AMIRHOSSEIN GHAZISAEIDI
Advanced Numerical Techniques for Design and
Optimization of Optical Links Employing Nonlinear
Semiconductor Optical Amplifiers
Thèse présentée
à la Faculté des études supérieures de l’Université Laval
dans le cadre du programme de doctorat en génie électrique
pour l’obtention du grade de Philosophiæ Doctor (Ph.D.)
Faculté des science et de génie
UNIVERSITÉ LAVAL
QUÉBEC
2011
c
Amirhossein
Ghazisaeidi, 2011
To my parents Afsar and Saeid,
to my aunt Parvaneh Bahadori,
and to my sisters, Maryam and Shahrzad.
“The most incomprehensible thing about the universe is that it is comprehensible.”
A. Einstein
Contents
Contents
i
Résumé
iv
Abstract
vi
Foreword
viii
Acknowledgment
x
Abbreviations
xi
List of Symbols
xv
List of Figures
xxvii
List of Tables
xxxii
1 Introduction
1.1 SOA-based signal processing in future advanced optical networks
1.1.1 The perspective . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Intensity noise suppression in SS-WDM . . . . . . . . . .
1.1.3 Design challenges of SOA-assisted SS-WDM . . . . . . .
1.2 Performance evaluation techniques in optical communications .
1.2.1 Sources of impairment . . . . . . . . . . . . . . . . . . .
1.2.2 Analytical methods . . . . . . . . . . . . . . . . . . . . .
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Contents
1.3
1.4
1.5
1.2.3 Monte Carlo simulations . . . . . . .
Multicanonical Monte Carlo . . . . . . . . .
1.3.1 Introduction . . . . . . . . . . . . . .
1.3.2 Importance Sampling . . . . . . . . .
1.3.3 Flat Histogram Importance Sampling
1.3.4 Multicanonical Monte-Carlo . . . . .
1.3.5 Complete MMC simulations . . . . .
1.3.6 An elementary example . . . . . . . .
The Semiconductor Optical Amplifier . . . .
1.4.1 Introduction . . . . . . . . . . . . . .
1.4.2 Choice of SOA model . . . . . . . . .
1.4.3 Propagation equation with ASE . . .
1.4.4 Reservoir model . . . . . . . . . . . .
Outline of the thesis . . . . . . . . . . . . .
2 Noise Suppression
2.1 Introduction . . . . . . . .
2.2 System Simulator . . . . .
2.2.1 The MMC platform
2.2.2 The System Model
2.2.3 The SOA Model . .
2.3 Numerical Results . . . .
2.4 Summary . . . . . . . . .
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3 Patterning Effect
3.1 Introduction . . . . . . . . . .
3.2 SOA Modeling . . . . . . . . .
3.2.1 Small-Signal Analytical
3.2.2 Large Signal Numerical
3.3 The Simulator . . . . . . . . .
3.3.1 Link Model . . . . . .
3.3.1.1 TX Model . .
3.3.1.2 RX Model . .
3.3.2 MMC Platform . . . .
3.4 Experimental Results . . . . .
3.4.1 Conditional PDFs . . .
3.5 Summary . . . . . . . . . . .
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Model
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iii
Contents
4 Filter Design
4.1 Introduction . . . . . . . . . . . . . . . . . .
4.2 Intensity Noise in the CW Regime . . . . . .
4.2.1 Experimental Validation of Simulator
4.2.2 Impact of SF and CSF on EIN . . . .
4.3 Multichannel PMMC Simulator . . . . . . .
4.3.1 Multi-channel MMC platform . . . .
4.3.2 Parallelization of MMC . . . . . . . .
4.4 BER Results . . . . . . . . . . . . . . . . . .
4.5 Spectrally Efficient Scenarios . . . . . . . . .
4.6 Cross validations . . . . . . . . . . . . . . .
4.7 Conclusion . . . . . . . . . . . . . . . . . . .
5 SAC OCDMA
5.1 Introduction . . . . .
5.2 System description .
5.3 Numerical model and
5.4 Conclusion . . . . . .
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results
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6 Conclusions and Future Work
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A Metropolis-Hastings Algorithm
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B SOA parameters for simulations of Chapter
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C Publication List
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Bibliography
114
Résumé
Les systèmes de communications optiques avancées futurs vont largement utiliser
les modules de traitement de signaux optiques. Une composante importante dans la
construction de blocs dans plusieurs schémas de traitement du signal optique modernes
est l’amplificateur optique à semi-conducteurs (SOA) en raison de son comportement
non-linéaire. Afin de concevoir et optimiser les systèmes comprenant la construction de
blocs, tels non-linéaire, des outils d’analyse de performances efficaces sont nécessaires.
Dans la présente thèse, un simulateur basé sur l’algorithme de Monte Carlo multicanonique (MMC) a été développé et utilisé pour analyser une importante fonctionnalité de
traitement de signaux optiques, à savoir la suppression du bruit d’intensité du SOA dans
SS-WDM PONs. L’algorithme de MMC a été introduit en début des années 90 dans la
communauté de physique statistique, et depuis 2003 il a été utilisé par les chercheurs
dans la communauté de communication optique. Dans la présente thèse, une brève introduction à la suppression du bruit d’intensité du SOA dans SS-WDM, l’algorithme
MMC, et la modélisation du SOA, sera présentée dans le chapitre 1.
Pour le chapitre 2, j’ai utilisé, pour la première fois, l’algorithme MMC pour estimer
les fonctions de densités de probabilité conditionnelles (PDFs) de marques et d’espaces
au niveau du récepteur d’un lien SS-WDM mono-utilisateur assisté par un SOA. En
exploitant les PDFs, j’ai estimé le taux d’erreur binaire (BER) pour les systèmes SSWDM classiques, les systèmes SS-WDM avec régime de suppression de bruit d’intensité
du SOA, et finalement les systèmes SS-WDM assisté par SOA, en tenant compte de
l’effet de sélection des filtres par le canal. Une nouvelle technique de déformation de
patron est également introduite, pour traiter les interférences inter-symboles (ISI) en
raison de la mèmoire du lien. Grace à cette technique, j’ai pu vérifier, par le moyen des
simulations, que pour les conditions supposées dans le chapitre 1, ISI découle simplement
Résumé
v
du filtre électrique que ce soit la mèmoire efficace du canal de communication est 1 ou
zéro (pas de mè moire). Les estimations de PDFs et de BERs ont été validées par des
mesures expérimentales.
Le chapitre 3 est consacré entièrement à la question de l’ISI, en particulier celui
dû à la dynamique de la SOA, qui est aussi appelé l’effet patterning. Un lien avec une
source laser à 10 Gb/s a été supposée, donc, la suppression de bruit d’intensité du SOA
n’est pas un problème dans ce chapitre. L’objectif principal est de démontrer la fiabilité
du simulateur pour estimer correctement les PDFs conditionnelles des marques et des
espaces reçus dans la présence de l’effet patterning. Une nouvelle méthode pour mesurer
directement les PDFs a été proposée. Les PDFs conditionnelles et les BER simulées ont
été comparées avec les valeurs correspondantes mesurées.
Le chapitre 4 est un suivi des systèmes SS-WDM basés sur des SOAs. Un système
réaliste à multi-canaux a été supposé. L’objectif principal est d’abord d’étudier l’impact
de la forme et les bandes passantes des filtres optiques (partage et sélection de canal) sur
la performance du système et, deuxièmement, de montrer comment choisir les largeurs
de bande des filtres, les formes, et l’espacement entre les canaux, afin de maximiser l’efficacité spectrale, lorsque les techniques de suppression du bruit d’intensité du SOA et de
correction d’erreur sont utilisées. Ces deux questions sont abordées pour la première fois
dans cette thèse. Pour ce faire, un nouveau module incluant l’asynchronisme inter-canal
est ajouté à l’algorithme MMC. En effet, on montre pour la première fois, que MMC
peut être facilement utilisé en parallélisme, contrairement aux affirmations précèdentes
dans la littérature, mais que le prix à payer c’est perdre une fraction des échantillons par
cycle MMC et par nœud de calcul. Les mesures de BER simulées croisent les mesures
déjà publiées par d’autres groupes de recherche.
Dans le chapitre 5, les performances des codes d’amplitude spectrale pour les systèmes à division à accés multiple optique (SAC-OCDMA), avec et sans la suppression
de bruit d’intensité du SOA, sont analysées pour la première fois. Les résultats simulés
pour le cas de 2 et 3 utilisateurs actifs sont validés par rapport aux mesures déjà réalisès
et publiés par notre groupe de recherche.
Abstract
Future advanced optical communication systems will widely use optical signal processing modules. A key building block in many modern optical signal processing schemes
is the semiconductor optical amplifier (SOA) due to its nonlinear behavior. In order
to design and optimize systems exploiting such nonlinear building blocks, efficient performance analysis tools are necessary. In the present thesis, a simulator based on the
multicanonical Monte Carlo (MMC) algorithm is developed and is used to analyze an
important optical signal processing functionality, namely, the SOA-based intensity noise
suppression in spectrum-sliced wavelength division multiplexed passive optical networks
(SS-WDM PON).
The MMC algorithm was introduced in early 90s in the statistical physics community. Since 2003, MMC has been used by researchers in the optical communication
community. In Chapter 1, I provide a brief introduction to SOA-based intensity noise
suppression in SS-WDM, the MMC algorithm, and SOA modeling.
In Chapter 2, I use, the MMC to estimate the conditional probability density functions (PDF) of marks and spaces at the receiver of a SOA-assisted single-user SS-WDM
link. Having obtained the conditional PDFs, I estimate the bit error rate (BER) for
conventional SS-WDM systems, SS-WDM systems with SOA-based intensity noise suppression, and SOA-assisted SS-WDM including the effect of a channel selecting filter.
A new pattern warping technique is also introduced, to deal with the inter-symbol interference (ISI) due to link memory. The PDF estimations and BERs are validated
with the experimental measurements.
Chapter 3 is devoted wholly to the issue of ISI arising from the SOA nonlinear
dynamics, which is sometimes called the patterning effect. A link with a laser source
at 10 Gb/s is assumed, in contrast to the broadband thermal sources considered in
Abstract
vii
chapter 2. The main goal is to demonstrate the reliability of the simulator to correctly
estimate the conditional PDFs of received marks and spaces in the presence of the
pattering effect. A new method to directly measure the conditional PDFs is proposed.
The conditional PDFs and the BER are simulated, and verified with the corresponding
measured values.
Chapter 4 is the followup on SOA-based SS-WDM. A realistic multi-channel system
is assumed. The main goal is to study the impact of the shape and bandwidths of the
optical filters (both slicing and channel selecting filters) on the system performance. We
also show how to choose filter bandwidths, shapes, and channel spacings to maximize
the spectral efficiency; both SOA-based noise suppression and forward error correcting
are assumed. These two issues are addressed for the first time in this dissertation. It
has been shown for the first time, that MMC can be easily parallelized, contrary to
previous statements in the literature. The cost of parallelization is the loss of a small
fraction of the samples per MMC cycle per computing node. The simulated BERs are
cross-validated against the measurements already published by other research groups.
In Chapter 5, the performance of spectral amplitude coded optical division multiple access (SAC-OCDMA), with and without a SOA-based intensity noise suppression
module, is analyzed for the first time. The simulated results in the case of 2 and 3
active users are validated against the previously published measurements, done by our
research group.
Foreword
Four chapters of the present dissertation are based on four different IEEE journal papers. The papers are fully presented; however, minor modifications have been
made to enhance the uniformity of the whole document. In what follows, I detail my
contributions to each of those papers.
Paper 1: A. Ghazisaeidi, F. Vacondio, A. Bononi, and L. A. Rusch, “SOA Intensity Noise Suppression in Spectrum Sliced Systems: A Multicanonical Monte Carlo
Simulator of Extremely Low BER", IEEE J. Lightwave Technol., vol. 27, no. 14, pp.
2667-2677, July 2009.
The problem was defined by L. A. Rusch. The major theme of this paper is to
apply the multicanonical Monte Carlo simulation for exact statistical characterization
of the SOA-based intensity noise suppression in the single-channel case. The whole
simulator was developed by me during summer 2007. Knowledge of MMC was based
on the lecture notes of A. Bononi, prepared by him for his PhD course at the Università
di Parma. I had collaborated with F. Vacondio concerning the MMC simulation engine
of generic application. The application of MMC to spectrum-sliced systems is my
work. The measurements of the intensity noise spectra and the PDFs were done by
me. The measurement of the bit error rates where jointly with F. Vacondio. The
analysis of the data was my work. Finally, I wrote the entire paper, with many helpful
comments/suggestions from all coauthors.
Paper 2: A. Ghazisaeidi, F. Vacondio, A. Bononi, and L. A. Rusch, “Bit Patterning in SOAs: Statistical Characterization Through Multicanonical Monte Carlo
Simulations”, IEEE J. Quantum Electron. vol. 46, pp. 570-578, April 2010.
The problem, and the research plan was suggested by L. A. Rusch. I wrote all simulations, as well as proposing and implementing the technique to measure the conditional
Foreword
ix
PDFs. F. Vacondio helped me in measuring the BER. The paper was written by me.
All coauthors, especially A. Bononi, provided valuable comments/suggestions during
the writing phase.
Paper 3: A. Ghazisaeidi, F. Vacondio, and L. A. Rusch, “Filter Design for SOAAssisted SS-WDM Systems Using Parallel Multicanonical Monte Carlo", IEEE J. Lightwave Technol., vol. 28, no. 1, pp. 79 - 90, January 2010.
The problem and the research plan was suggested by me and L. A. Rusch. This
paper has two contributions: the parallel implementation of the MMC, and using the
(accelerated) parallel multichannel MMC simulator to evaluate the impact of the shapes
and bandwidths of channel selecting and slicing filters, as well as the channel spacing, on
the system performance. I proposed the research and the methodology, and measured
the noise suppression ratio spectra. I developed the simulator code, and wrote the
entire paper. I had fruitful discussions with F. Vacondio during the research work that
finally resulted in this paper. All coauthors provided valuable comments/suggestions
on the text.
Paper 4: A. Ghazisaeidi and L. A. Rusch, “Capacity of SOA-Assisted SACOCDMA", IEEE Photonic Technol. Lett., vol. 22, pp. 441-443, 1 April 2010.
The problem was suggested by L. A. Rusch. I evaluated the performance of SACOCDMA with/without SOA-based noise suppression module per user, for different
number of active users in the network. I wrote all simulation code. J. Penon provided
the fiber Bragg grating profiles he had designed for his experimental work on the same
subject. In the case of 2 and 3 users, the simulation results were validated against
the previously published experimental results by J. Penon, et al. The manuscript
was written by me. L. A. Rusch provided valuable comments/suggestions over the
manuscript.
The material in subsection 1.2.3 and section 1.3 is based on the lecture notes of A.
Bononi. This novel development of the multicanonical Monte Carlo algorithm is also
described in the following conference paper at GLOBECOM 2009.
A. Bononi, L. A. Rusch, A. Ghazisaeidi, F. Vacondio, and N. Rossi, “A Fresh Look
at Multicanonical Monte Carlo from a Telecom Perspective", GLOBECOM2009.
Acknowledgment
The research work presented in this thesis, could not have been done, if I had not
been lucky to have the constant support of a few individuals. First, and foremost, I
would like to thank Leslie, my supervisor, for three years of close and devoted guidance.
Not only she inculcated me with a rigorous culture of scientific research, but also, she
taught me how in practice we can combine professionalism with a liberal mind, a tolerant
attitude, and a devotion to help people who depend on us. Leslie, Thanks!
During my PhD research, I was very lucky to know, and collaborate with Alberto.
His papers, lecture notes, and “short” reports, have always been, and still are, a major
source of first-class knowledge of optical communications for me.
Especial thanks to my friend, and colleague, Francesco, for helping me in the measurements, as well as for our never-ending scientific discussions, among many other
things. A sharp mind, dexterous hands, combined with a deep sense of humor makes
him an ideal colleague for any body. I thank Walid for many fruitful discussions we had,
especially in the first year of my PhD work, and Pegah, for helping me in my first days
in the Lab. I never forget the joyful moments I have had with Mehdi in Quebec. All my
other friends and colleagues in Laval, Mehrdad, Julien, Ziad, Mohammad, Habib, Jeff,
Simon, Mansour, Yousra, Serge, Philippe, and Patrick, enriched my moments during
the past years and/or assisted me in my research work. I thank my old friends Asie
and Afsaneh, for not forgetting me during hard times.
Finally, I would like to thank my family: Maryam and Shahrzaad, my sisters, my
aunt Parvaneh, and my parents. To be honest, I do not know how I can thank you
in a few words. As far as I remember, I have always received support from you, and
in return asked for even more!....Let me repeat the same buzz word encore une fois:
"thank you".
Abbreviations
ASE
Amplified spontaneous emission
AWG
Arrayed waveguide grating
AWGN
Additive white Gaussian noise
BIBD
Balanced incomplete block design
BBS
Broadband source
BER
Bit error rate
BERT
Bit error rate tester
BPG
Binary pattern generator
BT4
Bessel-Thompson of order 4 (Electrical filter type)
CBR
Conventional balanced receiver (for SAC-OCDMA)
CSF
Channel selecting filter
CH
Carrier heating
CW
Continuous-wave
DCF
Dispersion compensating fiber
DSP
Digital signal processing
xii
Abbreviations
EDFA
Erbium doped fiber amplifier
EF
Electrical filter
EIN
Excess intensity noise
EINP
Excess intensity noise penalty
FEC
Forward error-correcting code
FIR
Finite impulse response
FTTH
Fiber-to-the-home
FHIS
Flat histogram importance sampling
FWM
Four-wave mixing
GVD
Group velocity dispersion
i.i.d.
independent identically distributed
IS
Importing sampling
ISI
Inter-symbol interference
ISO
(Optical) Isolator
LD
Laser diode
MAI
Multiple access interference
MC
Monte Carlo
MH
Metropolis-Hastings
MCMC
Markov chain Monte Carlo
MGF
Moment generating function
MMC
Multicanonical Monte Carlo
MOD
Modulator
MZM
Mach Zehnder Modulator
NSR
Noise suppression ratio
xiii
Abbreviations
NVG
Noise vector generator
ODE
Ordinary differential equation
OF
Optical filter
OLT
Optical line terminal
ONU
Optical network unit
OCDMA
Optical code division multiple access
OOK
On-off keying
OSP
Optical signal processing
OSNR
Optical signal to noise ratio
OTDM
Optical time division multiplexing
PBS
Polarization beam splitter
PC
Polarization controller
PD
Photodetector
PDF
Probability density function
PG
Pattern generator
PMF
Probability mass function
PMMC
Parallel MMC
PNG
Pattern number generator
PON
Passive optical network
PRBS
Pseudo random bit sequence
RBR
Reduced balanced receiver (for SAC-OCDMA)
RIN
Relative intensity noise
RV
Random variable
RVG
Random vector generator
xiv
Abbreviations
RX
Receiver
SAC − OCDMA Spectral amplitude coded optical code division multiple access
SE
Spectral efficiency
SF
Slicing filter
SGM
Self gain modulation
SHB
Spectral hole burning
SMF
Single mode fiber
SOA
Semiconductor optical amplifier
SPM
Self phase modulation
SS − WDM
Spectrum sliced wavelength division multiplexing
SUT
System under test
TX
Transmitter
VOA
Variable optical attenuator
WDM
Wavelength division multiplexing
WDM − PON
Wavelength division multiplexing passive optical network
List of Symbols
a
Differential gain
a1
Wavelength to gain coupling coeficient
a2
Carrier density to peak wavelength coeficient
A (z, t)
Optical field envelope at point z and time t
inside the SOA waveguide
Ain (t)
Optical field envelope at the SOA input
Aout (t)
Optical field envelope at the SOA output
Ai,out (t)
Optical field envelope at port i of the MachZehnder modulator
Ai,I (t)
Optical field envelope at the input of ith section
of the Cassioli-Mecozzi model
Ai,O (t)
Optical field envelope at the output of ith section of the Cassioli-Mecozzi model
Anrad
Nonradiative recombination coefficient
b
Index for output bins addressing
be (t)
Impulse response of the receiver lowpass electrical filter
xvi
List of Symbols
bi
The ith element of the bit vector
bin(·)
The bin function
Bi
The ith bin in the output space
B
−
The bit pattern vector
B
−
i
The bit pattern subvector of the ith user
p
B
−
The proposed bit pattern vector
p
B
−
The proposed bit pattern subvector of the ith
user
Bbimol
The bimolecular recombination coefficient
BER(Gr , Gl )
BER as function of left-channel and rightchannel relative gains
BW3dB,CSF
The 3dB bandwidth of the channel selecting
filter
BW3dB,SF
The 3dB bandwidth of the slicing filter
c
MMC cycle counter
C
Number of MMC cycles
Cauger
The Auger recombination coefficient
Cn
The normalization constant for the warped input PDF at the nth MMC cycle
i
CM
C
Relative error of MC estimation of the ith output bin
d
SOA wvaguide active region thickness
ds
Number of independent data sources
Di
The inverse image of bin Bi in the input space
DX
The input space
i
xvii
List of Symbols
DY
The output space
ê
Unit polarization vector
E
− (r
−, t)
Optical field vector in space and time
f
Frequency
f (·)
Arbitrary function
g(·)
The abstract system mapping
g(z, t)
The material gain of the SOA at longitudinal
coordinate z inside SOA waveguide, at time t
gn [·]
Smoothing exponent at the nth MMC cycle
ĝn [·]
Normalized smoothing exponent at the nth
MMC cycle
gss
SOA small-signal material gain
G(t)
Power gain of the SOA
Ḡ
Average power gain of the SOA
G1
Gain parameter to set the SOA average input
power
G2
Gain parameter to set the receiver noise power
G2
Gain parameter to set the receiver average input power
Gi (t)
Power gain of the ith section in the CassioliMecozzi model
Gl
Relative power gain of the left channel
Gr
Relative power gain of the right channel
GR
Receiver gain
h(t)
SOA total integrated gain
xviii
List of Symbols
h(z, t)
Integrated gain from SOA input to the longitudinal point z of SOA waveguide at time t
δh(t)
SOA total integrated gain fluctuation
h̄(t)
SOA total integrated gain average
hss (t)
SOA small-signal total integrated gain
hi (t)
Total integrated gain of the ith section in the
Cassioli-Mecozzi model
h1 [·]
Digital impulse response of the slicing filter
h2 [·]
Digital impulse response of the channel selecting filter
~
Planck constant over 2π
HT X (f )
Frequency response of the transmitter
HRX (f )
Frequency response of the receiver
HCSF (f )
Frequency response of the channel selecting filter
HEF (f )
Frequency response of the electrical filter
HOF (f )
Frequency response of the optical filter
HP D (f )
Frequency response of the photodetector
HSF (f )
Frequency response of the slicing filter
HY (·)
Histogram at of the output random variable
HY∗ (·)
Histogram of the output random variable when
the input is drawn from the warped PDF in
importance sampling
(n)
Estimated histogram of the output random
variable when the input is drawn from the
warped PDF in the nth MMC cycle
HY [·]
xix
List of Symbols
I
SOA bias current
Iss
SOA transparency current
I0
Mean value of spaces
I1
Mean value of marks
K
Gain factor in the small-signal equivalent SOA
gain fluctuations RC filter
K0
SOA carrier-independent loss coefficient
K1
SOA carrier-dependent loss coefficient
l(t)
Impulse response of the small-signal SOA filter
L
SOA length
ms
system memory in terms of number of temporal
samples
M
link memory in terms of number of bits
Mn
Radius of the uniform random walk for generating each component of the noise vector in the
input space
Mp
Radius of the uniform random walk for generating the pattern number in the input space
Mseq
Exponent of the pseudo-random bit sequence
n
Counter of the MMC cycle
nE
The real Gaussian random variable of zero
mean and unit variance, which is the last element of the noise vector. It is scaled and used
to simulate the receiver electrical noise.
ns
Number of independent noise sources
n(z, t)
Complex Gaussian white spatio-temporal noise
to model the ASE inside the SOA waveguide
xx
List of Symbols
nr (z, t)
The real part of n(z, t)
ni (z, t)
The imaginary part of n(z, t)
ñ
−
BBS
The noise subvector to model the broadband
source emission
Rec
The noise subvector to model the ASE emission
of the receiver pre-amplifier EDFA
ñ
−
(SOA)
ñ
−
The noise subvector to model the ASE emission
of the SOA
ns
Length of the noise vector
nCSF
SuperGaussian order of the frequency reponse
of the channel selecting filter
nSF
SuperGaussian order of the frequency reponse
of the slicing filter
N
Number of samples per MMC cycle
N (z, t)
Carrier density at point z at time t
N
−
Noise vector
ASE
N
−
i
Noise subvector of the ith user
p
N
−
Proposed noise vector
p
N
−
Proposed noise subvector of the ith user
N0
Number of zeros in the header of the packetized
De Bruijn sequence
N1
Number of ones in the header of the packetized
De Bruijn sequence
NB
Number of output bins
NIS
Number of samples in importance sampling
NM C
Number of samples in Monte Carlo
i
xxi
List of Symbols
NM M C
Number of samples in MMC
Ns
Number of temporal samples per bit duration
Nsec
Number of section in an SOA space-resolved
model
NT
SOA carrier density at transparency
Nw
Total number of temporal samples in a packetized De Bruijn
N SR(f )
Noise suppression ratio function
pX (·)
PDF of each element of the input vector
pY (·)
PDF of the output variable
p̂Y (·)
(n)
Estimated PDF of the output variable at the
end of the nth MMC cycle
p∗X (·)
The warped input PDF in importance sampling
p∗X,opt (·)
The optimum warped input PDF
pr0 |b0 (r0 |b0 = i)
The PDF of received marks (i = 1), or spaces
(i = 0)
pr0 |M (r0 )
The PDF of the received signal, conditioned on
the previous M bits
Pi (r0 )
The PDF of received marks (i = 1), or spaces
(i = 0)
Pin (t)
Optical power at the SOA input
δPin (t)
Optical power fluctuations at the SOA input
P̄in
Average optical power at the SOA input
pin (t)
Optical power at the SOA input normalized to
Psat
xxii
List of Symbols
δpin (t)
Optical power fluctuations at the SOA input
normalized to Psat
p̄in
Average optical power at the SOA input normalized to Psat
Pout (t)
Optical power at the SOA output
δPout (t)
Optical power fluctuations at the SOA output
P̄out
Average optical power at the SOA output
pout (t)
Optical power at the SOA output normalized
to Psat
δpout (t)
Optical power fluctuations at the SOA output
normalized to Psat
p̄out
Average optical power at the SOA output normalized to Psat
psw
Small-world probability
Psat
SOA saturation power
P old
Old pattern number
P new
New pattern number
P−
Pattern numbers vector
Pi
Pattern number of the ith user
P− p
Proposed pattern numbers vector
Pip
Proposed pattern number for the ith user q
Q
Q-factor
Q(Si , Sj )
Markov chain transition probability from state
Si to state Sj
r(t)
Received signal waveform
r
Coordinate vector
−
Electron charge
xxiii
List of Symbols
R
Rejection ratio for Gaussian Markov chains per
noise subvector element
Rb
Bit rate
RIN (f )
Relative intensity noise spectrum
s
Normalization factor
Sf
Current state of the Markov chain
Si
Previous state of the Markov chain
t
Time
t
The vector of the relative delays between other
users and the desired user
tp
The proposed vector of the relative delays between other users and the desired user
tD
i
Relative delay between the ith user and the desired user
Tb
The bit duration
u
A realization of the uniform random variable
between 0 and 1
U [a, b]
Uniform distribution between a and b
Um
A realization of the uniform random variable
between 0 and 1
V
Volume of the active region
V (t)
Voltage applied to the MZM
Vb
The bias voltage of the MZM
vg
The group velocity of the signal in the SOA
waveguide
w
Width of the SOA waveguide
−
−
xxiv
List of Symbols
w(·)
Weight function in IS
Ŵi
Weight coefficient corresponding to the ith output bin
Ŵi∗
warped weight estimate of the ith bin
X
−
Input vector
p
X
−
Proposed input vector
old
X
−
Old input vector
new
X
−
New input vector
Xi
Input subvector of the ith user
Xip
Proposed input subvector for the ith user
y
Output variable
Y
Output variable
Y old
Old output variable
Y new
New output variable
z
Coordinate along the longitudinal direction of
the SOA waveguide
α
SOA linewidth enhancement factor
αij
The transition probability of the MetropolisHastings Markov chain
α1
The coupling loss ratio of the input 1-by-2 coupler of the Mach-Zehnder modulator
α2
The coupling loss ratio of the output 2-by-1
coupler of the Mach-Zehnder modulator
β
SOA-carrier lifetime to bit-rate ratio
β(z, t)
loss coefficient of the SOA waveguide
i
xxv
List of Symbols
β0
The independent loss coefficient
β1
The dependent loss coefficient
Γ
SOA waveguide confinement factor
δ(·)
Dirac delta function
∆t
Simulation temporal step
∆y
Bin width of at the output space
∆ω
Channel spacing in radians per second
(z, t)
The ASE optical filed inside the SOA waveguide
ηF EC
The BER threshold of the forward error correcting code
λ
Optical wavelength
λpeak
SOA peak gain wavelength
µIS
Estimated average of importance sampling
µM C
Estimated average of Monte Carlo
ν
fitting parameter
πi
The steady-state probability of visiting the ith
state of the Markov chain
π (n) (x)
The warped PDF of any element of the input
random vector during the nth MMC cycle
ϕ (x, y)
Transverse profile of the SOA waveguide optical mode
σ0
Standard deviation of the received spaces
σ1
Standard deviation of the received marks
σIS
Standard deviation of importance sampling estimation
xxvi
List of Symbols
σM C
Standard deviation of Monte Carlo estimation
τc
SOA carrier lifetime
τef f
SOA effective carrier lifetime
ω0
Optical carrier frequency
List of Figures
1.1
SOA-assisted SS-WDM architecture. OLT: optical line terminal, BBS:
broadband source, AWG: Arrayed waveguide grating, MZM: Mach-Zehnder
Modulator, SF: slicing filter, CSF: channel selecting filter, ONU: optical
network unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The simplified single-channel SOA-assisted SS-WDM link. . . . . . . .
1.3 A waveform simulation of a sample function of a CW spectrum sliced
source intensity at the SOA input (dashed blue), and the instantaneous
power gain of the SOA (red). . . . . . . . . . . . . . . . . . . . . . . .
1.4 RIN spectra of SS source i) without SOA, ii) with SOA noise cleaning
and no post-filtering, iii) with SOA noise cleaning post-filtered for several
linewidth enhancement factors α; identical 30 GHz Gaussian filters were
used for SF and CSF. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 The state-space model of Monte Carlo simulation. . . . . . . . . . . . .
1.6 The scenario of Monte Carlo Estimation of the output PDF: a) the PDF
in the input space and the exact and estimated PDFs at the output space,
b) Confidence intervals vs. total number of samples, for estimating the
probability of an output bin with probability 10−10 . . . . . . . . . . . .
1.7 Importance sampling estimation of the output PDF. . . . . . . . . . . .
1.8 Iterative steps of MMC. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 The block-diagram of MMC algorithm. The main elements of MMC are
presented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Chi-square order 10 PDF estimated by MMC. In the inset is shown the
warped histograms, at the end of MMC cycles n = 1, 5, 10, 15. . . . . .
4
5
5
7
12
13
16
19
21
22
xxviii
List of Figures
1.11 Warped PDFs of each component of
space at MMC cycles 1, 2, and 3. . .
1.12 Architecture of the SOA waveguide .
1.13 Simplified band structure of the SOA
2.1
the
. .
. .
. .
random
. . . . .
. . . . .
. . . . .
vectors in
. . . . . .
. . . . . .
. . . . . .
the input
. . . . . .
. . . . . .
. . . . . .
23
24
25
2.13
SS-WDM link equipped with a pre-modulator noise suppressing SOA.
MZM: Mach-Zehnder modulator, Y : sampled received voltage. . . . . .
MMC platform. RVG: random vector generator, SUT: system under test.
Detailed block-diagram of the MMC platform. RVG: random vector
generator, NVG: noise vector generator, PNG: pattern number generator,
SUT: system under test, D represents unit delay. . . . . . . . . . . . . .
Flowchart of the PDF Warper. U [0, 1] is a uniform RV on [0,1]. . . . .
Flowchart of the RVG. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flowchart of the MMC. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model of the SS-WDM link of Fig. 2.1 as a SUT inside the MMC platform
of Fig. 2.2; BPG, bit pattern generator, MOD, modulator, SF, slicing
filter, CSF, channel selecting filter, PD, photodetector, EF, electrical
filter; 4th BT stands for fourth-order (lowpass) Bessel Thompson. Gain
blocks are explained in the text. . . . . . . . . . . . . . . . . . . . . . .
measured and simulated spectrum slices at the SOA input and output.
The input-output definition of the SOA spatially-resolved model . . . .
Measured (dots) and simulated PDFs of the received voltage in a SSWDM link equipped with pre-modulator nonlinear SOA, (a) no electrical
filter, no CSF, (b) no electrical filter, with CSF, (c) electrical filter of
bandwidth 1.87 GHz, no CSF, and (d) electrical filter at 1.87 GHz and
CSF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulated PDF of marks corresponding to different values of system
memory. (a) The pre-modulator setup with the parameters coming from
the experiment. (b) The post-modulator setup with a hypothetical SOA
slower than what we used in the measurements . . . . . . . . . . . . . .
Simulated conditional PDFs of marks and spaces corresponding to: SSWDM (label “SS-WDM”), SS-WDM with pre-modulator SOA (label
“SOA”), and SS-WDM with pre-modulator SOA and CSF (label “SOA
and CSF”). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured and simulated BERs. . . . . . . . . . . . . . . . . . . . . . .
3.1
a) Basic setup, and b) block-diagram of the equivalent lowpass SOA model 56
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
36
37
39
40
41
43
44
45
47
48
49
51
52
List of Figures
3.2
3.3
Large signal SOA model . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured and simulated SOA waveforms; blue trace is the measured
TX output, red waveform is the SOA model output using measured TX
output as input, green waveform is simulation. . . . . . . . . . . . . . .
3.4 Eye diagrams at the SOA output for various operational conditions. Bitrate increases from left-to right, and average input power increases from
top to bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 a) Transmitter (TX) configuration, (b) TX numerical model; PBS: polarization beam splitter, PC: polarization controller, MZM: Mach-Zehnder
modulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Optical intensities at the output of the transmitter, measured (blue) and
simulated (red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 a) Numerical models of receivers used in measurements; WNG: white
noise generator, b) frequency domain characterization of RX2 . . . . .
3.8 Block diagram of the simulator; NVG: random vector generator, PNG:
pattern number generator . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Experimental setup to measure conditional PDFs (RX1) and BER (RX2);
PG: pattern generator, MZM: Mach-Zehnder modulator, PC: polarization controller, VOA: variable optical attenuator, ISO: isolator, OF: optical filter, PD: photodetector, BERT: BER tester . . . . . . . . . . . .
3.10 a) Steps to measure the conditional PDF using the packetized, b) conditional PDFs of marks and spaces measured for three different length De
Bruijn sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Measured and simulated conditional PDFs of marks and spaces . . . . .
3.12 Measured and simulated BERs at RX2; upper inset shows the conditional
PDFs used to estimate the BER curve (one pair per BER curve point),
lower inset is eye diagram for lowest BER estimated . . . . . . . . . .
4.1
4.2
4.3
Measured and simulated noise suppression ratios (NSR) of CW intensitysmoothed light by the SOA, with and without post-filtering. When postfiltering is absent, the analytical approximation is also plotted. . . . . .
Contour plots of log(EIN P ) vs. orders of SF and CSF super-Gaussian
filters with flat phase response. . . . . . . . . . . . . . . . . . . . . . .
The block diagram of the three-user SOA-assisted SS-WDM MMC platform. NVG: noise vector generator, PNG: pattern number generator,
IDG: interferer delay generator. D: programmable temporal delay element. The rest of variables are defined in 4.3.1. . . . . . . . . . . . . .
xxix
59
60
61
62
64
65
66
67
68
70
70
77
78
79
List of Figures
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
Parallelization of MMC: (a) Random walk in a 1-dimensional input space
perturbed by periodic re-initializations. (b) Sections of the perturbed
Markov chain are mapped to various computing nodes, (c) the flowchart
of the parallel MMC; k counts the MMC cycles, Nc is the pre-specified
(k)
number of cycles, HY,j is the histogram computed by node j at the end
of cycle k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Left: Table defining (SF, CSF) combinations. SF filter types are distinguished by markers and CSF types are distinguished by line type (also
colors). Right: the frequency response of the filter types used for BER
simulations; n is the super-Gaussian order. . . . . . . . . . . . . . . . .
BER of the multi-channel system as predicted by PMMC simulations.
SF filter types are distinguished by markers and CSF filter types are
distinguished by line type (also color). . . . . . . . . . . . . . . . . . .
BERs of the single-channel system as predicted by PMMC simulations.
SF filter types are distinguished by markers and CSF filter types are
distinguished by line type (also color). . . . . . . . . . . . . . . . . . .
Comparison of BERs of SS-WDM and SOA-assisted SS-WDM; nSF =
nCSF = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contour plot of 10log(BER(Gl , Gr )/BER(0, 0));BER(Gl , Gr ) is the BER
of the desired (central) channel, as a function of the relative gain of the
left (right) channel interferer Gl (Gr ). . . . . . . . . . . . . . . . . . . .
MMC BER estimations (empty markers), and Q-factor approximated
BERs (filled marker), of three representative cases in the multi-channel
scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BER vs. normalized channel spacing, for four different SF bandwidths,
for the first scenario. The spectral efficiency (in bits/s/Hz) is given next
to each point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
All BER curves estimated by PMMC during the SE optimization process
for the second scenario. Each curve corresponds to a different channel
separation, as described in the text. . . . . . . . . . . . . . . . . . . . .
Minimum BER (CSF bandwidth optimized) vs. normalized channel
spacing, corresponding to four systems with different SF bandwidths,
for the second scenario. The spectral efficiency (in bits/s/Hz) is given
next to each point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BER vs. received power simulations, and measurements taken from Fig.
3 of [90]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxx
83
84
85
86
87
88
89
90
91
92
93
List of Figures
4.15 Back-to-back BER vs. received power simulations, and measurements
taken from Fig. 4a of [94]. . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
The N -user SOA-assisted SAC-OCDMA setup with reduced balanced
receiver after ref. [97], BBS: broadband source, OF: optical filter, Enc:
SAC-OCDMA encoder, VOA: variable optical attenuator, MZM: MachZehnder modulator, EF: electrical filter. The polarization beam splitters,
and polarization controllers at the SOA input and MZM input, as well
as optical isolators are not shown for simplicity. . . . . . . . . . . . . .
(a): the spectral codes of the desired user and two interferers, (b): A
snapshot of a three-user simulation of the system. . . . . . . . . . . . .
BER estimations by MMC of SOA-assisted SAC-OCDMA. . . . . . . .
MMC vs. Q-factor BER estimations of SOA-assisted SAC-OCDMA . .
xxxi
94
98
100
101
103
List of Tables
2.1
SOA Parameters used in simulations . . . . . . . . . . . . . . . . . . .
46
5.1
BIBD Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
Chapter 1
Introduction
In the present chapter we outline the framework of the research work presented in the
remaining chapters of this thesis. We begin our discussion by highlighting the increasing
importance of emerging optical signal processing techniques in future optical communication
technology, and the usefulness of having powerful design tools to analyze the performance of
advanced optical systems employing optical signal processing modules. We are specifically
concerned with SOA-based intensity noise suppression in SS-WDM passive optical networks.
We discuss the basic concepts of SOA intensity noise cleaning and the post-filtering effect,
and comment on the prohibitive complexity of either analytical or Monte Carlo approaches to
evaluate the bit error rate of these systems. We propose to develop an multicanonical Monte
Carlo simulator to overcome the complexity that is insurmountable when using strictly Monte
Carlo numerical simulations.
Realizing an MMC-based simulator of SOA-assisted SS-WDM systems requires mastering
two different subjects: MMC method, and SOA device modeling. Section 2 of the present
chapter is devoted to introduce fundamental concepts of MMC, while Section 3 focuses on
standard ways of modeling light amplification by the SOA. In the last section of this chapter
the road map of the remaining chapters is outlined.
Chapter 1. Introduction
1.1
2
SOA-based signal processing in future
advanced optical networks
1.1.1
The perspective
Both electronic and optical signal processing (OSP) techniques continue to be of
active interest in optical communications [1]. Rapid electronics fuels the development
of digital signal processing (DSP) for the realization of highly spectral-efficient optical systems employing advanced modulation formats. Novel photonic components and
subsystems provide the impetus to unprecedented OSP operations, which is the focus
of this thesis. Important examples of OSP include wavelength conversion, 2R and 3R
regeneration, optical clock recovery, demultiplexing of optical time division multiplexed
(OTDM) streams, and intensity noise mitigation. A major building block of many modern OSP modules is the semiconductor optical amplifier (SOA) [2]. This is mainly due
to the rich nonlinear behavior of the SOA. The rapid progress of fabrication techniques,
as well as anti-reflection coating technologies, have resulted in robust and long-life SOA
devices. We now enjoy integrated and efficient OSP modules with competitive pricing.
The nonlinear behavior of most OSP subsystems leads to non-Gaussian noise statistics
and signal-noise interaction at various points in the transmission system. These scenarios are analytically intractable and present serious challenges for the system designer.
The present work develops efficient and reliable numerical techniques for performance
analysis of optical transmission systems employing SOA-based OSP modules, and applies these tools to study specific applications, including the SOA-based intensity noise
suppression in spectrum-sliced wavelength division multiple access (SS-WDM) systems.
1.1.2
Intensity noise suppression in SS-WDM
SS-WDM is an important candidate for developing the wavelength division multiplexed passive optical networks (WDM-PON) for last-mile fiber-to-the-home (FTTH)
applications [3]. The major benefit of SS-WDM is that it allows the thermal-like emission of a shared incoherent broadband source to be sliced in the frequency domain by a
set of optical slicing filters (SF), and then to be distributed among the users to be used
as light sources; therefore, it is no longer necessary to use one highly stabilized laser
3
Chapter 1. Introduction
source per user, and total system cost is reduced. However, the performance of SSWDM is severely curtailed by the high excess intensity noise (EIN) of the thermal-like
source.
Let A(t) represent the envelope of the output field of a wide-sense stationary continuouswave (CW) light source. The corresponding intensity is denoted by P (t) = |A(t)|2 . The
relative intensity noise spectrum (RIN) spectrum of the optical field is defined to be the
power spectral density of its intensity normalized to its squared time average intensity
[4]
1
(1.1)
RIN (f ) , 2 F.T. {RδP (τ )}
P̄
where P̄ is the average power, δP (t) = P (t) − P̄ is the zero-average intensity fluctuations, F.T. denotes Fourier transform, and RδP (τ ) denotes the autocorrelation function
of the random process δP . The EIN is calculated per
EIN =
Z ∞
−∞
RIN (f ) HEF (f )df
(1.2)
where HEF (f ) denotes all post-detection electrical filtering. When examining the EIN
of the source, as opposed to the photodetected EIN, HEF (f ) is identically one, and EIN
becomes
EINsource = var(δP )/P̄ 2
(1.3)
The EIN of a thermal source induces unacceptably high bit error rate (BER) floors
[5] (also cf. Fig. 2.13). It is therefore crucial to complement SS-WDM solutions with
efficient mechanisms to mitigate the EIN.
During the past years several solutions to mitigate the EIN in SS-WDM have been
proposed. Morkel, et al., [6], and Keating, et al., [7], introduced electronic noise cancelation schemes. Han, et al. [8], employed nonlinear broadening offered by highly
nonlinear fibers at the receiver side to suppress the EIN. The intensity noise suppression offered by an SOA operating in deep saturation turns out to surpass these previous
techniques due to its efficiency and ease of implementation, as well as its potential for
an integrated optics solution. This has been the subject of many studies in recent years
[9]-[20].
Figure 1.1 illustrates a typical SOA-assisted WDM-PON proposal. The incoherent
broadband source (BBS) emits a thermal-like light. Commercial broadband sources
with 40 nm bandwidth are available. An arrayed waveguide grating (AWG) cuts spectrum slices out of the shared BBS and distributes them to the users. At the optical
Chapter 1. Introduction
4
Figure 1.1: SOA-assisted SS-WDM architecture. OLT: optical line terminal, BBS:
broadband source, AWG: Arrayed waveguide grating, MZM: Mach-Zehnder Modulator,
SF: slicing filter, CSF: channel selecting filter, ONU: optical network unit.
line terminal (OLT), the CW spectrum-sliced source is first fed into a saturated noisecleaning SOA. The noise-suppressed light at the SOA output is directly modulated
using a Mach-Zehnder modulator (MZM). Data streams from all users are wavelength
multiplexed at the OLT egress (distribution network ingress) by a second AWG, and are
propagated. At the end of the distribution network another AWG is used to demultiplex
the channels and to distribute them among the optical network units (ONU).
As illustrated in Fig. 1.1, SOA-based EIN suppression is implemented by placing an
SOA after the spectrum-sliced light source and before data modulation. Slicing filters
(SF) are implemented via AWG1, while channel selecting filtering (CSF) is accomplished
by AWG3. Note that a signal at the ONU sees the corresponding data signal filtered by
AWG2 and AWG3. Hence the cascade of AWG2 and AWG3 is the effective CSF. While
only the downlink is discussed, for the uplink a SOA and MZM external modulator
would be located at the ONU.
For our introductory analysis in this chapter, we adopt the simplified version of the
SOA-assisted SS-WDM system system shown in Fig. 1.1, as shown in Figure 1.2. We
focus on a single-channel link; the multi-channel case will be studied in chapter 4. All
post-modulation optical filters are lumped into a single channel selecting filter.
Due to the large linewidth of the thermal source, SS-WDM is extremely sensitive to
group velocity dispersion (GVD); therefore, we assume throughout this dissertation that
the single mode fiber (SMF) GVD is completely compensated before photodetection by
5
Chapter 1. Introduction
CSF
SOA
SF
EF
BBS
A
B
Current
C
PD
D
Post-filtering
G (t ) (Arbitrary Units)
(Arbitrary Units)
Figure 1.2: The simplified single-channel SOA-assisted SS-WDM link.
Intensity peak
Ain (t )
2
Gain trough
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
Time Samples
Figure 1.3: A waveform simulation of a sample function of a CW spectrum sliced source
intensity at the SOA input (dashed blue), and the instantaneous power gain of the SOA
(red).
dispersion compensation fiber (DCF), and that we are in the linear propagation regime.
We simulate the system of Fig. 1.2. The SOA numerical model used is described in
Section 1.4.3. The input intensity is traced in blue dashed lines, while the instantaneous
SOA gain is traced in solid red. We can see that input intensity peaks heavily saturate
the SOA, so that low gain is seen at an input peak. When input intensity swings low,
the SOA has the opportunity to partially recover from saturation, and higher gain is
seen at input minima. This phenomenon is known as self gain modulation (SGM). The
instantaneous gain varies inversely with the input fluctuations. The output power is
therefore smoothed, i.e., the output suffers fewer fluctuations than the input, reducing
the EIN.
Chapter 1. Introduction
1.1.3
6
Design challenges of SOA-assisted SS-WDM
Due to the presence of optical nonlinear elements, the performance analysis of optical communication links including OSP modules in general is not straightforward. For
SOA-assisted SS-WDM, performance analysis is difficult due to two factors. First, the
SGM of the noise-cleaning SOA results in non-Gaussian statistics at the SOA output.
Second, optical filtering of this non-Gaussian stochastic process changes the light statistics in a way that is intractable analytically. It has been observed experimentally that
optical filtering of the noise-suppressed modulated signal greatly neutralizes noise suppression [18]-[20](see also Fig. 2.13). We will next consider this so-called post-filtering
effect.
We examine the noise spectrum for the system described in Fig. 1.2 at these points:
point A where no SOA is present, point B where only the SOA is present (no CSF), and
then at point C when the post-filtering effect is present. Figure 1.4 shows numerically
computed RIN spectra, based on the SOA model to be presented in (1.45), of the
signals at various points in Fig. 1.2. The SOA parameters are identical to those used in
Chapter 4. Details can be found in section B. Both SF and CSF are identical Gaussian
filters with 3 dB bandwidth of 30 GHz.
Solid black markers correspond to when no SOA is present (point A in Fig. 1.2).
Empty black markers show signal RIN at point B in Fig. 1.2, where the RIN is modified
only due to nonlinear amplification by the SOA. All the other curves correspond to
point C in Fig. 1.2, where simulations are repeated for SOAs with various α−factors
(linewidth enhancement factor), a parameter which couples the phase of the optical field
at SOA output to its intensity as will be seen in (1.45). The low frequency depression in
the RIN spectra after nonlinear SOA amplification is an indication of the intensity noise
suppression. The post-filtering effect reduces the depth of the low frequency depression,
an indication that the intensity noise suppression efficiency decreases.
As we will develop analytically in (1.45), the optical field at the SOA input is
amplified by a complex instantaneous gain; the real part of the exponent of SOA instantaneous gain is responsible for SGM, whereas the imaginary part models the self
phase modulation (SPM) parameterized by α. Both SGM and SPM contribute to increasing signal linewidth. Post-filtering the spectrum-broadend signal by a filter whose
bandwidth is comparable to the signal linewidth induces distortions, and results in
7
Chapter 1. Introduction
-104
-106
No SOA
-108
PSD [dBm/Hz]
SOA + CSF
-110
a=0
a=3
a=5
a=8
-112
-114
SOA only
-116
-5
-4
-3
-2
-1
0
1
2
3
4
5
Frequency [GHz]
Figure 1.4: RIN spectra of SS source i) without SOA, ii) with SOA noise cleaning
and no post-filtering, iii) with SOA noise cleaning post-filtered for several linewidth
enhancement factors α; identical 30 GHz Gaussian filters were used for SF and CSF.
increased intensity noise. In simulations of post-filtered RINs in Fig.1.4, α is swept
through three typical values 3, 5, and 8. Larger α is equivalent to more SPM-induced
spectrum-broadening, and results in more noise-suppression degradation. In the limit
case of α = 0, the broadening is only due to SGM; even for α = 0 we observe that
post-filtering still curtails noise suppression.
The filter-induced performance degradation, (i.e., post-filtering effect) can be reduced by increasing the bandwidth of the optical CSF. In a multi-channel system there
is a tradeoff between the filtering effect and adjacent channel crosstalk [19]. Efficient
tools are a major design challenge in analyzing SOA-assisted SS-WDM systems to account for the impact of SOA nonlinearity, filtering effect, and adjacent channel crosstalk.
While RIN gives some indication of noise cleaning efficiency, the best performance measure is bit error rate (BER). A dramatic reduction in RIN will not necessarily translate
into equally dramatic BER improvement. The system, and particularly the noise statistics, have complex dynamics. As analytical treatment is intractable, we developed an
efficient numerical routine encompassing a detailed large-signal dynamic model for the
SOA, that can estimate BER as low as 10−10 in a reasonable time.
8
Chapter 1. Introduction
To develop the numerical tool, we examine in the next two sections two independent
topics, that of accelerated Monte Carlo algorithms suitable for BER estimation, and
that of numerical modeling of SOAs. We will combine techniques from these fields to
develop a numerical simulator for the SOA-assisted SS-WDM link. We validate the
resulting tool with experimental data. In the last section of the chapter we will sketch
the road map for the rest of the thesis.
1.2
Performance evaluation techniques in optical
communications
1.2.1
Sources of impairment
In the physical layer, the performance of a communication system is usually quantified by the BER. The BER can be found from the PDFs of the decision statistic Y ,
when conditioned on marks (logical 1s) and spaces (logical 0s). In other words
1
1
BER = P (Y > η |0) + P (Y 6 η |1)
2
2
(1.4)
where η is the detection threshold and we assume equiprobable data, and P (·) is the
probability (conditional in this case).
The decision variable Y will vary with the detection type. In coherent detection,
the receiver is linear since the optical field is recovered from the beat between the
received signal and the local oscillator. In incoherent or direct detection the receiver
is nonlinear; the received signal is determined by magnitude squaring operation of the
photodetector. We are only concerned with direct detection. The randomness of the
decision statistic is due to the following factors: noise, inter-symbol-interference (ISI),
receiver structure, and multiple-access interference (MAI), also known as crosstalk.
The important noise sources in fiber-optic communication systems include receiver
thermal noise, receiver shot noise, source intensity noise and phase noise, and the amplified spontaneous emission noise due to optical amplification [4]. Thermal noise and
shot noise are often assumed to have Gaussian statistics. The light sources of interest
in this paper are thermal in nature. A good model for a thermal source is the amplified
Chapter 1. Introduction
9
spontaneous emission (ASE) from a light emitting diode, an erbium doped fiber amplifier (EDFA) or a semiconductor amplifier. ASE can be modeled as a complex Gaussian
random field that is white in both space and time. A CW optical field from a filtered
thermal source is thus a complex Gaussian process, and its intensity and phase noise
can be derived from the Gaussian process [21]. The intensity noise has a negative exponential distribution, while the phase noise is uniform distributed over (0, 2π). When
such a signal is amplified, additional wideband ASE is accrued by the signal.
ISI is an impairment inherent to any link with memory. ISI refers to the interdependency created among the originally independent bits. In optical communication
literature ISI is sometimes referred to as the patterning effect. ISI can have several
origins such as multipath reflections, a limited receiver and/or channel bandwidth, or
nonlinear systems with memory.
Finally, MAI is an important source of performance degradation in any multiuser
communication transmission system. MAI can be avoided by assigning distinct and
well separated wavelength bands to each user. Realistic communication system design
packs channels to create a careful tradeoff between tolerable MAI and spectral efficiency
(SE). MAI must be evaluated for all possible simultaneous transmissions to arrive at
statistics for the symbol of interest. In WDM systems we frequently refer to MAI as
crosstalk.
Optical communication systems might involve nonlinear elements. The most important nonlinear element is the optical fiber, where, depending on the launched power
and the transmission length, various intra- and inter-channel nonlinear impairments like
self- and cross-phase modulation, cross-polarization modulation, and four-wave mixing
can degrade system performance [22]. The saturated SOA, in which gain and input
signal are coupled through certain differential equations (cf. section 1.4), is another
example of a nonlinear system with memory. The nonlinearities along the link cause
signal-noise interactions, hence signal-dependent non-Gaussian noise statistics, while
the finite memory of the link enhances the patterning effect. In the following subsections we review analytical and numerical methods for performance analysis of linear
and nonlinear optical communication systems.
10
Chapter 1. Introduction
1.2.2
Analytical methods
For linear coherent receivers and linear optics, classical BER calculations can be used
for performance analysis. These methods can be modified to include source intensity and
phase noise [23]. For conventional incoherent receivers and linear optics, where the only
nonlinearity is the square-law photodetector, various techniques with different degrees
of sophistication are available. If the modulation format is on-off keying (OOK), and
the dominant noise is the signal-spontaneous beating (i.e., beating between the signal
and the wideband ASE), the Gaussian assumption for the received signal is a good
approximation [4]. In this case the BER is given by
√
1
BER = √ erf c(Q/ 2)
2
(1.5)
The erf c(·) is the complementary error function, Q is the Q-factor, defined to be
Q,
I1 − I0
σ1 + σ0
(1.6)
where I1 , and I0 are the average intensity values for marks and spaces and σ1 , and σ0
are the standard deviations of the intensity if the received marks and spaces. When
the modulation format is not OOK, and/or when we want to study the impact of
optical and/or electrical filters on the system performance, we can no longer employ
the simplistic BER calculation using only the Q-factor.
When the only nonlinear element in the link is the photoreceptor, a simple technique
can be used for BER. In this case, a modal expansion technique (e.g., Karhunen-Loeve)
is used to calculate the BER for arbitrary optical and electrical filters. This method
takes into account both signal-spontaneous and spontaneous-spontaneous beat terms in
the square-law photodetection process [25]. The modal expansion method was originally
proposed for OOK, but can be applied to other modulation formats.
We briefly describe the modal technique. The signal is assumed to be a periodic
pseudo-random bit sequence (PRBS), and the ASE is assumed to be a complex white
Gaussian random process. Both signal and noise waveforms are expanded into their
Fourier series, where the fundamental harmonic is the reciprocal of the bit duration. In
Chapter 1. Introduction
11
the second step, the Fourier series of each is truncated, and signal and noise are represented by finite-dimensional vectors in the space spanned by the harmonic functions.
In the third step, optical filtering is applied, by multiplying signal and noise vectors
by diagonal matrices, whose diagonal elements are the sampled frequency responses of
the associated filters. In the the fourth step, the impact of photodetector magnitude
squaring in the time domain is found via a discrete convolution. Finally, the electrical
filter is applied to the photodetected signal. All photodetector terms, i.e., signal-signal,
signal-noise, and noise-noise terms are retained. Since noise is Gaussian before photodetection, it is possible to derive a closed form expression for the moment generating
function (MGF) of the decision variable. The last step is to numerically calculate the
error probability by applying the saddle-point integration technique to the MGF of the
decision variable.
If the noise before photodetection is not Gaussian, the modal expansion is not applicable, as closed form expressions for the MGF no longer exist. Even in the case of
Gaussian statistics, the impact of source impairments, such as intensity and/or phase
noise, are not captured with the modal expansion method. In SOA-assisted SS-WDM
the intensity noise of the spectrum sliced source is the most dominant noise source,
not the signal-spontaneous beating; secondly, the presence of the nonlinear SOA invalidates the Gaussian assumption. As the modal expansion method cannot capture these
features, we resort to numerical simulations for performance analysis of SOA-assisted
SS-WDM systems.
1.2.3
Monte Carlo simulations
In the absence of analytical methods to derive the PDF of the decision statistic,
Monte Carlo simulations are frequently used to evaluate the performance of optical communication systems. Monte Carlo simulation can be cast into an abstract state-space
framework which is very useful in further discussions. The elements of this framework
are illustrated in Figure 1.5 where X = [N , B]T is a vector in the input state-space,
DX , consisting of a vector of noise samples, N , and a vector of data samples B.
The statistical properties of the input random variables are assumed to be known.
The system under study is represented by a mapping g : DX 7→ DY from the input
to the output space. The output waveform of the system is denoted by the random
12
Chapter 1. Introduction
X
B
Signal
Source
N
g (X )
Y = g(X )
g(X )
Di

Bi
State-space model of the system
{bi }
{N i }
Noise Samples
X Space
Random Bits
Y Space
Figure 1.5: The state-space model of Monte Carlo simulation.
process Y , which we assume to be scalar. In a Monte Carlo simulation NM C input
o
n
state-vectors, X (1) , X (2) , · · · X (NM C ) , are generated and fed to the system. The output
n
o
samples, Y (1) , · · · , Y (NM C ) , are calculated; statistical properties are unknown due to
the complexity of the mapping. The objective is to estimate the output statistics by
examining the set of output samples.
Since the output samples are generated by applying the mapping g independently to
each input sample, Y (i) = g X (i) , the dimensionality of the input state-space should
be sufficient to capture all statistical dependencies, i.e., each input vector contains
necessary and sufficient information for producing a realistic output sample. Let ds be
the number of independent input signals and ns be the number of input noise sources;
in order to obtain a single output sample each of the inputs should be sampled ms
times, where ms is the memory length of the system in terms of the number of samples.
The dimension of the input-space is thus dim (DX ) = ms (ds + ns ).
The most useful statistical property of the output samples is their probability density function (PDF). We suppose that the sampled output random process is a onedimensional continuous random variable, i.e., DY = R. The PDF of random variable
(RV) Y is denoted by pY (y). In order to estimate pY (y) the output space DY is divided
into NB bins {B1 , · · · , BNB }. The histogram function HY : {Bi }i=1...NB 7→ Z + is defined over the collection of output bins, onto the set of nonzero integers, where HY (Bi )
is the number of output samples falling in Bi . The Monte Carlo (MC), estimate of the
probability of the ith bin of the output space is
p̂Y (y) =
1
H
NM C Y
(Bi ) ∀y ∈ Bi
(1.7)
where p̂Y (y) is the estimate of the true PDF pY (y). The Monte Carlo estimation
(histogram) of the output PDF is illustrated on the left in Figure 1.6a for a scalar
13
Chapter 1. Introduction
pY  y 
pX  x 
g  
x
y
pˆ Y  y  Bi 
Y
X
Di
NMC samples
10
Bi
(a)
-9
Probability
pY ( yi )  1010
I0.99
-10
10



= 0.9
= 0.95
= 0.99
-11
10
11
10
12
10
Number of samples
13
10
(b)
Figure 1.6: The scenario of Monte Carlo Estimation of the output PDF: a) the PDF in
the input space and the exact and estimated PDFs at the output space, b) Confidence
intervals vs. total number of samples, for estimating the probability of an output bin
with probability 10−10 .
input.
The estimated probability of the ith bin is itself a random variable, with the following
mean and standard deviation for the ith bin
(i)
µM C = E {p̂Y (y)} = pY (y)
(i)
σM C ,
var {p̂Y (y)} ∼
=
q
q
pY (y)(1−pY (y))
NM C
(1.8)
∀y ∈ Bi
(1.9)
The relative error in estimating the probability of the ith bin is
(i)
M C =
v
u
u
t
var {p̂Y (yi )}
[E {p̂Y (yi )}]2
(1.10)
and (1.10) quantifies the error in estimating the PDF of the output of the system vs. the
number of samples that fall within each bin. The error is bin-dependent. When NM C
Chapter 1. Introduction
14
is large, the law of large numbers dictates p̂Y (y ∈ Bi ) ' pY (y ∈ Bi ) for all Bi ’s; output
bins located in the most probable regions in DY , i.e., where pY (y) is large, collect most
of the samples and are estimated with excellent accuracy. Output bins placed at the
tails of pY (y) either contain very small number of samples or zero samples. The only
way to force the MC simulator to estimate low probability bins is to increase NM C ,
which is quite inefficient in terms of simulation time.
Figure 1.6.b shows the η% confidence interval Iη for estimating the probability of
Bi , where the true (unknown) probability of that bin is pY (yi ) = 10−10 . The confidence
interval is defined as
¶(y ∈ Iη ) > 1 − η
(1.11)
From Fig.1.6.b we observe that for a reliable estimate of the bin probability, we need
about 100 bin visits.
1.3
1.3.1
Multicanonical Monte Carlo
Introduction
Monte Carlo simulation involves generating random samples from all input random
variables, calculating the system output, and computing appropriate averages to estimate desired statistical characteristics such as the mean, variance or the PDF of the
system output. The accuracy of such estimates scales with the total number of the
random samples generated. The link simulator, including the SOA simulator, is computationally expensive and a bottleneck to employing the MC technique. We require
accelerated Monte Carlo techniques such a multicanonical MC (MMC). MMC has recently proven to be a quite efficient tool in several problems in optical communications
(cf. [24] and references therein). In this section we briefly introduce MMC and discuss
how to implement it in practice. To put the MMC algorithm in proper context we begin
with a description of importance sampling and flat histogram methods.
15
Chapter 1. Introduction
1.3.2
Importance Sampling
Importance sampling, IS, is a statistical technique to increase the performance of
conventional MC simulations. In order to accelerate simulations we use a warped version
p∗X (x) of the input PDF pX (x). The function w(x) is called the weight function and is
given by
pX (x)
(1.12)
w (x) , ∗
pX (x)
For importance sampling:
(1)
(2)
(NIS )
1. Input samples are generated according to the pdf p∗X (x): X ∗ , X ∗ , · · · X ∗
where NIS is the total number of samples generated in the importance sampling
simulation.
2. Generated input samples pass through the system to obtain output samples:
(i) (i)
Y ∗ = g X∗ .
(1)
3. The histogram, HY∗ of the data set Y ∗ , · · · , Y ∗
set of bins as for MC simulations.
(NIS )
, is formed over the same
4. The estimate of the bin probability is formed p̂∗Y (yi ) = HY∗ (yi ) /NIS .
5. The estimate of the target PDF is formed by:
p̂Y (y) = Ŵi × p̂∗Y (y) ∀y ∈ Bi
(1.13)
where Ŵi is the average weight estimate corresponding to Bi and is given by:
(1.14)
µIS , E {p̂Y (y)} = pY (y) ∀y ∈ Bi
(1.15)
X
Ŵi =
(j)
w X∗
n o
(j)
j g X ∗
∈Bi
The mean of the PDF estimate using IS is given by
(i)
hence it is unbiased. The standard deviation of the IS PDF estimate is
(i)
σIS ,
q
var {p̂Y (y)} '
q
pY (y)(Wi −pY (y))
NIS
∀y ∈ Bi
(1.16)
where Wi = E {w (x) |x ∈ Di }, and Di = g −1 (Bi ) is the inverse image of the ith bin in
the output space. Wi is the average weight of all the samples that hit Bi , and Ŵi is its
estimate.
16
Chapter 1. Introduction
H Y  Bi 
fX  x
pˆ Y  y  
y  Bi
g  
Di
Ni
y
y
Bi
x
f X*  x 
H
*
Y
 Bi 
UNWARPING
x
Ni
N
pˆ Y  y   Wi  pˆ Y*  y 
*
i
N
WARPING
x
g  
Di
y
y
pˆ Y*  y  
x
N i*
N
Figure 1.7: Importance sampling estimation of the output PDF.
A proper choice of the weight function can greatly reduce the variance in the tail
region. The ratio of the variance of the MC and IS estimators is:
r
(i)
σIS
(i)
σM C
=
NM C Wi −pY (y)
NIS 1−pY (y)
∀y ∈ Bi
(1.17)
Equation (1.17) implies that if we find a warped PDF p∗X in the input space such
that
Wi −pY (y)
(1.18)
1 ∀y ∈ Bi
1−pY (y)
a simulation gain is obtained.We define the simulation gain to be NM C /NIS when fixing
the ratio in (1.17) to one.
Figure 1.7 illustrates the basic concepts discussed. The top row of Fig. 1.7 shows the
MC estimate of a low-probability output bin. The bottom row is the warped system
where the estimate is calculated. The input PDF is warped such that sampling the
new random variable generates more hits of the desired bin. The new estimate is then
deterministically “unwarped” per (1.13) and (1.14).
The major problem with IS is the difficulty in identifying a good warping. A random
warping might result in increased simulation time. Several academic problems have
17
Chapter 1. Introduction
optimum warps resulting in huge simulation gain. Finding the good warps for complex
real-world problems is extremely difficult, and for this reason the simple formulation of
IS has limited application in stochastic simulation of real problems.
1.3.3
Flat Histogram Importance Sampling
In this subsection we briefly discuss flat-histogram methods used to find optimal
warpings adaptively. Multicanonical Monte-Carlo is a flat-histogram technique, and
will be discussed in the next section.
If we temporarily assume that our problem is to estimate the probability of just
one output bin, say Bi , then (1.16) shows that the best warping is the one which gives
Wi = pY (y) for ∀y ∈ Bi , i.e., a uniform weight for all occurrences in Bi . The best
warping for estimating the probability of bin Bi is thus
∗(i)
pX,optimum
(x) =


pX (x)
pY (y)∆y
x ∈ Di (equivalently : y ∈ Bi )
0 otherwise

(1.19)
where ∆y is the bin width in the output space, DY . If ∆y is sufficiently small, pY (y)∆y
approximates the probability mass of hitting bin Bi . Clearly pY (y) is the unknown
quantity to estimate, so this formulation is not useful in the present form.
To estimate the PDF over the whole set of output bins {B1 , · · · , BNB }, the optimum
warp is obtained by summing the optimum warps (1.19) for each bin. The correctly
normalized PDF estimate
p∗X,F HIS =
NB
1 X
∗(i)
pX,opt (x)
NB i=1
(1.20)
is impractical, but optimal.
It can be shown that if input samples are generated according to the warped PDF
in (1.20), the histogram of the output samples will be flat over output bins. If the total
number of samples is denoted by NF HIS , the histogram will on average be
HYF HIS (Bi ) =
NF HIS
NB
∀i
(1.21)
Intuitively, (1.21) tells us that if we have limited resources to produce a fixed number
of samples, and if all the output bins are equally important, the best we can do to
Chapter 1. Introduction
18
evaluate all the bins as accurately as possible is to evenly distribute the samples among
all the output bins. However, this heuristic motivation for flat histogram methods is
not to be misconstrued as a solution. In order to estimate pY (y) we would need to know
pY (y)!!
1.3.4
Multicanonical Monte-Carlo
In this section we introduce the Multicanonical Monte-Carlo (MMC), a method to
iteratively reach a flat histogram (optimum warp) in the output space without prior
knowledge of pY (y). From (1.20), the metric to measure the goodness of the input warp
is the flatness of the resulting output histogram. MMC is one of several flat histogram
techniques. MMC has the important advantage of not requiring a priori information
about the system under study. It learns by iteratively exciting the system. This system
independence makes MMC quite an attractive tool to explore complicated nonlinear
optical communication systems.
The basic idea is to generate a collection of input samples from an initial warping,
apply the system mapping to them, observe the resulting output histogram, and propose
a new warping guided by the output histogram. The next run of simulation generates
input samples from the new warping and emphasizes the inverse image of the less
frequently visited bins and de-emphasizes those of the more frequently visited bins. By
performing this adaptive importance sampling, the visit histogram calculated at the
end of each run tends to the ideal flat histogram. The simulation iterations continue
until the visit histogram is sufficiently flattened over all the output bins of interest. The
first iteration of MMC is a conventional MC run. Figure 1.8 illustrates the iterative
steps of MMC.
We denote the warped input PDF, the resulting output histogram, and the resulting
(n)
(n)
PDF estimate at the end of the nth cycle of the simulation by pX (x), HY (y), and
(n)
(0)
p̂Y (y). The input and output PDFs are initialized according to: pX (x) = pX (x) and
(0)
p̂Y (y) = 1/NB . A total of C cycles, are performed, i.e., n = 1, · · · , Nc . At each cycle
a simulation run with NM M C samples drawn from the properly warped input PDF are
generated. At the beginning of the nth cycle
19
Chapter 1. Introduction
pY (y )
p*X (0) ( x)  p X ( x)
X
g (.)
Y
HY(1)  pˆY(1)

x
y
HY(2)  pˆY(2)

p*X (1) ( x)
X
g (.)
Y
y
p
* (2)
X
HY(3)  pˆY(3)
X
( x)
g (.)
Y
y
Figure 1.8: Iterative steps of MMC.
1. The new warping is applied:
(n)
pX (x) =
pX (x)
Cn p̂(n−1)
(g (x))
Y
(1.22)
where Cn is the normalization constant. Although we include Cn to be mathematically correct, we will see that an important property of MMC is that it is not
necessary to explicitly calculate Cn .
2. NM M C input samples are generated according to the PDF given by (1.22).
3. These samples are mapped to the output space and the new visit histogram,
(n)
HY (y), is formed.
4. The output PDF estimate is updated by
(n)
(n)
(n−1)
p̂Y (y) = Cn HY (Bi ) × p̂Y
(y) ∀y ∈ Bi
(1.23)
5. n = n + 1 and the control is transferred to step 1) until we reach n = Nc .
In practice a modified version of the preceding algorithm, which is called the smoothed
MMC, is often used. In the MMC algorithm presented in the previous paragraph, the
20
Chapter 1. Introduction
(n)
PDF estimates p̂Y (y) are prone to stochastic fluctuations. In the smoothed MMC algorithm proposed by Berg, the update law given by (1.23) is replaced by the following
relation
(n)
ĝn (y)
(n−1)
(n)
HY (Bi+1 )
p̂Y
(ỹ)
p̂Y (ỹ)
(1.24)
∀y ∈ Bi ; ỹ ∈ Bi+1
= (n−1) ×
(n)
(n)
p̂Y (y)
p̂Y
(y)
HY (Bi )
where ĝn (y) is given by
gn (y)
ĝn (y) = P
n
∀y ∈ Bi
(1.25)
gn (y)
p=1
and g is given by:
gn (y) ,
1
(n)
HY (Bi+1 )
+
1
(n)
HY (Bi )
−1
∀y ∈ Bi
(1.26)
At each step of the MMC, there are output bins that are not visited and the visit
histogram is zero over them. We follow Berg’s suggestion that in each step the output
histogram be modified by putting one sample in all empty output bins. In this way,
gn (·) remains well defined for unvisited bins.
As can be seen in Figure 1.8, the input space warped PDFs are complicated functions. Drawing samples from these PDFs is not trivial since the standard methods like
inverse cumulative distribution function and Von Neuman rejection methods are quite
inefficient. A well-known solution is to apply Markov chain Monte Carlo (MCMC)
techniques such as the Metropolis-Hastings (MH) algorithm. We discuss this technique
in the appendix.
1.3.5
Complete MMC simulations
Figure 1.9 illustrates the block diagram of the MMC algorithm, including the warping mechanism and the Markov chain to secure random samples. The left part of
Fig. 1.9 shows the proposal-generating Markov chain. At time-step m a proposed vector in the input space is generated, and denoted by X prop
m . This vector is mapped by
prop
g to the output proposal y ; this proposal is either rejected or accepted. If rejected,
the previously generated output is retained as the correct output. This continues for
Nc samples to complete the cycle. At the end of each MMC cycle, the histogram of all
the output samples is formed and used to update the estimate of pY (y). Then another
MMC cycle starts, until the machine reaches the pre-specified last cycle, i.e., cycle Nc .
At the last cycle the last update becomes the estimate of the output PDF pY (y).
21
Chapter 1. Introduction
prop
ymprop g
Xm
prop
Xm
Markov
Chain
g.
Yes
Xm
Yes
No
ym
Histogram Update
No
Z
1
ym
n
pˆY
?
Um
Um
1
n
pˆY
Z
1
1
1
pY n
1
ym 1
pYn
1
yprop
PDF Update
Figure 1.9: The block-diagram of MMC algorithm. The main elements of MMC are
presented.
1.3.6
An elementary example
As an elementary example to demonstrate how MMC behaves, we assume the input space is the ten dimensional real space R10 , i.e., DX = R10 , the components of
which are independent identically distributed Gaussian RVs with zero average and unit
variance. An arbitrary vector in DX is denoted by X, and is explicitly written as
X = [X1 , . . . , X10 ] where Xi ∼ N (0, 1). We suppose that the output is related to the
input by the relation y = g (X) where
y = g (X) =
10
X
Xi2
(1.27)
i=1
We know that the distribution of y is χ2 (10). We compare the results of the MC and
MMC with the true PDF in Figure 1.10. The histograms at the end of the nth MMC
cycle, where n is 1, 5, 10, and 15 are shown. In each cycle 104 iterations are performed.
The solid black curve is the theoretical output PDF. In the inset is shown the output
variable histograms, when the input vectors’ elements are drawn from the warped PDF.
We can see the histogram goes from strongly unimodal for n = 1, to more and more
flat. The flat line artifacts for each cycle are the result of a single occurrence allocated
to bins which were in fact unvisited.
22
Chapter 1. Introduction
0
-2
n 1
-4
-6
-8
log10 [ pˆ Y( n ) ( y )]
-10
n5
-12
-14
103
-16
4.5
-18
3.5
4
n 1
n  10
3
-20
-22
2.5
Hˆ n* 2
n5
1.5
-24
n  10
1
n  15
n  15
0.5
-26
0
0
50 100 150 200 250 300 350 400 450
Analytical
-28
-30
0
50
100
150
200
250
300
350
400
450
y
Figure 1.10: Chi-square order 10 PDF estimated by MMC. In the inset is shown the
warped histograms, at the end of MMC cycles n = 1, 5, 10, 15.
The warped PDFs of each element of the input states after cycles 1, 2, and 3 are
shown in Figure 1.11, where the initial PDF is a smooth Gaussian. The PDF becomes
more and more ill-behaved with growing n. The Metropolis-Hastings (MH) method
described in the appendix, nonetheless, efficiently generates samples from the warped
input PDFs.
Having demonstrated the implementation of the MMC algorithm, in the next section
we switch gears, and briefly introduce the semiconductor optical amplifier. Out goal is
to create the mapping g(·) that accurately models a SOA-assisted SS-WDM system.
23
Chapter 1. Introduction
-3
x 10
4
cycle1
cycle2
cycle3
Number of Samples
3.5
3
2.5
2
1.5
1
0.5
0
-5
0
X
5
Figure 1.11: Warped PDFs of each component of the random vectors in the input space
at MMC cycles 1, 2, and 3.
1.4
1.4.1
The Semiconductor Optical Amplifier
Introduction
Shortly after the invention of GaAs laser diode (LD), the first studies were reported
on the semiconductor optical amplifier (SOA) [26]. In principle the SOA and the LD are
very similar; their main difference lies in the fact that in the LD, the lasing condition
is achieved by pumping the device above the threshold, whereas in the SOA the facet
reflectivities are dramatically reduced, and the device is pumped below threshold [27].
Seminal papers on device modeling and noise properties of the SOA were published
in the 80’s [28, 29, 30]. While the erbium doped fiber amplifier (EDFA) has emerged as
the in-line amplifier of choice in optical communication systems due to its highly linear
nature, the SOA, in contrast, is highly nonlinear. This essential difference in behavior is
due to disparate fluorescence time (1 ms in EDFAs, tens of picoseconds to nanoseconds
in SOAs).
24
Chapter 1. Introduction
Longitudinal axis
Pump Current
Current Strip
O
Heterojunctions
Transversal plane
p-type
Current
Spreading
n-type
d
Fundamental
Mode Pattern
Input Optical
Field
Active Region
Intrinsic
material
L
Figure 1.12: Architecture of the SOA waveguide
Although the SOA cannot compete with the EDFA as a repeater in long-haul transmission links, the classical deployment of optical amplifiers, there is increasing interest in
studying and exploiting SOA nonlinear properties. Nonlinear characteristics make the
SOA a versatile functional block capable of achieving various all-optical signal processing tasks vital to emerging advanced all-optical networks. Besides its signal processing
potential, the SOA is preferred over the EDFA for integrated optics.
Semiconductor optical amplifiers used in optical communications work in the 1.3 µm
and 1.55 µm windows, and, as such, are made from III-V compound semiconductor
crystals like InP, InGaAs, InGaAsP, AlGaAs and InAlGaAs. The most important
materials are the InP-InGaAsP and InGaAs-InGaAsP quaternary systems. Figure 1.12
illustrates the waveguide structure of an SOA. The physics of semiconductor lasers and
amplifiers has been extensively discussed in many references [31]-[37]. Our discussion
in this subsection is limited to the equations we need for modeling the SOA in later
chapters. Note from Fig. 1.12 that the gain region is three-dimensional, with the optical
field propagating along the longitudinal axis, z.
25
Chapter 1. Introduction
S tm
i u lated
Em iss ion
S pontaneous
Em iss ion
S tim ulated
Absorp tion
Conduction Band
g
Valence band
Figure 1.13: Simplified band structure of the SOA
Light amplification by the SOA is the result of interaction between light and matter inside the device. The characteristics of the interaction of light with a specific
semiconductor material depends on the optical properties of the material.
The optical properties of semiconductors can be derived from their energy band
structure. The energy bands of the SOA, shown in Figure 1.13 [31], consist of a lower
level valence band (VB), and an upper level conduction band (CB). In III-V semiconductors, the VB is further subdivided to three subbands as is shown in Fig.1.13. At
absolute zero all the electrons are in the VB, leaving the conduction band completely
empty. At any nonzero temperature some electrons are thermally excited to the CB
leaving behind holes in the VB. We refer to electrons in the CB and holes in the VB as
carriers. Light photons interact with atoms through three major processes: stimulated
emission, stimulated absorption and spontaneous emission.
Chapter 1. Introduction
26
In thermodynamic equilibrium, the distribution of carriers in the energy levels obey
Fermi-Dirac statistics. Pumping the semiconductor by electric current causes more
electrons to be injected into the CB. In thermodynamic equilibrium higher energy states
are less populated; however, electrically pumping the semiconductor can establish the
population inversion condition (density of carriers in the upper level greater than that
of the lower level).
Photons input to the SOA interact with electron-hole pairs whose energy separation
match the photon energy; the result is that electrons combine with holes thus freeing
a photon with exactly the same energy as that of the incoming photon through the
process of stimulated emission. Some electrons spontaneously jump to VB and free
photons with random frequency and polarization, through spontaneous emission. The
spontaneous emission is amplified along with the signal, and results in noise at the SOA
output which is called amplified spontaneous emission (ASE).
In equilibrium, stimulated emission is balanced by stimulated absorption, where
electrons absorb input photons and jump from VB to CB. Under population inversion,
however, the carrier reservoir is not in equilibrium, and the rate of photons generated
due to stimulated emission is much higher than those which are lost due to stimulated
absorption; therefore the SOA can provide optical gain to the incoming signal.
1.4.2
Choice of SOA model
Besides the basic emission and absorption mechanisms discussed in the previous
subsection, other physical processes are involved in light amplification by the SOA.
For subpicosecond and/or multi-wavelength SOA inputs, where the optical signals are
wideband, the frequency dependence of the material gain [40], carrier heating (CH) and
spectral hole burning (SHB), group velocity dispersion in SOA waveguide, two-photon
absorption (TPA) loss, and spatial hole burning should be included in the model [40, 43].
Moreover, linewidth enhancement factor dynamics manifests itself in the subpicosecond
regime [41]. For multi-wavelength inputs, beating among input channels results in
signals generated at new frequencies due to four-wave mixing (FWM) [42].
The loss coefficient of the SOA has both carrier-independent and carrier-dependent
parts [31]. The ASE can be modeled as a spatio-temporal random process [39], an
Chapter 1. Introduction
27
equivalent input-referred noise source [44], or can be simply neglected [37]. If both coand counter-propagating signals co-exist at the same time, we have to deal with two
propagation equations for forward and backward fields, and we have to solve them in
the rest frame [45].
For SOA-based noise suppression, we can safely neglect the physical mechanisms
that are important only in the subpicosecond and/or multi-wavelength regimes. We
assume the input light is a single spectrum slice of bandwidth less than 100 GHz. We
are interested in bit rates less than or equal to 10 Gb/s. Since each noise-cleaning
SOA amplifies a single narrow channel, we neglect ultrafast effects like CH, SHB, TPA,
group velocity dispersion, linewidth enhancement factor dynamics, and spatial whole
burning.
We assume the material gain is wavelength-independent as per (1.29). The propagation inside the SOA waveguide is assumed unidirectional, and the carrier diffusion
is neglected. The carrier lifetime will be assumed to have a constant effective value.
The effects to be included in the model are therefore: the carrier-dependent material
gain, carrier-dependent, and carrier-independent loss, effective linewidth enhancement
factor, and the ASE. Finally, we do not address quantum-dot SOAs in this thesis.
1.4.3
Propagation equation with ASE
Let r = (x, y, z) refer to the spatial coordinates for the SOA, such as those shown in
Fig. 1.12, where z is the coordinate along the SOA waveguide longitudinal axis, x, and
y are coordinates in the transversal plane. The optical field inside the SOA waveguide
is written per
(1.28)
E(r, t) = êA(z, t)φ(x, y)exp[j(ω0 t − βz)]
where ê is the unit polarization vector, t is time, φ(x, y) is the fundamental mode
transversal field profile of the SOA waveguide, A (z, t) is the slowly varying envelope of
the optical field, ω0 is the angular frequency at the center of the signal bandwidth, and
β is the propagation constant.
The transversal coordinates are integrated out following the standard procedure in
28
Chapter 1. Introduction
analysis of laser diodes and SOAs [32]. The material gain is defined as
g (z, t) = Γa (N (z, t) − N0 )
(1.29)
where Γ is the confinement factor, a is the differential gain, N (z, t) is the carrier density
at point z along the longitudinal axis inside SOA waveguide and at time t, and N0 is
carrier density at transparency. The gain dynamics equation becomes
gss − g (z, t) g (z, t) |A (z, t)|2
∂g (z, t)
=
−
∂t
τc
Esat
(1.30)
where gss is the small-signal gain, τc is the carrier life-time, and Esat is the saturation
energy given by Esat = ~ω0 σ/a, where σ is the mode cross-section, which is calculated
as σ = wd/Γ, where w and d denote the thickness and the width of the SOA waveguide.
The small signal gain is given by
gss = ΓaN0
I
−1
Iss
(1.31)
where I is the bias current and Iss is the transparency current given by
Iss =
qV N0
τc
(1.32)
where q is the electron charge and V is the SOA waveguide volume.
We refer to (1.30) as the gain dynamics equation. The gain dynamics equation is
sometimes written in alternative forms. One can define the saturation power, Psat to
be Psat = Esat /τc . The gain dynamics equation can be rewritten as
gss − g (z, t) g (z, t) |A (z, t)|2
∂g (z, t)
=
−
∂t
τc
τc Psat
(1.33)
where vg is the group velocity, αint is the internal loss per length, and α is the linewidth
enhancement factor 1 .
In studying traveling-wave amplifiers we examine the slowly varying envelope A(z, t)
in the moving frame [37, 38], where, the temporal variable is referenced per t → t−z/vg .
The propagation equation is given by
∂A (z, t)
1
1
= g (z, t) (1 − jα) A (z, t) − β(z, t)A (z, t) + ε (z, t)
∂z
2
2
1. It is also referred to as the Henry factor, Henry α-factor, or α-factor.
(1.34)
29
Chapter 1. Introduction
where β(z, t) includes both the carrier-independent, and carrier-dependent SOA loss
coefficients. The loss coefficient is
β (z, t) = K0 + ΓK1 N (z, t)
(1.35)
where K0 and K1 are respectively the carrier-independent and carrier-dependent loss
coefficients [31]. After substituting (1.29) in (1.35) we get
β (z, t) = β0 + β1 g (z, t)
(1.36)
where β0 , K0 + ΓK1 NT and β1 , ΓK1 /a.
In (1.34) (z, t) is the amplified spontaneous emission modeled as a complex Gaussian random process
ε (z, t) =
q
g (z, t) + ΓaN0 [nr (z, t) + jni (z, t)]
(1.37)
where nr (z, t) and ni (z, t) are independent Gaussian white noise random processes
mean zero and variance one-half.
In chapter 2, following [39], we assume the space-resolved model, based on (1.30)
and (1.34), where the material gain and the optical field are functions of both z and t.
The carrier-independent loss and carrier-dependent loss coefficients are included, and
the ASE is modeled by its spatio-temporal stochastic process representation given by
(1.37).
1.4.4
Reservoir model
In [37] a simple, yet powerful model for SOA is derived by neglecting the amplified
spontaneous emission term and the distributed loss (1.34). The reduced propagation
equation becomes
1
∂A (z, t)
= g (z, t) (1 − jα) A (z, t)
(1.38)
∂z
2
We define the integrated gain from the input to any position along the longitudinal
axis of the SOA as
Z z
h (z, t) =
g (z 0 , t) dz 0
(1.39)
0
30
Chapter 1. Introduction
We define the total integrated gain htot (t), to be the integrated gain evaluated at the
SOA output
h (t) = h (z, t) |z=L
(1.40)
The input and output fields and powers are explicitly defined as:
Ain (t) = A (z, t) |z=0 , Aout (t) = A (z, t) |z=L
Pin (t) = P (z, t) |z=0 , Pout (t) = P (z, t) |z=L
(1.41)
At any position along the longitudinal axis the optical intensity and field are related to
the corresponding input by
P (z, t) = Pin (t) exp [h (z, t)]
(1.42)
1
(1.43)
A (z, t) = Ain (t) exp (1 − jα) h (z, t)
2
In particular, fields and powers at the input and output of the SOA are related by the
following
Pout (t) = Pin (t) exp [h (t)]
(1.44)
1
Aout (t) = Ain (t) exp (1 − jα) h (t)
(1.45)
2
Finally, substituting (1.42), or (1.44) in (1.33), and using (1.39), the following equations
for the time evolution of the integrated and the total integrated gain are obtained
i P (t)
gss z − h (z, t) h h(z,t)
∂h (z, t)
in
=
− e
−1
∂t
τc
τc
(1.46)
i P (t)
dh (t)
gss L − h (t) h h(t)
in
=
− e −1
dt
τc
τc
(1.47)
In contrast to the space-resolved model of (1.34), we refer to (1.47) as the single
reservoir. An alternative model, falling in between the space-resolved and the reservoir
models both in terms of accuracy and of complexity, was introduced in [44], where the
SOA is modeled by multiple reservoirs. In this model, a chain of SOA sections, each
modeled by a single-reservoir, with a lumped loss elements between successive sections.
This model is referred to as multi-reservoir.
It is shown in [44] that ASE can be effectively modeled as an input noise source.
The simulations of Chapter 2 are based on (??. Later on, we compared the resulting
Chapter 1. Introduction
31
estimated PDFs of the received spaces, and marks, of the received signal, where either
the space-resolved or the multi-reservoir models are used for the SOA. We concluded
that both models result in the same PDF curves; therefore, in Chapters 3, 4, and 5, the
multi-reservoir model are used. Since the light source in Chapter 3 is coherent, we had
to keep the input-referred ASE noise source as suggested in [44], Later on, we verified
that for incoherent spectrum-sliced sources, when we are in the intensity-limited regime,
neglecting SOA ASE does not impact the received signal PDF, therefor in simulations
of Chapters 4 and 5, we used the multi-reservoir model without input-referred noise
source.
1.5
Outline of the thesis
In the previous sections of this chapter we presented the motivation for studying
SOA-assisted SS-WDM systems. For performance analysis of such systems we need
efficient numerical techniques. These techniques are also useful for design and optimization of a wide variety of other optical communication systems employing nonlinear
SOAs as building blocks for OSP purposes. We proposed to use the MMC simulation
technique in order to exactly estimate the conditional PDFs of the SOA-assisted link,
down to arbitrarily small values in a feasible time. We reviewed the theory of light
amplification by the SOA, and commented on the level of complexity to be met in SOA
modeling for the specific problems that are addressed in the following chapters of this
document.
In the following chapters of the thesis, we build upon the two the fundamental
material presented in chapter 1, namely, MMC algorithm, and SOA models, and develop
a complete simulator to analyze the performance of SOA-assisted SS-WDM systems.
More importantly, we demonstrate how to use the simulator to design spectrally efficient
SOA-assisted SS-WDM transmission systems.
Chapter 2 is devoted to the performance analysis of a single channel SOA-assisted
SS-WDM link by MMC. Our contributions are the following:
– a bit pattern warping technique for MMC to account for the ISI,
– a complete time-domain model of a single-channel SOA-assisted SS-WDM,
Chapter 1. Introduction
32
– simulated PDFs of the received signal, as well as BERs (at 1.25 Gb/s and 2.5
Gb/s),
– comparison of simulated and measured PDFs and BERs.
As will be explained, the bit patterning effect from the SOA is not an issue in the single
channel SOA-assisted SS-WDM system considered in Chapter 2;all the ISI is due to
electrical filtering.
In Chapter 3 we introduce a general-purpose simulator for studying various systems
employing nonlinear SOAs. This chapter is not restricted to SOA-assisted SS-WDM.
We take as an example system a 10 Gb/s modulated laser output amplified by an in-line
SOA and examine bit patterning. Our contributions are:
– a measurement technique for conditional PDFs of marks and spaces at the receiver
side,
– simulated and measured conditional PDFs and BER for a 10 Gb/s on-off keying
externally modulated laser amplified by a nonlinear SOA.
In Chapter 4, we return to SOA-assisted SS-WDM systems. Our goal is to study the
multi-channel SOA-assisted SS-WDM system, where performance is set by the tradeoff
between the filtering effect and crosstalk from adjacent channels. The shape, bandwidths, and the separation of optical channels are selected to maximize performance.
Our contributions are:
– simulated and measured saturated RIN spectra,
– simulations of the RIN spectra after post-filtering, highlighting the importance of
1) SOA α−factor, and 2) SF and CSF roll-off ates,
– a multiple-channel MMC simulator for SS-WDM.
– a parallel implementation of the MMC simulator,
– simulations of BER for a three-user SOA-assisted SS-WDM link operating at 5
Gb/s, when SF and CSF filter type and CSF bandwidth are varied,
– comparison of optimum SOA-assisted SS-WDM performance to that of SS-WDM
without noise suppression,
– the impact of channel power imbalance,
– the accuracy of the Q-factor BER approximation,
– design of channel spacing and channel bandwidth to maximize the spectral efficiency in a coded SOA-assisted SS-WDM system,
Chapter 1. Introduction
33
– cross-validated of the multi-channel simulation against previously published measurements available in the literature.
In Chapter 5, we address performance analysis of SOA-assisted spectral amplitude
coded optical code division multiple access (SAC-OCDMA) systems. Our contributions
are:
– simulation of BER of conventional and SOA-assisted SAC-OCDMA system for 2,
3, 5, and 7 users,
– comparison of simulation and measurement for 2- and 3-users BERs. 2- and 3-user
estimated BERs are compared with the measured results,
– quantification of the efficiency of SOA intensity noise suppression for OCDMA,
– accuracy of the Q-factor BER approximation.
Finally, in Chapter 6, conclusions are drawn, and some of the possible future research
plans, based on the material developed in this dissertation will be suggested.
Chapter 2
Noise Suppression
We present a thorough numerical study of intensity noise mitigation of spectrum sliced
wavelength-division multiplexing (SS-WDM) systems employing a nonlinear semiconductor
optical amplifier (SOA) before the modulator. Our simulator of the SS-WDM link, embedded
inside a multicanonical Monte Carlo (MMC) platform, estimates the tails of the probability
density functions of the received signals down to probabilities smaller than 10−16 . We introduce a new, simple and efficient technique to handle intersymbol interference (ISI) in MMC
simulations. We address the impact of optical post-filtering on SOA noise suppression performance. While previous research experimentally observed the SOA-induced noise cleaning
in SS-WDM systems, this is the first complete simulator able to correctly predict the ensuing
BER improvement. We measure the BER at different bit-rates and validate predicted BERs
with and without post filtering.
Chapter 2. Noise Suppression
2.1
35
Introduction
While SOA-based intensity noise mitigation has been studied extensively both experimentally and theoretically, no general-purpose design tool exists to optimize the
BER performance for a desired SOA-based system.
The exact form of the photon statistics at the output of a nonlinear amplifier is
extremely complicated to derive [46]. In the case of saturated SOAs two approximate
approaches exist: 1) characterizing the noise spectra at the SOA output whether the
source is incoherent [12] or coherent [47], and 2) analytical approximations of the PDF
of the output intensity when the source is coherent [48]. In the first approach the
noise spectrum at the SOA output is calculated for CW input light. This method is
useful in that it determines the suppression bandwidth, and provides an estimate of the
relative intensity noise (RIN) reduction. However, it does not provide the indispensable
knowledge of the PDF of the output intensity for a complete statistical analysis.
In [48] a PDF is obtained when the source is coherent using perturbation theory; the
ASE field added to the coherent signal is treated as a perturbation. Analytical expressions for the PDF are derived using path-integral methods. However, the perturbation
approach to find the PDF cannot be extended to incoherent sources whose optical field
is a zero mean process. In summary, to date we can find the RIN spectrum of the photodetected signals when the source is CW, either coherent or incoherent; or we can find
the PDF of the SOA output light intensity (and hence find the BER) when the SOA
input is coherent CW. In either case the analysis is limited to CW only. The impact of
modulation, for example through the induced patterning effects in the amplifier [49], is
not captured. Our simulator fills these gaps.
Our interest in noise statistics is applied in particular to noise suppression properties
of a SOA on incoherent light in SS-WDM systems. In those systems, as discussed in
Section 1.1.3, optically filtering the received signal by a CSF at the receiver side results
in significant neutralization of the intensity noise suppression. Although the phase-tointensity conversion due to optical filtering signals with noisy phase is treated in [50] for
coherent sources, no quantitative analysis of the post-filtering in the case of SS-WDM
exists in the literature. Our simulator includes not just the SOA, but the complete SSWDM system in order to capture this important phenomenon. We model incoherent
36
Chapter 2. Noise Suppression
SOA
Data
Current
MZM
Y
Broadband
Source
Slicing
Filter
Channel Detector
selecting
Filter
Electrical
Filter
Figure 2.1: SS-WDM link equipped with a pre-modulator noise suppressing SOA. MZM:
Mach-Zehnder modulator, Y : sampled received voltage.
light, in the time domain, as a signal whose complex envelope is a zero-mean Gaussian
process [21]. This process is filtered optically in our numerical simulations; the level of
coherence of the output light depends on the spectral characteristics of the filter.
The single-channel SOA-assisted SS-WDM link that will be analyzed throughout
this chapter was illustrated in Fig. 1.2, and is reproduced in Figure 2.1 now with modulation. The optical field after the slicing filter is a band-limited, complex Gaussian
random process. The source intensity at each instant has a negative exponential distribution, resulting in 0 dB of EIN as given in (1.3). Such a large intensity noise introduces
a BER floor that severely limits the performance of SS-WDM. As discussed in Section
1.1.2, an SOA operating in saturation placed before the modulator offers considerable
intensity noise mitigation due to SGM. More details can be found in [12, 19, 47].
2.2
System Simulator
In this section we give a top-down detailed description of the MMC-based singlechannel SOA-assisted SS-WDM simulator. Basic concepts of MMC are reviewed in
Section 1.3 and Appendix A. In the next subsection we give the detailed flowcharts of
our implementation of the MMC algorithm. In subsection 2.2.2 we discuss the model
of the optical link, and in subsection 2.2.3 we discuss the SOA model.
2.2.1
The MMC platform
Figure 2.2 illustrates the block-diagram of the MMC platform. Following the con-
37
Chapter 2. Noise Suppression
Y = g(X )
X
Y
SUT
Histogram
Update
PDF Warper
PDF Update
RVG
MMC Platform
Figure 2.2: MMC platform. RVG: random vector generator, SUT: system under test.
ventions of Section 1.3, the MMC simulation consists of Nc MMC cycles. At each cycle
N random vectors are serially generated by the random vector generator (RVG) unit.
A random input vector is denoted by X. The action of the SUT on the input vector
is abstractly shown by a mapping g(·) from the ds -dimensional input space of random
vectors to the one-dimensional output space of the test statistic Y . The SUT is the
SS-WDM link equipped with the SOA, whose corresponding g (·) is described in the
next subsection. The test statistic is the sampled voltage at the receiver. The input
vector X has the following form
(2.1)
X , [N , P ]
N is a vector of independent identically distributed (i.i.d.), continuous random variables
called the noise vector. P is a nonnegative integer between 0 and 2M − 1, where M is
the SUT memory in terms of number of bit intervals. The binary representation of P
is the bit pattern loaded in the SUT 1 . The noise vector is written as
N
− , ñ
−
BBS
; ñ
−
(SOA)
ASE
; nE
(2.2)
(SOA)
The noise vector consists of two subvectors, ñ
, and ñ
, and one scalar nE .
−
−
BBS
ASE
Subvector ñ
contains random variables used to synthesize the input spectrum sliced
−
BBS
(SOA)
random process. Subvector ñ
is passed to the SOA model inside the SUT to
−
ASE
simulate the spontaneous emission events in the SOA. It contains samples of nr and ni ,
the Gaussian white noise processes used to synthesize the ASE term per (1.37), at all
sampled space-time points. The scalar nE represents the receiver noise voltage.
1. In the system state-space picture presented in Section 1.2.3, the input space vector X was
composed of the noise vector N , and the bit pattern vector B. Here, the whole bit pattern is generated
by first generating the single integer P , and then casting P into its binary repreresentation, which is
easier to program, since only one Markov chain is necessary for generating all patterns.
38
Chapter 2. Noise Suppression
At the first MMC cycle, the elements of N are independent Gaussian random variables with zero mean and unit variance. 2 P is distributed uniformly among integers
between 0 and 2M − 1. At each cycle, after all samples are generated and passed to the
SUT, the output histogram is formed. The histogram calculated in cycle k is denoted
(k)
(k)
by HY . The PDF estimate of Y is updated and denoted by p̂Y . At each cycle the PDF
warper unit uses the latest PDF update, calculated at the end of the previous cycle,
to warp the PDF of the random input vectors X
− such that the corresponding output
values are a driven toward rare events. The first MMC cycle is an MC simulation of the
SUT. In subsequent cycles, the joint PDF of the spectrum sliced light, SOA amplified
spontaneous emission (ASE), the receiver noise, and the bit pattern is warped and the
PDF of Y is estimated down to very low probabilities. At the last cycle, the latest PDF
(N )
update, p̂Y c (·), is output and the simulator stops. To estimate the BER, conditional
(N )
(N )
PDFs of marks, p̂Y c (y |1), and spaces, p̂Y c (y |0), are separately estimated. The intersection of the conditional PDF yields the optimal threshold η. The area under the
crossing tails is computed to yield the BER per (1.4).
We suppose the output space is divided into NB bins indexed by integer b = 1, ..., NB .
The MMC cycle number is denoted by k = 1, ..., Nc , where Nc is the number of MMC
(k)
cycles. The PDF estimate obtained at the end of the k th cycle is denoted by p̂Y [·].
The relation between the output histogram and the output PDF estimation is
(k)
(k)
p̂Y [b] ,
HY (Bb )
N
(2.3)
where N is the number of samples generated per MMC cycle. the output PDF is
initialized to
(0)
(2.4)
p̂Y [b] = N1 ; b = 1, . . . , NB
(k)
We denote the ith input sample, generated within the k th cycle by X i . The corre(k)
(k)
sponding output sample is yi = g(X i ), with i = 1, . . . , N .
The normalized histogram of the set of output samples over the pre-specified output
(k)
bins is denoted by HY . Details of MMC algorithm are given in Sections 1.3.4 and 1.3.5.
Figure 2.3 shows the details of the subsystems of the MMC platform, which, compared to Fig.1.9 contains more details on RVG. We have divided the MMC platform into
2. More precisely, element nE is not actually zero mean, but rather its mean is selected to match
measurements.
39
Chapter 2. Noise Suppression
RVG
k
X i-1
k
N i-1
X
N
prop
SUT
NVG
Yes
k
Pi-1
prop
y
prop
PNG
P
Z
yk
i
Yes
prop
No
X
k
i
k
i-1
y
-1
k 1
Y
pˆ k
pˆ
k 1
U
? pˆY
i
Histogram
Update
No
k 1
pˆY
Y
1
Z -1
Z-N
PDF
Update
yi-1k
y
prop
PDF Warper
MMC Platform
Figure 2.3: Detailed block-diagram of the MMC platform. RVG: random vector generator, NVG: noise vector generator, PNG: pattern number generator, SUT: system
under test, D represents unit delay.
four basic subsystems, i.e., PDF warper, random vector generator (RVG), histogram
update, and PDF update. We briefly discuss each subsystem here.
We start with the PDF warper subsystem, the most important subsystem of the
MMC platform. The adaptation in the MMC approach requires generation of realizations in the multi-dimensional input space following the statistics of the very irregular
multidimensional warped PDF that is fixed for the cycle (cf. (1.22)). We refer to the
(k)
warped PDF of the k th cycle by pX (·). The input realizations (or samples) are generated using the MH algorithm introduced in Appendix A. The idea is to propose input
samples according to their unwarped distribution pX (X), which is known, regular and
well-behaved, and then either reject or accept the proposed samples (rejection means
that the previous realization is reused) per a specified, randomized criterion. Proposal
of new samples is done by RVG, and will be discussed later. The resulting dependent
(k)
sequence of samples from pX (X) will asymptotically have the desired warped PDF pX ,
provided proper selection of the randomized rejection criterion. For an entire cycle, a
MH algorithm runs within the (input) PDF warper. The flowchart of the PDF warper
is shown in Figure 2.4.
Now we consider RVG. The RVG uses Markov chain Monte Carlo (MCMC) techniques to facilitate the generation of samples by the RVG, as illustrated in the flowchart
40
Chapter 2. Noise Suppression
Ui
Ui
yi k
yi-1k
k
Xi-1
Xi
?
pˆYk
0,1
1
k 1
Y
pˆ
yi-1k
y prop
yi k
k
Xi
k
y prop
X
prop
Figure 2.4: Flowchart of the PDF Warper. U [0, 1] is a uniform RV on [0,1].
in Figure 2.5 (also cf. appendix. A). While the proposed input vectors are now correlated, the net effect is to lead to a better overall convergence of the MMC adaptation.
Consider first the PNG. Suppose we are in the k th MMC cycle, and we want to generate the ith sample. PNG generates P uniformly distributed over the set of integers
n
o
0, . . . , 2M − 1 . With probability psw an independent sample P prop is generated (using
PNGind ), while with probability 1 − psw the sample is constrained to fall in a certain
(k)
neighborhood of the previous sample Pi−1 [71]. We used psw of 0.1 in our simulations.
PNGind calls a standard random integer generator routine to generate the new proposal
(k)
independent from the past. On the other hand, given Pi−1 , PNGMCMC proposes a new
(k)
pattern through P prop = Pi−1 ⊕2M up , where ⊕2M denotes modulo 2M addition. The
innovation up is constrained to permit only a limited number of bits to flip. up is a
zero-mean, discrete, uniformly distributed random variable taking integer values from
−Mp to Mp . In our simulations we used Mp = 1, i.e., from pattern P we go either to
P + 1, or to P − 1, or to P .
Consider next the RNG which generates a vector of Gaussian random variables.
The proposed noise vector is denoted by N prop , and the ith accepted noise vector at k th
(k)
MMC cycle is denoted by N i . These noise vectors are written as
h
N prop , N1prop , . . . , Ndprop
s
i
(2.5)
41
Chapter 2. Noise Suppression
START
Generate
u  U [0,1)
YES
NO
u ? psw
PNGind
PNGMCMC
prop
using
Generate
an integer random
number generator
uP  U P
P
P
RNGind
N
RNGMCMC
prop
using
Generate vector
a Gaussian random
number generator routine
(k)
prop
Pi-1
j
2M
uP
1
u N UN
N
prop
j
(k)
Ni-1,j u N
2
(k) 2
R min 1, exp -0.5 N jprop + 0.5 Ni-1,j
u U 0,1
NO
?
u R
k
N prop
 N i1, j YES
j
j ¬ j +1
YES
?
j ds
NO
X
prop
X
prop
;P
prop
STOP
Figure 2.5: Flowchart of the RVG.
42
Chapter 2. Noise Suppression
(k)
Ni
h
(k)
(k)
, Ni,1 , . . . , Ni,ds
i
(2.6)
With probability psw a sample N prop of i.i.d. zero-mean, unit variance Gaussian
elements is generated (using RNGind ), while with probability 1−psw the sample is either
(k)
constrained to fall in a certain neighborhood of the old sample N i−1 or N prop is simply
recycled (using RNGMCMC ). RNGMCMC consists of ds independent Markov chains. Each
chain generates an innovation uN that is uniformly distributed over [−MN , MN ]. We
used MN = 1.5 in our simulations. Note that parameters psw , Mp , and MN were chosen
(k)
by trial and error. With probability R the previous sample Ni−1,j is reused with no
innovation where
(k)
R = min(exp(−0.5(Njprop )2 + 0.5(Ni−1,j )2 ), 1).
(2.7)
(k)
With probability 1−R, the innovation is added to the previous sample Ni−1,j to generate
the Njprop . The RVG flowchart is given in Figure 2.5.
Having now described all the component parts, we give in Figure 2.6 the overall
flowchart of the MMC algorithm and discuss how the input warped PDF is generated
after each cycle. The histogram update subsystem collects accepted output samples
(k)
and calculates HY , over the output bins.
(k)
The PDF update subsystem uses HY and the latest output PDF estimate to make
(k)
a new estimate; p̂Y [b] is the new probability that the output will fall in the bth bin.
(k)
(k−1)
The probability of the first bin is p̂Y [1] = p̂Y
[1], and we then use [57, 61]
(k)
p̂Y [b+1]
(k)
p̂Y [b]
(k−1)
=
p̂Y
[b+1]
(k−1)
p̂Y
[b]
where
(k)
HY (Bb+1 )
(k)
HY (Bb )
(k)
g̃k [b] =
ĝk [b]
b ∈ [1, . . . , NB ]
(k)
HY (Bb )HY (Bb+1 )
(k)
(k)
HY (Bb ) + HY (Bb+1 )
and
ĝk [b] =
(2.8)
g̃k [b]
k
P
s=1
g̃s [b]
(k)
The resulting PMF p̂Y is normalized to assure the total probability is one 3 .
3. g̃ and ĝ should not be confused with the SUT mapping g (·).
(2.9)
(2.10)
43
Chapter 2. Noise Suppression
START
0
pˆ Y  b   1/ N B
b  1, , N B
k =1
Initialize:
X 0(k)
i =1
RVG
y prop  g  X prop 
PDF Warper
i ¬ i +1
YES
?
i <N
NO
Calculate
Update
HY
k
pˆYk
?
k  Nc
YES
k ¬ k +1
NO
STOP
Figure 2.6: Flowchart of the MMC.
2.2.2
The System Model
The block-diagram of the SUT is shown in Figure 2.7. This block diagram corresponds to the SS-WDM link of Fig. 2.1. At the input, the random vector X is
(SOA)
decomposed into its subcomponents: ñ
, ñ
, and nE . The gain parameter G1 is
−
−
BBS
ASE
44
Chapter 2. Noise Suppression
 SOA
n ASE
nBBS
X
h1 [⋅]
G1
Ain
SOA
Model
SF
P
BPG
B
nE
Aout
MOD
h2 [⋅]
⋅2
CSF
PD
4th BT G3
y
EF
G2
SUT
Figure 2.7: Model of the SS-WDM link of Fig. 2.1 as a SUT inside the MMC platform of
Fig. 2.2; BPG, bit pattern generator, MOD, modulator, SF, slicing filter, CSF, channel
selecting filter, PD, photodetector, EF, electrical filter; 4th BT stands for fourth-order
(lowpass) Bessel Thompson. Gain blocks are explained in the text.
used to set the average input power to the SOA, since all filters in the simulator are
normalized such that the vector of the impulse response has unit norm. G2 , and G3 are
used to adjust the noise and received signal power, respectively.
We model the thermal light source as having a lowpass equivalent optical field that
is a complex Gaussian random process [19]. Experimentally, the BBS used had a
33.580 nm 3 dB bandwidth, as directly measured by the optical spectrum analyzer. In
our SS-WDM experiment we filtered this BBS source using a 0.24 nm optical slicing
filter (SF). Experimentally the BBS spectrum is flat over the narrow band of the slicing
filter, thus we model the output optical field of the BBS in the time domain by a white
complex Gaussian noise, and filter it with a digital version of the SF described in the
next paragraph. The output light will be partially coherent, with temporal coherence
determined by the SF.
To synthesize the slicing filter, we measured the optical spectrum from the setup
in Fig. 2.1, and then used the Remez exchange method, implemented in MATLAB,
to extract the tap weights of an equivalent FIR filter, h1 [·], whose frequency response
matches the measured optical spectrum after the slicing filter (see Fig. 2.1). The
spectrum-sliced optical field is then obtained by filtering a complex white Gaussian
noise by h1 [·]. Figure 2.8 shows the measured and simulated optical spectra at the SOA
input and output. Fig. 2.8 shows measured and simulated PSDs of optical fields both
at the input and output of the SOA. The excellent correspondence of the measurement
45
Chapter 2. Noise Suppression
30
Simulated
20
PSD [dBm/nm]
10
Measured
Output
0
-10
-20
-30
-40
Input
-50
-60
-70
1549.5
1550
1550.5
1551
Wavelength [nm]
Figure 2.8: measured and simulated spectrum slices at the SOA input and output.
and simulation of the output light over the band of interest confirms that we have well
modeled the coherency introduced by filtering, and validates our use of ideal, incoherent
light as an input to the MMC simulator.
The SOA model is discussed in the next subsection. As illustrated in Fig. 2.8, we
have chosen h1 such that the measured and simulated spectrum slices match over a 30 dB
range, which is sufficiently accurate for the simulations of this Chapter. The FIR filter
had 10 taps. Matching over wider bandwidths can be achieved, if needed, at the expense
of increasing the number of taps. The binary pattern generator (BPG) subsystem
accepts the integer P , and outputs a vector m
− , which is the binary representation of
P . The modulator (MOD) subsystem shapes and upsamples bits m
− , and adjusts the
extinction ratio of the modulating waveform, for instance to match the experimental
values, and finally multiplies the modulating waveform by the output vector of the SOA
model.
The optical channel selecting filter (CSF) is modeled similarly to the SF, and the
impulse response of its digital equivalent is h2 [·]. The equivalent FIR filters synthesized
by the Remez method have flat group delay. We verified that SF and CSF filters used
experimentally also have flat group delay over their passbands. The photodetector (PD)
is an ideal square-law element, and the electrical filter (EF) is obtained as the bilinear
implementation of an analog fourth-order lowpass Bessel-Thompson filter [68].
46
Chapter 2. Noise Suppression
Table 2.1: SOA Parameters used in simulations
Carrier lifetime, τc
Saturation power, Psat
Linewidth enhancement factor, α
small signal gain, gss
Carrier independent loss coefficient, β0
Carrier dependent loss coefficient, β1
SOA length, L
~ω0
ΓaNT
2.2.3
170 ps
14 dBm
3.5
14500 1/m
2180 1/m
1600 1/m
650 µm
1.28e-19 J
70001/m
The SOA Model
The propagation equation of the optical field inside a traveling-wave SOA in the
moving frame was given in (1.34), and is repeated here for reference
∂A (z, t)
1
= [(1 − jα) g (z, t) − β (z, t)] A (z, t) + ε (z, t)
∂z
2
(2.11)
The dynamic gain equation is given by (1.33). The parameters of the SOA we used
in this work are listed in Table 2.1. Here we discuss how the SOA model fits, as a
subsystem, into the simulator. The behavioral block diagram of the SOA model, as a
subsystem in Fig. 2.7, is depicted in Figure 2.9. We denote by M the memory of the
link in terms of number of bits. Optical and electrical filters and dispersive elements
contribute to system memory. A SOA located before a modulator does not contribute
to the memory, while a post-modulator SOA with a carrier lifetime comparable to the
modulation bandwidth induces memory, i.e., the patterning effect.
We suppose each bit is upsampled Ns times. To calculate the SUT output at each
instant, the past M Ns time samples (called the memory window) of the input waveform
are needed. Since all waveforms are in the complex lowpass equivalent form, the length
of input (output) vector Ain (Aout ) is 2M Ns . In the spatially-resolved SOA model
that is used, the SOA cavity is divided into Nsec sections. The spontaneous emission
generated in each section over the memory window contributes to the SOA output.
(SOA)
The subvector ñASE contains samples of spontaneous emission events affecting the
SUT output and it has 2M Ns Nsec elements; the factor 2 exists because (z, t) in (1.37)
is a complex quantity. Given that nE is a single element, the dimension of the input
47
Chapter 2. Noise Suppression
out
 SOA 
n ASE
Figure 2.9: The input-output definition of the SOA spatially-resolved model
random vector is
ds = 2M Ns + 2M Ns Nsec + 1.
2.3
(2.12)
Numerical Results
In this section we report our numerical and experimental results on statistical properties of the SS-WDM received signals in the presence of a nonlinear SOA and the
CSF. Figure 2.10 shows the experimental and simulated PDFs of the received voltage
of the SS-WDM when a SOA was employed and the SOA input was a CW signal. The
slicing filter (SF) was 0.24 nm wide, and CSF was identical to SF. The electrical filter
bandwidth was 1.87 GHz. The DC-coupled receiver was an Agilent sampling scope.
The power to voltage conversion ratio was 0.75 V/W. Fig. 2.10a is the PDF of the SOA
output without electrical filtering. This PDF in fact corresponds to the light intensity at the SOA output; since the slice bandwidth was 30 GHz, and the photodetector
bandwidth was 50 GHz, the distortion induced by the finite bandwidth photodetector
was not significant. The receiver noise standard deviation, when the optical input was
turned off, was 8.8 µV as read from the scope.
Fig. 2.10b corresponds to when a CSF is placed after SOA, but no electrical filtering
is applied. In Fig. 2.10c the internal electrical filter of the sampling scope (bandwidth
48
Cycle 1 (MC)
-5
Cycle 2
no EF
-10
Cycle 3
no CSF
-5
log (PDF)
log (PDF)
Chapter 2. Noise Suppression
no EF
-10
with CSF (0.24nm)
Cycle 4
-15
Cycle 5
1
2
3
4
-0.5
0
0.5
1
Voltage [mV]
Voltage [mV]
(a)
(b)
-5
log (PDF)
log (PDF)
0
with EF (1.87 GHz)
-10
1.5
-5
with EF (1.87 GHz)
-10
with CSF (0.24 nm)
no CSF
-15
-15
0
0.05
0.1
0.15
Voltage [mV]
(c)
0.2
0.25
-0.04
0
0.04
0.08
0.12
0.16
Voltage [mV]
(d)
Figure 2.10: Measured (dots) and simulated PDFs of the received voltage in a SS-WDM
link equipped with pre-modulator nonlinear SOA, (a) no electrical filter, no CSF, (b)
no electrical filter, with CSF, (c) electrical filter of bandwidth 1.87 GHz, no CSF, and
(d) electrical filter at 1.87 GHz and CSF.
1.87 GHz) is applied, but the CSF is removed. In Fig. 2.10d both CSF and electrical
filter are present.
We can see that in all receiver configurations the fit of MMC and experiment is quite
satisfactory. The major conclusion from Fig. 2.10 is that the link model is accurate
enough to generate valid statistics, and 2) the MMC platform provides PDF estimation
down to very low probabilities with reasonable computation time: the MMC simulations
consisted of 5 cycles, and at each cycle 106 random vectors were generated. The SOA
was divided into 50 sections, and the simulation time-step was 4 ps. The slowest
simulation (Fig. 6d) took 1.5 hours per MMC cycle.
In all cases the average optical power input to SOA was 0 dBm, corresponding to
deep saturation, and the bias current was 495 mA. Optical attenuators at SOA output
were used to ensure the receiver electronics is not damaged. Adding optical and electrical filters led to extra insertion losses. We did not separately characterized the insertion
losses of optical and electrical filters; instead, in each measurement, we recorded the
sampled waveforms together with the histogram, and calculated the waveform mean
voltage. Since the receiver noise had been separately characterized, we could account
49
Chapter 2. Noise Suppression
pY (y 1)
-2
M=1
M=2
-4
log (PDF)
M=3
-6
-8
-10
M = 2, and M = 3 coincide.
-12
-14
0
0.05
0.1
0.15
0.2
0.25
Voltage [V]
(a)
-2
pY (y 1)
M=1
M=2
-4
M=3
log (PDF)
-6
M=4
-8
-10
-12
-14
M = 3, and M = 4 coincide.
-16
0.02
0.07
0.12
0.17
0.22
Voltage [V]
(b)
Figure 2.11: Simulated PDF of marks corresponding to different values of system memory. (a) The pre-modulator setup with the parameters coming from the experiment.
(b) The post-modulator setup with a hypothetical SOA slower than what we used in
the measurements
for the losses in our simulation. In each case, we manually set the histogram window
of the scope, and recorded their limit values. These numbers, together with the length
of the measured histogram, were used to define the output bins in simulations.
To compute the BERs we need to estimate the conditional PDFs of marks and
spaces. To quantify system memory, we performed a set of MMC simulations, with
increasing values of the system memory, and continued the simulations until the PDF
estimates converged. Figure 2.11a shows the PDF estimates at the last cycle in three
separate MMC simulations with increasing M , when the SOA is placed before the
modulator. The small mismatch in the tails is due to the ISI introduced by the electrical
Chapter 2. Noise Suppression
50
filter. Although not the focus of our paper, the case in which the SOA follows the
modulator provides an interesting contrast in the PDF of marks, as shown in Fig. 7b.
In the post-modulator case much larger ISI is visible in the multimodal structure of the
“true” PDF, obtained by increasing the system memory up to M = 4. Note that for
M = 1 the MMC routine is not able to reproduce the second ISI-induced mark “rail”
on the eye diagram, and thus a single-mode PDF is produced, much as in Fig. 7a.
In the case of Fig. 7b the bit-rate was set to 2.5 Gb/s, and the SOA carrier lifetime
was set to one bit duration, i.e., 400 ps. Note this is faster response than the SOA we
characterized and used in our pre-modulator measurements with 170 ps lifetime. The
extinction ratio was set to 20 dB to exaggerate the patterning effect. The conclusion
of Fig. 7b is that our simulation tool can capture a possible link memory enhancement
due to SOA nonlinear operation.
To predict the BER of our SS-WDM link equipped with pre-modulator SOA-based
noise suppression, we set the SUT memory to M = 2. Figure 2.12 shows the conditional
PDFs on both marks and spaces at a received power of -8 dBm in the following cases: (a)
with neither SOA nor CSF (label “SS-WDM”); (b) with noise cleaning SOA but without
CSF (label “SOA”); (c) with both SOA and CSF (label “SOA and CSF”). In each case,
the BER at optimal threshold is the area under the crossing tails of the conditional
PDFs. Both BER improvement due to SOA noise cleaning, and BER degradation due
to post-filtering are visible in Fig. 8. In this example, the BER degradation ensuing
from post-filtering is not severe, due to the rather low linewidth enhancement factor (α
= 3.5).
We next compare simulated BERs with measured BERs. The measured conversion
ratio of the Agilent 11982A PD was 320 V/W. The extinction ratio of the external MachZehnder modulator was used as a fitting parameter to match the floors of SS-WDM BER
curves; an 11.2 dB extinction ratio was used in all simulations. For simulated BERs we
swept the input power, found the optimal threshold (intersection of conditional PDFs),
and calculated the BER from the conditional PDFs. Both 1.25 Gb/s and 2.5 Gb/s BERs
were investigated. Figure 2.13 reports both measured and simulated BERs for the three
cases already illustrated in Fig. 2.12. The receiver and BERT noises were characterized
using the techniques discussed in [69]. We note the excellent match between MMC
simulation and experiments, clearly illustrating the performance estimation accuracy
of the MMC method when a reliable simulator of the SUT is available.
51
Chapter 2. Noise Suppression
pY (y 0)
pY (y 1)
Figure 2.12: Simulated conditional PDFs of marks and spaces corresponding to: SSWDM (label “SS-WDM”), SS-WDM with pre-modulator SOA (label “SOA”), and SSWDM with pre-modulator SOA and CSF (label “SOA and CSF”).
Finally, we comment on possible extensions of the presented work. The SOA model
used in this study included neither SOA ultrafast processes nor polarization effects.
Neglecting ultrafast dynamics is justified for SS-WDM, as the optical field input to
the SOA has a linewidth (0.24 nm) set by the SF. The signal variations at the SOA
input are much slower than typical time constants of carrier heating and spectral hole
burning [42], hence we neglected these processes in our study. By replacing the present
SOA model with one of the well-known models that include ultrafast dynamics, we
could investigate these effects. Similarly, neglecting polarization effects was not critical
for our experimental validation. We used a polarization beam splitter after the BBS,
and controlled the polarization state of the light both at the input of the SOA, and
at the MZM input using polarization controllers. The measurements were recorded
after adjusting the SOA input polarization for maximum gain. The impact of cross
polarization on the light statistics can be studied by replacing the SOA model in our
simulator with a one including polarization effects, e.g., [70], and enlarging the input
vector space to produce random input vectors for TE, and TM polarization states.
Chapter 2. Noise Suppression
52
Figure 2.13: Measured and simulated BERs.
2.4
Summary
In this chapter we described a simulation tool to evaluate the performance of optical
links employing nonlinear SOAs. We applied our simulator to study noise mitigation of
SS-WDM systems by a pre-modulator SOA. We modeled the broadband source, slicing,
channel selecting, and electrical filters all in the time domain. We used a spatiallyresolved SOA model including distributed carrier dependent, and carrier independent
loss mechanisms and ASE. We completed the standard MMC simulation algorithm with
a fast and efficient pattern warping technique to capture the ISI. We showed that both
the statistics of the CW slices, and the BERs at various bit-rates can be predicted
with our simulation tool. In particular, we are able to quantify the impact of receiver
optical filtering effect on system performance. The simulator can be useful as a design
tool to optimize SS-WDM systems, as well as studying various SOA-based regenerative
systems.
Chapter 3
Patterning Effect
In Chapter 2 we found the ISI due to the SOA was negligible, only ISI contributions due
to electrical filtering were significant. In this chapter we focus on ISI induced by nonlinear
effects in the SOA. We present a simulation tool based on the Multicanonical Monte Carlo
method to characterize the statistical properties of bit patterning in semiconductor optical
amplifiers. Our tool estimates the conditional probability density functions of marks and
spaces of the received signal. We introduce an experimental technique to directly measure
the conditional PDFs of the received marks and spaces using a high bandwidth sampling
scope. We demonstrate that predictions from our simulation tool match experimental data.
We measure the bit error rate (BER) of a SOA-based preamplified receiver, where the SOA
operates in the nonlinear regime, and demonstrate that our simulation tool can predict the
measured BER.
Chapter 3. Patterning Effect
3.1
54
Introduction
All-optical signal processing techniques for future advanced optical networks are now
among the key research topics in the optical communication society. The semiconductor
optical amplifier is instrumental in this context due to its compactness, integrability
and rich nonlinear functionality. Some major examples of SOA-based optical signal
processing applications include wavelength conversion [72], 2R and 3R all-optical signal
regeneration [73, 74], intensity noise suppression [19], inline amplification [75], and this
list is by no means exhaustive.
These emerging applications pose new challenges in design and optimization of future optical networks. From the viewpoint of communication systems engineering we
need efficient tools to evaluate the performance of optical links via calculation of the
BER. The KL-based method introduced in Section 1.2.2 requires Gaussian noise statistics before photodetection. Although the Gaussian assumption can be retained in the
presence of moderate fiber nonlinearity in special cases [76], the signal-noise interdependency in general limits the applicability of the KL-based method. An example of
where the KL-based method is of limited value is the presence of a saturated SOA in
the link.
The SOA is a nonlinear element with memory [56]. The nonlinearity of the SOA is
mainly due to carrier depletion induced saturation (typical saturation power of SOAs
is around 1-10 mW ), whereas its memory is due to its finite carrier lifetime (typically
about 100-500 ps) [77]. The signal-dependent, instantaneous gain of the saturated SOA
results in non-Gaussian statistics at the output, and the finite memory of the SOA
leads to bit patterning effects, thus resulting in “nonlinear”, i.e., signal-dependent, enhancement of the intersymbol interference (ISI), on top of the “linear” ISI enhancement
stemming from fiber dispersion, optical and electrical filters.
Analytical treatments of light statistics at the SOA output are not numerous in the
literature, to our knowledge, due to the inherent complexity of the problem. An exact
analysis encompassing all the physical mechanisms does not exist. In a recent study,
Ohman and Mork apply second-order regular perturbation theory and path integrals to
derive analytical expressions for the received signal probability density function (PDF)
when the link is composed of a continuous-wave (CW) laser, an SOA, an ideal pho-
Chapter 3. Patterning Effect
55
todetector, and an arbitrary electrical filter [48]. The carrier-independent loss and the
ASE generated inside the SOA are included, and the SOA operates in saturation. Since
the analysis is limited to the CW regime, the resulting expressions are useful for BER
prediction only when the nonlinear ISI due to bit-patterning is negligible. In a different
approach, Saleh and Habbab [49] consider a typical optical link consisting of an ideal
On-Off Keying (OOK) transmitter (TX), emitting square pulses for marks, and zero
power for spaces, an SOA, an ideal photodetector and an integrate-and-dump or an RC
electrical filter. The SOA model includes only saturation and finite carrier lifetime. By
performing simulations on this model they are able to determine the range of bit-rates
and power levels where SOA-induced nonlinear ISI enhancement is considerable; however, the only noise source considered in their model for BER evaluation is the Gaussian
receiver noise.
Another approach to predicting SOA noise statistics is using computer simulations.
Due to the computational complexity of SOA dynamic models, conventional Monte
Carlo simulations are of limited value. Bilenca and Eisenstein used MMC to study the
PDF of the peak power of a single pulse amplified by the SOA [65, 66]. Their model
included ASE generated inside the SOA, and the input pulses where assumed noiseless.
In Chapter 2 we applied MMC to study intensity noise-suppression of spectrum-sliced
wavelength division multiplexed (SS-WDM) systems by an SOA. We also described a
simple pattern-warping method to improve MMC to jointly warp the bit pattern and
the continuous noise sources; however, in Chapter 2 the dominant source of error was
noise redistribution of the thermal light source both after the noise-cleaning SOA and
after the channel selecting filter, not the small residual linear ISI from optical and
electrical filters.
In this chapter we describe in detail how our simulation tool can be used in practice
to predict the BER of optical links including nonlinear SOAs, where ISI is a significant
source of error, and provide an experimental validation. In Section 3.2 we review the
existing theory of the SOA bit patterning. In Section 3.3 we introduce our simulation
tool. In Section 3.4 we present our experimental technique to probe the memory depth of
the SOA, and show that both conditional PDFs of marks and spaces, directly measured
in the lab, and the BER can be accurately predicted by our simulation tool.We draw
conclusions in 3.5
56
Chapter 3. Patterning Effect
{bi }
Ain
Pout
Aout
pin (t )
G (t )
d pin (t )
r (t )
r (t )
dh (t )
ò
+¥
-¥
l (t - t )[⋅]d t
Figure 3.1: a) Basic setup, and b) block-diagram of the equivalent lowpass SOA model
3.2
SOA Modeling
In this section we discuss modeling the SOA dynamics and study its impact on
bit patterning. The typical link under study is shown in Figure 5.1a, where bi are
the information bits, Ain and Aout are the optical fields at the SOA input and output
respectively, Pout (t) = |Aout (t)|2 is the detected optical power, and r(t) is the received
signal. Our ultimate goal is to study the PDF of r(t) sampled at the decision instant,
taking into account the memory and nonlinearity of the channel represented in Fig.
5.1a., and to do so, we need to model SOA dynamics.
The departure point of our study of SOA dynamics is the model consisting of (1.45)
and (1.46), reproduced here for reference, which expresses the SOA input and output
optical fields through following relations which neglect internal losses:
1
Aout (t) = Ain (t) e 2 (1−jα)h(t)
(3.1)
|A (t)|2
dh (t)
in
h(t)
τc
= h0 − h (t) − e − 1
dt
Psat
(3.2)
Starting from (3.1) and (3.2), the analysis can be conducted in two disparate di-
57
Chapter 3. Patterning Effect
rections: we can further simplify the model presented in (3.1) and (3.2) by applying
first-order perturbation, hence deriving small-signal approximations for the received
signal [49, 80, 81]; on the other hand, we can use (3.1) and (3.2) to build more elaborate models and study the dynamics numerically [44]. The small-signal model provides
insight on the bit patterning mechanism whereas the numerical method provides accuracy. We discuss these two methods in the following subsections.
3.2.1
Small-Signal Analytical Model
In the small signal model, the total integrated gain is written as h (t) = h̄ + δh (t),
where h̄ is the average total integrated gain, and δh(t) is the zero-mean fluctuations. The
input optical power is Pin (t) , |Ain (t)|2 = P̄in +δPin (t), where P̄in is the average input
power, and δPin is the zero-mean input power fluctuations. Similar definitions hold for
the output optical powers: Pout (t) , |Aout (t)|2 = P̄out + δPout (t). Furthermore, we normalize all powers to Psat : pin (t) , Pin (t) /Psat , p̄in , P̄in /Psat , δpin (t) , δPin (t) /Psat ,
with similar definitions for the normalized output powers. The following model for the
SOA operation results
pout (t) = G(t)pin (t)
(3.3)
where
G (t) ' Ḡ (1 + δh (t)) .
(3.4)
In (3.4) we have Ḡ , eh̄ ; and h̄ satisfies
h0 − h̄
= p̄in
eh̄ − 1
(3.5)
δh (t) = l (t) ⊗ δpin (t) .
(3.6)
and
In (3.6) ⊗ denotes convolution in time and l(t) is
l (t) , Ke−t/τef f u (t)
(3.7)
In (3.7) u(t) is the unit step function, and K and τef f are given by the following
K,
τef f ,
1 − eh̄
τc
(3.8)
τc
1 + p̄out
(3.9)
58
Chapter 3. Patterning Effect
The equivalent block-diagram of the first-order model is shown in Fig. 3.1b. The nonlinearity of the SOA is due to signal-dependent gain, and the memory is due to the
impulse response l(.). The input optical power to the SOA is assumed to be an OOK
signal
pin (t) = 2p̄in
∞
X
bn p (t − nTb )
(3.10)
n=−∞
where bn ∈ {0, 1} are the information bits, and p (·) is the ideal rectangular pulse:
p (t) = 1 for 0 6 t 6 Tb and p (t) = 0 otherwise, and Tb is the bit duration. Substituting
(3.10) into (3.3), and using (3.4), (3.6), and (3.7) we obtain


pout (Tb ) = 
2p̄out (1 + 2τef f K p̄in ) + 4Kτef f p̄in p̄out θ b0 = 1
0
b0 = 0
(3.11)
where we have assumed bit b0 starts at t = 0. In (3.11) θ is given by
θ , (1 − ξ)
∞
X
b̄−(j+1) ξ j
(3.12)
j=0
where
ξ , e−Tb /τef f
(3.13)
and b̄i , 1 − bi .
The quantity θ is a random geometric series [49, 82], whose exact distribution for
arbitrary ξ is not known. The bit patterning effect resulting from all the preceding bits
is captured in θ; if all the preceding bits are zero θ = 1, and if all are one, θ = 0. For
ξ > 0.5, Saleh and Habbab [49] use numerical simulations to show θ has approximately
the following beta distribution:
Γ (2ν)
[θ (1 − θ)]ν−1
pΘ (θ) ∼
= 2
Γ (ν)
(3.14)
where ν , ξ/ (1 − ξ).
This small-signal analysis provides a tangible explanation of the bit patterning in
SOA; we can isolate in a single random variable θ the ISI contribution. Nonetheless,
analysis of this random variable is problematic. Furthermore, this analysis lacks precision, since 1) large signal behavior is not included, and 2) many important phenomena,
notably the ASE and the distributed loss, are excluded. For these reasons, we turn our
attention to more accurate numerical models, as is described in the next subsection.
59
Chapter 3. Patterning Effect
E1,I (t )
Ein (t )
E1,O (t )
G1 (t )
E2,I (t ) E2,O (t )
L
G2 (t )
L
EN,I (t ) EN,O (t )
GN (t )
L
Eout (t )
SOA
nASE
Figure 3.2: Large signal SOA model
3.2.2
Large Signal Numerical Model
As a fair compromise between computational complexity and completeness, we use
the multi-reservoir model presented in [44] to model the SOA (cf. also Section 1.4.4).
As represented in Figure 3.2, in this model the SOA cavity is divided into Nsec sections.
The instantaneous gain of the ith section is denoted by Gi (t), and we have Gi (t) ,
exp [hi (t)]. The input (output) optical field to the j th section is denoted by Ej,I (t)
(Ej,O (t)). The input field to the first section is written as
E1,I (t) = Ein (t) + ñSOA
ASE (t)
(3.15)
where Ein (t) is the optical field input to the SOA, and ñSOA
ASE (t) models the SOA ASE, as
described in [44]. The ASE term is a complex Gaussian noise, white over the simulation
bandwidth; the variance of this term is treated as a fitting parameter to match the
measured PDFs in section 3.4.
The input-output optical fields of other sections are related per
Ej,I (t) = LEj−1,O (t)
(3.16)
j = 2, . . . , Nsec
where L is a lumped loss modeling the distributed loss of each section, and is given by
L , exp [−β0 D/(2Nsec )]
(3.17)
where D is the SOA length. The SOA output field Eout (t) is
Eout (t) = LEN −1,O (t)
(3.18)
and the total integrated gain of the j th section follows
τc
|E (t)|2
dhj (t)
j,I
= h0 − hj (t) − ehj (t) − 1
dt
Psat
j = 1, . . . , Nsec
(3.19)
Chapter 3. Patterning Effect
60
Figure 3.3: Measured and simulated SOA waveforms; blue trace is the measured TX
output, red waveform is the SOA model output using measured TX output as input,
green waveform is simulation.
A unique feature of the model presented in [44] is that (3.19) can be extended to include
SHB and CH if necessary; however, since in this work we will examine NRZ signals at
10 Gb/s, we could safely neglect the ultrafast effects.
Figure 3.3 illustrates the measured and simulated optical intensities at the SOA
output, using 10 sections. The parameters of the SOA that we used in the experiments
are given in Table 2.1.
As mentioned in the introduction, the nonlinearity of the SOA is mainly due to
carrier depletion induced saturation, whereas its memory is due to its finite carrier
lifetime. To highlight these dependencies, we vary saturation level and the speed of the
SOA response (carrier lifetime) as referenced to the bit rate; results are presented in
Figure 3.4. The eyediagrams are computed using the SOA numerical method described
in this subsection. The TX and RX models used in Fig. 3.4 are described later. Such
results numerically support the general trends predicted in the previous subsection.
In particular, when the SOA carrier lifetime and the bit duration widely mismatch,
i.e., ξ → 0 corresponding to very low bit rate, and ξ → 1 for very high bit rates, the
patterning effect vanishes. This trend is predicted by (3.11) and (3.12). On the other
hand, at any bit-rate, if the SOA is driven more into saturation the term multiplying θ
in (3.11) increases, and patterning effect is enhanced.
To summarize, bit patterning is only important when two situations occur. The
61
Chapter 3. Patterning Effect
tc /Tb = 2
tc /Tb = 8
tc /Tb = 16
Pin / Psat = 2
Pin / Psat = 1
Pin / Psat = 0.5
Pin / Psat = 0.1
tc /Tb = 0.5
Figure 3.4: Eye diagrams at the SOA output for various operational conditions. Bit-rate
increases from left-to right, and average input power increases from top to bottom.
SOA must be in saturation, e.g., as a booster amplifier, following in-line amplification
in 2R, or in 3R regenerators. Also, the bit-rate must be comparable with the effective
carrier lifetime: when the bit-rate is extremely high, or when the carrier lifetimes are
very low (for example, novel quantum dot SOAs with high saturation power [83]), the
patterning effect becomes less important. In the case of typical commercially available
SOAs, and at bit-rates up to 40 Gb/s some residual patterning effect will exist in
SOA-based 2R regenerators [74].
62
Chapter 3. Patterning Effect
{bi }
1 0 0 1 1
PG
Bit Pattern
Driver
Ain (t )
V (t )
Light Source
PBS
PC
A1,out (t )
MZM
(a)
¥
å b p (t - kT )
k
b
HTX ( f )
k =-¥
Light Source
V (t )
éA1,out (t )ù
éA t ù
ê
ú = Z (a1 , a2 ,V (t ) ,Vb ) ê in ( )ú
ê 0 ú
Ain (t ) ëêA2,out (t )ûú
ë
û
A1,out (t )
(b)
Figure 3.5: a) Transmitter (TX) configuration, (b) TX numerical model; PBS: polarization beam splitter, PC: polarization controller, MZM: Mach-Zehnder modulator.
3.3
The Simulator
Having described the SOA model to be exploited, we now describe the model of the
system where the SOA is to be tested (3.3.1). Following that, we describe in 3.3.2 the
MMC simulator that allows us to test system performance down to very low bit error
rates with realistic, accurate SOA models.
3.3.1
Link Model
3.3.1.1
TX Model
Figure 3.5a illustrates the lab setup of the transmitter, and Fig. 3.5b shows its numerical model. Logical bits enter the TX subsystem and produce a realistic modulated
optical field. Ain (t) and A1,out (t) are respectively the optical fields at the output of the
63
Chapter 3. Patterning Effect
laser, and Mach-Zehnder
modulator (MZM), and V (t) is the RF data driving the MZM.
q
Note that Ain (t) = P̄in where P̄in will be treated as a fitting parameter including both
the laser power and the MZM loss. A lowpass fourth-order Bessel-Thompson (BT4)
filter, HT X (f ), in Fig. 3.5b smooths the logical bits. This filter is used for its small
overshoot, and as it gives good fit with the measured traces; the bandwidth of HT X (f )
is set by trial and error, and we normalize to have HT X (0) = 1. We use the well-known
two-port model of the MZM [4]




A
(t) 
Ain (t) 
 1,out
= Z (α1 , α2 , V (t), Vb ) 
A2,out (t)
0
(3.20)
where
Z (α1 , α2 , V (t) , Vb ) ,



√
√
√
√
j(V (t)−Vb )/2
α
j
1
−
α
e
0
α
j
1
−
α
1
1
2
2

 √

 √
√
√
j 1 − α1
α1
0
e−j(V (t)−Vb )/2
j 1 − α2
α2
(3.21)
and α1 and α2 are the power split ratios of the MZM couplers, and Vb is the bias voltage.
All voltages in (3.20) are normalized to Vπ /π, where Vπ is the voltage inducing a π phase
shift in the MZM.

Figure 3.6 shows the measured waveform at the output of the transmitter and the
simulated result. To achieve this correspondence between experiment and numerical
simulation we needed to extract several parameters. For electrical filter BT4, we set
the 3dB bandwidth to 0.8Rb where Rb is the bit rate, and then exhaustively searched
the 4-dimensional parameter space of all α1 , α2 , Vb , P̄in . We adopted the parameter
set where the Euclidean distance between simulated and measured waveforms of a prespecified sequence is minimum.
The measurements were taken by a high bandwidth sampling scope, the mean noise
of which was characterized, and the averaging option was enabled to suppress noise.
3.3.1.2
RX Model
Two receivers were employed (cf. Fig. 3.9): RX1 to measure the conditional PDFs
and RX2 to measure the BER. RX1 was an Agilent high bandwidth sampling scope,
and RX2 a bit error rate tester. Block diagrams of these receivers are given in Fig. 3.7.a.
In the case of RX1, we assume the receiver is an ideal square-law device. The receiver
64
Chapter 3. Patterning Effect
Figure 3.6: Optical intensities at the output of the transmitter, measured (blue) and
simulated (red)
noise is denoted by nR , and all the coupling losses either from VOAs or from optical or
RF couplings are lumped into GR . In the case of RX2, GR contains the RF amplifier
gain and all the losses. A white complex Gaussian process, ñRec
ASE (t), models the noise
generated by the broadband source. In Fig. 3.7b the measured frequency responses of
the optical filter HOF (f ), the electrical filter HEF (f ), and the Agilent photoreceiver
HP D (f ) are shown.
3.3.2
MMC Platform
Referring to Fig. 5.1, the received signal is
r (t) = be (t) ⊗ Pout (t)
(3.22)
where be (t) is the impulse response of the electrical lowpass filter. The sampled received
signal, corresponding to the current bit b0 is r0 , r (ts ), where ts is the optimum
sampling time between 0 and Tb . The conditional PDFs of marks and spaces are written
as
Pi (r0 ) , pr0 |b0 (r0 |b0 = i)
(3.23)
where i = 0 (i = 1) corresponds to the conditional PDF of spaces (marks). Assuming
that the “effective” memory of the link is M bits, the truncated conditional PDF of
marks and spaces is
Pi,M (r0 ) =
1
2M
X
{b−1 ,...,b−M }
pr0 |b0 (r0 |b0 = i, b−1 , . . . , b−M )
(3.24)
65
Chapter 3. Patterning Effect
GR

2
nR
GR

H PD ( f ) H EF ( f )
2
nR
Rec
nASE
WNG
HOF ( f )
-10
0
-20
HOF ( f )
dB
A.U.
-60
20
15
-10
-40
-50
H PD ( f )
dB[V/W]
-30
H EF ( f )
10
-20
-70
-80
-0.2 -0.4 -0.6 -0.8 0 0.2 0.4
Wavelength [nm]
-30
-40 -30 -20 -10 0 10 20
Frequency [GHz]
30
40
Figure 3.7: a) Numerical models of receivers used in measurements; WNG: white noise
generator, b) frequency domain characterization of RX2
where summation is over all possible patterns of the past M bits. By effective memory
we mean kPi,M (r0 ) − Pi,M +1 (r0 )k to be sufficiently small for some metric k·k. We
propose to use the MMC method to estimate the effective memory length, and the
conditional PDF Pi,M (r0 ). To determine memory length, we gradually increase M
until successively estimated conditional PDFs coincide. The block-diagram of our MMC
simulator is shown in Fig. 3.8. The numerical system model is composed of three
parts (TX, SOA, and RX), all described previously. The details of our MMC platform
are presented in sections 1.3.4 and 2.2.1. Here we briefly review them. We denote
the simulation time step by ∆t, and the number of time samples per bit by Ns , i.e.,
Tb = Ns ∆t. Assuming the effective memory is M , the past M Ns time samples of all
independent noise sources have an impact on the distribution of r0 . The vector of all
66
Chapter 3. Patterning Effect
y
Pp
Bp
yp
Figure 3.8: Block diagram of the simulator; NVG: random vector generator, PNG:
pattern number generator
noise samples is denoted by N , which is explicitly written as
h
Rec
N , ñSOA
ASE , ñASE , nR
i
(3.25)
Rec
where ñSOA
ASE and ñASE are vectors of independent identically distributed white complex
Gaussian noise samples each of length M Ns ; the former accounts for ASE noise from
the SOA (cf. Fig. 3.2), and the latter accounts the ASE of the pre-amplified receiver
(cf. Fig. 3.7), and nR is a real Gaussian random variable with proper mean and variance
modeling the receiver noise (cf. Fig. 3.7). The vector B
− contains all the past bits falling
in the effective memory of the link
B
− , [b−1 , . . . , b−M ]
(3.26)
The noise vector generator (NVG) subsystems in Fig. 3.8 is a Metropolis-Hastings map
chine, which proposes noise vector samples X
− . The pattern number generator (PNG)
subsystem in Fig. 3.8 is an other Metropolis-Hastings machine, proposing pattern numbers P p ; the binary representation of a pattern number is the
The PDF
bit pattern.
p
p
warper accepts or rejects the proposals from NVG and PNG X
according to the
− ;P
MMC algorithm. Consequently, the PNG performs a random walk over the index in the
summation of (3.24), while the NVG performs a random walk to explore the conditional
PDFs within the sum.
Chapter 3. Patterning Effect
67
Figure 3.9: Experimental setup to measure conditional PDFs (RX1) and BER (RX2);
PG: pattern generator, MZM: Mach-Zehnder modulator, PC: polarization controller,
VOA: variable optical attenuator, ISO: isolator, OF: optical filter, PD: photodetector,
BERT: BER tester
3.4
Experimental Results
The experimental setup is shown in Figure 3.9. We performed two different measurements: RX1 to directly measure the conditional PDFs of marks and spaces, and
RX2 to measure the BER.
3.4.1
Conditional PDFs
We developed an experimental technique to directly measure the conditional PDFs
of marks and spaces using a PC-controlled 50 GHz Agilent 86116A sampling scope.
We must unambiguously determine the samples corresponding to marks and spaces
at the receiver; then the conditional histograms can be computed. To this end we
transmit many packets consisting of a De Bruijn sequence preceded by a header of
N1 marks followed by N0 spaces. Processing consists of filtering the zero-averaged
packet by a moving average filter of length N1 and detecting the peak. The location
of the peak coincides with the one-to-zero transition in the header; once the header is
synchronized, the transmitted sequence can be identified without error. The principles
of this technique are illustrated in Fig. 3.10a.
Several practical considerations enter
into setting various parameters. The sequence length 2Mseq should have Mseq bigger than
the (unknown) effective memory of the link. De Bruijn sequences end with a series of
68
Chapter 3. Patterning Effect
Received packet
Header
De Bruijn Sequence
Filtered zero-averaged packet
Reconstructed packet
(a)
0
M=3
-1
M=5
M=7
log (PDF)
-2
-3
-4
-5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Voltage [mV]
(b)
Figure 3.10: a) Steps to measure the conditional PDF using the packetized, b) conditional PDFs of marks and spaces measured for three different length De Bruijn sequences
69
Chapter 3. Patterning Effect
zeros; to distinguish the one-to-zero transition in the header we require N1 > Mseq .
Increasing N1 results in more pronounced peaks in the filtered zero-averaged packet; if
the OSNR is low, the conditional PDFs of marks and spaces overlap considerably, and
we must use a large N1 .
Finally, the temporal setting of the scope, the bit-rate, and the packet length should
satisfy two necessary conditions for the measurements to be stable. Suppose the buffer
length of the scope is denoted by Nw (for the Agilent 86116A the maximum is 4096). If
we want Ns samples per bit, the sampling time is dt = Tb /Ns , and the temporal width
of the scope’s buffer is Tw = Nw dt; on the other hand, if the time per division of scope
is ∆T , we have Tw = 10∆T . Therefore the following condition should hold
10∆T
Nw
=
.
Ns
Tb
(3.27)
Clearly the packet must fit in the buffer so
Nw > N1 + N0 + 2Mseq
(3.28)
We do not know the length of the SOA effective memory a priori; but we can sound
the memory depth of SOA as is depicted in Fig. 3.10b. We measured the conditional
PDFs for various lengths of the De Bruijn sequence: Mseq = 3, 5, 7. When Mseq is
smaller than the link effective memory, the measured PDFs are meaningless; as Mseq
approaches the effective memory, the measured conditional PDFs approach the true
PDFs. Note that the memory sounding can also be done in software by estimating the
truncated conditional PDFs for increasing values of M .
Figure 3.11 illustrates measured and simulated conditional PDFs of marks and
spaces when Mseq = 7. The SOA input power was -2.65 dBm, resulting in deep saturation; the bit-rate was 10 Gb/s. The PDFs were calculated at the middle of the
bit. Simulation results are for five MMC cycles of 105 samples each; each cycle took
71 seconds to execute. In Fig. 3.11, only the PDFs corresponding to the last cycle are
shown.
Using RX2, we measured the BER as a function of the received OSNR and present
these results in Fig. 3.12. The SOA average input power and the bit-rate were set as
in the previous case. MMC simulations (one for conditional PDF of marks, the other
for spaces) were required at each BER point; the BER was computed by numerically
integrating the overlapping tails of estimated conditional PDFs of marks and spaces.
Chapter 3. Patterning Effect
70
Figure 3.11: Measured and simulated conditional PDFs of marks and spaces
Figure 3.12: Measured and simulated BERs at RX2; upper inset shows the conditional
PDFs used to estimate the BER curve (one pair per BER curve point), lower inset is
eye diagram for lowest BER estimated
Each PDF estimation included seven MMC iterations, to improve the accuracy. In the
lower inset of Fig. 3.12 we show an eye diagram for high OSNR that clearly depicts
the strong patterning effect from the SOA. The upper inset is the set of estimated
conditional PDFs used to calculate one BER point.
Chapter 3. Patterning Effect
3.5
71
Summary
We presented a new simulation tool based on MMC to accurately model an optical
link containing a nonlinear SOA. We verified experimentally that our simulator can
accurately characterize the received signal statistics in the presence of high bit patterning due to the saturated SOA. We introduced an experimental technique to directly
measure the conditional PDFs using a sampling scope, experimentally probed the effective memory of SOA, and were able to accurately predict the measured PDFs and
BER with our simulator. The purely numerical nature of the simulator allows for the
exact nonlinear dynamics of the SOA to be captured, and the use of MMC makes it
fast and efficient. Besides being a design and optimization tool per se, it can be used
to 1) examine the accuracy of analytical approximations, 2) reduce computation time,
and 3) study the impact of changing the modulation format on performance. Moreover, by adding features to the SOA model of (3.19), the impact of ultrafast processes,
nonlinear polarization rotation, enhanced phase dynamics, and interchannel effects in
the multichannel regime can in principle be assessed.
Chapter 4
Filter Design
In this chapter we return again to discussion of SOA-assisted SS-WDM. We address design
and optimization of optical filters for SS-WDM systems employing saturated SOAs to suppress
intensity noise. We study the impact of the shape of both slicing and channel selecting
optical filters vis-à-vis two important impairments: the filtering effect and the crosstalk. The
quantification of BER is made possible by a parallel implementation of the MMC algorithm.
The intensity noise suppression by the SOA and signal degradation by subsequent optical
filtering are studied both numerically and experimentally. We find optical filter shape and
bandwidth that minimizes BER.
By varying channel spacing and width, we estimate the achievable spectral efficiency when
using both noise-cleaning SOA and forward error correction. We show that when constrained
to use a symmetric architecture, i.e., identical filters for both slicing and channel selecting
filters, there is a degradation in achievable spectral efficiency. We show that noise suppression
is robust to variations in relative channel powers in multichannel systems.
Chapter 4. Filter Design
4.1
73
Introduction
Noise suppression in SOA-assisted SS-WDM is due to the nonlinear operation of
the saturated SOA. Various device parameters influence the SOA dynamics, most notably the carrier lifetime and the saturation power, as well as the loss and the linewidth
enhancement factor, and directly impact the noise cleaning performance [12, 19]. Moreover, it has been observed that optical filtering of the noise-suppressed light significantly
degrades noise suppression [18, 19], a phenomenon which is referred to as the filtering
effect or post-filtering effect (cf. also Section 1.1.3). For instance, when a Gaussian
slicing filter (SF) of 21 GHz 3 dB bandwidth, and a Gaussian channel selecting filter (CSF) of 25 GHz 3 dB bandwidth were used in the single-channel SOA-assisted
SS-WDM experiments of Chapter 2, the filtering effect degraded the BER floor of the
SOA-assisted system from 10−10 to 10−7 (cf. Fig. 2.13).
Fiber group velocity dispersion (GVD) has a great impact on SS-WDM systems
due to the large linewidth of the sliced source [5]. In SOA-assisted SS-WDM the GVD
penalty is even higher, since the nonuniform phase profile of the dispersive fiber over
the signal bandwidth induces partial loss of noise suppression [19, 86]. Throughout this
work we assume dispersion is fully compensated.
In this chapter we focus on the impact of the shape and bandwidth of optical filters
in the transmitter (SF), and receiver (CSF) on the overall performance of multi-channel
SOA-assisted SS-WDM systems. Similar studies for coherent WDM have already appeared in the literature [87].
First we consider CW operation of a single-channel system, and focus on RIN spectrum of the received signal, and the impact of the filter shape on the intensity noise.
Since CW RIN simulations are fast, we can obtain a global qualitative picture of how
noise suppression degradation behaves in the filter space, where the only impairment is
the filtering effect. This analysis does not capture dynamics in the modulated systems,
so next we turn to MMC techniques.
In the second step, we choose a limited set of filter shapes (four cases) and estimate
both the single-channel and the multi-channel BERs, when SF and CSF are independently varied. To estimate the BER, we extend the single-channel MMC simulator of
Chapter 4. Filter Design
74
Chapter 2 to a multi-channel version.
MMC offers a tremendous acceleration of simulation compared to conventional
Monte Carlo (MC). To reliably estimate BERs as small as 10−10 by MC, 1012 bits
should be simulated, whereas using MMC, we can typically get as accurate BER estimates with only ' 106 bits. The remarkable speedup by MMC enables us to perform
simulations that would need astronomic run times when using MC. However, to develop
a practical design and optimization tool for complex optical systems, MMC should be
further accelerated. To this end, we introduced a novel parallelized implementation of
the MMC (PMMC).
The non-Gaussian nature of noise statics lead to BER calculations from MMC that
outperform the simple Q-factor approximation. Section 4.6 compares published experimental studies, and find our simulation results are still in good agreement with the
published measurements.
Forward error correction (FEC) offers an alternative approach to cope with the
BER floor in the SS-WDM [88]. FECs are especially manageable in the metro-access
applications, where bit-rates are below 10 Gb/s. In SOA-assisted SS-WDM applications
we can combine FEC and SOA noise suppression to achieve high spectral efficiency (SE).
By varying the channel spacing and channel width, we calculate the optimal spectral
efficiency attainable when combining FEC and SOA. We consider two scenarios: one
where the designer is constrained to use identical optical filters throughout the network,
(a lower cost solution), and one where optimal SF and CSF can be used.
Our contributions are: 1) we quantify the exact BER of multi-channel SOA-assisted
SS-WDM systems for the first time, and study the impact of the optical filters, 2) we introduce the PMMC, a parallelized implementation of the MMC, and 3) we calculate the
optimum spectral efficiency that we can obtain for SS-WDM. This chapter is organized
as follows. In Section 4.2 we discuss the RIN spectra and the excess intensity noise in
the CW regime. In section 4.3 we introduce the PMMC simulator. In section 4.4 we
present the BER results. In section 4.5 we present results concerning the achievable
SE by using FEC. In section 5.4 we conclude. The cross-validations of the simulator
against published measurements are presented in the Section 4.6.
Chapter 4. Filter Design
4.2
75
Intensity Noise in the CW Regime
Consider our experimental setup for the single-channel SOA-assisted SS-WDM system given in Figure 2.1. Our goal in this section is to study the impact of the choice
of SF and CSF on the RIN spectrum of the electrical signal after the photodiode, and
on the intensity noise of the signal after the electrical filter at the receiver. To this
end, first we measure the RIN spectrum for a specific pair of SF and CSF, when data
modulation is not applied, and use the experimental data to calibrate and validate our
link simulator. Then we use the link simulator to sweep through various choices of SF
and CSF and calculate the resultant intensity noise following electrical filtering. The
amplitude and phase of SF and CSF are allowed to independently vary over a limited
range.
4.2.1
Experimental Validation of Simulator
In our experiment, we used a broadband source (BBS) with 38 nm bandwidth,
which was sliced, amplified by an erbium-doped fiber amplifier (EDFA), and sliced
again, to provide a -3 dBm source, enough to drive the SOA into deep saturation. The
unpolarized emission of the BBS source was passed through a polarization beam splitter
(PBS), and the stronger PBS output was fed through a polarization controller (PC) to
the SOA input. The SOA output was coupled to the Mach-Zehnder modulator through
a second PC. The EDFA, PBS, and the PCs are not shown in Fig. 2.1, and the two
optical filters in the transmitter side are lumped into the SF block. The two optical
filters at the transmitter, as well as the CSF at the receiver, were identical JDS TB9
filters with a 3 dB bandwidth of 0.24 nm. Since the BBS spectrum was much wider than
the filter bandwidths, the BBS was modeled as complex white noise over the simulation
bandwidth. The frequency responses of all optical filters were well modeled as having
a Gaussian profile over their 30 dB bandwidth; their group delays were flat over the
passband. The SOA parameters were the same as those found in Table 2.1.
We model optical filters in the frequency domain. The spectrum-sliced source was
implemented with two identical .25 nm bandwidth Gaussian-like optical filters. The
equivalent SF model is a single Gaussian filter of 21 GHz 3 dB bandwidth; the CSF
has a 25 GHz bandwidth (single filter). The phase response of SF and CSF were
Chapter 4. Filter Design
76
assumed flat in simulations. Since we are in the CW regime, any electrical bandwidth
is permissible, but to be consistent with the later BER simulation results at 5 Gb/s, in
our calculations of the intensity noise we used a 4th -order Bessel-Thompson electrical
filter with 3.75 GHz 3 dB bandwidth.
We used the multi-reservoir model introduced in Section 3.2.2 parsing the SOA into
10 sections. We compared estimated PDFs for marks and spaces resulting from the
multi-reservoir with ASE and ultrafast terms, and without them; the two predictions
coincided. The SOA is deeply saturated, so ASE is negligible; input optical signal
bandwidth is 21 GHz, hence ultrafast features can be neglected. For the balance of this
article we neglect ASE an ultrafast terms.
The definition of RIN was given in (1.1). The RIN of a spectrum-sliced source can
be calculated as
RINss (f ) = |HSF (f )|2 ⊗ |HSF (f )|2
(4.1)
where ⊗ represents convolution in the frequency domain, and HSF (f ) is the SF frequency response with HSF (0) = 1. The same normalization is used for HCSF (f ), the
CSF frequency response. The RIN at the SOA output can be approximated semianalytically using first-order perturbation in intensity [12]. We validate our numerical
model both with experiment and the perturbation theory approximation.
To measure the intensity noise suppression efficiency, we define the noise suppression
ratio (NSR)
N SRx (f ) = 10 log [RINx (f )/RINss (f )]
(4.2)
where x = SOA indicates SOA noise cleaning, but no CSF; x = pf is the “post-filtered”
case with a CSF present. Figure 4.1 shows the results of measured and simulated
N SRSOA (f ) and N SRpf (f ), as well as the analytical (perturbation theory) approximation of the N SRSOA (f ).
The good match between measured and simulated NSRs in Fig. 4.1 confirms that
our link simulator is well-calibrated, and that it captures the filtering effect. As can be
seen in Fig. 4.1, the RIN attains a minimum at DC: a 14 dB reduction when no CSF
is present, and a 6 dB reduction when using a CSF.
All components of the SOA-assisted system, i.e., SF, SOA, CSF, and electrical filter
(EF), play a role in determining the overall performance, and in principle, should be
77
Chapter 4. Filter Design
4
2
0
-2
-4
-6
-8
-10
-12
-14
-16
0
2
4
8
6
10
12
Figure 4.1: Measured and simulated noise suppression ratios (NSR) of CW intensitysmoothed light by the SOA, with and without post-filtering. When post-filtering is
absent, the analytical approximation is also plotted.
jointly optimized to achieve the maximum performance. However, to make the problem
tractable, the focus of our effort in this chapter will be only on the impact of SF and
CSF filter shape and bandwidth. Details of SOA parameters are given in appendix B.
4.2.2
Impact of SF and CSF on EIN
The excess intensity noise (EIN) of the electrically filtered voltage y(t) at the receiver
is defined in (1.2) and (1.3) and is repeated here for reference
.
EIN , var [y (t)] E2 [y (t)]
(4.3)
where E[·] stands for expectation. In direct detection, the EIN is calculated by integrating RIN over the receiver electrical bandwidth. For a specific choice of SF and CSF,
we define the EIN penalty (EINP) as follows
EIN P ,
#
" R +∞
−∞ RINpf (f )HEF (f )df
10 log R +∞
−∞
RINSOA (f )HEF (f )df
(4.4)
where HEF (f ) is the electrical filter frequency response. We assume the following
general form for the optical filters
h
Hx (f ) = exp − (|f |/f0,x )2nx
i
(4.5)
78
Chapter 4. Filter Design
4
1.1
0. 4
1
3.5
0.9
0.3
3
0.8
0.4
0.5
0.7
0.3
nSF
2.5
0 .5
2
0.4
0.6
0.6
0.5
1.5
0.5
0 .6
0.7
0.4
0.7
1
0.5
0.8
0 .6
0.8
0.7
0 .8
0.5
1
1.5
0.3
0.9
0.9
1
1
1.1
2
2.5
nCSF
3
3.5
4
0.2
Figure 4.2: Contour plots of log(EIN P ) vs. orders of SF and CSF super-Gaussian
filters with flat phase response.
where x ∈ SF, CSF . We confine our examination to amplitude response, and assume
linear phase. The parameter nx is the super-Gaussian order controlling the filter roll-off,
and f0,x is given by
f0,x , 0.5BW3cB,x /(0.5 ln 2)1/(2nx )
(4.6)
where BW3dB,x is the filter 3 dB bandwidth.
We examine the impact of SF and CSF roll-off on EINP for BW3dB,SF = BW3dB,CSF
= 30 GHz, when all filters have flat phase response, i.e., β2,CSF = β3,CSF = 0. Figure 4.2
gives log(EIN P ) when the super-Gaussian orders of the SF and CSF are independently
swept over the range [0.4,4]. The average input power to the SOA is set to 0 dBm in
all simulations.
For given SF and CSF filter bandwidths, the shape (roll-off) of these filters has
significant impact. The upper left corner of Fig. 4.2 corresponds to the SF flat-topped
79
.2
MMC Platform
IDG
tp
t
PNG
P
p
P
N
NVG
Np
Xp
X
B2
N2p
Interferer #2
p
PDF
Warper
e  jt
MZM
SOA
N3p BBS
SF
Desired User
Interferer #1
N1p
B1p
B3p
e  jt
D
D
t2p
t1p
CSF
PD
EF
yp
N rp
y
PDF
Update
Hist.
Update
Chapter 4. Filter Design
Figure 4.3: The block diagram of the three-user SOA-assisted SS-WDM MMC platform.
NVG: noise vector generator, PNG: pattern number generator, IDG: interferer delay
generator. D: programmable temporal delay element. The rest of variables are defined
in 4.3.1.
Chapter 4. Filter Design
80
with the steepest roll-off (nSF = 4) and the CSF as heavy-tailed as possible (nCSF =
0.4). In this case, the tails of the SF and CSF have minimal overlap and we see the
smallest penalty. The worst penalty (lower right region) occurs when the overlap is
greatest, i.e., when SF is heavy-tailed, and CSF is flat-topped (nSF = 0.4, nCSF = 4).
Due to the intractability of analysis of modulated systems, small signal perturbation analysis of CW signals is often used. Using either these methods, or numerical
simulations of CW signals, we would conclude that high order (flat-topped) slicing filters, and low order (heavy-tailed) channel select filters would be optimal. When using
the simulation tools introduced in the next section, we can examine modulated signals
and BER for performance. Most importantly this will allow modeling of cross-talk in
modulation.
4.3
Multichannel PMMC Simulator
To evaluate the BER we run the link model of the previous section inside the MMC
platform. The sequential single-channel MMC platform is described in detail in sections
1.3 and 2.2.1. In this section we describe the extension of the single-channel sequential
MMC to the multi-channel parallel MMC. The BER results are presented in the next
section.
4.3.1
Multi-channel MMC platform
The block diagram of the multi-channel MMC platform, used to estimate the conditional PDFs of the received marks and spaces and thereby the system BER, is shown in
Fig. 4.3. Throughout this paper, we study a three-channel scenario where the central
channel is the desired channel. Per [90], a three-channel system is sufficient to capture
the crosstalk effect for a larger SS-WDM system.
Three replicas of the link model described in the previous section are used to model
the desired channel and two adjacent channels. Since the link model is baseband, the
adjacent channels are up-, and down-converted. The channel-spacing is denoted by
81
Chapter 4. Filter Design
p
p
p
p
∆ω. The proposed vectors in the input space are X
, N
t , which map to
−
− ;P
− ;−
p
p
output samples y , g X
, where g(·) is an abstract mapping formally represent−
ing the system. The superscript indicates a proposed sample that may or may not be
rejected within the MMC algorithm. To indicate an accepted proposal we drop the
superscript in Fig. 4.3. The proposed input vector consists of three parts. The noise
vector N p , [N p1 , N p2 , N p3 , Nrp ] contains identical independent Gaussian random variables of zero mean and unit variance; the sub-vector N pj is used to model the incoherent
spectrum-sliced source of the j th user, and Nrp is a scalar modeling receiver electrical
noise. The noise vectors are generated by a Metropolis-Hastings machine called the
noise vector generator (NVG). The proposed bit pattern vector is P− p , [P1p , P2p , P3p ],
where Pjp is the decimal representation of the binary bit pattern of the j th channel.
The bit pattern proposed for the j th channel is denoted by B pj . The pattern numbers
are proposed by a Metropolis-Hastings machine called the pattern number generator
(PNG). The relative delay vector is −t p , [tp1 , tp2 ], which is composed of random variables
representing the time delays between the desired channel and the adjacent interfering
channels. The vector of relative delays is proposed by a Metropolis-Hastings machine
called the interferer delay generator (IDG). The effective memory of the single-user system is assumed to be M − 1 bits. To estimate the conditional PDF of marks (spaces)
of the desired user, the current bit of the center channel is set to 1 (0), and the past
M − 1 bits are adaptively changed by the MMC platform; therefore P2p is an integer
random variable (rv) uniformly distributed between 0 and 2M −1 . P1p and P3p are integer
rvs uniform between 0 and 2M +1 . The relative delays tp1 and tp2 are integer RVs uniform
over 0 and Ns − 1, where Ns is the number of time samples per bit duration.
4.3.2
Parallelization of MMC
Conventional MC for PDF estimation of RV’s is “embarrassingly” parallelizable,
as random samples can be independently generated by different cluster nodes. At the
end of the simulation, all samples are collected and the histogram is calculated over all
collected samples. In the case of MMC, the proposed samples are generated by Markov
chains using the Metropolis-Hastings algorithm (cf. appendix. A), a process which is
sequential in nature. While at first blush MMC does not appear parallelizable, we show
that, fortunately, this is not the case.
Chapter 4. Filter Design
82
Consider a 1-dimensional input space where sequential MMC is used to estimate the
output PDF. During each MMC cycle, the Metropolis-Hastings module of the MMC
generates a random walk in the 1-dimensional input space. Suppose we periodically
perturb the random walk in the input space by re-initializing it, as shown in Fig. 4.4a.
Each random walk is generated by the same Metropolis-Hastings submodule as before,
but at time instants T , 2T , 3T , and 4T , we select a new random state in the input
space. The initial states are assumed independent and uniformly distributed over the
input space.
The perturbed Markov chain is not statistically equivalent to the original unperturbed Markov chain, required by the MMC platform, as the forced jumps induce
transients. If, however, the MMC platform discards the transient samples after each
forced jump, the remaining samples of the perturbed Markov chain will lead the MMC
to the same solution as the single Markov chain case. The perturbed random walk
provides the transition from sequential to parallel implementations of the MMC. The
generation of each segment of the perturbed random walk can be assigned to a different
computing node, as shown in Fig. 4.4b, allowing for parallel processing.
During each MMC cycle, all nodes run exactly the same code to propose new samples, and perform an accept/reject operation accordingly. At the end of each MMC
cycle, all the output samples are collected by a pre-specified head node, the PDF update and smoothing are executed, and the updated PDF is broadcast to all nodes
for the next MMC cycle. We call this parallel implementation of MMC the PMMC.
The flowchart of PMMC is shown in Fig. 4.4c. The PMMC follows the paradigm
of SPMD (single program multiple data). In [93] another parallel implementation of
MMC is introduced; however, as explained by the author, the resulting algorithm is
a problem-dependent, modified MMC without the important PDF smoothing feature.
Our PMMC, on the other hand, is a natural parallelization of the MMC, without any
modification to the original algorithm.
Note that even in sequential MMC, we discard transient elements at the beginning
of each MMC cycle. The length of the transient period is problem-dependent, and
is fixed during the code development and fine-tuning of the simulator. We discarded
the first 100 samples at the beginning of each MMC cycle per node. We parallelized
four cores of a Quad Intel processor, and obtained a three-fold speedup. The rigorous
convergence analysis and optimization of PMMC will be addressed in future work.
83
Chapter 4. Filter Design
Start
Serial MCMC
Initialization
Restarting
the chain
k=0
k=k+1
0
T
Parallel
MCMC
2T
time
3T
(a)
Node 1
4T
Node 1
Node 2
...
Node K
(k)
HY,1
H Y k,2
...
H Y ,3
k
Node 2
Node 3
PDF Update
k = Nc ?
Node 4
0
time
T
(b)
No
Yes
End
(c)
Figure 4.4: Parallelization of MMC: (a) Random walk in a 1-dimensional input space
perturbed by periodic re-initializations. (b) Sections of the perturbed Markov chain are
mapped to various computing nodes, (c) the flowchart of the parallel MMC; k counts the
(k)
MMC cycles, Nc is the pre-specified number of cycles, HY,j is the histogram computed
by node j at the end of cycle k.
4.4
BER Results
We introduce in Fig. 4.5 the set of filter types to be examined, covering a representative collection of realizable optical filters based on the super-Gaussian shape. The
filters are labeled as heavy-tailed (HT), Gaussian (GA), S2, and S4. These filters are
all super-Gaussian, with orders nHT = 0.4, nGA = 1, nS2 = 2, and nS4 = 4. The
phase response of all filters was assumed to be flat. Each entry of the table in Fig. 4.5
corresponds to a specific combination of SF (rows), and CSF (columns) filter types for
a three-channel SOA-assisted SS-WDM link. Simulations are run at Rb = 5 Gb/s and
channel spacing ∆CH = 100 GHz. We separately simulated all sixteen cases. In each
simulation, the 3 dB bandwidth of the SF, BWSF , was set to 30 GHz, and the 3 dB
bandwidth of the CSF, BWCSF , was varied from BWSF to 2∆CH − 2BWSF , i.e., 30 to
140 GHz. The output of each case was a 10-point curve of BER vs. BWCSF . The input
power to the receiver was fixed to 0 dBm, so that all BER values correspond to the
intensity-noise limited regime. Markers in Fig. 4.5 distinguish different SFs, whereas
color and line-type discriminate CSF types.
84
Chapter 4. Filter Design
SF
HT
GA
0
S2
S4
HT
n = 0.4
-5
2
GA
n=1
Hf
CSF
GA
 [dB]
HT
S2
-10
S2
n=2
S4
n=4
S4
-15
2
-40 -30 -20 -10 0 10
0
Frequency [GHz]
30 40
Figure 4.5: Left: Table defining (SF, CSF) combinations. SF filter types are distinguished by markers and CSF types are distinguished by line type (also colors). Right:
the frequency response of the filter types used for BER simulations; n is the superGaussian order.
Figure 4.6 reports results of the multi-channel BER simulations for all 16 cases.
For each BER point, two MMC simulations were performed to estimate the conditional
PDFs of marks and spaces; the BER was calculated by integrating the overlapping
tails of the two conditional PDFs. Each MMC simulation consisted of 12 cycles; 50,000
samples were generated per cycle. We assumed M = 3 bits of effective channel memory.
After parallelization, each BER point was calculated in 25 minutes.
In order to better understand the relative importance of filtering effect vis-à-vis
crosstalk, simulations were repeated for the single channel scenario, with results in
Fig. 4.7. The number of samples per cycle was raised to 150,000 in order to estimate
much lower BERs; other parameters were unchanged.
The relative importance of the filtering effect vis-à-vis the crosstalk can be understood by comparing the multi-channel BERs of Fig. 4.6 to single-channel BERs of
Fig. 4.7. In the multi-channel scenario CSFs with steep roll-offs (nCSF = 4) are favored.
In the single-channel case, where the only impairment is the filtering effect, an optimal
(SF, CSF) pair does not exist. The relative performance of (SF,CSF) pairs follow EINP
trends for the CW analysis, as seen in Fig. 4.2. At the rightmost points of BER curves
of Fig. 4.7, the filtering effect is negligible, as the CSF bandwidth far exceeds that of
85
Chapter 4. Filter Design
0
-1
nSF = 0.4 → Crosstalk enhancement
-2
-3
nCSF = 1
nCSF = 0.4
log (BER)
-4
nCSF = 2
-5
-6
nCSF = 4
-7
-8
-9
Best performance
for nSF, nCSF = 2,4
-10
-11
20
40
60
80
100
120
CSF 3 dB Bandwidth [GHz]
140
Figure 4.6: BER of the multi-channel system as predicted by PMMC simulations. SF
filter types are distinguished by markers and CSF filter types are distinguished by line
type (also color).
the SF. In this region the performance is set by the source intensity noise; smoother SF
roll-off results in lower M-factors [21] and thus less noise.
In the multi-channel scenario, as seen in Fig. 4.6, there is a tremendous difference
in system performance when the super-Gaussian order varies from 0.4 to 1, and a large
difference when it is varied from 1 to 2, but the performance only slightly changes from
super-Gaussian order 2 to 4. From a practical point of view, realizing super-Gaussian
filters of lower orders is easier. Although the optimum BER varies by less than one
order of magnitude when the CSF order is changed from 1 to 2, the optimum CSF
bandwidth varies. The order 1 CSF is much more sensitive to the CSF bandwidth than
order 2 or 4. We conclude the roll-off steepness offered by the super-Gaussian order 2
is sufficient for SOA-assisted SS-WDM systems.
86
Chapter 4. Filter Design
-2
-3
-5
-6
BWCSF → ,
lower nSF is better.
8
BWCSF ≈ BWSF :
trends are like Fig. 3.
-4
No clear trend
-7
-8
nSF = 4
log (BER)
-9
-10
-11
-12
nSF = 2
-13
-14
nSF = 1
-15
-16
-17
nSF = 0.4
-18
-19
-20
-21
20
40
60
80
100
120
CSF 3 dB bandwidth [GHz]
140
Figure 4.7: BERs of the single-channel system as predicted by PMMC simulations. SF
filter types are distinguished by markers and CSF filter types are distinguished by line
type (also color).
To investigate the efficiency of SOA-assisted noise suppression, we compared the
BERs for nSF = nCSF = 2; other parameters are unchanged. Results are reported in
Fig. 4.8, for single and multi-channel cases. In the optimum multi-channel case, employing the SOA-assisted scheme decreases the BER from 10−2 to 10−10 . The threshold
of powerful forward error correcting codes is at 10−3 .
To this point we have assumed all channels have equal average power. In practice,
imbalances along the fiber link for different channels, e.g., variation of optical loss of
different ports of the arrayed-waveguide gratings used as SF, and/or CSF, can cause
variations in received power. We calculate (via simulation) the performance penalty
when adjacent channels are more powerful. To quantify this penalty, we simulate order
2 super-Gaussian SF and CSF. The SF bandwidth is 30 GHz, and the CSF bandwidth
is the optimum value under the equal-power assumption in Fig. 4.6. The left and
87
Chapter 4. Filter Design
nSF = 2
nCSF = 2
SS-WDM
Multi-channel
log (BER)
-2
Single-channel
-3
-4
SOA-assisted SS-WDM
-5
-6
-7
Multi-channel
-8
-9
-10
-11
-12
Single-channel
20
40
60
80
100
120
140
CSF 3 dB Bandwidth [GHz]
Figure 4.8: Comparison of BERs of SS-WDM and SOA-assisted SS-WDM; nSF = nCSF
= 2.
right interfering channels were amplified by factors Gl and Gr , with respect to the
desired (center) channel. Gl and Gr were independently varied from 0 to 3 dB; for each
combination, the BER for the desired channel BER(Gl , Gr ) was computed. Figure 4.9
presents the contour plot of 10log(BER(Gl , Gr )/BER(0, 0)). We see that in the worst
case, where both interferers are 3 dB stronger than the desired users, using a CSF
optimized under equal-power condition results in one order-of-magnitude penalty in
BER. Hence the significant gain from SOA noise suppression (10−2 to 10−10 ) is largely
retained (10−2 to 10−9 ).
Finally, we examined the usefulness of Q-factor in BER approximation in Figure 4.10. For each BER point, the exact conditional PDFs of marks and spaces are
estimated, therefore the Q-factor can be calculated per (1.6) at each BER curve point.
Having extracted the Q-factor, we compare the exact BERs with the Q-factor approximation using (1.5) in Fig. 4.10 for three representative cases in the multi-channel
88
Chapter 4. Filter Design
11
10
9
9
10
8
7
9
9
8
8
10
7
2
6
7
6
5
8
5
9
7
6
1
4
4
3
5
3
2
2
0
0
1
4
3
9
1
8
6
7
Relative gain of left-channel interferer [dBm]
3
5
1
2
3
0
Figure 4.9: Contour plot of 10log(BER(Gl , Gr )/BER(0, 0));BER(Gl , Gr ) is the BER
of the desired (central) channel, as a function of the relative gain of the left (right)
channel interferer Gl (Gr ).
scenario. Two important observations can be made. First, the Q-factor is more accurate in the left part of the BER curves, where the filtering effect is dominant, than
in the right part, where the crosstalk is dominant. The divergence between exact and
Q-factor approximated BERs in the crosstalk limited region increases as the crosstalk
decreases by choosing steeper roll-off CSF shapes. Second, the Q-factor approximation
always fails in providing the optimum CSF bandwidth, thus the Q-factor is not a useful
performance metric for designing SOA-assisted SS-WDM systems.
4.5
Spectrally Efficient Scenarios
FEC and SOA-based noise suppression can be used in concert to enable dense SSWDM optical networks. SOA suppression of intensity noise results in crosstalk limited
operation, thus directly impacting spectral efficiency. A FEC with a given BER threshold ηFEC can be used to combat crosstalk; typical thresholds range from 10−3 to 10−5 .
89
Chapter 4. Filter Design
-2
nSF, nCSF = 1
nSF, nCSF = 2
nSF, nCSF = 4
-3
-4
Empty marker: MMC
Filled marker: Q-factor
-5
log (BER)
-6
-7
-8
-9
-10
-11
-12
20
40
60
80
100
120
CSF 3 dB Bandwidth [GHz]
140
Figure 4.10: MMC BER estimations (empty markers), and Q-factor approximated
BERs (filled marker), of three representative cases in the multi-channel scenario.
We jointly optimize the SF, CSF, and the channel spacing ∆CH , under the constraint BERno-FEC = ηF EC to maximize the spectral efficiency SE = Rb /∆CH . The
application of FEC enables error-free operation even for tightly spaced channels with
large crosstalk. This optimization requires significant simulation power, and would be
impossible without recourse to the PMMC, which brings simulation durations down
from months to a reasonable time. The spectral efficiency is optimized for the architecture shown in Fig. 1.1, under two hypothesis. In the first scenario, we assume use
of identical arrayed waveguide gratings (AWG), and a 2.5 Gb/s bit-rate. In the second
scenario we assume the system designer is free to independently vary SF and CSF, and
that the operating bit-rate is 5 Gb/s. All channels employ identical SOAs, with the
same parameters as the previous simulations in this paper.
Guided by the results of the previous sections, we assume all AWGs have superGaussian frequency responses of order 2. In the first scenario, all AWGs are identical
90
Chapter 4. Filter Design
0
-1
-2
0.139
0.104
0.114
0.083
0.096
0.085
0.083
-3
0.069
0.057
0.072
Log10 (BER)
0.063
0.049
0.034
0.043
0.048
-5
0.050
-6
0.038
FEC Region
0.041
0.029
0.036
0.032
0.042
-7
SF 18 GHz
SF 22 GHz
SF 26 GHz
SF 30 GHz
-8
-9
-10
0.5
0.042
0.068
0.058
-4
0.060 0.052 0.046
1
0.036
0.031
0.028
1.5
2
2.5
3
Channel spacing/SF bandwidth
0.025
3.5
Figure 4.11: BER vs. normalized channel spacing, for four different SF bandwidths,
for the first scenario. The spectral efficiency (in bits/s/Hz) is given next to each point.
and their 3 dB bandwidth is denoted by BWAW G . We have BWSF = BWAW G , and
√
BWCSF = BWAW G / 4 2, since the equivalent CSF consists of the cascade of two AWGs.
We consider four values for BWSF , 14, 22, 26, and 30 GHz. The channel spacing is
varied over the scaled range [s ∗ 30 GHz, ..., s ∗ 100 GHz], where the scaling factor s is
defined as BWSF /30 GHz.
For each combination of SF bandwidth and channel spacing, we calculate BER and
SE. BER is reported in Fig. 4.11, where the corresponding SE is given next to each
BER point. Each BER curve in Fig. 4.11 corresponds to a fixed BWSF , therefore the
range of channel spacings examined differs from one curve to other; however, the ratio
of channel spacing to SF bandwidth sweeps over the same range for all curves.
As can be seen in Fig. 4.11, for a fixed value of BER, wider SF bandwidths yield
higher spectral efficiency. Without FEC 10−10 BER is achieved at 30 GHz SF 3 dB
bandwidth with 100 GHz channel spacing, resulting in 0.025 bits/s/Hz spectral efficiency at 2.5 Gb/s. Employing a FEC with ηF EC = 10−5 increases the spectral efficiency to 0.05 bits/s/Hz, whereas a more aggressive FEC with ηF EC = 10−3 results in
91
Chapter 4. Filter Design
BWSF =
log (BER)
log (BER)
BWSF =
Increasing ∆CH
BWSF =
BWSF =
log (BER)
log (BER)
∆CH = 60 GHz
∆CH = 100 GHz
2BWSF
2DCH - 2BWSF
BWCSF
2BWSF
2DCH - 2BWSF
BWCSF
Figure 4.12: All BER curves estimated by PMMC during the SE optimization process
for the second scenario. Each curve corresponds to a different channel separation, as
described in the text.
a spectral efficiency greater than 0.072 bits/s/Hz. Note that in the first scenario, the
effective bandwidth of CSF is smaller than that of SF, and the system is in fact filtering
effect-limited rather than crosstalk-limited.
In the second scenario, we assume the CSF bandwidth can be set independently
from the SF bandwidth. The optimization procedure is as before, except that for each
combination (BWSF , ∆CH ) the BWCSF is swept through the range [2BWSF , ..., 2∆CH −
2BWSF ]. To increase resolution, the channel spacing covers [s∗60 GHz, ..., s∗100 GHz].
BER curves presented in Fig. 4.12 are used to select the CSF bandwidth yielding the
minimum BER for each (BWSF , ∆CH ), and producing the performance map in Fig. 4.13.
As can be seen in Fig. 4.13, at a fixed BER, the narrower SFs are favorable, although
variations of SE vs. BWSF are not significant. Employing a FEC with ηF EC = 10−5
increases the SE from 0.025 bits/s/Hz to 0.12 bits/s/Hz when BWSF = 14 GHz. This
should be compared to 0.072 bits/s/Hz in the first scenario. A FEC with ηF EC = 10−3
92
Chapter 4. Filter Design
-2
0.38
0.28
-3
0.22
0.18
0.15
-4
0.11
FEC Region
0.15
0.13
0.13
0.12
0.11
0.10
log (BER)
-5
0.08
0.12
0.10
0.09
-6
0.09
0.07
0.07
0.07
-7
0.08
0.08
0.08
0.07
0.07
0.065
-8
SF 14 GHz
0.06
0.06
0.06
SF 22 GHz
-9
SF 26 GHz
0.057
SF 30 GHz
-10
2
2.2
2.4
2.6
2.8
3
Channel Spacing/ SF Bandwidth
0.05
3.2
0.05
3.4
Figure 4.13: Minimum BER (CSF bandwidth optimized) vs. normalized channel spacing, corresponding to four systems with different SF bandwidths, for the second scenario. The spectral efficiency (in bits/s/Hz) is given next to each point.
would result in SE = 0.28 bits/s/Hz, when BWSF = 14 GHz, and still higher spectral
efficiencies are possible by lowering the SF bandwidth. The second scenario allows the
noise cleaning to have its full effect, so that overall spectral efficiency sees a significant
increase. Combining efficient noise cleaning with FEC is an effective tool to enhance
spectral efficiency. Our tool allows for design and optimization, once the architecture
and the FEC type are known. BER points in Fig. 4.12 required 25 minutes, as MMC
parameters are like those of the multi-channel BER simulations of the previous section.
Generating all results of Fig. 4.12 took 5.5 days our computing cluster was limited to
four nodes.
4.6
Cross validations
In this section we demonstrate the accuracy of our simulator by cross-validating it
against published measurements of two different multi-channel SOA-based SS-WDM
93
Chapter 4. Filter Design
0
Bit-rate = 5 GB/s
SF 3 dB Bandwidth = 0.24 nm
CSF 3 dB Bandwidth = 0.8 nm
Channel spacing = 0.8 nm
nSF = nCSF = 2
-1
-2
log (BER)
-3
-4
-5
-6
-7
-8
-9
-10
-20 -18
Simulation
Measurement (ref. [33])
-16
-14 -12 -10 -8
-6
Received Power [dBm]
-4
-2
0
Figure 4.14: BER vs. received power simulations, and measurements taken from Fig.
3 of [90].
systems [90, 94]. Good agreement of our simulated results with the published measurements, despite the lack of exact characterizations, indicates the reliability of our
simulator.
In [90], Mathlouthi, et al., reported BER measurements for various bit-rates and
SOA-assisted SS-WDM receiver architectures. One case corresponds to a three-channel
system operating at 5 Gb/s, with 30 GHz 3 dB bandwidth for SF, 100 GHz channel
spacing, and 100 GHz CSF 3 dB bandwidth. The “conventional receiver” (CR) reported
in [90] is the receiver we used throughout this paper; SF and CSF were both flat-topped.
We present our simulations results in Fig. 4.14, as well as reproducing the measured
values reported in Fig. 3 of [90]. For the BER curve of Fig. 4.14, only one pair of PMMC
simulations were executed to estimated the conditional PDFs in the intensity limited
regime. Each point of the BER curve was then calculated by rescaling the original
conditional PDFs, and convolving them with the receiver Gaussian noise PDF. Again a
good correspondence is observed between our simulations and the experimental results.
Finally, Lee, et al., published experiments comparing the impact of filter shapes
in a single-channel SOA-based SS-WDM system [94]. They measured BERs at 1.25
94
Chapter 4. Filter Design
-4
Bit-rate = 1.25 GB/s
: Measurement (ref. [35])
log (BER)
-5
Solid lines: Simulation
-6
nSF = nCSF = 1
1 dB Bandwidth (SF and CSF) = 0.3 nm
-7
-8
-9
-10
nSF = nCSF = 2
1 dB Bandwidth (SF and CSF)
= 0.5 nm
-12
-29
-28
-26
-25
-27
Received Power [dBm]
-11
-24
-23
Figure 4.15: Back-to-back BER vs. received power simulations, and measurements
taken from Fig. 4a of [94].
Gb/s, for two scenarios: when both SF and CSF are Gaussian with 1 dB bandwidth of
0.3 nm, and when both SF and CSF are super-Gaussian with 1 dB bandwidth of 0.5
nm. Using the filter parameters in [94], we simulated the BERs of these two scenarios
in back-to-back operation. We present our simulations results in Fig. 4.15, as well as
reproducing in markers the measured values reported in Fig. 4a of [94]. Once again,
good correspondence is achieved.
4.7
Conclusion
We introduced a multi-channel parallel MMC simulator to study SOA-assisted SSWDM systems, which enables us to evaluate extremely low BERs in spite of the extreme
complexity of the system model. We used this simulator to optimize the performance of
SOA-assisted system with respect to optical filters at transmitter and receiver. Specifically, we looked into the impact of the bandwidth and roll-off of SF and CSF in single
and multi-channel cases. The excess intensity noise was studied for a large number of
filter combinations; BER curves were estimated for a limited set of filters.
Chapter 4. Filter Design
95
Although conventional analysis of the filtering effect suggests favoring CSFs with
smooth roll-off, the SOA-assisted system is ultimately crosstalk limited. Our simulation
of BER in modualted systems found the best performance is achieved by using flattopped optical filters at TX and RX. The Q-factor approximation of the BER was
shown to poorly predict the BER, highlighting the non-Gaussian nature of the noise in
SOA-assisted SS-WDM systems.
The simulation power offered by the parallel implementation of MMC allowed optimization of spectral efficiency when both SOA and FEC are employed to counter the
intensity noise. The reliability of our link simulator was demonstrated by matching
the measured noise suppression ratio spectra in the CW regime. The reliability of the
BER simulator was cross-validated against published BER measurements. The parallel MMC we introduce is a powerful simulation paradigm, whose applicability goes far
beyond the specific problem addressed in the present work.
Chapter 5
SAC OCDMA
We present, for the first time, the performance analysis of spectral amplitude coded optical
code division multiple access (SAC-OCDMA) systems using a multicanonical Monte Carlo algorithm, and examine the efficiency of SOA-based intensity noise suppression in SAC-OCDMA
as the number of users is increased. We find that noise mitigation offered by SOA rapidly
degrades as the number of users increase, contrary to behavior in spectrum-sliced wavelength
division multiplexed systems. We examine the Gaussian approximation for the noise distribution and find it inaccurate for predicting BER. We validate our numerical results against
published experiments.
Chapter 5. SAC OCDMA
5.1
97
Introduction
The performance of spectral amplitude coded optical code division multiple access
(SAC-OCDMA) systems is severely curtailed by the excess intensity noise of the incoherent thermal-like light source. To mitigate the intensity noise, various solutions
have been proposed. In one approach, the optical bandwidth of the frequency bins of
all codes are optimized to trade-off noise reduction (due to optical bandwidth enlargement) when using wide overlapping optical bins against enhanced beat noise due to bin
overlapping [95]. We have already examined in this thesis intensity noise suppression
offered by saturated SOAs for SS-WDM systems. SOA-based noise suppression in SACOCDMA has been validated experimentally, however, the solution is costly with one
SOA per occupied frequency bin per user [96]. A scheme with one SOA per user was
proposed by Penon et al. [97] exploits the optimally-designed code profiles in [95]. A
reduced balanced receiver (RBR) is used for SOA-assisted SAC-OCDMA, replacing the
standard conventional balanced receiver (CBR) [97]. The RBR was shown to be more
robust than the CBR to noise-cleaning degradation due to post-SOA optical filtering
[90]. The approach adopted in [97] was purely experimental; due to the system complexity, the experiments were limited to three active users in a seven user system. The
noise mitigation steadily degraded as users went from one to three, and no conclusion
could be drawn regarding the ultimate capacity of this system.
We investigate quantitatively via simulation the extent to which one SOA per user
can mitigate intensity noise in a SAC-OCDMA system in terms of bit error rate. Simulations in [96] are limited to RIN which gives good indication of trends, but is a poor
predictor of BER. Several factors contribute to the difficulty of this simulation: the numerically complex SOA model to capture noise cleaning, multiple optical filters for each
user (transmitter and receiver), and the asynchronicity of the OCDMA signals. To this
end we adapted our parallel MMC simulator to SAC-OCDMA systems, and employed
it to analyze the performance of SOA-based noise suppression in such systems.
We present MMC simulations of the SOA-assisted SAC-OCDMA systems, and validate our simulator against measurements in [97]. After establishing the reliability of
our simulator, we examine systems with increasing numbers of users. We show that
SOA-based noise mitigation efficiency rapidly degrades as the number of users increases,
despite exploiting both optimally-designed spectral codes and the RBR. Our simulator
98
Chapter 5. SAC OCDMA
EDFA
Enc. #1
SOA
VOA MZM
EDFA
BBS
7.2 nm
Enc. #2
SOA
VOA MZM
.
.
.
1×N
OF
EDFA
Enc. #N
K
EDFA
f
N×1
Enc. #1
EF
SOA
VOA MZM
Figure 5.1: The N -user SOA-assisted SAC-OCDMA setup with reduced balanced receiver after ref. [97], BBS: broadband source, OF: optical filter, Enc: SAC-OCDMA
encoder, VOA: variable optical attenuator, MZM: Mach-Zehnder modulator, EF: electrical filter. The polarization beam splitters, and polarization controllers at the SOA
input and MZM input, as well as optical isolators are not shown for simplicity.
can be useful in studying other OCDMA systems, especially when optical nonlinearites
result in non-Gaussian signal statistics. In Section 5.2 we describe the SAC-OCDMA
receiver and transmitter, simulator and numerical results appear in Section 5.3, and we
conclude in Section 5.4.
5.2
System description
Figure 5.1 gives the N -user SOA-assisted SAC-OCDMA experimental setup of [97],
including an RBR. The thermal-like radiation from an incoherent BBS is sliced by an
optical filter (OF) with 7.2 nm 3-dB bandwidth and distributed to all users by a coupler.
In simulations the incoherent BBS, the 7.2 nm optical filter and the EDFA are modeled
by a single complex Gaussian optical field whose spectrum is determined by the optical
filter and whose power is the optical power at the EDFA output.
At each transmitter, the incoherent slice is amplified by an EDFA to ensure saturation of the noise-cleaning SOA. The SAC-OCDMA code is applied (Enc#i, for user
i, Table 5.1) before entering the noise-cleaning SOA; data is imprinted via a MZM. All
signals are combined by an coupler, further amplified by a booster EDFA, and launched
into 20 km of SMF fiber; a dispersion compensating fiber (DCF) assures full dispersion
compensation. The booster EDFA adds extra ASE noise to all the users, however, we
verified by separate simulations that the system is intensity-noise limited and the EDFA
can be assumed noiseless in the simulations. The SAC-OCDMA codes are BIBD codes
99
Chapter 5. SAC OCDMA
Table 5.1: BIBD Codes
Desired User 0 0 0 1 0 1 1
User #2
0010110
User #3
0101100
User #4
1011000
User #5
0110001
User #6
1100010
User #7
1000101
(Table 5.1) of length 7, weight 3, and cross-correlation 1 [97].
The RBR is illustrated in Fig. 5.1. The RBR is different from the CBR in that the
desired user encoder is removed from the upper arm, and the variable optical attenuator
(VOA) is moved from the lower to the upper arm. The removal of the desired user
encoder in the upper arm results in cancelation of the post-SOA filtering penalty in
the upper arm. The VOA is adjusted such that the average detected optical powers in
both arms of the RBR are equal whenever the desired user is absent. Moving the VOA
from the lower to the upper branch of the balanced receiver results in '1.4 dB power
penalty in SAC-OCDMA performance compared to the CBR for the BIBD codes used;
however, this small penalty is compensated by gain achieved by significantly reducing
the post-SOA filtering penalty [97].
5.3
Numerical model and results
To numerically estimate the BER of the SOA-assisted SAC-OCDMA system in
reasonable run-time, we used a parallel implementation of the MMC algorithm, similar
to that for SS-WDM in Chapter 3. The extension of the simulator to SAC-OCDMA
must take into account the randomness in SAC-OCDMA systems stemming from four
different origins: 1) the CW waveform of the light source utilized by each user - the
optical field is a sample function of a complex Gaussian stochastic random process
where the power spectral density of it is set by the specific spectral code assigned to
that user, 2) the bit pattern transmitted by each user, 3) the relative temporal delays
between the desired user and all other active users, and 4) the electronic noise at the
receiver side of the desired user.
100
Chapter 5. SAC OCDMA
M
-5
-15
-25
1538
1540 1542 1544 1546
Tb
td1
-5
-15
-25
1538 1540 1542
1544 1546
td2
-5
-15
-25
1538
1540 1542
1544 1546
Figure 5.2: (a): the spectral codes of the desired user and two interferers, (b): A
snapshot of a three-user simulation of the system.
Two MMC simulations estimate the conditional PDFs of marks and spaces when the
receiver noise is set to zero. Each conditional PDF is convolved with the Gaussian PDF
of the receiver noise, and finally the bit error rate (BER) is calculated by integrating
the overlapping tails of the conditional PDFs of marks and spaces. Proper scaling of
the horizontal axes of the two conditional PDFs yields the performance at a specific
received power. Each MMC simulation consisted of 6 adaptation cycles. During each
MMC cycle, k = 20000 time-domain simulations of the whole system were executed.
Fig. 5.2a shows three optimized spectral codes according to [95], realized by fiber
Bragg gratings and used in the experiments reported in [97]. Figure 5.2b illustrates one
system realization during an MMC cycle, giving a snapshot of the temporal waveforms
generated. We assume M = 3 bits of memory are introduced by nonlinearities in
the SOA. The final bit of the desired user is set to one for marks (zero for spaces),
while the preceding bits of the desired user, and all bits of the interferers, are random.
The light source sample functions are synthesized by proposing vectors of independent
identically distributed Gaussian random variables of zero mean and unit variance, and
filtering them by the corresponding spectral codes.
To realistically estimate the OCDMA performance, the asynchronicity of users
should be properly modeled. To do so, during each of k simulations within each MMC
cycle, we generate an (M + 1)-bit waveform for the desired user, where the final bit
is forced to be either always one or always zero, and (M + 2)-bit waveforms for all
101
Chapter 5. SAC OCDMA
N=7
N=2
N=5
N=3
N:
Figure 5.3: BER estimations by MMC of SOA-assisted SAC-OCDMA.
interferers. The reference timing is that of the desired user, and the temporal sample
used to calculate the conditional PDF is taken at the end of the final bit of the desired
user. The waveform of interferer i is temporally shifted with respect to the desired
user waveform by a random time delay denoted by tdi , which is uniformly distributed
over (0, Ns − 1), where Ns is the number of time samples within a bit duration. If
the bit-rate is denoted by Rb , the bit duration is Tb = 1/Rb , and Ns , Tb /∆t, where
∆t is the simulation time step. The multiple access time delays are proposed by a
Metropolis-Hastings module called the interferer delay generator as in Section 4.3. The
SOA was modeled as described in 3.2.2. The key SOA parameters are saturation power,
carrier life-time, line width enhancement factor, small-signal gain, and distributed loss.
The parameters we used in our simulations are exactly the values obtained by careful
characterization of the same components used by our colleagues who authored [5]. The
electrical filter was a 4th -order Bessel-Thompson with 0.75Rb 3 dB bandwidth.
Figure 5.3 shows the MMC estimated BER vs. received power for 2, 3, 5, and 7
active users at Rb = 2.5 Gb/s. The solid curves show the BERs of the SAC-OCDMA,
while the dashed curves show those of the SOA-assisted SAC-OCDMA. For 2 and 3
users, simulated BER floors for SAC-OCDMA are 5.0 × 10−10 and 1.1 × 10−6 ; the
corresponding measured values are 5.4 × 10−10 and 5.2 × 10−6 . The simulated BER
floors for 2 and 3 users of SOA-assisted SAC-OCDMA are 2.8 × 10−16 and 5.7 × 10−7 ;
the corresponding 3-user measured value is 2.8 × 10−7 . The SOA-assisted 2-user BER
Chapter 5. SAC OCDMA
102
floor was unmeasurable.
The good match between simulation and measurement, in spite of lack of full characterization of the experimental setup, attests to the reliability of the simulator. From
Fig. 5.3 we see diminishing returns in BER improvement as the number of active users
grows. The BER floor improvement due to SOA-based noise suppression is 4 orders of
magnitude for 2 users, less than 2 orders of magnitude for 3 users, and less than 1 order
of magnitude for 5 and 7 users.
Comparing SOA-based intensity noise suppression efficiency in SAC-OCDMA and
SS-WDM, we observe that noise-cleaning efficiency is not degraded as the number of
active users is increased in SS-WDM, whereas it rapidly becomes ineffective in SACOCDMA. Note that in SAC-OCDMA, there is always a strong filtering effect in the
lower arm of the receiver balanced detector; as the number of users increases, so does
the number of noisy interferers that are added to the decision variable through the
lower arm. Contrary to this, in SS-WDM the performance is only slightly degraded due
to the crosstalk induced by two adjacent channels. Further increases in the number of
SS-WDM users does not impact BER [90].
Figure 5.4 represents the MMC estimated 2- and 3-user BER curves of SOA-assisted
SS-WDM, together with the Q-factor approximations of the BERs directly calculated
from the estimated conditional PDFs by MMC using (1.5) and (1.6). The observation
in [97] that the true BER is underestimated when using Q-factor approximation is
confirmed by simulation. This indicates that signal statistics is highly non-Gaussian,
and justifies the need of numerical tools to study such systems.
5.4
Conclusion
An MMC simulator for modeling SOA-assisted SAC-OCDMA systems was developed, and was used to estimate the BERs of SAC-OCDMA when SOA-based noise
suppression scheme is optionally employed in the transmitter side. The BER improvement vs. number of active users were traced, and it was shown that, contrary to
SOA-assisted SS-WDM, the noise mitigation offered by SOA-based noise suppression
scheme rapidly degrades as the number of active users passes three. The inadequacy
103
Chapter 5. SAC OCDMA
N=3
N=2
N:
Figure 5.4: MMC vs. Q-factor BER estimations of SOA-assisted SAC-OCDMA
of the Gaussian assumption for signal statistics was demonstrated by comparing BER
predictions from MMC by those obtained from Q-factor approximation. The numerical
results were validated against the previously published experimental results.
Chapter 6
Conclusions and Future Work
The major subject of this thesis was to develop a reliable and efficient numerical tool
to estimate the exact BER of the optical links employing nonlinear semiconductor optical amplifiers, especially for SOA-assisted SS-WDM systems. To achieve this goal, we
proposed and implemented an MMC-based simulator. Our simulator correctly handled
all sources of randomness inherent in SOA-assisted system: the noise-like incoherent
spectrum-sliced source, receiver noise, SOA and/or EDFA ASE, random bit patterns of
all users, and random relative delays between the desired user and adjacent channels.
In Chapter 2, we developed a single-channel SOA-assisted MMC-based simulator,
and verified its predictions against measured the PDFs of SOA output light in the
CW regime, and the BERs in the modulated case. We also introduced a pattern
warping technique to handle the impact of bit patterns on the distribution of the decision
statistic. The SOA was not responsible for patterning in this application.
Chapter 6. Conclusions and Future Work
105
In Chapter 3, we focused on exploring SOA bit patterning with our pattern warping
method. We developed a measurement technique to directly measure the conditional
PDFs of marks and spaces, and verified our simulator tool experimentally. In Chapter
3, the SOA input was a 10 Gb/s externally modulated laser source, since our major
concern was the nonlinear ISI stemming from the nonlinear dynamics of the SOA.
In Chapter 4, we reconsidered performance analysis of SOA-assisted SS-WDM systems. We found the SF and CSF filter shapes and bandwidths in a multi-channel SOAassisted SS-WDM system that optimize performance. We upgraded the single-channel
simulator of chapter 2, to a multi-channel simulator to capture cross-talk. We introduced an efficient parallelized MMC algorithm to reduce simulation time. We used
our simulator to design user channels to maximize the spectral efficiency of a coded
SOA-assisted SS-WDM system. The impact of channel power imbalance on BER was
studied, and the accuracy of the Gaussian assumption for signal statistics was examined. We cross validated the multi-channel simulator against published experimental
results.
Finally in Chapter 5, we focused on the scalability of SOA-assisted SAC-OCDMA
systems. We found, as opposed to SS-WDM where SOA-based intensity noise suppression is very efficient, in SAC-OCDMA SOA noise cleaning yields diminishing returns
when more than three users exist in the network.
The contributions of this thesis are two-fold. First, we created a powerful tool
to design and optimize spectrally efficient WDM schemes employing incoherent light
sources. SS-WDM is a potentially winning candidate for next generation WDM PONs,
due to the economic transmitter. Secondly, our numerical tool can be used to investigate
many other problems involving nonlinear behavior in SOAs.
Several possible research projects could exploit material presented in this thesis. In
the context of SS-WDM, our simulator could be used to optimize network topology, by
examining the best configuration for placing noise cleaning SOAs, and CSFs. The impact of SOA parameters on BER can be studied to design SOAs. The RBR introduced
in Chapter 5 for SAC-OCDMA was originally proposed to boost the spectral efficiency
of SS-WDM. Our simulator could be used to optimize the RBR structure for SS-WDM.
Besides SS-WDM, we can use our simulator to do exact statistical characterization of
Chapter 6. Conclusions and Future Work
106
SOA-MZI structures used in optical 2R regenerators and wavelength conversion units.
Another option is to study the impact of SOA ultrafast dynamics on the statistical
distribution of amplified short pulses (100 Gb/s and beyond). Moreover, the pattern
warping technique allows us to use the simulator to study ISI in a broad range of
problems.
Appendix A
Metropolis-Hastings Algorithm
In the Metropolis-Hastings (MH) algorithm, random variables from a specified PDF
are generated by running a Markov chain. Suppose we have a continuous real RV X with
a given PDF pX (x). We divide the real line into uniformly-sized bins {B1 , · · · , BNX }
of width ∆x. The probability that a randomly generated sample falls in the ith bin is
πi = Pr {x ∈ Bi } ∼
= pX (xi ) ∆x where xi lies at the center of Bi . Now we build a Markov
chain with states {S1 , · · · , SNB }, such that the steady state probability of finding the
chain in state Si is equal to πi . Running the Markov chain produces random samples
with the desired PDF pX (x), provided the chain has passed its transient phase.
MH is a method to construct a Markov chain with any prespecified steady-state
distribution {π1 , · · · , πNB }. 1 We begin with an arbitrary Markov chain with NB states.
1. The chains discussed in the text are discrete Markov chains. All the arguments can be extended
to the continuous case. In fact, each bin can be a single point and πi becomes pX (x). In the simulations
of the next chapter the chains are continuous.
108
Appendix A. Metropolis-Hastings Algorithm
The kernel, or the transition probability matrix, of this Markov chain is denoted by
Q(Si , Sj ) which is define by
Q(Si , Sj ) = Pr {xn = Si |xn−1 = Sj }
(A.1)
where xn is the state of the Markov chain at the nth time step. This kernel is then
transformed to another kernel
QM H (Si , Sj ) = αij Q(Si , Sj )
(A.2)
where αij is given by
(
)
πi Q(Sj , Si )
αij = min
,1
πj Q(Si , Sj )
(A.3)
The important property of the new kernel QM H is that it satisfies the following balance
condition
πi QM H (Sj , Si ) = πj QM H (Si , Sj )
(A.4)
This condition guarantees that {πi |i=1,··· ,NB } is in fact the steady-state distribution of
the Markov chain with kernel QM H . This can be shown by the following calculation:
NB
X
j=1
πj QM H (Si , Sj ) =
NB
X
πi QM H (Sj , Si ) = πi
NB
X
QM H (Sj , Si ) = πi
(A.5)
j=1
j=1
|
{z
=1
}
To implement the MH algorithm within the framework of MMC, we need a method
to generate samples from the original distribution pX (x). The algorithm is as follows:
suppose the QM H chain is at state Si at the beginning of the current time step, (the nth
step). Given the current state is Si , first the Q chain is one-step advanced to produce
a proposal state: Sprop . Then a random number, Un , with uniform distribution in the
interval [0, 1) is generated and the following condition is tested:
Un 6
πi Q(Sprop , Si )
πprop Q(Si , Sprop )
(A.6)
If Sprop passes the above test, it is accepted as the new state, Sj = Sprop , otherwise it
is rejected and the new state will be the same as the old state, Sj = Si . The probability
of the transition: Si → Sprop is determined by Q(Sprop , Si ), while the probability of the
transition: Si → Sj is QM H (Sj , Si ).
Appendix A. Metropolis-Hastings Algorithm
109
For MMC a wise choice of Q(·, ·) leads to considerable simplification:
Q(Si , Sj ) = pX (xi ) ∆x
(A.7)
the ratio appearing in defining αij can be simplified as:
πi Q (Sj ,Si )
πj Q Si ,Sj
=
pX (xi )
(n−1)
Cn p̂Y
(g(xi ))
pX (xj )
(n−1)
Cn p̂Y
(g(xj ))
×
p̂(n−1) (g (xj ))
pX (xj ) ∆x
= Y(n−1)
pX (xi ) ∆x
p̂Y
(g (xi ))
(A.8)
When the Markov chain is switched on, it passes a transient phase before reaching
the steady-state. The statistical properties of the samples during this transient are
generally unknown, and in addition it is very difficult to estimate how long it takes for
a chain to reach the steady-state. The practical way to avoid errors induced by the
transients is to observe the chain during the design phase and to decide by trial and
error how many samples should be discarded as transients.
Using (A.7) as the kernel of the proposal generating Markov chain is problematic, as
most of the input proposal vectors, X prop s are, by construction, in the modal zones of
pX (·). If the system mapping were such that modal zones at the input were mapped to
tails at the outputs; the proposals would be accepted most of the time, since pY (g(xi ))
and pY (g(xj )) are both small, due to the fact that g(xi ), and g(xj ) are in the tails
of pY (y). In this case, the tails of pY (y) would be explored; however, modes of the
target PDF would not be explored frequently, since the image of the new proposal xi ,
i.e., g(xi ), falls only rarely in the modal zones of pY (y). The best solution is to force
the proposal-generating Markov chain to make proposals uniformly from all the input
space. The kernel given in (A.7) is therefore replaced by the following update law
xi = xj + u
(A.9)
where u ∼ uniform[−δ, δ] is a symmetric uniform random variable. The parameter δ
should be adjusted by trial and error. The necessary condition for this technique to
work properly is that the components of the input vector samples should be independent
RV’s.
Appendix B
SOA parameters for simulations of
Chapter
The EF is fixed to the 4th -order Bessel-Thompson, and its bandwidth is fixed to
0.75Rb , where Rb is the bit-rate. The SOA parameters carrier lifetime τc , saturation
power Psat , loss coefficient, and small-signal gain directly impact the noise suppression
by the SOA when no CSF is present [12]. In the presence of a CSF, the SOA linewidth
enhancement factor α also has an impact. Generally speaking, SOAs with lower carrier
lifetimes, “fast SOAs", have wider noise suppression bandwidths. Lower saturation
power, resulting in deeper saturation for a fixed input average power, results in better
noise suppression performance. For the simulations of Fig. 4.1, we used τc = 170 ps,
coming from the characterization of the SOA used in the experiments, whereas for the
rest of the paper, we instead use τc = 100 ps, corresponding to commercially available
faster SOAs. We assumed Psat = 10 dBm. As demonstrated in [90], cascading two SOAs
results in better noise suppression performance due to the overall speedup offered by the
Appendix B. SOA parameters for simulations of Chapter
111
“turbo" structure [91]. We assume the noise cleaning module consists of the cascade
of two identical SOAs, each having 25 dB small-signal gain. Optimization of SOA
parameters can be the subject of a separate study.
In the multi-channel scenario, high linewidth enhancement factors result in more
spectral broadening of the adjacent channel SOAs, and consequently, enhanced crosstalk.
Thus SOAs with low enhancement factors are preferred for SOA-assisted SS-WDM systems. Throughout this paper, unless explicitly stated otherwise, we fix the enhancement
factor to α = 2.5, corresponding to the characterization of the SOA used in the experiments of Chapter 2.
Appendix C
Publication List
1. A. Ghazisaeidi, F. Vacondio, A. Bononi, and L. A. Rusch, “SOA Intensity Noise
Suppression in Spectrum Sliced Systems: A Multicanonical Monte Carlo
Simulator of Extremely Low BER”, IEEE J. Lightwave Technol.vol. 27, no. 14,
pp. 2667-2677, July 2009.
2. A. Ghazisaeidi, F. Vacondio, A. Bononi, and L. A. Rusch, “Bit Patterning in
SOAs: Statistical Characterization Through Multicanonical Monte Carlo
Simulations”, IEEE J. Quantum Electron. vol. 46, pp. 570-578, April 2010.
3. A. Ghazisaeidi, F. Vacondio, and L. A. Rusch, “Filter Design for SOA-Assisted
SS-WDM Systems Using Parallel Multicanonical Monte Carlo”, IEEE J.
Lightwave Technol. vol. 28, pp. 79-90, Jan. 1, 2010.
4. A. Ghazisaeidi, and L. A. Rusch, “Capacity of SOA-Assisted SAC-OCDMA”,
IEEE Photonic Technol. Lett. vol. 22, pp. 441-443, 1 April 2010.
Appendix C. Publication List
113
5. A. Ghazisaeidi, and L. A. Rusch, “On the capacity of SOA-assisted
SAC-OCDMA systems: A numerical approach using Multicanonical Monte
Carlo”, Summer Topical Meeting LEOSST, July 2009.
6. A. Ghazisaeidi, F. Vacondio, A. Bononi, and L. A. Rusch, “Statistical
characterization of bit patterning in SOAs: BER prediction and experimental
validation”, OWE7, OFC 2009.
7. A. Ghazisaeidi, F. Vacondio, and L. A. Rusch, “Evaluation of the Impact of
Filter Shape on the Performance of SOA-assisted SS-WDM Systems Using
Parallelized Multicanonical Monte Carlo”, GLOBECOM2009.
8. A. Bononi, L. A. Rusch, A. Ghazisaeidi, F. Vacondio, and N. Rossi,“A Fresh
Look at Multicanonical Monte Carlo from a Telecom Perspective”,
GLOBECOM2009.
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