulses in channels m and k is weighted by the local pulse energy
, and the accumulated dispersion between position (where the col-
lision is evaluated) and (the final destination of pulse propagation). Inte-
gration over the bit crossing distance interval , gives a
normalized value for the time position deviation of a single collision between
channels m and k. Then, a summation of all collisions between the same
pulses is performed. After squaring, the final summation is performed over
all possible distance intervals between channels m and k, and over all
neighboring channels k of channel m.
4.4.2.2 Numerical implementation
Eq. (4-32) and Eq. (4-37) give a complete set of equations for determining
the average and variance of the pulse central time. In order to calculate CITJ
using Eq. (4-32) and Eq. (4-37), the following procedure needs to be followed.
1. Calculate the propagation of the pulse shape by numerically solving
the GNLS given in Eq. (2-19) for one signal pulse per channel. Consider-
ing no dependence between the influences of ASE-noise and interchannel
pulse collisions, ideal lumped amplifiers are assumed along the fiber.1
2. Substitute the calculated pulse shape into Eq. (4-31) and calculate the
collision integral for each m.
3. Substitute and the local pulse energy into Eq. (4-32) and calcu-
late the average of the pulse central time for each channel by
numerical integration.
4. Finally, after substituting , , and into Eq. (4-37), and apply-
ing a simple numerical algorithm for calculating the sum of integrals
over the bit crossing distance intervals as defined in
Eq. (4-36), the variance of the pulse central time is evaluated.
1. The impact of ASE-noise induced timing jitter can be estimated using linearization approx-
imation as derived in Chapter 4, p. 51.
Ψmk
UmDmz1
z
zJmk,()
lzJmk,()
l1+
,()
l
Jmk
q
q
Ψmk
Ψmk Um
µtm,
Ψmk Umµm
zJmk,()
lzJmk,()
l1+
,()
σtm,
2
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 73
This model is implemented in an iterative algorithm, where steps 2., 3., and
4. of the procedure are performed after each split-step applied to solve the
GNLS.
The calculation of CITJ is very sensitive to the precise capturing of local
propagation characteristics of the signal pulses. Thus, it is important to
choose the step-sizes such that a certain dispersion-length product per step
is not exceeded. In estimating pulse width changes for the linear case, the
graph shown in Figure 4-8 depicts the dependence of convergence on the
minimum number of calculation steps per fiber span for various disper-
sion-length-products. The figure is plotted for the case that the maximum
change of pulse width per step does not exceed = 0.075 ps.
Figure 4-8. Minimum number of split-steps per fiber span
versus dispersion-length-product.
The main advantage of the presented algorithm is that it reduces drasti-
cally the computational effort compared to full numerical simulations mainly
due to two factors.
1. It is not necessary to process the frequency bandwidth of the whole WDM
signal comb, but only the bandwidth of a single channel. However, the
time resolution should stay reasonably large to be able to model rapid
pulse shape changes.
∆τ
20
40
60
80
0
100
120
160
100 200 300 500
D * L [ps/nm]
400
140
74 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
2. It is not necessary to simulate the propagation of a long stream of
pseudo-random bits in order to find numerical estimates of the central
time variance. Instead, propagation of only one mark per channel is sim-
ulated.
In particular the second factor allows for reduction of computational effort by
two orders of magnitude compared to a full numerical simulation.
4.4.2.3 Validation
In order to validate the elastic collision approximation (ECA) method, it is
compared with full numerical simulations of several typical WDM systems.
One comparison is presented here, while others are given throughout
Chapter 5.
The first scenario under consideration is a 10-channel WDM system with
10 Gbit/s bitrate per channel and 100 GHz channel spacing. Table 4-2 lists
the parameters of the dispersion map including amplifier positioning.
Figure 4-9 illustrates graphically the meaning of the parameters listed in
Table 4-2. The dispersion map consists of a span of anomalous NZDSF with
dispersion D1 and length L1, and a span of DCF with dispersion D2 and
length L2.1 The amplifier spacing is LA; they are placed at the midpoints of
each span. The pulses are launched in each channel from the middle of the
anomalous dispersion span.
Table 4-2. Parameters of dispersion map B.
L195.6 km D14.0 ps/nm-km
L24.4 km D2-85.0 ps/nm-km
Llaunch 47.8 km
LA50.0 km
1. The dispersion slope parameters S1, and S2 are set, such that the slope of the chromatic
dispersion vanishes. See Chapter 2, p. 9 for details.
Modeling of timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 75
Figure 4-9. Schematic of dispersion map B.
Figure 4-10 presents results of comparison for an initial time shift of
pulses in neighboring channels of 50 ps. The peak power of the initially
Gaussian pulses is 3.13 mW, the FWHM duration is set to 16.74 ps.1 As seen
in Figure 4-10 the middle channel experiences a larger accumulation of tim-
ing jitter compared to the outer channel. This can be explained by the larger
number of close neighbors the middle channel sees compared to the outer
one. Thus, it experiences more collisions with slowly moving neighboring
channels. The outermost channels 1 and 10 experience the same timing jit-
ter, the values for all intermediate channels lie in between the timing jitter
of channel 5 and 10. Figure 4-10 shows good agreement between the elastic
collision approximation (ECA) method and full numerical simulations.
1. Pulse width and peak power are selected such that they define a pulse, which is similar in
energy and width to a DMS solution for the given dispersion map.
L1-Llaunch Llaunch
D1
D2
L2
LA
L1-Llaunch Llaunch
D1
D2
L2
LA
76 A. Richter - Timing Jitter in WDM RZ Systems
Modeling of timing jitter
Figure 4-10. CITJ versus distance for the middle (m = 5) and outer (m = 10)
channel in a 10 channel system. Results of the ECA method (dashed line) are
compared with results using full numerical simulations (solid line). Dispersion
map B was applied (see Table 4-2) [61].
In summary, a semi-analytical model has been presented that allows cre-
ation of an efficient algorithm for the accurate estimation of XPM-induced
timing jitter in WDM systems applying arbitrary RZ pulse shapes. The com-
putational effort is reduced by two orders of magnitude, as the functional
dependence of CITJ on the shape of the signal pulses is derived analytically.
Example results agree well with reference data gathered from full numerical
simulations. Additional validations and detailed application examples evalu-
ating the dependence of CITJ on different system parameters are presented
in Chapter 5.
A. Richter - Timing Jitter in WDM RZ Systems 77
Chapter 5
WDM system simulations -
timing jitter
5.1 Overview
Various results of system simulations are presented in this chapter.
Firstly, ASE-noise induced timing jitter (ANTJ) is investigated. During the
first part of this chapter, it is shown that ANTJ depends strongly on the dis-
persion map and the amplifier positioning. As there is already a numerous
amount of literature published on this topic, their findings are briefly dis-
cussed and summarized here.
In the following part, the accumulation of XPM-induced timing jitter1 in
dispersion-managed WDM transmission systems is investigated. It will be
shown that CITJ depends strongly on the average dispersion, channel spac-
ing and number of channels interacting nonlinearly during propagation.
Additionally, it is shown that initial pulse positioning in the bit interval
plays a crucial role on the accumulation of XPM-induced timing jitter.
Most simulations were performed using the VPIsystems product
VPItransmissionMakerWDM, and its predecessor Photonic Transmission
Design Suite, which are both synthesis and analysis tools to explore WDM
system and link designs [153]. They apply diverse types of signal representa-
tions, which allow discrimination throughout the fiber transmission between
data signals, optical noise and crosstalk signals [101]. The two semi-analyti-
1. Also noted as collision-induced timing jitter (CITJ).
78 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
cal algorithms for estimating ANTJ and CITJ1 have been incorporated in the
tools. They are part of today’s standard releases, frequently undergoing rig-
orous validation tests. However, there exist some additional features, which
are not part of the commercial release yet. Unless otherwise stated, idealized
transmitter and receiver models are applied, as their distortions are of negli-
gible influence or of no importance to the discussed subjects. Several, mainly
post-processing tasks, have been performed using MATLAB.
5.2 ANTJ in dispersion-managed systems
In this section, results of system simulations estimating ASE-noise
induced timing jitter are presented.
Consider Eq. (4-12), which describes the three contributions to ASE-noise
induced timing jitter. They are mainly determined by the local pulse power
and shape , as well as the accumulated dispersion from the point
where noise is added to the point of final destination. Thus, ANTJ is strongly
influenced by the applied dispersion map and the amplifier positioning
inside the dispersion map. These influences are investigated in this section.
Firstly, the dependence of dispersion and amplifier spacing is investigated
for classical soliton systems. Figure 5-1 shows simulation results of ANTJ
versus distance for six different cases of soliton systems. The upper group of
curves represents a set of cases, where the dispersion is set to
D= 1.0 ps/nm-km, whereas the lower group of curves represents results for
D= 0.078 ps/nm-km. The three curves for each set are gathered for three dif-
ferent amplifier spacings, namely LA= 1 km, 20 km and 50 km. The curve
LA= 1 km corresponds to the case of distributed amplification and serves as
reference. A noise figure of 4 dB of the amplifiers is assumed for all the cases.
1. See Chapter 4, p. 51, Chapter 4, p. 59 for details.
qzt,()
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 79
Figure 5-1. ANTJ versus distance for two different dispersion values and
three different amplifier spacings, NF = 4 dB.
Figure 5-1 shows that ANTJ is growing with increased amplifier spacing.
This is mainly due to the fact that the ratio between amplifier spacing LA
and effective fiber length Leff is growing with increased amplifier spacing.
Hence, the power enhancement of solitons in the average soliton regime1 is
raising.
A second study is performed on the dependence of ANTJ accumulation on
the dispersion map. For this, the results of a soliton system are compared
with a dispersion-managed soliton (DMS) system. It is important to select
carefully the values for pulse widths and peak powers in order to be able to
compare the behavior of both systems. The FWHM duration is 20 ps for the
soliton, and 16.74 ps for the DMS. The durations are set such that both
pulses have the same energy. The dispersion of the soliton system is
D= 0.078 ps/nm-km. This corresponds to the average dispersion of the DMS
system. See Table 4-1 for details on the dispersion map A, which is applied
here again [132]. The amplifier spacing is set to LA=50km.
Figure 5-2 shows the results of three different scenarios. Curve A refers to
the results for the average soliton system with LA= 50 km. Curves B1 and B2
refer to the results of the DMS system. B1 shows ANTJ after each amplifier,
whereas B2 shows ANTJ after each dispersion map only. For comparison,
curve C shows the accumulated timing jitter for the equivalent soliton sys-
1. See Chapter 3, p. 35 for details.
D=0.078ps/nm-km
D=1ps/nm-km
LA=50km
LA=20km
LA= 1km
80 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
tem with distributed amplification1. Figure 5-2 (a) shows the accumulated
ANTJ values for the case that amplifiers have a noise figure of 4 dB, and
Figure 5-2 (b) for the case that they have a noise figure of 6 dB.
Figure 5-2. ANTJ versus distance for different dispersion maps, A: average
soliton system with LA= 50 km, B: DMS system (see Table 4-1), snap shots
after each amplifier (B1), and after each dispersion map (B2), C: soliton system,
distributed amplification; (a) NF = 4 dB, (b) NF = 6 dB.
As shown in Figure 5-2, the dispersion-management of the DMS system
reduces the accumulated ANTJ by an amount of 20% (curve B2 is below
curve A by approximately this amount). This is mainly due to the power
enhancement of the DMS system compared to the soliton system2. While the
accumulation of ANTJ inside the dispersion map of the DMS system is rap-
1. Distributed amplification is modeled by placing an amplifier every one kilometer.
2. See Chapter 3, p. 37 for details.
A
B1
C
B2
A
B1
C
B2
(a)
(b)
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 81
idly changing (curve B1), it is comparable in average to the soliton system
with distributed amplification (curve B2 is almost matching curve C). The
increase of the noise figure from 4 dB to 6 dB yields also an 20% increase in
ANTJ (compare results of Figure 5-2 (a) and (b)).
5.3 CITJ in WDM transmission systems
In this section, results of various system simulations estimating
XPM-induced timing jitter are presented.
Consider Eq. (4-31), which describes the collision-integral defining the
strength of the collision between a pulse in channel m with its closest pulse
in channel k at a certain distance point. The collision-integral is mainly
determined by the local pulse power and shape , and the local over-
lap of the colliding pulses, given by the relative time delay of both pulses
. As outlined in Chapter 4, p. 66, there are three contributions to
accounting for the dispersion-induced timing difference, the initial pulse
position difference, and the average position deviation due to collisions. Since
the third contribution is usually small, is strongly influenced by the
applied dispersion map and the initial settings of the pulses inside the bit
interval. The pulse power and shape are determined by the amplifier posi-
tioning inside the dispersion map. The following system design parameters
are investigated:
•dispersion map
• amplifier positioning
• dispersion slope
• RZ modulation scheme
• channel spacing
• initial pulse positioning.
qmzT,()
Θmk z() Θmk
Θmk
82 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
5.3.1 Dependence of CITJ on dispersion map and
amplifier positioning
XPM-induced timing jitter varies for different WDM channels even when
each channel is experiencing the same dispersion. Firstly, systems with a
zero dispersion slope are investigated to evaluate this difference in timing
jitter. The impact of dispersion slope on the accumulation of XPM-induced
timing jitter will be investigated in Chapter 5, p. 85.
The growth of CITJ with distance is not a monotonous function as shown
in Figure 4-10. There are distinctly observable humps, which are periodically
spaced. For this example, these humps are clearly visible up to 6,000 km,
after this they start to degrade. The analysis of several other dispersion
maps later in this chapter shows that the period of these humps is pro-
portional to the bit passing distance for neighboring channels with respect to
average dispersion, i.e.,
(5-1)
where is the bit duration, is the spacing between channel m and k,
is the average dispersion of the dispersion map.
In the case of Figure 4-10, is calculated to be 1,488 km. It follows from
Eq. (5-1) that the period of humps will scale in inverse proportion to the aver-
age dispersion for different dispersion maps and equal channel spacing.
Eq. (5-1) is validated by using two other examples of a
10 channel x 10 Gbit/s system illustrated in Figure 5-3 and Figure 5-4. The
dispersion map for both graphs in Figure 5-3 is described in Table 5-1.
The peak power of the initially Gaussian-shaped pulses was 2.9 mW. The
FWHM duration was set to 16.75 ps; no additional chirp was applied. Note
that the pulse width and peak power are selected such that they define a
Table 5-1. Parameters of dispersion map C.
L193.0 km D13.0 ps/nm-km
L27.0 km D2-38.74 ps/nm-km
Llaunch 46.5 km
LA50.0 km
Lh
Lh
Tb
Dave∆λmk
----------------------------=
Tb∆λmk
Dave
Lh
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 83
pulse, which is similar in energy and width to a DMS solution for the given
dispersion map. The initial time shift of pulses in neighboring channels was
50 ps to reduce the effects from initial pulse overlaps1. The channel spacing
is 100 GHz (Figure 5-3 (a)) and 75 GHz (Figure 5-3 (b)). Amplifiers are
located at the midpoints of the normal and anomalous spans [132].
Figure 5-3. CITJ versus distance for the middle (m = 5) and outer (m = 10)
channel in a 10 channel system. Results of the ECA method (dashed line) are
compared with results using full numerical simulations (solid line). Dispersion
map C was applied (see Table 5-1). Channel spacing was (a) 100 GHz [59], (b)
75 GHz [61].
Figure 5-3 shows that the accumulated timing jitter is almost twice as
large for 75 GHz channel spacing than it is for 100 GHz channel spacing.
Using Eq. (5-1), the period of humps is calculated to be approximately
Lh= 1,600 km for 100 GHz channel spacing and approximately
1. See Chapter 5, p. 92 for details on the influence of initial pulse positioning.
(a)
(b)
84 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
Lh= 2,133 km for 75 GHz channel spacing. Both values agree well with
results presented in Figure 5-3.
In Figure 5-4 results for the dispersion map A are shown, which has
already been introduced in Chapter 4, p. 57. The parameters of this disper-
sion map are listed in Table 4-1. The channel spacing was 75 GHz as in the
lower graph of Figure 5-3. The peak power of the initially Gaussian pulses
was 2.8 mW, the initial time shift of pulses in the neighboring channels was
50 ps.
The calculated period of humps is Lh = 2,133 km, as in Figure 5-3 for
75 GHz channel spacing. Note that the local dispersion values of the disper-
sion maps B and C are very different, which results in considerably different
pulse dynamics along the map. However, the average dispersion in
Figure 5-3 and Figure 5-4 is the same. It is remarkable that the accumulated
timing jitter values match as well, suggesting that CITJ is mainly deter-
mined by the average dispersion and channel spacing for these DMS scenar-
ios.
Note that in all the three cases presented in Figure 5-3 and Figure 5-4 the
calculated period of humps agrees well with the results using full numerical
simulations up to 5000 km. For larger distances there is some discrepancy
between the curves, however, the agreement remains reasonably good.
Figure 5-4. CITJ versus distance for the middle (m = 5) and outer (m = 10)
channel in a 10 channel system. Results of the ECA method (dashed line) are
compared with results using full numerical simulations (solid line). Dispersion
map A was applied (see Table 4-1). Channel spacing was 75 GHz [131].
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 85
5.3.2 Dependence of CITJ on dispersion slope
Up to this point, the dispersion slope was set such that the slope of GVD
becomes zero1. This is a reasonable assumption for higher order dispersion
management (HODM), as employed in modern long-haul transmission sys-
tems (see Table 1-1). The influence of the dispersion slope is analyzed in the
next example for the case that no HODM is applied. The considered disper-
sion map is very similar to the dispersion map A described in Table 4-1. So
the results presented in this section are comparable with the results pre-
sented in Figure 4-10 already. Only that now, a non-zero dispersion slope
was considered. The dispersion map parameters are presented in Table 5-2.
The channel spacing is 100 GHz. The initial time shift of neighboring chan-
nels was 50 ps. The FWHM duration of the Gaussian pulses at the input was
set to 16.74 ps. Due to the non-zero slope of the GVD, channel 1 experiences
an average dispersion of 0.244 ps/nm-km, while channel 10 experiences an
average dispersion of only 0.03 ps/nm-km. To ensure stable pulse propaga-
tion for all the 10 channels, increased average dispersion can be compen-
sated by an increased nonlinear phase shift, which arises for higher peak
power values, as outlined in Chapter 3, p. 37. Thus, pulses are launched with
different peak powers in different channels (see Table 5-3).
1. With Eq. (2-6), this means that .
Table 5-2. Parameters of dispersion map D.
L195.66 km D14.0 ps/nm-km
S10.04 ps/nm2-km
L24.34 km D2-85.0 ps/nm-km
S2-0.2 ps/nm2-km
Llaunch 47.83 km
LA50.0 km
Table 5-3. Peak power values of channel 1 to 10 at the input of the fiber.
channel12345678910
P [mW] 8.0 7.2 6.4 5.6 4.8 4.0 3.3 2.5 1.7 1.0
S
S2–Dλ⁄=
86 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
Figure 5-5 shows the resulting CITJ versus transmission distance. It is
seen that the timing jitter in channel 1 is significantly larger than it is for
channel 10. This is due to the fact that pulses in channel 1 experience a
larger amount of XPM because of their enhanced peak power values. There is
a good agreement of results obtained from the ECA method and from full
numerical simulations for channels 1 and 10, and a reasonably good agree-
ment for channel 6. Note that the humps are not visible in Figure 5-5 since
the periodicity of bit passing distances is violated as different channels have
different average dispersions.
Figure 5-5. CITJ versus distance for a system with non-zero dispersion slope.
Results of the ECA method (dashed line) are compared with results using full
numerical simulations (solid line). Dispersion map D was applied (see
Table 5-2) [61].
5.3.3 Dependence of CITJ on RZ modulation scheme
In all cases considered so far RZ pulses did not expand beyond their bit
intervals. In the following, timing jitter in CRZ systems is investigated,
where pulses spread over more than one bit duration1. When RZ pulses
expand beyond their bit intervals, one would expect that pulse powers
decrease, which reduces the effect of XPM, in turn suggesting that CITJ
becomes insignificant. Hence, one can expect that the elastic collision
1. As introduced in Chapter 3, p. 41.
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 87
approximation (ECA) as presented in Chapter 4, p. 59 will still be valid for
systems with large stretching factors.
In order to investigate the relevance of the ECA method for CRZ systems,
CITJ is calculated for a 10-channel CRZ system with parameters similar to
those in [20], [50]. The bitrate per channel was 10 Gbit/s; the FWHM pulse
duration of the initially raised-cosine shaped pulses was 50 ps. The peak
power was set to 0.6 mW; slight sinusoidal phase modulation of
∆fmax =18.5GHz (mp= 0.6) was applied. The dispersion map is described in
Table 5-4.
Figure 5-6 shows pulse evolution over a single dispersion map length of
495 km [133]. Snap shots were taken every 45 km, right after the inline
amplifiers. The pulse spreading is clearly visible. It can be seen that pulses
widen over the two neighboring bit intervals at their maximum, and shrink
to a fraction of their original width developing very large local peak powers of
about three times the initial peak power. At the end of each span, pulses are
back to their original shape. Isolated marks do not suffer significantly from
intersymbol interferences. Streams of marks, however, develop large ghost
pulses right at the edges of the bit interval.
Figure 5-6. Pulse evolution over one dispersion map length for an isolated
mark (left) and a stream of marks (right), snap shots were taken after each
amplifier (every 45 km).
Table 5-4. Parameters of dispersion map E.
L1450.0 km D1-2.0 ps/nm-km
L245.0 km D219.8 ps/nm-km
Llaunch 225.0 km
LA45.0 km
0.6
1.2
[mW]
1.8
50 150-150 -50 [ps] 50 150-150 -50 [ps]
0 1 0 1 1 1
0.6
1.2
[mW]
1.8
50 150-150 -50 [ps] 50 150-150 -50 [ps]
0 1 0 1 1 1
88 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
Figure 5-7 compares CITJ results from the ECA method with data from
full numerical simulations versus distance. Timing jitter values of a middle
channel are presented for 100 and 75 GHz channel spacing. For both cases,
very good agreement is observed. Thus, the assumption made above holds −
that when RZ pulses expand significantly, pulse peak powers decrease and
XPM effects are reduced such that contributions of collisions between these
pulses to accumulated CITJ is small. Consequently, the elastic collision
approximation remains valid. Note that even the relatively large ghost
pulses, which develop between neighboring marks during propagation, have
no significant influence on accumulation of XPM-induced timing jitter.
Figure 5-7. CITJ versus distance. Results of the ECA method (solid line) are
compared with results using full numerical simulations (crosses). Upper curve
is for 75 GHz channel spacing, lower curve is for 100 GHz channel spacing.
Dispersion map E was applied (see Table 5-4) [133].
It is noticeable that the CRZ system is more robust against CITJ build up
than the DMS systems analyzed so far. This is mainly due to the fact that in
DMS systems, pulses maintain their pulse shape over the dispersion map,
and thus, their relative large peak power values. In the CRZ system pulses
spread drastically, and so, reduce their sensitivity to XPM effects.
02000 4000 6000 8000 10000
0
1
2
3
4
5
Distance [km]
Timing Jitter [ps]
∆f= 75 GHz
∆f = 100 GHz
02000 4000 6000 8000 10000
0
1
2
3
4
5
Distance [km]
Timing Jitter [ps]
∆f= 75 GHz
∆f = 100 GHz
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 89
5.3.4 Dependence of CITJ on channel spacing
It follows from Eq. (5-1) that the period of humps will scale in inverse pro-
portion to channel spacing for different dispersion maps with the same aver-
age dispersion.
From soliton theory it is derived that the frequency shift of a two-soliton
collision is inversely proportional to channel spacing [27]. This is intuitively
clear as an increased frequency separation of the two colliding pulses reduces
their nonlinear interaction length, i.e., the collision length 1. However,
these results have been derived for transmission links with constant disper-
sion, where two pulses undergo only one collision with each other during
propagation.
For DMS or other practical RZ modulation schemes employing dispersion
management, two pulses undergo multiple collisions with each other. So a
simple relation might not hold for them. In order to investigate the depen-
dence of accumulated CITJ on channel spacing more closely, two previously
discussed systems are investigated here again. These are the DMS system,
which was studied in Chapter 5, p. 82 (dispersion map C), and the CRZ sys-
tem, which was analyzed in Chapter 5, p. 87 (dispersion map E).
Figure 5-8 shows the XPM-induced timing jitter versus channel spacing at
different distances for the DMS system. The CITJ is monotonically decreas-
ing for increased channel spacing with the following approximate relation
. (5-2)
The two dashed reference curves drawn in Figure 5-8 visualize this behavior.
1. For definition of , see Chapter 2, p. 19.
Lc
Lc
∆f
σt
21
∆f3
--------
∼
90 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
Figure 5-8. CITJ versus channel spacing for different transmission distances,
starting from the bottom curve, each curve represents an increase in distance of
400 km. Dispersion map C was applied (see Table 5-1).
Figure 5-9 shows the XPM-induced timing jitter versus channel spacing at
different distances for the CRZ system [133]. It is remarkable that CITJ is
not monotonically decreasing with increased channel spacing. XPM-induced
timing jitter seems to accumulate at certain channel spacings faster than at
others. However, only small resonances are visible. In general, the variance
of the pulse postion is approximately in inverse proportion to the third power
of the channel spacing.
Figure 5-9. CITJ versus channel spacing for different transmission distances,
starting from the bottom curve, each curve represents an increase in distance of
495 km. Dispersion map E was applied (see Table 5-4).
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 91
Resonances of the XPM efficiency versus channel spacing have also been
observed experimentally [127]. For a two channel RZ system, the periodicity
of such resonances could be related to the bit duration divided by the disper-
sion-length product of one amplifier span [42]:
(5-3)
where is the channel spacing, is the bit duration,
is the dispersion-length product of the dominating span.
For channel spacings just falling on these points of resonance, XPM effects
tend to add in a more coherent manner. For channel spacings just falling in
the middle of these resonances, XPM effects add in a more incoherent man-
ner.
However, the small resonances observed in Figure 5-9 can not be directly
related to Eq. (5-3). The reason for this could be that in [42], separable pulse
shapes are considered, whereas for the dispersion map considered here,
pulses smear out rapidly over the complete bit duration.
In order to investigate the resonance behavior more closely, another dis-
persion map was investigated (see Table 5-5). This dispersion map applied
the same initial pulse shape (CRZ) and power. However, its dispersion map
incorporates more potential resonance distances.
Figure 5-10 shows the dependence of CITJ on channel spacing for differ-
ent distances. It is remarkable to note that several clear resonances are
observable in this case. Also, note that these resonances are dependent on
the transmission distance, which is indicated by the lines in Figure 5-10.
With increased distance (most upper curve corresponds to about 20,000 km)
the resonances shift towards lower channel spacing, and move closer
together. However, this trend is only visible after approximately 13,500 km.
Before this, no real trend can be determined.
Table 5-5. Parameters of dispersion map F.
L1243.0 km D14.1 ps/nm-km
L227.0 km D2-36.7 ps/nm-km
Llaunch 108.0 km
LA54.0 km
∆λkl
Tb
LD()
span
-------------------------=
∆λkl Tb
LD()
span
92 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
Figure 5-10. CITJ versus channel spacing for different transmission dis-
tances, starting from the bottom curve, each curve represents an increase in dis-
tance of 540 km. Dispersion map F was applied (see Table 5-5).
An important question to address is why is there such a big difference in
CITJ dependence on channel spacing between Figure 5-9 and Figure 5-10.
Both dispersion maps have approximately the same average dispersion. The
main differences between both setups is the amplifier spacing (45 km versus
54 km) and the absolute value of accumulated dispersion per amplifier spac-
ing (90 ps/nm versus 221.4 ps/nm). Additionally, dispersion map E (see
Table 5-4) has the only resonance distance at LA= 45 km, whereas disper-
sion map F (see Table 5-5) has several potential resonance distances:
L2=27km, LA=54km, L1=273km.
In summary, CITJ is in general decreasing with increased channel spac-
ing according to Eq. (5-2). Slight to moderate resonances might be observ-
able. The locations of these resonances depend on the periodic distances in
the dispersion map, over which complete bit crossings could happen for cer-
tain channel spacings, leading to coherent addition of XPM effects.
5.3.5 Dependence of CITJ on initial pulse positioning in
bit interval
It has been shown in [118] that the temporal overlap of pulses in different
WDM channels at the input of the fibers causes unbalanced collisions, and
thus, initial frequency shifts. Figure 5-11 shows the frequency shift due to
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 93
the collision of two solitons in a lossless fiber versus distance [118]. The
channel spacing of the two sech-shaped pulses with FWHM duration of 20 ps
was 100 GHz, the dispersion was D= 0.4 ps/nm-km.
Figure 5-11. Dependence of frequency shift due to the collision of two solitons
in a lossless fiber. Channel spacing was 100 GHz, FWHM duration was 20 ps,
dispersion was 0.4 ps/nm-km.
The overlap of pulses at the input is determined by the initial pulse posi-
tioning (IPP). If pulses are initially overlapping, for instance, are placed at
the same position in the bit interval, they experience an incomplete collision
right at the beginning of the propagation. According to Figure 5-11, this
results in a strong frequency shift.
These frequency shifts have been studied in detail for the propagation of
classical solitons [13], [95] and periodically amplified soliton systems [1].
However, only single two-soliton collisions have been considered. Beside add-
ing a large contribution to CITJ right at the transmitter, the overlap of
pulses at the fiber input also impacts the balance of all the consecutive colli-
sions in the fiber. This effect has not been addressed yet.
In real WDM systems IPP can hardly be controlled since different signal
pulse sources, which are typically used for different channels, are mutually
never synchronized allowing a slow relative drift in time. This drift results in
a variation of the XPM-induced timing jitter. In this section, the impact of
random IPP on the accumulated pulse timing jitter for an arbitrary number
of collisions is determined.
94 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
The 10 x 10 Gbit/s WDM transmission system under investigation is the
same as the one introduced in Chapter 4, p. 74 (dispersion map B). Only the
amplifier spacing is increased to a value of 100 km (see Table 5-6 for details).
The channel spacing was varied between 100 GHz and 75 GHz. Gaussian
pulses were launched at the input with FWHM duration of 16.75 ps and peak
power of 5.78 mW.
The evaluation of statistical CITJ variations using full numerical simula-
tions in a reasonable amount of time is very memory intensive, if possible at
all, as one must repeat CITJ calculations very often with different settings of
the IPP. However, one can explore the impact of random IPP fluctuations
efficiently when using the elastic collision approximation. It is assumed that
the IPP of all the channels is randomly varying, mutually independent, and
uniformly distributed within the bit interval. For each simulation setup 180
runs were performed1. At each run, the initial pulse central times were
selected randomly. As a measure of the dependence of CITJ on IPP, the aver-
age and variation (± standard deviation) of CITJ values are calculated for
each setup.
Figure 5-12 shows results for a middle channel. It shows the average (cen-
ter solid line) and the variation of CITJ (shaded area around the solid line
between the two dashed lines). Figure 5-12 shows that for 100 GHz channel
spacing the spread in CITJ after 10,000 km is about 3 ps, while the average
timing jitter is 7.6 ps. This average CITJ increases by a factor of 1.5 while
the spread doubles and becomes even more significant when the channel
spacing is reduced to 75 GHz. Similar curves, as depicted in Figure 5-12, can
be obtained for all channels.
Table 5-6. Parameters of dispersion map G.
L195.6 km D14.0 ps/nm-km
L24.4 km D2-85.0 ps/nm-km
Llaunch 47.8 km
LA100.0 km
1. Convergence of the calculated values is already reached for 100 simulation runs.
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 95
.
Figure 5-12. Average and variation of CITJ of the middle channel due to IPP.
Dispersion map G was applied (see Table 5-6) [61].
Note that the humps in the evolution of CITJ for individual IPP settings as
observed in Chapter 5, p. 82 have vanished. CITJ is growing almost linearly
for increased distance.
Figure 5-13 shows the statistics of CITJ for all the 10 channels after
10,000 km. For each simulation setup, variations (length of bars) of the final
timing jitter values are about the same for all channels, namely 3.0 ps (chan-
nel spacing 100 GHz) and 5.9 ps (channel spacing 75 GHz), respectively.
While variations are almost constant for each channel, the average timing
jitter values of the two outermost channels are smaller compared to the val-
ues of the other channels. This means that for the investigated configura-
tions here, random variations of the CITJ per channel due to random IPP
96 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
settings are mainly determined by the collisions with pulses in neighboring
channels, which is due to the dispersion-induced walk-off between channels.
Figure 5-13. Average and variation of CITJ due to IPP for all
channels after 10,000 km.
Figure 5-13 implies that the observed variations inside each channel also
lead to a remarkable variation in the accumulated timing jitter values
between the channels. Figure 5-14 presents the average and variation of the
maximum timing jitter differences between the 8 middle channels against
distance. The two outermost channels are not regarded in this analysis, since
their smaller average timing jitter values would falsify the results. After
10,000 km, the difference of the accumulated timing jitter between the chan-
nels varies statistically between around 3.0 ps and 5.0 ps for a channel spac-
ing of 100 GHz, and between around 5.5 ps and 9.5 ps for a channel spacing
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 97
of 75 GHz. Thus, while the phase margin is acceptable for some channels, it
can be inacceptable for other channels.
.
Figure 5-14. Average and variation of maximum difference of
CITJ between channels 2 to 9.
In summary, random fluctuations of IPP may have a significant impact on
the degradation of phase margins in WDM RZ systems, as they introduce
large variations of timing jitter from channel to channel. Even if a setup is
optimized to yield acceptable phase margins, timing jitter caused by a
non-ideal laser source, or pre-propagation effects inside a complex network,
can degrade the performance significantly.
98 A. Richter - Timing Jitter in WDM RZ Systems
WDM system simulations - timing jitter
A. Richter - Timing Jitter in WDM RZ Systems 99
Chapter 6
Estimation of system
performance
6.1 Overview
In the Chapter 4 sources of distortions of pulse positions were discussed.
This chapter is dedicated to the estimation of system performance.
Firstly, the main performance measures of a WDM transmission system
are presented. One aspect that these measures all have in common is that
timing jitter influences are rather loosely reflected.
Therefore, an estimation model is presented, which determines the influ-
ence of amplitude and position fluctuations on system performance.
Finally, the influence of timing jitter on the detected BER is investigated
for various system examples. It is shown that, depending on the system
under consideration, timing jitter might be the major reason for detection
errors. It is shown that in the presence of timing jitter, BER estimation using
the Gaussian approximation for amplitude fluctuations at the detector might
deliver strongly misleading results.
100 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
6.2 Performance measures
6.2.1 Optical signal-to-noise ratio (OSNR)
The optical signal-to-noise ratio (OSNR) describes the ratio of optical sig-
nal power to optical noise power. For systems limited by ASE-noise from
EDFAs, it gives a good estimate of the system performance. It is not suited
for the performance estimation of systems where dynamic propagation
effects such as dispersion and phase modulation due to the Kerr nonlinearity
play an important role. With Eq. (2-33), the OSNR for a single EDFA with
constant output power, Pout, is given by1
(6-1)
where is the amplifier noise figure, is the amplifier gain,
is the photon energy,
is the optical measurement bandwidth.
To cover long transmission distances, amplifiers are placed along the line
such that they compensate for the total loss along the link. For a periodically
amplified fiber link, with N gain-controlled amplifiers compensating for the
span loss, the OSNR simply becomes [89]
.(6-2)
Usually, signal and noise power values are measured directly from the
optical spectrum analyzer (OSA), which displays power versus wavelength or
frequency for a fixed resolution bandwidth of typically ∆λ =0.1 nm
(∆f=12.5 GHz).
1. ASE-noise is measured here over both polarizations.
OSNR Pout
PASE
---------------Pout
NF G⋅1–()hf∆f
---------------------------------------------==
NF G
hf
∆f
OSNR Pout
NF G⋅1–()hf∆fN⋅
------------------------------------------------------=
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 101
6.2.2 Eye-opening penalty (EOP)
The OSNR is a time-averaged performance measure, only the average
power of optical carriers and noise are considered. However, dynamic propa-
gation effects such as dispersion, nonlinearities and filtering might strongly
influence the pulse shape leaving the average power constant.
The eye-opening penalty (EOP) is a performance measure considering
dynamic propagation effects. It is defined as the ratio between a reference
eye-opening , typically gathered from a back-to-back measurement,
and the eye-opening after transmission .
(6-3)
where is the sample time within the bit interval,
is the difference between the amplitudes of the lowest mark and
the highest space.
The EOP is especially useful for noise-free system evaluations, as it gives
a useful measure of deterministic pulse distortion effects. Note that it is
dependent on the sample time, where the eye-opening is measured within
the bit interval. When the detected eye shows timing jitter resulting from
intra- or interchannel ISI effects, the optimum EOP might not be affected.
However, the sampling width over which an EOP below a certain threshold
level can be achieved will be.
If noise is present, determination of the EOP can become ambiguous as
the definition of lowest mark and highest space depends on the simulated bit
stream1. A better method of determining the eye-opening is given by [33]
(6-4)
where are the mean values of the spaces and marks, respectively,
are the standard deviations of the spaces and marks,
respectively.
1. If an infinite number of bits could be collected to plot the eye diagram, it would be totally
closed.
EOref
EOdec
EOP t() 10log EOref t()
EOdec t()
------------------------
=
t
EO
EO µ13σ1
–()µ
03σ0
+()–=
µ01,
σ01,
102 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
For Gaussian amplitude fluctuations, the value range incorporates
approximately 99.7% of all possible cases.
6.2.3 Q-factor
Another parameter that determines system performance is the Q-factor. It
is defined as the difference between mean level of marks and spaces, normal-
ized by the sum of their standard deviations [18].
.(6-5)
The Q-factor can be related to the eye-opening as depicted in Figure 6-1.
Figure 6-1. Relationship between eye-opening and Q-factor.
When amplitude fluctuations of the marks and spaces follow a Gaussian
probability density function (PDF), given by [140]
(6-6)
the threshold corresponds to the value, where the arguments of the expo-
nentials in both Gaussian PDFs are identical. It is given by
.(6-7)
µµ,3±σ()
Qµ1µ0
–
σ1σ0
+
-------------------=
µ0
µ1
Qσ1
Qσ0
DQ
σ0
σ1
µ0
µ1
Qσ1
Qσ0
DQ
σ0
σ1
fXx() 1
2πσ
---------------exp xµ–()
2
2σ2
--------------------–
=
DQ
DQ
σ0µ1σ1µ0
+
σ0σ1
+
---------------------------------=
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 103
The OSNR can be used to derive an approximate value for the Q-factor, if
the system under investigation shows no significant performance degrada-
tion from propagation effects other than noise (such as dispersion and non-
linearities), and an infinite extinction ratio is assumed. The Q-factor can be
related to the OSNR at the output of the receiver for a system using an opti-
cal pre-amplifier, which ensures that the input power to the receiver is high
so that thermal and shot noise are negligible. The Q-factor in dB1 is then
given by [154]
(6-8)
where is the bandwidth of the optical receiver,
is the bandwidth of the electrical post-detection filter.
The relationship assumes that signal-ASE beat noise and ASE-ASE beat
noise are the main contributions to electrical noise at the detector.
6.2.4 Bit error rate (BER)
Ultimate measure of system performance is the bit error rate (BER). It is
defined as probability of faulty detected bits, e.g, marks detected as spaces
and vice versa.
6.2.4.1 Monte Carlo (MC) experiment
The direct way of measuring BER is to calculate the ratio of corrupt
detected bits to total number of transmitted bits over a statistically signifi-
cant time period in so-called Monte Carlo (MC) experiments. No assumptions
about the probability distributions need to be applied. The BER found in MC
experiments is the system performance measure, which is most closely
related to realistic system conditions.
However, for fiber-optic communication systems BER values of interest
are usually very small (10-9 −10-12). This makes it difficult to measure the
BER directly as error counting needs to be performed over very long, some-
1. Logarithmic Q is related to linear Q by QdB =20log(Q).
QdB 20 2OSNR BoBe
⁄
114OSNR++
----------------------------------------------
log=
Bo
Be
104 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
times impossibly long time intervals. It is even more difficult to estimate the
BER via MC experiments from computer simulations. The variance of the
estimated BER in a MC experiment is given by [157]
(6-9)
where is the probability of error,
is the number of test samples in the MC experiment.
So to estimate a BER of 10-9, errors over at least 1011 bits need to be counted
to achieve an estimation accuracy1 of about 10%. This is barely possible in
laboratory experiments, however, totally unfeasible in computer simulations.
Therefore, techniques for approximating the BER are usually applied in
computer simulations. All these techniques make certain assumptions about
the PDF of amplitude fluctuations of marks and spaces at the detector (sta-
tistical techniques) or about the considered class of systems (analytical or
semi-analytical techniques).
6.2.4.2 Gaussian approximation (GA)
One statistical approach commonly applied is the so-called Gaussian
approximation (GA) technique. It is based on the assumption that amplitude
fluctuations of marks and spaces are Gaussian distributed. Under this
assumption, the BER is derived as
(6-10)
where is the probability that a is received while b was sent,
denotes the number of marks and spaces, respectively,
denotes the complementary error function,
is the decision threshold.
1. Measured as the width of the standard deviation.
σMC
2Pe1P
e
–()
N
---------------------------=
Pe
N
BER 1
N0N1
+
----------------------N1P01()N0P10()+[]=
1
N0N1
+
----------------------N1
2
------- erfc µ1D–
2σ1
--------------------
N0
2
------- erfc µ0D–
2σ0
--------------------
+=
Pab()
N10,
erfc x() 2
π
------- et2
–td
x
∞
∫
=
D
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 105
The optimum decision threshold , i.e., the amplitude value for which
the detected BER becomes minimum, is given at the point where the Gauss-
ian PDFs for marks and spaces cross each other, namely where
. (6-11)
can be derived to
. (6-12)
Under certain presumptions, the BER can be estimated indirectly by cal-
culating the Q-factor. Both measures can be exactly related, for the case that
the decision threshold is set as given in Eq. (6-7) and that the number of
transmitted marks equals the number of transmitted spaces.
(6-13)
The approximate solution in Eq. (6-13) is derived from an upper bound
approximation of the integration by parts of the erfc-function [99]. Figure 6-2
plots BER versus Q-factor highlighting several reference values.
Dopt
1
σ0
------ exp Dopt µ0
–()
2
2σ0
2
--------------------------------–
1
σ1
------ exp µ1Dopt
–()
2
2σ1
2
--------------------------------–
=
Dopt
Dopt
µ1µ0
–
2
----------------for σ0σ1
=
σ0
2µ1σ1
2µ0
–σ1σ0µ1µ0
–()
2σ0
2σ1
2
–()ln σ0
2σ1
2
⁄()+–
σ0
2σ1
2
–
-----------------------------------------------------------------------------------------------------------------------------------------for σ0σ1
≠
=
BER 1
2
---erfc Q
2
-------
1
Q2π
----------------Q2
2
-------–
exp≈=
106 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
Figure 6-2. BER versus Q-factor according to Eq. (6-13).
Note that the optimum decision threshold, e.g., the amplitude where the
PDFs for amplitude fluctuations of marks and spaces cross each other, is not
the same as applied in Eq. (6-13). To visualize the difference, Figure 6-3 plots
both decision thresholds for different ratios of the variances for marks and
spaces, . The mean of marks and spaces are set to one and zero, respec-
tively.
Figure 6-3. Decision thresholds for different ratios of , with and
. Solid lines: Decision threshold according to Eq. (6-12), with
. Dashed line: Decision threshold according to
Eq. (6-7).
σ0
2σ1
2
⁄
σ0
2σ1
2
⁄µ
00=
µ11=
σ1µ1
⁄0.5 0.4 0.3 0.2 0.1,,,,=
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 107
Figure 6-3 clearly shows the difference between both decision thresholds
for large numbers of , and thus the misinterpretation of the estimated
BER. Note, however, that for realistic applications in fiber-optic communica-
tions, where BER estimates of about 10-9 to 10-15 are of interest, the differ-
ence becomes marginal, and the approximate formula for the decision
threshold as given in Eq. (6-7) is good enough1.
Although the GA allows BER estimation to be feasible in computer simu-
lations, it still requires a large number of test bits to estimate the mean and
variance of the amplitude fluctuations with a good degree of numerical confi-
dence. For example, 800 bits need to be simulated to achieve 95% estimation
accuracy for the linear Q-factor. However, as the BER is related via the com-
plementary error function with the estimated standard deviation, and not
linearly like the Q-factor, the same number of bits will lead to a fluctuation
of 1.5 orders of magnitude for BER estimation around 10-9. An example sce-
nario is presented in Chapter 6, p. 112.
Beside numerical uncertainties, the validity of the GA needs to be investi-
gated very carefully. Even though disturbances tend to be Gaussian for a
reasonably large number of sources (central limit theorem [142]), and thus
would imply the validity of the GA, it fails for several practical cases. One
reason is that initially Gaussian disturbances are passed through nonlinear
elements (fiber, receiver) altering the statistics. Also intra- and interchannel
interferences might result in degrading effects, which could not be approxi-
mated by additive Gaussian disturbances, mainly because the critical num-
ber of interacting sources is not obtained.
Gaussian approximation including intersymbol interferences (GA-ISI)
An alternate approach is based on the GA technique, but considers intrac-
hannel intersymbol interferences (ISI) from neighboring bits [11]. Instead of
relating the Gaussian PDF of amplitude fluctuations to marks and spaces
only, all three-bit patterns are considered separately. That is, four patterns
are associated for each mark (010, 110, 011, 111) and for each space (101,
001, 100, 000). The BER is given as the superposition of the BER calculated
1. For cases, where Forward-Error Correction (FEC) schemes are used, BER values of up to
10-3 are of interest, and thus a proper selection of the decision threshold is important.
σ1
2
108 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
for the center bit of all eight patterns, weighted with respect to the number of
occurrences each pattern is propagated in the bit sequence,
(6-14)
where is the number of occurrences of the pattern k, and .
Note that the price for obtaining higher precision when including ISI
effects is that this technique requires a sample set of bits approximately four
times larger than the GA technique in order to reach the same numerical
accuracy as the GA technique.
6.2.4.3 Deterministic noise approximation (DNA)
If phase interactions between the noise from optical amplifiers and the
WDM signal comb can be neglected, noise can be modeled deterministically
at the optical receiver. This is possible, for instance, when the noise and sig-
nal streams are modeled to propagate separately from each other over the
transmission link using multiple signal representations [101]. The optical
signal can be represented by its full time dynamics. It is passed to the
receiver as a noise-free signal, however, taking deterministic impairments,
such as dispersion, crosstalk due to FWM and due to non-ideal filtering along
the optical path into account. Noise is described by wavelength-sensitive val-
ues of its power spectral density (PSD) directly. It is passed along the optical
path to obtain a deterministic measure of accumulated noise PSD versus
wavelength at the receiver input.
The variance of amplitude fluctuations for each bit is determined from the
noise accumulated along the optical path and the noise from the receiver.
The variance of the pulse amplitude can be written as
(6-15)
BER 1
8N
-------- Nkerfc µkD–
2σk
--------------------
k1=
8
∑
=
NkNN
k
k1=
8
∑
=
σa
2σsignal ASE,
2σASE ASE,
2σsh
2σth
2
+++=
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 109
where is the Signal-ASE beat noise, as defined in Eq. (2-38),
is the ASE-ASE beat noise, as defined in Eq. (2-39),
is the shot noise of the receiver unit, as defined in Eq. (2-36),
is the thermal noise of the receiver unit including the electrical
preamplifier, as defined in Eq. (2-37).
The main advantage of the DNA technique over the GA technique lies in
the fact that the demand for simulations of long bit streams is drastically
reduced, as there is no numerical uncertainty in determining the amplitude
variances. An example comparison with the GA technique is presented in
Chapter 6, p. 112.
However, the DNA technique still suffers from assumptions made about
the underlying PDFs of fluctuations for marks and spaces. It has, for
instance, been derived theoretically [105], and shown experimentally [28]
that the PDF of ASE-ASE beat noise is not Gaussian. Systems which are lim-
ited by ASE-noise contributions are better modeled by a chi-squared PDF for
the marks and spaces [104], [3]. Other modeling techniques are based on
finding the moment generating function1 for marks and spaces at the detec-
tor considering the influence of the nonlinear receiver characteristic and the
filtering on Gaussian noise sources [86], [87].
6.3 Impact of pulse timing jitter on BER
6.3.1 Motivation
To detect pulses correctly, the minimum amplitude as well as phase mar-
gins need to be considered. The reduction of the phase margin might be
induced by timing jitter, e.g., statistical fluctuations of the pulse positions.
Timing jitter manifests itself at the decision circuitry in fluctuations of the
1. Which is the Laplace-transform of the PDF.
σsignal ASE,
2
σASE ASE,
2
σsh
2
σth
2
110 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
detected amplitude. Even though the received pulse position might be
regarded as following a Gaussian distribution [77], [108], the resulting
amplitude fluctuations at the decision circuitry are not. They are strongly
dependent on the received pulse shape, which might be affected by fiber
propagation effects as well as the optical and electrical filter transfer func-
tions at the receiver. As with noise contributions, a large number of bits
would be required to be propagated over the WDM system link in order to
investigate the timing jitter by calculating mean and variance of pulse posi-
tions directly and derive from them the phase margin1.
System performance in the presence of timing jitter has been discussed in
some detail in [81] and others. The derived solutions, however, are either of
general nature without applicable solutions, or are very specific, making
severe assumptions about the pulse shape, e.g, are only valid for NRZ trans-
mission, or rectangular-shaped RZ pulses. In [108], joint measurement of
timing jitter and Q-factor was used to predict the performance for a single
channel soliton system. It has been shown that when the decision variable is
not Gaussian, the Q-factor is not sufficient to describe the system perfor-
mance.
In the following section, a technique is presented that considers fluctua-
tions of pulse amplitudes and positions for arbitrary pulse shapes, without
the need to simulate extensively long bit sequences. The BER is estimated
from the superposition of both effects.
6.3.2 Modeling
In Chapter 6, p. 108, the deterministic noise approximation technique was
introduced that allowed a fast and efficient estimation of the statistical
amplitude fluctuations due to noise and crosstalk sources. The total PDF of
amplitude fluctuations was assumed to be Gaussian with variance defined in
Eq. (6-15).
In Chapter 4, efficient techniques were developed for estimating the accu-
mulation of pulse timing jitter due to the main sources of phase margin deg-
radations in long-haul transmission systems, namely XPM and ASE-noise.
These techniques are restricted to finding estimates of the pulse position
1. See Chapter 4, p. 59 for a more detailed discussion.
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 111
average and variance only, as it has been widely accepted that pulse position
deviations due to these random processes are sufficiently described by a
Gaussian distribution [77], [108].
At the detector, the effects of all stochastic disturbances are summarized.
In order to investigate the impact of the different sources of timing jitter on
the total BER, their influence needs to be translated through the optical
receiver onto the electrical pulse shape seen by the decision circuitry. After
the optical receiver, the pulse shape is changed, depending on the response of
the photodiode and the electrical filter. However, it is assumed that the rela-
tive phase information of individual bits is not changed, such that individual
pulse positions after the receiver are assumed to still follow a Gaussian dis-
tribution.
The translation of pulse position statistics to pulse amplitude statistics is
strongly dependent on the pulse shape at the detector. To illustrate this, the
inverse of the average pulse shape is approximated using a Taylor
series expansion at an arbitrary power level
(6-16)
where , and are the bias, slope and curvature, respectively, of the
inverse of the pulse shape at .
For pure timing jitter disturbances, the PDF of the pulse amplitude can
then be written for an arbitrary sample time as [134]
(6-17)
where denotes a pulse shape related correction factor,
is the continuos Dirac-delta function,
is the normalized variance of the timing jitter.
g1–p() t=
p0
g1–p0
() t0a0a1p0a2p0
2Op
0
〈〉
3
()+++==
a0a1a2
p0
p
t
ftp() Bt() 1
2σ
˜t
2
----------p0
ta
0
–
a1
--------------–
–δpp
0
–()
p0
2σ
˜t
2
----------a2
a1
------p0
2
2a2
a1
------p0
ta
0
–
a1
--------------–
+–
exp
×exp=
Bt()
δxy–()
σ
˜t
2σta1
⁄=
112 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
Eq. (6-17) suggests that if the curvature could be neglected , Gaus-
sian distributed pulse position fluctuations would result in Gaussian distrib-
uted amplitude deviations with scaled variance. However, for many RZ
transmission systems, this cannot be assumed. So depending on the received
pulse shape, pulse timing jitter might lead to strongly non-Gaussian ampli-
tude fluctuations. Comparisons of the technique presented here with the GA
technique are presented in the following chapter.
The total PDF, which determines the BER, is given by the weighted sum
of all individual PDFs gathered from the pulse energy and position fluctua-
tions with respect to amplitude (see Figure 6-4).
Figure 6-4. Total PDF as superposition of PDFs from amplitude and timing
fluctuations.
Note that this technique considers deterministic pattern effects due to ampli-
tude fluctuations, but ignores ISI effects resulting from timing jitter. How-
ever, this might be important for soliton systems where close temporal
placement of pulses invokes soliton interactions1, which influence the posi-
tion of neighboring pulses [130].
6.4 WDM system simulations - BER
Diverse system scenarios are investigated in this section to illustrate the
effectiveness of the different BER estimation techniques proposed in the pre-
vious sections. Results have been published in [134].
1. See Chapter 3, p. 34 for details.
a20≈()
p
t t t
+=
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 113
6.4.1 RZ system over dispersion-managed link with
mainly SSMF
In the first example, a single-channel transmission system is investigated,
where timing jitter effects are negligible. Pulses are propagated with a
bitrate of 40 Gbit/s. The transmission link consists mainly of SSMF with a
nominal dispersion of 16 ps/nm-km. DCF spans with D = -90 ps/nm-km are
inserted to provide 100% dispersion compensation. The RZ pulses are
launched with 25% rise time and FWHM duration of 12.5 ps. No extra phase
modulation is applied. The receiver consists of a PIN photodiode and a 3rd
order Bessel filter with a 3-dB bandwidth of 32 GHz.
Firstly, the BER calculation method based on the deterministic noise
description (DNA method)1 is compared with the standard method of esti-
mating the mean and variance of pulse amplitude fluctuations due to noise
contributions directly (GA method)2. For this, a single RZ signal stream is
transmitted over 2,000 km, applying optical inline amplifiers with a noise
figure of 6.0 dB and a periodic spacing of about 94 km.
The modeled system example is limited by pulse energy fluctuations due
to signal-ASE and ASE-ASE beat noise; timing jitter effects can be neglected.
Figure 6-5 shows the calculated BER values versus distance. Lines with
crosses correspond to the DNA method using a signal stream of 128 bits,
whereas lines with squares correspond to the GA method based on 128 bits
in the upper graph and 1024 bits in the lower graph.
1. See Chapter 6, p. 108 for details.
2. See Chapter 6, p. 104 for details.
114 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
Figure 6-5. BER versus distance for the DNA technique (crosses) and the GA
technique (squares) using 128 bits (upper) and 1024 bits (lower) [134].
The upper graph shows strong fluctuation of BER values calculated with
the GA technique because of poor numerical estimation accuracy. It is clearly
visible that 128 bits are not sufficient to give a reasonably good BER estima-
tion. The smooth, straight increase with distance of BER values calculated
with the DNA technique suggests that all important stochastic effects had
been sufficiently modeled using the separable noise representation [101].
The lower graph suggests that results from both techniques converge if
the uncertainty of moment estimation is eliminated by increasing the bit
stream length sufficiently. This also implies that timing jitter effects are,
indeed, not of system limiting importance for the modeled setup. The advan-
tage of the DNA technique is obvious in this case as it allows an increase of
simulation speed by about one order of magnitude.
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 115
6.4.2 Dispersion-managed soliton system
In the next example a WDM transmission system is considered using dis-
persion-managed solitons. Ten channels are transmitted with a bitrate of
10 Gbit/s per channel and a channel spacing of 100 GHz. The dispersion map
is the same as described in Table 4-1 (dispersion map A). The optical inline
amplifiers have a noise figure of 4.0 dB and are spaced by 50 km. Gaussian
pulses were launched with a FWHM width of 16.75 ps and a peak power of
2.8 mW. Pulses in neighboring channels were misplaced by half a bit interval
to avoid initial interchannel pulse collisions. The receiver unit consists of a
PIN photodiode, and a 3rd order Bessel filter with a 3-dB bandwidth of
7.5 GHz.
After 5,000 km the BER is estimated using the estimation technique
described in Chapter 6, p. 109, which considers energy fluctuations and tim-
ing jitter statistics. The accumulated timing jitter is approximately 4 ps (4%
of bit duration), where ASE-noise induced timing jitter (ANTJ) is approxi-
mately 1.75 ps, and collision-induced timing jitter (CITJ) is approximately
3.59 ps. The amplitude fluctuations due to the direct overlay of ASE-noise
and pulse-to-pulse interactions could be estimated at the detector to be
approximately 4% for the marks and 1% for the spaces with respect to the
average peak power value.
Figure 6-6 shows an eye-mask with surface lines denoting constant BER
values. The outermost line corresponds to the average pulse shape after the
electrical post-detection filter, the innermost one corresponds to a BER of
10-9. The figure clearly shows that the BER is limited by timing jitter, which
in this case, is mainly caused by interchannel XPM effects.
116 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
Figure 6-6. BER eye-mask after 5,000 km WDM transmission using DMSs
considering timing jitter and amplitude noise [134].
Figure 6-7 shows the BER versus decision threshold for two different sam-
ple points, namely in the middle and 10% off the middle of the bit interval.
The solid curves refer to the approach of considering the statistics of ampli-
tude fluctuations due to timing jitter. The dashed lines are drawn for the
assumption that amplitude fluctuations are Gaussian in distribution (GA
technique). The graph shows clearly that the BER would have been underes-
timated by orders of magnitude without considering the timing jitter prop-
erly.
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 117
Figure 6-7. BER versus decision threshold after 5,000 km WDM transmission
using DMSs considering timing jitter and amplitude noise at two different
sample points: t = 0.0 (middle of the bit interval) and t = 0.1 (10% from the
middle of the bit interval); pure Gaussian amplitude approximation (dashed),
inclusion of timing jitter statistics (solid) [134].
6.4.3 Chirped RZ system
In the final example, the BER was calculated for a 10-channel WDM
transmission system using chirped RZ pulses. The pulses launched initially
were raised-cosine shaped with a peak power of 0.6 mW. Sinusoidal phase
modulation was applied to each pulse with an initial phase offset of –90° and
an index of approximately 108° (mp= 0.6). The extinction ratio at the trans-
mitter was 15 dB. The dispersion map is described in Table 5-4 (dispersion
map E). The inline amplifiers have a nominal noise figure of 4.0 dB and were
placed every 45 km. The same receiver unit is used as described in
Chapter 6, p. 115.
The BER was calculated after 5,000 km right at the output of the 10th dis-
persion map. The accumulated timing jitter was estimated to be approxi-
mately 5.2 ps. The major contribution was due to ASE-noise with
approximately 5.0 ps, whereas XPM effects contribute to only 1.2 ps. Timing
jitter accumulation due to ASE-noise and interchannel XPM effects differ
between the presented DMS and CRZ system scenarios because of the differ-
ent dispersion maps. For the DMS transmission scenario, pulses do not
118 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
broaden significantly outside a bit interval, whereas for the CRZ transmis-
sion, pulses broaden over more than two bit durations experiencing strong
pulse shaping effects over the dispersion map. The standard deviation of
amplitude fluctuations is also increased: for marks to approximately 7% of
the average peak power value, and for spaces to 2%.
Figure 6-8 shows the eye-mask with surface lines denoting constant BER
values. Figure 6-9 shows BER versus decision threshold for two different
sample points. Both figures show clearly that system performance is limited
by timing jitter due to interactions between signal and ASE-noise. Even
though amplitude fluctuations are also increased they are not of system lim-
iting importance. Figure 6-9 shows that overly confident BER values would
be calculated without considering the statistics of timing jitter properly.
Figure 6-8. BER eye-mask after 5,000 km WDM transmission using CRZs
considering timing jitter and amplitude noise [134].
Investigating the eye-masks of the DMS and CRZ cases, the BER is
strongly dependent on the detected pulse shape. This is of significant impor-
tance for RZ modulation formats with narrow pulse peak plateaus, as was for
the cases discussed here.
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 119
Figure 6-9. BER versus decision threshold after 5,000 km WDM transmission
using CRZs considering timing jitter and amplitude noise at two different sam-
ple points: t = 0.0 (middle of the bit interval) and t = 0.1 (10% from the middle
of the bit interval); pure Gaussian amplitude approximation (dashed), inclu-
sion of timing jitter statistics (solid) [134].
In summary, BER has been estimated for diverse system examples using
fast and memory efficient techniques. The aspect these techniques all have
in common is that they increase estimation speed and accuracy by orders of
magnitude. However, special care needs to be applied when using these tech-
niques as they are not universally applicable.
120 A. Richter - Timing Jitter in WDM RZ Systems
Estimation of system performance
A. Richter - Timing Jitter in WDM RZ Systems 121
Chapter 7
Summary
The presented work summarizes a contribution to the investigation of tim-
ing jitter in long-haul WDM transmission systems applying return-to-zero
(RZ) modulation formats. Efficient modeling techniques for calculating tim-
ing jitter due to optical amplifier noise and interchannel cross-phase modula-
tion were presented and validated. The influence of various system
parameters on timing jitter accumulation was analyzed.
This work was started with a presentation of recent trends in the design of
long-haul WDM systems. Design considerations of the optical transmitter,
the optical fiber propagation including optical amplifiers and the optical
receiver were discussed. Different RZ modulation formats were presented,
which are of great interest for long-haul propagation. The time and fre-
quency dynamics of dispersion-managed soliton (DMS) and chirped
return-to-zero (CRZ) pulses were discussed in detail. Trends of convergence
between DMS- and CRZ-based transmission systems and other new modula-
tion formats were presented as well.
After a brief overview of the main distortions that occur in long-haul
WDM transmission, investigations were concentrated on the two main
sources of pulse timing jitter in long-haul fiber-optic transmission systems.
These jitter sources are noise generated from optical amplifiers, and inter-
channel cross-phase modulation. A recently reported approach for the calcu-
lation of ASE-noise induced timing jitter (ANTJ) was presented and
implemented in a semi-analytical algorithm. As one of the main achieve-
ments of this work, a new semi-analytical algorithm was developed for calcu-
lating collision-induced timing jitter (CITJ) considering RZ pulses of
arbitrary shapes. Both algorithms solve an analytically derived solution for
the variance of the pulse central time considering local pulse shape informa-
122 A. Richter - Timing Jitter in WDM RZ Systems
Summary
tion, which is obtained as numerical solution of the generalized nonlinear
Schrödinger equation (GNLS).
The main advantage of the newly developed algorithms is that they do not
make any assumption about the propagated pulse shapes, and that they
reduce the computational effort for calculating timing jitter by orders of mag-
nitude compared to full numerical simulations. This allows a systematic opti-
mization of parameters for many long-haul WDM RZ transmission systems.
This is not possible, if only full numerical simulations would be available.
The algorithms were implemented in an commercially available software
package from VPIsystems for modeling and analyzing WDM transmission
systems [153]. The applicability of the algorithms was validated for various
system conditions.
Another main achievement of this work is the in-depth analysis of timing
jitter accumulation in typical long-haul WDM system scenarios employing
the most common modulation formats, namely DMS and CRZ. It was shown
that timing jitter accumulation depends strongly on the dispersion map, and
amplifier positioning. Together with the channel spacing, these parameters
determine the number of pulses interacting nonlinearly during propagation.
It was found that the variance of the pulse central time is proportional to the
inverse of the cube of the channel spacing, and that initial pulse positioning
inside the bit interval plays a crucial role on the accumulation of CITJ.
Finally, the influence of timing jitter on the detected BER was investigated
for various system examples employing RZ modulation formats. It was
shown that timing jitter might be the limiting factor in long-haul propaga-
tions, and that an BER estimation technique using the approximation that
amplitude fluctuations at the detector are Gaussian distributed when timing
jitter is present, might deliver strongly misleading results.
As outlook, schemes for reducing or even suppressing the effect of
XPM-induced timing jitter could be investigated. It should be worth applying
solutions derived for NRZ systems, such as [17], to RZ systems. Also, it would
be of great interest to find a closed description for the impact of XPM on sys-
tem performance, which combines the gathered knowledge for RZ systems,
where XPM results mainly in timing jitter, and NRZ systems, where XPM
results mainly in amplitude jitter. Finally, the accumulation of CITJ in the
presence of PMD, and the accumulation of ANTJ and CITJ in systems
employing distributed amplification could be investigated.
A. Richter - Timing Jitter in WDM RZ Systems 123
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A. Richter - Timing Jitter in WDM RZ Systems 135
Appendix A
List of Acronyms
ACF AutoCorrelation Function
ANTJ ASE-Noise induced Timing Jitter
ASE Amplified Spontaneous Emission
BER Bit Error Rate
CITJ Collision-Induced Timing Jitter
CRZ Chirped Return-to-Zero
DCF Dispersion Compensating Fiber
DD Direct Detection
DMS Dispersion-Managed Soliton
DNA Deterministic Noise Approximation
DSF Dispersion Shifted Fiber
EAM Electro-Absorption Modulator
ECA Elastic Collision Approximation
EDFA Erbium Doped Fiber Amplifier
EOP Eye Opening Penalty
FEC Forward Error Correction
FFT Fast Fourier Transform
FWHM Full Width Half Maximum
FWM Four Wave Mixing
GA Gaussian Approximation
GNLS Generalized Nonlinear Schrödinger Equation
136 A. Richter - Timing Jitter in WDM RZ Systems
List of Acronyms
GVD Group Velocity Dispersion
HODM Higher Order Dispersion Management
I-FWM Intrachannel Four-Wave Mixing
IM Intensity Modulation
IPP Initial Pulse Positioning
ISI InterSymbol Interferences
I-XPM Intrachannel Cross-Phase Modulation
LCF Large Core Fiber
MC Monte Carlo
MZM Mach-Zehnder Modulator
NF Noise Figure
NRZ Non Return-to-Zero
NZDSF Non-Zero Dispersion Shifted Fiber
OSNR Optical Signal-to-Noise Ratio
PDF Probability Density Function
PSD Power Spectral Density
RZ Return-to-Zero
SPM Self-Phase Modulation
SSMF Standard Single Mode Fiber
WDM Wavelength Division Multiplexing
XPM Cross-Phase Modulation
A. Richter - Timing Jitter in WDM RZ Systems 137
Appendix B
List of Symbols
complex field envelope, slowly varying field amplitude
effective core area
bandwidth
optical bandwidth
electrical bandwidth
speed of light, in a vacuum
dispersion coefficient
electric field vector
transversal field distribution
,probability density function with respect to time, amplitude
A: net gain, combines fiber loss and amplifier gain
B: gain per distance inside an EDFA
total amplifier gain
Planck’s constant,
field intensity
Boltzmann’s constant,
spacing between two lumped amplifiers
doped fiber length in optical amplifier
collision length of two colliding pulses
Azt,()
Aeff
B
B
o
Be
cc02.99792458 108
×ms⁄[]=
D
E
Fxy,()
ftfa
gz()
G
hh6 62607 10 34–
×Ws2
[],=
It()
kBkB1.38054 10 23–
×JK⁄[]=
LA
LAmp
Lc
138 A. Richter - Timing Jitter in WDM RZ Systems
List of Symbols
collapse length of two interacting solitons
dispersion length
effective fiber length
nonlinear length
soliton period for initial shape recovery
walk-off length
A: number of samples
B: number of trials
random noise process
refractive index
linear refractive index
nonlinear refractive index
soliton order
spontaneous emission factor, population inversion factor
electric polarization vector
initial, peak power
ASE-noise power
electron charge,
normalized field amplitude
slope of dispersion coefficient
time, eventually normalized, retarded; central time of pulse
pulse duration: half-width at the 1/e-intensity
pulse duration: full-width at half the max intensity (FWHM)
pulse energy
group velocity
coordinate in propagation direction, could be normalized
fiber attenuation
mode propagation constant
LCollapse
LD
Leff
LNL
LS
Lw
N
Nzt,()
nω()
n0
n2
Ns
nsp
P
P0
PASE
qq1.60218 10 19–
×C[]=
qzt,()
SD
tT,
T0
TF
U
vg
zZ,
α
βω()
List of Symbols
A. Richter - Timing Jitter in WDM RZ Systems 139
inverse of group velocity
group velocity dispersion (GVD), chromatic dispersion
slope of GVD, 2nd order GVD
nonlinear coefficient
FWM efficiency
wavelength
effective mode radius of the fiber
variance of pulse energy
variance of pulse position
permeability, in a vacuum
initial phase
phase increment
circular frequency
central frequency of pulse
A: pulse broadening
B: pulse separation
step size in split-step Fourier method
phase mismatch
β1vg
β2
β3
γ
η
λ
ρm
σa
2
σt
2
µµ04π10 7–
×NA
2
⁄[]=
φ0
∆φ
ω
Ω
∆T
∆z
∆β
140 A. Richter - Timing Jitter in WDM RZ Systems
List of Symbols
A. Richter - Timing Jitter in WDM RZ Systems 141
Acknowledgements
I would like to take this opportunity to thank several people for the sup-
port that they provided me during the last four years in getting this work
done.
I am especially grateful to Dr. Vladimir Grigoryan, as he introduced me to
the foundations of pulse dynamics in nonlinear fiber propagation. This work
was inspired and greatly influenced by Dr. Vladimir Grigoryan. He derived
the mathematical foundations for the developed semi-analytical algorithms,
which calculate timing jitter due to optical amplifier noise and interchannel
cross-phase modulation. I regard Dr. Vladimir Grigoryan as both, a good
teacher and valuable friend. I am also very grateful to Prof. Curtis Menyuk,
who invited me to stay with his research group at the University of Mary-
land, Baltimore County for three months in 1998. During this time, I had
many fruitful discussions with members of his team. I would especially like
to mention here Dr. Edem Ibragimov and Dr. Tao Yu, whom I learned to
appreciate as friends as well.
I am also very grateful to my fellow colleagues at VPIsystems, who
allowed me the support and necessary freedom to achieve this work.
Dr. Kay Iversen and Dr. Rudolf Moosburger provided me with creative free-
dom and also gave me their support as supervising colleagues. I am also
grateful for the knowledge and understanding regarding physics and model-
ing techniques of fiber-optic communication systems that I gained in discus-
sions with Dr. Arthur Lowery, Dr. Igor Koltchanov, Dr. Rudolf Moosburger,
Ronald Freund, and Dr. Dirk Breuer.
I am grateful to Prof. Klaus Petermann from the Technische Universität
Berlin for supporting me and allowing me to write this thesis as an external
student of his research team.
I would also like to thank Marcia Bascombe, for introducing to me the
deep secrets of AdobeFrameMaker, and giving me valuable comments
and ideas regarding the layout. I thank her and Dr. Peter Moar for
proof-reading the text.
Acknowledgements
A. Richter - Timing Jitter in WDM RZ Systems 142
Finally, I would like to thank my wife, Synke and my sons Tim and Jan for
their patience and tender support.
André Richter
Berlin, 10. 10. 2001
