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University of Pisa and Scuola

Superiore Sant’Anna

Master’s Thesis

Performance Evaluation of a

Low-Complexity Digital

Backpropagation Method for

Optical Fiber Communications

Author:

Menelaos Ralli

Camara

Supervisor:

Marco Secondini

Academic Year 2015/2016

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University of Pisa and Scuola Superiore Sant’Anna

Abstract

MSc in Computer Science and Networking

Master’s Thesis

Performance Evaluation of a Low-Complexity Digital Backpropagation Method for Optical Fiber Communications

by Menelaos Ralli Camara

To compensate for linear and nonlinear distortions, several optical and dig-ital techniques have been proposed. However, the significant performance improvement due to the combination of coherent detection with Digital Sig-nal Processing has resulted in an almost complete abandonment of optical techniques. Consequently, digital nonlinear compensation has prevailed; this technique includes digital back propagation, perturbation-based tech-niques, and a combination of these two. All of these digital methods rely on numerically solving the nonlinear Schrödinger equation.

Despite the well-known potential of digital back propagation, this method has an Achilles heel: its computational complexity. The aim of the current study is to analyze and investigate on the performance obtained using a low-complexity numerical method that can be used in DBP to solve the NLSE. Such a method is the Enhanced split-step Fourier method suggested by Secondini, Marsella and Forestieri [54].

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v

Acknowledgements

I would like to express my profound gratitude to my supervisors Marco Secondini for the useful comments, remarks and engagement through the learning process of this master thesis.. . .

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vii

Contents

Abstract iii

Acknowledgements v

1 Introduction 1

1.1 Motivation and Contribution . . . 3

1.2 Organization of the Current Study . . . 3

2 Background: Overview of the Fiber Impairments and of the Corresponding Mitigation Techniques 5 2.1 Channel Model . . . 5

2.2 Fiber Optic Impairments . . . 9

2.2.1 Fiber Loss and Amplification . . . 10

Attenuation Coefficient . . . 10

Amplification and Noise . . . 11

2.2.2 Chromatic Dispersion. . . 12 2.2.3 Polarization-Mode Dispersion . . . 15 2.2.4 Kerr Effects . . . 15 Self-Phase Modulation . . . 17 Cross-Phase Modulation . . . 19 Four-Wave Mixing . . . 19 2.3 Impairments Mitigation . . . 21 2.3.1 Mitigation Methods . . . 22

Optical Domain Techniques . . . 22

Electrical Domain Techniques . . . 22

2.3.2 Digital Backpropagation . . . 23

Digital Backpropagation Noise Limitation . . . 24

Digital Bacpropagation PMD Limitation . . . 26

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formance Analysis 29

3.1 Digital Back Propagation and Split-Step Fourier Method

and Variants. . . 29

3.1.1 Asymmetric and Symmetric Split-Step Fourier Method 30 3.1.2 Filtered Split-Step Fourier Method . . . 33

3.1.3 Split-Step Fourier Method Implementation and Costs 34 3.2 Digital Back Propagation and Enhanced Split-Step Fourier Method . . . 35

3.2.1 Enhanced Split-Step Fourier Method . . . 36

3.2.2 Enhanced Split-Step Fourier Method Complexity Anal-ysis . . . 37

3.3 Simulation Results . . . 38

3.3.1 Implementation Choices . . . 38

3.3.2 Simulation Set-up . . . 40

3.3.3 Single Channel Variable Symbol Rate . . . 42

3.3.4 Single-Channel Variable Distance Test . . . 48

3.3.5 Enhanced Split-Step Fourier Method versus Filtered-SSFM . . . 51

3.3.6 Discussion of Simulation Results. . . 55

4 Conclusions and Outlook 57 4.1 Summary . . . 57

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ix

List of Figures

2.1 Schematic Representation of the Fiber Attenuation . . . 11

2.2 Schematic Representation of the Amplification Schemes . . . 12

2.3 Schematic Representation of the Fiber Chromatic Dispersion 13

2.4 Schematic Representation of the Fiber Polarization Mode Dispersion . . . 16

2.5 Schematic Representation of the Self-Phase Modulation . . . 17

2.6 Schematic Representation of the Fiber Four-Wave Mixing . . 20

2.7 Block diagram of linear and nonlinear channel impairments . 21

2.8 Block diagram of forward propagation (FP) and digital back-ward propagation (DBP). . . 24

2.9 Block diagram that describe the nonlinear signal-noise inter-action . . . 25

3.1 Digital Backpropagation Model . . . 29

3.2 Schematic Representation of a Fibre and its Simulation through ssfm . . . 31

3.3 Nonlinear System Models. . . 31

3.4 Schematic Representation of ssfm with Improved Accuracy . 32

3.5 Block Diagram of the Filtered DBP (F-DBP) . . . 34

3.6 Schematic Representation of the Simulated System . . . 40

3.7 Backward Propagation Models . . . 41

3.8 Max SNR vs Backward Steps for Low Medium and High Symbol Rate System . . . 45

3.9 Enhanced Split-Step Fourier Method Backward Steps Gain . 48

3.10 Enhanced Split-Step Fourier Method Performance Gain for Low Symbol Rate System . . . 50

3.11 Enhanced Step Fourier Method Versus Filtered Split-Step Fourier Method . . . 52

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List of Tables

3.1 System Parameters . . . 43

3.2 Performance Gain of the ESSFM for the Low Symbol Rate System . . . 44

3.3 Performance Gain of the ESSFM with Nc = 64 in the 50

GBd System . . . 46

3.4 Performance Gain of the ESSFM for Low Symbol Rate System 47

3.5 Percentage Gain of the ESSFM as a Function of the Distance 51

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List of Abbreviations

DSF Dispersion Shifted Fiber

WDM Wavelength-Division Multiplexing

SPM Self-Phase Modulation

XPM Cross-Phase Modulation

FWM Four-Wave Mixing

SBS Stimulated Brilliun Scattering

SRS Stimulated Raman Scattering

CD Chromatic Dispersion

DSP Digital Signal Processing

NLSE Nonlinear Schrödinger Equation

SMF Single Mode Fiber

ASE Amplified Spontaneous Emission

PMD Polarization Mode Dispersion

CNLSE Coupled Nonlinear Schrödinger Equation EDFA Erbium-Doped Fiber Amplifier

ISI InterSymbol Interference

GVD Group Velocity Dispersion

SNR Signal-to-Noise Ratio

DCF Dispersion Compensation Fiber PMF Polarization Maintaining Fiber

OPC Optical Phase Conjuction

TDE Time Domain Equalizer

FDE Frequency Domain Equalizer

SSFM Split-Step Fourier Method

ESSFM Enhanced Split-Step Fourier Method NSNI Nonlinear Signal-Noise Interaction

MI Modulation Instability

PG Parametric Gain

NLPN Nonlinear-Phase Noise

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1

Chapter 1

Introduction

Optical communication systems have revolutionized the technology behind telecommunications. The lightwave technology, along with microelectron-ics, is a major factor in the advent of the Information Age. Fiber-optic com-munication systems now represent the load-bearing structure of high-speed and long-distance communication. In fact, lightwave systems have reached a certain stage of maturity, even though this kind of technology is barely 40 years old. During these 40 years, improvements obtained by means of research have allowed classifying this kind of communication systems into five different generations, each of which has brought a fundamental change that helps to improve the systems’ performance.

First generation of lightwave systems were able to transmit at a bit rate of up to 45 Mb/s and were limited principally by fiber dispersion. Sec-ond generation became available thanks to the availability of the InGaAsP semiconductor laser that allowed this generation to operate at a bit rate of up to 1 Gb/s and to increase the spacing between repeaters by approxi-mately 50 km. Thanks to the combination of dispersion shifted fiber (DSF) with lasers oscillating in a single longitudinal mode, the third generation overcame dispersion problems of the previous generations and was able to operate at a maximum bit rate of 10 Gb/s. FThe fourth generation of lightwave systems operates on the order of terabits per second; this out-standing result was obtained by combining wavelength-division multiplexing (WDM) and optical amplification techniques that enable increasing the bit rate capacity of a system by simply increasing the number of optical car-rier signals onto a single optical fiber and amplifying the entire signal in the optical domain without demultiplexing. Given that the first genera-tion had a capacity of 45 Mb/s in 1970, it is remarkable that the capacity

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The current fifth generation of fiber-optic communication systems is con-cerned with extending the wavelength range over which a WDM system can operate.

The technological achievements in fiber-optic communication systems over the last 40 years, have resulted in the demand for an increase in transmis-sion capacity being met. However, the continuous increase in the capacity has augmented the amount of power in the fiber; consequently, channel nonlinear effects have begun to play an important role in telecommuni-cation systems. The relation between the power and nonlinear effects in optical fibers derives from two phenomena: the change in the refractive index of a medium with optical intensity and inelastic scattering. Intensity dependence of the refractive index, also known as the optical Kerr effect, manifests itself as three different effects, namely Self-Phase Modulation (SPM), Cross-Phase Modulation (XPM) and Four-Wave Mixing (FWM). Conversely, the inelastic scattering phenomenon can induce, at a high power level, stimulated undesirable effects such as Stimulated Brillouin-Scattering (SBS) and Stimulated Raman-Brillouin-Scattering (SRS). Nevertheless, effective techniques have been found to suppress SBS [1] [20], and SRS is of only minor concern, even in wavelength-division multiplexing WDM systems with a large number of channels [20]. On the contrary, the in-terplay between Kerr nonlinearity and chromatic dispersion (CD) is the reason for complicated signal distortions that occur and are accumulated along the fiber distance. To compensate for linear and nonlinear distor-tions, several optical and digital techniques have been proposed. However, the significant performance improvement due to the combination of co-herent detection with Digital Signal Processing (DSP) has resulted in an almost complete abandonment of optical techniques. Consequently, digital nonlinear compensation has prevailed; this technique includes digital back propagation, perturbation-based techniques, and a combination of these two. All of these digital methods rely on numerically solving the non-linear Schrödinger equation (NLSE), which allows for the description of deterministic effects in a single-mode fiber (SMF). Therefore, deterministic distortion can be theoretically compensated for by solving a reverse NLSE in the digital domain using DBP.

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1.1. Motivation and Contribution 3

1.1

Motivation and Contribution

Despite the well-known potential of DBP, this method has two Achilles heels that are not negligible:

• Though the NLSE well describes CD and the signal-to-signal nonlin-ear interaction, which allow for numerically solving the equation and counteracting nonlinearities by channel inversion through backward propagation, the NLSE includes only deterministic effects. Conse-quently, the NLSE does not include the nonlinear interaction be-tween the signal and Amplified Spontaneous Emission (ASE) noise and between the signal and Polarization Mode Dispersion (PMD). Both effects have a significant impact on the effectiveness of DBP. • Digital back propagation allows for mitigating various channel

impair-ments through the use of a numerical method that allows approximat-ing a solution of the NLSE. However, the principal drawback of such a method is the high computational complexity that they require to solve the NLSE; such a high complexity is reflected in latency and power consumption. For that reason, a difficult challenge for real-time implementation of DBP is keeping its complexity, latency, and power consumption within feasible values.

In the current study, we focus on the DBP computational complexity prob-lem . Particularly, the aim of the current study is to analyze and investigate the performance obtained by a low-complexity numerical method that can be used in DBP to solve the NLSE. Such a method is the enhanced split-step Fourier method suggested by Secondini, Marsella and Forestieri[54]. To better understand the potential improvement of this method, the en-hanced split-step Fourier method is compared with the classical split-step Fourier method and with a filtered split-step Fourier method [9].

1.2

Organization of the Current Study

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to describe signal propagation within fiber optics is described. There-after, channel impairments and their interaction are illustrated. Fi-nally, an overview of the mitigation techniques is given with a partic-ular focus on DBP.

Chapter 3 discusses the split-step , the enhanced split-step Fourier method and their computational complexity. Subsequently, the results ob-tained through numerical simulations are illustrated.

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5

Chapter 2

Background: Overview of the

Fiber Impairments and of the

Corresponding Mitigation

Techniques

In this chapter, different channel impairments are described, and mitiga-tion techniques are illustrated, with particular attenmitiga-tion to DBP and the limitation that is induced by the ASE noise and PMD. To this end, the current study is organized as follows: In Section 2.1, the equation that models the propagation of the signal envelope inside the fiber is described. In Section 2.2, the channel impairments are illustrated. Finally, a general overview of different mitigation techniques, and the limitation of DBP that is caused by ASE noise and PMD are discussed in Section 2.3

2.1

Channel Model

Light in a single mode optical fiber propagates in two eigenmodes that are distinguished from each other by the polarization states. The light propagation is modeled in the (x, y, z)-plane by making z to correspond to the distance along the fiber and the (x, y)-plane to correspond to the transverse dimension. The electric field of light, E, that propagates at a carrier frequency, ω0, can be expressed as [43]

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Here t denotes the time , and β(ω0, z) is the wavenumber, which through

the frequency dependence determines the dispersion relation of the fiber. The vector fields R1 and R2 are two eigenfunctions that describe the (x,

y)-dependence of the electric field, E. These vector fields satisfy the relation ˆ

ez × R1 = R2, which means that they are orthogonal to each other and

to the propagation direction z. The functions U1 and U2 describe the slow

variation of the envelope of the electric field with respect to the rapidly varying carrier wave given by the exponential factor in Eq. (2.1). The factor κ is chosen so that |U1| + |U2| equals the optical signal power.

The degenerate nature of the two orthogonally polarized modes holds only for an ideal single-mode fiber with a perfectly cylindrical core of uniform diameter. Real fibers present asymmetry and anisotropy; therefore, the de-generacy between the orthogonally polarized fiber modes is removed. Con-sequently, real fibers exhibit a small birefringence: Light travelling in the two different eigenmodes propagates at slightly different group velocities. Because of the properties of real fibers, eigenfunctions R1 and R2 rotate

rapidly and randomly in the (x, y)-plane along the fiber distance, z. The propagation of an electric field within SMF can be described by the coupled nonlinear Schrödinger equation (CNLS). The CNLS is derived by applying the multiple-length-scale method to Maxwell’s equation in a di-electric medium and therefore, averaging over the rapid variations of the carrier wave, exp[iβ(ω0, z)z − iω0τ ], and over the eigenfunctions R1 and

R2 [43]. Such an equation describes the z-evolution of the vector function

U(z, t) = (U1(z, t), U2(z, t)T), which is the Jones vector of the light that

models the polarization state of the light at (z, t). The variable t from now on refers to the retarded time, tretarded = t −∂ω∂β(ω0, z)z, which defines a

co-ordinate system that moves at the group velocity ∂β∂ω(ω0, z)

−1

and allows for viewing the signal evolution in a time window of limited duration. The CNLS states that [43] ∂U ∂z − αU − i∆BU + ∆B 0∂U ∂t + i β2 2 ∂2U ∂t2 = iγ  |U|2U − 1 3 U † σ2U σ2U  (2.2)

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2.1. Channel Model 7

the Kerr nonlinearity is accounted for by the scalar coefficient γ. The fac-tor ∆B = ∆B(ω0, z) is the birefringence matrix that models the

asymme-try and anisotropy of the fiber’s linear dielectric response tensor, averaged with respect to the eigenfunctions R1 and R2. Under the assumption of

no polarization-dependent loss, which is the case in optical-fiber commu-nication systems [2], ∆B can be assumed to be a 2 × 2 Hermitian matrix. The matrix ∆B0 is defined by ∆B0 = ∂∆B∂ω (ω0, z) whose contribution to the

evolution of U is very important since it leads to PMD. The scalar coeffi-cient β2 = ∂

2β

∂ω2(ω0, z) governs the strength of the CD phenomenon. Finally,

† denotes the conjugate transpose, and σ2 is the second of the three Pauli

spin matrices: σ1 = 0 1 1 0 ! , σ2 = 0 −i i 0 ! , σ3 = 1 0 0 −1 ! . (2.3)

Under the assumption of linear birefringence,which is possible because of the glass’s polarization properties [50], the matrix ∆B can be defined as

∆B = ∆β(cos(θ)σ3+ sin(θ)σ1), (2.4)

where ∆β is the magnitude of birefringence, and θ is the angle that models the rotation of the birefringence axes as light passes a small section of the fiber. Since birefringence is due to anisotropies and asymmetries that are very weak functions of frequency, the orientation of the matrix ∆B0 can be assumed to have the same direction as that of ∆B so that [43]

∆B0 = ∆β0(cos(θ)σ3+ sin(θ)σ1). (2.5)

The fundamental difficulty in solving Eq. (2.2) is that the birefringence co-efficient θ varies over the length scale of 1–100 m, whereas the length scale for PMD, CD, and the Kerr nonlinearity are hundreds or even thousands of kilometers. Therefore, if one wishes to solve Eq. (2.2) accurately, then one should use a numerical method that is able to track the rapid changes in the polarization state of the light of the field, which is computation-ally expensive. Moreover, the physical effects that contribute to the signal evolution are mixed together in a complicated manner in Eq. (2.2). For instance, when Kerr effects and CD can be ignored, the signal undergoes

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PMD [47]. Conversely, when birefringence is rapidly and randomly vary-ing and PMD is small, the Manakov equation applies. However, it is not immediately apparent where in Eq. (2.2) these effects are embedded. To overcome this problem, the multiple-scale technique come to one’s aid again and transforms the CNLS by averaging over the rapid and random vari-ations of the birefringence to obtain the Manakov-PMD equation [43][44]. Such an equation not only allows developing a rapid numerical scheme for determining the signal evolution with no loss of accuracy, but the different physical effects are isolated in separate terms of the equation that have a transparent physical meaning.

∂W ∂z − αW + ∆β 0 ¯ σ3 ∂W ∂t + i β2 2 ∂2W ∂t2 = i8 9γ|W| 2W − i1 3γ[(W † ¯ σ2W) ¯σ2W − 1 3|W| 2W] (2.6)

The last term of Eq. (2.6) is referred to as the nonlinear PMD and contains the effect of incomplete mixing on the Poincaré sphere [41]. Nevertheless, in fiber optic communication systems, this term is always negligible [41]. Therefore, the ultimate form of Eq. (2.6) that is used in telecommunication systems assumes the following form:

∂W ∂z = αW − ∆β 0 ¯ σ3 ∂W ∂t − i β2 2 ∂2W ∂t2 + i 8 9γ|W| 2W. (2.7)

Here W (z, t) = Q(z)U (z, t), Q(z) is a unitary transformation and ¯σ3 =

T (z)−1σ3T (z) for a matrix T (z) that is determined by the birefringence

parameters ∆β and θ. The transformation made by Q(z) can be explained by noting that the Fourier conjugate of the retarded time, t, is the fre-quency, ω, measured relative to the carrier frequency of the optical signal, ω0. Therefore, even if the Fourier transform, ˆU(z, ω), of U(z, t) varies

rapidly with the propagation distance z, it only has a very weak depen-dence on the frequency, ω. Therefore, it makes sense to transform U in a manner that the motion of the center frequency, ω0, freezes, so that ˆW(z, 0)

is constant in z. Thus, rapid changes of the polarization state of the signal at the carrier frequency are exactly followed in the new coordinate system;

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2.2. Fiber Optic Impairments 9

consequently, the Fourier transform, ˆW(z, ω), of the the Manakov-PMD solution measures the relatively slow motion of the polarization state of the light at other frequencies, ω, with respect to he polarization state of the light at the carrier frequency, ω0.

When PMD can be neglected, the Manakov-PMD equation can be reduced to the simple Manakov equation [43]

∂U ∂z = αU − i β2 2 ∂2U ∂t2 + i 8 9γ|U| 2U (2.8)

where we have replaced W with U while taking into account that the meaning of the Jones vector U has been changed; the evolution of the polarization state at the carrier frequency, ω0, has been removed. If we

assume that at z = 0 the envelope U is in a single polarization state, then there is no more coupling to the orthogonal polarization state. In that case, Eq. (2.8) can be reduced to the well-known NLSE, which is basically the scalar representation of Eq. (2.8).

∂u ∂z = αu − i β2 2 ∂2u ∂t2 + i 8 9γ|u| 2u (2.9)

Therefore, the conditions for the NLSE to hold are that PMD is negligible and the signal is initialized in a single polarization state. Of course, as the signal propagates along the fiber, the polarization state of the optical signal varies rapidly as a function of z. These rapid variations are only hidden by the Q(z) transformation. However, when the NLSE applies, they are the same at every point in time and do not affect the evolution of the envelope.

2.2

Fiber Optic Impairments

In a fiber-optic communication system, we can distinguish between linear and nonlinear impairments. Attenuation, chromatic dispersion and polar-ization mode dispersion belong to the first category. Conversely, the Kerr effects and the inelastic scattering effects are all nonlinear degrading effects due to fiber nonlinearity. In this section, the discussion focuses entirely on attenuation, chromatic dispersion, polarization mode dispersion and Kerr effects.

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2.2.1

Fiber Loss and Amplification

Fiber loss is a channel impairment that reduces the power of the signal that reaches the receiver, consequently limiting the transmission distance of the channel. To overcome the loss limitation, optoelectronic repeaters were used until 1995; however, this solution turned out to be unsuitable for complex systems such as WDM. Therefore, to eliminate the limitation of the optoelectronic conversion, repeaters have been replaced by optical am-plifiers, which allow direct amplification of the entire WDM signal without converting each channel into the electric domain. Conversely, an optical amplifier adds noise that becomes high when multiple amplifiers are used, such as in long-haul systems in which the optical signal is amplified peri-odically.

The following section explains the most important factors that cause fiber loss. Next are descriptions of different amplification schemes and modifica-tions that should be made to the NLSE to consider the gain and the noise that are introduced by amplifiers.

Attenuation Coefficient

During the propagation of a bit stream inside an optical fiber, changes in the average optical power P are governed by Beer’s law [4]

dP

dz = −αP (2.10)

where α is the attenuation coefficient, which typically expressed in units of dB/km and referred to as the fiber loss parameter, a graphic representation of the power attenuation is showed in Figure (2.1). The total amount of losses is a consequence of several factors, the most important of which are the following three:

Material absorption: The loss of the material absorption is the result of two phenomena: the absorption of by the fused silica, called intrinsic absorption, and losses caused by the presence of impurities within silica, called extrinsic absorption.

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2.2. Fiber Optic Impairments 11

Figure 2.1: Schematic Representation of the Fiber Attenuation

Rayleigh scattering: Rayleigh scattering in a fiber is due to microscopic fluctuation in density that leads to a random fluctuation of the re-fractive index on a scale smaller than the optical wavelength λ. The loss of silica fibers from Rayleigh scattering can be written as [4]

αR= C/λ4 (2.11)

where C depends on the constituents of the fiber core.

Waveguide imperfections: Waveguide imperfection can also generate index inhomogeneities on a scale longer than the optical wavelength. Such a fluctuation generates Mie scattering that leads to additional losses that contribute to the net fiber loss.

Amplification and Noise

Amplifiers can be divided into two categories known as lumped and dis-tributed amplifiers. The most employed amplifiers under the lumped-amplifier category are erbium-doped fiber lumped-amplifiers (EDFAs); the Raman amplifier falls under the distributed-amplifier category. These two cate-gories differ by the manner in which the fiber loss is managed. In the lumped scheme, losses accumulated over the fiber lengths are compensated using short lengths of erbium-doped fibers, whereas in distributed case, loss compensation is obtained by taking advantage of the SRS nonlinear effect caused by a periodical optical power injection ( Figure2.2 compare lumped with EDFAs distributed scheme). Independently of the scheme, the optical amplified signal undergoes SNR degradation due to the noise added by the

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100 km 100 km

(a) Lumped Scheme

300 km LASER PUMP POWER PROFILE Fiber attenuation Distributed Raman Gain (b) Distributed Scheme

Figure 2.2: Schematic Representation of the Amplifi-cation Schemes

amplifier. Such noise can be modeled by adding a noise-and-gain term to the NLSE, which results in the following form: [4]

∂U ∂z = [g0(z) − α]U − i β2 2 ∂2U ∂t2 + i 8 9γ|U| 2U + f n(z, t) (2.12)

where g0(z) is the amplifier gain coefficient, which depends on the

ampli-fication scheme used. Amplifier-induced noise is accounted for by the last term, fn(z, t).

2.2.2

Chromatic Dispersion

Chromatic dispersion is the dominant dispersion mechanism in SMF. Dis-persion is the phenomena in which the phase velocity of a wave depends on its frequency. Therefore, because a modulated lightwave is not monochro-matic, different frequency components of the lightwave travel through the fiber in generally different phase shifts. Consequently, short pulses be-come longer, thus leading to significant intersymbol interference (ISI), a

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2.2. Fiber Optic Impairments 13

schematic representation of the fiber CD is illustrated in Figure (2.3). By

Pulse broa

dening

1 0 1 Spectral width 1 1 1 Spectral width

Figure 2.3: Schematic Representation of the Fiber Chromatic Dispersion

expanding the mode-propagation constant β in the Taylor series the carrier frequency ω0, it is possible to find

β(ω) = n(ω)ω c = β0+ β1(ω − ω0) + 1 2β2(ω − ω0) 2+ ..., (2.13) where βm =  dmβ dωm  ω=ω0 (m = 0, 1, 2, ...). (2.14)

Here, β0 is the value of β at the carrier frequency, β1 is the inverse of the

group velocity vg, and β2and higher-order terms lead to envelope distortion.

Particularly, β2 is known as the first-order dispersion coefficient or group

velocity dispersion (GVD) coefficient, and β3 is known as the second-order

dispersion coefficient. The relation between the pulse broadening, ∆T , and the β2 parameter is governed by [4]

∆T = Ld

2β

dω2δω = Lβ2∆ω (2.15)

where ∆ω is the spectral width of the pulse. Another way to characterize the chromatic dispersion is through the dispersion parameter D that is related to β2 by [4] D = d dλ  1 vg  = −2πc λ2 β2. (2.16)

The dispersion parameter can be written as the sum of two terms, the ma-terial dispersion DM and the waveguide dispersion DW, in which the former

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occurs because the refractive index of silica changes with the optical fre-quency ω, and the latter takes place because the propagation constant, β, depends on the light frequency, ω, in a nonlinear fashion. It has been showed that the dispersion interaction between DM and DW, for certain

wavelengths, can bring the total dispersion near to zero: D = DM + DW;

the wavelength at which this happens is known as the zero-dispersion wave-length, λZD. Nevertheless, even for λZD, dispersion effects do not

dis-appear completely; optical pulses still experience broadening because of higher-order dispersion effects, which are governed by the dispersion slope, S = dD/dλ, that has the following proportionality with β3 and β2 [4]:.

S = 2πc λ2 2 β3+  4πc λ3  β2 (2.17)

As mentioned earlier, the consequence of GVD is pulse broadening. This effect is undesirable since it interferes with the detection process, thus lead-ing to errors in the received bit pattern. Therefore, GVD limits the bit rate B and the transmission distance L of a communication. An estimation of the bit rate degradation due to dispersion effects can be done by relating the pulse broadening, ∆T , to the information-carrying capacity of the fiber measured through the bit rate B. Although a precise relation between B and ∆T depends on many details - i.e. pulse shape, spectral width etc. - it is intuitive that ∆T should be less than the allocated bit slot (TB= 1/B).

Thus, a rough estimation of the bit rate is obtained from the condition B∆T < 1 combined with Eq. (2.15) and 2.16)

BL|D|∆λ < 1 (2.18)

which clearly describes how the BL product is limited by GVD in opti-cal fibers. By contrast, in case of a channel operating at zero-dispersion wavelength λZD, the limiting bit rate can be retrieved by noting that for

a source of spectral width ∆λ, the effective dispersion parameter value becomes D = S∆λ. The resulting condition becomes

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2.2. Fiber Optic Impairments 15

2.2.3

Polarization-Mode Dispersion

As mentioned in Section 2.1, SMF support two modes of propagation dis-tinguished by their polarization. Only when a fiber has a perfectly isotropic material and a perfectly symmetric core and cladding geometry, these two modes have the same group delay. However, in practice, fibers exhibit considerable variation in the shape of their core along the fiber length. Furthermore, they may also experience non-uniform stress such that the cylindrical symmetry of the fiber is broken. Therefore, degeneracy of the two orthogonally polarized modes is broken, and the fiber acquires birefrin-gence. The degree of modal birefringence can be written as[4]

Bm = |¯nx− ¯ny| (2.20)

Here the two indices ¯nx and ¯ny represent the refractive indices of the two

orthogonally polarized fiber modes. Moreover, the degree of modal bire-fringence, Bm, is not constant but, due to variations in the core shape and

the anisotropic stress acting on the core along the fiber, the degree of bire-fringence changes randomly both in magnitude and direction. Although birefringence is small in absolute terms in communication systems, typi-cally Bm ∼ 10−7 and the corresponding beat length1, LB, ∼ 10m, it can be

devastating in communication systems. Even if the rapid variation of the birefringence orientation tends to make the effect of the birefringence av-erage to zero. The residual effect causes a phenomenon by which different frequency components of a pulse acquire different polarization states that lead to pulse spreading, which is referred to as polarization mode dispersion, a schematic representation of the PMD is illustrated if Figure 2.4.

2.2.4

Kerr Effects

In this section, only the Kerr effects are discussed because the inelastic scattering effects are negligible or can be completely suppressed [20], [1].

1the beat length, L

B= λ/Bm, is the length after which the phase relation between

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Figure 2.4: Schematic Representation of the Fiber Polarization Mode Dispersion

The optical-fiber medium can only be approximated as a linear medium when the lunch power is sufficiently low. However, for the long-haul fiber-optic transmission system and wideband WDM systems to combat accumu-lated noise added by the amplifier chain along the transmission fiber link, the launch power must be increased to keep signal-to-noise ratio (SNR) high enough the error-free detection at the receiver.

As the launch power increases, the nonlinearity of the fiber becomes sig-nificant and leads to severe performance degradation. Nonlinear effects in optical fibers are mainly due to two causes. The first cause lies in the fact that the index of refraction of many materials, including glass, is a func-tion of light intensity. This phenomenon is called the Kerr effect, and it was discovered in 1875 by John Kerr. The second cause is the nonelas-tic scattering of photons in fibers, which results in stimulated Raman and stimulated Brillouin scattering phenomena.

In general, nonlinearity lies in inharmonic motion of bound electrons under the influence of an applied field. Due to this inharmonic motion, the total polarization P induced by electric dipoles is not linear but satisfies a more general relation as

P = 0χ(1)E + 0χ(2)E2+ 0χ(3)E3+ ..., (2.21)

where 0 is the permittivity of vacuum and χ(k)(k = 1, 2, ...) is kth-order

susceptibility. The coefficient χ(1) represents the linear effects and is ac-counted for by the refractive index n and the attenuation coefficient α in the discussion of fiber. Instead, the second-order susceptibility χ(2) can be

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2.2. Fiber Optic Impairments 17

avoided in optical fibers due to their inversion symmetry [3]. The lowest-order nonlinear effects observed in optical fibers are thus those that result from χ(3) [3] and include third-harmonic generation, FWM and nonlinear

refraction. Particularly, most of the nonlinear effects in optical fibers orig-inate from nonlinear refraction, a phenomenon that refers to the intensity dependence of the refractive index. To include nonlinear refraction, the refractive index may be rewritten as [3]

˜ n ω, |E|2 = n(ω) + n2  P Aef f  (2.22)

Here n(ω) is the linear part derived from the Sellmeier equation [3], P is the optical power, n2is the nonlinear-index coefficient derived from χ(3) [3],

Aef f is the effective cross-sectional area2 . We have substituted |E|2 with

P/Aef f.

Self-Phase Modulation

Distance Non linear medium Distance

Figure 2.5: Schematic Representation of the Self-Phase Modulation

In a medium with an intensity-dependent refractive index, the signal-intensity variation leads to a time-varying refractive index. This temporally vary-ing index changes results in a temporally varyvary-ing phase change. Since this nonlinear phase modulation is self-induced, the nonlinear phenomenon re-sponsible for it is called self-phase modulation. Under SPM, the intensity-dependence-phase fluctuations make different parts of the pulse undergo different phase shifts. This results in frequency chirping. The rising edge

2effect of nonlinearity grows with intensity in fiber and the intensity is inversely

proportional to area of the core. Since the power is not uniformly distributed within the cross-section of the fiber, it is reasonable to use effective cross-sectional area Aef f.

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of the pulse finds a frequency shift in the upper side, whereas the trailing edge experiences a shift in the lower side. Hence, the primary effect of SPM is to broaden the pulse’s spectrum [35], a representation of the SPM effect is illustrated in Figure 2.5. SPM effects, are more pronounced in systems with high-transmitted power because the chirping effect is proportional to the transmitted signal power. The nonlinear dependence of the phase (φ) introduced by a field E over a fiber that contains a high-transmitted power can be written as

φ = 2π

λ (n + n2(P/Aef f)) Lef f (2.23)

φN L = γP LLef f (2.24)

Here, Lef f is the effective length 3, which defines the length up to which

power is assumed to be constant [49], and γ is the nonlinear parameter. Lef f and γ can be written as

γ = 2πn2 Aef fλ (2.25) Lef f = 1 − e−αL α (2.26)

The SPM by itself leads only to chirping, which causes broadening of the spectrum without any change in temporal distribution. Dispersion is re-sponsible for pulse broadening in the time domain, whereas spectral com-ponents are unaltered. In general, the damage on system performance due to SPM-induced pulse broadening depends on the pulse shape, the power transmitted and length of the link. However, in some cases, by proper choice of both pulse shape and pulse power carried, the interaction between CD and SPM that usually damages the system allows an undistorted propaga-tion of the pulse by means of mutual compensapropaga-tion of dispersion and SPM. Under these particular conditions, the pulse is called a soliton and broaden neither in the frequency domain (as in SPM) nor in the time domain (as

3The nonlinear effects depend on transmission length. The longer the fiber link

length, the more the light interaction and greater the nonlinear effect. As the optical beam propagates along the link length, its power decreases because of fiber attenuation. For that reason the effective length is important

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2.2. Fiber Optic Impairments 19

in linear dispersion) [24] [37]. Since soliton pulse does not broaden during its propagation; it has tremendous potential for applications in super-high-bandwidth optical communication systems.

Cross-Phase Modulation

Another nonlinear phenomenon caused by the intensity dependence of the refractive index is the so-called XPM. Essentially, the nonlinear refractive index seen by two or more simultaneously co-propagating optical beams depends not only on the intensity of the isolated beam but also on the intensity of the other co-propagating beams [36]. Therefore, for an N -channel transmission system, the nonlinear phase shift derived for the ith channel can be given as [4]

φinl = γLef f Pi+ 2 N X n6=i Pn ! . (2.27)

The first term in the Eq. (2.27) represents the contribution of SPM, whereas the second term represents the XPM contribution. The Eq. (2.27) indicates that XPM converts power fluctuations of a particular wavelength channel to phase fluctuations in other co-propagating channels. This inter-channel in-teraction may lead to asymmetric spectral broadening and distortion of the pulse shape and, even though the XPM mechanism that hinders the system performance is the same as that of SPM (chirping frequency and chromatic dispersion), can damage the system performance more than SPM. For in-stance, XPM influences the system severely when the number of channels is large.

Four-Wave Mixing

In the quantum-mechanical context, FWM occurs when photons from one or more waves are annihilated and new photons are created at different frequencies such that net energy and momentum are conserved during the interaction. In practice, if three optical fields with carrier frequencies ω1,

ω2 and ω3 co-propagate simultaneously inside a fiber, (χ(3)) generates a

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Figure 2.6: Schematic Representation of the Fiber Four-Wave Mixing

relation ω4 = ω1 ± ω2 ± ω3, a schematic representation of FWM is given

in Figure 2.6. The result of FWM is the power transfer between channels; such a power transfer leads to power depletion of some channels, which consequently, leads to performance degradation of those channels. However, if a system is not working in the extremely high nonlinear regime, energy transfer between the channels requires a certain length to be enough to cause a significant amount of distortion. Furthermore, the energy transfer magnitude and direction are determined by the phase relation between the four interacting channels; therefore, if the phase relation between channels is not preserved for long distances, the distortion caused by FWM do not grow and can be neglected.

Due to phase matching, the power transfer efficiency of FWM strongly depends on fiber dispersion. Since dispersion varies with wavelength, the signal generated waves have different group velocities. This destroys the phase matching of interacting waves and lowers the efficiency of power transfer to newly generated frequencies4.

Differently from SPM and XPM, FWM does not depend on bit rate but is critically dependent on the channel spacing and fiber dispersion, such

4 FWM was seen as a serious impairment in systems using DSF to overcome

chro-matic dispersion, for that reasons, methods to prevent it impacting system performance were investigated. However, today systems maintain a minimum chromatic dispersion coefficient of 4 ps/(nm·km) through the use of non-zero dispersion shifted fiber or stan-dard single mode fiber and then sufficient mismatch of the phase is achieved.

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2.3. Impairments Mitigation 21

that decreasing the channel spacing increases the FWM effect and so does decreasing the dispersion.

2.3

Impairments Mitigation

The effects described in the previous sections can be classified into two macro groups. In the first group, the linear effects such as CD and PMD are contained, whereas the nonlinear effects or the Kerr effect, belong to the second group. An additional classification is possible inside a group of linear effects, which we can distinguish into deterministic, CD, and non-deterministic (PMD) impairments. Furthermore, the ASE noise introduced by the amplifiers during the propagation adds a further distortion of the signal due to the nonlinear interaction between the signal and noise through the Kerr effect (in Figure 2.7 are sketched the various impairments).

Fiber Impairments

Non linear impairments

Figure 2.7: Block diagram of linear and nonlinear channel impairments

Optical communication systems are affected by any of these impairments, although the amount of degradation caused by these effects depends on the characteristics of a particular optical system. Hence, different techniques have been developed in the last years with the aim of restoring the optical received signal.

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2.3.1

Mitigation Methods

As well as the fiber impairments, the mitigation techniques can be divided in two groups. Depending on the domain in which the compensation is performed, it is possible to distinguish between electrical domain techniques that perform compensation after the signal reception and optical domain techniques that operate during or after propagation.

Optical Domain Techniques

Among the optical techniques we can find that based on a dispersion com-pensation fiber (DCF) that allows for perfectly restoring the signal in the presence of CD and in absence of other effects [21]. Another technique, which is used for PMD compensation, is the one based on PMD compen-sator that through a polarization controller followed by a birefringent ele-ment such as a polarization-maintaining fiber (PMF) and a feedback loop, measures the degree of polarization and compensate for PMD effects. The feedback signals could be various, each one with its pros and cons, some examples of feedback signals are the degree of polarization (DOP) [53], the spectral line [25] [31] [46] or the eye-opening or Q factor estimation [13]. Between the various compensation techniques operating in the optical do-main, a particular one, named the optical phase conjugation (OPC), some-times referred to as spectral inversion, has been and is still deeply studied. This technique was initially proposed only for chromatic dispersion compen-sation in [58] but was latter also employed to cancel nonlinear impairments that result from the Kerr effect such as SPM [56] [40], intrachannel nonlin-ear effects [7] and nonlinear phase noise ( also called Gordon-Mollenauer phase noise), [33] [32]. OPC is based on spectrally inverting the optical signal at one or multiple points in the transmission path, thus making transmission after the spectral inversion effectively appear as transmission in a medium with inverted parameters [4].

Electrical Domain Techniques

The introduction of a coherent receiver has allowed for improving the poten-tiality of electronic domain techniques. In a coherent receiver, the optical

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2.3. Impairments Mitigation 23

signal is down-converted to the electronic domain to preserve all relevant information on the optical electric fields in the two polarizations. Therefore, the baseband equivalent electrical signal can be sampled and processed by a digital signal processor to enable software algorithms to compensate for channel impairments. Digital equalization has been deeply used to mitigate linear impairments. Particularly, CD can be compensated without a loss by a linear equalizer through a time domain equalizer (TDE) or a frequency domain equalizer (FDE). Because of PMD fluctuations on a millisecond time scale, a linear equalizer needs to be adaptive, which means that the received samples processed by the linear equalizer are subtracted from the desired signal to obtain an error signal that is used to adjust the equalizer taps through a gradient estimation block that computes the coefficient ad-justment to minimize the cost function. In practice, due to the static nature of CD and the dynamic nature of a PMD, a hybrid equalizer structure is used to combine a non-adaptive equalizer with an adaptive equalizer [29]. Recently, a particular digital technique has been suggested as a universal method for the compensation of both linear and nonlinear effects. Such a technique consists of post-compensation based on DBP. Electrical back-propagation was first studied in [51] as a pre-compensation method, since in absence of coherent detection, manipulation of the field is only possi-ble at the modulator. However, the possibility of recovering the received electric field, which is allowed by the coherent receiver, has permitted post-compensation backpropagation that was first studied in [42]. The receiver side’s backpropagation is preferred due to the great flexibility offered, since adaptive compensation can be incorporated without the need for a feedback channel.

2.3.2

Digital Backpropagation

Backpropagation involves solving an inverse NLSE through the fiber to es-timate the transmitted signal, as illustrated in Figure2.8. Since no general exact solution for the NLSE is known [11], DBP uses a numerical method known as the step Fourier method (SSFM) to solve NLSE. The split-step Fourier method solves the NLSE by dividing the fiber into small split-steps such that after each step, both the change in the spectrum through nonlin-earity and the change in the instantaneous power profile through dispersion

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Forward Propagation (FP) Digital Back Propagation (DBP)

Figure 2.8: Block diagram of forward propagation (FP) and digital backward propagation (DBP).

are small. Then in one step the NLSE is solved by ignoring the nonlinear term, and in the next step, it is solved by ignoring the dispersion term. One of the main drawbacks of DBP is the computational requirement; therefore, an intensive research in low-complexity variation has been con-ducted, including perturbation DBP [26], filtered DBP [9] [16], weighted DBP [48], Volterra-based DBP [22], iterative symmetrics SSFM [38] and non-iterative asymmetric SSFM [29].

The great focus on DBP comes from the fact that theoretically, with the exact knowledge of the channel characteristics, all the deterministic inter-actions that occur during the transmission can be completely removed with the right number of backpropagation steps. Therefore, the ultimate limita-tion to the fiber capacity arise from non-deterministic effects that include the stochastic polarization-dependent nonlinearity interaction [28] [14] [15] and the nonlinear signal-noise interaction (NSNI)[10].

Digital Backpropagation Noise Limitation

In the absence of PMD, NSNI has been regarded as the fundamental lim-itation to SMF system capacity [48]. Such an interaction can be classified as follows (Figure2.9):

Modulation instability The modulation instability (MI) refers to the case in which the noise grows considerably [27]. Such effect derives directly from the parametric gain (PG) phenomenon, which causes the creation of a spectral region around the propagated signal where another small signal can experience gain. The effect of PG on ASE

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2.3. Impairments Mitigation 25

Signal - Noise - Interaction

MI

Noise incrementation Phase noise due to

amplitude fluctuations Time jitter

NLPN Gordon - Haus effects

Figure 2.9: Block diagram that describe the nonlinear signal-noise interaction

noise has different connotations according to the fiber dispersion char-acteristics at the pump frequency. Particularly, the modulation insta-bility phenomena leads in the case of an anomalous dispersion region (β2 < 0) [34]

Nonlinear-phase noise The nonlinear-phase noise (NLPN) also called the Gordon-Mollenauer effect derives from the interaction between SPM and the introduced noise during each amplification. The ASE noise generates random fluctuations of the signal amplitude; such a signal distortion is transformed by SPM into phase noise [18]. A sim-plified description of the Gordon-Mollenauer effect can be derived in the case of zero dispersion (β = 0) by assuming a lumped amplifica-tion scheme with a fiber span of a constant length. In this case, the total nonlinear phase added to the signal can be written as follows [4]: φN L= γLef f N X k=1 [|A(0, t) + k X j=1 nj|2] ! (2.28) where nj is the noise added by the jth amplifier.

Gordon-Haus The Gordon-Haus effect [17] refers to the fluctuation of the signal pulses’ time due to ASE noise and interchannel interfer-ence. Physically, the ASE-noise-induced time jitter derives from a time-dependent variation of the optical phase induced by the ASE noise. Such a phase variation leads to small frequency shifts after

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each amplification that, due to chromatic dispersion, causes random variation in the speed of the pulse propagation after each amplifier. It is quite intuitive that the NSNI is the source of several kinds of degrada-tion effects that strongly depend on the optical system characteristics such as the symbol rate and modulation formats [5]. Furthermore, the sensitiv-ity of a DSP-enhanced coherent receiver, coupled with the reaches that are possible when DBP is employed, would result in a significant increase in the impact of nonlinear impairments. An extension of DBP that also takes care for the NSNI is suggested in [30]. To the best of our knowledge, few studies have been conducted with the aim of extending DBP to account for NSNI, as it is often argued that one cannot compensate for NSNI in DSP due to the non-deterministic nature of ASE noise [12].

Digital Bacpropagation PMD Limitation

Besides NSNI, DBP also suffers due to PMD effects. Moreover, it has been illustrated that PMD significantly affects the effectiveness of DBP [28] [57], even more than NSNI [14] and, therefore, can be considered to be the fundamental limitation of the SMF channel capacity.

The problem of PMD is the random rotation of the state of polarization due to the frequency-dependent birefringence. Such a random rotation is particularly important for WDM systems with many channels in which, even though the baud rate of the individual channels could be small, PMD can still induce random rotations of distant WDM channels with respect to one another and affect the nonlinear interaction among them. It is possible to notice from Eq. (2.8) that the strength of nonlinear processes such as SPM, XPM and FWM depends on the state of polarization. For instance, for densely spaced OFDM systems in which all the third-order nonlinear interactions can be considered as FWM [6] it is possible to notice that the transfer of power between different channels depends on the polarization states of interfering subcarriers [14]. Another example is illustrated in the case of a linearly polarized channel in which the nonlinear phase caused by SPM accumulates more than in another channel with elliptical polarization [39].

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2.3. Impairments Mitigation 27

Aside from the dependence of the nonlinear process on the polarization state, another problem is caused by the nonlinear polarization rotation due to the nonlinear birefringence phenomenon. In the case of nonlinear bire-fringence, the refractive index depends on the intensity and the polarization state of the incident light [3]. Therefore, in a WDM system, in which fiber nonlinearities that dominate are the cross-talk, and in particular, XPM is the dominating nonlinear impairment, it has been indicated that a bit-pattern-dependent phase change in an optical signal, whenever an opti-cal power is present at other wavelengths, causes a bit-pattern-dependent variation in the state of polarization [44]: this effect is also called the cross-polarization modulation (XPolM). A XPolM-induced rotation can be described through the model derived in [45] such that given an ith channel and an ~Stot =

P

iSi, the sum of the Stokes vectors of all the channels, the

XPolM leads to a nonlinear rotation of the ith channel around the collec-tive vector at a rate proportional to the part of the colleccollec-tive vector that is orthogonal to the ith channel [57].

Therefore, the random rotation induced by the PMD combined with the nonlinear effects and the nonlinear birefringence leads to several problems during the backward propagation. Random rotations caused by PMD, should be followed frequently enough in the backward direction that at each step, the amount of relative rotations should be small; thus, nonlin-ear processes may be compensated effectively. However, because both the magnitude and direction of residual birefringence fluctuate along the fiber and in time, the magnitude and direction of PMD also vary randomly in time, which leads to the PMD description and its mitigation in DBP being a non-trivial problem.

Therefore, like the ASE noise, the non-deterministic nature of PMD makes its compensation a problem, though, unlike ASE noise, PMD can be charac-terized fairly accurately through a series of measurable link characteristics such as Jones matrix, the PMD coefficient of the total accumulated DGD and PSP. Even if the knowledge of such metrics is not sufficient to describe the exact evolution of the local birefringence along the link, PMD metrics can still be used to limit the mismatch between the forward and backward propagation. For instance, it has been indicated numerically that, assum-ing priori PMD knowledge for every span, the impact of PMD on DBP can significantly decrease through a PMD compensation at each span [57],

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though a receiver typically does not have such knowledge.

Some techniques have been proposed to account for PMD during DBP. For instance, the modified DBP algorithm proposed in [19] allows for accounting for PMD by introducing an appropriate DGD at link PSP at each span such that the accumulation of DGD in the forward propagation is reversed. Another method has been suggested in [8], in which the DBP is modified by adding PMD sections during backward propagation in which sections are chosen such that they match the overall PMD accumulated over the entire link, but they an opposite sign to that of the one observed by the receiver.

2.4

Concluding Remarks

The mitigation of the fiber impairments is a topic of fundamental impor-tance in any kind of fiber-optic telecommunication system. It can be de-duced from the brief description of the previous sections that the mitigation of various channel effects and their interaction represents a big challenge for any mitigation techniques that typically deal with one impairment per time. For that reason, the DBP represents an interesting universal solution that aims to simultaneously mitigate various channel impairments while considering their interactions.

However, although DBP is very promising, it has two important limita-tions. The first is due to the non-deterministic effects (i.e. PMD and NSNI ), and the second is due to high computational cost required to approximate a solution of the NLSE. The problem of computational cost is particularly important because it is directly related with the latency and power con-sumption of the system. Since DBP is used in real-time systems, it should be designed in a manner that guarantees that these two metrics do not exceed a fixed bound.

The objective of the current study is precisely to analyze and investigate the performance of DBP implemented through the ESSFM , which is a low-complexity numerical method based on the classical SSFM that allows for solving the NLSE and thus mitigating linear and nonlinear channel impairments.

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29

Chapter 3

Enhanced Split-Step Fourier

Method - Description and

Performance Analysis

This chapter illustrates the ESSFM by first expanding the discussion of the SSFM in Section3.1. After the discussion of the SSFM, which is a method on which the ESSFM is based, the ESSFM is discussed in Section 3.2 by focusing on the complexity of the algorithm and comparing the complexity with the classical SSFM. In Section 3.3, various tests and their results are presented.

3.1

Digital Back Propagation and Split-Step

Fourier Method and Variants

Section

Optical fiber

Dispersion only Nonlinearity only

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As introduced in the previous chapter, DBP typically makes use of the SSFM to solve the inverted NLSE, i.e. to propagate the signal backward, since no known general exact solution for the NLSE [11]. The idea behind the SSFM is to divide NLSE into its linear and nonlinear sub-parts and al-ternatively solve it, as illustrated in Figure (3.1). To simplify the discussion that follows the NLSE is represented in a more compact form:

∂U ∂z =  ˆD + ˆN (3.1) ˆ D = iβ2 2 ∂2 ∂t2 − α 2 ˆ N = −iγ|U|2 (3.2)

Here, the operator ˆD contains the linear effects (i.e. dispersion and loss), whereas the operator ˆN considers the nonlinear effects. Starting from this representation, the general solution of the NLSE can be written as

U(z) = exp−ih ˆD + ˆNizU(0) (3.3)

3.1.1

Asymmetric and Symmetric Split-Step Fourier

Method

The problem with the general solution in Eq. (3.3) is that the fact that ex-ponential operators do not commute; therefore, it cannot be evaluated. However, an approximation of the solution can be obtained by solving individual terms that describe the linear and nonlinear effects. Such an approximation can be realized by employing the Baker-Campbell-Hausorf theorem, which can be expressed as follows: [23]

exp−ih ˆD + ˆNiz= exp−i ˆDz+exp−i ˆN z+exp  h ˆD, ˆNiz2 2  +· · · . (3.4) By combining Eqs. (3.3) and (3.4) the solution of the Eq. (3.1) can be written as follows:

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3.1. Digital Back Propagation and Split-Step Fourier Method and

Variants 31

Step n

Channel Model

Step 1 Step 2 Step 3

(a) Actual Fibre

Simulation Model

Step n

Step 1 Step 2 Step 3

(b) ssfm (Wiener Model)

Figure 3.2: Schematic Representation of a Fibre and its Simulation through ssfm

(a) Wiener Model (b) Hammerstein Model

(c) Wiener-Hammerstein Model

Figure 3.3: Nonlinear System Models

Here, the commutator h ˆD, ˆNi = ˆD ˆN − ˆN ˆD and higher-order commuta-tors, which are associated with the square power or higher power of h, can be eliminated because h is made sufficiently small to make them negligible. Physically, to simulate the propagation into an arbitrary long system, the fiber is divided into n small blocks of length h, such that at each block, a piece of nonlinear step of length h is followed by a piece of linear sub-step of length h, or vice versa. Therefore, to propagate a signal through a fiber consisting of n steps of length h, the signal is actually propagated through 2n blocks, as illustrated in Figure (3.2) (where Figure (3.2a) repre-sent the actual fiber whereas Figure (3.2b) represent the simulated model), in which each pair of linear and nonlinear blocks forms a nonlinear sys-tem of either the Wiener or Hammerstein model (Figure (3.3a) and Figure (3.3c) respectively) depending on their sequential arrangement.

Due to the elimination of the terms associated to h2 or to higher order of

h in Eq. (3.5), a local error of O(h2) is introduced; thus, the global error is

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Step n Step 1 Step 2

(a) Symmetric ssfm (S-ssfm)

Step n Step 1 Step 2

(b) S-ssfm with Effective Length

Figure 3.4: Schematic Representation of ssfm with Improved Accuracy

(i.e. the term h2h ˆD, ˆNiis no more negligible); thus, a global error of O(h2) can be constructed. Since the sign of the local error depends on the order in which individual operators are applied, as illustrated in Eq. (3.6):

h ˆD, ˆNi

= ˆD ˆN − ˆN ˆD = − ˆN ˆD − ˆD ˆN= −h ˆN , ˆDi (3.6) a local error of second order in two adjacent sections of fiber can be elimi-nated by applying, for sufficiently small lengths, linearity and nonlinearity in opposite order. Therefore, to apply such a more accurate approximation, the propagation from z to z + h should be performed in two steps of length

h

2, which in terms of the operator representation becomes:

exp−ihh ˆD + ˆNi= exp  −ih 2h ˆD + ˆN i exp  −ih 2h ˆN + ˆD i ≈ exp −ih ˆD 2 ! exp −ih ˆN 2 ! exp −ih ˆN 2 ! exp −ih ˆD 2 ! = exp −ih ˆD 2 !

exp−ih ˆNexp −ih ˆD 2

!

(3.7)

The step of length h is thus now modeled as a Wiener-Hammerstein non-linear system (Figure3.3c). When the form in Eq. (3.7) is applied again in the next blocks, the result obtained is that similarly to the second equality of Eq. (3.7), the final operator of one length can be combined (exactly) with the first operator of the subsequent length. This yields a fiber of length nh that is modeled in 2n + 1 blocks of which the central n nonlinear and n − 1 linear blocks are of length h and the linear blocks on either end are of length h2; the final result is illustrated in Fig. (3.4a). Therefore, at the cost of a single additional step in passing from Eq. (3.5) to Eq. (3.7),

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3.1. Digital Back Propagation and Split-Step Fourier Method and

Variants 33

a significant improvement in accuracy has been gained.

Finally, further optimization can be achieved by considering the attenuation during the nonlinear step and thus by introducing the nonlinear step size as

hef f =

1 − e−αh

α (3.8)

Then, by combining Eq. (3.5) with Eq. (3.7) and Eq. (3.8), the final solution can be written as follows:

U(z + h, T ) ≈ exp −ih ˆD 2 ! exp−ihef fNˆ  exp −ih ˆD 2 ! U(z, T ) (3.9)

which is the typically employed variant of the SSFM (Figure 3.4b).

3.1.2

Filtered Split-Step Fourier Method

In [9] introduced the concept of filtered DBP (F-DBP) along with the opti-mization of non-linear operator calculation point. It is observed that during each DBP step, the intensity of the out-of-band distortion becomes higher. The distortion is produced by high-frequency intensity fluctuations that modulate the outer sub-carriers in the non-linear sections of DBP. This lim-its the performance of DBP in the form of noise. To overcome this problem, a low pass filter (LPF), as shown in Figure (3.5), is introduced in each DBP step. The digital LPF limits the bandwidth of the compensating waveform, so it is possible to optimize the compensation for low-frequency intensity fluctuations without overcompensating the high-frequency intensity fluctu-ations. This filtering also reduces the required oversampling factor. The bandwidth of the LPF has to be optimized according to the DBP stages used to compensate for fiber transmission impairments, i.e. bandwidth is much narrow when very few BP steps are used, and bandwidth increases accordingly when more DBP stages are used.

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Phase Modulator

LPF

Input Output

Figure 3.5: Block Diagram of the Filtered DBP (F-DBP)

3.1.3

Split-Step Fourier Method Implementation and

Costs

In a coherent optical system, the SSFM is employed to backward-propagate the digital version of the optical field. The discrete version of the optical field is composed by N samples {xk}Nk=1, where xk= [xkX xkY]T. Thus, by

assuming a fiber of length L = N Sh blocks, the digital signal is backward-propagated through 2N S + 1 blocks as follows [54]:

Xk= [F {xkX} F {xkY}]T k = 1, · · · , N (3.10) Yk= Xkei2π 2β 2fk2h k = 1, · · · , N (3.11) yk=F−1{YkX} F−1{YkY}  k = 1, · · · , N (3.12) zk= ykejξγ 8 9|yk| 2h ef f k = 1, · · · , N (3.13)

Here, the operator F is the Fourier transform, ξ is a coefficient that depends on the channel characteristics, and fk is the frequency of the kth sample.

As it can be noticed, the linear sub-step is performed in the frequency do-main, whereas the nonlinear step is performed in the time domain. From this implementation, follows one of the most important drawbacks of the SSFM: the computational cost. It is clear from Eqs. (3.10) and (3.12) that the computational cost is dominated by the fast Fourier transform (FFT), and by even assuming an FFT algorithm such as the Cooley-Turkey FFT algorithm (the most common FFT algorithm), the required complex mul-tiplications and complex additions necessary to simulate the propagation of N samples into a fiber block are respectively N2log2(N ) and N log2(N ).

Therefore, the overall numbers of complex multiplications and additions per sample needed to propagate a sequence of N samples through a fiber

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3.2. Digital Back Propagation and Enhanced Split-Step Fourier Method35

of length L = NSh scale as follows [52]:

× : NSlog2(N ) = 2NS 1 N N 2log2(N ) (3.14) + : 2NSlog2(N ) = 2NS 1 NN log2(N ) (3.15) It is clear from Eqs. (3.14) and (3.15) that the trade-off between the com-putational cost, in terms of the number of operations per sample, and the accuracy, in terms of step size. In fact, on one hand the lower the sub-step size, h, (i.e. increasing NS), the greater the accuracy of the SSFM. On the

other hand, the lower the sub-step size, h, the greater the computational cost. Therefore, a compromise must be found by properly selecting the size of the propagation step. One can notice that any approach that increases the accuracy without increasing the number of steps, NS, or that reduce

the computational cost while keeping the accuracy intact would be highly desirable. To this end, an approach that is based on the logarithmic per-turbation method [11] allows for increasing the accuracy without changing the number of steps. Such an improvement of the SSFM has been proposed by Secondini, Marsella and Forestieri [54] and is discussed and evaluated in the next sections.

3.2

Digital Back Propagation and Enhanced

Split-Step Fourier Method

In this section, the enhanced split-step Fourier method (ESSFM), suggested by Secondini, Marsella and Forestieri [54], for mitigation of linear and non-linear impairments with back propagation is discussed, and its effectiveness in case of dually polarized signal is numerically investigated. To indicate the gain in accuracy, the performance and the computational complexity (i.e. number of steps) of the ESSFM are compared with the performance and complexity of the SSFM.

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3.2.1

Enhanced Split-Step Fourier Method

The method suggested by Secondini, Marsella and Forestieri is based on the SSFM discussed in Section3.1. As discussed in Section 3.1, the separation of the operators ˆD and ˆN allows the separation of linear and nonlinear steps, as illustrated in Eqs. (3.11) and (3.13) by introducing an error unless the step size h is kept small. The enhanced split-step Fourier method allows improvement of the accuracy through a more accurate approximation of the nonlinear sub-step. In fact, the error introduced by the separation of the linear and nonlinear operators may be reduced by incorporating the interaction between dispersion and nonlinearity within h and thus allowing an increase of the step size without affecting overall accuracy. The more accurate expression for the nonlinear sub-step introduced by the ESSFM is based on the fact that in the nonlinear step, the signal at discrete time k undergoes a frequency-dependent nonlinear phase rotation [55]:

θk(f ) = ∞ X m=−∞ ∞ X n=−∞ cnm(f )yk−myk−n∗ . (3.16)

Here, the coefficients cnm depend on the link characteristics. Starting from

this nonlinear phase rotation, Eq. (3.16), and truncating the channel mem-ory to the first Nc past and future samples, retaining only the diagonal

terms of the quadratic form, and averaging the frequency-dependent coef-ficients cmn(f ) over the signal bandwidth results in

θk= c0|yk| + Nc

X

i=1

ci(|yk−i|2+ |yk+i|2), (3.17)

where {ci}Ni=0c are Nc+ 1 real coefficients that correspond to the average

values of the coefficients {cii(f )}Ni=0c over the signal bandwidth. Equation

(3.17) gives an accurate approximation of the frequency-dependent non-linear phase rotation; therefore, combining Eq. (3.17) with the nonlinear step of Eq. (3.13) results in a more accurate nonlinear step, which can be written as

zk = ykejγ

8

9hef f(c0|yk|+

PNc

i=1ci(|yk−i|2+|yk+i|2)) k = 1, · · · , N (3.18)

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