FRANCESCO VACONDIO

On the benefits of phase shift keying to optical

telecommunication systems.

Thèse présentéeà la Faculté des études supérieures de l’Université Laval

dans le cadre du programme de doctorat en génie électriquepour l’obtention du grade de Philosophiæ Doctor (Ph.D.)

Faculté des sciences et de génieUNIVERSITÉ LAVAL

QUÉBEC

2011

c©Francesco Vacondio, 2011

Contents

Contents i

Acknowledgements iv

Résumé vi

Abstract viii

Foreword x

List of Figures xiii

List of Tables xviii

Acronyms and Abbreviations xix

List of Symbols xxii

1 Meeting the Needs of a Bandwidth Hungry World 1

1.1 Why Phase Modulation? . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 PSK Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 DPSK Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1.3 DPSK Performance . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2 SOAs and phase modulation . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 Optical amplification in semiconductor diodes . . . . . . . . . . 19

1.2.2 SOA basic equations . . . . . . . . . . . . . . . . . . . . . . . . 21

Contents ii

1.2.3 Nonlinear Effects in SOAs . . . . . . . . . . . . . . . . . . . . . 23

1.2.4 Phase modulation to mitigate SOA nonlinearity . . . . . . . . . 29

1.3 Renewed interest in coherent detection . . . . . . . . . . . . . . . . . . 31

1.3.1 Coherent reception . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.3.2 Flagging interest in 1990s . . . . . . . . . . . . . . . . . . . . . 33

1.3.3 Coherent reception in 2010s . . . . . . . . . . . . . . . . . . . . 34

1.4 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 DQPSK: When is a Narrow Filter Receiver Good Enough? 37

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2 Linear Impairments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.3 Back-to-back operation . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.4 Chromatic dispersion . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.5 Polarization mode dispersion . . . . . . . . . . . . . . . . . . . . 51

2.3 Nonlinear Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3.1 Experimental Results and discussion . . . . . . . . . . . . . . . 55

2.3.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.4 Multichannel operation of the narrow filter receiver . . . . . . . . . . . 58

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 SOA nonlinearity postcompensation for PSK signals 64

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Exploiting the low-pass nature of SOAs . . . . . . . . . . . . . . . . . . 66

3.3 Phase Noise Variance at SOA Output . . . . . . . . . . . . . . . . . . . 69

3.4 Large vs. small signal models . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 NLPN compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5.1 NPLN Reduction: noisy CW laser . . . . . . . . . . . . . . . . . 74

3.5.2 NPLN reduction: DPSK modulation . . . . . . . . . . . . . . . 79

3.6 Nonlinear distortion compensation for intensity modulated signals . . . 82

3.6.1 Dependence on Saturation Level . . . . . . . . . . . . . . . . . . 83

3.6.2 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . 85

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 An introduction to the digital coherent receiver 88

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Contents iii

4.2 Electro optical circuit of the dual polarization downconverter . . . . . . 90

4.3 Digital signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3.1 Normalization and resampling . . . . . . . . . . . . . . . . . . . 93

4.3.2 CD compensation . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3.3 Polarization demultiplexing and equalization . . . . . . . . . . . 97

4.3.4 Carrier frequency and phase estimation . . . . . . . . . . . . . . 99

5 Dual versus single carrier for 40 Gb/s coherent BPSK systems 101

5.1 Coherent Detection in Core Networks . . . . . . . . . . . . . . . . . . . 102

5.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3 Single channel back-to-back measurements . . . . . . . . . . . . . . . . 107

5.4 WDM transmission over 2400 km . . . . . . . . . . . . . . . . . . . . . 110

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Conclusions and Future Work 114

A Amplification in semiconductor diodes 117

Bibliography 120

List of publications 134

Acknowledgements

I am greatly in debt to my supervisor, Leslie Ann Rusch. Leslie, I have learned a lot

from you; your guidance and good advices have always been there for me when I needed

them. My gratitude also goes to my cosupervisor Alberto Bononi. You have been a

true inspiration for the work of this thesis. I have learned (and continue learning) a lot

from you. It has been my honor to work with you both during the past few years.

I thank the members of the jury Prof. Roberto Gaudino, Dr. Leo Spiekman and

Dr. Andrew Chraplyvy, for taking the time to read my thesis and for their very fruitful

comments.

I was lucky enough to have had great colleagues who became great friends. Amir,

Walid, Simon, Julien: the time spent in Canada for this thesis would not have been the

same without you.

I have fond memories of the technical and non-technical discussions with the other

members of Leslie and Sophie Larochelle group. I want to particularly thank Mehrdad,

Mehdi, Marco, Jeffrey, José, Pegah and Ziad. Being in COPL with you all has been a

pleasure.

I want to express my gratitude to Dr. Sébastien Bigo, the director of the WDM

Dynamic Networks department in Alcatel-Lucent Bell Labs France. He gave me the

opportunity to spend a very fruitful internship in his laboratories during the time of

this thesis. Gabriel, Jeremie, Oriol, Max, Haik and Patrice welcomed me as one of their

team from the very beginning, I am grateful to you for this.

A Ph.D. is a life experience which goes way beyond this dissertation. I very simply

would not have done this without Veronica: you make me the person I am. Also, I want

to thank my lifelong friends for their support and for always treating me like I never

left, during my short but intense trips back in Italy. I cannot mention all the wonderful

Acknowledgements v

people I met in Québec, who made me feel at home away from home: that’s something

I’ll always be grateful for.

Finally, all the credit goes to my family, to whom I owe everything.

Résumé

Les avantages de la modulation de phase vis-à-vis la modulation d’intensité pour les

réseaux optiques sont claires et accepté par la communauté scientifique des télécommu-

nications optiques. Surtout, la modulation de phase montre une meilleure sensibilité

au bruit, ainsi qu’une plus grande tolérance aux effets non-linéaires que la modulation

d’intensité.

Nous présentons dans cette thése un étude qui vise à développer les avantages de la

modulation de phase. Nous attaquons d’abord la complexité du récepteur en détection

directe, en proposant une nouvelle configuration dont la complexité est comparable à

celle du récepteur pour la modulation d’intensité traditionnel, mais avec des meilleures

performances. Cette solution pourrait convenir pour les réseaux métropolitains (et

même d’accès) à haut débit binaire. Nous passons ensuite à l’examen de la possibilité

d’utiliser des amplificateur à semi-conducteur (SOA) au lieu des amplificateurs à fibre

dopée à l’erbium pour fournir amplification optique aux signaux modulés en phase.

Les non-linéarité des SOA sont étudiées, et un compensateur simple et très efficace est

proposé. Les avantages des amplificateurs à semi-conducteur par rapport à ceux à fibre

sont bien connus. Surtout, la méthode que nous proposons permettrait l’integrabilité

des SOA avec d’autres composants de réseau (par exemple, le récepteur nommé ci-

dessus), menant à des solutions technologiques de petite taille et efficaces d’un point de

vue énergétique.

Il y a deux types de systèmes pour signaux modulés en phase: basé sur la détection

directe, ou sur les récepteurs cohérents. Dans le dernière partie de ce travail, nous nous

concentrons sur cette dernière catégorie, et nous comparons deux solutions possibles

pour la mise à niveau des réseaux terrestres actuel. Nous comparons deux configurations

dont les performances sont très comparables en termes de sensibilité au bruit, mais nous

Résumé vii

montrons comment la meilleure tolérance aux effets non linéaires (en particuliers dans

les systèmes à débit mixte) fait que une solution soit bien plus efficace que l’autre.

Abstract

The advantages of phase modulation (PM) vis-à-vis intensity modulation for optical

networks are accepted by the optical telecommunication community. PM exhibits a

higher noise sensitivity than intensity modulation, and it is more tolerant to the effects

of fiber nonlinearity. In this thesis we examine the challenges and the benefits of working

with different aspects of phase modulation.

Our first contribution tackles the complexity of the direct detection noncoherent

receiver for differentially encoded quadrature phase shift keying. We examine a novel

configuration whose complexity is comparable to that of traditional receivers for inten-

sity modulation, yet outperforming it. We show that under severe nonlinear impair-

ments, our proposed receiver works almost as well as the conventional receiver, with

the advantage of being much less complex. We also show that the proposed receiver is

tolerant to chromatic dispersion, and to detuning of the carrier frequency. This solution

might be suitable for high-bit rates metro (and even access) networks.

Our second contribution deals with the challenges of using semiconductor optical

amplifiers (SOAs) instead of typical erbium doped fiber amplifiers (EDFAs) to provide

amplification to phase modulated signals. SOAs nonlinearities are investigated, and we

propose a simple and very effective feed-forward compensator. Above all, the method

we propose would permit the integrability of SOAs with other network components (for

example, the aforementioned receiver) achieving small size, power efficient sub-systems.

Phase modulation paves the way to high spectral efficiency, especially when paired

with digital coherent receivers. With the digital coherent receiver, the degree of free-

dom offered by polarization can be exploited to increase the channel bit rate without

increasing its spectral occupancy. In the last part of this work we focus on polarization

multiplexed signaling paired with coherent reception and digital signal processing.

Abstract ix

Our third contribution provides insight on the strategies for upgrading current ter-

restrial core networks to high bit rates. This is a particularly challenging scenario, as

phase modulation has to coexist with previously installed intensity modulated channels.

We compare two configurations which have received much attention in the literature.

These solutions show comparable performance in terms of back-to-back noise sensitiv-

ity, and yet are not equivalent. We show how the superior tolerance to nonlinear fiber

propagation (and particularly to cross phase modulation induced by the presence of

intensity modulated channels) makes one of them much more effective than the other.

Foreword

Three chapters of this thesis are composed of material already published in techni-

cal journals or conference papers. In the thesis, text and figures have been modified in

order to be consistent with the rest of the document. Additionally, some material which

did not find place in the original papers has been added. The introduction sections have

been most heavily modified. Here, I detail my contributions to these published papers.

5106 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15,

DQPSK: When Is a Narrow FilterReceiver Good Enough?

Francesco Vacondio, Amirhossein Ghazisaeidi, Alberto Bononi, Member, IEEE, andLeslie A. Rusch, Senior Member, IEEE

Abstract—In this paper, we investigate experimentally and viasimulation the pros and cons of a narrow filter receiver for differ-ential quadrature phase-shift keying based on a single optical filterand eschewing the conventional asymmetrical Mach–Zehnder in-terferometer structure. We quantify the performance differencesbetween the two receivers, allowing system designers and oper-ators to determine when the less complex narrow filter receivermight be the appropriate choice. We numerically optimize the 3-dBbandwidth and center frequency of the narrow filter and showit is more robust to carrier frequency detuning than the conven-tional solution. We show that the narrow filter receiver is moretolerant to chromatic dispersion (CD) than the conventional one,and equally tolerant to first-order polarization-mode dispersion.We show the impact of the 3-dB bandwidth on the receiver per-formance when CD accumulates. Finally, we show via experimentsand simulations that the 3 dB advantage of the conventional re-ceiver vanishes when the nonlinear impairments are fiber nonlin-earities; comparing the two receivers at the optimum launch powerfor a 25 80 km system, the difference in optical SNR margin isreduced to 1.6 dB. Experiments are conducted at 42 Gb/s usinga commercially available narrow filter for reception.

Index Terms—Differential phase-shift keying, modeling, opticalfiber communication, optical receivers.

I. INTRODUCTION

PHASE modulation has drawn much attention in the last

few years for next-generation spectrally efficient optical

networks [1]. In particular, optical differential quadrature

phase-shift keying (DQPSK) is emerging as a promising so-

lution, and the technology is today mature enough to permit

validations outside the research labs [2]. The conventional

receiver for DQPSK, depicted in Fig. 1(a), is composed of

an optical filter, two asymmetrical Mach–Zehnder interfer-

ometer (AMZI) structures, and two balanced photodiodes.

Its complexity is twice the complexity of a binary DPSK

(DBPSK) receiver. In order to increase the cost/benefit ratio of

phase-modulated formats and to eventually avoid the interfer-

ometric structure, much effort has been focused in proposing

Manuscript received January 30, 2009; revised June 16, 2009 and July 29,2009. First published August 07, 2009; current version published September 25,2009. This work was supported in part by 2007–2009 Québec–Italy ExecutiveProgram for Scientific Development, Project 13.

F. Vacondio, A. Ghazisaeidi, and L. A. Rusch are with the Center of Optics,Photonics and Laser (COPL) and with the Department of Electrical and Com-puter Engineering, Université Laval, Québec, QC G1V0A6, Canada (e-mail:francesco.vacondio.1@ulaval.ca).

A. Bononi is with Dipartimento di Ingegneria dell’Informazione, Universitàdegli Studi di Parma, Parma 43100, Italy.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2009.2029538

Fig. 1. (a) Conventional receiver and (b) narrow filter receiver for DQPSK sig-nals. is a symbol time, is the carrier frequency, , and .

alternative, lower complexity receivers for both binary and

quadrature phase-shift keying.

One such reduced complexity receiver [3], [4] is based on po-

larization, where the two arms of the AMZI are replaced by the

slow and the fast axis of a polarization maintaining fiber, whose

differential group delay (DGD) equals one symbol interval; the

signals out of the two axes are then mixed before photodetec-

tion. This receiver is extended to DQPSK in [5], demonstrating

experimentally the main advantage of such a receiver, which

is the relative ease of implementation and the wide range over

which the DGD can be tuned, hence permitting the use of the

same receiver for different bit rates. The main drawback is that

the signal polarization needs to be controlled with very high pre-

cision and stability, thus increasing cost and complexity.

We turn our attention to two other promising receivers that

use a single photodetector for DBPSK; in the case of DQPSK,

these receivers have one photodetector in each of the I and Q

arms. In the first receiver, the AMZI structure is maintained as

in Fig. 1(a); however, only one port (either the constructive or

the destructive port) is populated with a photo detector [6]. In

the second receiver, the AMZI structure is not used and instead

only a narrowband filter is found in each branch, as illustrated

in Fig. 1(b) [7]; the narrowband filters replace the channel se-

lect filter and the AMZI structure. It is well known that the

conventional receiver for DQPSK has cosine-shaped equivalent

transfer functions in the frequency domain. They are plotted in

Fig. 2 versus the frequency normalized to the symbol rate .

The frequency responses are periodic and shifted with respect

to the carrier frequency. Thick lines indicate the main lobes of

0733-8724/$26.00 © 2009 IEEE

Francesco Vacondio, Amirhossein Ghazisaeidi, Al-

berto Bononi, Leslie A. Rusch, “DQPSK: when is a

narrow filter receiver good enough?,” IEEE Jour-

nal of Lightwave Technology, Vol. 27, Issue 22, pp.

5106-5114 (2009)

This journal paper is devoted to the study of

the narrow filter receiver for DQPSK signals. The

original idea is proposed by myself, the experi-

ments were conducted at CMC labs at Queen’s

University in Kinston, Ontario the weeks of 15-

20 September 2008 and 1-5 December 2008. I

was assisted in the experiments by Amirhossein

Ghazisaeidi, a PhD student within our group. The

simulations were done by me. My thesis advisors provided insightful suggestions and

invaluable guidance. The manuscript was prepared by me and revised by the other

authors before submission.

Foreword xi

Francesco Vacondio, Amirhossein Ghazisaeidi,

Leslie A. Rusch, “Simultaneous WDM-DQPSK

Demodulation With a Single AWG,” European

Conference on Optical Communications (ECOC)

2009, oral presentation, Wien, 20-24 September

2009.

This conference paper deals with the multi-

channel operation of the narrow filter receiver for

DQPSK signals. It is a purely experimental pa-

per, and the experiments were conducted at CMC

labs at Queen’s University in Kinston, Ontario the

weeks of 15-20 September 2008 and 1-5 Decem-

ber 2008. I was assisted in the experiments by

Amirhossein Ghazisaeidi, a PhD student within our group. The manuscript was pre-

pared by me and reviewed by the other authors before submission.

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 3, FEBRUARY 1, 2010 277

Low-Complexity Compensation of SOA Nonlinearityfor Single-Channel PSK and OOK

Francesco Vacondio, Amirhossein Ghazisaeidi, Alberto Bononi, and Leslie Ann Rusch, Senior Member, IEEE

Abstract—Carrier density fluctuations in semiconductor opticalamplifiers (SOAs) impose penalties on phase-shift keying (PSK)signals due to nonlinear phase noise (NLPN), and on-off keying(OOK) signals due to self-gain modulation. In this paper, we pro-pose a simple scheme to equalize the impairments induced by SOAnonlinearities, derived from the small signal analysis of carrierdensity fluctuations. We demonstrate via simulation almost com-plete cancelation of the NLPN added by a saturated SOA on adifferential PSK signal. We demonstrate via both simulations andexperiment the effectiveness of the method for mitigation of non-linear distortions imposed by SOAs on an OOK signal.

I. INTRODUCTION

SEMICONDUCTOR OPTICAL AMPLIFIERS are a very

interesting alternative to erbium-doped fiber amplifiers

(EDFAs) due to their wide spectral gain, compact size, inte-

grability, and cost effectiveness [1]. The nonlinear behavior of

semiconductor optical amplifiers (SOAs) can be another attrac-

tive feature. For example, SOAs are used as building blocks

for 2R regenerators [2], wavelength converters [3], nonlinear

media for four-wave mixing [4], or intensity noise suppressors

[5]. EDFAs remain the amplifiers of choice when linearity

is essential. When SOAs are used strictly as amplifiers, gain

variations induce distortions and impose performance penalties.

We will discuss how these nonlinearities can be overcome in

SOAs.

SOA gain dynamics are determined by carrier recombina-

tion and few ultrafast processes (spectral hole burning, carrier

heating, Kerr effect, and two photon absorption) [6]. Due to their

fast response, these processes become important when working

with subpicosecond pulses. In this paper, we focus rather on the

gain fluctuations caused by carrier recombination, whose time

scale is dominated by the carrier lifetime, typically between a

few picoseconds and several hundred picoseconds.

The carriers inside the SOA can be thought as one single

reservoir of carriers replenished by the dc current and from

which the optical signals drain carriers for amplification [7].

Manuscript received June 11, 2009; revised September 08, 2009. First pub-lished November 24, 2009; current version published January 15, 2010. Thiswork was supported by the 2007–2009 Quebec–Italy Executive Program forScientific Development under Project 13.

F. Vacondio, A. Ghazisaeidi, and L. A. Rusch are with the Center ofOptics, Photonics and Laser and the Department of Electrical and Com-puter Engineering, Université Laval, Quebec, QC G1V0A6, Canada (e-mail:rusch@ulaval.ca).

A. Bononi is with the Dipartimento di Ingegneria dell’Informazione, Schoolof Engineering, Università degli Studi di Parma, 43100 Parma, Italy.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2009.2036868

The gain is a function of the amount of carriers in the reservoir.

When the SOA is saturated, any variation of the input power

translates into a variation of the level of the reservoir, which,

in turn, the reservoir variations cause variations of the instan-

taneous gain. For typical SOA carrier lifetimes, the gain can

follow the variations of the input power over a bandwidth of

the order of gigahertz.

In the case of ON-OFFkeying (OOK), this behavior leads to

nonlinear distortions being imposed on the waveform [8]. The

problem is less important for very low bit rates, i.e., when the

bit time is much smaller than the SOA carrier lifetime ,

or at very high bit rates, i.e., when . For typical SOAs,

significant bit pattern distortions are imposed on the signals with

a bit rate in the range from 1 to 30 Gb/s.

To overcome this problem, SOAs have been used with

differential phase-shift keying (DPSK) [9], quadrature differ-

ential phase-shift keying (DQPSK) [10], and other constant

or quasi-constant envelope modulation formats [11]. When

phase-modulated signals are amplified with SOAs, the limiting

nonlinear impairment becomes nonlinear phase noise (NLPN).

NLPN arises whenever intensity noise is translated into phase

noise through self-phase modulation (SPM). This is a very well

known and widely studied phenomenon in optical fiber, named

the Gordon–Mollenauer effect [12]. Fiber NLPN is due to the

Kerr effect, whereas NLPN in SOAs arises from the refractive

index modulation due to carrier density fluctuations [13]. In

nonlinear fiber links, the NLPN bandwidth is limited to the

nonlinear diffusion bandwidth, inversely proportional to the

map strength [14]. In SOAs, on the other hand, the NLPN has

the same bandwidth as the gain fluctuations.

NLPN in SOAs was investigated, for example, in [15] and

[16]. A good approximation of the relation between the output

and the input phases to the SOA [17] is

(1)

where is the linewidth enhancement factor of the SOA and

is the integrated gain. When the input to the SOA is a con-

stant (or quasi-constant) envelope signal with finite optical SNR

(OSNR), its intensity suffers from random fluctuations due to

the noise. If the SOA is saturated, these fluctuations modulate

the reservoir of carriers and hence the SOA gain . The mech-

anism by which these fluctuations translate into phase noise is

given in (1); note that an SOA with zero linewidth enhance-

ment factor would exhibit no NLPN. The linewidth enhance-

ment factor is an unavoidable, inherent characteristic of SOAs,

deriving from changes in the real part of the refractive index

0733-8724/$26.00 © 2010 IEEE

Francesco Vacondio, Amirhossein Ghazisaeidi,

Alberto Bononi, Leslie A. Rusch, “Low Complex-

ity compensation of SOA Nonlinearity for Single-

Channel PSK and OOK,” IEEE Journal of Light-

wave Technology, Vol. 27, Issue 22, pp. 5106-5114

(2009)

This journal paper explores a novel post

compensation method for SOA-induced nonlinear

penalties that I proposed. Very fruitful discus-

sions about SOA linearization with Amirhossein

Ghazisaeidi and Prof. Alberto Bononi are at the

basis of this idea. I performed the simulations.

The experiment was conducted by me with the

help of Amirhossein Ghazisaeidi. I prepared the manuscript and all the authors re-

vised it before submission.

Foreword xii

Dual- versus single-carrier configuration for 40 Gb/s coherent BPSK-based solutions over low dispersion fibers

F. Vacondio(1,2), J. Renaudier(1), O. Bertran-Pardo(1), P. Tran(1), H. Mardoyan(1), G. Charlet(1), S. Bigo(1) (1) Alcatel Lucent Bell Labs, Centre de Vill arceaux, Route de Vill ejust, 91620, Nozay, France

(2) Center for optics, photonics and lasers (COPL), Dept. of ECE, Laval University, G1V 0A6, Canada Francesco.Vacondio@alcatel-lucent.com

Abstract: We experimentall y compare the performance of BPSK-based dual- and single-carrier coherent solutions at 40 Gb/s over low-dispersion fiber. We show that the single-carrier solution is more tolerant to cross-nonlinearities induced by 10 Gb/s neighbors. ©2010 Optical Society of America OCIS codes: (060.1660) coherent communication; (060.4510) optical communication

1. Introduction Recent trends in optical networks demonstrate much interest of the scientific community about coherent detection associated with digital signal processing (DSP). This technology appears as a key enabler for next-generation high bit rate solutions based on advanced modulation formats and polarization division multiplexing (PDM) [1,2]. In order to reduce the required bandwidth and sampling rates of transponders, much interest has also been devoted to using two sub-carriers inside the same 50 GHz slot [3,4]. Such a solution indeed decreases the baud rate per carrier, but it requires a doubling in the number of components both at the transmitter and receiver sides. The performance of the dual-carrier (DC) solution using PDM quadrature phase shift keying (QPSK) format paired with coherent detection has recently been investigated at 100 Gb/s [5], where it was shown that DC configuration is more sensiti ve to cross-nonlinear impairments induced by 10 Gb/s non return to zero (NRZ) neighbors, when compared to the single carrier (SC) solution. In this paper, we focus our investigation on the performance of the 40 Gb/s DC solution based on binary phase shift keying (BPSK) format and coherent detection, which could be attractive for 10 Gb/s infrastructure upgrades. Over a nonzero dispersion shifted fiber transmission link, we assess the performance of one 40 Gb/s channel inserted in one wavelength slot with 50 GHz spacing, originall y designed for non-return to zero on-off keying (NRZ-OOK) channels at 10 Gb/s. For comparison, the SC solution is also investigated to serve as a reference. We thus compare the tolerance to interchannel nonlinear effects of both solutions to assess if the DC solution can be beneficial at 40 Gb/s, despite its unavoidable increase in complexity of the transceivers.

2. Experimental setup Figure 1 (left) shows the experimental setup of the two transmitters used in this experiment. In both cases we have 79 distributed feedback lasers spaced by 50 GHz in the spectral range [1530.31-1562.61nm] and separated in even and odd combs. The combs are intensity-modulated with two independent Mach-Zehnder modulators driven by two 10.7 Gb/s pattern generators to produce NRZ signals.

For all configurations, the channel under test is located in the slot centered on λs = 1546.92 nm. In SC configuration (Fig.1b), the laser source at λs is modulated by a BPSK modulator fed with a pseudo-random bit sequence at 21.4 Gb/s. Polarization multiplexing is emulated by splitti ng the signal along two paths, delaying one of

a

even

odd NRZ

NRZ

10.7Gb/s

λd1 BPSK

BPSKλd2

10.7Gb/s

10.7Gb/s

10.7Gb/s

b

even

odd NRZ

NRZ

21.4Gb/s

λS BPSK

10.7Gb/s

10.7Gb/s

a

even

odd NRZ

NRZ

10.7Gb/s

λd1λd1 BPSK

BPSKλd2λd2

10.7Gb/s

10.7Gb/s

10.7Gb/s

b

even

odd NRZ

NRZ

21.4Gb/s

λSλS BPSK

10.7Gb/s

10.7Gb/s

Tunablefilter

6

EDFA

Pre-compensation

100 km

Co

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nt

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er

Sam

plin

g (s

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)

DS

P (o

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Local oscill ator

DCF

A.O.

100 km 100 km

100 kmNZDSF

A.O.

DCFTx

WSS

Rx

Tunablefilter

66

EDFA

Pre-compensation

100 km

Co

here

nt

Mix

erC

ohe

rent

M

ixer

Sam

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ampl

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Local oscill ator

Local oscill ator

DCF

A.O.A.O.

100 km 100 km

100 kmNZDSF

A.O.A.O.

DCFTx

WSS

Rx

Fig. 1 Left: transmitter setups for dual carrier (a) and single carrier (b) configuration. Right: setup of the recirculating loop experiment.

Francesco Vacondio, Jeremie Renaudier, Oriol

Bertran-Pardo, Patrice Tran, Haik Mardoyan,

Gabriel Charlet, Sebastien Bigo, “Dual- Versus

Single-Carrier Configuration for 40 Gb/s Coher-

ent BPSK-Based Solutions over Low Dispersion

Fibers,” Optical Fiber Conference (OFC) 2010,

oral presentation, San Diego (CA), 2010, paper

OTuL.

This is an experimental paper comparing two

possible signaling methods based on coherent de-

tection for the upgrade of legacy terrestrial systems

to higher bit rates. The work is the fruit of my in-

ternship in the WDM Dynamic Networks department led by Dr. Sebastien Bigo, in

Alcatel-Lucent Bell Labs France between June and December 2009. The idea of the

comparison has been proposed by Gabriel Charlet. I completed the experimental work,

with the invaluable help of Dr. Jeremie Renaudier and Oriol Bertran-Pardo. Patrice

Tran and Haik Mardoyan were fundamental when it came to debugging the experimen-

tal setup. I ran the signal processing on the measured data for obtaining the results. I

have prepared the manuscript, and all the authors revised it before submission.

List of Figures

1.1 Aggregated (measured 2001-2009 and predicted 2010-2013) Internet traf-

fic in Exabit per month. Inset shows the repartition per category since

2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Capacity×distance product of a single fiber in km·petabit/s as a function

of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Signal constellations for three PSK formats: (a) BPSK (b) QPSK (c)

8-PSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Optimum BER for coherently demodulated BPSK and noncoherently

demodulated DBPSK in an AWGN channel. . . . . . . . . . . . . . . . 7

1.5 (a) Schematic of Mach-Zehnder modulator used for generation of OOK

and BPSK signals; (b) nested Mach-Zehnder structure of a IQ modulator

for generating QPSK signals. PG: (electrical) pattern generator. . . . . 8

1.6 Mach-Zehnder transfer function for intensity (dashed) and field (solid).

The bias points and required peak to peak voltage swings are also indi-

cated, for OOK and BPSK. . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 DBPSK symbols and effect of phase modulator and MZ modulator. . . 9

1.8 (a) Direct receiver for intensity modulation, (b) balanced receiver for

DBPSK, (c) balanced receiver for DQPSK. Ts is a symbol time. Optical

bandpass filter and electrical lowpass are also shown. . . . . . . . . . . 11

1.9 Asymmetric Mach-Zehnder interferometric structure. Ts is a symbol

time, ϕ0 = ±π/4 for DQPSK, and ϕ0 = 0 for binary DPSK . . . . . . . 11

1.10 Equivalent periodic filter of DBPSK demodulator at the constructive

(dashed line) and destructive (solid line) ports. . . . . . . . . . . . . . . 13

List of Figures xiv

1.11 Four equivalent periodic filters of the DQPSK receiver. Dashed lines

for constructive ports, and solid lines for destructive ports. Thick lines

represents the lobes that the narrow filter receiver approximates, for (a)

the in phase component and (b) the quadrature component. . . . . . . 15

1.12 Constellations in the complex plane for signals with the same optical

average power P for (a) DPSK, and (b) OOK. . . . . . . . . . . . . . . 16

1.13 Back to back bit error rates for OOK, DBPSK and DQPSK at a bit rate

of 42.7 Gb/s as function of the OSNR in 0.1 nm. Simulation details are

summarized in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.14 Energy vs. wave vector schematic of the energy bands of a direct-gap

semiconductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.15 Schematic of a double-heterostructure SOA. . . . . . . . . . . . . . . . 20

1.16 Input and amplified pulses by the SOA in the time domain, for different

values of the small signal gain. . . . . . . . . . . . . . . . . . . . . . . . 25

1.17 Input and amplified pulses by the SOA in the frequency domain, for

different values of the small signal gain. . . . . . . . . . . . . . . . . . . 25

1.18 SPM-induced chirp on the output pulse. . . . . . . . . . . . . . . . . . 26

1.19 Integrated gain in short-pulse amplification. . . . . . . . . . . . . . . . 30

1.20 Principle of (a) direct detection, (b) differential and (c) coherent receivers. 32

2.1 (a) Conventional receiver and (b) Narrow filter receiver for DQPSK sig-

nals. Ts is a symbol time, fs is the carrier frequency, fI = fs + δf and

fQ = fs − δf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Experimental setup for the investigation of chromatic dispersion and first

order PMD tolerance. PC: polarization controller, MZM: Mach-Zehnder

modulator, PMD: polarization mode dispersion, VOA: variable optical

attenuator, EDFA: erbium doped fiber amplifier, OSA: optical spectrum

analyzer, AWG: arrayed waveguide grating, OBPF: optical bandpass fil-

ter, CR: clock recovery, ED: error detector. . . . . . . . . . . . . . . . . 42

2.3 Measured transmission and group delay profiles of the narrow filter em-

ployed in the experiments. The theoretical shapes are also reported,

along with the transmission spectrum of a Gaussian filter with 3-dB

bandwidth equal to 0.6R, where R is the symbol rate. . . . . . . . . . . 45

2.4 Measured (left column) and simulated (right column) eye diagrams for

the three tested receivers. . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5 Experimental and simulated back to back bit error rate for both receivers

under investigation. Symbol rate is 21 Gbaud, bit rate is 42 Gb/s. . . . 48

List of Figures xv

2.6 OSNR penalty at BER=10−5 for the narrow filter receiver as a function of

filter 3-dB bandwidth and filter center frequency for (a) the quadrature

component and (b) the in-phase component. Stars indicate minimal

penalty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.7 OSNR penalty for a BER=10−5 for the narrow filter receiver and for the

conventional receiver as a function of the detuning frequency. . . . . . . 50

2.8 Simulated OSNR penalties at BER=10−5 as a function of accumulated

chromatic dispersion for both receivers under investigation. Two experi-

mental points are also provided for comparison. . . . . . . . . . . . . . 52

2.9 Simulated OSNR penalties at BER=10−5 as a function of accumulated

chromatic dispersion and filter bandwidth. . . . . . . . . . . . . . . . . 53

2.10 Experimental OSNR penalties at a BER=10−5 due to first-order PMD

as a function of differential group delay for both receivers under investi-

gation, for the worst case 50/50 power split. . . . . . . . . . . . . . . . 54

2.11 Experimental setup for the nonlinear transmission experiment. . . . . . 55

2.12 BER for both receivers in the nonlinear regime. . . . . . . . . . . . . . 56

2.13 BER when KL method is used or Monte Carlo method is used. Fully

compensated DQPSK 21 Gbaud single-channel 25x80km TW system,

conventional receiver only. . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.14 Simulated tolerance to fiber nonlinearities. Fully compensated DQPSK

21 Gbaud single-channel 25x80km system. . . . . . . . . . . . . . . . . 59

2.15 Setup for the WDM-DQPSK experiment. PC: polarization controller,

CR: clock recovery, ED: error detector. The inset shows a WDM-DQPSK

spectrum measured on the optical spectrum analyzer. . . . . . . . . . . 61

2.16 Bit error ratio for conventional and narrow filter receiver. In the WDM

case, different output ports of the same demultiplexer are used for the

different channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.1 3-dB bandwidth of M(ω) as function of the two parameters which com-

pletely determine its time constant. . . . . . . . . . . . . . . . . . . . . 68

3.2 Signal (either CW or phase modulated DPSK) plus additive white Gaus-

sian noise pass through an optical bandpass filter before entering a SOA.

The post-compensation stage includes a photodiode, electrical filter and

phase modulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3 SOA model includes Nsec sections along the propagation direction, each

with a lossless “short” SOA and a lumped loss. . . . . . . . . . . . . . 71

List of Figures xvi

3.4 Logarithmic plot of the normalized standard deviation as function of the

SOA intrinsic losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 Predictions of integrated gain from large δhT (t) vs. small δhl(t) signal

models; NSD is 0.018 in this example. . . . . . . . . . . . . . . . . . . . 74

3.6 PSD of intensity and phase noise at the input and output of the SOA. The

input signal is a CW laser plus AWGN. OSNR is 23-dB, noise is filtered

with a supergaussian filter of order 2, with 15 GHz 3-dB bandwidth. . . 75

3.7 Logarithmic two-dimensional probability density functions of the com-

plex electric fields; input power is 0 dBm, OSNR=23 dB, supergaussian

optical filter of order 2 and 15 GHz bandwidth. (a), (b) and (c) are

respectively the signals at the input of the SOA, output from the SOA,

and after post-compensation. . . . . . . . . . . . . . . . . . . . . . . . 76

3.8 Phase variance vs. linewidth enhancement factor, PSOA,in = 0 dBm. . . 78

3.9 Phase variance vs. SOA input power, α = 7. . . . . . . . . . . . . . . . 78

3.10 DPSK eye diagrams after demodulation; input power is 0 dBm, OSNR=23-

dB, supergaussian optical filter of order 2 and 15 GHz bandwidth. . . . 80

3.11 On the left: differential phase Q in dB vs. m(t) 3-dB bandwidth; abscissa

are normalized to the optimal value derived in the previous section. On

the right: eye diagrams of the differential phase at f3dB = 1/(2πτeff).

The input power is 0 dBm, OSNR 23-dB, and optical filter bandwidth

15 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.12 OOK post-compensation. . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.13 Qeye for OOK signal at SOA input, output and after post-compensation. 84

3.14 Experimental setup for OOK postcompensation. . . . . . . . . . . . . . 85

3.15 Measured waveforms of OOK signal at the SOA input (a) and output

(b). The result of offline processing are also reported: (c) is hl(t), and

(d) is the signal after post-compensation. . . . . . . . . . . . . . . . . . 86

4.1 Structure of the digital coherent receiver. . . . . . . . . . . . . . . . . . 89

4.2 Scheme of one possible implementation of the dual polarization downcon-

verter. It includes two 90 degrees optical hybrids (one per polarization),

and four balanced photodiodes. QWP: quarter wave plate; HM: half

mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3 DSP chain and two examples with measured data: PDM-QPSK on the

left, and PDM-BPSK on the right. . . . . . . . . . . . . . . . . . . . . 94

4.4 FIR filter structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

List of Figures xvii

4.5 The 2×2 butterfly structure of FIR filters used for polarization demulti-

plexing and channel equalization. . . . . . . . . . . . . . . . . . . . . . 98

4.6 Carrier phase estimation and correction. The delay block compensates

for the eventual delay introduced by the filter W [k]. . . . . . . . . . . . 99

5.1 Performance comparison of BPSK and QPSK over NZDSF fiber with 10G

NRZ OOK neighbors. QPSK measurements courtesy of Oriol Bertran-

Pardo et al. [71] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2 Schematic illustration of the different hardware complexity between SC

and DC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3 Experimental configuration of the two transmitters used in the experiment.105

5.4 Experimental setup of the fiber recirculating loop and the receiver. . . . 106

5.5 Correspondence of QBER and bit error ratio. . . . . . . . . . . . . . . . 108

5.6 Measured QBER for DC as function of the subcarrier frequency offset ∆f . 108

5.7 Measured QBER for DC and SC as function of OSNR. A single channel

is measured in back-to-back. . . . . . . . . . . . . . . . . . . . . . . . . 109

5.8 Measured QBER as function of the 3-dB bandwidth of an optical flat-top

filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.9 Measured spectra of (a) dual carrier and (b),(c) single carrier solutions.

Surrounding 10 Gb/s NRZ-OOK channels can also be seen. Resolution

is 100 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.10 Measured QBER penalty after transmission for the single carrier solution. 112

5.11 Measured QBER penalty after transmission for the dual carrier solution. 112

List of Tables

1.1 Short comparison of some important characteristics of SOAs and EDFAs.

Values are typical (not records), and taken from literature. . . . . . . . 21

3.1 Parameters used in the SOA simulation (unless specified otherwise in the

text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Acronyms and Abbreviations

Acronyms

ADC analog to digital converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

AMZI asymmetrical Mach-Zehnder interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

ASE amplified spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

AWG arrayed waveguide grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

AWGN additive white Gaussian noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

BER bit error rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

BPSK binary phase shift keying. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

CD chromatic dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

CH carrier heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

CMA constant modulus algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97

CW continuous wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

DC dual carrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

DCF dispersion compensation fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

DDMZM dual-drive Mach-Zehnder modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

DFB distributed feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

DGD differential group delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

DM dispersion managed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

DSP digital signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

DPSK differentially encoded phase shift keying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

either binary or quaternary

Acronyms and Abbreviations xx

DBPSK differential binary phase shift keying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

DQPSK differential quadrature phase shift keying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

EDFA erbium doped fiber amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

FBG fiber Bragg grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

FEC forward error correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

FIR finite impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

FSR free spectral range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

FWM four wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ISI intersymbol interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

ISO isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

LO local oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

LPF low pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

KLSE Karhunen Loève series expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

MC Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

MGF moment generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

MIMO multiple-input multiple-output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

MPSK m-ary phase shift keying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

MQAM m-ary quadrature amplitude modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

MZM Mach-Zehnder modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

NDA non-data aided. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99

NLPN nonlinear phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

NLT nonlinear threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

NRZ non-return to zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

NSD normalized standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

NZDSF non-zero dispersion shifted fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38

ODE ordinary differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

OOK on-off keying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

OSA optical spectrum analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

OSNR optical signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

PC polarization controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

PDM polarization division multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

PDF probability density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

PDL polarization dependent loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Acronyms and Abbreviations xxi

PLL phase locked loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

PM phase modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

PMD polarization mode dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

PSD power spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

PSK phase shift keying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

p2p peak-to-peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

PRQS pseudo random quaternary sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

QPSK quadrature phase shift keying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

ROADM reconfigurable optical add-drop multiplexer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98

RZ return to zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

SC single carrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103

SGM self gain modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

SHB spectral hole burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

SNR signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

SSMF standard single mode fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

SOA semiconductor optical amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

SOP state of polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

SPM self phase modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

VOA variable optical attenuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

WDM wavelength division multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

WSS wavelength selective switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

XGM cross gain modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

XPM cross phase modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Abbreviations

A.U. : Arbitrary units.

et al. : From latin et alii, means and others or and colleagues.

i.e. : From latin id est, means that is.

e.g. : From latin exempli gratia, means for example.

List of Symbols

a SOA gain coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

ALO Local oscillator electric field magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

As(t) Magnitude of the noiseless signal electric field. . . . . . . . . . . . . . . . . . . . . .32

As,p(t) Amplitude of electric field on polarization p . . . . . . . . . . . . . . . . . . . . . . . .91

Be Front end receiver electrical bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Bo(ω) Frequency response of optical filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

C Multiplicative constant for post-compensation . . . . . . . . . . . . . . . . . . . . . 68

c Speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

cn Values of the taps of a generic FIR filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

D Fiber dispersion coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

e Electron charge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

e1 Error function for the Godard algorithm in the equalizer . . . . . . . . . . . 98

e2 Error function for the Godard algorithm in the equalizer . . . . . . . . . . . 98

Ec Minimum of conduction band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117

ECB Energy in the conduction band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

EV B Energy in the valence band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Ev Maximum of valence band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Efc Quasi-Fermi level for the conduction band . . . . . . . . . . . . . . . . . . . . . . . . 118

Efv Quasi-Fermi level for the valence band. . . . . . . . . . . . . . . . . . . . . . . . . . . .118

Esat SOA saturation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Epump,in Pump signal for FWM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

Eprobe,in Probe signal for FWM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

Econj Conjugate signal produced by FWM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

EAMZI,in(f) Vector of input fields to AMZI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

List of Symbols xxiii

EAMZI,out(f) Vector of output fields by AMZI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

E′

AMZI,in(f) First component of EAMZI,in(f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

E′′

AMZI,in(f) Second component of EAMZI,in(f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

E′

AMZI,out(f) First component of EAMZI,out(f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

E′′

AMZI,out(f) Second component of EAMZI,out(f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

EA(t) Electric field at test point A (see Fig. 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . 91

EB(t) Electric field at test point B (see Fig. 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . 91

EC(t) Electric field at test point C (see Fig. 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . 91

ED(t) Electric field at test point D (see Fig. 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . 91

ESOA,in(t) Electric field input to SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

ESOA,out(t) Electric field output from SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

ELO(t) Local oscillator electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Es(t) Noiseless signal electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

EMZ,in Input field to a Mach Zehnder modulator . . . . . . . . . . . . . . . . . . . . . . . . . . 44

EMZ,out Output field to a Mach Zehnder modulator . . . . . . . . . . . . . . . . . . . . . . . . 44

EO,i(t) Electric field output from the ith SOA section . . . . . . . . . . . . . . . . . . . . . .72

EI,i(t) Electric field input to the ith SOA section. . . . . . . . . . . . . . . . . . . . . . . . . .72

Es(t) Polarization multiplexed signal received by the coherent detector. . .90

Es,p(t) Electric field on polarization p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

EDi,1[k] Input field to the DSP 2×2 butterfly equalizer on pol. 1 . . . . . . . . . . . 97

EDi,2[k] Input field to the DSP 2×2 butterfly equalizer on pol. 2 . . . . . . . . . . . 97

EDo,1[k] Output field from the DSP 2×2 butterfly equalizer on pol. 1 . . . . . . . 97

EDo,2[k] Output field from the DSP 2×2 butterfly equalizer on pol. 2 . . . . . . . 97

f Frequency (typically, normalized to the carrier frequency) . . . . . . . . . . 10

F Amplifier noise figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

fc Occupation probability of an electron with energy ECB . . . . . . . . . . . 118

fI Center frequency of the narrow filter for the I component . . . . . . . . . . 41

fQ Center frequency of the narrow filter for the Q component . . . . . . . . . 41

fs Carrier frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

fv Occupation probability of an electron with energy EV B . . . . . . . . . . . 118

G0 SOA small signal gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

g0 SOA small signal material gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

g(t, z) SOA material gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

G(z, ω) Dispersion all-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Ginv(z, t) Inverse Fourier transform of Ginv(z, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Ginv[k] Digital FIR filter approximating Ginv(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . 96

List of Symbols xxiv

Ginv(z, ω) Inverse filter of G(z, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

~ Reduced Planck constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

h Average of integrated gain h(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

h0 SOA Small signal integrated gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

h11 FIR direct filter between polarizations 1 → 1 . . . . . . . . . . . . . . . . . . . . . . 97

h12 FIR cross filter between polarizations 2 → 1 . . . . . . . . . . . . . . . . . . . . . . . 97

h21 FIR cross filter between polarizations 1 → 2 . . . . . . . . . . . . . . . . . . . . . . . 97

h22 FIR direct filter between polarizations 2 → 2 . . . . . . . . . . . . . . . . . . . . . . 97

h(t) SOA integrated gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

hi(t) Integrated gain of the ith SOA section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

hl(t) Linearized version of h(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

hT (t) Total integrated gain (including intrinsic losses). . . . . . . . . . . . . . . . . . . .72

HAMZI(f) AMZI transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

HC Coupler transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

HD AMZI delay stage transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

I1(t) One photocurrent output from coherent receiver . . . . . . . . . . . . . . . . . . . 92

I2(t) One photocurrent output from coherent receiver . . . . . . . . . . . . . . . . . . . 92

ISOA SOA bias current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

ISOA,0 SOA bias current at transparency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

K Constant defined by Eq. 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

kB Boltzmann constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118~kw Wave vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

kw Amplitude of wave vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

L SOA length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71

M Cardinality of the transmission symbol alphabet . . . . . . . . . . . . . . . . . . . . 5

mc Effective mass of electrons in conduction band . . . . . . . . . . . . . . . . . . . . 117

mh Effective mass of holes in valence band . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

m(t) Impulse response of RC filter with time constant τeff . . . . . . . . . . . . . . 67

m[k] Signal-noise cross term in the M-th power algorithm. . . . . . . . . . . . . .100

N Number of taps of a generic FIR filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

N0 SOA carrier density at transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

n[k] Noise term sample affecting the signal EEQout,1[k] . . . . . . . . . . . . . . . . . 100

Nch Number of input channels to SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Nd Number of taps of the FIR filters 2×2 butterfly equalizer . . . . . . . . . . 97

Nsec Number of sections in the spatially resolved SOA model . . . . . . . . . . . 71

Nt Number of taps of the FIR filter Ginv[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

List of Symbols xxv

NW Number of taps of W [k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

n(t) Noisy component of the signal at the input of an optical receiver . . . 17

np(t) Real part of Gaussian random noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

nq(t) Imaginary part of Gaussian random noise . . . . . . . . . . . . . . . . . . . . . . . . . . 69

N(t, z) SOA carrier density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

p Subscript for polarization (p = 1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Ppeak Gaussian pulse peak power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Psat SOA saturation power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

pT Transition probability between valence and conduction band . . . . . . 118

PSOA(t, z) Power of optical signal inside the SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

PSOA,in(t) Power of optical signal input to the SOA . . . . . . . . . . . . . . . . . . . . . . . . . . 23

PSOA,in Average power input to SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

pSOA,in(t) Normalized power of optical signal input to the SOA. . . . . . . . . . . . . . .23

pSOA,in Average of pSOA,in(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

PSOA,out(t) Power of optical signal output from the SOA . . . . . . . . . . . . . . . . . . . . . . 23

pSOA,out(t) Normalized power of optical signal output from the SOA. . . . . . . . . . .67

pSOA,out Average of pSOA,out(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

pSOA,post(t) Normalized power of optical signal after post compensation . . . . . . . . 83

Q1(t) One photocurrent output from coherent receiver . . . . . . . . . . . . . . . . . . . 92

Q2(t) One photocurrent output from coherent receiver . . . . . . . . . . . . . . . . . . . 92

QBER BER-equivalent Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Qeye Q-factor evaluated by the eye diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Q∆ϕ Differential phase Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

R Symbol rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Rabs Absorption rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

rc Resampling factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

rD Integer downsampling factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Rstim Stimulated emission rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

rU Integer upsampling factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95

r(t) Total signal at the input of an optical receiver . . . . . . . . . . . . . . . . . . . . . 17

s(t) Useful part of the signal at the input of an optical receiver . . . . . . . . . 17

sm(t) Possible signal waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Snp,q(ω) Power spectral density of np(t) and nq(t) . . . . . . . . . . . . . . . . . . . . . . . . . . 70

sin[k] Input signal to a generic FIR filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96

sout[k] Output signal from a generic FIR filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

T Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

List of Symbols xxvi

T0 Gaussian pulse temporal parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Tc Sampling time of the ADC in the coherent receiver . . . . . . . . . . . . . . . . 95

Ts Symbol duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

U(z, t) General electric field propagating in optical fiber . . . . . . . . . . . . . . . . . . . 95

U(z, ω) Fourier transform of U(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Vπ Voltage needed to obtain a phase shift of π . . . . . . . . . . . . . . . . . . . . . . . . . 7

W [k] FIR digital smoothing filter for the phase estimation. . . . . . . . . . . . . .100

α SOA linewidth enhancement factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

αint SOA intrinsic losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

αi,j Alternative definition of linewidth enhancement factor . . . . . . . . . . . . . 27

Γ SOA confinement factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

γb Signal to noise ratio per bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

γ Fiber nonlinear coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

δf Detuning from the carrier frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

δh(t) Zero-mean fluctuations of h(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

δpSOA,out(t) Zero-mean fluctuations of pSOA,out(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

δhl(t) Zero-mean fluctuations of hl(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

δφ Error in phase adjustment of a DQPSK receiver . . . . . . . . . . . . . . . . . . . 48

∆nre(ωi) Change of the real part of the refractive index at frequency ωi . . . . . 27

∆νSOA,out Chirp of optical signal at SOA output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

∆f Frequency detuning of a DQPSK receiver due to δφ . . . . . . . . . . . . . . . .48

∆RB Resolution bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

δhT (t) Zero-mean fluctuations of hT (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

ǫ Mach-Zehnder DC extinction ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

ε1 Unit vector indicating polarization 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

ε2 Unit vector indicating polarization 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

ηFWM FWM efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

κ Coupler splitting ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

λ Wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95

µ1 µ0 Voltage average of logical 1 and 0 on eye diagram. . . . . . . . . . . . . . . . . . 16

µ(t) Function representing the filter effect of SOA . . . . . . . . . . . . . . . . . . . . . . 70

ξ Normalized linear OSNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

ρ(hf) Density of the incoming photons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118

Mode cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

σ1 σ0 Noise std deviation of logical 1 and 0 on eye diagram . . . . . . . . . . . . . . 16

σ∆ϕ,1 σ∆ϕ,0 Noise std deviation of π and 0 on differential eye diagram. . . . . . . . . . 16

List of Symbols xxvii

σ2pq of np(t) (equal variance for nq(t)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

σ2sn,LO Local oscillator shot noise variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

ς Convergence parameter for the Godard algorithm in the equalizer . . 98

τc SOA carrier lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

τeff time constant of the filter m(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ϕ0 Phase rotation in one of AMZI arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

ϕSOA(t, z) Phase of optical signal inside the SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

ϕSOA,out(t) Phase of optical signal output from the SOA. . . . . . . . . . . . . . . . . . . . . . .23

ϕSOA,in(t) Power of optical signal input to the SOA . . . . . . . . . . . . . . . . . . . . . . . . . . 23

ϕSOA,post(t) Phase of optical signal after post-compensation . . . . . . . . . . . . . . . . . . . . 68

φs(t) Phase of the noiseless signal electric field . . . . . . . . . . . . . . . . . . . . . . . . . . 32

φ′

MZ Phase shifts in the first Mach-Zehnder modulator arm . . . . . . . . . . . . . 44

φ′′

MZ Phase shifts in the second Mach-Zehnder modulator arm. . . . . . . . . . .44

φs,p(t) Phase of signal on polarization p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91

φn,p(t) Phase noise of the signal on polarization p . . . . . . . . . . . . . . . . . . . . . . . . . 91

φLO(t) Phase noise of the local oscillator laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91

φp(t) Overall phase to be estimated for the polarization p. . . . . . . . . . . . . . . .92

φnLO,1[k] Effective phase noise samples to be evaluated by CPE . . . . . . . . . . . . 100

φnLO,1[k] Estimation of the phase noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100

Ω Frequency separation between pump and probe for FWM . . . . . . . . . . 28

ωLO Local oscillator carrier angular frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 32

ωs Angular frequency of the noiseless signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

ωIF Intermediate angular frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter 1

Meeting the Needs of a Bandwidth

Hungry World

Optical fiber telecommunications have been a key enabler of the huge increase in

Internet traffic over the last decade. Figure 1.1 shows the total monthly Internet traffic,

both current and predicted. Note that traffic is expressed in exabit (=1018 bits) per

month. Data are taken from a study made by Cisco Sysyems [1], which also predicts the

future traffic until 2013. The most interesting feature of this study is that the amount

of transmitted and received data has been growing exponentially, and this trend is

predicted to continue in the years to come. The increase is mostly driven by bandwidth

hungry applications such as Internet delivery of video, for viewing either on personal

computers, as well as standard and high-definition televisions.

Optical communications is the only technology capable of sustaining such intense

traffic, and phenomenal growth. Bandwidth requirements are growing (and will con-

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 2

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

0

10

20

30

40

50

Agg

regat

ed inte

rnet

tra

ffic

[E

xab

it/m

onth

]

6020

08

2009

2010

2011

2012

2013

0

10

20

30

40

50

60

[Exabit/m

onth

]

Video

Business

Web and mail

Non-IP

Mobile

Peer to peer

Figure 1.1: Aggregated (measured 2001-2009 and predicted 2010-2013) Internet traffic

in Exabit per month. Inset shows the repartition per category since 2008.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 3

1

10

100

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

Cap

acity

of

sing

le o

ptic

al f

iber

[km

Pb/

s]

direct detection

coherent detectiondifferential detection

Figure 1.2: Capacity×distance product of a single fiber in km·petabit/s as a function

of time.

tinue to do so) exponentially; optical bit rates must continue to advance to keep up

with the market demands. Figure 1.2 shows the evolving capacity of a single optical

link over the past 15 years. Those results represent state-of-the-art “hero” experiments

in laboratories around the world. Laboratory demonstrations have capacities that ex-

ceed 100 petabit (=1015 bits) of information per second per km in a single optical fiber.

We group these achievements by the technology used. Capacity increases as we go

from on-off keying (OOK), i.e., intensity modulation with direct detection (circles), to

differential phase modulation with noncoherent detection (triangles), to coherent sys-

tems (squares) using m-ary phase shift keying (MPSK) or m-ary quadrature amplitude

modulation (MQAM). Deployed systems still fall for the most part in the first category.

Note that the vertical axis is in logarithmic scale.

As we can see, increasing throughput in optical communications systems requires

higher modulation formats. In this thesis we will examine the challenges of working with

phase modulation at various stages of adoption. The general objective of the thesis is to

investigate novel configurations for high bit rate, spectrally efficient phase modulated

systems allowing optical communications to keep pace with of our bandwidth-hungry

world.

Only recently have commercial systems based on differential phase modulation sur-

faced. The main reason is the high technological complexity (hence high cost) of more

advanced systems based on PM. First adoption of phase modulation involves the use

of noncoherent detection. The simplest form of higher order modulation is quadra-

ture phase shift keying (QPSK) with two bits per symbol; it doubles spectral efficiency

compared to simple OOK. In chapters 2 and 3, we describe our first contributions,

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 4

proposing techniques to render noncoherent detection of differential quadrature phase

shift keying (DQPSK) (and differential binary phase shift keying (DBPSK)) robust and

even more cost-effective. While now confined to long haul networks, reducing costs can

make phase modulation appropriate for access and metro networks. We investigate two

important sub systems: the receiver (optical to electrical conversion) and the optical

amplifier.

We begin this chapter with an introduction to differential phase shift keying and

conventional noncoherent detectors. While OOK receivers provide channel selection (by

optical filtering) and photodetection, differentially encoded phase shift keying (DPSK)

receivers must also accomplish differential demodulation and phase-to-intensity conver-

sion. We discuss a DPSK receiver whose complexity is comparable to that of standard

intensity modulated OOK receivers, and achieves the additional functionality with es-

sentially no added hardware. Our first contribution, in chapter 2, is the examination

of system performance for the reduced complexity noncoherent detector for differential

QPSK. We probe receiver performance experimentally and via simulation, increasing

our understanding of this receiver in many operating regimes and for practical systems.

In section 1.2 we provide a review of models for SOA gain dynamics, and discuss

how phase shift keyed signals are perturbed by the SOA. Advantages of SOAs over

EDFAs are widely accepted, and include gain over a wide spectrum, compact size,

integrability and, above all, cost-effectiveness [2]. On the other hand, EDFAs generally

have a lower noise figure, higher saturation power, and For OOK, performance is limited

by the phenomena of self gain modulation (SGM) and cross gain modulation (XGM)

in the SOA. Phase modulation alleviates the impact of these effects, but imposes new

challenges, mainly due to two other SOA phenomena, self phase modulation (SPM)-

induced nonlinear phase noise (NLPN). In chapter 3 we present our second contribution,

a mitigation strategy for impairments introduced by the SOA. We propose a simple yet

very effective post compensation method to combat SPM-induced NLPN for DPSK.

We demonstrate via simulation almost complete cancellation of the NLPN added by a

saturated SOA on a differential phase shift keying (PSK) signal. This technique is also

effective to mitigate nonlinear distortions imposed by SOAs on an OOK signal, as we

demonstrate via both simulations and experiment.

As phase modulation is adopted, it will have to coexist with other legacy systems.

This becomes very challenging as we migrate from simple noncoherent detection to

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 5

coherent systems. Coherent systems have the potential to further increase spectral

efficiency. For instance, simply adding multiplexing across two polarization states can

again double the spectral efficiency. However, exploiting polarization diversity cannot

be accomplished cost effectively with noncoherent detection.

The second stage of adoption of phase shift keying will, therefore, see the rollout

of higher-performance, coherent optical systems co-existing with legacy OOK commu-

nications systems in a wavelength division multiplexing environment. At certain baud

rates, coherent detection is extremely sensitive to perturbations due to the presence of

the legacy OOK signals. Our final contribution in Chapter 5 examines two implemen-

tation strategies for introducing 40 Gb/s coherent systems in the presence of legacy

OOK signals. We assess their cost/performance tradeoffs. In order to enhance the flow

of this thesis, we defer a detailed introduction to chapter 4. However, in this chapter

we explain the basics concepts for coherent reception, and how it has evolved since the

birth of the coherent receiver in the 1980s.

In summary, this introductory chapter discusses how phase shift keying can help

meet growing demand for increased throughput. We cover the basics of phase modula-

tion with noncoherent detection. Next we introduce semiconductor optical amplifiers,

focusing on gain dynamics and the impairments introduced to a phase modulated sig-

nal. We give a brief overview of coherent detection of phase shift keying, providing

an historical perspective on renewed interest in coherent optical systems. Finally, we

describe the organization of the remaining chapters of this thesis.

1.1 Why Phase Modulation?

Phase shift keying is a digital modulation format in which information is carried by

the phase of the signal. The complex envelope of the M possible signal waveforms is

sm(t) = exp

j2π[2π

M(m − 1)

]

for m = 1, 2, ...M (1.1)

where fs is the carrier frequency and M is the cardinality of the transmission symbol

alphabet, i.e., log2 M bits per symbol. Fig. 1.3 shows several possible constellations

in the complex plane: binary phase shift keying (M=2), quadrature phase shift keying

(M=4) and octal phase shift keying (M=8). Numbers beside the constellation points

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 6

00

01

10

11

000

001011

010

110

111 101

100

1 0

(a) (b) (c)

Figure 1.3: Signal constellations for three PSK formats: (a) BPSK (b) QPSK (c) 8-PSK.

specify the association between symbols and binary information when using Gray cod-

ing.

The demodulation of PSK signals requires knowledge of the carrier phase to recover

the information bits. If the channel induces phase rotations or distortions, a technique

is needed for estimating the phase of the carrier at reception. This is accomplished by

a synchronous optical carrier that is recovered at the receiver so that absolute phase

information can be correctly extracted at the receiver side without ambiguity. This is

referred to as coherent detection.

Provided that the channel-induced phase rotations are slower than the symbol rate

(i.e., the effects of the channel can be considered constant over at least two symbols),

demodulation can be simplified by use of differential encoding of the information. The

carrier then need not be regenerated, and we can instead use noncoherent detection.

Data is coded in the phase jumps of the signal, rather than the absolute phase. As an

example, in DBPSK a logical “1” is transmitted by shifting the previous phase by π/2,

and a logical “0” is transmitted by a zero phase shift. In DQPSK, the symbols “00”,

“01”, “11”, “10” are transmitted as phase shifts of 0, π/2, π and 3π/2, respectively.

The bit error rate (BER) for optimal coherent detection with perfect carrier recovery

and matched filtering of binary phase shift keying (BPSK) signals in additive white

Gaussian noise (AWGN) channel is [3]

BER =12

erfc (√

γb)

whereas the BER for noncoherent detection of DBPSK in the same conditions is

BER =12

e−γb

where erfc(.) is the complementary error function defined as erfc(x) = 2√π

∫∞x exp(−x2)dx,

and γb is the energy per bit to noise power spectral density ratio. Figure 1.4 presents the

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14-12-11-10-9-8

-7

-6

-5

-4

-3

-2

-1

SNR [dB]

Log

10(

BE

R)

CoherentBPSK

DifferentialDBPSK

Figure 1.4: Optimum BER for coherently demodulated BPSK and noncoherently de-

modulated DBPSK in an AWGN channel.

two BERs. Please note that the bit error rates in Fig. 1.4 refer to ideal matched filter

systems, and are to be considered as the ultimate upper bound in a benign channel. The

optical channel might significantly differ from the AWGN channel, and therefore more

sophisticated tools are needed for the BER calculation. We will forgo discussion of BER

performance of PSK in optical systems until after the description of PSK transmitters

and receivers.

1.1.1 PSK Transmitters

There are two ways to encode binary information in the phase of an optical carrier:

the first and most obvious is to use a phase modulator. The second method, see

schematic in Fig. 1.5(a), employs a Mach-Zehnder modulator (MZM). The ideal MZM

structure has a perfect sinusoidal transfer function, depicted in Fig. 1.6. Note that Vπ

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 8

/2

(b)(a)

OOK

QPSKBPSK

orPG

PG for IPG for Q

Figure 1.5: (a) Schematic of Mach-Zehnder modulator used for generation of OOK and

BPSK signals; (b) nested Mach-Zehnder structure of a IQ modulator for generating

QPSK signals. PG: (electrical) pattern generator.

is the required voltage to induce a π phase shift on one of the arms of the ideal MZM.

The transfer function for the intensity (proportional to a square of a sine) is shown by

the black dashed curve, and the transfer function for the electrical field (proportional

to a sine) is shown by the solid blue line.

Consider how a MZM is used for an OOK transmitter. The electrical signal has

peak-to-peak (p2p) voltage Vπ; the average voltage (i.e., the bias of the MZM) is set

halfway between a maximum and a minimum of the intensity transfer function. For

phase modulation, in contrast, the electrical p2p swing should be 2Vπ, and the bias

should be set to one of the zero crossings of the transfer function.

The main difference between employing an ideal MZM or an ideal phase modulator

is the transition between 0 and π phase values. If the electrical signal was a sequence

of perfectly square bits, there would be no difference. Realistic electrical signals are

bandwidth limited, and thus have rising and falling edges, so a difference exists. When

using an ideal MZM the amplitude of the signal passes through the origin of the complex

plane at every phase transition (see Fig. 1.7). On the other hand, a phase modulator

leads to signals with a constant envelope (the amplitude of the signal is constant); only

the phase rotates. This difference might be overcome with return to zero (RZ) pulse

shaping. The modulators also behave differently in response to non-ideal peak to peak

electrical voltages. When using a MZM, if the input electrical signal swing is smaller

than the required 2Vπ, the output signal will nevertheless have a phase of either 0 or

π. When using a phase modulator, the input signal swing is directly proportional to

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 9

+1

0

-1

MZ input voltage

BPSK p2p

OOK p2p

Tra

nsf

er funct

ion

2V

OOKbias

BPSK bias

V

Figure 1.6: Mach-Zehnder transfer function for intensity (dashed) and field (solid). The

bias points and required peak to peak voltage swings are also indicated, for OOK and

BPSK.

In-phase

Quadrature

+1-1

Phase modulator

MZ modulator

+j

Figure 1.7: DBPSK symbols and effect of phase modulator and MZ modulator.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 10

the phase of the optical signal and a reduction of the ideal voltage swing results in a

reduction of the phase difference between logical “1” and logical “0”.

Fig. 1.5(b) shows the schematic of a QPSK modulator, where two MZM are nested

in a interferometric superstructure, often called an IQ modulator. The two MZM

independently modulate the in-phase and the quadrature components of the optical

signal. One of them is rotated by 90 degrees with respect to the other before being

coupled together. Two binary pattern generators can feed the two QPSK inputs, or the

signal from one generator can be split and one copy delayed to form the two inputs.

1.1.2 DPSK Receivers

We consider first noncoherent reception of DPSK. The typical noncoherent receiver

for OOK is shown in Fig. 1.8(a). The noncoherent differential receiver for DPSK involves

a balanced photodetector and an asymmetrical Mach-Zehnder interferometer (AMZI),

and is shown in Fig. 1.8(b).The optical band-pass filters are inserted prior to photode-

tection and an electrical low-pass filter follows photodetection. Ts is the duration of a

symbol, which in the case of DBPSK is equal to the duration of a bit. The DQPSK

receiver shown in Fig. 1.8(c) basically uses two DBPSK receiver structures, slightly

modified with a ±π/4 phase shift in the lower arms.

The key component in the DPSK receiver is an asymmetrical Mach-Zehnder inter-

ferometer (asymmetric as the signal is delayed in one arm by a symbol time). The signal

interferes either constructively or destructively with a version of itself delayed by one

symbol time. Being an interferometric structure, the DPSK receiver can be represented

in the frequency domain as a pass-band periodic filter. We next give equations for such

a filter and demonstrate its equivalence to the standard receiver.

Let us consider the generic AMZI structure depicted in Fig. 1.9. We seek the AMZI

transfer function HAMZI(f) such that the input and output electrical field to the AMZI

are related by the following equation: EAMZI,out(f) = HAMZI(f) ∗ EAMZI,in(f) where ∗is the matrix multiplication operator. This equation can be written as

E′

AMZI,out

E′′

AMZI,out

=

HAMZI,11 HAMZI,12

HAMZI,21 HAMZI,22

·

E′

AMZI,in

E′′

AMZI,in

(1.2)

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 11

fs

AMZI

fsQ

fs

+/4

Ts

AMZI

-/4

Ts

AMZI

ITs

I(c)

(b)

(a)

I

Figure 1.8: (a) Direct receiver for intensity modulation, (b) balanced receiver for

DBPSK, (c) balanced receiver for DQPSK. Ts is a symbol time. Optical bandpass

filter and electrical lowpass are also shown.

30

HC

TSE'MZ,in

E''MZ,in

HD HCconstructive

destructive

E'MZ,out

E''MZ,out

Figure 1.9: Asymmetric Mach-Zehnder interferometric structure. Ts is a symbol time,

ϕ0 = ±π/4 for DQPSK, and ϕ0 = 0 for binary DPSK

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 12

The AMZI is composed of three stages (coupler, delay stage and a second coupler), so

that the total frequency response matrix can be written as the product

HAMZI(f) = HC · HD(f) · HC (1.3)

where HC is the transfer function matrix of a two by two coupler with splitting ratio

κ

HC =

√κ j

√1 − κ

j√

1 − κ√

κ

(1.4)

and HD(f) is the transfer function matrix of two delay lines with

HD(f) =

e−j2πTSf 0

0 e−jϕ0

. (1.5)

Recall that Ts is the fixed symbol time, and ϕ0 is the desired phase rotation for the

modulation format (ϕ0=0 for DBPSK, ϕ0 = ±π/4 for DQPSK). Considering κ = 0.5

for an ideal 3-dB coupler and typical demodulation where E′′

AMZI,in = 0 , we can re-write

the terms of interest as

HAMZI,11(f, ϕ0) = sin(πTsf − ϕ0/2)e−j(2πTsf+ϕ0+π)/2 (1.6)

HAMZI,21(f, ϕ0) = cos(πTsf − ϕ0/2)e−j(2πTsf+ϕ0−π)/2 (1.7)

By convention, we refer to the upper branch output E′

AMZI,out (affected by HAMZI,11)

as coming from the constructive port, and E′′

AMZI,out (affected by HAMZI,21) as coming

from the destructive port.

For DBPSK, we plot the frequency domain representation of the receiver for the con-

structive and destructive ports in Fig. 1.10 (i.e., |HAMZI,11(f, ϕ0 = 0)|2 and |HAMZI,21(f, ϕ0 =

0)|2, respectively). The abscissa frequency is normalized to the symbol rate R . Note

that each lobe has 3-dB bandwidth of ∼ 0.6R. The equivalent filter is sinusoidal and

periodic. Most of the signal energy will be captured by the first lobe (thick line in

Fig. 1.10), so an optical filter equivalent to the first lobe will yield performance close

to that of the AMZI filter, i.e. DBPSK signals.

This leads to an alternative implementation of the AMZI as an optical bandpass

filter with a 3-dB bandwidth narrower than the signal bit rate. Originally proposed as

a low-cost implementation of optical duo binary coding [4], this concept was soon ap-

plied as a DBPSK demodulator [5]. In [6] the authors demonstrate experimentally that

narrow filtered DBPSK achieves a 1.2 dB advantage against OOK. The improvement

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 13

-1 -0.5 0 0.5 1-40

-30

-20

-10

0

Normalized frequency f/R

Nor

mal

ized

tran

smis

sion

[dB

]

|HAMZI,11|2

|HAMZI,21|2

Figure 1.10: Equivalent periodic filter of DBPSK demodulator at the constructive

(dashed line) and destructive (solid line) ports.

is attributed to the smaller optical bandwidth leading to smaller receiver noise equiva-

lent bandwidth. In [7] the authors demonstrate an array of dispersionless fiber Bragg

gratings (FBGs) that both demodulates and demultiplexes a 16-channel DBPSK trans-

mission. This technique leads to extremely interesting savings in component count:

one single array of FBGs replaces 16 filters, complete AMZI receivers, and balanced

detectors.

The first simulations of the system, to our knowledge, are reported in [8], in which

different filter shapes are compared (among those considered, the Gaussian shape gives

the best performance). This technique is also shown to have stronger tolerance to dis-

persion than OOK. Narrowband optical bandpass filtering can be extended to balanced

reception by exploiting both transmission and reflection of the FBG, as illustrated in [9].

The transmission spectral shape of a FBG is complementary to its reflection, so a single

FBG (and a circulator) approximates both the constructive and destructive arms of the

AMZI, allowing for balanced detection. A disadvantage of this technique, apart from

requiring a circulator which increases the component count, is the additional wideband

optical filter required before the receiver, otherwise the noise equivalent bandwidth of

the destructive port (filter used in transmission) would be infinite.

For DQPSK, the transfer functions for the constructive and destructive ports of the

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 14

in-phase component are

Hcon,I(f) =1√2

HAMZI,11(f, π/4) (1.8)

Hdes,I(f) =1√2

HAMZI,21(f, π/4) (1.9)

For the quadrature component we have

Hcon,Q(f) =j√2

HAMZI,11(f, −π/4) (1.10)

Hdes,Q(f) =j√2

HAMZI,21(f, −π/4) (1.11)

These equivalent transfer functions are plotted in Fig. 1.11 vs. frequency normalized

to the symbol rate R. The frequency responses are periodic and shifted with respect to

the carrier frequency f = 0. Thick lines indicate the main lobes of the destructive ports

in both the I and Q cases. One narrowband filter approximating the main lobes can be

used for each branch in order to demodulate the DQPSK signal. The main lobes should

be centered at +1/8 units of normalized frequency for the in-phase component, and

−1/8 units for the quadrature component. This configuration eliminates the channel

selective optical filter, the two AMZIs, and replaces the balanced photodiode with a

single-ended one. We will explore this proposal more in detail in Chapter 2.

1.1.3 DPSK Performance

This section deals with the most important issue with DPSK, which is its superior

BER performance with respect to the more common OOK. The signal space repre-

sentation of DPSK and OOK in Fig. 1.12 shows greater Euclidian distance between

the constellation points for DPSK compared to OOK for the same average optical

power P . A factor of√

2 in distance corresponds to 3-dB higher optical signal-to-noise

ratio (OSNR) tolerance. Of course this reasoning only applies to a linear AWGN chan-

nel. The optical channel is quite different, first and foremost due to the presence of at

least one important nonlinear device, which is the photodetector. Furthermore, optical

propagation in fiber also is nonlinear, due to the Kerr effect. We must therefore turn

to more sophisticated tools for performance predictions.

In OOK systems, a widely used technique to estimate the BER is by supposing

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 15

-1 -0.5 0 0.5 1-40

-30

-20

-10

0

Normalized frequency f/R

Norm

alize

d t

ransm

issi

on [dB

]

|Hcon,I|2

|Hdes,I|2

(a) In phase

-1 -0.5 0 0.5 1-40

-30

-20

-10

0

Normalized frequency f/R

Norm

alize

d t

ransm

issi

on [dB

]

|Hcon,Q|2

|Hdes,Q|2

(b) Quadrature

Figure 1.11: Four equivalent periodic filters of the DQPSK receiver. Dashed lines for

constructive ports, and solid lines for destructive ports. Thick lines represents the lobes

that the narrow filter receiver approximates, for (a) the in phase component and (b)

the quadrature component.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 16

1 0P 10 2 P

(a) (b)

Figure 1.12: Constellations in the complex plane for signals with the same optical

average power P for (a) DPSK, and (b) OOK.

Gaussian statistics for the noise; the BER is calculated as [10]:

BER(Qeye) = 0.5erfc

(

Qeye√2

)

where erfc denotes the complementary error function, and

Qeye =µ1 + µ0

σ1 + σ0(1.12)

The variables µ1,µ0,σ1 and σ0 represent means and standard deviations of the received

voltage for a logical “1” and “0”, respectively. Noise statistics are not really Gaussian

in optical OOK, but the approximation can be shown to work with sufficient precision,

due to a numerical coincidence [11]. Such relation is extremely helpful when computer

simulations assess performance. The error rate can be evaluated directly from Qeye,

which converges within the first few hundreds or thousands of bits.

BER evaluation in DPSK systems is more complicated. Even supposing the noise

at the input to the receiver is perfectly white and Gaussian, after balanced reception

the statistics are no longer Gaussian due to the nonlinearity inherent in the receiver

structure and square law-detection. The numerical coincidence which occurred for OOK

does not occur for DPSK. Therefore, the relation between BER and Qeye can be shown

to be inaccurate for DPSK [12]. In [13] and [14] two different group suggest a novel

definition of “Q-factor”, called differential phase Q-factor:

Q∆ϕ = π/(σ∆ϕ,1 + σ∆ϕ,0) (1.13)

where σ∆ϕ,1 and σ∆ϕ,0 are the standard deviations of the high and low rails of the differ-

ential eye diagrams, respectively. The BER is then calculated as BER = erfc(Q∆ϕ/√

2).

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 17

OSNR/0.1nm [dB]

8 10 12 14 16 18 20 22-12-11-10-9

-8

-7

-6

-5

-4

-3

-2

Log

10(B

ER

)

OOK

DQPSK

DBPSK

16 dB12.1 dB 13.4 dB

42.7 Gb/s

Figure 1.13: Back to back bit error rates for OOK, DBPSK and DQPSK at a bit rate

of 42.7 Gb/s as function of the OSNR in 0.1 nm. Simulation details are summarized in

the text.

This formulation essentially applies the Gaussian approximation to the optical phase,

instead of applying it to the received photocurrent. This new Q-factor solves qual-

itatively the problem (it catches the nonlinear regime imposed by NLPN), but not

quantitatively: this is due to the fact that phase statistics differ widely from Gaussian.

The BER performance of DPSK can be found via a semi-analytical approach origi-

nally proposed for OOK. A number of different versions of the algorithm exist [15, 16,

17, 18, 19], but they are all based on the following.

1. The received signal is written as r(t) = s(t) + n(t), i.e., the sum of the useful

signal and a noise term due to amplification. Noise n(t) is always a Gaussian,

possibly colored, random process.

2. The signal s(t) is expanded in a suitable basis, e.g., the Karhunen Loève series

expansion (KLSE), which is very practical for taking into account nonideal pre-

and post-reception filters.

3. The moment generating function (MGF) of the decision variable (a quadratic form

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 18

of a sum of possibly correlated Gaussian random variables) is derived analytically.

4. Finally, the probability density function (PDF) is obtained as the inverse Laplace

transform of the MGF. A practical numerical procedure for such an inversion is

the so called saddle-point integration.

A major advantage of this technique is the ability to account for optical and electrical

filters, as well as intersymbol interference (ISI) on the signal waveform.

We present in Fig. 1.13 the result of a simulation of back-to-back transmission at

42.7 Gb/s for DPSK, DQPSK and OOK. We show BER as a function of OSNR. Here

and throughout the document, the OSNR is referred to the standard 0.1 nm bandwidth.

We include realistic optical and electrical filters: an optical supergaussian of order 2

with 3-dB bandwidth equal to 85 GHz (typical for optical communication systems based

on 100 GHz channel spacing), and an electrical 5th order Bessel filter (bandwidth of

30 GHz for the binary formats and 15 GHz for DQPSK). Transmitters and receivers

were ideal, as presented in the previous section. The MZM characteristics were chosen

such that the extinction ratio of an OOK signal would be 13 dB.

As can be seen, DBPSK is the best performing modulation format, with a sensitivity

of 12.1 dB for a BER=10−3. OOK is the format with the poorest sensitivity at 16 dB.

Sensitivity of DQPSK is 13.4 dB, only 1.3 dB worse than DBPSK, but twice as spectrally

efficient. Absolute numbers depend on the filter choice, but in general the sensitivity

penalty of OOK with respect to DBPSK is between 3 and 4 dB. Typically DQPSK

has very low penalty with respect to DBPSK in the high BER region, with a tendency

towards OOK performance in the low BER region. Since we are mostly interested in

BERs around 10−3 (where forward error correction (FEC) codes work), DQPSK is a

very attractive modulation format which combines good performance and a fairly high

spectral efficiency. We will investigate in great detail the performance of DQPSK and

the narrow filter receiver in Chapter 2.

1.2 SOAs and phase modulation

Research on SOAs started soon after the invention of semiconductor lasers in 1962.

However, it was only during the 1980s that SOAs were developed for practical applica-

tions, motivated largely by multiple applications in lightwave systems [20]. SOAs are in

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 19

fact very versatile components thanks to their peculiar properties, which we will quickly

review in this section. A complete treatment of the subject can be found in [21, 22];

our objective is to recall the basics of SOA operation, and to describe the aspects which

are most relevant to the following chapters. In particular, we describe SOAs for PSK

amplification. We will see impairments resulting from amplification. In Chapter 3 our

contribution is a method to mitigate these impairments.

1.2.1 Optical amplification in semiconductor diodes

Fig. 1.14 sketches the (approximate) energy vs. wave vector diagram of a direct

band-gap bulk semiconductor. As opposed to indirect band-gap semiconductors, the

minimum of the conduction band curve and the maximum of the valence band curve

occur at roughly the same wave vector. The three basic optical processes of an ampli-

fier (stimulated emission, spontaneous emission and absorption) occur in this energy

system. For stimulated emission, an incoming photon having the right energy (i.e.,

the right frequency) may cause an electron in the excited state to radiatively decay to

the fundamental level. The electron would emit a photon of the same energy as the

incoming photon, with the same polarization, frequency and phase.

A common approximation of the exact band-structure of direct-gap semiconductors

is the parabolic band model (the relation between the energy and the wave vector is

parabolic, as sketched in Fig. 1.14). As we detail in Appendix A, optical gain can be

achieved only under a bias condition that fills the conduction band while emptying

the valence band (or, equivalently, the filling of the valence band with holes). This is

achieved in SOAs by injecting a current.

In Fig. 1.15 we have sketched a double-heterostructure SOA with bulk active region.

The active region consists of a layer of semiconductor material between two cladding

layers of higher band-gap energy [23]. SOAs are basically laser diodes without the

feedback cavity, i.e. without reflections at the facets. In SOAs the facets are covered

with anti-reflection coatings, whose residual reflectivity is in the order of 0.2%. The

residual reflectivity translates into oscillations of the gain spectrum, usually negligible.

SOA spectrum is typically 60 nm wide; by changing composition and dopants the

central wavelength can be tuned from ∼ 1300 nm to ∼ 1600 nm. SOAs, unlike EDFAs,

are polarization dependent, because of the waveguide structure and the gain material.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 20

Conduction Band (CB)

Holes

Electrons

E

Valence Band (VB)

Ec

Ev

Eg

k

Figure 1.14: Energy vs. wave vector schematic of the energy bands of a direct-gap

semiconductor.

z

0

L

Current Injection

Active Region

Current Spreading

p-typ

e

n-typ

e

Heterojunctions

intrinsic

Electrical Contact

Propag

ation

direc

tion

Figure 1.15: Schematic of a double-heterostructure SOA.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 21

SOA EDFA

Gain >30 dB >40 dB

Wavelength 1280-1650 nm 1530-1570 nm or 1560-1604 nm

Bandwidth 60 nm 30-60 nm

Sat. Power 15 dBm 20 dBm

Polarization <1 dB 0 dB

Noise Figure 8 dB 5 dB

Pump 400 mA 25 dBm

Time Constant ∼0.2 ns 10 ms

Size compact rack module

Integrability yes no

Cost low medium to high

Table 1.1: Short comparison of some important characteristics of SOAs and EDFAs.

Values are typical (not records), and taken from literature.

Nevertheless, engineering the waveguide and the strain of the semiconductors has led

to polarization sensitivity lower than 1 dB in typical SOAs. Table 1.1 is a summary

comparison of various important characteristics of SOAs and EDFAs.

1.2.2 SOA basic equations

A plethora of different models have been presented in the literature with the purpose

of describing semiconductor optical amplifier physics. They can be divided into two

broad categories: those which take into account the spatial dimension of the SOA, and

those which do not. We will refer to the first category as spatially resolved models

[24, 25], and to the second category as reservoir models [26, 27, 28].

The spatially resolved models are more sophisticated and computationally intense.

They solve a collection of partial differential equations as a function of both time and

space. They permit an analysis of the signal amplification as well as the interaction

of the signal with noise along propagation. Most also take into account the amplified

spontaneous emission (ASE) noise build-up inside the amplifier.

Reservoir models average over the spatial dimension, and the system is described

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 22

by a single ordinary differential equation in time only. The computational burden is

lessened in reservoir models while still capturing many important phenomena, such as

SGM, SPM, NLPN, and other nonlinear effects. The interaction between noise and

signal generated inside the amplifier is neglected, a fair assumption if the dominant

noise is ASE accumulated before the amplifier. In general, the choice of model must be

carefully weighed for the application under study.

For our purposes, we work mostly with a reservoir model [26, 27]. We must capture

the most important (inter-band) nonlinear phenomena while maintaining analytical

insight, which will lead us to our post compensation scheme explained in Chapter 3.

Our use of a reservoir model means we lack the spatial description of the SOA along

the propagation direction, and we therefore we neglect intrinsic losses and spontaneous

emission. In Chapter 3 we will compare results with a more computationally-intensive,

spatially-resolved model and see that for our purposes the first model is very accurate.

The reservoir model also does not include intra-band ultra fast phenomena. We will

discuss these phenomena later in this section, and find their time constants are so low

they may be neglected. Their effect would be more important when dealing with, for

example, ultra-short pulses.

We quickly present the basic equations for the reservoir model. The following set of

equations govern amplification inside a SOA

∂PSOA(t, z)∂z

= [g(t, z) − αint] PSOA(t, z) (1.14)

∂ϕSOA(t, z)∂z

= −12

αg(t, z) (1.15)

∂g(t, z)∂t

=g0 − g(t, z)

τc− g(t, z)P (t, z)

Esat(1.16)

where z is propagation direction and t is time. PSOA(t, z) is the optical power inside the

amplifier, g(t, z) is the material gain which can be approximated as a linear function of

the carrier density: g(t, z) = Γa [N(t, z) − N0], where Γ is the confinement factor, a the

gain coefficient, N(t, z) the carrier density and N0 the carrier density at transparency.

αint are the intrinsic losses of the amplifier, ϕSOA is the phase of the optical signal, α is

the linewidth enhancement factor, g0 is the small-signal material gain defined by

g0 = ΓaN0(ISOA/ISOA,0 − 1)

where ISOA is the electrical current and ISOA,0 is the electrical current at transparency.

τc is the carrier lifetime, and Esat is the saturation energy of the amplifier defined as

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 23

Esat = ~ω0/a where ~ω0 is the photon energy, and is the mode cross section.

In general (1.14)-(1.16) must be solved numerically, except the special case where

we neglect the intrinsic losses αint. We therefore take αint = 0 to develop the reservoir

model. Let us define the integrated gain along the z-direction h(t) as

h(t) =∫ L

0g(z, t)dz (1.17)

Since g(z, t) is proportional to the carrier density, the quantity h(t) can be thought

of as a properly normalized reservoir of carriers available for amplification. We can

write the output power and the output phase as a function of input signal and h(t) as

PSOA,out(t) = PSOA,in(t)eh(t) (1.18)

ϕSOA,out(t) = ϕSOA,in(t) + 0.5αh(t) (1.19)

By integrating (1.16) over the length of the amplifier and using (1.14) we have a single

ordinary differential equation (ODE) in h(t)

τcd

dth(t) = h0 − h(t) −

[

eh(t) − 1]

pSOA,in(t) (1.20)

where h0= g0L is the unsaturated integrated gain of the SOA, and pSOA,in(t) is defined

as the input power normalized to the saturation power of the amplifier: pSOA,in(t)=

|ESOA,in(t)|2/Psat. We can solve (1.20) with a numerical algorithm. The output optical

power and phase are calculated from (1.18) and (1.19). The output field can therefore

be expressed as

ESOA,out(t) = ESOA,in(t) exp [h(t)(1 + jα)/2] (1.21)

where ESOA,in is the input field.

1.2.3 Nonlinear Effects in SOAs

SOAs, unlike EDFAs, cannot be considered linear amplifiers. The most important

nonlinear effects are

– Self gain modulation (SGM)

– Self phase modulation (SPM)

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 24

– Cross gain modulation (XGM)

– Cross phase modulation (XPM)

– Four wave mixing (FWM)

In the following we present them one by one, and discuss nonlinear effects arising from

inter-band modulation of the carrier density. We also discuss the introduction of carrier

density modulation due to intra-band processes, such as spectral hole burning (SHB)

and carrier heating (CH), and its implications.

Self gain modulation (SGM)

SGM is the consequence of gain saturation on the power of the signal traveling

through the amplifier. The ODE (1.20) is the link between input power variations and

gain variations; (1.18) leads to gain variations being imprinted on the output power. The

gain variations are nonnegligible only when PSOA,in(t) is large enough for the amplifier

to be out of its linear region (in other words, when the SOA is saturated) and when

the variations of PSOA,in(t) are slower than or comparable to the carrier lifetime τc.

Self phase modulation (SPM)

Input power variations induce gain variations; the gain is proportional to the reser-

voir of carriers available for amplification, or the carrier density. The refractive index

is a function of the carrier density, so it also is modulated by input power variations.

The accumulated phase is a function of the refractive index, hence there is an intensity-

to-phase conversion which is called SPM. This phenomenon is captured in (1.19): any

fluctuation of the reservoir h(t) is imprinted on the output phase. As the underlying

phenomenon is the same, the conditions under which SPM becomes nonnegligible are

the same for SGM. SPM becomes important as the carrier lifetime τc decreases, and

as the the input power Pin(t) approaches the saturation power of the amplifier Psat.

The linewidth enhancement factor α also strongly affects SPM. Typical SOAs have α

between 3 and 8. The amount of SPM is directly proportional to α per (1.19).

For visualizing the effects of SGM and SPM we numerically solve (1.20) with a fourth

order Runge-Kutta method, and we calculate the output with (1.21). The input to the

SOA (as in [27]) is a Gaussian unchirped pulse, ESOA,in(t) =√

Ppeak exp[−0.5(t/T0)2],

whose normalized intensity in the time domain is the green dotted curve in Fig. 1.16.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 25

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t / To

Nor

mal

ized

Inte

nsity

InputG

o = 10 dB

Go = 20 dB

Go = 30 dB

Figure 1.16: Input and amplified pulses by the SOA in the time domain, for different

values of the small signal gain.

−1.5 −1 −0.5 0 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized frequency

Nor

mal

ized

Inte

nsity

inputG

o = 10 dB

Go = 20 dB

Go = 30 dB

Figure 1.17: Input and amplified pulses by the SOA in the frequency domain, for

different values of the small signal gain.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 26

−3 −2 −1 0 1 2 3

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

t / To

To∆ν

Go = 10 dB

Go = 20 dB

Go = 30 dB

Figure 1.18: SPM-induced chirp on the output pulse.

The simulation parameters are the same as in [27], and in particular the saturation

energy of the amplifier has been set to ten times the input pulse energy, Esat = Ein/0.1.

The linewidth enhancement factor is α = 5. We vary the small signal gain of the am-

plifier G0 = exp(g0L) from 10 to 30 dB, and the time-domain results are reported in

Fig. 1.16. The input pulse becomes distorted, and especially the leading edge becomes

sharper, since this part of the pulse experiences unsaturated gain. The amplifier be-

comes saturated as the pulse travels, so that the available gain towards the end of the

pulse is smaller. This behavior is more pronounced for high small-signal gains, as we

have fixed the saturation level.

Figure 1.17 shows what happens to the pulse in the frequency domain. For high

small-signal gains the spectrum becomes multi-peaked, and the main peak is shifted

towards the low frequencies, a phenomenon known as a “red shift” due to the SPM-

induced intensity-to-phase conversion. The pulse actually acquires a chirp ∆νSOA,out

defined as

∆νSOA,out ,1

2π

∂ϕSOA,out

∂t

which we calculate and plot in Fig. 1.18 for the same simulations.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 27

Cross gain modulation (XGM)

The presence of multiple wavelengths entering the SOA complicates the reservoir

dynamics. The reservoir level depends not just on a single channel input power (as

with SGM), but rather on all input powers of all channels. Cross gain modulation

refers tot the gain variations on one channels created by the presence of variations

on other channels. Numerically, XGM is accounted for automatically in the model

previously described if the input electric field is written as the sum of the Nch channel

fields: ESOA,in =∑Nch

k=1 ESOA,in,k(λk). Some analytic works exist on XGM as the key

enabler of wavelength conversion [29, 30, 31]. As with SGM, XGM is stronger when

the SOA is saturated and τc is small.

Cross phase modulation (XPM)

Due to (1.19), variations in gain induced by multiple wavelengths (XGM) will also

induce variations in phase. This effect is known as cross phase modulation, and is

present whenever XGM is present. An extension of the reservoir model [27] to include

cross phase modulation can be found in [32]. Note that (1.19) should be modified to take

into account multiple channels, but the coupling of intensity and phase derives from this

equation. Instead of the classic linewidth enhancement factor α, a new enhancement

factor is defined as

αi,j=

∆nre(ωi)∆nre(ωj)

where ∆nre(ωi) is the change of the real part of the refractive index (which governs

phase rotation) at frequency ωi and ∆nre(ωj) is the change of the imaginary part of the

refractive index (which governs the gain) at frequency ωj. The traditional linewidth

enhancement factor corresponds to α ≡ αi,i, for a signal at ωi. In addition to signal

power and frequency and SOA carrier lifetime, a very important factor for cross phase

modulation (XPM) is αi,j, which plays the same role as α in SPM.

Four wave mixing (FWM)

Four wave mixing is treated (among many other papers) in [33, 34]. To explain its

origin, suppose that the input of the SOA is composed of a strong pump at frequency

ω0/2π and a weaker probe at (ω0 − Ω)/2π, then

ESOA,in = Epump,inejω0t + Eprobe,inej(ω0−Ω)t

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 28

Equivalently

|ESOA,in|2 = |Epump,in|2 + |Eprobe,in|2 + 2ℜEpump,inE∗probe,inejΩt

The gain is influenced by |ESOA,in|2 via the rate equation. The gain will have a

term oscillating at the beat frequency Ω/2π. This oscillation will produce a new field

at (ω0 + Ω), generated by the amplification of the probe by the modulated gain. The

generated field will have opposite phase with respect to the probe, so that its spectrum

will be an inverted copy of the probe, at a different frequency (symmetric with respect

to ω0/2π). If we write the SOA output as

ESOA,out = Epump,outejω0t + Eprobe,oute

j(ω0−Ω)t + Econjej(ω0+Ω)t

we can define the four wave mixing (FWM) efficiency as ηFWM ,|Eprobe,in|2

|Econj|2 . The FWM

efficiency for a typical set of SOA parameters is strongly dependent on the detuning

frequency [34]; the further apart the channels are spaced, the smaller the efficiency.

FWM for dense wavelength division multiplexed systems with SOA amplification leads

to interchannel interference.

Ultrafast Effects

All nonlinearities discussed so far have their origin in carrier density dynamics.

These dynamics arise in part from inter-band processes, which involve the exchange of

electrons from the valence band to the conduction band, and vice versa. The main inter-

band process in amplification is stimulated emission, which provides gain by producing

a copy of an incoming photon.

Inter-band processes are not the only ones determining the carrier dynamics. There

also are intraband processes, which do not involve exchange of electrons between bands,

but rather rearrangements of electrons inside the band. As the electrons need not change

band, the time constants of these processes are much shorter than those of inter-band

processes, hence they are referred to as ultrafast effects. There are two main ultrafast

effect in SOAs: spectral hole burning (SHB) and carrier heating (CH).

In SHB a hole is “burned” in the gain spectrum. A signal at a specific frequency

arrives at the SOA; electrons used in amplification are those closest in frequency to the

incoming photon. A finite time is required for electrons to redistribute in the valence

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 29

band and “fill the hole” burned by the amplified signal. The typical time associated

with SHB is around 120 fs.

Carrier heating is a consequence of the perturbation of the effective temperature of

the carriers. Since the lower energy states are occupied with the highest probability,

carriers used for amplification come mostly from the lower states. Therefore, when a

signal is amplified, the temperature of the reservoir of carriers tends to rise, because the

“coldest” carriers (i.e. the ones in the lower states) just left. A finite time is required

for the carriers to redistribute to again fill the lower states and reach equilibrium. The

typical time associated with CH is 480 fs.

Given the time constants of these effects, they only tend to be important (compared

with inter-band dynamics) when the spectrum of the signal is very wide, as in short pulse

amplification, or wavelength division multiplexing (WDM) amplification with many

channels. Detailed analytical treatment of ultrafast phenomena is given in [35, 36], and

a few models exist to take them into account in numerical simulations: from simpler

models [37] to more complicated models [38, 39]

To visualize the effect of ultrafast phenomena, we have simulate with model de-

scribed in [37]. A short pulse (1 ps full width half maximum) is injected into the SOA;

Fig. 1.19 shows the induced integrated gain h(t) =∫

z g(ξ, t)dξ. In the gain recovery two

regimes can be clearly seen. An ultrafast regime right after gain depletion (in which

intra-band effects are dominant), where the gain starts to recover very fast. In the sec-

ond regime the gain recovery is much slower, because the time constant of this process

is given by the carrier lifetime.

1.2.4 Phase modulation to mitigate SOA nonlinearity

The nonlinear behavior of SOAs can actually be an attractive feature. Due to

this behavior, SOAs are used as building blocks for 2R regenerators [40], wavelength

converters [41], nonlinear media for four wave mixing [42], or intensity noise suppressors

[43]. EDFAs remain the amplifiers of choice when linearity is essential. As we have seen,

when SOAs are used strictly as amplifiers, gain variations induce distortions and impose

performance penalties.

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 30

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10−10

0.8

0.85

0.9

0.95

1

time [s]

Inte

grat

ed g

ain

h(t)

UltrafastRecovery

Figure 1.19: Integrated gain in short-pulse amplification.

In the case of OOK, SGM leads to nonlinear distortions being imprinted on the

waveform [44]. The problem is less important for very low bit rates, i.e., when the bit

time is much smaller than the SOA carrier lifetime TB ≪ τc, or at very high bit rates,

i.e., when TB ≫ τc. For typical SOAs, significant bit pattern distortions are imposed

on signals with a bit rate in the range from 1 to 40 Gb/s.

To overcome this problem, SOAs have been used with DPSK [45], DQPSK [46],

and other constant or quasi-constant envelope modulation formats [47]. When phase-

modulated signals are amplified with SOAs, the limiting nonlinear impairment becomes

nonlinear phase noise (NLPN). NLPN arises whenever intensity noise is translated into

phase noise though SPM. This phenomenon is very well known and widely studied

in optical fiber, and referred to as the Gordon-Mollenauer effect [48]. Fiber NLPN

is due to the Kerr effect, whereas NLPN in SOAs arises from the refractive index

modulation due to carrier density fluctuations [49]. In nonlinear fiber links, the NLPN

bandwidth is limited to the nonlinear diffusion bandwidth, inversely proportional to the

map strength [50]. In SOAs, on the other hand, the NLPN has the same bandwidth as

the gain fluctuations.

NLPN in SOAs was investigated, for example, in [51, 52]. When the input to the

SOA is a constant (or quasi-constant) envelope signal with finite OSNR, its intensity

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 31

suffers from random fluctuations due to noise. If the SOA is saturated, those fluc-

tuations modulate the reservoir of carriers and, therefore, the SOA gain h(t). The

mechanism by which those fluctuations translate into phase noise is given in (1.19); a

SOA with zero linewidth enhancement factor would exhibit no NLPN. The linewidth

enhancement factor is an unavoidable, inherent characteristic of SOAs, deriving from

changes in the real part of the refractive index [53]. In this work we consider static α;

we neglect the complex dynamics of α, as they only become important at very high bit

rates [54].

We will propose in Chapter 3 a method derived from the small signal approximation

of the reservoir model we have presented in this section, which significantly mitigates the

effect of NLPN when PSK signals are amplified by SOAs. SOAs using this mitigation

technique offer viable amplification device for PSK signals, reducing the complexity

of the whole system. The method we propose is general enough to be adapted to

mitigate the intensity distortions induced on OOK signals, thereby extending its range

of usefulness, as will be discussed in Chapter 3.

1.3 Renewed interest in coherent detection

So far, we have covered two important topics, namely PSK modulation in optics,

and the possibility of using SOAs for amplification of those signals. Chapters 2 and 3

will delve into performance of phase shift keying with noncoherent reception. Chapter

4 will give a detailed description of coherent detection of phase modulated signals, and

Chapter 5 will assess performance of a coherent system. In this section we give a brief

introduction to coherent detection. We will discuss the original concepts behind optical

coherent reception, dating back to the 1980s [55, 56]. We will explain the revived

interest in coherent reception in recent years.

1.3.1 Coherent reception

Figure 1.20 sketches the fundamental difference between (a) direct, (b) differential,

and (c) coherent detection. In direct detection the signal is simply photodetected, and

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 32

LO

Signal

(c)

delay

(b)

SignalSignal

(a)

Figure 1.20: Principle of (a) direct detection, (b) differential and (c) coherent receivers.

in differential detection the signal beats against a delayed version of itself. These are

both noncoherent, square law detectors. With coherent detection, the incoming signal

is detected by beating against a laser, which is typically called a local oscillator (LO).

This approach leads to an electrical signal that preserves information on the optical

phase, unlike noncoherent detection (absolute, not differential).

The LO electric field is ELO(t) = ALOej(ωLOt), where ALO is the amplitude of the field

and ωLO is the angular frequency of the optical carrier. The incoming signal (assumed to

be noiseless and co-polarized with the LO) is Es(t) = As(t)ej(ωst+φs(t)) where, similarly,

As(t) is the field amplitude, ωs is the carrier angular frequency, and φs(t) is the signal

phase. The output of the photodetector is

I(t) = κA2s(t) + (1 − κ)A2

LO + 2As(t)ALO

√

κ(1 − κ) cos [ωIFφs(t)] (1.22)

where κ is the coupler splitting ratio (0 < κ < 1), and ωIF = ωs −ωLO is an intermediate

angular frequency. In the simpler case of homodyne reception, the LO is tuned on

the exact same frequency as the signal, i.e. ωIF = 0. Typically, the LO power and

κ are chosen such that the first term in (1.22) is negligible with respect to the others

[κA2s(t) ≪ (1 − κ)A2

LO]. The second term is filtered out by a DC-block. We are therefore

left with only the third term; the LO amplitude acts as a gain for the incoming signal.

The LO power can be chosen strong enough to have the output signal dominate over

the receiver thermal noise. In that case, the limiting noise source would be the shot

noise coming from the LO. The shot noise variance is σ2sn,LO = 2e(1 − κ)A2

LOBe, where

e is the electron charge, and Be the bandwidth of the receiver electronics. In the case

where σ2sn,LO is the dominant noise term, optimal receiver sensitivity can be achieved

[57]. It can be shown that achievable sensitivity for homodyne detection with BPSK

transmission can be as small as 9 photons/bit for BER=10−9[58]. Impressive sensitivity

records were achieved in laboratory experiments in the early 90s, such as 20 photons/bit

at 565 Mbit/s [59], 46 photons/bit at 1 Gb/s [60] and 83 photons/bit at 4 Gb/s [61].

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 33

In addition to impressive sensitivity, coherent detection also yields information on both

the amplitude and the phase of the electric field per (1.22). Therefore, information can

be coded in any combination of amplitude and phase, and the receiver is suitable for

complex modulation formats.

There are a number of technical challenges with the practical implementation of this

receiver. The first and foremost is dictated by the homodyne condition ωIF = 0, which

requires phase locking between the LO and the incoming signal requiring a phase locked

loop (PLL). Equivalently, in the case of heterodyne receivers, the phase of the LO must

be locked to that of the signal plus an intermediate frequency difference. Additionally,

the state of polarization (SOP) of the incoming light must match that of the LO. As

fiber propagation imposes random fluctuations of the signal SOP, a tracking mechanism

is required at the receiver. For higher order modulation formats, a number of additional

operations of varying complexity levels are required to recover the original information.

1.3.2 Flagging interest in 1990s

The elegance and complexity of shot-noise limited coherent systems were defeated

in the marketplace by the simplicity, cost-effectiveness and performance of preamplified

direct detection. In particular, the invention of EDFAs in 1987 [62, 63] introduced af-

fordable multi-wavelength optical amplification, paving the way for optically amplified

intensity modulated direct detection WDM systems. This made the sensitivity advan-

tage less crucial, since reasonably good performance became possible with preamplified

ASE-limited receivers. The ideal sensitivity limit is 38 photons/bit for OOK, and 20

photons/bit with balanced DBPSK [64, 11, 58, 65] (to be compared with the aforemen-

tioned 9 photons/bit of homodyne BPSK). Experiments have demonstrated, for OOK,

a sensitivity as low as 43 photons/bit at 5 Gb/s [66] and 52 photons/bit at 10 Gb/s [67].

For DBPSK, record sensitivities are 30 photons/bit at 10 Gb/s [68], and 45 photons/bit

at 42.7 Gb/s [69].

Another often cited advantage of the coherent receiver is its inherent frequency

selectivity. This refers to the fact that the beating between the LO and the signal

actually acts as a channel selector, because any signal whose carrier frequency is far

from the one of the LO would not fall in the bandwidth of the receiver. As optical filter

technology developed and matured [70], all-optical signal filtering was affordable and

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 34

this motivation for coherent reception was diminished.

Due to these market realities, research on the coherent receiver was put aside for al-

most two decades, until very recently. We will next explain renewed interest in coherent

reception, and how it differs significantly from the originally proposed versions.

1.3.3 Coherent reception in 2010s

The terahertz bandwidth of fiber once seemed an immense territory never to be

depleted. Commercial systems with multiple wavelength operation, 40 Gb/s electronics,

and forward error correction have begun to probe the limits of that bandwidth. Today,

we have the first harbingers of our encroaching on the capacity of the optical fiber

within the next 10 years. This evolution of optical communications unfolded using

simple OOK. Continued expansion of fiber information carrying capacity requires we

migrate away from on-off keying into the use of higher modulation formats, and coherent

detection. The combination of highly spectrally efficient modulation and increasingly

aggressive forward error correction will allow us to approach the capacity of fiber.

In the 1980s, the expense of electrical and optical components for coherent detection

could not be justified. Coherent detection need no longer be so expensive, as we can

exploit digital signal processing (DSP) to a much greater extent. An important bottle-

neck in the adoption of coherent optical communications has been the speed of analog

to digital converter (ADC); success depends on getting the optical signals into the DSP

engines. The state-of-the-art for analog-to-digital conversion for telecom equipment

rests today in the vicinity of a 16 GHz radio frequency (RF) frontend, and 56 Gsamples

per second (GS/s). This has been demonstrated to be enough for 100 Gb/s transmission

over a 50 GHz grid at a symbol rate of 28 Gbaud [71].

Commercialization of coherent detection has been announced by Alcatel-Lucent in

June 2010, and is targeting the use of polarization division multiplexing (PDM) phase

modulation involving four parallel channels: two polarization states, and in-phase and

quadrature states in the electrical field. In order to divide the single incoming, mul-

tiplexed optical signal into four separate signals, we use an optical hybrid. A local

oscillator introduced at the hybrid interferes with the incoming signal to enable chan-

nel separation. Each channel is converted to electrical format and processed by separate

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 35

ADCs.

A significant paradigm change with coherent detection (with respect to the tradi-

tional coherent detection we have illustrated in Sec. 1.4.1) is the heavy reliance on

digital signal processing. The recovery of polarization state information is impractical

without DSP, as is frequency and phase recovery. The presence of DSP also enables

functionalities that are currently performed optically. To some extent these functions

come with little or no cost; the added DSP complexity is slight. Chromatic disper-

sion is easily accomplished today in the optical domain using dispersion compensation

fiber (DCF). When using coherent detection a simple digital filter could accomplish

the same function, without the disadvantages of optical dispersion compensation (power

losses and nonlinearity due to DCF). The cost of the DCF is eliminated, and without

dispersion compensation modules the span loss is decreased (enabling longer spans or

higher amplifier noise figure for the same span length).

Digital signal processing also provides for reconfigurable receivers. Channels can

be equalized in an adaptive manner, vastly increasing the information bearing capacity

of the link. Two states of polarization are input to the optical fiber; they are mixed

together during propagation. Separating these polarization states at reception is diffi-

cult to accomplish in the optical domain and was a major impediment to adoption of

coherent detection. DSP, however, can separate, or demultiplex, the two polarization

states. For instance, a 2x2 adaptive channel equalizer (digital filters arranged in a 2x2

butterfly configuration), can invert the effects of the fiber channel. Significantly, the

filter can equalize not only polarization rotation, but also any other linear impairment

such as polarization mode dispersion, filter-induced distortions, etc.

The received signal and the local oscillator (free running laser) will necessarily have

frequency offset that must be estimated and compensated. Carrier phase must also

be recovered. Many digital algorithms exist for frequency and phase estimation, and

their complexity/benefit ratio varies by application. All lasers have phase noise that

eventually limits the performance of coherent detection.

In Chapter 4 we will provide a more in-depth discussion of the digital coherent

receiver, in preparation of our discussion of results in Chapter 5. We will investigate

coherent solutions for terrestrial core networks. For channels at 100 Gb/s, the unan-

imous solution has been identified by the research community in PDM-QPSK paired

Chapter 1. Meeting the Needs of a Bandwidth Hungry World 36

with digital coherent reception. For channels at 40 Gb/s, the scientific debate is still

open.

It is known that BPSK is more resilient to XPM than QPSK [72]. It is therefore more

suited for the upgrade of legacy systems, where the phase modulated channels must

share the line with the old 10 Gb/s intensity modulated ones. Our final contribution

in Chapter 5 is the comparison of two technological solutions for the upgrade of legacy

terrestrial core networks to 40 Gb/s based on PDM-BPSK. We will show that despite

having the same sensitivity to additive white Gaussian noise, their different tolerance

to optical nonlinear effects (especially XPM) makes one of them much more suitable

than the other for the upgrade of legacy terrestrial systems to 40 Gb/s.

1.4 Thesis organization

This document is organized as follows. In Chapter 2, we present the results of

our investigation of the narrow filter receiver for DQPSK. We test it throughly via

simulations and experiment against linear and nonlinear impairments, and we show

advantages and disadvantages with respect to the conventional AMZI-based balanced

receiver.

In Chapter 3, we deal with SOA-based amplification of DPSK-based systems. We

derive a rather simple but very effective post compensation scheme that can be used

to overcome limitations induced by NLPN on the amplification of DPSK signals. We

extend the idea also to intensity modulation, and we give an experimental proof of

concept.

In Chapter 4, we detail the structure and algorithms of the digital coherent receiver

that we used for the results in the following chapter. In Chapter 5, we show the results

of an experimental comparison between two promising solutions based on coherent

receiver for the upgrade of legacy terrestrial systems to higher bit rates.

Chapter 2

DQPSK: When is a Narrow Filter

Receiver Good Enough?

In section 1.2.2 we discussed two noncoherent receivers for DQPSK, the conven-

tional AMZI structure and a reduced complexity narrowband filter receiver. In this

chapter, we describe our first contribution: an investigation, both experimental and

via simulation of the pros and cons of a narrow filter receiver for DQPSK. We quan-

tify the performance differences between the two receivers, allowing system designers

and operators to determine when the less complex narrow filter receiver might be the

appropriate choice.

We numerically optimize the 3-dB bandwidth and center frequency of the narrow

filter, and show it is more tolerant to carrier frequency detuning than the conventional

solution. We show that the narrow filter receiver is also more tolerant to chromatic

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 38

dispersion (CD) than the conventional one, and equally tolerant to first order polar-

ization mode dispersion (PMD). We show the impact of the 3-dB bandwidth on the

receiver performance when chromatic dispersion accumulates. We show via experiments

and simulations that the 3-dB advantage of the conventional receiver almost vanishes

when the nonlinear impairments are fiber nonlinearities; comparing the two receivers

at the optimum launch power for a 25×80 km dispersion-managed non-zero dispersion

shifted fiber (NZDSF) system, the difference in OSNR margin is reduced to ∼1.6 dB.

Experiments are conducted at 42 Gb/s using a commercially available narrow filter

for reception. We extend the narrow filter receiver to the multi channel scenario by

experimentally demonstrating simultaneous demodulation of WDM DQPSK signals

with a single device. We discuss the possible savings in components count that our

solution could generate.

2.1 Introduction

We have seen in the introduction how phase modulation has drawn much attention

in the last few years for next-generation, spectrally-efficient optical networks [73]. In

particular optical DQPSK is emerging as a promising solution, and the technology is

today mature enough to permit validations outside the research labs [74]. The con-

ventional receiver for DQPSK, depicted in Fig. 1.8 and for convenience also in 2.1(a),

is composed of an optical filter, two AMZIs, and two balanced photodiodes. Its com-

plexity is twice the complexity of a DBPSK receiver. One issue with this receiver is

that the AMZI, being an interferometric structure, needs a very careful control of the

relative delay between its two outputs. This is typically achieved at the expense of a

feedback electric circuit, and temperature stabilization, which contribute to increase the

cost and complexity of the solution. In order to increase the cost/benefit ratio of phase

modulated formats and to eventually avoid the AMZI, much effort has been focused in

proposing alternative, lower complexity receivers for both binary and quadrature phase

shift keying.

One such reduced complexity receiver [75, 76] is based on polarization, where the

two arms of the AMZI are replaced by the slow and the fast axis of a polarization

maintaining fiber, whose differential group delay (DGD) equals one symbol interval;

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 39

the signals out of the two axes are then mixed before photodetection. This receiver is

extended to DQPSK in [77], demonstrating experimentally the main advantage of such

a receiver, which is the relative ease of implementation and the wide range over which

the DGD can be tuned, hence permitting the use of the same receiver for different bit

rates. The main drawback is that the signal polarization needs to be controlled with

very high precision and stability, thus increasing cost and complexity.

We turn our attention to two other promising receivers that use a single photode-

tector for DBPSK; in the case of DQPSK, these receivers have one photodetector in

each of the I and Q arms. In the first receiver, the AMZI structure is maintained as in

Fig. 2.1(a), however only one port (either the constructive or the destructive port) is

populated with a photo detector [78]. In the second receiver, see section 1.2.2, the AMZI

structure is not used, and instead only a narrowband filter is found in each branch, as

illustrated in Fig. 2.1(b) [5]. The narrowband filters replace the channel select filter

and the AMZI structure.

Researchers have examined the DBPSK version (single branch) of the narrow filter

receiver in Fig. 2.1(b) [6, 7, 8, 9, 79], with only very recent results for the DQPSK

version [80]. Since the narrow filter receiver (without AMZI) approximates the single-

ended AMZI with only one photodiode, we expect them to show similar performance.

We confine our attention thus to the narrow filter receiver and compare it to the con-

ventional AMZI balanced receiver of Fig. 2.1(a).

We demonstrate via BER measurements and simulations that the narrow filter re-

ceiver has a 3-dB penalty in the linear, dispersion-free regime, compared with the

conventional AMZI balanced receiver. By avoiding the interferometric structure and

using a single-ended photodiode instead of a balanced one, the narrow filter receiver

enjoys reduced cost and complexity, as well as greater flexibility. We will show that the

tolerance to non-optimal tuning of its center frequency is greater than the tolerance of

the conventional receiver to frequency detuning. We compare quantitatively the per-

formance of the narrow filter to that of the conventional receiver. In this chapter we

present a complete analysis of advantages and drawbacks of the narrow filter receiver,

assessing its performance in simulations and experiments vis-à-vis linear and nonlinear

transmission impairments.

The narrow filter receiver is by construction spectrally efficient; channels in a WDM

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 40

system can be packed more closely than when using the wider band, conventional AMZI

receiver of Fig. 2.1. Typically when packing channels more tightly to improve spectral

efficiency, we expect the narrower channel select filters to induce greater ISI. When

using an AMZI receiver, we can tweak the free spectral range (FSR) such that the overall

frequency response 1 tends to equalize the line, to combat ISI. The spectral efficiency of

a narrow filter WDM system can be further increased using the same techniques used

with the AMZI receiver. The narrow filter receiver is simply chosen to match the main

lobe of the AMZI destructive port when using the most advantageous FSR. Note that

for both binary and quadrature DPSK, narrowing the channel selective filter makes the

conventional AMZI receiver more resilient to accumulated chromatic dispersion, as well

as more spectrally efficient [81, 82, 83, 84, 85]. It is not surprising then that we find the

narrow filter receiver exhibits the same greater tolerance to intersymbol interference as

the conventional AMZI with partial D(Q)PSK.

For DBPSK it has already been reported that, in the nonlinear regime, the perfor-

mance of the conventional receiver tends to that of the single-ended receiver (where the

balanced receiver is replaced with a single ended photodiode on one of the two output

arms of the AMZI). The advantage of balanced reception disappears as the dominant

noise is nonlinear phase noise, and not amplitude noise [86] [87]. Since the narrow filter

receiver emulates the single-ended one, it is expected that under nonlinear propagation

in fiber its performance should be equivalent to those of the conventional receiver. We

will show that this is the case.

In the rest of the chapter, Section 2.2 is devoted to the comparative performance

analysis, both experimental and numerical, of the two receivers in back-to-back oper-

ation, and in the presence of chromatic dispersion and first order polarization mode

dispersion. In Section 2.3 we present measurements and simulations of the behavior of

the two receivers in the nonlinear regime. Finally, we wrap up the chapter in Section

2.4.

1. The overall frequency response is determined by both the channel selection optical filter and the

AMZI structure, as well as the photo-detection process.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 41

fI

(b)

AMZI

fsQ

+/4

Ts

AMZI

-/4

Ts

(a)

I

fQ

Q

I

Figure 2.1: (a) Conventional receiver and (b) Narrow filter receiver for DQPSK signals.

Ts is a symbol time, fs is the carrier frequency, fI = fs + δf and fQ = fs − δf

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 42

Tunable laser

IQ

VOAPolar. mode disp.

Chromatic disp.or

PMD EmulatorPC

PC

MZM

/2

D A T A

D A T A21Gbps

Bias control

MZM

SHF 46213

SHF 47210

Conventional receiver

Narrow-filter receiver

OSA

90

10

1x32 de

mux

CR and ED

OBPFBias control

Preampor

/4

Ts

Figure 2.2: Experimental setup for the investigation of chromatic dispersion and first

order PMD tolerance. PC: polarization controller, MZM: Mach-Zehnder modulator,

PMD: polarization mode dispersion, VOA: variable optical attenuator, EDFA: erbium

doped fiber amplifier, OSA: optical spectrum analyzer, AWG: arrayed waveguide grat-

ing, OBPF: optical bandpass filter, CR: clock recovery, ED: error detector.

2.2 Linear Impairments

2.2.1 Experimental Setup

The experimental setup for the investigation of linear impairments is sketched in

Fig. 2.2. The experiments in sections 2.2 and 2.3 are limited to a single channel system,

while multichannel systems are investigated in section 2.4. These experiments were

performed in late 2008 at the advanced photonic systems lab at Queen’s University,

which is a shared test facility available to all Canadian academic researchers. There,

we could access test equipment which was not available in our laboratories at the time,

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 43

in particular an I-Q modulator and a BER tester up to 42 Gb/s.

A polarization controlled tunable laser is modulated with a commercial I-Q modula-

tor (SHF 46213). The modulator is composed of two MZMs nested in an interferometric

structure [88]. We use the data and data bar (logical inverse) outputs of a R=21 Gb/s

pattern generator for our I and Q components. Two challenges are imposed. First, we

must delay the two data outputs to decorrelate them so that the overall data rate is

42 Gb/s (that is, to achieve two independent data streams). However, the delay must

be a multiple of the bit duration in order to successfully demodulate the streams (our

phase-matched RF cables introduced a relative delay of precisely 21 information bits).

The π/2 phase shift is manually adjusted and constantly monitored.

Two transmission media are considered. To investigate the effects of chromatic

dispersion, the signal enters a spool of optical fiber. To investigate the effects of PMD,

the signal passes through a first-order PMD emulator. The emulator consists of a

polarization beam splitter (PBS) to split the signal into a slow and a fast component.

The slow component is delayed by an optical delay line, and recombined with the fast

component by means of a second PBS. A polarization controller in front of the PMD

emulator is set to assure that the signal is aligned at 45 with the first PBS in the PMD

emulator.

At the receiver side, the signal power is controlled with a variable optical attenuator

(VOA). A preamplifier stage (composed of an EDFA, a 1.4 nm optical bandpass filter,

and a second EDFA) sets the OSNR. The spectrum of the noisy signal is constantly

monitored with an optical spectrum analyzer (OSA) via the 10/90 power splitter. The

signal can be received by either of the two available receivers, narrow filter or conven-

tional.

The narrow filter receiver is implemented with one channel of a commercially avail-

able 1×32 demultiplexer; the measured demux spectrum is shown in Fig. 2.3 along with

the theoretical AMZI spectrum and a Gaussian spectrum with 3-dB bandwidth equal

to 0.6 R. As we can see, our narrow filter closely approximates the Gaussian shape.

The photodiode incorporates a transimpedance amplifier.

The conventional receiver is formed by a channel select filter followed by a commer-

cial optical DPSK demodulator. The channel is selected by an optical bandpass filter

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 44

with a 3-dB bandwidth of ∼ 50 GHz. The balanced photoreceiver also incorporates a

transimpedance amplifier.

Following photodetection, the clock is reconstructed with a commercial clock recov-

ery circuit and fed to a bit error rate tester together with the received data. Note that

no differential encoder (at the transmitter) or differential decoder (at the receiver) is

used, so the received pattern is different from the transmitted pattern. For experimen-

tal convenience, we chose a 27 DeBrujin sequence at the transmitter, leading to the

following received sequences [88]

uk = IkIk−1Qk−1 + QkIk−1Qk−1 + IkIk−1Qk−1 + QkIk−1Qk−1 (2.1)

vk = QkIk−1Qk−1 + IkIk−1Qk−1 + QkIk−1Qk−1 + IkIk−1Qk−1 (2.2)

where u is the 21 Gbps data stream on the in-phase branch of the receiver and v is the

data stream on the quadrature branch. These resulting patterns were programmed into

the bit error rate tester. The two streams were accessed sequentially using, in the case

of the conventional receiver, a commercial (SHF model 47210) DBPSK demodulator

or, in the case of the narrow filter receiver, by manually tuning the central frequency

of the demultiplexer.

2.2.2 Numerical Model

In addition to the experimental investigation of the performance of the narrow

filter and the conventional receivers, we also examined their performance via numerical

simulation. We model the optical source as an ideal laser. The Mach-Zehnder modulator

is modeled as described in [89, 90], where the output field of a single MZM is

EMZ,out =EMZ,in

2

(

ejφ′

MZ +(√

ǫ − 1)(√

ǫ + 1)ejφ

′′

MZ

)

(2.3)

where EMZ,in is the input field to the MZM, φ′

MZ and φ′′

MZ are the phase shifts in the

two Mach-Zehnder arms, and ǫ is the MZM DC extinction ratio. The pattern generator

is modeled as an ideal signal source filtered with a fifth order lowpass Bessel filter. The

3-dB bandwidth of this filter and the extinction ratio of the MZM are chosen to best fit

experimental results. The measured frequency responses are used for the optical filters

in the simulation. Photodetection is supposed ideal, and another fifth order lowpass

Bessel filter models the finite electric bandwidth of the receiver (3-dB bandwidth is

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 45

-50 0 50-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Normalized frequency [GHz]

Nor

mal

ized

tra

nsm

issi

on [dB

]

Theor. conventionalMeasuredTheor. Gauss bw=0.6R

-50 0 50-70

-60

-50

-40

-30

-20

-10

0

10

20

Normalized frequency [GHz]

Gro

up d

elay

[ps]

Theoretical

Measured

(a) Intensity profile

(b) Group delay profile

Figure 2.3: Measured transmission and group delay profiles of the narrow filter em-

ployed in the experiments. The theoretical shapes are also reported, along with the

transmission spectrum of a Gaussian filter with 3-dB bandwidth equal to 0.6R, where

R is the symbol rate.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 46

-1 -0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

Normalized time

Am

plitu

de [A

U]

20 ps

(a) DQPSK optical waveform

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Normalized time

Am

plitu

de [A

U]

20 ps

(b) Conventional receiver

-1 -0.5 0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

Normalized time

Am

plitu

de [A

U]

20 ps

(c) Narrow filter receiver

Figure 2.4: Measured (left column) and simulated (right column) eye diagrams for the

three tested receivers.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 47

again chosen to best fit the experimental back-to-back results). Figure 2.4 shows the

measured and simulated eye diagrams of the optical DQPSK signal and of the electrical

signals output from the two receivers.

The bit patterns in the simulations are pseudo random quaternary sequences (PRQSs)

[91] of length 48. The BER is simulated with the semi-analytical approach accounting

for the memory of the system as well as optical and electrical non-ideal filter shapes

that we described in the introduction (Section 1.1.3). Details of the algorithm we used

can be found in [15].

2.2.3 Back-to-back operation

Figure 2.5 shows the back-to-back bit error rate measurements, along with simu-

lation results. Simulated performance very closely matches measured performance; in

particular, a penalty of ∼ 3.2 dB for the narrow filter is correctly predicted by our sim-

ulator. Such a penalty is in agreement with both the simulation results in [80] and the

intuition that the narrow filter emulates the single-ended DQPSK receiver. The novelty

in our work is threefold: 1) we give experimental validation of bit error rate (not only

eye diagrams); 2) we provide simulation results which are both more accurate (using

realistic filter models), and more precise as we investigate also the low BER regime,

including good agreement with the experimental data; and 3) we use a realistic network

element (demultiplexer) as an optical filter, instead of the monochromator stage of an

OSA as in [80].

We next numerically investigate performance dependence on filter center frequency

and 3-dB bandwidth. We vary both parameters over a wide range of values and measure

the OSNR penalty at BER 10−5. Contour plots in Fig. 2.6(a) give results for the

quadrature component (the center frequency is negative) and Fig. 2.6(b) the in-phase

component (the center frequency is positive). Frequencies in the contour plots are

normalized to the bit rate; stars indicate the minimal penalty (optimal parameters).

As in [80], the optimal center frequency does not correspond to R/8, but to a slightly

higher value. We find the optimal center frequency to be close to 0.18R, for a 3-dB

bandwidth of 0.6R. In [80] an optimal center frequency of 0.173R was reported. The

3-dB bandwidth was not swept in [80], which might explain the small discrepancy. The

bowl around the minimal OSNR penalty is quite shallow for the narrow filter receiver:

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 48

14 16 18 20 22 24 26-10

-9

-8

-7

-6

-5

-4

-3

OSNR [dB/0.1nm]

Log

(BE

R)

Conventional (sim)

Conventional (meas)

Narrow filter (sim)

Narrow filter (meas)

Figure 2.5: Experimental and simulated back to back bit error rate for both receivers

under investigation. Symbol rate is 21 Gbaud, bit rate is 42 Gb/s.

the OSNR penalty is confined to less than 1 dB (2 dB) for a range of more than 0.15R

(0.25R) around the optimal value. At 21 Gbps, this corresponds to a 1 dB tolerance of

±1.57 GHz (2 dB tolerance of ±2.62 GHz).

A major concern in direct detection DQPSK systems is the robustness to frequency

offset (or detuning) between the carrier and the receiver [92, 93, 94]. To explore toler-

ance to detuning for both receivers, we introduce δφ, the error in phase adjustment of

the receiver. For instance, the phase shift of the in-phase arm would be π/4 + δφ. Such

an error induces a frequency detuning of ∆f = δφR/(4π). In the case of the narrow

filter receiver, the detuning is simply a mismatch between the filter center frequency

and the carrier frequency. We investigate numerically the tolerance of the receivers to

this effect. Figure 2.7 shows the results in terms of OSNR penalties at BER=10−5.

The penalties are relative to the performance of each receiver when ∆f = 0. We can

see that the narrow filter receiver is more tolerant (by slightly more than a factor 2) to

frequency detuning.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 49

1Center frequency

(normalized to symbol rate)

3 dB

ban

dw

idth

(nor

mal

ized

to

sym

bol

rat

e)

0.1 0.15 0.2

0.4

0.5

0.6

0.7

0.8

0.9

1

Center frequency (normalized to symbol rate)

3 dB

ban

dw

idth

(nor

mal

ized

to

sym

bol

rat

e)

-0.2 -0.15 -0.1

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.24 0.24

(a) (b)

[dB]

0

1

2

3

4

5

6

7

8

9

2

2

2

2

2

3

3

3

3

3

4

4

4

4

4

5

5

5

5

5

6

6

6

1

1

1

1

7

7

7

6

6

8

8

89

9

9

7

7

8 9

2

2

2

2

2

3

3

3

3

3

4

4

4

4

4

5

5

5

5

5

6

6

6

1

1

1

1

77

7

6

6

88

8

9 9

9

7

78

9

Figure 2.6: OSNR penalty at BER=10−5 for the narrow filter receiver as a function of

filter 3-dB bandwidth and filter center frequency for (a) the quadrature component and

(b) the in-phase component. Stars indicate minimal penalty.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 50

-0.02 -0.01 0 0.01 0.020

0.5

1

1.5

2

2.5

3

Frequency detuning ûf/R

OSN

R p

enal

ty for

BE

R =

10-

5 [d

B]

Conventional

Narrow filter

Figure 2.7: OSNR penalty for a BER=10−5 for the narrow filter receiver and for the

conventional receiver as a function of the detuning frequency.

In the next subsections we examine the impact of linear fiber impairments (chromatic

dispersion and polarization mode dispersion) via experiment and simulation.

2.2.4 Chromatic dispersion

Simulations of BER-equivalent Q-factor for DQPSK [80] found the narrow filter

receiver has a higher tolerance to accumulated CD by a factor of ∼2. Two definitions

of Q-factor can be found in the literature. One is based on the mean and variance of

logical ones and zeros. We refer to this as Qeye, and its definition is given in (1.1.3). A

Gaussian approximation estimates the BER from Qeye as in (1.1.3). A second definition

is to take a BER measurement and numerically find the corresponding signal-to-noise

ratio (SNR) that in an AWGN channel would have yielded the measured BER. This is

the BER-equivalent Q-factor which in units of dB is given by

QBER = 20 log10

[√2inverfc (2BERmeas)

]

(2.4)

where inverfc is the inverse error function and BERmeas the measured BER.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 51

We extend that work by simulating OSNR penalties. Two experimental measure-

ments were captured and found to coincide with simulation results. We also numerically

investigate the influence of the filter 3-dB bandwidth on the system performance in the

presence of chromatic dispersion. While OSNR penalties are more difficult to calculate 2

they provide a very useful performance metric for the network engineer. Experimen-

tally, we use one of two spools of single mode fiber (136 and 216 ps/nm of accumulated

dispersion respectively). Input power to the fiber is -3-dBm, ensuring linear propaga-

tion. Simulations are also carried out in this linear regime; the fiber is modeled as a

simple all pass filter. We use as a reference point the OSNR required to achieve BER

= 10−5 when no dispersion is present (back-to-back). We report the OSNR penalty at

BER = 10−5 as a function of the accumulated dispersion. Please note that the OSNR

is always referred to a standard bandwidth of 0.1 nm.

Chromatic dispersion results are summarized in Fig. 2.8. The improved resilience

of the narrow filter receiver to accumulated chromatic dispersion is clearly visible. The

results of two measurements (at 136 and 216 ps/nm of accumulated dispersion) are

indicated by square markers for the conventional detector and circle markers for the

narrow filter detector. The measurements can be seen to fall very close to the simulated

data. The narrow filter receiver shows a factor of ∼2 improvement, i.e., the same OSNR

penalty is reached by accumulating about twice the total chromatic dispersion.

Figure 2.9 shows the simulated OSNR penalties at BER= 10âĹŠ5 as a function of

accumulated chromatic dispersion and filter 3 dB bandwidth. One can see that the

optimal value of the 3 dB bandwidth only slightly depends on accumulated dispersion.

2.2.5 Polarization mode dispersion

Previous results on polarization mode dispersion and DQPSK signals focused on

the performance of the AMZI receiver as a function of DGD [95]. We report the

performance of the narrow filter receiver, in particular as it compares to the performance

of the conventional AMZI receiver, when first order polarization mode dispersion is

present leading to DGD. We vary the amount of DGD via the PMD emulator with

the input state of polarization giving a 50/50 power split on the two principal states of

2. We calculate BER down to 10−5 vs. OSNR for each CD value; note that [80] fixed the OSNR to

a regime where BER was assured to be above 10−4 and only one BER point was found per CD value.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 52

-400 -200 0 200 4000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Accumulated dispersion [ps/nm]

x2

OSN

R p

enal

ty for

BE

R =

10-

5 [d

B]

Conventional (sim)Conventional (meas)Narrow filter (sim)Narrow filter (meas)

Figure 2.8: Simulated OSNR penalties at BER=10−5 as a function of accumulated

chromatic dispersion for both receivers under investigation. Two experimental points

are also provided for comparison.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 53

1

1

1

1

1

2

22

2

3

33

3

3

4

44

4

55

5

66

6

2

7

78 8

Filter 3 dB bandwidth (normalized to bit rate)

Acc

um

ula

ted d

isper

sion

[ps/

nm

]

0.5 0.6 0.7 0.80

100

200

300

400

500

0

1

2

3

4

5

6

7

8

dB

Figure 2.9: Simulated OSNR penalties at BER=10−5 as a function of accumulated

chromatic dispersion and filter bandwidth.

polarization, and we again record the OSNR penalty at BER=10−5 for the two receivers.

Results are summarized in Fig. 2.10. There is a negligible difference in the penalties,

hence the two receivers have the same tolerance to first order PMD. We also show,

in dashed and dotted lines, the results of simulations of first order PMD for the two

receivers. Results are in good qualitative agreement with the measured values.

2.3 Nonlinear Phase Noise

In this section we evaluate the tolerance of the narrow filter receiver to fiber non-

linearities, i.e., intrachannel effects and NLPN. NLPN arises from amplifier ASE that

is translated to the phase of the signal by self-phase modulation in fiber [96].

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 54

5 10 15 20 25 30 35-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

DGD [ps]

OSN

R p

enal

ty for

BE

R=

10-5

Conventional (sim)Narrow filter (sim)Conventional (meas)Narrow filter (meas)

Figure 2.10: Experimental OSNR penalties at a BER=10−5 due to first-order PMD

as a function of differential group delay for both receivers under investigation, for the

worst case 50/50 power split.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 55

Tunable laser

IQ

PC

MZM

/2

D A T A

D A T A21Gbps

Bias control

MZM

SHF 46213

Conventional receiver

Narrow-filter receiver

OSA90

10

CR and ED

or

VOA

DCF

80 km NZDSF

Booster EDFA

Gain controlled EDFA

1.4 nm OBPF

VOA

Figure 2.11: Experimental setup for the nonlinear transmission experiment.

2.3.1 Experimental Results and discussion

The experimental setup used to probe nonlinear phase noise effects is depicted in

Fig. 2.11. After the DQPSK modulator a first variable optical attenuator controls the

power of the useful signal. An amplification stage built with a booster EDFA, a 1.4 nm

optical filter and a gain controlled EDFA provides sufficient power and sets the OSNR.

With a 10/90 splitter, the OSNR is monitored after the transmitter. A signal power

of ∼+13-dBm is injected into the fiber, in order to trigger nonlinearities. The DQPSK

signal and the noise interact when traveling in a spool of 80 km of NZDSF fiber, and

a matched DCF module provides full compensation of the chromatic dispersion. A

second VOA controls the power on the receiver, to keep it constant. The narrow filter

receiver and the conventional receiver are used in turn, and the BER is measured as

per the previous section.

The results are presented in Fig. 2.12. As we expected, the performance of the

receivers now overlap: the 3-dB advantage of the conventional receiver has vanished.

A similar result, but for DBPSK, was shown in [86], where the single-ended receiver

and the conventional receiver were shown to have the same BER in the nonlinear

regime, notwithstanding the use of a simple photodiode instead of the balanced one.

We demonstrate here for the first time that for DQPSK the narrow filter receiver also

has the same performance as the conventional one, obviating not only the balanced

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 56

18 20 22 24 26 28-10

-9

-8

-7

-6

-5

-4

-3

Log

(BE

R)

OSNR [dB]

Conventional (fit)Conventional (meas)Narrow filter (fit)Narrow filter (meas)

Figure 2.12: BER for both receivers in the nonlinear regime.

photodetection, but also the AMZI itself.

2.3.2 Numerical Model

The KLSE method with white ASE noise, described in section 1.1.3 and used in

simulations in section 2.2 accounts for intra-channel nonlinearities, but not for NLPN,

since the nonlinear signal-noise interaction alters the statistics of the received ASE in

a complex way. To appreciate the impact of NLPN in a realistic dispersion managed

(DM) scenario, we performed Monte-Carlo simulations of a 21 Gbaud DQPSK single-

channel 25x80 km DM line with NZDSF transmission fiber (dispersion coefficient D =

8 ps/nm/km, nonlinear coefficient γ = 1.7 W/km) with full in-line compensation. No

pre- and post-compensaton was added, and the EDFA noise figure was F = 6 dB. In

Fig. 2.13 we show the BER versus input power for the conventional receiver, calculated

with both the KLSE method and with brute-force, error-counting Monte Carlo (MC)

estimation with at least 100 errors collected for each point [97].

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 57

-8 -6 -4 -2 0 2-5

-4

-3

-2

Input power to line [dBm]

Log

(BE

R)

KL method (ASE at Rx)

MC (ASE along the line)

Figure 2.13: BER when KL method is used or Monte Carlo method is used. Fully com-

pensated DQPSK 21 Gbaud single-channel 25x80km TW system, conventional receiver

only.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 58

Both KLSE and MC use the split-step Fourier method [98] for field propagation, but

in the MC case the noise is added along the line and propagated with the signal, thus

making simulations much longer. We note that KLSE estimation becomes overly opti-

mistic at powers larger than -5 dBm, since NLPN is becoming the dominant impairment.

Having established the significance of NLPN contribution to the fiber nonlinearity for

the scenario under investigation, we will thus use only MC estimation to assess the

relative performance of the two receivers of Fig. 2.1 in the nonlinear regime. In order

to do so, we simulated the same DM line as before, but with either NZDSF or standard

single mode fiber (SSMF) (D = 17 ps/nm/km, nonlinear coefficient γ = 1.3 W/km)

transmission fiber. The comparison of the two receivers is provided in Fig. 2.14(a) in

terms of the required OSNR to achieve BER=10−3 versus signal input power (at BER

= 10−5 such error-counting MC simulations are impractical).

We observe that the 3-dB advantage of the conventional receiver is confirmed in

the linear regime (low input power), but it diminishes as nonlinearities become the

dominant source of errors. At large launch powers the performance difference between

the two receivers is negligible both for NZDSF and for SSMF, confirming the conclusion

we drew from our experimental setup. In Fig. 2.14(b) we report the OSNR margin for

a BER = 10−3, i.e., the difference between the actual OSNR of the line at F = 6 dB,

and the OSNR required for BER = 10−3.

The optimum launch power achieving the largest OSNR margin, i.e., the nonlinear

threshold (NLT), is higher for SSMF than for NZDSF. This is due to the larger local

dispersion of SSMF (NLT of +0.5 dBm instead of -1.5 dBm for conventional receiver,

and of +1 dBm instead of -0.5 dBm in narrow-filter receiver). The difference in OSNR

margin at the NLT between the narrow filter receiver and the conventional receiver is

limited to approximately 1.6 dB for both SSMF and NZDSF fiber.

2.4 Multichannel operation of the narrow filter

receiver

As we mentioned, the narrow filter receiver can play the role not only of the AMZI,

but also that of the channel selection optical filter. Therefore, in a WDM scenario,

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 59

-10 -8 -6 -4 -2 0 2 412

14

16

18

20

22

24

26

Input power to the line [dBm]

Requ

ired

OS

NR

fo

r B

ER

=10

-3

Conventional, NZDSF

Narrow filter, NZDSF

Conventional, SMF

Narrow filter, SMF

(a)

-10 -8 -6 -4 -2 0 2 4-6

-5

-4

-3

-2

-1

0

1

2

3

4

Input power to the line [dBm]

Conventional, NZDSF

Narrow filter, NZDSF

Conventional, SMF

Narrow filter, SMF

OS

NR

ma

rgin

fo

r B

ER

=10

-3

(b)

Figure 2.14: Simulated tolerance to fiber nonlinearities. Fully compensated DQPSK 21

Gbaud single-channel 25x80km system.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 60

the mandatory demultiplexer (if intelligently designed) can be used as a narrow filter

receiver as well. Hereafter, we experimentally demonstrate that a single demultiplexer

can be used to simultaneously demodulate the in-phase (or quadrature) component of

a comb of DQPSK signals. The use of the demultiplexer for demodulation of WDM-

DQPSK signals represents a considerable advantage in terms of component count. With

Nch multiplexed channels, this solution only requires two demultiplexers instead of Nch

channel selection filters and 2Nch AMZIs. Moreover, 2Nch standard photodetectors are

needed, instead of 2Nch balanced photodetectors. We will show that the penalty with

respect to the single-channel performance of the conventional receiver is below 4 dB.

The experimental setup for the WDM-DQPSK demodulation is sketched in Fig. 2.15.

Five distributed feedbacks (DFBs) are multiplexed with an arrayed waveguide grating

(AWG). The channel spacing is 100 GHz. The resulting WDM signal is controlled in

polarization and modulated with a commercial I/Q modulator (SHF 46213), similarly

to the previous section. The bit rate per channel is again 42 Gb/s. As is customary in

WDM experiments, we use a spool of fiber after the modulator, for signal decorrelation.

The total accumulated dispersion in the spool is ∼ 210 ps/nm. This value is enough for

delaying adjacent channels of around 4 bits. We have shown in the previous sections

that when the narrow filter receiver is used with DQPSK signals at 21 Gbaud, the

OSNR penalty for a bit error rate (BER) of 10−5 with ∼ 210 ps/nm of accumulated

dispersion is less than 0.5 dB.

After channel decorrelation, the signal power is controlled with a VOA. An EDFA

sets the OSNR. The OSNR is always referred to the conventional bandwidth of 0.1 nm.

The spectrum of the noisy signal is constantly monitored with an OSA via the 10/90

power splitter. We verified that, due to the high signal power, the EDFA is working in

saturation. When the input power decreases by 1 dB, the output signal power is almost

constant, whereas the noise floor increases by ∼1 dB. Therefore, we are assured that

we work in a OSNR-limited regime. The narrow filter receiver is implemented with

one demultiplexer for all the WDM channels. The frequency response of the demux

channels are very similar, and one instance was reported in Fig. 2.3. By tuning the

demultiplexer we can select the I or the Q component of the DQPSK signals. As in the

previous section, a photodiode and a transimpedance amplifier are followed by a clock

recovery circuit and an error detector.

BER results are reported in Fig. 2.16, as a function of the OSNR at the receiver.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 61

IQ

VOA

OSA

90

PC

MZM

/2

Bias control

MZM

SHF 46213

Preamp

100GAWG

D A T A

D A T A21Gbps

CR and ED

Narrow-filter rx

1x32 de

mux

10 dB

0.8 nm

10

#1 #2 #3 #4 #5

Figure 2.15: Setup for the WDM-DQPSK experiment. PC: polarization controller,

CR: clock recovery, ED: error detector. The inset shows a WDM-DQPSK spectrum

measured on the optical spectrum analyzer.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 62

-10

log(

BE

R)

14 16 18 20 22 24 26

-9

-8

-7

-6

-5

-4

-3

OSNR [dB/0.1nm]

Single chan (conv. Rx)

WDM chan. 2 (NF Rx)

Single chan (NF Rx)

WDM chan. 3 (NF Rx)

WDM chan. 4 (NF Rx)

Figure 2.16: Bit error ratio for conventional and narrow filter receiver. In the WDM

case, different output ports of the same demultiplexer are used for the different channels.

The square markers represent the BER of a single-channel DQPSK signal measured

with a conventional receiver, and are repeated from Fig. 2.5. The circle markers are the

BER results obtained with the narrow filter receiver when only one channel is present

(and no decorrelating fiber). The other markers are the BER of channels 2, 3 and 4,

measured on three different demultiplexer ports. First, we notice that (with a 100 GHz

grid) the performance of the WDM signals have a small penalty with respect to the

single-channel signal (∼1 dB at low BER, partially due to the decorrelating fiber).

Secondly, the OSNR penalty with respect to the single-channel conventional receiver is

smaller than 4 dB, in the WDM scenario.

Chapter 2. DQPSK: When is a Narrow Filter Receiver Good Enough? 63

2.5 Conclusion

In this chapter, we have investigated a simplified receiver for DQPSK signals based

on a single optical filter slightly offset from the carrier frequency, with 3-dB bandwidth

smaller than the symbol rate. We have compared it to the conventional DQPSK receiver

with respect to the major optical fiber impairments. The filter is implemented with one

of the 32 channels of a standard demultiplexer, a mature technology naturally enabling

simultaneous demodulation of wavelength division multiplexed DQPSK signals. We

have reported experiments at 42 Gb/s, showing ∼ 3 dB sensitivity penalty with respect

to the conventional receiver in the linear regime. We have also shown that the narrow

filter receiver has a stronger tolerance to accumulated chromatic dispersion. We have

compared simulation results against the experimental data, finding good agreement.

We have discussed the impact of fiber nonlinearities on the performance of the receiver.

We have shown in experiments and simulations that the advantage of the conventional

receiver over the narrow filter receiver diminishes when the fiber nonlinearities are

taken into account. Finally, we experimentally demonstrated, for the first time to our

knowledge, simultaneous demodulation of WDM-DQPSK signals with a single device.

Chapter 3

SOA nonlinearity postcompensation

for PSK signals

The last chapter dealt with a novel simple receiver for DQPSK. This chapter tackles

another important aspect of PM systems, namely optical amplification. We discussed

the advantages of SOAs over traditional EDFAs, despite the penalties imposed on PM

signals due to SOA nonlinearities (in particular NLPN). In this chapter we propose

a simple scheme to equalize the impairments induced by SOA nonlinearities. This

technique is derived from the small signal analysis of carrier density fluctuations. We

demonstrate via simulation almost complete cancellation of the NLPN added by a

saturated SOA on a differential PSK signal. We demonstrate via both simulations

and experiment the effectiveness of the method for mitigation of nonlinear distortions

imposed by SOAs on an OOK signal.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 65

3.1 Introduction

Variations of the input power to a SOA translate into variations of the carrier density,

as we have seen in Chapter 1. In turn, the carrier density variations cause variations of

the instantaneous gain and refractive index. For typical SOA carrier lifetimes, the gain

can follow the variations of the input power over a bandwidth on the order of GHz.

We have also seen how this physical phenomenon leads to the main nonlinear effects

that a SOA typically exhibits. We have discussed the nonlinear effect which limits PSK

signals when SOA are used, namely NLPN. In this chapter, we propose a method to

mitigate such nonlinear effect.

The literature is rich with proposals for the compensation of NLPN in fiber [99,

93, 100, 101, 102, 103]. There are also several studies for the compensation of SOA

nonlinearities. For instance, SOA bias current can be modulated to counter-balance the

reservoir fluctuations by injecting more or less carriers according to the instantaneous

input power [104]. The authors in [105] propose an all-optical, linear equalizer made

by two Mach-Zehnder interferometers to combat gain variation-induced distortions. In

[106] the reduction of SOA-induced nonlinear impairments is achieved with the use

of electronic maximum-likelihood sequence estimation at the receiver. A very recent

paper [107] proposes digital signal processing following coherent detection; the SOA

dynamic gain equation is solved numerically, implementing SOA backpropagation in an

integrated circuit.

In this work, we use small signal analysis of the SOA dynamic equation to estimate

the gain fluctuations, and to compensate SOA nonlinearity. Our solution is simpler

and more cost efficient than those previously proposed. We focus on differential phase

modulation but we will also consider intensity modulation. For phase modulation we

alleviate NLPN via an opto-electronic feed forward mechanism. In the case of intensity

modulation, an electrical filter is used to derive a signal to correct for the distortion of

the received waveform.

As in [103] for fiber-induced NLPN, we also exploit the correlation between intensity

and phase noise due to nonlinear interaction. The novelty of our solution stems from

the different nature of NLPN in fiber and in SOAs. For SOA-induced NLPN, we can

compensate virtually all the phase noise introduced by the SOA with an electrical filter

Chapter 3. SOA nonlinearity postcompensation for PSK signals 66

whose optimal characteristic we derive from SOA equations. This is contrary to fiber

NLPN; as discussed in [103], post-compensation cannot remove all the fiber NLPN,

since the nonlinear phase shift is proportional to the path-averaged intensity, and not

the intensity at the end of the link.

In this chapter we focus on single-channel systems, i.e. when SOAs are used as

building blocks for monolithically integrated modulators [108], receivers [109], or other

agile network components such as wavelength converters [110, 111]. On the other hand,

our solution might be extended to the WDM scenario, where SOAs would amplify

a comb of signals. When more than one channel is present at the SOA input, the

reservoir of carriers fluctuates as a function of the total WDM signal power, causing

nonlinearities. Given the low pass nature of the SOA, the reservoir fluctuations have

a bandwidth in the order of a few GHz. These fluctuations can therefore be estimated

and corrected with the schemes we propose. Nevertheless, the extension of our work to

the WDM case is not straight forward, since gain might vary from channel to channel,

depending on the number of channels, their spacing, and the spectral shape of the SOA

gain. We leave this matter as a subject of future investigations.

This chapter is organized as follows. In Sec. 3.2 we introduce our proposal for SOA

nonlinearity post-compensation based on a small signal model for the SOA. In Sec. 3.3,

we quantify the problem by analytically calculating the phase noise at the SOA output.

In Sec. 3.4 we describe a large-signal SOA numerical model that we will use to assess

the validity of our NLPN compensation method in Sec. 3.5. In Sec. 3.6, we investigate

via simulation and experiment the equalization of distortions induced by the SOA on

a 10 Gb/s OOK signal. We conclude in Sec. 3.7.

3.2 Exploiting the low-pass nature of SOAs

As we have seen, according to the model presented in [26, 27], the relation between

the output and the input phase to the SOA is given by (1.19), repeated here

ϕSOA,out(t) = ϕSOA,in(t) + 0.5αh(t) (1.19)

and the equation governing the dynamics of h(t) is (1.20), which reads

τcd

dth(t) = h0 − h(t) −

[

eh(t) − 1]

pSOA,in(t) (1.20)

Chapter 3. SOA nonlinearity postcompensation for PSK signals 67

Finally, the relation between the output and the input field to the SOA is given by

ESOA,out(t) = ESOA,in(t) exp [h(t)(1 + jα)/2] (1.21)

In this model, the SOA internal losses are neglected, so that the spatial distribution

of the carriers along the propagation direction z does not affect the total gain.

It is trivial from (1.21) and (1.20) to derive the differential equation relating h(t)

and the output power pSOA,out(t) , |ESOA,out(t)|2/Psat

τcd

dth(t) = h0 − h(t) +

[

e−h(t) − 1]

pSOA,out(t) (3.1)

We can now write the gain and the output power as their mean values (indicated

by a bar) plus a zero-average term (indicated by δ), as follows h(t) = h + δh(t) and

pSOA,out(t) = pSOA,out + δpSOA,out(t). It is well known that the SOA acts, in the small

signal approximation (i.e., when δh(t) ≪ h), as a low pass filter between the optical

intensity and the gain fluctuations [26]. In the case of output power we can write

δhl(t) ≃ KδpSOA,out(t) ⊗ m(t) (3.2)

where the subscript l reminds us that a linearization of (3.1) was performed, and ⊗denotes the convolution; K is

K =e−h − 1

1 + pSOA,oute−h(3.3)

and m(t) is a single-pole lowpass filter with time constant

τeff =τc

1 + pSOA,oute−h(3.4)

We report in Fig. 3.1 a contour plot of the 3-dB bandwidth of the filter as function of

the SOA carrier lifetime τc, and of the input average power pSOA,in. The 3-dB bandwidth

is smaller than a few GHz for a wide range of realistic values of input power, when the

carrier lifetime is larger than 50 ps. The carrier lifetime τc, an intrinsic parameter of

every SOA, is swept over the wide range of 10 ps to 1 ns. At carrier lifetimes smaller

than approximately 20 ps the bandwidth of the filter becomes as large as 10 GHz.

In this work, we will exploit (3.2) to estimate the gain variations by observing the

output power. We will then compensate for the SOA nonlinearities both for phase

Chapter 3. SOA nonlinearity postcompensation for PSK signals 68

1 1

1

2 22

3 33

4 4

4

5 55

6 66

7 77

8 88

9 99

10 1011 1112 1213 1314 1415 15

15

16 16

1617181920

21

pin = Pin/Psat [dB]

log

10(τ

c/1

ps)

−20 −15 −10 −51

1.5

2

2.5

3

GH

z

2

4

6

8

10

12

14

16

18

20

22

Figure 3.1: 3-dB bandwidth of M(ω) as function of the two parameters which com-

pletely determine its time constant.

modulated and intensity modulated signals. We begin with phase modulation and

compensate the SOA-induced nonlinear phase noise. The reference link is sketched in

Fig. 3.2. The input signal (either a CW laser or a phase-modulated signal) is optically

filtered and injected in a SOA. After amplification, there is a compensation stage

composed of an AC-coupled photodiode, an electrical filter and a phase modulator. The

optimal bandwidth of the electrical filter depends on the saturation level of the SOA,

and therefore on the average optical input power (see, e.g., Fig. 3.1). The average input

power to the line amplifiers is an important design parameter of an optical network,

and it should not vary much with time. Nevertheless, simple low-frequency circuits

can be devised to monitor the input average power and adjust the filter bandwidth

accordingly.

As can be seen in Fig. 3.2, and using (1.19), the phase after postcompensation can

be written as

ϕSOA,post(t) = ϕSOA,out(t) + CδpSOA,out(t) ⊗ m(t)

= ϕSOA,in(t) +α

2h(t) + CδpSOA,out(t) ⊗ m(t) (3.5)

where we have taken the SOA lowpass filter m(t) impulse response and implemented it

Chapter 3. SOA nonlinearity postcompensation for PSK signals 69

SOA PMSignal + AWGN

Subscript used:

OBPF

filter

in out post

Postcompensation stage

Figure 3.2: Signal (either CW or phase modulated DPSK) plus additive white Gaus-

sian noise pass through an optical bandpass filter before entering a SOA. The post-

compensation stage includes a photodiode, electrical filter and phase modulator.

in an electrical filter. The constant C is used to take into account the effect of splitting

loss, photodiode responsivity, filter losses, and phase modulator efficiency.

By writing h(t) = h+δh(t) and using (3.2), we can reformulate the postcompensated

phase as

ϕSOA,post(t) = ϕSOA,in(t) +α

2h +

α

2δh(t) + Cδhl(t)/K

≃ ϕSOA,in(t) +α

2h (3.6)

where in the last passage we choose C = −Kα/2. Thus, under the small signal as-

sumption, we can perfectly suppress the NLPN introduced by the SOA, leaving only an

offset to the input phase.

3.3 Phase Noise Variance at SOA Output

In order to better understand and to quantify the impact of phase noise, we find

an analytical result for the variance of the phase noise ϕSOA,out(t) due to signal-noise

nonlinear interaction in the SOA. The calculation is straightforward, and a similar one

was performed in [52]. We examine a continuous wave (CW) laser plus filtered AWGN

at the SOA intup, and make the small signal assumption to find var [ϕSOA,out(t)]. The

system is shown in Fig. 3.2. The input field can be written as

ESOA,in(t) =√

PSOA,in 1 + ξ [np(t) + jnq(t)] (3.7)

where j is the imaginary unit, and ξ is defined as

ξ , 10−0.05OSNR/√

2 (3.8)

Chapter 3. SOA nonlinearity postcompensation for PSK signals 70

where OSNR is the signal optical signal-to-noise ratio in dB on the typical resolution

bandwidth ∆RB = 0.1 nm. The real and imaginary part of the zero-mean Gaussian

random noise are np(t) and nq(t), with equal power spectral density (PSD) given by

Snp,q(ω) =

1∆RB

|Bo(ω)|2 (3.9)

where Bo(ω) is the frequency response of the optical filter.

We will calculate the variance of the output phase

var [ϕSOA,out(t)] =⟨

ϕ2SOA,out(t)

⟩

− 〈ϕSOA,out(t)〉2 (3.10)

Under the small signal assumption, the average of the output phase is straightforward

from (1.19)

〈ϕSOA,out(t)〉2 =(

α

2h)2

(3.11)

where h is the average of the integrated gain h(t). The second moment can be written

as⟨

ϕ2SOA,out(t)

⟩

= ξ2[

σ2pq + α2p2

SOA,inK2µ(0)]

+(

α

2h)2

(3.12)

where σ2pq is the variance of np(t) (equal variance for nq(t)), and µ(t) gives the filtering

effect of the SOA, defined as

µ(t) , m(−t) ⊗ m(t) ⊗ Rnp,q(t) (3.13)

where Rnp,q(t) is the autocorrelation function of np(t) and nq(t), equal to the inverse

Fourier transform of (3.9) due to the Wiener-Khinchin relation. At t = 0 we have

µ(0) =1

∆RB

∫ ∞

−∞

|Bo(ω)|21 + ω2τ 2

eff

dω (3.14)

Noting that the variance of the input phase can be written (in the small signal

regime) as

var [ϕSOA,in(t)] = ξ2σ2 (3.15)

and the output variance can be written as

var [ϕSOA,out(t)] = var [ϕSOA,in(t)] + α2ξ2p2SOA,inK2µ(0) (3.16)

This clearly shows the SOA leads to an increase in phase noise variance, with explicit

dependence on linewidth enhancement factor and saturation level. Fig. 3.8 and Fig. 3.9

show good match between (3.16) and simulation results.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 71

...

Section 1 Section Nsec

Lossles SOA Lumped Loss

Section 2

ESOA,outEO,1 EI,2 EO,2 EI,3 ... ESOA,in

Figure 3.3: SOA model includes Nsec sections along the propagation direction, each

with a lossless “short” SOA and a lumped loss.

Table 3.1: Parameters used in the SOA simulation (unless specified otherwise in the

text).Parameter Value Units

SOA length L 650 µm

α 7

τc 175 ps

h0 6.22

Psat 13 dBm

Intrinsic losses αint 2180 1/m

Number of sections 10

3.4 Large vs. small signal models

The small signal model allowed us to develop our mitigation technique and to find

an analytic expression for the phase noise variance at the SOA output. We next turn to

more accurate large signal models to validate our results are useful even outside regimes

where the small signal and large signal models coincide. Our simulations will always

be based on the more accurate large signal model. We use the space-resolved model

described in [112] that captures the distributed nature of the carrier density along the

propagation direction z, as well as the distributed intrinsic losses of the SOA. Notice

that neglecting these features was the main drawback of the small-signal analysis of the

previous sections.

The schematic of the large signal model is sketched in Fig. 3.3. A SOA of length L

is divided into Nsec sections. We used Nsec = 10, as results were unchanged with larger

Nsec. Each section is composed of a lossless SOA of length L/Nsec followed by a lumped

Chapter 3. SOA nonlinearity postcompensation for PSK signals 72

500 1000 1500 2000 2500 3000−3.5

−3

−2.5

−2

−1.5

SOA intrinsic losses αint [1/m]

log

10(N

SD

)

OSNR = 10 dB/0.1nm

OSNR = 26 dB/0.1nm

Figure 3.4: Logarithmic plot of the normalized standard deviation as function of the

SOA intrinsic losses.

loss. The equations governing the ith section are

EO,i(t) = EI,i(t)ehi(t)(1+jα)/2 (3.17)

τcd

dthi(t) = h0 − hi(t) −

[

ehi(t) − 1] |EI,i|2

Psat(3.18)

where EI,i(t) and EO,i(t) are the input and output fields of the ith section, respectively.

The unsaturated integrated gain of each SOA section is h0, and hi(t) is the integrated

gain of the ith section. The initial condition is EI,1(t) = ESOA,in(t), and for each section

after the first the input field is the attenuated output of the previous section per

EI,i(t) = EO,i−1(t)e(−αintL/Nsec) i ≥ 2 (3.19)

The output field is therefore ESOA,out(t) = EI,Nsec+1(t).

In our simulations, we neglect internal ASE. In [112] the ASE is treated as a white

noise added to the input signal. Hence, including the ASE would translate into a

different OSNR at the SOA input. We will show that the accuracy of the proposed

method shows very little dependence on the input OSNR and, therefore, results would

not change when including ASE as in [112].

The total integrated gain hT (t), as distinguished from the lossless gain h(t) of the

Chapter 3. SOA nonlinearity postcompensation for PSK signals 73

single-reservoir model of the previous section, is defined as

hT (t) = ln (pSOA,out/pSOA,in) =Nsec∑

i=1

hi(t) − αintL. (3.20)

The fluctuation, not necessarily small, of the gain around its average is δhT (t) , hT (t)−〈hT (t)〉. In order to establish the effectiveness of (3.2), we compare the ”true“ results of

the large signal model δhT (t) against the result of the small signal approximation (3.2)

in terms of the normalized standard deviation (NSD)

NSD =

∣

∣

∣

∫+∞−∞ [δhT (t) − δhl(t)]

∣

∣

∣

2

∣

∣

∣

∫ +∞−∞ δhT (t)

∣

∣

∣

2 . (3.21)

We next present simulations of the reference system in Fig. 3.2 using parameters

in Table 3.1. SOA parameters were chosen to give realistic results when compared

with experimental data [113, 114]. The input signal to the SOA is a CW laser plus

additive white Gaussian noise filtered by a second order supergaussian filter with a

3-dB bandwidth of 15 GHz. The total signal power is 3-dBm, and two disparate optical

signal-to-noise ratios are considered, 10 and 26 dB. Fig. 3.4 shows the NSD calculated

from (3.21), where δhT (t) is obtained numerically with the large signal SOA model,

and δhl(t) is obtained from (3.2). We sweep a wide range of intrinsic loss αint; NSD

is always below 1.8 × 10−2, regardless of the signal OSNR. To visualize the accuracy

of the small signal model, we plot in Fig. 3.5 a portion of the waveforms for OSNR =

10 dB and αint = 3000.

We take advantage of these simulations to examine the PSD of the intensity (Fig. 3.6(a))

and phase (Fig. 3.6(b)) of the CW signal both at the input and output of the SOA. In

Fig. 3.6(b) we see the advantageous reduction in intensity noise at low frequencies. In

Fig. 3.6(a) we have instead an enhancement of phase noise at low frequencies, a man-

ifestation of the NLPN problem [49]. The SOA gives an advantage in intensity noise

suppression we wish to maintain, but an enhancement of phase noise that we wish to

combat. Predictions from large and small signal models coincide.

The bandwidth of the noise redistribution (a few GHz) indicates the speed with

which the reservoir can follow the input power fluctuations. The novel, important

message of Fig. 3.6 is that numerical PSD calculated with the large signal model and

the small signal PSD are very similar, validating our exploitation of small signal analysis

of the single-reservoir model that neglects SOA intrinsic losses.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 74

201 201.5 202 202.5 203 203.5 204 204.5 205

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time [ns]

Inte

grat

edga

in

δhl(t) δhT (t)

Figure 3.5: Predictions of integrated gain from large δhT (t) vs. small δhl(t) signal

models; NSD is 0.018 in this example.

3.5 NLPN compensation

In this section we will numerically implement the compensation scheme presented in

Fig. 3.2 and evaluate its effectiveness. For the sake of generality, and in order to clarify

the principle of the compensator, we will begin with the simple case where the signal

is a noisy CW laser and evaluate the phase noise variance suppression. The proposed

method would work with any phase modulated scheme; we will show as an example

the case of binary DPSK modulation, and calculate the noise suppression in terms of

differential phase Q factor Q∆ϕ as defined in (1.1.3).

3.5.1 NPLN Reduction: noisy CW laser

Consider a noiseless continuous laser light to which white Gaussian noise is added,

then filtered by a supergaussian optical filter (order 2 and 15 GHz 3-dB bandwidth),

and injected into a SOA. The SOA is simulated with the large signal model of the

previous section, again with parameters listed in Table 3.1. The nonlinear phase noise

generated by the SOA is compensated with the scheme presented in Fig. 3.2.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 75

Phas

e noi

se P

SD

[dB

]

Frequency [GHz]

Inte

nsi

ty n

ois

e P

SD

[dB

]

(a)Increase in phase noise

-10 -5 0 5 10-10

-5

0

5

10

15

InputOutput(Large signal)

Output(Small signal)

3 dB bw

(b)

-10 -5 0 5 10

-10

-5

0

Reduction of intensity noise

3 dB bw

Figure 3.6: PSD of intensity and phase noise at the input and output of the SOA.

The input signal is a CW laser plus AWGN. OSNR is 23-dB, noise is filtered with a

supergaussian filter of order 2, with 15 GHz 3-dB bandwidth.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 76

RealIm

ag

0 0.5 1 1.5−1

−0.5

0

0.5

1

log 1

0(P

DF)

−6

−5.5

−5

−4.5

−4

−3.5

−3

(a)

Real

Imag

0 0.5 1 1.5−1

−0.5

0

0.5

1

log 1

0(P

DF)

−6

−5.5

−5

−4.5

−4

−3.5

(b)

Real

Imag

0 0.5 1 1.5−1

−0.5

0

0.5

1

log 1

0(P

DF)

−6

−5.5

−5

−4.5

−4

−3.5

−3

(c)

Figure 3.7: Logarithmic two-dimensional probability density functions of the complex

electric fields; input power is 0 dBm, OSNR=23 dB, supergaussian optical filter of order

2 and 15 GHz bandwidth. (a), (b) and (c) are respectively the signals at the input of

the SOA, output from the SOA, and after post-compensation.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 77

In this first round of simulations, the input power to the SOA is fixed at 0 dBm.

In Fig. 3.7(a) we report the two dimensional PDFs of the electric field in the complex

plane at the input of the SOA, where the Gaussian noise gives a bell-shaped PDF in two

dimensions (circular contours). Fig. 3.7(b) is the PDF of the signal at the output of the

SOA. Any constant phase shift has been ignored, and all PDFs are centered at (1,0).

As predicted by the PSDs of Fig. 3.6, we can see a clear enhancement of the phase

noise, and a reduction (squeezing) of the intensity noise. Notice the real and imaginary

parts of the electric fields are now strongly correlated due to the NLPN process. Finally,

Fig. 3.7(c) shows the PDF of the signal at the output of the postcompensation stage

described in Fig. 3.2 when using the optimal C in (3.5). The phase noise has clearly

been reduced, whereas the intensity noise remains suppressed. In other words, this

compensator preserves the reduction in the intensity noise, and drastically reduces the

phase noise introduced by the SOA. The net effect is that of having a SOA with almost

zero linewidth enhancement factor, i.e., no NLPN.

To better quantify our results, we report in Fig. 3.8 the variance of the optical phase

versus SOA linewidth enhancement factor α at the SOA input, the SOA output and

after the compensation stage. The variance at the SOA output is found both numer-

ically and via (3.16). The postcompensation stage is very effective, almost perfectly

compensating the NLPN added by the SOA; the input and post-compensation variance

are indistinguishable in Fig. 3.8. The discrepancy between the small signal approxima-

tion and the true signal is acerbated by α per (3.6); compensation is expected to be

less accurate for very high values of α. Nonetheless, for realistic values of α in Fig. 3.8,

the discrepancy is barely noticeable.

We also expect the small signal approximation (and, therefore, the compensator

itself) to be less accurate when working in deep saturation. To investigate this case,

we sweep the SOA input power up to +10 dBm, with resulting phase noise variances

at the input, output and following post-compensation given in Fig. 3.9. The method is

very effective even at very high saturation levels: the gain compression at +10 dBm of

input power is ∼ 15 dB.

In Sec. 3.3 we developed an analytic expression for the phase noise variance at the

SOA output, showing the quadratic dependence of the output phase noise on α and

input power. In Fig. 3.8 and 3.9 we show (3.16) agrees with the large signal model

numerical prediction.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 78

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Linewidth Enhancement Factor α

Phase

variance

[rad

2]

in

out, sim

out, eq. (3.16)

post

Figure 3.8: Phase variance vs. linewidth enhancement factor, PSOA,in = 0 dBm.

−30 −20 −10 0 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Input power [dBm]

Phase

variance

[rad

2]

in

out, sim

out, eq. 3.16

post

Figure 3.9: Phase variance vs. SOA input power, α = 7.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 79

3.5.2 NPLN reduction: DPSK modulation

Our study of the CW input signal allowed us to unambiguously interpret input/output

behavior of the SOA and our compensator, however this is not the situation where

NLPN is problematic. In this subsection, we show the effect of NLPN post-compensation

on the more realistic case of a binary differential phase shift keying signal. The signal

is generated by an ideal non-return to zero (NRZ)-DPSK transmitter at a bit rate of

R = 10 Gb/s. We use De Brujin sequences of length 213 for the bit pattern. The

signal is loaded with additive white Gaussian noise, and then optically filtered to limit

out-of-band ASE. Simulation parameters are unchanged from the previous subsection.

The scheme for NLPN post-compensation is again the one presented in Fig. 3.2. The

DPSK signal is demodulated incoherently with an ideal one bit delay interferometer

followed by a balanced detector. In the electronic domain, the photocurrent is then

filtered with a 5th order Bessel-Thompson filter with a 3-dB bandwidth of 0.65R.

Fig. 3.10 shows the eye diagrams of the received signal after demodulation at the

three test points (before the SOA, after amplification and after post-compensation).

The eye diagram after the SOA is closed due to excess phase noise; the eye becomes

very open after post-compensation. The eye is actually better after post-compensation

than at the SOA input due to the intensity noise reduction introduced by the SOA.

One of the major differences of SOA post-compensation with respect to compen-

sation of fiber-generated NLPN [103] is the role of the electrical filter. We have

shown in Sec. 3.2 that an RC filter with time constant τeff completely removes

SOA NLPN in the small signal regime. We now investigate the importance of the

bandwidth of post-compensation electrical filter. The differential phase is defined as

∆ϕ(t) , ϕ(t) − ϕ(t − TB), where TB is the bit duration and ϕ(t) is the instantaneous

phase of the optical field. We recall the definition of differential phase Q factor [115]

we gave in section 1.1.3

Q∆ϕ ,π

(σ∆ϕ0+ σ∆ϕπ

)1.1.3

where σ∆ϕ0and σ∆ϕπ

are the standard deviations of the sampled differential phase on

the 0 and π rails, respectively. The curve to the left in Fig. 3.11 shows the dependence

of Q∆ϕ on the bandwidth of the electrical filter. The input power to the SOA is 0 dBm,

the optical signal to noise ratio is 23-dB, and the 3-dB bandwidth of the second order

supergaussian filter is 15 GHz. The x-axis is the 3-dB bandwidth normalized to the

Chapter 3. SOA nonlinearity postcompensation for PSK signals 80

−1.5 −1 −0.5 0 0.5 1 1.5−2

−1

0

1

2

Time (normalized to bit rate)

Am

plitu

de

[AU

]

(a) Input to SOA

−1.5 −1 −0.5 0 0.5 1 1.5−2

−1

0

1

2

Time (normalized to bit rate)

Am

plitu

de

[AU

]

(b) Output

−1.5 −1 −0.5 0 0.5 1 1.5−2

−1

0

1

2

Time (normalized to bit rate)

Am

plitu

de

[AU

]

(c) Postcompensated

Figure 3.10: DPSK eye diagrams after demodulation; input power is 0 dBm, OSNR=23-

dB, supergaussian optical filter of order 2 and 15 GHz bandwidth.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 81

f3dB×22eff

Diffe

rential

phas

e Q

: 20

log(

Qû3)

[dB

]

-1 0 1

0

1

û3 [r

ad/

]-1 0 1

0

1

û3 [r

ad/

]Time/R

-1 0 1

0

1

Time/R

û3 [r

ad/

]

Time/R

inoutpost

0.5 1 1.5 212

14

16

18

20

22

24

26

28

Figure 3.11: On the left: differential phase Q in dB vs. m(t) 3-dB bandwidth; abscissa

are normalized to the optimal value derived in the previous section. On the right: eye

diagrams of the differential phase at f3dB = 1/(2πτeff). The input power is 0 dBm,

OSNR 23-dB, and optical filter bandwidth 15 GHz.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 82

SOAOOK signal

OBPFfilter exp(.)

Postcompensation stage

LPF

Figure 3.12: OOK post-compensation.

optimal value derived in Sec. 3.2. To be consistent with [103], the differential phase

Q is expressed in dB, i.e., 20 log Q∆ϕ. With this convention, the differential phase Q

required for BER< 10−9 is ∼15.6 dB.

To the right in Fig. 3.11 we present differential phase eye diagrams, where the

differential phase is traced in the [−π/2, 3π/2] interval. The eye diagrams are reported

for the optimal value of filter bandwidth. The figure demonstrates that, as we expected

from the results of the previous section, the post-compensation is very effective at

suppressing the NLPN introduced by the SOA, even when the differential phase Q

is degraded by as much as ∼11 dB before compensation. Moreover, we see a clear

maximum at the optimal filter bandwidth. For this specific case, Q∆ϕ after post-

compensation is within 1 dB of the unimpaired Q over a bandwidth range of 200 MHz

(from 0.9 × 2πτeff to 1.1 × 2πτeff), attesting the robustness of this solution.

3.6 Nonlinear distortion compensation for

intensity modulated signals

We can also exploit the correlation between the SOA output and the gain pertur-

bations to develop a compensation strategy for intensity modulated signals. Consider

the setup presented in Fig. 3.12 where the SOA output is again filtered; in this case

the control signal multiplies the output, essentially inverting the gain perturbations.

We will evaluate the importance of including or not the exponential function inside the

gray dotted box. Without the exponential function, the setup can be built by analog

microwave devices (one splitter, one filter, one mixer).

Chapter 3. SOA nonlinearity postcompensation for PSK signals 83

The post-compensated intensity pSOA,post is given by

pSOA,post(t) = pSOA,out(t)e[Cδhl(t)]

= pSOA,in(t)e[h+δh(t)+Cδhl(t)]

≃ pSOA,in(t)e[h] (3.22)

where C is a properly chosen constant, and we ignored the effect of the low pass

filter (LPF) for the sake of clarity. The filter will be included in the simulations.

Since the input signal to the SOA is an intensity modulated signal, possibly with high

extinction ratio, in principle δhl(t) differs from the true gain δh(t), as (3.2) is the

result of a small signal approximation. As we will see via a numerical example and an

experimental validation, the method nonetheless yields very good results.

Please note that the method as it is cannot compensate for the chirp induced by

the SOA on an intensity modulated signal. This is not an issue if the SOA is used as

a preamplifier, since photodetection is not sensitive to chirp. It could be a problem if

SOAs are used before fiber transmission. SOA-induced chirp could then be compensated

with the scheme of Fig. 3.2. Chirp and intensity-induced distortions could actually be

compensated by cascading the two proposed schemes, or even at the same time with

the scheme of Fig. 3.2 by substituting the phase modulator with a dual-drive Mach-

Zehnder modulator (DDMZM), and deriving appropriate filters for the two arms of the

DDMZM. We leave this matter for future investigations.

3.6.1 Dependence on Saturation Level

Once again, we expect the small signal approximation to be less effective when

increasing the depth of gain saturation. In this subsection, we show through a numerical

example that the proposed method in Fig. 3.12 works very well up to high values of

gain compression. The OOK signal simulated has an extinction ratio of 13-dB at

a bit rate of R = 10 Gb/s. Noise is added such that its OSNR is 23-dB over the

typical 0.1 nm bandwidth. Even though in this section we are mainly interested in

the deterministic distortions induced on the modulated waveform, we include the noise

(with a rather high OSNR) in order to also take into account realistic signal-noise

interactions inside the SOA. The noisy signal is then filtered with a second order

supergaussian filter with a 3-dB bandwidth equal to 1.5R. The SOA nonlinear gain

Chapter 3. SOA nonlinearity postcompensation for PSK signals 84

-30 -25 -20 -15 -10 -5 08

10

12

14

16

18

20

22

Input power [dBm]

20lo

g(Q

eye)

inout

post [exp(/r l)]

post [1+/r l]

Figure 3.13: Qeye for OOK signal at SOA input, output and after post-compensation.

fluctuations distort the signal. The SOA output is photodetected and low pass filtered

with a 5th order Bessel-Thompson filter with 3-dB bandwidth equal to R, in order to

emulate a bandwidth-limited detector. The post-compensation stage follows, composed

of a splitter, the same RC filter presented in the previous sections with impulse response

m(t), and a mixer. The output of the RC filter is an estimate of the integrated gain

δhl(t), therefore we should apply an exponential function. We explore suppressing the

exponential function, allowing implementation in analog microwave devices (splitter,

filter and mixer). In place of the exponential eδhl(t), we use its linearized counterpart

[1 + δhl(t)].

Figure 3.13 illustrates the behavior of the simulated Q factor Qeye as function of

the average input power to the SOA. As can be seen, the method is very effective up

to high input powers. Insets show the eye diagrams at the SOA input, output, and

after post-compensation for 0 dBm input power, demonstrating the effectiveness of the

method. The maximum gain in Qeye is ∼ 9 dB. Without the exponential block, the

post-compensator performance deteriorates slightly, and the maximum gain in Qeye is

∼ 8 dB.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 85

MZM

Laser

BPG

SOA

Agilent 86116A

PC PC ISO ISOVOA

Sampling scope

Figure 3.14: Experimental setup for OOK postcompensation.

3.6.2 Experimental Validation

In order to further establish the effectiveness of the method, we present measured

SOA output waveforms, rather than simulated. The experimental setup is sketched

in Fig.3.14. A polarization controlled distributed feedback laser is modulated by a

MZM driven with a length 27 De Brujin sequence. No significant difference in the

signal statistics was observed when using a longer sequence. The modulated signal

passes another polarization controller (PC) and an isolator (ISO) before entering the

SOA with average input power of -2.65 dBm. The SOA output is again isolated to

avoid backreflections, and a VOA is used to control the power on the photodiode. An

Agilent 86116A wideband sampling oscilloscope captures the output, with the averaging

function invoked; the waveform is the average of 100 realizations. This minimizes the

impact of random noise, as we are mostly interested in the deterministic distortions

induced by the SOA.

Our simulator was extensively verified to assure good predictions of measured wave-

forms, as we reported in [113, 114]. The parameters we used are the same as for the

numerical predictions of the previous sections, and reported in Table 3.1. The resulting

optimal bandwidth of the low pass filter is smaller than 1 GHz. Figure 3.15 summa-

rizes the results of the experiment. Figure 3.15(a) shows the measured waveform at

the output of the MZM (input to SOA), whereas Fig. 3.15(b) is the waveform at the

output of the SOA. With this signal we can estimate (off-line) hl(t) from (3.2). The

result is shown in Fig. 3.15(c). Using the scheme proposed in Fig. 3.12 we obtain

the waveform shown in Fig. 3.15(d), where the distortions introduced by the SOA are

almost perfectly canceled.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 86

-0.5

0

0.5

Est

imat

ed /

h l(t

)

3 3.5 4 4.5 5 5.5 60

1

2

3

Time [ns]

Inte

nsi

ty [A

U] (a)

(b)

(c)

3 3.5 4 4.5 5 5.5 60

1

2

3

Time [ns]

Inte

nsi

ty [A

U]

3 3.5 4 4.5 5 5.5 6Time [ns]

(d)

3 3.5 4 4.5 5 5.5 60

1

2

3

Time [ns]

Inte

nsi

ty [A

U]

Figure 3.15: Measured waveforms of OOK signal at the SOA input (a) and output (b).

The result of offline processing are also reported: (c) is hl(t), and (d) is the signal after

post-compensation.

Chapter 3. SOA nonlinearity postcompensation for PSK signals 87

3.7 Conclusion

Starting from the small signal analysis of the SOA gain dynamic equation, we have

presented a simple yet effective method for the post-compensation of SOA-induced

nonlinearities. We have shown via numerical simulations performance of the nonlin-

ear phase noise compensation for phase modulated signals, and investigated the range

of saturation level and other SOA parameters covered. We have demonstrated both

via simulation and experiment its effectiveness for the compensation of SOA-induced

waveform distortions on OOK signals. The post-compensation stage in Fig. 3.2 could

be integrated into one device, where the new component is the equivalent of a zero

linewidth enhancement factor SOA for phase modulation. The post-compensator of

Fig. 3.12 could be exploited at the receiver side, alleviating the restrictions imposed by

SOAs to the use of intensity modulated signals.

Chapter 4

An introduction to the digital

coherent receiver

In this chapter we explain the basic theoretical concepts behind the optical coherent

receiver. This chapter therefore contains a synthesis of existing literature. We first

investigate its general structure, and then focus on its most important building blocks:

the 90 degrees optical hybrid, and the digital signal processing. We illustrate one of

the possible physical implementations for the hybrid, deriving the equation for the

generic phase and polarization diversity coherent receiver. We then describe, block by

block, the signal processing chain used to recover the data and measure bit error ratios

starting from the received signal. We therefore review the basics of digital chromatic

dispersion compensation, polarization demultiplexing, channel equalization and carrier

phase estimation.

Chapter 4. An introduction to the digital coherent receiver 89

Dual polar. down-

converter

LO

Incoming Signal

I1(t)

Anal

og t

o D

igital

Digital Signal

Processing

Q1(t)

I2(t)

Q2(t)

I1[k]

Q1[k]

I2[k]

Q2[k]

Figure 4.1: Structure of the digital coherent receiver.

4.1 Overview

As explained in Chapter 1, the fundamental objective of the optical coherent re-

ception is to provide information on the linear electric field of the signal hitting the

receiver. The clear advantage over direct detection receivers is to have information on

both amplitude and phase of the field, therefore opening a realm of possible modulation

formats (by exploiting the degree of freedom offered by the phase information of the

field), and signal processing capabilities.

The general structure of the optical coherent receiver is sketched in Fig. 4.1. The

incoming signal and a local oscillator are fed to a dual polarization downconverter,

along with the light coming from a local oscillator. Four electrical currents are output,

carrying information on the real and imaginary parts of the two polarizations of the

incoming signal. The currents are then converted to the digital domain with an analog-

to-digital converter stage. Afterwards, digital signal processing recovers the original

information. In the following subsections we detail both the electro-optical circuit of

the downconverter, and the functions performed by the DSP block. In our experiments,

the ADC operation is realized by a real-time sampling fast oscilloscope, and the digital

signal processing is performed off-line on a computer with Matlab and C programs.

Chapter 4. An introduction to the digital coherent receiver 90

QWPHM

PBSLO

Incoming Signal

QWPHM

Pol.

Pol.

Pol.

Pol.

I1

Q1

I2

Q2

A

B

C

D

E

F

G

H

Hybrid 1

+

-

+

-

+

-

+

-

Hybrid 2

Figure 4.2: Scheme of one possible implementation of the dual polarization downcon-

verter. It includes two 90 degrees optical hybrids (one per polarization), and four

balanced photodiodes. QWP: quarter wave plate; HM: half mirror.

4.2 Electro optical circuit of the dual polarization

downconverter

The scheme of the dual polarization downconverter we used in our experiments

is reported in Fig. 4.2. Other physical realizations are possible, e.g. [116], but the

underlying principle is the same: mixing a local oscillator with the incoming signal to

extract the information on the incoming signal’s amplitude and phase (or, equivalently,

its real and imaginary part). In the following, we detail how the device works.

Let us write the received signal electric field as the sum of two orthogonal polariza-

tion components

Es(t) = Es,1(t)ε1 + Es,2(t)ε2 (4.1)

where ε1 and ε2 are two orthogonal unit vectors, and each of the components can be

Chapter 4. An introduction to the digital coherent receiver 91

written as

Es,p(t) = As,p(t)ej(ωst+φs,p(t)+φn,p(t)) p ∈ 1, 2 (4.2)

where As,p(t) is the amplitude of the electrical field, ωs is the optical carrier frequency,

φs,p(t) is the useful signal phase where the information is encoded, and finally φn,p(t) is

the phase noise of the signal, all on polarization p. The local oscillator electric field is

ELO(t) = ALOej(ωLOt+φLO(t)) (4.3)

where ALO is the amplitude of the field, ωLO is the local oscillator optical frequency,

and φLO(t) is the phase noise of the local oscillator. Note that we have supposed the

local oscillator intensity noise is negligible, i.e., its amplitude ALO does not depend on

time.

As can be seen in Fig. 4.2, the (polarization multiplexed) incoming signal is split by

a polarization beam splitter, whereas the linearly polarized local oscillator is split by a

polarization maintaining 3-dB power splitter. Signal and local oscillator then enter two

optical circuits called 90 degrees optical hybrids. In each hybrid, the local oscillator

travels through a polarizer and a quarter wave plate. The quarter wave plate makes the

local oscillator circularly polarized, i.e. one polarization state is shifted by π/2 with

respect to the other one. Signals then go through a half mirror (50% reflectivity), and

finally two other PBS split their polarization components.

We can write the electric fields at points A, B, C and D in the scheme of Fig. 4.2 as

EA(t) =1

2√

2

(

Es,1(t) + ELO(t)ejπ)

(4.4)

EB(t) =1

2√

2

(

Es,1(t)ejπ/2 + ELO(t)ejπ)

(4.5)

EC(t) =1

2√

2

(

Es,1(t)ejπ + ELO(t)ejπ/2)

(4.6)

ED(t) =1

2√

2

(

Es,1(t)ejπ/2 + ELO(t)ejπ/2)

(4.7)

where, without loss of generality, we have supposed that the signal polarizations along

ε1 and ε2 are aligned with the principal axes of the first PBS (i.e., the upper mixer deals

with Es,1, and the lower one deals with Es,2). In the second part of the chapter, when we

discuss the signal processing, we see how to recover the two transmitted polarizations

starting from the received ones, which are in general misaligned. Let us now define

φ1(t)= [(ωs − ωLO)t + φs,1(t) + φn,1(t) − φLO(t)] (4.8)

Chapter 4. An introduction to the digital coherent receiver 92

Due to the photodetection, we are only interested in the intensities of the optical fields

we just calculated, which are

|EA(t)|2 =18

|As,1(t)|2 + |ALO|2 − 2As,1(t)ALO cos [φ1(t)]

(4.9)

|EB(t)|2 =18

|As,1(t)|2 + |ALO|2 − 2As,1(t)ALO sin [φ1(t)]

(4.10)

|EC(t)|2 =18

|As,1(t)|2 + |ALO|2 + 2As,1(t)ALO sin [φ1(t)]

(4.11)

|ED(t)|2 =18

|As,1(t)|2 + |ALO|2 + 2As,1(t)ALO cos [φ1(t)]

(4.12)

And finally (assuming unit responsivity for the photodiode) the photocurrents out-

put of the upper balanced photodiodes are

I1(t) = |ED(t)|2 − |EA(t)|2 =12

As,1(t)ALO cos [φ1(t)] (4.13)

Q1(t) = |EC(t)|2 − |EB(t)|2 =12

As,1(t)ALO sin [φ1(t)] (4.14)

Note that I1(t) and Q1(t) are the real and imaginary part (respectively) of the

complex signal 12As,1(t)ALOejφ1(t). Going back to the definition of φ1(t) it is clear that,

if one is capable of estimating the difference between the signal and local oscillator

frequencies ωs − ωLO, and the residual phase noise φn,1(t) − φLO(t), then one has access

to the amplitude and phase of one of the polarizations of the incoming signal. When

using higher modulation the information is coded in both amplitude and phase.

We define φ2(t) as

φ2(t)= [(ωs − ωLO)t + φs,2(t) + φn,2(t) − φLO(t)] (4.15)

where φn,2(t) is the phase noise of the second polarization of the incoming signal and

φs,2(t) is the information bearing phase. With the same calculations as before, the

photocurrents I2(t) and Q2(t) can be written as

I2(t) =12

As,2(t)ALO cos [φ2(t)] (4.16)

Q2(t) =12

As,2(t)ALO sin [φ2(t)] (4.17)

Chapter 4. An introduction to the digital coherent receiver 93

The four outputs from the dual polarization downconverter, I1(t), Q1(t), I2(t) and

Q2(t) provide all components necessary for reconstruction of the complex field of the

incoming signal. However, the four outputs require significant processing to compensate

for unavoidable perturbations of the signal. At a minimum, digital signal processing

must:

– recover the original polarization axis from the received ones,

– estimate and correct the frequency offset (ωs − ωLO) between the local oscillator

and the incoming signal, and

– estimate and correct the accumulated phase noise [φn,p(t) − φLO(t)]

4.3 Digital signal processing

The block diagram of the digital signal processing chain we explain (and use) is

sketched in Fig. 4.3. Two examples are also provided: on the left, we report samples

of a PDM-QPSK signal (in the complex plane, for the two polarizations) at different

points along the chain. On the right, we do the same for a PDM-BPSK signal.

4.3.1 Normalization and resampling

We begin by normalizing the four output signals to compensate for different path

delays and attenuations of the four lanes. The normalization coefficients are found

during calibration of the receiver non-ideal components.

A resampling function also must be performed, and depends on the specifications of

the ADC stage. Typical state-of-the-art analog-to-digital converters have bandwidths

up to 16 GHz and sampling rates up to 50 Gsamples/second. These numbers apply to

the ADC we used for experimental results presented in the next sections and chapters.

The electrical transfer function of the ADC converters typically cuts sharply at the

nominal bandwidth. The Nyquist sampling theorem states that we should sample at

a rate which is twice as high as the highest frequency component of the signal to be

digitalized. In the case of the ADC that we used we sample at 50 Gsamples/second,

therefore the signal must not have frequency components above 25 GHz. This condition

Chapter 4. An introduction to the digital coherent receiver 94

Resampling

j j

CD comp. CD comp.

FIR-based polarization demux and equalizer

CFE/CPE CFE/CPE

Decision and BER

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2

-1

0

1

2

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2Re [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

-2 0 2-2

-1

0

1

2

Re [au]

Im [au]

PDM-QPSK example PDM-BPSK example

I1[k] Q1[k] I2[k] Q2[k]

EDi,1[k]

EDi,2[k]

EDo,1[k]

EDo,2[k]

Ep,1[k] Ep,2[k] Ep,1[k] Ep,2[k]Ep,1[k]

Ep,2[k]

EDi,1[k] EDi,1[k]EDi,2[k] EDi,2[k]

EDo,1[k] EDo,2[k] EDo,1[k] EDo,2[k]

Figure 4.3: DSP chain and two examples with measured data: PDM-QPSK on the left,

and PDM-BPSK on the right.

Chapter 4. An introduction to the digital coherent receiver 95

is fulfilled due to the low-pass nature of the ADC converter, which starts to cut the

signal at 16 GHz.

The resampling function in the DSP assures our useful signal has exactly two samples

per symbol (sps). If the analog signal has a symbol rate of R Gbaud, and the sampling

rate is 1/Tc, then the output of the DSP has sps = 1/RTc samples/symbol. In the

typical cases of R = 21 Gbaud or R = 28 Gbaud and 1/Tc =50 Gsamples/s, the

digitalized signal has sps = 2.38 or sps = 1.78 samples/symbol, respectively. We

therefore need to resample the signal by a factor rc = 2/sps. There are many algorithms

to resample a signal, the simplest of which is based on upsampling the signal by an

integer factor rU , filtering it to prevent aliasing, and finally downsampling it (also by

an integer factor, called rD), where rU and rD are such that rU/rD = rc [117].

4.3.2 CD compensation

Chromatic dispersion is one of the most important linear impairments imposed by

fiber propagation. In the absence of fiber nonlinearity, the scalar nonlinear Shroedinger

equation can be written as

∂U(z, t)∂z

= jDλ2

4πc

∂2U(z, t)∂t2

(4.18)

where U(z, t) is the signal electric field, z and t the fiber length and the time in the

moving frame, D the fiber dispersion coefficient, λ the signal wavelength and c the

speed of light. Equation (4.18) is a partial differential equation whose solution is a

filtering function U(z, ω) = G(z, ω)U(0, ω), where the resulting filter is all pass, and

can be written as

G(z, ω) = exp

(

−jDλ2

4πcω2

)

(4.19)

To ideally undo the effects of chromatic dispersion, it is sufficient to filter the signal

after propagation with an inverse filter whose frequency response is

Ginv(z, ω) =1

G(z, ω)= exp

(

jDλ2

4πcω2

)

(4.20)

and whose impulse response is

Ginv(z, t) =√

jc

Dλ2zexp

(

jπc

Dλ2zt2)

(4.21)

Chapter 4. An introduction to the digital coherent receiver 96

As can be seen, Ginv(z, t) is infinite in duration with an all-pass spectrum. This

leads to aliasing when implemented digitally in the time domain, no matter how high

the sampling rate. The impulse response must be truncated for a DSP implementation.

If we sample every Tc seconds, we need to truncate the impulse response for ω > π/Tc

for aliasing not to occur. The impulse response Ginv(z, t) is basically a linearly chirped

signal, whose angular frequency is linearly increasing with time

ω = j2πc

Dλ2zt (4.22)

The condition for absence of aliasing (with fixed sampling rate) reduces to

− |D|λ2z

2cTc< t <

|D|λ2z

2cTc(4.23)

Once the impulse response is truncated, we approximate it with a finite impulse

response (FIR) filter Ginv[k] whose Nt taps are samples of the continuous-time response

[118]

Ginv[k] =

√

jcT 2

c

Dλ2zexp

(

jπcT 2

c

Dλ2zk2

)

−⌊

Nt

2

⌋

< k <⌊

Nt

2

⌋

(4.24)

where the total number of taps is Nt =⌊

2|D|λ2z2cT 2

⌋

+ 1 , where ⌊x⌋ indicates the smallest

integer number of the argument x.

As a rule of thumb, the number of taps needed in the 1550 nm window for |D|z =1000 ps/nm

of dispersion to be compensated scales as Nt = 0.032R2, where R is the symbol rate in

Gbaud. For 21 Gbaud, Nt = 14 taps are required for 1000 ps/nm. Fewer taps can be

used by truncating the impulse response sooner than the limit imposed to avoid alias-

ing. Results show that the number of taps can be reduced by (roughly) 60% without

significant loss in performance [118].

For the sake of completeness, Fig. 4.4 shows the reference structure of a digital FIR

filter made of N taps, which filters the signal sin[k − n] to produce the signal sout[k]:

sout[k] =N−1∑

n=0

cnsin[k − n] (4.25)

where cn are called the taps of the filter.

Chapter 4. An introduction to the digital coherent receiver 97

z-1z-1 ......

c0 c1 c2 cN-2 cN-1

sin[k] sin[k-1]

sout[k]

sin[k-2]z-1

sin[k-N-2] sin[k-N-1]

Figure 4.4: FIR filter structure.

4.3.3 Polarization demultiplexing and equalization

After chromatic dispersion compensation, implemented with two simple FIR filters,

the polarizations are still mixed, as can be seen in the examples of Fig. 4.3.

While polarizations are separated at transmission, propagation mixes them. The

polarization mixing of an optical fiber channel can be modeled as a 2×2 Jones matrix.

This matrix is in general not unitary due to polarization dependent loss (PDL), and

is frequency dependent due to PMD. The polarization demultiplexing block aims to

estimate and equalize the Jones matrix of the channel, thus separating the two polar-

izations as they were at transmission. The Jones matrix is time dependent, therefore

the equalizer must be adaptive.

One of the most widely used schemes for adaptive equalization is based on FIR

filters arranged in a 2×2 butterfly structure, shown in Fig. 4.5. We can express the

output as

EDo,1[k] = h11 ⊗ EDi,1[k] + h12 ⊗ EDi,2[k] (4.26)

EDo,2[k] = h21 ⊗ EDi,1[k] + h22 ⊗ EDi,2[k] (4.27)

where h11, h12, h21, h22 are adaptive filters with Nd taps each. The literature on adap-

tive filters is huge [119]. For the specific case of coherent optics with phase modulation,

one of the most popular algorithms for adapting the filter coefficient is called the con-

stant modulus algorithm. The idea behind the constant modulus algorithm (CMA) is

to exploit a common property to any PSK format: the fact that the intensity of the

signal is constant (since the modulation is on the phase only). Therefore, the CMA

adjusts the filter taps in order to push the output signals EDo,1 and EDo,2 onto the unit

circle, as can be seen in the examples in Fig. 4.3.

Chapter 4. An introduction to the digital coherent receiver 98

h11

h21

h12

h22

EDi,1[k]

EDi,2[k]

EDo,1[k]

EDo,2[k]

feedback

feedback

Figure 4.5: The 2×2 butterfly structure of FIR filters used for polarization demulti-

plexing and channel equalization.

Defining the error functions ep = 1 − |EDo,p|2 for p = 1, 2, the equalizer tries to

minimize the gradient of the mean square of the error functions with respect to the

taps (which ideally would be zero)

∂〈e21〉

∂h11= 0;

∂〈e21〉

∂h12= 0;

∂〈e22〉

∂h21= 0;

∂〈e22〉

∂h22= 0; (4.28)

The equalizer tap weights are iteratively updated according to the stochastic gradi-

ent algorithm [120] as follows

h1p −→ h1p + ςe1EDo,1E∗Di,p p = 1, 2 (4.29)

h2p −→ h2p + ςe2EDo,2E∗Di,p p = 1, 2 (4.30)

where ς is a adjustable convergence parameter, the star superscript denotes complex

conjugate, and the arrow shows the update mechanism. This corresponds to a blind,

adaptive, feed-forward, multiple-input multiple-output (MIMO) channel equalizer.

It is clear that the complexity of the equalizer grows with the number of taps, as

does its correction capability. With a single tap, the equalizer can only compensate for

a simple polarization rotation (by converging to the opposite rotation matrix). With

more taps, the impulse response of the equalizer covers over more than one sample,

and makes it capable of correcting any form of intersymbol interference may come from

PMD, residual CD from the previous DSP block, narrow filtering along the optical line,

misalignment between optical carrier or reconfigurable optical add-drop multiplexer

(ROADM) filters, and so forth. With as few as 7 taps per filter an impressive 100 ps

Chapter 4. An introduction to the digital coherent receiver 99

(.)M arg(.)/M unwrap(.)filter W[k]

e-j(.)

delay

Phase estimation

Figure 4.6: Carrier phase estimation and correction. The delay block compensates for

the eventual delay introduced by the filter W [k].

of DGD tolerance has been demonstrated [121], yielding a factor of ∼ 5 improvement

as compared with direct detection.

As in the examples of Fig. 4.3, at this stage the two polarizations are successfully

demultiplexed (including ISI mitigation), but a decision on the symbols cannot be taken

yet. The frequency offset between the local oscillator and the signal, as well as the phase

noise, still must be estimated and corrected.

4.3.4 Carrier frequency and phase estimation

Typical temperature stabilized DFB lasers with wavelength locking have a frequency

accuracy in the order of ± 1.25 GHz [122]. Therefore, even though the free running LO

is tuned to the same nominal frequency as the signal laser, we have to expect a nonzero

frequency offset between the two. This frequency mismatch can be estimated digitally

with well known methods such as those reported in [123, 124]. The algorithms for this

purpose require limited amount of DSP resources, as the frequency mismatch is a slowly

varying quantity. Once the frequency mismatch is evaluated, it can be corrected 1) in

DSP by a rotating phasor with opposite sign, or 2) by providing a (slow) feedback loop

to the local oscillator to tune the right frequency.

Carrier phase estimation in a digital coherent receiver is a much more thorny issue,

due to the rapid changes of the phase noise. The linewidth of semiconductor DFB

lasers used as the transmitter and LO typically ranges from 100 kHz to 10 MHz. Many

algorithms for phase estimation exist, and research is very active in this area. We

focus on a feed-forward non-data aided (NDA) approach widely used due to its balance

between the ease of implementation (and therefore low DSP resource consumption) and

good performance.

Chapter 4. An introduction to the digital coherent receiver 100

The technique is based on the so called M-th power phase estimation originally

proposed in [125], and sketched in Fig. 4.6. Let us rewrite one polarization of the signal

Eout[k] as

EDo,1[k] = ej[φs,1[k]+φnLO,1[k]] + n[k] (4.31)

where φnLO,1[k]= φn,1[k] − φLO[k] is the effective noise phase to be evaluated, and

n[k] is the additive noise sample. We have supposed the modulus of the signal field is

constant and equal to one. By raising the signal Eout,1 to the M-th power, where M is

the constellation order (M = 2 for BPSK, M = 4 for QPSK), we get

(EDo,1[k])M = ejMφnLO,1[k] + m[k] (4.32)

where we have exploited the fact that for M-PSK ejMφs,1[k] = 1, i.e., raising to the M-

th power removes data modulation. The term m[k] contains all the cross-terms between

signal and noise that are the source of perturbation in the phase estimation. The noise

can be reduced by applying a filter W [k]. Let us take W [k] as a simple equal-tap-weight

FIR filter with NW taps, so we can write an estimate of the phase noise as

φnLO,1[k] = unwrap

1M

arg

NW∑

k=1

(EDo,1[k])M

(4.33)

Per estimation theory, the optimal choice for W [k] is a Wiener filter [117, 126, 127].

The estimate of (4.33) is used to de-rotate the signal so that transmitted symbols

can be recognized, as seen in Fig. 4.3. Having completed our recovery of the optical

field, we can use standard procedures for detection of the transmitted symbols.

We have reviewed in this chapter the hardware implementation of the digital coher-

ent receiver. We have detailed the digital signal processing that is required to extract

information from the incoming signal. Next chapter will make use of this material

and present experimental results with a digital coherent receiver in the context of the

upgrade of current terrestrial optical networks.

Chapter 5

Dual versus single carrier for

40 Gb/s coherent BPSK systems

Current core terrestrial optical networks are mostly working with 10 Gb/s OOK

channels on a 50 GHz grid, and optimized dispersion maps. A promising modulation

format for reaching higher bit-rates (40 Gb/s and 100 Gb/s) is polarization division

multiplexed MPSK, paired with digital coherent detection. When transitioning from

10 Gb/s on a 50 GHz grid to 40 Gb/s and more per wavelength slot, the two solutions

(10 Gb/s and coherently detected MPSK) must coexist on different wavelengths in the

same fiber. For 40 Gb/s, it has been recently shown how this can be best achieved

with PDM-BPSK [128, 129, 130]. Our contribution described in this chapter is to

experimentally evaluate the performance of two competing implementations of 40 Gb/s

PDM-BPSK employing coherent detection. We experimentally compare BPSK-based

dual- and single-carrier coherent solutions at 40 Gb/s. For transmission over 2400 km

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 102

of low-dispersion fiber, the single-carrier solution is shown to be more tolerant to cross-

nonlinearities induced by 10 Gb/s neighbors, due to its higher baud rate. The single

carrier solution is therefore more appropriate for the upgrade of legacy terrestrial core

networks.

5.1 Coherent Detection in Core Networks

The optical telecommunications community has expressed great interest in the up-

grade of the currently installed intensity modulated 10 Gb/s systems. Progressive

updates are particularly attractive, where some wavelength channels at 10 Gb/s are re-

placed with 40 Gb/s (and even 100 Gb/s) channels in a drop-in fashion. Such a rollout

is economical as only the affected transponders need be changed, and the underlying

system remains untouched (including any neighboring 10 Gb/s channels). The great

challenge in this scenario is the coexistence of the newly inserted channels with the

former OOK signals, all on the same 50 GHz grid and with the legacy dispersion map

that was designed and optimized for intensity modulation.

For 100 Gb/s transmission, industry has reached a consensus for adoption of PDM-

QPSK at a symbol rate of 28 Gbaud (for a gross bit rate of 112 Gb/s, including 12%

overhead for error correction). For 40 Gb/s transmission, the debate is still open.

Much attention has been devoted to PDM-QPSK at 10.7 Gbaud (for a gross bit rate of

42.7 Gb/s, including 7% overhead for error correction). Unfortunately, PDM-QPSK at

42.7 Gbaud is greatly perturbed by co-propagating intensity modulated signals [131].

The main problem is the XPM induced by the strong intensity variations of the OOK

neighbor channels. In this chapter we focus on this impairment, and possible solutions.

BPSK is known to have a higher tolerance to XPM than QPSK [72], thanks to

the double angular distance between the symbols in the complex plane (π for BPSK,

π/2 for QPSK). As an example, we report in Fig. 5.1 the results of a measurement

performed by Bertran-Pardo et al. in [71] for PDM-QPSK, and our measurement on

the same transmission link. We report QBER-penalty relative to an OSNR limited

transmission. QBER is defined as in (2.2.4): QBER = 20 log10

[√2inverfc (2BERmeas)

]

.

The link is a 800 km typical dispersion managed terrestrial link of NZDSF. The PSK

channel was embedded in 10 Gb/s OOK neighbors. Details of the experiment will be

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 103

-1

0

1

2

3

4

5

6

7

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

Power per chan [dBm]

QB

ER-p

enal

ty [dB

] 40 Gb/s PDM-QPSK

40 Gb/s PDM-BPSK

Figure 5.1: Performance comparison of BPSK and QPSK over NZDSF fiber with 10G

NRZ OOK neighbors. QPSK measurements courtesy of Oriol Bertran-Pardo et al. [71]

explained later in the chapter. Most importantly, we note that PDM-BPSK enhances

the nonlinear threshold (defined as the power per channel at which the solutions reach

the same penalty of 1 dB) by roughly 5 dB, providing greater robustness to perturbation

from OOK neighbors (XPM in particular). We will therefore explore solutions based

on BPSK.

To achieve 42.8 Gb/s, BPSK imposes higher requirements (by a factor of 2) on

bandwidth and sampling rates of transponders, compared to QPSK. To reduce such

requirements, much interest has been devoted to using two sub-carriers inside the same

50 GHz slot [132, 133]. Each 50 GHz slot is divided in two sub-slots, half as wide,

each carrying half the data rate. Doing so decreases the baud rate per carrier, but

requires a doubling in the number of components. We will refer to this solution as dual

carrier (DC), as opposed to the single carrier (SC) where we have only one carrier per

50 GHz slot. Figure 5.2 depicts the complexity and hardware requirements of the two

solutions. For DC, at the transmitter we need two laser sources with tight wavelength

control, four modulators and two polarization multiplexers. At the receiver, we require

two coherent detectors, one per carrier.

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 104

d1

R/2

R/2

d2

R/2

R/2

ADC (R)

90° hybrid

DSPLO #1

90° hybrid

DSPLO #2

ADC (2R)

90° hybrid

DSPLO

s

R

R

Rx

Tx

TxRx

Single Carrier

Dual Carrier

Figure 5.2: Schematic illustration of the different hardware complexity between SC and

DC

The performance of the DC solution using PDM-QPSK format paired with coherent

detection has recently been investigated at 100 Gb/s [134], where it was shown that the

DC configuration is more sensitive to cross-nonlinear impairments induced by 10 Gb/s

NRZ neighbors, as compared to the SC solution. Our contribution is the experimen-

tal characterization of the performance of single carrier vs. dual carrier PDM-BPSK

solutions.

The work presented in this chapter was accomplished during an internship at the

WDM Dynamic Networks Department of Alcatel-Lucent Bell Labs in Paris, France.

The test facilities at our disposal included multiformat transmitters, a NZDSF recircu-

lating loop based on a typical terrestrial dispersion map, and a state-of-the-art coherent

receiver. At this facility we were able to assess the performance of one 40 Gb/s channel

inserted in a wavelength slot with 50 GHz spacing, originally designed for NRZ-OOK

channels at 10 Gb/s. We compare it with the SC solution in terms of tolerance to linear

and nonlinear impairments. Our goal is to assess if the DC solution can be beneficial

at 40 Gb/s, despite its unavoidable increase in complexity of the transceivers.

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 105

BPSKd1

10.7 Gb/s

BPSKd210.7

Gb/s

even

odd

OOK10.7

Gb/s

OOK10.7

Gb/s

even

odd

OOK10.7

Gb/s

OOK10.7

Gb/s

BPSKs

21.4 Gb/s

Figure 5.3: Experimental configuration of the two transmitters used in the experiment.

5.2 Experimental setup

Figure 5.3 shows the experimental setup of the two transmitters used to investigate

the relative performance of SC and DC PDM-BPSK at 40 Gb/s. In both cases we have

79 distributed feedback lasers spaced by 50 GHz in the spectral range [1530.31 nm –

1562.61 nm] and separated in even channel and odd channel combs. The combs are

intensity modulated with two independent Mach-Zehnder modulators driven by two

10.7 Gb/s pattern generators to produce NRZ signals. All the binary sequences used

in the experiment are 215-1 bits long.

For all configurations, the channel under test is located in the slot centered on

λs = 1546.92 nm. In the SC configuration (Fig. 5.3, right), the laser source at λs is

modulated by a BPSK modulator fed with a pseudo random bit sequence of length

215 −1 at 21.4 Gb/s. Polarization multiplexing is emulated by splitting the signal along

two paths, delaying one of the two with a section of polarization maintaining fiber, and

recombining the two paths using a polarization beam combiner. The test channel is

then coupled into the modulated even comb. The resulting sub-multiplexed signal is

passed into a polarization scrambler and then it is spectrally interleaved with the odd

comb (also polarization scrambled).

In the DC configuration (Fig. 5.3 left), data is modulated onto two lasers and using

two separate BPSK modulators operating at 10.7 Gb/s. We denote with λd1 and λd2

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 106

Tx A.O.DCF

DCF100 km NZDSF

DCF100 km

DCF100 km

DCF100 km

WSS A.O.

OBPF

90°

hybr

id

Sam

plin

g

DS

P (

offli

ne)

LO

EDFA

Rx

Pre-compensation

6

Figure 5.4: Experimental setup of the fiber recirculating loop and the receiver.

the wavelengths of the two subcarriers, separated by the frequency offset ∆f . As

in the previous case, the output 21.4 Gb/s DC signal is polarization multiplexed to

42.8 Gb/s, subsequently combined with the modulated even comb and, through the

interleaver, with the modulated odd comb.

The experimental setup for fiber propagation is described in Fig. 5.4. The transmit-

ter output is boosted through a dual-stage EDFA, including optical pre-compensation,

and sent into a recirculating loop. The loop incorporates four 100 km-long spans of

NZDSF with 4 ps/nm.km dispersion at 1550 nm and 72 µm2 effective area. Each span

is followed by dual-stage EDFAs. Each EDFA includes a DCF in order to partially

compensate CD. The dispersion map, including pre-compensation, was optimized ac-

cording to an optimal terrestrial transmission dispersion map at 10 Gb/s as in [135].

A wavelength selective switch (WSS) is inserted at the end of the fourth span in order

to equalize channel power. In these experiments, we vary the power per channel at

each fiber input from -7 to -1 dBm by increasing the output power of each dual-stage

EDFA from 12 to 18 dBm (the WSS was recalibrated for each input power level). This

allows us to sweep the end-of-link optical signal to noise ratio. We measure the BER

after six loop roundtrips, corresponding to a transmission distance of 2400 km. In

all measurements, the power of the test channel is set at the same level as the OOK

channels.

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 107

At the receiver side, the channel under test is selected by a tunable filter and sent to

the coherent receiver. The coherent receiver has the structure we explained in section

4.1. The dual polarization downconverter has the structure we illustrated in section

4.2. It is composed of a polarization beam splitter followed by two 90 degree hybrids,

one for each polarization state. The hybrids combine the incoming signal with a nar-

row linewidth (∼100kHz) local oscillator so as to supply the in-phase and quadrature

components of the signal to the balanced photodiodes.

The four balanced photodiode outputs are digitized by the ADCs of a real-time

oscilloscope. The oscilloscope works at 50 Gsamples/s, with an equivalent front-end

electrical bandwidth of 16 GHz. The samples are finally stored by sets of 2 million

(corresponding to a time slot of 40µs). Acquired waveforms are processed off-line. The

DSP we apply is the same we explained in section 4.3. At first, we resample the signal

at twice the symbol rate and we compensate the residual chromatic dispersion with

an FIR filter. We then use a 2×2 butterfly adaptive equalizer to demultiplex the two

polarizations [118]. Carrier phase follows, using the M-th power algorithm [125] with

optimized tap length. Finally, symbols can be identified and BER is computed.

BER is subsequently converted into QBER, as per (2.2.4). For clarity, Fig. 5.5

visualizes QBER for a wide range of BERs. In the DC case, the local oscillator is

sequentially tuned on the two sub-carriers, signals are independently processed, and

the results shown in this chapter are the average of the BER measured on each sub-

carrier.

5.3 Single channel back-to-back measurements

In Fig. 5.6, we show the back-to-back QBER of the dual carrier configuration as a

function of the subcarrier frequency offset ∆f , for two values of OSNR. As we can

see, cross-talk between subcarriers has visible impact when the frequency offset is lower

than 15 GHz. Performance is optimal for a wide range of frequency offsets beyond

15 GHz, regardless of OSNR. We chose to work in the middle of the plateau where

QBER is roughly constant, and the frequency offset ∆f for the rest of the experiment

will be fixed to 21.4 GHz.

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 108

-12-11-10-9-8-7-6-5-4-3-2-12

4

6

8

10

12

14

16

18

Log10(BER)

QB

ER [dB

]

Figure 5.5: Correspondence of QBER and bit error ratio.

12 14 16 18 20 22 24

OSNR = 12 dB

OSNR = 9 dB

7

8

9

10

11

12

13

QB

ER [dB

]

Separation between subcarriers [GHz]

Figure 5.6: Measured QBER for DC as function of the subcarrier frequency offset ∆f .

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 109

8 9 10 11 12 13 14 157

8

9

10

11

12

13

QB

ER [dB

]

7

OSNR [dB/0.1nm]

Dual Carrier

Single Carrier

Figure 5.7: Measured QBER for DC and SC as function of OSNR. A single channel is

measured in back-to-back.

We can now measure the BER in back-to-back operation. Figure 5.7 shows the

back-to-back performance of the investigated schemes as a function of the OSNR at

the receiver. At a given channel power, the energy per symbol is the same for the two

configurations, therefore we theoretically expect the same BER. In fact the DC solution

turns out to be slightly better (by a fraction of a dB), which we attribute to the better

performance of our 10 Gb/s modulators compared to the 20 Gb/s ones. back-to-back

performance is limited by additive white Gaussian noise. Our results indicate that there

is no difference between the two solutions with respect to sensitivity to AWGN. While

the two solutions appear equivalent, we will show in the rest of the chapter that this is

not the case, as the tolerance to nonlinear effects is very different.

The third and last back-to-back test assesses the robustness of the two configurations

to optical filtering. Figure 5.8 shows the measured QBER as a function of the 3-dB

bandwidth of a flat-top optical filter. As it can be seen, the SC solution turns out to be

much more tolerant to narrowband optical filtering. The reason is that the SC optical

spectrum is much more compact around the carrier frequency than the DC spectrum,

which is more “distributed” in the 50 GHz slot. Figure 5.9 shows measured spectra

where this characteristic can be observed.

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 110

45

3 dB bandwidth [GHz]

7

8

9

10

11

12

13

QB

ER [dB

]

Dual Carrier

Single Carrier

25 4937 4129 3317 21

Figure 5.8: Measured QBER as function of the 3-dB bandwidth of an optical flat-top

filter.

5.4 WDM transmission over 2400 km

In order to test the tolerance to nonlinear effects of the solutions, we transmit over

our 2400 km of NZDSF. The neighbor channels are OOK modulated, at the same

average power as the coherent signal. We measure BER, and Fig. 5.10 and 5.11 show

the experimental results obtained for SC and DC solutions, respectively. The curves

present the QBER-penalty as a function of the power per channel. The penalty is relative

to an OSNR limited transmission. Whenever the QBER-penalty is different from zero,

it means that some nonlinear effect has been triggered. Solid curves refer to WDM

measurements, and dashed curves to single channel measurements. We can see that

the penalty associated with WDM transmission is much higher than that of the single

channel. This support our hypothesis that the inter-channel effects are dominant in

this scenario, due to the presence of intensity modulated channels.

The channels most responsible for the BER penalty in a dispersion managed WDM

environment are presumably the closest neighbors [136, 131]. We therefore did another

measurements turning off the two closest neighbors, leaving one empty slot on both sides

of the test channel. This creates a 100 GHz guardband on each side, and the measured

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 111

(a)

1545.5 1546 1546.5 1547 1547.5 1548Wavelength [nm]

10 dB0.4 nm

1545.5 1546 1546.5 1547 1547.5 1548Wavelength [nm]

10 dB0.4 nm

1545.5 1546 1546.5 1547 1547.5 1548Wavelength [nm]

10 dB0.4 nm

(b)

(c)

Figure 5.9: Measured spectra of (a) dual carrier and (b),(c) single carrier solutions.

Surrounding 10 Gb/s NRZ-OOK channels can also be seen. Resolution is 100 MHz.

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 112

-1

0

1

2

3

4

5

6

7

-8 -7 -6 -5 -4 -3 -2 -1 0

Power per chan [dBm]

QB

ER-p

enal

ty [dB

]

Single Chan

SC WDM

Figure 5.10: Measured QBER penalty after transmission for the single carrier solution.

-1

0

1

2

3

4

5

6

7

-8 -7 -6 -5 -4 -3 -2 -1 0Power per chan [dBm]

QB

ER-p

enal

ty [dB

]

Single Chan

DC WDM

DC and guardband

Figure 5.11: Measured QBER penalty after transmission for the dual carrier solution.

Chapter 5. Dual versus single carrier for 40 Gb/s coherent BPSK systems 113

spectrum is shown in Fig. 5.9(c). The dashed curve diamond markers in Fig. 5.11 refers

to the measurement done with a 100 GHz guardband for the DC solution (i.e., the two

adjacent channels removed).

At (for example) a power per channel of -3 dBm, the QBER-penalty of the DC

solution is ∼6 dB, whereas the QBER-penalty for SC is ∼3.5 dB. This indicates that

DC is more affected by cross nonlinear effects induced by intensity modulated signals.

Cross phase modulation induced penalties generated by NRZ-OOK channels depend on

the symbol rate of the test channel. In particular, XPM arising from 10 Gb/s NRZ-

OOK channels is less detrimental when the symbol rate of the phase modulated test

channel is higher than the bit rate of the intensity modulated channels [136, 131].

In our measurement of DC with the 100 GHz guardband, the QBER-penalty falls

to ∼3.5 dB. The penalty becomes close to that of the SC solution without guardband.

This means that the DC solution requires a 100 GHz guardband to achieve roughly the

same performance of the SC solution.

In conclusion, the SC solution shows a higher tolerance to inter-channel effects

than the DC one, in addition to having simpler hardware architecture. Figure 5.2

depicts the hardware requirements of the transmitter and receiver for the two solutions.

Comparing the two rows of the figure, it can be seen that dual carrier requires almost

twice the number of components (operated at half the speed) than the single carrier,

hence increasing the complexity and footprint of the solution.

5.5 Conclusion

We have investigated the tolerance to inter-channel nonlinear impairments of the

dual-carrier coherent BPSK configuration paired with coherent detection over a typical

terrestrial link of 24×100 km of low dispersion fibers. In addition to doubling the

component count for transmitter and receiver, the dual-carrier solution turns out to be

more sensitive than single carrier to inter-channel cross-nonlinearities stemming from

co-propagating intensity modulated channels at 10 Gb/s.

Conclusions and Future Work

We have investigated novel methods for phase shift keying to help meet the growing

demand for increased throughput in optical communications. We have covered both

differential and coherent detection.

Our first contribution was a thorough investigation of a simplified receiver for

DQPSK signals. The receiver is based on a single optical filter, whose 3 dB bandwidth

is narrower than the signal symbol rate. We have shown its performance vis-à-vis lin-

ear and nonlinear impairments. We have also shown that only two network devices,

if properly designed, are required for simultaneously demultiplexing and demodulating

a comb of DQPSK signals. There is a penalty with respect to the conventional full

receiver, roughly 3 dB in back-to-back operation, and less than 3 dB (1.2 dB in the

case that we have investigated) after transmission. Moreover, the narrow filter receiver

is more tolerant than the conventional receiver to both the detuning of the carrier from

the central frequency, and to uncompensated chromatic dispersion. The trade-off is

therefore between complexity/cost and performance, and the less complex proposed

receiver might be interesting in cost-driven networks looking to increase their spectral

efficiency with DQPSK.

Our second contribution also aims at reducing the complexity of PM systems. It

focuses on the use of SOAs (instead of EDFAs) to provide amplification to phase mod-

ulated signals. SOAs and phase modulation are generally perceived as a good com-

bination, due to the quasi-constant intensity of PM signals. If the signal envelope is

Conclusions and Future Work 115

constant, SGM and XGM become much less of an issue. The dominant nonlinearity

in this scenario becomes SPM, which comes from the nonzero linewidth enhancement

factor of SOAs. We propose and verify a method to postcompensate NLPN induced

by the SOA. This technique is derived from the small signal analysis of carrier density

fluctuations, and we demonstrate via simulation almost complete cancellation of the

NLPN added by a saturated SOA on a differential PSK signal.

For our third contribution we move our focus to the more performance-driven sce-

nario of terrestrial core networks, and we investigate possible solutions for the upgrade

of such networks to 40 Gb/s and beyond. The technology of choice is the digital coher-

ent receiver, a paradigm shift with respect to differential detection. We compare two

solution for 40 Gb/s networks, namely single and dual carrier PDM-BPSK. Despite

having the same noise sensitivity, we show that using only one carrier is beneficial for

the resilience to nonlinear effects, and in particular to XPM, which is the dominant

nonlinearity in hybrid networks where phase modulated signals share the fiber with

previously installed intensity modulated channels.

In summary, this thesis covered many aspects of the applicability of phase mod-

ulation to optical communications. On one hand we offer solutions to overcome its

higher cost and complexity (as compared to intensity modulation), in order to permit

lower complexity networks to enjoy the benefits that phase modulation brings in terms

of nonlinear resilience and noise sensitivity. On the other hand, we assess the perfor-

mance of a promising solution for 40 Gb/s high-end networks, using digital coherent

receivers.

Our work on the narrow filter receiver presented in Chapter 2 might be extended

by an optimization of the filter shape in the WDM case. We know that the optimal

shape for the single channel case is Gaussian, and we used the same one for the WDM

experiment. Especially when the channel spacing is tight, the filter shape becomes an

important free parameter to be optimized. The tradeoff is between the crosstalk between

adjacent channels (if the filter skirts are not steep enough), and signal distortions (if the

filter skirts are too steep). An optimal shape, possibly different from Gaussian, might

increase the interest of the proposed receiver.

The SOA phase noise mitigation technique presented in Chapter 3 for a single chan-

nel might be extended to the WDM scenario, where SOAs would amplify a comb of

Conclusions and Future Work 116

signals. When more than one channel is present at the SOA input, the reservoir of

carriers fluctuates as a function of the total WDM signal power, causing nonlinearities.

Given the low pass nature of the SOA, the reservoir fluctuations have a bandwidth in

the order of a few GHz. These fluctuations can therefore be estimated and corrected

with the schemes we propose. Nevertheless, the extension of our work to the WDM

case is not straight forward, since gain might vary from channel to channel, depending

on the number of channels, their spacing, and the spectral shape of the SOA gain. We

leave this matter as a subject of future investigations. Please note that the method as

it is cannot compensate for the chirp induced by the SOA on an intensity modulated

signal. This is not an issue if the SOA is used as a preamplifier, since photodetection is

not sensitive to chirp. It could be a problem if SOAs are used before fiber transmission.

SOA-induced chirp could then be compensated with the alternative scheme of Fig. 3.2.

Chirp and intensity-induced distortions could actually be compensated by cascading

the two proposed schemes, or even at the same time with the scheme of Fig. 3.2 by

substituting the phase modulator with a DDMZM, and deriving appropriate filters for

the two arms of the DDMZM. We also leave this matter for future investigation.

Finally, our comparison of single and dual carrier of Chapter 5 neglects one impor-

tant scenario: when the subcarrier spacing satisfies the orthogonality principle (i.e., the

two subcarriers are spaced by an amount equal to the symbol rate). In this case, the

technological constraints are different: two sources (ideally frequency-locked) and two

transmitters are needed, but only one receiver, with a different digital signal processing

with respect to the single carrier case. A more complete investigation would include

this implementation as well, even though the complexity comparison would be more

difficult due to the inherently different technological implementation.

Appendix A

Amplification in semiconductor

diodes

In this parabolic band model the relation between the energy and the wave vector

is parabolic [137]:

ECB(k) = Ec +~

2k2w

2mc, for the conduction band (A.1)

EV B(k) = Ev − ~2k2

w

2mh, for the valence band (A.2)

where ECB(k) and EV B(k) are the energies in the conduction band (CB) and valence

band (VB) respectively, Ec and Ev are the minimum and maximum of conduction and

valence bands respectively (so that the bandgap energy is Eg = Ec − Ev), ~ is the

reduced Planck’s constant, kw is the magnitude of the wave vector ~kw and mc and mh

are the effective masses of electrons and holes.

Appendix A. Amplification in semiconductor diodes 118

The Fermi-Dirac statistics give us the occupation probability of an electron with

energy ECB [138, 139]

fc(ECB) =1

e[(ECB − Efc)/kBT ] + 1(A.3)

where Efc is the quasi-Fermi level for the conduction band, kB is Boltzmann’s constant,

and T is the temperature of the medium. Similarly, the occupation probability of an

electron with energy EV B is

fv(EV B) =1

e[(EV B − Efv)/kBT ] + 1(A.4)

where Efc is the quasi-Fermi level for the valence band. Note that the occupation

probability of a hole with energy EV B would be (1 − fv(EV B)).

Consider the case of incoming photons of energy hf = ECB −EV B, as shown in Fig.

1.14. The absorption rate for such a photon is

Rabs = pT (1 − fc)fvρ(hf) (A.5)

where pT is the transition probability, ρ(hf) is the density of the incoming photons,

and (1 − fc) and fv are the probabilities that the electron and holes states ECB and

EV B are not occupied. The stimulated emission rate is

Rstim = pT fc(1 − fv)ρ(hf) (A.6)

where fc and (1 − fv) are the occupation probabilities of the electron and hole states

ECB and EV B.

For optical gain the stimulated emission needs to be (much) greater than the ab-

sorption: Rstim > Rabs, which can be written with (A.5) and (A.6)

fc > fv (A.7)

Substituting the values for fc and fv given by (A.3) and (A.4) one can write the

condition as

EV B − Efv > ECB − Efc (A.8)

and rearranging, we finally have

Efc − Efv > hf (A.9)

Appendix A. Amplification in semiconductor diodes 119

Using (A.1) and (A.2), the above equation is equivalent to

Efc − Efv > Eg +~

2k2w

2mc+

~2k2

w

2mv(A.10)

Thus the separation between the quasi-Fermi levels has to be bigger than the energy

of the incoming photon. Such a condition can be respected only if the quasi-Fermi

levels Efc and Efv lie respectively inside the conduction and valence bands, which for

a semiconductor happens only under strong bias condition.

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List of publications

During the time of my PhD, I have had the privilege of contributing to the following

works, some of which I have presented in great details in the present document:

1. Francesco Vacondio, Amirhossein Ghazisaeidi, Alberto Bononi, Leslie A. Rusch,

"Low Complexity compensation of SOA Nonlinearity for Single-Channel PSK and

OOK," IEEE J. Lightwave Technol., vol. 28 Issue 3, pp.277-286 (2010)

2. Francesco Vacondio, Mehrdad Mirshafiei, Ansheng Liu, Ling Liao, Juthika Basak,

Mario Paniccia, Leslie A. Rusch, "Silicon Modulator Enabling RF over Fiber for

802.11 OFDM Signals," IEEE Journal of Selected Topics in Quantum Electronics,

vol. 16 Issue 1, pp.141-148 (2010)

3. Amirhossein Ghazisaeidi, Francesco Vacondio, Leslie A. Rusch, "Filter Design

for SOA-assisted SS-WDM Systems Using Parallel Multicanonical Monte Carlo,"

IEEE J. Lightwave Technol., vol. 28, Issue 1, pp. 79-90 (2010).

4. Francesco Vacondio, Amirhossein Ghazisaeidi, Alberto Bononi, Leslie A. Rusch,

"DQPSK: When is a Narrow Filter Receiver Good Enough?," IEEE J. Lightwave

Technol., vol. 27, Issue 22, pp. 5106-5114 (2009).

5. Amirhossein Ghazisaeidi, Francesco Vacondio, Alberto Bononi, Leslie A. Rusch,

"Bit Patterning in SOAs: Statistical Characterization through Multicanonical

Monte Carlo Simulations," IEEE Journal of Quantum Electronics, vol. 46, Issue

4, pp.570-578, April 2010.

6. Walid Mathlouthi, Francesco Vacondio, and Leslie A. Rusch, "High-Bit-Rate

Dense SS-WDM PON Using SOA-Based Noise Reduction With a Novel Balanced

List of publications 135

Detection," IEEE Journal of Lightwave Technology, vol. 27, Issue 22, pp. 5045-

5055 (2009).

7. Amirhossein Ghazisaeidi, Francesco Vacondio, Alberto Bononi, and Leslie A.

Rusch, "SOA Intensity Noise Suppression in Spectrum Sliced Systems: A Mul-

ticanonical Monte Carlo Simulator of Extremely Low BER," IEEE Journal of

Lightwave Technology vol. 27, no. 14, pp. 2667-2677, July 15 2009.

8. Walid Mathlouthi, Francesco Vacondio, Pascal Lemieux, and Leslie A. Rusch,

"SOA gain recovery wavelength dependence: simulation and measurement using

a single-color pump-probe technique," Optics Express 16, pp. 20656-20665, 2008.

Conferences

9. Francesco Vacondio, Jeremie Renaudier, Oriol Bertran-Pardo, Patrice Tran, Haik

Mardoyan, Gabriel Charlet, Sebastien Bigo, "Dual- Versus Single-Carrier Configu-

ration for 40 Gb/s Coherent BPSK-Based Solutions over Low Dispersion Fibers,"

Optical Fiber Conference (OFC) 2010, oral presentation, San Diego (CA), 21-25

March 2010, paper OTuL.

10. Francesco Vacondio, Amirhossein Ghazisaeidi, Leslie A. Rusch, "Simultaneous

WDM-DQPSK Demodulation With a Single AWG," European Conference on

Optical Communications (ECOC) 2009, oral presentation, Wien, 20-24 September

2009.

11. Alberto Bononi, Leslie A. Rusch, Amirhossein Ghazisaeidi, Francesco Vacondio,

and Nicola Rossi, "A Fresh Look at Multicanonical Monte Carlo from a Tele-

com Perspective," in Proc. Globecom 2009, paper CTS-14.1, Honolulu, Hawaii,

Nov./Dec. 2009.

12. Amirhossein Ghazisaeidi, Francesco Vacondio and Leslie A. Rusch, "Evaluation

of the Impact of Filter Shape on the Performance of SOA-assisted SS-WDM Sys-

tems Using Parallelized Multicanonical Monte Carlo," in Proc. Globecom 2009,

Honolulu, Hawaii, Nov./Dec. 2009.

13. Amirhossein Ghazisaeidi, Francesco Vacondio, Alberto Bononi, Leslie A. Rusch,

"Statistical Characterization of Bit Patterning in SOAs: BER Prediction and

Experimental Validation," Optical Fiber Conference (OFC) 2009, paper OWE7.

14. Walid Mathlouthi, Francesco Vacondio, J. Penon, Amirhossein Ghazisaeidi and

Leslie A. Rusch, "DWDM Achieved with Thermal Sources: a Future-proof PON

Solution," European conference in Optical Communications (ECOC) 2007, Berlin,

Germany, Sept. 2007.