OFDM for Fiber-Optic Data Transmission at High Spectral Eﬃciency

OFDM für die Datenübertragung über Glasfasern bei hoher spektraler Eﬃzienz

Der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Grades

Doktor-Ingenieur

vorgelegt von

Markus Mayrock

Erlangen, 2012 Als Dissertation genehmigt von der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der Einreichung: 09.01.2012 Tag der Promotion: 25.05.2012 Dekanin: Prof. Dr.-Ing. Marion Merklein Berichterstatter: Prof. Dr.-Ing. Herbert Haunstein Prof. Dr.-Ing. Bernhard Schmauß Danksagung

(Acknowledgments)

Diese Arbeit stellt Ergebnisse vor, die im Rahmen meiner Tätigkeit als wissenschaftlicher Mitarbeiter am Lehrstuhl für Informationsübertragung entstanden sind. An dieser Stelle möchte ich einigen Personen danken, die zum Gelingen dieser Arbeit beigetragen haben. In erster Linie gilt mein Dank Prof. Dr. Herbert Haunstein, der mich für die Thematik der optischen Nachrichtenübertragung begeistert hat, jederzeit für Diskussionen zur Ver- fügung stand und mich mit seinen Ratschlägen stets optimal unterstützt hat. Prof. Dr. Bernhard Schmauÿ möchte ich danken für das Interesse an meiner Arbeit und dafür, dass er sich bereit erklärt hat, die Aufgabe des Gutachters zu übernehmen. Ferner bedanke ich mich bei Prof. Dr. Johannes Huber und allen Kollegen vom LIT für die angenehme Arbeitsatmosphäre und die vielen anregenden fachlichen wie auch fachfremden Diskussionen. Schlieÿlich gilt mein Dank meinen Eltern und Geschwistern, die mich zu jeder Zeit bei meinen Plänen unterstützen.

Abstract

Data traffic within networks worldwide increases exponentially – different authors give different numbers; some mention growth rates up to 50 percent per year. This increasing traffic load gives rise for various kinds of challenges for the network operators. Especially, they have to offer higher bitrates in the core networks, which work on the basis of optical data transmission over silica fibers. There have been many major steps in technical progress which allowed for increasing the datarates per wavelength to 10 and 40 GBit/s within the last decade. Currently, transponders offering 100 GBit/s become available. The new equipment generation is facilitated by coherent receivers, which enable linear downconversion of the optical receive signal to the electrical baseband. Signal distortion can be compensated for with the help of digital signal processing. Furthermore, two data streams can be transmitted on orthogonally polarized optical carriers of the same frequency (polarization-division multiplexing, PDM ). In the future, a further raise of bitrates is enabled by increasing the spectral efficiency. In this context, orthogonal frequency-division multiplexing (OFDM) is a promising technique as it offers efficient compensation of linear signal distortion caused by dispersive channels. This work investigates the usage of OFDM for high bitrate data transmission over silica fibers at high spectral efficiency. For this purpose, at first the relevant sources of signal distortion in the optical waveguide are discussed. Linear dispersive effects determine how to choose the OFDM system parameters. Polarization dependent loss causes a difference between the signal-to-noise power ratios of both PDM channels. By means of polarization- time coding the influence on the bit error ratio can be delimited. When implementing OFDM special attention has to be payed to non-linear effects. Most dominantly the fiber channel is affected by the Kerr effect causing non-linear signal distortion along with crosstalk between channels at different wavelengths. In this work the transmission system shall be modeled as a weakly non-linear system implying that non-linear signal distortion is considered as a noise-like additive distortion. This approach allows for an estimation of the channel capacity – and thereby approximating the maximum achievable spectral efficiency. Moreover, in this system model sources of distortion can be included, which are caused by the non-ideal implementation of transmitters and receivers. Laser phase noise, clipping vi and quantization turn out to be crucial effects. In the weakly non-linear system model their contributions are regarded as further uncorrelated sources of noise, and thus, the estimated noise variances are increased by respective summands. Doing so, the estimation of the channel capacity can be repeated and it turns out that the mentioned impairments delimit the achievable spectral efficiency especially for short transmission distances. The power of the Kerr effect induced distortion can be separated into two contributions: The first one is only considering non-linear signal distortion of the useful signal. The second part corresponds to non-linear crosstalk of neighboring channels. System parameters are identified for which the first contribution dominates. For that scenario methods are discussed which help to compensate non-linear signal distortion leading to improved transmission performance. vii

Zusammenfassung

Der Datenverkehr in den weltweiten Netzen wächst exponentiell – verschiedene Quellen nennen unterschiedliche Zahlen, wobei Zuwächse von bis zu 50% pro Jahr ermittelt werden. Aus diesem zunehmenden Datenaufkommen ergeben sich viele Herausforderungen für die Netzbetreiber. Insbesondere müssen ständig höhere Datenraten in den Kernnet- zen, welche mit optischer Übertragung über Glasfasern arbeiten, zur Verfügung gestellt werden. Es gab viele technologische Entwicklungen, die im letzten Jahrzehnt eine Steigerung der Datenrate pro Wellenlänge auf 10 bzw. 40 GBit/s erlaubten. Aktuell werden die ersten Geräte mit 100 GBit/s verfügbar. Die letzte Erhöhung wird vor allem durch den Einsatz von kohärenten Empfängern ermöglicht. Dabei wird das optische Empfangssignal linear umgesetzt in ein elektrisches Basisbandsignal. Signalverzerrungen werden mittels digitaler Signalverarbeitung kompensiert. Ferner können zwei Datensignale auf orthogonal polari- sierten optischen Trägern gleicher Frequenz transportiert werden (Polarisationsmultiplex, PolMux). In der Zukunft wird eine weitere Erhöhung der Datenraten möglich, indem die spektrale Effizienz gesteigert wird. In diesem Kontext ist “orthogonal frequency-division multiplexing” (OFDM) eine interessante Option, da diese Technik eine effiziente Kompen- sation von Signalverzerrungen, die durch dispersive Kanäle hervorgerufen werden, erlaubt. Die vorliegende Arbeit untersucht den Einsatz von OFDM zur hochbitratigen Daten- übertragung über Glasfasern bei hoher spektraler Effizienz. Dazu werden zunächst die relevanten Fasereffekte diskutiert. Die linearen dispersiven Effekte bestimmen die System- parameterwahl. Es zeigt sich, dass polarisationsabhängige Dämpfung eine Verschiebung der Störabstände der beiden PolMux-Teilkanäle verursacht. Mittels “polarization-time coding” kann die Auswirkung auf die Fehlerrate eingegrenzt werden. Besondere Beachtung beim Einsatz von OFDM muss nichtlinearen Effekten gelten. Für den optischen Faserkanal ist insbesondere der Kerr-Effekt zu nennen, der zu nichtlinearen Signalverzerrung und Übersprechen zwischen Nachbarkanälen auf Trägern unter- schiedlicher Wellenlänge führt. In dieser Arbeit wird das OFDM-Übertragungssystem als schwach nichtlineares System modelliert und nichtlineare Signalverzerrungen als additive Störung charakterisiert. Dieser Ansatz erlaubt eine Abschätzung der Kanalkapazität und damit der maximal erreichbaren spektralen Effizienz. Auch gestattet das Systemmodell die Hinzunahme von Störungen, die bedingt sind viii durch nichtideale Implementierung von Sender und Empfänger. Als kritisch stellen sich Laserphasenrauschen, Amplitudenlimitierung und Quantisierung heraus. Die verursachte Signalstörungen werden in der Systemmodellierung als weitere Quelle für Rauschen be- trachtet und die geschätzten Störleistungen erhalten entsprechend weitere Beiträge. Damit kann die Abschätzung der Kanalkapazität wiederholt werden und es zeigt sich, dass die genannten Effekte die erreichbare spektrale Effizienz insbesondere auf kurzen Übertra- gungsstrecken limitieren. Die Leistung des vom Kerr-Effekt hervorgerufenen Störsignals lässt sich separieren in einen Anteil, der lediglich nichtlineare Verzerrungen des eigentlichen Nutzsignals betrifft und einen weiteren Beitrag, der durch nichtlineares Übersprechen hervorgerufen wird. Es werden Systemparameter identifiziert, bei denen der erstgenannte Effekt überwiegt. Für diesen Fall werden Methoden zur Kompensation der nichtlinearen Störung diskutiert, durch welche die Leistungsfähigkeit des Systems verbessert werden kann. ix

Contents

1 Introduction and Outline 1

2 Digital Data Transmission over Silica Fibers 5 2.1 Optical Transmission System ...... 5 2.2 Fiber Channel ...... 7 2.2.1 Attenuation ...... 8 2.2.2 Group Velocity Dispersion ...... 8 2.2.3 Polarization Mode Dispersion ...... 10 2.2.4 Polarization Dependent Loss ...... 12 2.2.5 Kerr Effect ...... 13 2.3 Optical Amplification and Concatenation of Fiber Spans ...... 14 2.3.1 Lumped Optical Amplification ...... 14 2.3.2 Multi-Span Fiber Link ...... 16 2.3.3 Distributed Amplification ...... 17 2.4 Polarization-Division Multiplexing ...... 17 2.5 Wavelength-Division Multiplexing ...... 19 2.6 Simulation Technique ...... 20 2.6.1 Split-Step Fourier Method ...... 20 2.6.2 ASE Noise Generation ...... 21

3 Orthogonal Frequency-Division Multiplexing for Optical Data Transmis- sion 23 3.1 OFDM Principle ...... 24 3.2 Optical Transmitter Architecture ...... 26 x Contents

3.3 Optical Receiver Architecture ...... 28 3.3.1 Single Polarization Coherent Detection ...... 28 3.3.2 Dual Polarization Coherent Detection ...... 31 3.4 System Simulation ...... 31 3.4.1 Oversampled OFDM Signal ...... 31 3.4.2 Multiple-Input/Multiple-Output System ...... 32 3.5 Direct Detection Optical OFDM ...... 32

4 Optical OFDM at the Presence of Linear Fiber Eﬀects 35 4.1 Chromatic Dispersion ...... 35 4.1.1 Pulse Broadening ...... 35 4.1.2 OFDM Design Rule ...... 36 4.1.3 Simulations ...... 37 4.2 Polarization Mode Dispersion ...... 43 4.2.1 Single Polarization Coherent Detection ...... 43 4.2.2 Dual Polarization Coherent Detection ...... 44 4.3 Polarization Dependent Loss ...... 46 4.3.1 Simpliﬁed Transmission Model ...... 46 4.3.2 Concatenation of PDL Elements ...... 49 4.3.3 PDL in Multi-Span Link ...... 50 4.3.4 Channel Capacity ...... 53 4.3.5 Polarization Time/Frequency Coding ...... 55

5 Impact of the Kerr Eﬀect 59 5.1 Weakly Non-linear System Approach ...... 60 5.1.1 Single Polarization Model ...... 61 5.1.2 Dual Polarization Model ...... 64 5.2 Interrelation of Optical Signal Power and Equivalent Noise Power ...... 66 5.2.1 Single OFDM Channel ...... 66 5.2.2 Multiple OFDM Channels ...... 68 5.3 Separation of Noise Power Contributions ...... 69 5.4 Channel Capacity ...... 73 Contents xi

5.4.1 Mutual Information ...... 73 5.4.2 Gaussian Signaling ...... 75 5.5 Achievable Spectral Eﬃciency ...... 75

6 Impairments through TX and RX Implementation 79 6.1 LaserPhase Noise ...... 79 6.2 Clipping ...... 85 6.3 Quantization Noise ...... 87 6.4 Transmission Simulations ...... 89 6.5 Optical Frontend ...... 93 6.6 Achievable Spectral Eﬃciency ...... 95 6.7 Outlook: Forward Error Correction ...... 98

7 Compensation of Non-linear Signal Distortion 101 7.1 Backpropagation ...... 101 7.2 Non-linear Phase Rotation ...... 102 7.2.1 Approach ...... 102 7.2.2 Phase Rotation per Span ...... 105 7.2.3 Simulation Results ...... 105 7.2.4 Frequency Domain Description ...... 109 7.3 Decision Directed Compensation ...... 110 7.4 SPM Compensation in WDM Scenarios ...... 111

8 Summary and Conclusions 117

A Abbreviations and Symbols 121 A.1 Abbreviations ...... 121 A.2 Mathematical Operations ...... 123 A.3 Symbols and Functions ...... 123

B Separation of Equivalent Noise Power 129

Bibliography 132 xii Contents 1

1 Introduction and Outline

In the core networks of worldwide data transmission systems, the datarates per wavelength have increased to 10 and 40 GBit/s within the last decade. Currently, transponders offering 100 GBit/s become available. While commercial receivers for 10 and 40GBit/s utilize direct detection, the latest transponder generation is based on coherent detection. Thereby, the received optical signal and a local laser signal are superposed. At the photo- detector, the received signal is down-converted to an electrical baseband signal in a linear fashion. Nowadays, powerful signal processors are available enabling complex operations, e.g., with respect to detection, equalization, advanced FEC coding schemes in the digital domain – even at currently required datarates. Several research groups have been working on theoretical aspects as well as on implementation issues of single carrier coherent optical systems [Sav08], [LKKC07]. Besides single carrier modulation, orthogonal frequency-division multiplexing has gained interest in the context of high bitrate optical systems [LA06], [SA06], [JMTT07]. OFDM allows for efficient equalization and flexible adaptation of the system’s spectral efficiency. Datarates can be increased by applying larger symbol alphabets on a sub-carrier basis – at the same time, the signal bandwidth can be kept constant. The achievable spectral efficiency is upper-bounded by the channel capacity. For linear channels, which are impaired by additive Gaussian noise, Shannon’s equation evaluates this measure. The signal propagation in silica fibers however is affected by non-linear distortion. Thus, the capacity evaluation technique has to be adapted. Several approaches have been published in the literature [MS01], [EFW+08], [Göb10]. In this work, OFDM transmission over non-linear fiber channels is investigated based on numerical simulations. 2 1. Introduction and Outline

For different channel scenarios, the maximum achievable spectral efficiency is estimated. Besides non-linear fiber effects, impairments through implementation limitations are included in the simulation model. Finally, the effectiveness of compensation techniques for non-linear signal distortion is addressed. The thesis is organized as follows:

Chapter 2 presents the mathematical description of an optical data transmission • system. The blocks for transmitter and receiver are kept rather general. They are basically intended to motivate the generation of the optical transmit signal, the detection operation at the receiver, and to introduce the corresponding notation. All the optical signals are described in the equivalent complex baseband. The focus of the chapter is on the characterization of the fiber channel which exhibits various linear and non-linear effects. In this context, the non-linear Schrödinger equation is elaborated describing the propagation of optical signals along the waveguide. Optical amplifiers are important devices which are necessary for regenerating the signal powers after certain transmission distances. We give the mathematical model used in the simulations for these devices as well as for techniques like polarization- division multiplexing and wavelength-division multiplexing. Finally, the chapter gives some remarks about techniques regarding the numerical simulation of the signal transmission.

Chapter 3 introduces the basics of orthogonal frequency-division multiplexing. This • technique is widely used in wireless and wireline communications. The OFDM principle can be directly transferred to the modulation of optical carriers. The chapter elaborates the required transmitter and receiver structures, which are just brieﬂy mentioned in Chapter 2. In the context of transmission simulations, oversampling and the modeling of polarization-division multiplexing are addressed. In addition, this chapter sketches an alternative optical OFDM system which is based on direct detection. Its advantages are discussed; however, despite its lower complexity from optical component point of view the system has a crucial drawback, i.e., it does not reach very high spectral eﬃciency.

Subsequently, Chapter 4 is dedicated to investigations on optical OFDM trans- • mission systems at the presence of linear fiber effects. Throughout this work we assume that there is no optical dispersion compensation. As a consequence, chromatic dispersion is the dominating source of linear signal distortion, which has to be considered for the definition of the OFDM system parameters. A design rule is derived and verified by simulations, which achieves inter-symbol interference free transmission over such channels. As a further aspect, the influence of PMD is studied. It turns out that this effect can be ideally compensated for, when polarization diversity detection is applied. Signal degradation through polarization dependent 1. Introduction and Outline 3

loss is investigated along with a coding scheme which leads to reduced SNR penalties. Parts of this chapter have already been published by the author in [MH07d], [MH07b], [MH07a], [MH09b], [MH09a], [MM10].

Following up, the impact of non-linear signal distortion through the Kerr effect • is investigated in Chapter 5. The OFDM system model is extended to a “weakly non-linear” system. By means of Monte Carlo simulations the interrelation between optical signal power and an equivalent noise power is determined – both for the single as well as dual polarization transmission case. Respective results allow for separating the overall observed noise-like distortion into three noise power contributions: Actual additive noise caused by optical amplifiers, distortion due to non-linear interaction between signal and fiber, as well as non-linear crosstalk from neighboring wavelengths. Moreover, this chapter discusses the effect of fiber non-linearity on the mutual information of transmitted and received signals. Assuming Gaussian signaling and noise, the weakly non-linear system approach delivers a straight-forward estimate of the maximum achievable spectral efficiency. Several results of this chapter have been published by the author in [MH08b], [MH08a], [MH09b], [MH09a], [MH09c], [HM10a], [HM10b], [May10], [MH11b], [MH11a].

The following Chapter 6 extends the model presented in Chapter 5 by incorporating • implementation impairments. Laser phase noise is described as a Wiener process; the resulting signal distortion can be approximated by an additive noise power contribution. The time domain OFDM signals exhibits a nearly Gaussian probability density. The practical limitation of signal amplitudes leads to clipping noise. Fur- thermore, the number of quantization intervals of high speed digital-to-analog and analog-to-digital converter devices is limited to less than 10 bit resolution. Conse- quently, quantization noise has to be taken into consideration. Finally, the optical receiver front-end has an influence on the signal quality through shot-noise and thermal noise of the transimpedance amplifier. The implementation impairment related noise contributions translate to a loss in achievable spectral efficiencies. The comparison with published transmission experiments shows a gap between achieved spectral efficiency and the estimated bound. Application of advanced FEC schemes may help for reducing this gap – some approaches are briefly discussed. Parts of this chapter have already been published in [MH09c], [HM10a], [HM10b], [MH11b], [MH11a]

Up to this point the Kerr eﬀect induced signal distortion is treated as an additive • noise-like contribution. However, as it stems from a non-random interaction between the signal and the ﬁber channel, it can in principle be estimated and compensated 4 1. Introduction and Outline

for. Chapter 7 addresses this aspect by reviewing several known compensation methods: Backpropagation, non-linear phase rotation, and decision directed compensation – especially, their interrelation is elaborated. By means of numerical simulations the compensation performance is investigated for single channel and WDM scenarios. The eﬀectiveness becomes evident along with the noise power separation approach introduced in Chapter 5. Several results have already been published by the author in [May10].

Finally, Chapter 8 summarizes the work and draws some conclusions. • 5

2 Digital Data Transmission over Silica Fibers

Optical data transmission over silica fibers is the basic technology for today’s communication networks, which provide very high bit-rates and – at the same time – bridge very long distances. The key techniques for these systems were the development of low loss optical waveguides and the invention of optical amplification. This chapter shall provide a generic description of optical transmission systems where the focus is on the fiber channel and its effects. The discussion leads to the formulation of the non-linear Schrödinger equation which describes the propagation of optical signals in single mode fibers. The following sections do not aim at a stringent derivation of the wave equations, but at some important steps along with considerations rendering the resulting differential equations plausible.

2.1 Optical Transmission System

The starting point of the system description is a very generic model for digital data transmission depicted in Figure 2.1. At the transmitter side there are complex valued symbols x[k] taken from a ﬁnite set , where k is the discrete time index. For example, A may denote the symbol alphabets 16QAM or 8PSK. A Usually, digital data to be transmitted is represented by binary symbols. Thus, in order to obtain complex valued symbols, these binary symbols have to be mapped to x[k]. This step along with further possible operations, like for example channel coding, shall not be considered here. In fact, the transmission model for symbols x[k] developed in this and further chapters can be the basis for the design and optimization of mappings and 6 2. Digital Data Transmission over Silica Fibers

x[k] TX baseband RX baseband y˜[k] processing E/O O/E processing

s(t) u(t, z = 0) u(t, z = L) r(t)

electrical domain optical domain electrical domain

Figure 2.1: Optical digital data transmission system. channel coding schemes along with respective decoding algorithms. The ﬁrst building block in Figure 2.1 then refers to the generation of an analog baseband signal in the electrical domain. In general, it comprises an inphase and quadrature component, which are expressed by a complex valued signal

s(t) = s (t) + j s (t). (2.1) I · Q This operation is often implemented as pulse amplitude modulation (PAM). In this case, s(t) is composed as a sequence of basic pulses g(t), which are weighted by x[k]. The shape of g(t) is usually chosen as √Nyquist-pulse, which enables inter-symbol-interference-free detection after matched ﬁltering [Kam04]. But there is no restriction to PAM – one can also think of other modulation techniques, e.g. orthogonal frequency-division multiplexing which is addressed in Chapter 3. The “TX baseband processing” block is followed by the conversion of the electrical signal to an optical signal (“E/O”). Equivalently, the baseband signal s(t) is up-converted to the frequency fc of the optical carrier. The ideal “E/O” conversion is described by

u (t, z = 0) = √2R [s (t) cos (2πf t) s (t) sin (2πf t)] . (2.2) optical · I c − Q c The purpose of the coefficient R is to scale the optical signal to the desired power. As a source for the optical carrier most commonly infrared lasers are applied, operating at wavelengths around λ = 1550 nm (minimum attenuation of the fiber). This value leads to a carrier frequency of approximately 193.5 THz. As further modeling of the fiber channel is carried out in the equivalent complex baseband, signals u(t, z) in the optical domain (c.f. Figure 2.1) shall also be assumed as baseband signals. Thus, the step of (2.2) is omitted; where necessary, non-ideal effects of the modulator are described in the complex baseband (c.f. section 3.2). 2.2. Fiber Channel 7

The optical signal is then launched into a silica fiber of length L. Throughout this thesis we assume the application of single mode fibers. This type of fiber exhibits a design with respect to geometry and refractive index profile such that only one mode can propagate through the waveguide. By this means modal dispersion is avoided; much larger signal bandwidths can be utilized compared to multi-mode fibers. An extensive derivation of propagation equations is omitted. In the following sections, the focus shall be on a phenomenological description of the main fiber effects, which characterize the interrelation of input and output signals u(t, z = 0) and u(t, z = L). At the receiver, the optical signal u(t, z = L) has to be converted back to an electrical baseband signal. For this purpose, the signal is filtered by an optical bandpass first, which ensures finite noise bandwidth. The conversion to an electrical signal itself is carried out by photo-diodes. The optical signal is converted to a photo current, which is proportional to the power of the incident RX signal. In a direct detection configuration, the optical receiver front-end retrieves a signal r(t) with

r(t) u(t, z = L) 2. (2.3) ∼ | | This non-linear operation yields a real valued, positive signal, i.e., phase information of the optical signal is lost. Hence, direct detection systems can utilize on-off keying (used in 10 GBit/s systems) or amplitude shift keying with higher numbers of signal power levels. In a modified configuration (differential pre-coding, interferometric detection at the receiver) direct detection is applied for 40 GBit/s systems [GW05]. An alternative receiver architecture is based on the superposition of the received optical signal and a local laser signal (coherent detection) [Ho05]. The non-linear mixing of the photo diode down-converts the received signal in a linear fashion

r(t) u(t, z = L). (2.4) ∼ The further steps which operate on the electrical baseband signal r(t) (may) include: Filtering, analog electrical equalization and data decision or analog-to-digital conversion followed by digital signal processing. The target is to obtain complex valued symbols y˜[k], which exhibit – compared to the source symbols x[k] – minimum amounts of noise and distortion.

2.2 Fiber Channel

In order to describe the characteristics of the fiber channel a mathematical tool is required which models the interrelation between the signal at the channel input to its output. In this connection, linear and non-linear effects have to be distinguished. In the following sub- sections the most relevant fiber effects are introduced, finally ending up in the formulation 8 2. Digital Data Transmission over Silica Fibers of a differential equation (set of two coupled differential equations, respectively) for the signal propagation in silica fibers.

2.2.1 Attenuation

The power of the optical signal propagating along the ﬁber decays exponentially with distance. Thus, the signal power at position z can be expressed in terms of the launch power by

2 2 αz u(t, z) = u(t, z = 0) e− . (2.5) | | | | · This observation can also be described by the diﬀerential equation

∂u(t, z) α + u(t, z) = 0. (2.6) ∂z 2 The attenuation coefficient α is given in units of Neper per distance, i.e., 1/α is the 1 distance where the signal power is attenuated to e− 36.8% of the original launch ≈ power. Often, the loss effect is expressed in decibels per fiber length, i.e.,

α /20 z u(t, z) = u(t, z = 0) 10− dB · . (2.7) ·

(2.5) and (2.7) yield an interrelation between α and αdB

α = α 10 log e = 4.343α. (2.8) dB · 10 In general, the attenuation coefficient varies for different signal wavelengths; however, the variation is quite small within typical signal bandwidths. Within the wavelength range of 1500 to 1600 nm, silica fibers reach minimum loss coefficients of approximately 0.2 dB/km.

Throughout this thesis we assume a constant value αdB = 0.2 dB/km.

2.2.2 Group Velocity Dispersion

The design of single mode fibers avoids modal dispersion. However, the optical transmission channel still exhibits dispersive effects. One observes that signal contributions of different frequencies have different group velocities (group velocity dispersion, GVD). Equivalently, this dispersive effect depends on the wavelength – therefore, this effect is often referred to as chromatic dispersion (CD). The mathematical description of GVD is done in the frequency domain by introducing a frequency dependent phase term β(f) [Agr92]

α z jβ(f)z U(f, z) = U(f, 0) e− 2 e− . (2.9) · · 2.2. Fiber Channel 9

In a next step the wave number is expanded in a Taylor series around the (angular) carrier frequency ωc

1 1 β(ω) = β + β (ω ω ) + β (ω ω )2 + β (ω ω )3 + ... (2.10) 0 1 − c 2 2 − c 6 3 − c with dmβ β = . (2.11) m dωm ωc=ω

In order to obtain an equivalent complex baseband expression (2.10) is shifted back to

ω ω f = − c . (2.12) 2π

The further model neglects a constant phase shift (set β0 = 0) and defines β1 = 0. The latter step causes zero propagation time for harmonic signals. The GVD model is intended to describe time delays between signal contributions of different frequencies. The absolute propagation time is of less significance. As a consequence, at zero mean signal group delay the system model usually becomes acausal. The expression for β(f) now reads

1 1 β(f) = β (2πf)2 + β (2πf)3 + ... (2.13) 2 2 · 6 3 ·

Higher order terms shall be neglected (often β3 can also be neglected). Now (2.13) is inserted in (2.9) and the diﬀerential ∂/∂z is evaluated

∂U(f, z) α j j = β (2πf)2 β (2πf)3 U(f, z). (2.14) ∂z − 2 − 2 2 · − 6 3 · · Application of the inverse Fourier transform along with the calculation rule [BSMM99]

∂nf(t) (jω)n f(t) = (2.15) F{ } F ∂tn yields ∂u(t, z) α j ∂2u(t, z) 1 ∂3u(t, z) + u(t, z) β β = 0. (2.16) ∂z 2 − 2 2 ∂t2 − 6 3 ∂t3 In order to find the chromatic dispersion properties in measurements one usually determines time delays with respect to wavelength differences. Hence, we can define the dispersion coefficient dβ ω2 dβ D = 1 = 1 . (2.17) dλ −2πc dω 10 2. Digital Data Transmission over Silica Fibers

The derivative with respect to λ was substituted with diﬀerentiation with respect to the angular frequency. As dβ1/dω = β2 we ﬁnd the expression

Dλ2 β = . (2.18) 2 − 2πc Repeating this operation results in

Dλ3 β = . (2.19) 3 2π2c2 A typical dispersion coeﬃcient for standard single mode ﬁbers is D = 17 ps/nm/km. 26 s2 41 s3 According to (2.18) and (2.19) we obtain β = 2.2 10− and β = 3.6 10− . 2 − · m 3 · m 2.2.3 Polarization Mode Dispersion

Within single mode fibers two modes are able to propagate, i.e., two waves which are orthogonally polarized. These modes have equal propagation properties in ideal waveguides; for this reason these two modes are said to be degenerate. Real fibers may differ from a perfect cylindrical geometry due to asymmetries during the fabrication process. Moreover, external effects like mechanical stress and temperature variation also have an impact on the propagation properties for both fundamental modes. This observation usually is referred to as birefringence [GK00]. The degenerate property is lost; thus, orthogonally polarized signal contributions have to be modeled separately. To do so, a vectorial notation is introduced

u (t, z) u(t, z) = 1 = e u(t, z). (2.20) " u2(t, z) # ·

In a ﬁrst order approximation there are two distinct states of polarization for which a maximum and minimum group delay can be observed [PW86]. These SOPs are called “slow axis” and “fast axis”. Without loss of generality these axes shall be aligned with the vector components of (2.20). Then the following matrix describes the diﬀerential group delay (DGD) ∆τ in the frequency domain [KL02]

jπf ∆τ e · 0 HDGD = jπf ∆τ . (2.21) " 0 e− · #

In a further step, the present propagation model in frequency domain (2.9) is extended to vectorial notation and the DGD transfer matrix is appended

α jβ(f)z U(f, z) = H U(f, 0) e− 2 e− . (2.22) DGD · · · 2.2. Fiber Channel 11

Phase parameters are introduced

∆τ ∆τ β = + , β = , (2.23) 1,1 2z 1,2 − 2z which describe group delays per transmission distance. The expressions are of opposite signs; the mean group delay still equals zero. The deﬁnitions in (2.23) yield

∆τ = z (β β ) . (2.24) · 1,1 − 1,2 Elimination of ∆τ in (2.22) yields two transmission equations in the frequency domain which are then diﬀerentiated with respect to the distance z, resulting in

∂U (f, z) α j j 1 = + jβ 2πf β (2πf)2 β (2πf)3 U (f, z) ∂z − 2 1,1 · − 2 2 · − 6 3 · · 1 (2.25) ∂U (f, z) α j j 2 = + jβ 2πf β (2πf)2 β (2πf)3 U (f, z). ∂z − 2 1,2 · − 2 2 · − 6 3 · · 2 Finally, inverse Fourier transform is applied and the two diﬀerential equations

2 3 ∂u1(t, z) α ∂u1(t, z) j ∂ u1(t, z) 1 ∂ u1(t, z) + u1(t, z) β1,1 β2 2 β3 3 = 0 ∂z 2 − ∂t − 2 ∂t − 6 ∂t (2.26) ∂u (t, z) α ∂u (t, z) j ∂2u (t, z) 1 ∂3u (t, z) 2 + u (t, z) β 2 β 2 β 2 = 0 ∂z 2 2 − 1,2 ∂t − 2 2 ∂t2 − 6 3 ∂t3 are obtained. The presented DGD model is valid for short fiber segments where the birefringence is constant. Long fibers are assumed to consist of a concatenation of short segments for which the above given description holds. The orientation of their slow and fast axes varies from segment to segment (c.f. sketch in Figure 2.2). This effect causes mode coupling, i.e., a certain signal portion of one axis at the end of a short fiber piece is coupled into the slow axis of the succeeding one. The remaining signal power fraction is coupled into

...

Figure 2.2: Concatenation of birefringent ﬁber pieces (mode coupling). 12 2. Digital Data Transmission over Silica Fibers the fast axis. Mathematically, mode coupling can be accounted for by unitary rotation matrices

θ θ cos 2 sin 2 R = θ − θ . (2.27) " sin 2 cos 2 #

Arbitrary SOP rotation including circular as well as elliptical states of polarization are realized by the matrix product [GK00]

cos θ sin θ cos φ j sin φ R = 2 2 2 − 2 . (2.28) sin θ cos θ · j sin φ cos φ " − 2 2 # " − 2 2 # comprising two free angle parameters. Given a sufficiently large number of segments this is an accurate model (usually referred to as wave-plate model) which also comprises higher order polarization mode dispersion (PMD). But also in this case one can still find a pair of orthogonal states of polarization in which the optical signal can propagate without distortion (to first order) [PW86]. These distinct SOPs are called principal states of polarization (PSP). The DGD of the long fiber then equals the group delay between the two PSP. A further important aspect of PMD is that mode coupling is not constant in time, but varies through, e.g., motion, bending, and temperature drift. Hence, DGD is a random process – it can be characterized by a Maxwellian distribution [FP91]. The mode coupling effect has another implication, i.e., the mean DGD is not proportional to the fiber length, but

E ∆τ √z. (2.29) { } ∼ Typical values are in the range of approximately 0.1 ps/√km to 2 ps/√km, where older installed ﬁbers exhibit the larger measures.

2.2.4 Polarization Dependent Loss

Polarization dependent loss (PDL) describes the observation that optical waveguides may exhibit different attenuation for different states of polarization. In the literature there is no agreed definition for PDL; there are different ways of introducing PDL into an optical waveguide. One possibility is to assume an additional attenuation contribution in one polarization – either as a multiplicative or additive value [GH97]. This operation causes some additional overall attenuation for the transmission link, which may not be desired. Therefore an alternative approach strives to model PDL as an “energy-preserving” effect. 2.2. Fiber Channel 13

In any case, usually PDL is deﬁned by the ratio of a maximum and minimum attenuation observed in distinct states of polarization

2 amax ρ = 2 . (2.30) amin

In contrast to [GH97] we omit the deﬁnition of PDL in Stokes space. Instead, a PDL element shall be described by a frequency independent 2x2 transfer function, which is equivalent with the PDL deﬁnition in Jones space

amax 0 Hpdl = . (2.31) " 0 amin #

For the model used in the sequel, the given transfer matrix shall only quantify the diﬀer- ence of the attenuation in both polarizations, but not incorporate the overall loss of the

ﬁber. Therefore, in order to keep Hpdl energy-preserving, we postulate

2 2 amax + amin = 2. (2.32)

With the help of (2.30) and (2.32), Hpdl can be expressed in terms of the polarization dependent loss parameter ρ

2ρ 1+ρ 0 Hpdl = . (2.33)  q 2  0 1+ρ  q  2.2.5 Kerr Eﬀect

The most important source of non-linear distortion in silica fibers is the Kerr effect. It describes the observation that the refractive index of the waveguide is influenced by the power of the optical signal [Agr95]. Hence, there is an interaction between the transmitted signal and the channel: The variation of the refractive index causes power dependent phase modulation of the signal. This fact is accounted for by extending (2.16) by a term which is proportional to u(t, z) 2. The propagation equation now becomes | |

∂u(t, z) α j ∂2u(t, z) 1 ∂3u(t, z) + u(t, z) β β = jγ u(t, z) 2u(t, z). (2.34) ∂z 2 − 2 2 ∂t2 − 6 3 ∂t3 − · | |

In the literature, it is often referred to as the “non-linear Schrödinger equation” (NLSE) [Agr95]. The parameter γ is the non-linearity coefficient and accounts for the intensity of the Kerr effect’s influence. It is determined by the fiber material and waveguide structure 14 2. Digital Data Transmission over Silica Fibers and is described by further parameters

2π n γ = · 2 , (2.35) λAeff where n2 and Aeff are the non-linear refractive index and the effective core area, re- 20 m2 spectively. Typical values for standard single mode fibers are n2 = 2.6 10− W and 2 1 · Aeff = 80 µm . As a result γ amounts to approximately 1.3 Wkm . No analytical solution is known for the NLSE in the general case1. For simulations numerical methods are utilized, e.g., the split-step Fourier method [Agr95] (c.f. section 2.6.1). When the modes of both polarizations shall be described separately, (2.34) has to be extended by a contribution which describes “cross polarization modulation”. The result is a set of two coupled differential equations [Agr95]

∂u α ∂u j ∂2u 1 ∂3u 2 1 + u β 1 β 1 β 1 = jγ u 2 + u 2 u ∂z 2 1 − 1,1 ∂t − 2 2 ∂t2 − 6 3 ∂t3 − | 1| 3| 2| 1 (2.36) ∂u α ∂u j ∂2u 1 ∂3u 2 2 + u β 2 β 2 β 2 = jγ u 2 + u 2 u . ∂z 2 2 − 1,2 ∂t − 2 2 ∂t2 − 6 3 ∂t3 − | 2| 3| 1| 2 2.3 Optical Ampliﬁcation and Concatenation of Fiber Spans

Due to the rather low attenuation of silica fibers optical transmission systems can reach large distances. For example, 100 km standard single mode fiber cause a power loss of approximately 20 dB. For the transmission over several hundreds of kilometers signal regeneration is required. Before the invention of optical amplifiers this task was carried out by optical to electrical conversion followed by an optical transmitter, which launches the regenerated signal into the subsequent fiber span. An important development came up in the 1990s: The availability of optical amplifiers [KBW96]. Along with this technique electrical conversion is avoided; the incoming optical signal is (coherently) amplified by stimulated emission. The most relevant implementation is the erbium doped fiber amplifier (EDFA), but also other techniques exist. Two basic principles shall be distinguished in the sequel: Lumped versus distributed optical amplification.

2.3.1 Lumped Optical Ampliﬁcation

The term “lumped” shall indicate that the optical ampliﬁcation takes place within a compact device. The main part of an EDFA is a piece (several 10 meters) of erbium doped

1Neglecting attenuation one can find solutions, which are referred to as “solitons” [Agr92]. For these special waveshapes linear dispersive effects and fiber non-linearity cancel out. 2.3. Optical Amplification and Concatenation of Fiber Spans 15

u(t) G uout(t) u(t) uout(t) ←→

√G nASE(t)

Figure 2.3: Model of an ideal optical ampliﬁer.

fiber. An alternative implementation is the semiconductor optical amplifier (SOA), which works similar to a semiconductor laser. Basic prerequisite for stimulated emission is the activation of excited energy levels within the active media of the amplifiers. In a SOA this “pumping energy” is provided electrically, while in EDFAs an optical signal excites erbium ions. An extensive physical model of optical amplifiers shall not be provided here. Instead the relevant gain and noise characteristic is described in the following. In accordance with Figure 2.3 the ideal optical amplifier model shall be expressed by

u (t) = √G u(t) + n (t). (2.37) out · ASE G denotes the power gain and shall be constant for all wavelengths for simplification reasons. Besides stimulated emission, which is the desired effect for optical amplification, spontaneous emission occurs. Excited ions occasionally return to their ground state spon- taneously and thereby emit photons which are of random phase and polarization. This effect introduces noise, which in turn gets amplified: Amplified spontaneous emission noise (ASE noise). Its power spectral density per polarization is given by

c Φ = h n (G 1). (2.38) NN · λ · sp − Here h and c denote Planck’s constant and the speed of light, respectively. The variable nsp is the spontaneous emission factor, which is greater than 1, but its value cannot be accessed directly. The noise characteristic is therefore preferably described by the noise

ﬁgure FN , which describes the ratio of the SNR at the ampliﬁer input and at its output.

For optical amplifiers FN is defined by electrical input and output SNR values after “ideal” detection. As a result we obtain an interrelation between the noise figure, the amplifier’s gain and nsp

SNRin G 1 FN = 2nsp − . (2.39) SNRout ≈ · G 16 2. Digital Data Transmission over Silica Fibers

Now the spontaneous emission factor in (2.38) can be eliminated and an expression for the OSNR (signal and noise in one polarization) is obtained

P 2P λ OSNR = o = o · , (2.40) 1 pol Φ B G F h c B NN · · N · · with Po denoting the optical signal power at the ampliﬁer’s output. Due to measurement technique reasons, the noise power is considered for both polarizations within the reference bandwidth Bref; hence, the OSNR deﬁnition becomes

P λ OSNR = o · . (2.41) G F h c B · N · · ref

Often Bref is expressed by the wavelength diﬀerence, which corresponds to the edge frequencies of the ﬁlter pass band, e.g. ∆λ = 0.1 nm, which is approximately Bref = 12.5 GHz at wavelengths around 1550 nm.

2.3.2 Multi-Span Fiber Link

In order to calculate the OSNR which is the result of a concatenation of multiple optically amplified fiber spans (c.f. Figure 2.4), we make some assumptions. The whole fiber link shall be built up by uniform fiber spans (same lengths, same fiber type) along with EDFAs in-between. The amplifiers’ gain shall equal the fiber span attenuation. Hence, the optical launch power Popt at the transmitter equals the signal power present at each EDFA output

Popt = Po. (2.42)

ASE noise contributions which are added by each optical amplifier are uncorrelated. Thus, the total ASE noise power becomes ns times the noise power of a single amplifier stage. The optical SNR at the end of the fiber link then reads

P λ OSNR = opt · . (2.43) G F h c n B · N · · · s · ref An equivalent logarithmic equation can be given

hc Bref Popt OSNRdB = 10 log + 10 log 10 log G − 10 λ 1 mW 10 1 mW − 10 (2.44) · 10 log F 10 log n . − 10 N − 10 s

Insertion of constants h and c along with setting λ = 1550 nm, Bref = 12.5 GHz yields 58 dBm for the ﬁrst summand in (2.44). In the sequel (2.44) is used for the computation of the OSNR at the receiver for optically ampliﬁed transmission paths. Some remarks on 2.4. Polarization-Division Multiplexing 17

u(t) GGG... uout(t)

Figure 2.4: Concatenation of ﬁber spans. the implementation of ASE noise generation in the simulation chain are given in section 2.6.2.

2.3.3 Distributed Ampliﬁcation

There is an alternative optical amplification technique, which is based on signal power regeneration within the fiber channel itself (associated with the term distributed amplification). A high-power signal (pump signal) is launched into the fiber – either at the transmitter side (“forward pumping”), the receiver side (“backward pumping”) or both. Stimulated Raman scattering is the effect, which describes the coherent power transfer from the pump signal to the data carrying signal, which shall be amplified. As with lumped optical amplification ASE noise is generated. Here it has its origin in spontaneous Raman scattering. The mathematical description of Raman amplification requires expressions for the local gain G(z) and the local ASE noise power spectral density

NASE(z). The result of the analysis is that less ASE noise is generated – compared to EDFA based systems. Equivalently an effective noise figure can be formulated, which yields values below that of lumped amplifiers (which are lower-bounded by 3 dB). While EDFAs are widely deployed in installed systems, Raman amplifiers are not (yet) widespread in commercial applications. But they play an important role for transmission experiments in the laboratory.

2.4 Polarization-Division Multiplexing

As long as there is no PMD or PDL the two orthogonally polarized modes within single mode fibers are affected by the channel in the same manner. Both orthogonally polarized modes can be utilized for the transmission of two independent source signals. This method is called Polarization-Division Multiplexing (PDM), which is an effective method to double the data rate while keeping the bandwidth of the electrical source signals constant. The PDM principle is depicted in Figure 2.5: Two independent streams of transmit symbols are both converted to optical signals u1(t, 0) and u2(t, 0). These signals have to 18 2. Digital Data Transmission over Silica Fibers

u1(t, 0) x1[k] TX1 RX 1 y˜1[k]

u2(t, 0) x2[k] TX2 RX 2 y˜2[k] u(t, z)

Figure 2.5: Polarization-division multiplexing. be orthogonally polarized and are fed to a “polarization beam combiner”. The resulting optical signal is then launched into the transmission ﬁber. At the receiver, corresponding operations in reversed order are required, i.e., a polarization beam splitter divides the received signal into two contributions which are then down-converted and processed in the electrical domain. As in section 2.2.3 the vectorial notation for optical signals is utilized; in (2.20) this notation was used to include the SOP. Now we describe the multiplexed signal by

u (t, z) u(t, z) = 1 . (2.45) " u2(t, z) #

The resulting signal u(t, z) is more or less unpolarized. Accordingly, the discrete time symbol vectors are deﬁned as

x [k] y˜ [k] x[k] = 1 and y˜[k] = 1 . (2.46) " x2[k] # " y˜2[k] #

In general, the fiber channel does not preserve the polarization properties of an optical signal. As a consequence, there is interference between both signals along the fiber channel. The axes of the polarization beam splitter is probably not aligned with the two optical carriers’ SOPs. Therefore, the two separated signals represent a superposition of the TX signals; PMD and PDL could have affected signal contributions in a different way. The requirements for the receiver are extraction of amplitude and phase information along with joint signal processing in order to determine equalized symbol vectors y˜[k]. 2.5. Wavelength-Division Multiplexing 19

λ1 λ1 DEMUX

λ2 MUX λ2 ......

λW λW

Figure 2.6: Wavelength-division multiplexing.

2.5 Wavelength-Division Multiplexing

The advantage of EDFA based optical amplification (besides overcoming the need for signal regeneration in the electrical domain) is the large operating wavelength range. Thereby the huge bandwidth of the fiber channel becomes available for high bitrate transmission over long distances. Several independent optical signals can be transmitted over a single fiber when the wavelength differences between the optical carriers are sufficiently large. Figure 2.6 sketches the basic principle of Wavelength-Division Multiplexing (WDM). At the transmitter side, several optical signals (wavelengths of their carriers: λ1 ... λW ) are combined with the help of an optical multiplexer. At the end of the transmission link the wavelengths are separated by a demultiplexer. From a system theoretic point of view, a frequency based description is more useful. According to Figure 2.7 we calculate the combined WDM signal

W j2π fwt u (t, z) = u (t, z) e · . (2.47) WDM w · w=1 X

Noteworthy, this description is still a baseband model: fw denote carrier frequencies relative to f = 0. One can observe interference between WDM channels, which is caused by linear and non-linear ﬁber eﬀects. Linear cross-talk can be kept small, when the frequency spacing is

j2π f1t e ·

u1(t, z)

j2π fW t e · uWDM(t, z) ...

uW (t, z)

Figure 2.7: Wavelength-division multiplexing – total field approach. 20 2. Digital Data Transmission over Silica Fibers chosen appropriately. But there may be a distinct influence through non-linear crosstalk caused by the Kerr effect.

The simulation of the propagation of uWDM(t, z) of (2.47) is referred to as total field approach. Alternatively, the separate field approach extends the non-linear Schrödinger equation by terms accounting for group delays between WDM channels along with non- linear interaction. After doing so, the actual propagation simulation is carried out for the WDM channels individually. Both approaches deliver equivalent results [Lei07]. In the sequel, only the total field approach is pursued.

2.6 Simulation Technique

For numerical evaluation, analog signals are expressed through oversampled discrete time signals. The simulation of the propagation of the optical signals then consists of the numerical solution of the non-linear Schrödinger equation. The most important technique is the split-step Fourier method, which is discussed in section 2.6.1. A further aspect related with the simulation of transmission over the non-linear ﬁber channel shall be discussed in a further section: ASE noise generation at the actual location of the optical ampliﬁers versus equivalent noise generation at the receiver input.

2.6.1 Split-Step Fourier Method

The basic idea of the split-step Fourier method is to divide the fiber span into short pieces of length ∆z. Then one assumes that the linear fiber effects and the non-linear impairment through the Kerr effect can be separated in this short region [Agr95]. Along with this separation approach, the non-linear signal distortion is accounted for through a power dependent phase rotation in the time domain

jγ u(t,z) 2∆z u′(t, z) = u(t, z) e− | | . (2.48) · This equation solves (2.34), when all dispersive eﬀects are neglected. After this operation, which takes place at distance z solely, the linear transfer function of the ﬁber piece of length ∆z is considered

α 2 2 4 3 3 ( +j2π β2f +j π β3f )∆z H∆z(f) = e− 2 3 . (2.49)

Both operations are carried out successively; hence, the Fourier transform is required for alternating between time domain and frequency domain computations. In a single expression, the output signal after passing one ﬁber segment reads

1 jγ u(t,z) 2∆z u(t, z + ∆z) = − u(t, z) e− | | H (f) . (2.50) F F · · ∆z n n o o 2.6. Simulation Technique 21

For the dual polarization description, these steps have to be applied to the coupled diﬀerential equations (2.36). The transfer function now becomes

α 2 2 4 3 3 ( jβ1,12πf+j2π β2f +j π β3f )∆z e− 2 − 3 0 H∆z(f) = α 2 2 4 3 3 . ( jβ1,22πf+j2π β2f +j π β3f )∆z " 0 e− 2 − 3 # (2.51) Finally, the computations for the signal propagation over a short ﬁber piece reads

2 2 2 jγ( u1(t,z) + u2(t,z) )∆z 1 u1(t, z) e− | | 3 | | u(t, z + ∆z) = − H∆z(f) · 2 2 2 . (2.52) jγ( u1(t,z) + u2(t,z) )∆z F ·F u (t, z) e− 3 | | | | ( (" 2 · #)) When PMD is present (β , β = 0) H (f) implements a DGD element according to 1,1 1,2 6 ∆z the waveplate model. In this case, a SOP rotation operation after each ﬁber piece of length ∆z shall be incorporated (again carried out with the help of a unitary rotation matrix R).

2.6.2 ASE Noise Generation

As pointed out in section 2.3, long transmission distances are bridged through the concatenation of fiber spans with EDFAs in-between. ASE noise is added by the optical amplifiers; thus, an accurate numerical simulation model works as depicted in Figure 2.8 a), where the position of the noise summation is set according to the actual position of the amplifier. Equal noise variances and uncorrelated contributions in both states of polarization shall be assumed

E n nH = ... = E n nH = σ2 E . (2.53) ASE,1 · ASE,1 ASE,ns · ASE,ns n · 2 From simulation complexity point of view, it is often advantageous to modify the simulation chain according to Figure 2.8 b). Now all the amplifiers deliver an ideal (noise-free) power gain. An equivalent noise signal is added at the end of the fiber chain. Given equal optical amplifier characteristics (gain and noise figure) its covariance reads

E n nH = n σ2 E . (2.54) ASE,equ · ASE,equ s · n · 2 A further prerequisite for this equation to hold is a linear fiber channel exhibiting an all- pass characteristic. With the help of this modification the propagation of a specific signal block over the fiber channel is simulated once, while a larger set of noisy representations can be generated at the receiver. Doing so, the number of computationally intensive simulation runs for modeling the propagation of signal blocks can be reduced while still maintaining reasonable numbers of received symbols. This method is also applicable for 22 2. Digital Data Transmission over Silica Fibers non-linear channel simulations. When the non-linear interaction between the propagating signal and ASE noise is small compared to the non-linear signal distortion of the signal by itself, both simulation methods can be assumed as approximately equal.

u(t) GGG uout(t)

nASE,1(t) nASE,2(t) nASE,ns (t)

a) Distributed ASE noise.

u(t) GGG uout(t)

nASE, equ(t)

b) Equivalent ASE noise generation at the receiver.

Figure 2.8: Two simulation methods for ASE noise generation in optically ampliﬁed links. 23

3 Orthogonal Frequency-Division Multiplexing for Optical Data Transmission

Chapter 2 revealed that optical transmission channels may exhibit considerable amounts of dispersion. Orthogonal Frequency-Division Multiplexing (OFDM) is an established transmission technique allowing for efficient equalization of signal distortion caused by the channel’s dispersive effects. OFDM was invented in the 1960s [Wei09]. It is based on block-wise processing at the transmitter and receiver – most importantly, the (inverse) discrete Fourier transform has to be mentioned in this context. By this means, the used frequency band is partitioned into a number of sub-bands. This is done in such a way that the narrow-band sub- channels have a scalar transfer characteristic, i.e., these sub-channels are supposed to be “flat-fading”. Despite the long time since the invention of OFDM it still took many years until it found its way into standardization, i.e., until powerful and affordable signal processing became available. However, nowadays OFDM is utilized in many wireline as well as wireless systems. For example, it was introduced in digital sub-scriber line (DSL)1 [Int99] and wireless local area networks (IEEE standard 802.11 [IEEa]). Also mobile internet access (WiMAX, IEEE standard 802.16 [IEEb]), digital video broadcast (DVB), and 4th generation wireless systems (downlink in long term evolution (LTE) [3GP]) are driven by OFDM. As an important advantage of OFDM, it allows for moderately complex transmis-

1As DSL operates on baseband signals, a real-valued OFDM variant has been introduced – often referred to as digital multi-tone (DMT) 24 3. Orthogonal Frequency-Division Multiplexing for Optical Data Transmission sion over dispersive channels, while the complexity scales favorably with larger signal bandwidths. Moreover, it gives the systems considerable flexibility, e.g., bitrates can be adjusted by choosing appropriate constellations on sub-carrier basis. Due to these advantages and the fact that the advances in microelectronics provide more and more powerful signal processing resources, OFDM was also proposed to be applied for optical transmission over single mode fibers [LA06], [SA06], [Arm09]. Since then, several research groups are working on various aspects of OFDM for optical transmission, e.g., theoretical investigations [SBT08], [JMST08], [YSM08a]. Moreover, the impact of linear [LA07], [Shi07] and non-linear channel effects is studied [DB09]. A further important aspect is the sensitivity on phase noise [YSM08b], [YST07], [JMTT07]. In addition there are publications dealing with real-time implementation aspects [YCMS09] and very high bitrate systems (1 TBit/s) [DBK10].

3.1 OFDM Principle

The principle of OFDM is depicted in Figure 3.1. The starting point is a series of transmit symbols X[q], which are taken from a M-ary alphabet . Their variance is AM

σ2 = E X[q] 2 . (3.1) x | | A block of Q symbols is collected after serial-to-parallel conversion (S/P) and interpreted as the coeﬃcients of a discrete Fourier spectrum. The corresponding time domain OFDM

x[k] x(cp)[k] s(t) X[q] S/PIDFT P/S CP DAC / O/E E/O

Y [q] P/S DFT S/P CP ADC r(t) y[k]

Figure 3.1: OFDM transmission model. 3.1. OFDM Principle 25

x[k] k

Qcp Q Figure 3.2: Cyclic extension of an OFDM symbol. frame is obtained by means of inverse discrete Fourier transform (IDFT)

Q 1 − q 1 j2πk x[k] = X[q]e Q , k = 0,...,Q 1. (3.2) √Q · − q=0 X In contrast to the common definition of the IDFT, (3.2) is scaled by √Q, which ensures that the signal energy is preserved. The next step is intended for avoiding inter-symbol interference in order to detect OFDM blocks independent from preceding and succeeding ones. Here, different ways of implementation are possible: Firstly, one can insert a guard time between OFDM symbols. An alternative is provided by the insertion of well defined samples (unique word OFDM [HHH10]). The most common method is the so-called “cyclic extension”, where a number of samples (Qcp) are copied from the end of the OFDM symbol and added before the block (cyclic prefix, CP; c.f. Figure 3.2). When Qcp is chosen sufficiently large, i.e., when the duration of the cyclic extension is larger than the delay spread of the channel, no inter-symbol interference occurs. The subsequent building block is the digital-to-analog conversion (DAC) of xcp[k] resulting in s(t). The electrical baseband signal is up-converted to the optical domain and launched into the fiber span. As explained in Chapter 2, the optical signal propagates over the waveguide; the analog baseband signal after down-conversion to the electrical domain is denoted by r(t). Further processing steps at the receiver correspond to the building blocks at the transmitter in reversed order: analog-to-digital conversion (ADC), then the samples of the cyclic prefix are discarded and the remaining Q samples are given to the discrete Fourier transform (DFT)

Q 1 − j2πq k Y [q] = Q y[k] e− Q , q = 0,...,Q 1. (3.3) · · − k=0 p X The cyclic preﬁx eﬀects that the linear convolution of the analog source signal with the channel impulse response becomes a circular convolution when the signal y[k] is observed. 26 3. Orthogonal Frequency-Division Multiplexing for Optical Data Transmission

Accordingly, we can write

y[k] = x[k] h[k]. (3.4) ⊗ By means of the convolution theorem of the discrete Fourier transform

DFT y[k] = DFT x[k] DFT h[k] (3.5) Q { } Q { }· Q { } we obtain the input-output relation on OFDM sub-carrier basis

Y [q] = X[q] H[q], q = 0 ...Q 1. (3.6) · − Thus, the orthogonality of the sub-carriers is preserved at the receiver. The transmission channel is eﬀectively used for the parallel transmission of Q symbols within Q sub-carriers.

3.2 Optical Transmitter Architecture

After TX signal processing and digital-to-analog conversion the electrical baseband signal s(t) has to be up-converted to the optical carrier frequency. Different transmitter layouts are applicable: At first, via modulating the TX laser’s current the amplitude of the optical signal can be influenced (“direct modulation”). Hence, this method allows for the upconversion of real-valued baseband signals. However, also the phase of the optical carrier is affected in an unwanted way. Better performance is provided by external modulation, where a continuous wave optical signal is generated by a laser subsequently modulated by a further device. Most relevantly, this functionality is given by Mach-Zehnder modulators (MZM) as sketched in Figure 3.3. The continuous wave laser signal is coupled into a waveguide structure, where it is split into two arms and electrical fields are applied with the help

v1(t)

u cw u(t) v2(t)

Figure 3.3: Mach-Zehnder modulator. 3.2. Optical Transmitter Architecture 27

of v1(t) and v2(t). In this medium the refractive index varies dependent on the applied voltage (Pockels eﬀect). One can observe a linear phase shift of the optical signal

∆Φ (t) v (t), ∆Φ (t) v (t). (3.7) 1 ∼ 1 2 ∼ 2

The signal in the equivalent complex baseband – assuming symmetric splitting/combining, and neglecting attenuation by the MZM waveguide material – can then be written as

j (∆Φ (t)+∆Φ (t)) ∆Φ1(t) ∆Φ2(t) u(t) = u e 2 1 2 cos − . (3.8) cw · · 2 In other words, this component allows for modulation of the amplitude and the phase of the optical carrier. The “push-pull mode” is used most often in actual implementations, i.e., complementary drive voltages are applied at the electrodes of the two arms. Hence,

v (t) = v (t) ∆Φ (t) = ∆Φ (t). (3.9) 2 − 1 → 2 − 1

For that mode of operation, (3.8) becomes

π v (t) u(t) = u cos (∆Φ (t)) = u cos 1 . (3.10) cw · 1 cw · 2 v s where vs describes the switching voltage, a characteristic parameter of the MZM device. Obviously, now the MZM is acting as an amplitude modulator. The baseband source signal is mapped to the optical signal in a non-linear fashion when v (t) s (t). However, through proper biasing along with scaling of the source signal’s 1 ∼ I amplitude the non-linear signal distortion can be kept low [TYSE07]. On the other hand, this mode of operation leads to low optical signal powers and therefore potential OSNR limitation has to be kept in mind. Nonetheless, in the sequel we assume

u(t) = p s(t). (3.11) ·

In order to achieve the up-conversion of complex valued baseband signals (I/Q modulation), usually a nested Mach-Zehnder modulator layout is applied [Gri05], [KSM+06] as depicted in Figure 3.4. The structure provides separate amplitude modulation of the inphase and quadrature component. After independent modulation of the optical carrier according to the inphase and quadrature signal amplitude, a 90 degree phase shift has to be implemented. 28 3. Orthogonal Frequency-Division Multiplexing for Optical Data Transmission

sI(t)

ucw u(t)

90◦

sQ(t)

Figure 3.4: I/Q modulation by nested Mach-Zehnder modulator structure (single polarization).

3.3 Optical Receiver Architecture

In general, at the receiver side full information about the OFDM signal is required, i.e., knowledge about the amplitude and phase. This task can be achieved by coherent detection. These receiver architectures have been an important concept in research and development before optical ampliﬁcation became available [Fra88]. As coherent receivers have higher sensitivity than direct detection, larger transmission distances could be realized. This advantage vanished as soon as EDFAs came up. But recently, there is a renewed interest in coherent receivers as they allow for high spectral eﬃciency by the application of large modulation alphabets, polarization division multiplexing, and electronic equalization [Ho05], [ILBK08], [KL09].

3.3.1 Single Polarization Coherent Detection

The basic principle of coherent optical detection is the superposition of the received signal with a local laser signal. The concept is the same as in RF down-conversion; thus the local laser is often referred to as local oscillator (LO). In order to obtain a baseband representation of the inphase and quadrature component of the received signal, four versions of signal-LO superpositions are required. This operation can be achieved by a six port device with the transfer matrix 1 1 1 1 1  −  . (3.12) 2 1 j    1 j   −    3.3. Optical Receiver Architecture 29

90◦

Figure 3.5: Implementations of 2 4-90◦ optical hybrids. ×

There are different ways of implementing these “2 4-90◦ optical hybrids”. Two of them are × depicted in Figure 3.5. One hybrid architecture utilizes splitters followed by two couplers. In one LO signal branch there is an additional 90 degrees phase shift. Alternatively one can use a single coupler with succeeding polarization beam splitters. It has to be mentioned that these hybrid structures are not equivalent with respect to polarization conditions. Whereas the first one (left hand side in Figure 3.5) requires the same state of polarization for the LO laser and the received signal, the second architecture assumes a circularly polarized LO signal. The RX wave must then be linearly polarized at 45◦. Hence, in both cases polarization control is required in order to provide aligned states of polarization of the superposed signals at the outputs of the hybrids. Mixing of the signals only works for same SOPs. The output optical signals are then fed to two balanced detectors finally generating electrical signals of the inphase and quadrature component of the received signal (c.f. Figure 3.6). Further architectures can be found in literature, e.g., utilization of alternative coupler structures. However, for the following mathematical description the focus shall be on hybrids which can be modeled according to the transfer matrix (3.12). The baseband signal of the local laser reads

j(2π ∆fLO t+φ(t)) u (t) =u ˆ e · · . (3.13) LO LO ·

It has the amplitude uˆLO; ∆fLO is the diﬀerence of the frequencies of the LO carrier and the received signal’s carrier. Finally, φ(t) corresponds to time dependent phase deviations, like for example laser phase noise. Applying the hybrid’s transfer characteristic (3.12) along with the magnitude square operation of the photo detectors yields

u(t) u (t) 2 u(t) u (t) 2 r (t) = R + LO R LO (3.14) I 2 2 − 2 − 2

30 3. Orthogonal Frequency-Division Multiplexing for Optical Data Transmission

u(t, z = L) rI(t)

2x4 90◦ optical hybrid uLO(t) rQ(t)

Figure 3.6: Coherent detection utilizing a 2 4-90◦ optical hybrid. ×

u(t) u (t) 2 u(t) u (t) 2 r (t) = R + j LO R j LO . (3.15) Q 2 2 − 2 − 2

Both signal components can be combined to a complex valued signal representation

r(t) = r (t) + j r (t) = R u∗ (t) u(t). (3.16) I · Q · LO ·

Apparently, the presented detector achieves a linear down-conversion of the optical signal; the term uLO∗ (t) corresponds to a residual carrier. Diﬀerent situations have to be distinguished: If the intermediate frequency ∆fLO is larger than the signal bandwidth, the receiver operates as a heterodyne detector. A further electrical mixer stage is required to down-convert the signal to the baseband. If there is no residual carrier the device is called a homodyne receiver. In the ideal case there is no phase noise and the down-converted signal becomes

r(t) = R uˆ u(t). (3.17) · LO ·

This model is assumed in the following; the impact of laser phase noise is investigated in Chapter 6. In practice, the implementation of a homodyne receiver requires complex optical phase- locked loops. This eﬀort is reduced with intradyne receivers, where there is a free running local laser with carrier frequency close to the carrier of the received signal. The residual carrier frequency shall be small compared to the signal bandwidth. It is estimated and compensated for in the electrical domain via digital signal processing. 3.4. System Simulation 31

3.3.2 Dual Polarization Coherent Detection

As mentioned above the states of polarization of the incoming optical signal as well as the LO laser have to be controlled. In the case of improper SOP alignment parts of the signal-LO beat products vanish. Figure 3.7 depicts a way of overcoming the need for polarization control of the received signal. A polarization beam splitter divides the signal into its two orthogonally polarized contributions, which are then down-converted by separate coherent receivers. We extend (3.16) to vectorial notation

R r(t) = u∗ (t) u(t). (3.18) √2 · LO ·

By this means (“polarization diversity detection”) the full signal information can be recon- structed through further signal processing, like for example maximum ratio combining. Furthermore, this receiver architecture can be applied for polarization division multiplexing, which has been introduced in section 2.4.

3.4 System Simulation

3.4.1 Oversampled OFDM Signal

All the system simulations are carried out on the basis of transmitter and receiver signal processing according to the description of this chapter. The signal propagation is modeled as shown in Chapter 2. In order to represent analog signals through discrete time numerical simulations, the signals are approximated through oversampling by factor ros.

u(t) coherent r1,I(t)

receiver r1,Q(t)

coherent r2,I(t)

receiver r2,Q(t)

Figure 3.7: Dual polarization coherent detection. 32 3. Orthogonal Frequency-Division Multiplexing for Optical Data Transmission

The resulting source signal then reads

Q 2 1 Q 1 − q − q+(r −1)Q (os) 1 j2πk j2πk os x [k] = X[q]e rosQ + X[q]e rosQ , k = 0, . . . , r Q 1. √Q   os · − q=0 q= Q X X2   (3.19) This equation assumes an ideal rectangular transfer characteristic of the interpolation filter in the frequency domain. It corresponds to a sin(x)/x behavior in the time domain. In practice, a set of several sub-carriers at the frequency band’s edges are not modulated. Consequently, there is a guard band between aliasing spectra. Hence, realistic low-pass filters are then sufficient for interpolation [Kam04].

3.4.2 Multiple-Input/Multiple-Output System

Along with OFDM for optical transmission, polarization division multiplexing is utilized in order to double the system’s spectral eﬃciency. The mathematical description is given by combining both signal components to two-dimensional vectors as described in Section 2.4. For both orthogonally polarized signals, OFDM signal processing is applied according to the considerations of this chapter. Assuming that the OFDM parameters are chosen appropriately, the channel is sub-divided into Q orthogonal frequency independent sub- carriers, which then are described in vector–matrix–notation through

Y [q] H [q] H [q] X [q] N [q] 1 = 11 12 1 + 1 , (3.20) " Y2[q] # " H21[q] H22[q] # · " X2[q] # " N2[q] # or short

Y[q] = H[q] X[q] + N[q], q = 0,...,Q 1. (3.21) · − These equations correspond to a 2 2-MIMO-OFDM system [Win04], assuming a linear × transmission channel. The matrix H[q] accounts for the channel characteristic including linear crosstalk between orthogonally polarized signal contributions within the respective sub-carrier.

3.5 Direct Detection Optical OFDM

An alternative concept for OFDM transmission along with optical signals has been proposed [LA06], mainly aiming at limiting complexity with respect to optical components. The setup operates with real valued electrical TX signals. Thus, a standard Mach-Zehnder modulator is applicable. Furthermore – through proper biasing and optical single side- band ﬁltering – direct detection can be utilized at the receiver. 3.5. Direct Detection Optical OFDM 33

r(t) ADC ... VSB

uVSB(t)

s(t) ... DAC

Figure 3.8: Direct detection based optical OFDM system.

The basic building blocks of the proposed setup are depicted in Figure 3.8. A certain number of sub-carriers are modulated using an arbitrary QAM-constellation. The modulated sub-carriers occupy the upper half of the upper side-band. Lower sub-carriers are not modulated; the spectral part of negative frequencies is fed by corresponding complex- conjugates. Hence, after inverse discrete Fourier transform, cyclic extension, and digital- to-analog conversion, a real valued signal s(t) is obtained. Alternatively, the signal can be generated by up-converting the original complex valued base-band signal in the electrical domain through RF mixing. The resulting signal is modulated onto an optical carrier by a Mach-Zehnder modulator; then the lower side-band of the optical signal is suppressed (c.f. Figure 3.9 (a), vestigial side-band (VSB) ﬁltering). The modulator’s bias and the optical ﬁlter are adjusted in such a way, that the power of the optical carrier is about 50% of the total signal power. The VSB signal then propagates over the optical link and is received through direct detection. Even after squaring there

→ 0 40 → 20 −20 [dB] | 0 [dB] −40 ) | f )

( −60 −20 f ( ssb

R −80 | U −40 | 10 10 −60 −100 −80 −120 20 log 20 log −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 f [GHz] f [GHz] → → (a) TX signal after VSB ﬁltering. (b) RX signal after direct detection.

Figure 3.9: Signal spectra in direct detection optical OFDM. 34 3. Orthogonal Frequency-Division Multiplexing for Optical Data Transmission are no beat-products within the used spectrum. Compared to the architecture based on coherent detection, laser phase noise requirements are relaxed. The received signal r(t) is sampled; the succeeding signal processing is like with other OFDM systems. The proposed system has the advantage of low complexity of the optical RX frontend along with relaxed phase noise requirements. However, the system offers just small spectral efficiency as a frequency gap between the optical carrier and the OFDM sub- carriers must be introduced. Furthermore, the implementation of polarization diversity receivers needs extra effort [MH07c]. Also polarization multiplexing becomes challenging. There are several publications on alleviating these disadvantages [PWA+09], [SRB+08], [APL+10]. However, throughout this work, solely OFDM systems with coherent detection shall be considered. 35

4 Optical OFDM at the Presence of Linear Fiber Eﬀects

The optical OFDM transmission system, which has been introduced in the preceding chapters, allows for efficient equalization of linear fiber effects in the digital domain. Therefore, in the following we consider transmission channels where no optical dispersion compensation shall be applied. In such a scenario chromatic dispersion is the dominating effect – the OFDM system parameters have to be chosen according to the expected chromatic dispersion induced pulse broadening. Further sections investigate the influence of PMD and PDL.

4.1 Chromatic Dispersion

4.1.1 Pulse Broadening

Signal contributions of different frequencies propagate along the fiber at different group velocities. Correspondingly, in the time domain chromatic dispersion causes pulse broadening. An estimate of this measure shall serve as an approximation for the delay spread of the transmission channel, which in turn is essential for a proper choice of OFDM system parameters. The starting point of the following considerations is the linear transfer characteristic of the waveguide. Based on (2.9) we additionally assume that the fiber is lossless (α = 0) and β2 is the only non-zero coefficient of the phase polynomial. Then the transfer function 36 4. Optical OFDM at the Presence of Linear Fiber Effects becomes U(f, z) H (f, z) = = exp j2β π2z f 2 . (4.1) β2 U(f, 0) − 2 · Application of the Fourier transform provides the corresponding impulse response

2 1 1 t h (t, z) = − H (f, z) = exp j . (4.2) β2 F { β2 } j2πβ z · 2β z r 2 2 Obviously, the impulse response has no finite length. In other words, when the system described by Hβ2 (f, z) is excited by a Dirac impulse – which has no limited bandwidth, the system’s response has an infinite duration. However, still a finite pulse broadening can be observed for band-limited input signals. The actual value depends on the input signal itself. General results cannot be derived, but it is possible to give an estimate based on the method, which has also been used for the design of dispersion compensating filters in single carrier optical transmission systems [Sav08]. The impulse response exhibits a time dependent phase rotation. Its angular frequency results from differentiation of the phase term in the complex exponential function in (4.2)

1 ωh = 2πfh = t. (4.3) β2z

Assuming that the input signal has the two-sided bandwidth B, the impulse response hβ2 (t, z) can be restricted to a time interval where

B B f (4.4) − 2 ≤ h ≤ 2 holds. Replacing fh by the corresponding expression from (4.3) yields

π β z B t π β z B. (4.5) − | 2| · ≤ ≤ | 2| · By this means, ﬁnally the truncated length of the impulse response is obtained

λ2 ∆t = 2π β z B = B D z. (4.6) | 2| · · c · | |

4.1.2 OFDM Design Rule

Inter-symbol interference and inter-carrier interference are avoided if each OFDM symbol is extended by a cyclic preﬁx of appropriate length. This length Tcp has to be chosen 4.1. Chromatic Dispersion 37 greater than the maximum delay spread of the channel’s impulse response. An approximate design criterion is therefore based on (4.6)

λ2 T > B D z. (4.7) cp · c · | |

In the following, Tcp shall be eliminated; therefore we deﬁne the relative cyclic extension duration T Q r = cp = cp , (4.8) cp T Q i.e., the ratio of the cyclic preﬁx length and the duration of the original OFDM symbol (before cyclic extension). The duration of the cyclic extension then reads

Q T = r T = r (4.9) cp cp · cp · B and is inserted into (4.7). The ﬁnal result is the OFDM design criterion for the minimum number of sub-carriers Q for a given signal bandwidth B, relative cyclic preﬁx duration and accumulated chromatic dispersion D z: | | B2 λ2 Q > D z (4.10) rcp · | | · c Noteworthy, the number of sub-carriers has to be scaled linearly with the accumulated chromatic dispersion and quadratically with the signal bandwidth [MH07a]. The latter implies, that the bandwidth Bsc of a single sub-carrier (Bsc = B/Q) has to be reduced when the total signal bandwidth is increased

1 B . (4.11) sc ∼ B 4.1.3 Simulations

In the following, these findings shall be verified through simulations. For this purpose, two system scenarios shall be defined, which operate with different signal bandwidths. In optical communications the frequency grids of WDM systems are well defined; the ITU-T recommendation G.694.1 [Int02] specifies values of 12.5 GHz, 25 GHz, 50 GHz, and 100 GHz. For that reason the simulated systems shall fit within standard grids; the signal bandwidths are chosen as 12.2 GHz and 48.7 GHz providing some guard bands between neighboring WDM channels. These values are kept constant which results in different gross bitrates when varying free OFDM system parameters. For the fiber channel we assume a target distance of 500 km; no optical dispersion compensation is applied. Hence, the chromatic dispersion accumulates to 8500 ps/nm. 38 4. Optical OFDM at the Presence of Linear Fiber Effects

System 1 B 12.2 GHz D z 8500 ps | | nm ∆t 0.83 ns

rcp 1/10 1/8 1/5 Q 102 82 52 bitrate (4QAM) 22.18 GBit/s 21.69 GBit/s 20.33 GBit/s bitrate (16QAM) 44.36 GBit/s 43.78 GBit/s 40.67 GBit/s

Bsc 120 MHz 149 MHz 235 MHz

Table 4.1: System parameters for operation in 12.5 GHz WDM grid.

System 2 B 48.7 GHz D z 8500 ps | | nm ∆t 3.3 ns

rcp 1/10 1/8 1/5 Q 1616 1292 808 bitrate (4QAM) 88.55 GBit/s 86.58 GBit/s 81.17 GBit/s bitrate (16QAM) 177.1 GBit/s 173.2 GBit/s 162.3 GBit/s

Bsc 30.1 MHz 37.7 MHz 60.3 MHz

Table 4.2: System parameters for operation in 50 GHz WDM grid.

Tables 4.1 and 4.2 summarize the expected pulse broadening according to (4.6). Fur- thermore the minimum numbers of sub-carriers are listed, where for both systems three values for the relative cyclic prefix duration are set. Further parameters resulting from this choice are the gross bitrates assuming 4QAM and 16QAM modulation along with the respective sub-carrier bandwidths. For the following simulations non-linear channel effects are neglected; the OSNR at the receiver is treated as a free parameter. The simulated transmission distance is varied from 2 to 12 fiber spans of 80 km length. For a fair comparison of different digital transmission techniques, the OSNR measure shall be converted to Eb/N0, which is the ratio of the received energy per bit over the one-sided noise power density [Pro01], [Hub08]. It is obtained by scaling the OSNR [EFWK09]

2Bref Eb/N0 = OSNR , (4.12) · Rb with Rb abbreviating the bitrate.

Figure 4.1 depicts the resulting bit error ratios (BER) versus Eb/N0 for various trans- 4.1. Chromatic Dispersion 39 mission distances for system 1 according to Table 4.1. BER values were determined for 4QAM (solid lines) and 16QAM (dashed lines) modulation on the sub-carriers. The diagrams correspond to three diﬀerent choices for the cyclic preﬁx length (1/10, 1/8, and 1/5). Optimum performance can be observed for transmission distances up to 480 km (which equals 6 spans, 80 km each). There is a shift with respect to the AWGN reference curves:

This Eb/N0-loss is caused by the insertion of the cyclic extension. Additional signal energy is spent for preserving the orthogonality of the sub-carriers, but these samples do not carry information. The loss amounts to [Kam04]

10 log10(1 + rcp) dB. (4.13)

Hence, for rcp = 1/10,..., 1/5 the efficiency degradation becomes 0.41 to 0.79 dB. More- over the results show a slight degradation for 640 km, especially for the case of 16QAM modulation. Beyond 8 fiber spans, there is a distinct BER increase. System 2, which is designed for the 50 GHz WDM grid exhibits similar results (Fig- ure 4.2). There is a larger degradation when the target distance (500 km) is exceeded. The best performance is obtained, when the relative cyclic extension overhead is small. How- ever, in that case the (absolute) OFDM symbol duration has to be set quite long in order to meet the pulse broadening requirement. This fact implies some disadvantages: The number of sub-carriers must be large (complexity). Moreover the sub-carrier bandwidth becomes small, which makes the system sensitive to phase noise (c.f. Chapter 6). Further simulations have been carried out after scaling the target transmission distance to 1000 km. According to (4.10) the number of sub-carriers has to be doubled in order to preserve inter-symbol-interference-free transmission over the doubled fiber length. Re- spective results in terms of BER performance for system 1 are summarized in Figure 4.3. The simulations reveal that the design criterion (4.10) is applicable. The performance degradation turns out to be smaller when the target distance is exceeded – compared to the 500 km scenario. For the investigated setups without optical dispersion compensation, D z is under- | | stood as the accumulated dispersion at the receiver; when optical dispersion compensation is applied, the expression has to be interpreted as the maximum residual chromatic dispersion at the receiver. The design rule given in (4.10) is still applicable for small amounts of accumulated chromatic dispersion. Then the effective pulse broadening is small, which allows for a low number of OFDM sub-carriers. However, in this situation chromatic dispersion is possibly not the dominating effect. The choice of system parameters has to be verified in conjunction with other effects like PMD. 40 4. Optical OFDM at the Presence of Linear Fiber Effects

0 10 rcp = 1/10

−1 10

−2 10 → −3 10 BER −4 10

−5 10

−6 10

0 10 rcp = 1/8

−1 10

−2 10 → −3 10 BER −4 10

−5 10

−6 10

0 10 rcp = 1/5

−1 10

−2 10 → −3 10 960 km BER −4 10 800 km 640 km 480 km −5 10 320 km 160 km −6 AWGN ref. 10 0 3 6 9 12 15 10 log (E /N )[dB] 10 b 0 → Figure 4.1: BER performance for various transmission distances; target distance: 500 km, 12.2 GHz total bandwidth; 4QAM (solid lines) and 16QAM (dashed lines) modulation. 4.1. Chromatic Dispersion 41

0 10 rcp = 1/10

−1 10

−2 10 → −3 10 BER −4 10

−5 10

−6 10

0 10 rcp = 1/8

−1 10

−2 10 → −3 10 BER −4 10

−5 10

−6 10

0 10 rcp = 1/5

−1 10

−2 10 → −3 10 960 km BER −4 10 800 km 640 km 480 km −5 10 320 km 160 km −6 AWGN ref. 10 0 3 6 9 12 15 10 log (E /N )[dB] 10 b 0 → Figure 4.2: BER performance for various transmission distances; target distance: 500 km, 48.7 GHz total bandwidth; 4QAM (solid lines) and 16QAM (dashed lines) modulation. 42 4. Optical OFDM at the Presence of Linear Fiber Eﬀects

0 10 rcp = 1/10

−1 10

−2 10 → −3 10 BER −4 10

−5 10

−6 10

0 10 rcp = 1/8

−1 10

−2 10 → −3 10 BER −4 10

−5 10

−6 10

0 10 rcp = 1/5

−1 10

−2 10 → −3 10 BER −4 10 1600 km 1440 km 1280 km −5 10 1120 km 960 km −6 AWGN ref. 10 0 3 6 9 12 15 10 log (E /N )[dB] 10 b 0 → Figure 4.3: BER performance for various transmission distances; target distance: 1000 km, 12.2 GHz total bandwidth; 4QAM (solid lines) and 16QAM (dashed lines) modulation. 4.2. Polarization Mode Dispersion 43

4.2 Polarization Mode Dispersion

This section is dedicated to an analysis of the system performance for the case of PMD channels. For this purpose an exemplary scenario is set up, which consists of six fiber spans (80 km length). Each span is taken from an ensemble of PMD fibers with a mean DGD value of 20 ps. The simulation model (wave-plate model) considers all order PMD. Hence, the fiber channel shows a PMD value of approximately 2.2 ps/√km. Modern fibers may achieve values 0.1 ps/√km. However, for simulations the rather high measure was ≤ chosen for highlighting the principal behavior of the optical OFDM system in a PMD scenario. Two receiver architectures are distinguished in the sequel: Single polarization and polarization diversity receivers (c.f. sections 3.3.1 and 3.3.2).

4.2.1 Single Polarization Coherent Detection

PMD causes depolarization [Kik01], i.e., whereas the optical signal at the transmitter is perfectly polarized, it comprises signal contributions in orthogonal states of polarization after propagation over a PMD affected fiber. As the coherent receiver solely down-converts those parts of the signal which are aligned with the SOP of the local laser, parts of the incoming signal are lost. From communications engineering point of view this effect is referred to as an origin for “fading”. In order to quantify the signal power loss, the magnitudes of the sub-channel coefficients H[q] are analyzed. | | Figure 4.4 depicts the magnitudes of the sub-channel coefficients in logarithmic scale.

5 realization 1 realization 2 0 → −5 [dB] | ] q [ −10 H | 10 −15 20 log −20

−25 1 100 300 500 700 900 1100 1292 sub-carrier q → Figure 4.4: Magnitudes of sub-channel coeﬃcients in logarithmic scale; total bandwidth: 48.7 GHz. 44 4. Optical OFDM at the Presence of Linear Fiber Eﬀects

0 10 realization 1 realization 2 −1 10 AWGN ref.

−2 10 → −3 10 BER −4 10

−5 10

−6 10 0 3 6 9 12 15 10 log (E /N )[dB] 10 b 0 → Figure 4.5: BER performance for two exemplary PMD channel realizations.

Two arbitrarily chosen ﬁber realizations have been investigated; the OFDM signal bandwidth is 48.7 GHz. The OFDM system parameters are the same as “system 2” in Table 4.1. The simulated coherent receiver implements polarization control in order to maximize the power of the detected signal. The magnitudes are mostly within the interval of 0 to -5 dB. Realization 2 shows several severely faded sub-carriers (-20 dB). It should be mentioned that 0 dB corresponds to the PMD free channel, i.e., the overall channel does not exhibit attenuation (EDFA gains are matched to the ﬁber spans’ attenuation). As a consequence, the SNR of several sub-carriers is reduced, which in turn degrades the performance in terms of BER (as illustrated in Figure 4.5).

4.2.2 Dual Polarization Coherent Detection

The channel transfer characteristic of chromatic dispersion combined with the PMD wave- plate model is described by

jπf∆τi jβ(f)z e 0 HCD,PMD(f) = e− Ri. (4.14) · 0 e jπf∆τi · i " − # Y Thus, the channel transfer matrix for sub-carrier q is obtained by sampling at the appropriate frequencies

H[q] = H (f = q B B/2), q = 0,...,Q 1. (4.15) CD,PMD · sc − − 4.2. Polarization Mode Dispersion 45

5 polarization 1 polarization 2 0

→ −5 [dB]

| −10 ] q [ H | −15 10

−20 20 log −25

−30 1 100 300 500 700 900 1100 1292 sub-carrier q → Figure 4.6: Magnitudes of sub-channel coeﬃcients in both receive branches.

H[q] is therefore obtained by a product of unitary matrices, ﬁnally yielding a unitary matrix.

Maximum Ratio Combining

The transmission in one polarization along with dual polarization detection is written by

X[q] H [q] Y[q] = H[q] + N[q] = 11 X[q] + N[q]. (4.16) · " 0 # " H21[q] # ·

Figure 4.6 shows magnitudes of channel coeﬃcients for both receive branches versus the sub-carrier index in logarithmic scale. For maximizing the resulting SNR the received symbols are weighted with the conjugates of the channel coeﬃcients [Bre59]

Y˜ [q] = H∗ [q] Y1[q] + H∗ [q] Y2[q] = 11 · 21 · (4.17) 2 2 = H + H X[q] + H∗ [q]N [q] + H∗ [q]N [q]. | 11| | 21| 11 1 21 2 The unitary sub-carrier matrix property makes the coeﬃcient before X[q] unity; moreover, the variance of the resulting noise contribution is not changed. Therefore, after maximum ratio combining AWGN channel performance is reached. 46 4. Optical OFDM at the Presence of Linear Fiber Eﬀects

Polarization Division Multiplexing

In the polarization division multiplexing scenario there is a unitary channel matrix per OFDM sub-carrier. Hence, the PMD effect can be compensated without degradation. In this special case, linear equalization with the zero-forcing condition equals maximum ratio combining in both states of polarization. As a prerequisite, knowledge of accurate channel state information at the receiver is required. Moreover, a sufficient update rate is necessary as more or less rapid SOP fluctuations may occur.

4.3 Polarization Dependent Loss

The investigations so far revealed that chromatic dispersion can be ideally compensated. Moreover, in the case of polarization diversity detection, this finding also holds for PMD. In a next step the influence of polarization dependent loss shall be examined. As PDL is a further linear channel effect, and as CD and PMD do not deteriorate the transmission performance, the following PDL investigations are based on a simplified transmission model.

4.3.1 Simpliﬁed Transmission Model

The reduced model considers solely polarization dependent loss and additive white Gaus- sian noise. PDL is modeled as a frequency independent transfer matrix as introduced in (2.33). The transmission of one complex valued symbol-pair per polarization is written as

y = H R x + n, (4.18) pdl · · where x and y denote 2x1 transmit and received symbol vectors. The matrix R accounts for random SOP rotation, i.e., the orthogonally polarized transmit signal contributions shall not be aligned with the main axes of the PDL element. As the model exhibits no frequency dependency, it is applicable to single carrier transmission as well as to the description of sub-carriers in OFDM. In the following, no discrete time index k or sub- carrier index q are added. The vector n denotes AWGN. The contributions in both states of polarization shall be mutually independent and of equal power. Hence, the covariance matrix is

E nnH = σ2 E . (4.19) n · 2 At the receiver, the transmitted symbols have to be estimated based on the observation of the distorted symbol vectors y. There are various detection strategies which exhibit diﬀerent performance but also diﬀerent computational complexity. 4.3. Polarization Dependent Loss 47

Maximum Likelihood Detection

On the basis of the mentioned assumptions the optimum receiver is looking for the transmit vector x, which leads to the maximum probability for the observation of y [Kam04]

xˆ = arg max p(y x). (4.20) x 2 | ∈A The AWGN assumption allows for an equivalent formulation in terms of minimizing squared Euclidean distances

xˆ = arg min y Hx 2. (4.21) x 2 || − || ∈A

Linear Equalization, Zero-Forcing

As the search space grows exponentially with the number of bits per symbol, less complex methods are of interest. A straight-forward equalization scheme consists of a linear operation which inverts the channel transfer characteristic (zero-forcing, ZF)

1 H = (H R)− , (4.22) ZF pdl ·

x˜ = H y = x + H n. (4.23) ZF · ZF · This method ideally reconstructs the transmitted symbols in the absence of noise. How- ever, it influences the effective noise variance, finally deteriorating – in general – the transmission performance compared to maximum likelihood detection. Simulations have been carried out for channels with PDL values of 3 and 5 dB (single PDL elements); a set of random channel realizations was created by choosing random SOP rotation matrices R. The resulting ergodic bit error ratios for symbols taken from a 4QAM constellation are shown in Figure 4.7. For both PDL scenarios linear ZF equalization and maximum likelihood (ML) detection was applied. As a reference, the BER versus Eb/N0 is given for the PDL-free AWGN channel. In contrast to CD and PMD, now PDL leads to a performance degradation. Linear equalization based on the ZF criterion results in average SNR penalties of 0.75 and 1.8 dB 3 for the 3 and 5 dB PDL case, respectively (at BER = 10− ). The curves also indicate that ML detection can reduce these gaps to approximately 0.45 and 1.2 dB loss with respect to the pure AWGN channel. For the purpose of a statistical analysis of the SNR penalties, we evaluate the complementary cumulative distribution (ccdf) function

ccdf (η ) = Pr η > η , (4.24) ηS S,th { S S,th} 48 4. Optical OFDM at the Presence of Linear Fiber Eﬀects

0 10 3 dB; ZF 3 dB; ML 5 dB; ZF 5 dB; ML −2 10 AWGN ref. →

−4 10 BER

−6 10

2 4 6 8 10 10 log (E /N )[dB] 10 b 0 → Figure 4.7: Ergodic bit error ratio for two PDL values; application of linear ZF equalization and ML detection.

0 10 3 dB PDL 5 dB PDL

−1 10 → )

S,th −2 η 10 ( S η ccdf −3 10

−4 10 0 0.5 1 1.5 2 2.5 10 log η [dB] 10 S,th → 3 Figure 4.8: Ccdf of SNR penalties at BER = 10− for two PDL values; application of linear ZF equalization.

i.e., the probability that the SNR penalty ηS exceeds a given threshold value ηS,th. Figure 4.8 summarizes the ccdfs for the SNR penalties for the case of linear ZF equalization. The main ﬁnding is that there are minimum and maximum penalties, which amount to 0.51 and 1.03 dB for the 3 dB PDL element. Correspondingly the SNR penal- 4.3. Polarization Dependent Loss 49 ties in the 5 dB PDL scenario are in the range of 1.37 to 2.20 dB. These extreme values are associated with the best-case and worst-case coupling into the PDL element.

4.3.2 Concatenation of PDL Elements

In a next step ns PDL elements shall be concatenated. They are assumed to have equal PDL, but random principal states of polarization – once again accounted for by random

SOP rotation matrices Ri. The overall transfer function now reads

ns H = H R . (4.25) pdl · i i=1 Y This matrix is diagonalized with the help of a singular value decomposition in order to extract the eﬀective PDL value of the concatenation of PDL elements

atot,1 0 H H = Ri Rj . (4.26) · " 0 atot,2 # ·

In numerical simulations ns = 10 randomly chosen PDL elements have been concatenated; the resulting transfer matrices have been diagonalized according to (4.26) and the probability density of the resulting PDL values, expressed in decibels

atot,1 ρdB = 20 log10 (4.27) atot,2 have been evaluated. Figure 4.9 depicts the resulting curves for two channel scenarios.

The overall mean PDL values µPDL were obtained as 3 dB and 5 dB, respectively. The total PDL values follow a Maxwellian distribution [MS02]

32 4ρ2 pdf(ρ ) = ρ2 exp dB . (4.28) dB π2µ3 · dB · −πµ2 PDL PDL For comparison the dashed lines in Figure 4.9 show the analytical probability density functions. The deviation between numerical results and analytical expressions is due to the restriction of the number of PDL elements (ns = 10). Like with the PMD model through birefringent wave-plates, the following relationship between the mean value of the overall PDL and the polarization dependent loss of a single element holds [MS02]

3π ρ = µ . (4.29) dB,span PDL · 8n r s 50 4. Optical OFDM at the Presence of Linear Fiber Eﬀects

0 10 3 dB mean PDL 5 dB mean PDL Maxwellian −1 10 →

) −2

dB 10 ρ ( pdf

−3 10

−4 10 0 5 10 15 ρ dB →

Figure 4.9: Pdf of PDL values after concatenating ns = 10 PDL elements; two scenarios with average PDL equaling 3 and 5 dB.

4.3.3 PDL in Multi-Span Link

The transmission model given in (4.18) is extended to a scenario of polarization multiplexed transmission over a multi-span link. There are ns fiber spans; the signal attenuation is compensated for by optical amplification. These amplifiers insert noise, which shall be described as additive white Gaussian noise. As with (4.18) the receiver is assumed to use polarization diverse coherent detection. Figure 4.10 sketches the multi-span link. Each (amplified) span is characterized by a transfer matrix, which solely accounts for PDL with random orientation of the main axes. The transfer characteristic of one span shall preserve the overall optical signal power. Hence, the modeling of the optical amplifier reduces to adding noise vectors at the respective positions within the transmission chain. The variance of ASE noise is assumed 2 to be equal for each amplifier and both polarizations and is denoted by σn. This multi- span PDL model neglects other fiber effects and assumes that there is a single device or component per span, which introduces PDL1. The used notation and the following derivation is based on the model published in [Sht08]. The input output relation is given by

y = T x + n , (4.30) 0 · tot 1in contrast to the approach in [Sht08], where each ﬁber span is modeled by a concatenation of many PDL elements; this assumption gives rise to altered attenuation coeﬃcients per PDL element in order to meet the energy preserving property of the whole span. 4.3. Polarization Dependent Loss 51

n1 n2 nns

x H0 H1 Hns 1 y −

Figure 4.10: PDL in a multi-span transmission link.

where the 2x2 matrix T0 describes the overall transfer function of the optical path. It is the result of the concatenation of the transfer matrices of the individual spans. Noise signal contributions of different amplifiers experience different channels; therefore we define

ns 1 − Tj = Hi, (4.31) i=j Y which corresponds to the ﬁber channel beginning after j spans and ending at the receiver. With the help of this term the eﬀective total noise at the receiver can be given

ns 1 − ntot = Tini + nns . (4.32) i=1 X The PDL transfer functions generate correlation between noise contributions in both states of polarization, which can be analyzed by the covariance matrix

ns 1 − E n nH = σ2 T TH + E . (4.33) tot · tot n · i i 2 i=1 ! X It is advantageous to deﬁne

1 Ψ n nH := 2 E tot tot , (4.34) nsσn · · which becomes an identity matrix for the PDL free case. In a next step the spatial 1/2 correlation of noise samples is removed by multiplying the received symbols by Ψ− (noise whitening)

1/2 1/2 1/2 ˜y = Ψ− y = Ψ− T0x + Ψ− ntot. (4.35)

This operation eﬀects that the resulting noise samples have no correlation between orthogonal states of polarization and have the same variance for both contributions. 52 4. Optical OFDM at the Presence of Linear Fiber Eﬀects

−1 10 3 dB PDL, ZF 3 dB PDL, ML 5 dB PDL, ZF −2 10 5 dB PDL, ML AWGN ref. → −3 10 BER

−4 10

−5 10 3 4 5 6 7 8 9 10 log (E /N )[dB] 10 b 0 → Figure 4.11: Ergodic bit error ratios in a 10 spans transmission link; application of linear ZF equalization and ML detection.

0 10 3 dB, ZF 3 dB, ML 5 dB, ZF 5 dB, ML −1 10 → ) S,th η (

S −2

η 10 ccdf

−3 10

0.5 1 1.5 2 10 log η [dB] 10 S,th → 3 Figure 4.12: Ccdf of SNR penalties at BER = 10− in a 10 spans transmission link; application of linear ZF equalization and ML detection.

Based on this transmission model, the above presented performance analysis has been repeated. The bit error ratios for two PDL scenarios have been evaluated. Here, the diﬀerence is that 10 PDL elements have been concatenated, which generate a set of channels with an average PDL value of 3 dB and 5 dB, respectively. In terms of detection 4.3. Polarization Dependent Loss 53 strategies, linear ZF equalization and maximum likelihood detection have been utilized. Figure 4.11 summarizes the ergodic BER results showing curves similar to the single PDL element channel. The average penalties are slightly smaller. The statistical analysis of 3 SNR penalties for BER = 10− yields results in terms of ccdfs as depicted in Figure 4.12. Compared with the single PDL element case there is not a very steep descent of the curves. This means that there is a certain probability for relatively large penalties which occur when the PDL axes are aligned.

4.3.4 Channel Capacity

The system model described by (4.35) can be used to modify the capacity of the pure 2x2 AWGN channel [Tel99] to

2 σx H 1 C = log det E + T T Ψ− . (4.36) 2 2 n σ2 · 0 0 · s n The investigated set of PDL channels can now be analyzed in terms of channel capacities. For the two ensembles with mean PDL values of 3 and 5 dB, in the sequel the SNR shall 2 σx be set to 10 dB, i.e., 2 = 10. The resulting probability densities of the capacities are nsσn given in Figure 4.13. For the PDL free channel (4.36) evaluates to 6.92 bit/channeluse. The capacities for the case of 3 dB mean PDL amount to values around 6.8 bit/channel- use. Hence, the inﬂuence of PDL is a slight reduction of the capacity by approximately 0.12 bit/channeluse. When the average PDL value is increased to 5 dB, one can observe a further (but small) reduction of the capacity to 6.58 bit/channeluse. Moreover, there is a slightly larger spread. In order to obtain a measure, which allows for a comparison with the SNR penalties obtained above, we introduce the deﬁnition of a capacity based SNR penalty [NMS09]

2 2 1 σx ! σx H 1 log det E + E = log det E + T T Ψ− . (4.37) 2 2 η · n σ2 2 2 2 n σ2 · 0 0 · C s n s n

The variable ηC describes the factor by which the AWGN channel SNR has to be reduced in order to achieve the same capacity as in the PDL scenario (for a speciﬁc noise variance).

Solving for ηC yields

2 σx 2 nsσn ηC = . (4.38) 2 σx H 1 det E2 + 2 T0T0 Ψ− 1 nsσn · · − r Respective values, which correspond to the probability densities in Figure 4.13 are plotted in Figure 4.14 in terms of ccdfs. Once again the SNR is set to 10 dB. The 54 4. Optical OFDM at the Presence of Linear Fiber Eﬀects probability for the SNR penalty to exceed 0.2 dB decays rapidly in the 3 dB PDL scenario; similarly for 5 dB PDL (mean) channels, the ccdf decreases for threshold penalties above 0.55 dB. This result indicates that – with the presented model – PDL causes relatively

45 3 dB PDL 40 5 dB PDL

→ 25 ) C

( 20

pdf 15

6.4 6.5 6.6 6.7 6.8 6.9 C [bit/channel-use] → Figure 4.13: Pdf of channel capacities in a transmission link consisting of 10 PDL elements; SNR: 10 dB.

0 10 3 dB PDL 5 dB PDL

−1 10 → ) th C, η

( −2 C 10 η ccdf

−3 10

0 0.2 0.4 0.6 0.8 1 10 log η [dB] 10 C,th → Figure 4.14: Ccdf of SNR penalties with respect to channel capacities in a transmission link consisting of 10 PDL elements; SNR: 10 dB. 4.3. Polarization Dependent Loss 55 small SNR penalties, when the channel realizations are analyzed and compared in terms of capacity. The degradation is much smaller than for determining BER based penalties as depicted in Figure 4.8 and Figure 4.12. In that case a certain number of symbols is severely distorted, causing bit errors which ﬁnally dominate the overall BER. A countermeasure is provided by “power loading” [Kal89]. Here the performance is improved by adapting the signal powers of the sub-channels to their expected attenuation. Apparently, this operation requires channel knowledge at the transmitter. An alternative method which avoids signaling of the channel state information to the transmitter is given by “polarization time coding”, which is addressed in the following section.

4.3.5 Polarization Time/Frequency Coding

In wireless communications, “space-time coding” has been introduced in order to improve the transmission performance in MIMO systems. The basic idea is to generate interrelation and (possibly) redundancy between data symbols, which are to be transmitted at diﬀerent TX antennas, instead of transmitting mutually independent data streams, e.g. [Ala98]. In the context of polarization multiplexing in optical systems this concept is referred to as “polarization time coding”. The OFDM scenario oﬀers a further dimension: The coding procedure can be done over sub-carrier symbols, which then may be called “polarization frequency coding”. Of special interest is the family of “2x2 full rate space time block codes”. Two vectors of source symbols (z , . . . , z ) are mapped to two vectors of the same size 1 4 ∈ A

z z x x 1 3 1 3 . (4.39) " z2 z4 # 7−→ " x2 x4 #

The resulting symbols x1, . . . , x4 of this encoding procedure stem from a set of discrete complex numbers, which in general diﬀers from the original set . A In the sequel, two codes out of this family shall be applied for transmission over PDL channels: The Golden code and the Silver code. Both codes are found to have good distance properties after transmission over a PDL element [MOJ10], i.e., the received code vectors still have a large minimum distance – even after several dBs of attenuation of one polarization component. The encoding rule of the Golden code reads [BRV05]

x1 x3 1 α(z1 + θz2) α(z3 + θz4) golden = = ¯ ¯ , (4.40) C (" x2 x4 # √5 " j¯α(z3 + θz4)α ¯(z1 + θz2) #) 56 4. Optical OFDM at the Presence of Linear Fiber Eﬀects

−1 10 3 dB, Golden C. 5 dB, Golden C. 3 dB, Silver C. −2 10 5 dB, Silver C. AWGN ref. → −3 10 BER

−4 10

−5 10 3 4 5 6 7 8 9 10 log (E /N )[dB] 10 b 0 → Figure 4.15: Ergodic bit error ratios in a 10 spans transmission link; application of Golden and Silver code along with ML detection. where the following parameters are used

1 + √5 θ = 2 1 √5 θ¯ = 1 θ = − (4.41) − 2 α = 1 + j jθ − α¯ = 1 + j jθ¯ = 1 + jθ. −

1+√5 Apparently, the code is named after the parameter θ = 2 , the Golden number. The encoding procedure for the Silver code [TH02] is done in two steps. At ﬁrst, one TX symbol vector is multiplied by a matrix

z¯ 1 1 + j 1 + 2j z 3 = − 3 . (4.42) z¯4 √7 · 1 + 2j 1 j · z4 " # " − # " # The resulting vector along with the second symbol vector are then combined according to

x1 x3 1 z1 +z ¯3 z2∗ z¯4∗ silver = = − − . (4.43) C x2 x4 √2 z2 z¯4 z∗ z¯∗ (" # " − 1 − 3 #) At the receiver, ML decoding is applied, which – in this scenario – equals the search for 4.3. Polarization Dependent Loss 57

0 10 3 dB, Golden C. 5 dB, Golden C. 3 dB, Silver C. 5 dB, Silver C.

−1

→ 10 ) S,th η ( S η −2 10 ccdf

−3 10

0 0.2 0.4 0.6 0.8 10 log η [dB] 10 S,th → 3 Figure 4.16: Ccdf of SNR penalties at BER = 10− in a 10 spans transmission link; application of Golden and Silver code along with ML detection. the nearest vector in the four dimensional complex space. The transmission in conjunction with the Golden and Silver code over the above presented PDL channels leads to distinctly reduced penalties in terms of the ergodic BER. Figure 4.15 depicts respective curves 3 showing penalties below 0.2 and 0.5 dB at BER = 10− for 3 and 5 dB mean PDL. Once more the statistical properties of these penalties are examined by means of ccdfs. Respective results are given in Figure 4.16 confirming the findings of the ergodic BER curves. Moreover, obviously the Silver code performs slightly better than the Golden code. Overall, the penalties are in the region of the capacity based penalties shown in Figure 4.14. As a summary, polarization time coding with the Golden or Silver code is an effective technique for distributing the impact of PDL over all symbols. Contrary to the uncoded scenario, where PDL may disturb one data stream severely, which then leads to a BER dominating degradation. One advantage of the technique is the fact that no channel state information is required at the transmitter. The full rate code does not introduce redundancy. As a drawback, the complexity of the ML decoder has to be mentioned. It is implemented as a full search algorithm. Due to the four-dimensional code vectors the cardinality of the code increases enormously when the size of the source symbol alphabet is increased. 58 4. Optical OFDM at the Presence of Linear Fiber Effects 59

5 Impact of the Kerr Eﬀect

In the previous chapter, the investigations on OFDM transmission were restricted to linear channels, i.e., to single-mode ﬁbers exhibiting attenuation, chromatic dispersion, and PMD, as well as polarization dependent loss. In that case, the overall channel can be described as a linear time invariant system which is characterized by its impulse response h(t). The input-output relation of the signals according to Figure 2.1 is given by the convolution integral

∞ r(t) = h(t) s(t) = h(τ) s(t τ)dτ. (5.1) ∗ · − Z−∞ When now non-linear eﬀects shall be included into the channel model, this equation is no longer applicable for an accurate description of the system. For the characterization of non-linear systems, one has to distinguish between memory-less systems and systems with memory. While for memory-less non-linear systems a polynomial approach like

n r(t) = p (s(t))i (5.2) i · i=1 X can be utilized, things become more complicated with non-linear systems exhibiting memory. In general, the Volterra series approach is capable of describing non-linear systems with memory. It can be used for modeling the input-output relation of the non-linear fiber channel and therefore can provide an approximate solution for the non-linear Schrödinger equation [XBP02], [Lou06]. However, the Volterra series approach introduces a large number of channel parameters, which might make the system identification process rather 60 5. Impact of the Kerr Effect complex. An alternative – and less complex – technique is offered by the “weakly non- linear” system approach, which shall be investigated in the sequel.

5.1 Weakly Non-linear System Approach

The term “weakly non-linear” reﬂects that one assumes that the characteristic of the system is dominated by a linear transfer function. The system’s non-linearity generates distortion which shall be modeled as an additive noise-like signal contribution (c.f. Fig- ure 5.1)

r(t) = h(t) s(t) +n(t). (5.3) ∗ rL(t)

The output signal r(t) is split into rL(|t) and{z n}(t) in such a way that both signals are uncorrelated

E r (t) n(t + τ) = 0, τ. (5.4) { L · } ∀ The corresponding system description in the frequency domain reads

R(f) = H(f) S(f) + N(f). (5.5) · In [Don90] a measurement technique is proposed for the characterization of communication systems utilizing digital signal processing. The author derives an estimation procedure for the extraction of H(f) and the power density of n(t) from corresponding input and output signals. The technique adapts the above presented model to discrete time periodic signals. The actual analysis is carried out in the frequency domain. Therefore, the technique is well suited in the context of OFDM transmission. The starting point are random discrete

H rL(t)

s(t) r(t)

n(t) SN

Figure 5.1: Weakly non-linear system model. 5.1. Weakly Non-linear System Approach 61

time periodic signals xξ[k] with zero-mean

E x [k] = 0, (5.6) { ξ } where ξ denotes the ensemble index. Subsequently, one period shall be considered. Then the response of the weakly non-linear system to the input xξ[k] is

y [k] = h[k] x [k] + n [k]. (5.7) ξ ⊗ ξ ξ This equation translates to the frequency domain

Y [q] = DFT y [k] = H[q] X [q] + N [q]. (5.8) ξ { ξ } · ξ ξ On the basis of corresponding spectra of source signals and measured outputs, the transfer function

E Y [q] X [q]∗ H[q] = ξ { ξ · ξ } (5.9) E X [q] 2 ξ {| ξ | } minimizes the power of the noise-like distortion which is caused by the system’s non- linearity [SD89], [Don90]. Estimates for H[q] can be obtained for frequencies q where X [q] = 0. In a next step (5.8) is solved for N [q], in order to estimate the power spectral ξ 6 ξ density of nξ[k] Φ [q] = E Y [q] H[q] X [q] 2 . (5.10) nn ξ | ξ − · ξ | 5.1.1 Single Polarization Model

In transmission simulations, estimates for H[q] are obtained by exchanging the expectation operator through averaging over nP representations

1 nP Yξ[q] Xξ[q]∗ Hˆ [q] = nP ξ=1 · . (5.11) 1 nP 2 Xξ[q] nPP ξ=1 | | The power density of the non-linearity inducedP signal distortion is estimated by

n 1 P 2 Φˆ nn[q] = Yξ[q] Hˆ [q] Xξ[q] . (5.12) nP − · ξ=1 X

The approach is readily applicable for OFDM (c.f. Chapter 3); after transmission of nP OFDM symbols which are known to the receiver, the evaluation of (5.11) and (5.12) delivers a numerical characterization of the optical transmission channel, which may incorporate non-linear eﬀects. 62 5. Impact of the Kerr Eﬀect

4 Popt = 1 dBm 2 P = −3 dBm opt − 0 → −2

[rad] −4 ]) q [ −6 ˆ H ( −8

phase −10

−12

−14 0 50 100 150 200 sub-carrier q → Figure 5.2: Phase of estimated linear transfer function for two optical power levels.

The simulation chain used in Chapter 4 has been extended for the following investigations. We choose the parameters of system 1 in Table 4.1 as a start. The number of sub-carriers is doubled to 204 allowing for the transmission over 1000 km standard single- mode fiber. Again, the fiber channel is composed of twelve spans with 80 km each (no dispersion compensation). Each span’s attenuation is compensated through an optical amplifier with 16 dB gain and 4 dB noise figure. At first, single polarization transmission shall be considered. Therefore, the numerical model of the signal propagation in the fiber channel is based on (2.34). The non-linear coefficient γ accounting for the Kerr effect is set to 1.3/(W km). · The transmission of nP = 60 OFDM symbols was simulated at different optical signal powers. Figure 5.2 shows the phase characteristic of the estimated linear transfer function for two signal power levels, i.e., -3 and -1 dBm. Due to the dominating chromatic dispersion, the diagram shows parabola. There is a slight vertical shift between the two curves. It corresponds with the difference of the mean phases for both points of operation. The parabolic phase characteristic is preserved when the optical signal power is varied. The corresponding estimated power densities of the signal distortion are depicted in Figure 5.3 in logarithmic scale. It turns out, that the additive distortion is not white. There is a modest decay at the edges of the signal spectrum. This observation agrees with the results in [Göb10]. In that work a combinatoric approach is introduced in order to determine the number of intra-channel four-wave-mixing products, finally yielding the power spectral density of Kerr effect induced signal distortion. These results hold for the single wavelength case; the noise characteristic changes in a WDM scenario. When five 5.1. Weakly Non-linear System Approach 63

−38 Popt = 1 dBm P = −3 dBm −40 opt −

→ −42

−44 [dB] ] q [ −46 nn ˆ Φ

10 −48

−50 10 log

−52

−54 0 50 100 150 200 sub-carrier q → Figure 5.3: Estimated power spectral density of noise-like distortion for two optical power levels.

Pch = 1 dBm −36 P = −3 dBm ch −

→ −38

[dB] −40 ] q [ nn ˆ Φ −42 10

−44 10 log

−46

0 50 100 150 200 sub-carrier q → Figure 5.4: Estimated power spectral density of noise-like distortion for two optical power levels; five WDM channels. channels are set up on a 12.5 GHz grid (the observed OFDM channel shall be located in the center), the power spectral density of the noise-like distortion becomes flat (see Figure 5.4). 64 5. Impact of the Kerr Effect

5.1.2 Dual Polarization Model

For dual polarization transmission, the weakly non-linear model has to be extended to 2 2-MIMO systems. A straight forward approach is based on the subsequent transmission × T T of pilot symbols XP[q] in just one polarization, i.e., [XP[q] 0] followed by [0 XP[q]] [JMST08]. The transmission of the ﬁrst training vector yields

X [q] H [q]X [q] N [q] Y[q] = H[q] P + N[q] = 11 P + 1 . (5.13) · " 0 # " H21[q]XP[q] # " N2[q] #

Hence, the ﬁrst column of H[q] is estimated by

nP ξ=1 Yi,ξ[q] XP∗,ξ[q] Hˆi1[q] = · , 1, 2 . (5.14) nP 2 ∈ { } P ξ=1 XP∗,ξ[q]

P T Analogously, the second column results from using the training vector [0 XP[q]] . Having obtained these estimates, the statistics of the noise-like signal distortion is evaluated with the help of the covariance matrix

n 1 P H Φˆ nn[q] = Yξ[q] Hˆ [q] Xξ[q] Yξ[q] Hˆ [q] Xξ[q] . (5.15) nP − · − · Xξ=1 As a drawback of this pilot symbol choice, the signal power is lowered by 3 dB during channel estimation. In this phase, the Kerr effect is not as prominent as in the normal transmission situation. Therefore, it is sensible to generate pilot symbol vectors with the same signal power as the data vectors. One could simply boost XP[q] by √2. However, despite of equal signal powers, the influence of the Kerr effect is not optimally accounted for as non-linear crosstalk between orthogonal polarized signal contributions is neglected by this channel estimation method. Therefore, an alternative set of two pilot symbols has been proposed [LB09]

X [q] X [q] P,1 , P,1 . (5.16) X [q] X [q] " P,2 # " − P,2 #

For a general formulation, nP arbitrary pilot symbol vectors are collected in a matrix

X [q] X [q] ...X [q] Y[q] = H[q] 1,1 1,2 1,nP +N[q]. (5.17) · " X2,1[q] X2,2[q] ...X2,nP [q] #

XP[q]

The least squares estimates for| the 2 2 sub-carrier{z matrices are} obtained utilizing the × 5.1. Weakly Non-linear System Approach 65

Moore-Penrose pseudo inverse

H H 1 Hˆ [q] = Y[q] X+[q] = Y[q] X [q] X [q]X [q] − . (5.18) · P · P P P In further simulations, both methods for the estimation of the linear transfer function of the weakly non-linear 2 2 system have been compared. Now, dual polarization × transmission and detection is applied. The OFDM parameters stay the same as above. The fiber channel is simulated by numerical evaluation of the dual polarization version of the non-linear Schrödinger equation (2.36). Again, the optical power levels are set to -1 and -3 dBm, i.e., the sum power of both states of polarization. The results of the estimated Φnn[q] are shown in Figure 5.5. No difference can be observed for both branches of the polarization diversity receiver (solid lines versus dashed lines); there is a distinct increment of the noise power when the signal power is increased from -3 to -1dBm. Most importantly, there is a larger distortion when the linear channel characteristic is estimated according (5.14) compared to (5.18). Using the first method, the non-linear channel is excited in a different point of operation during channel estimation in contrast to the normal transmission situation. As a consequence, the resulting channel estimate differs from the desired, most probable one. When then Φnn is calculated on the basis of dual polarization data vectors, larger deviations between the received signals from linearly distorted signals are observed, and therefore, the first channel estimation method ends up in larger estimates for the non-linear signal distortion.

−42 1 dBm, (5.14) − −44 3 dBm, (5.14) −1 dBm, (5.18) − → 3 dBm, (5.18) −46 −

[dB] −48 ] q [ −50 nn,ii ˆ Φ −52 10

−54 10 log −56

−58 0 50 100 150 200 sub-carrier q → Figure 5.5: Estimated power spectral density of noise-like distortion for two optical power levels; two estimation techniques are compared. 66 5. Impact of the Kerr Eﬀect

5.2 Interrelation of Optical Signal Power and Equivalent Noise Power

5.2.1 Single OFDM Channel

In the following, the powers of the Kerr eﬀect induced signal distortion shall be compared for various transmission scenarios. For this purpose, an average noise power measure is evaluated by integrating over the sub-carriers. Moreover, this value is normalized to the useful signal power

Q 1 ˆ i q=0− Φnn,ii[q] N = 2 , i 1, 2 . (5.19) Si 1 Q 1 nP ∈ { } − PHˆ [q]X [q] + Hˆ [q]X [q] nP q=0 ξ=1 i1 1,ξ i2 2,ξ

P P As a result, an inverse SNR measure – or equivalently, a relative noise power measure – is obtained. In Figure 5.6 respective simulation results are summarized for system 1 operating with a dual polarization OFDM signal at a bandwidth of 12.2 GHz. The optical signal power P is varied in the range from -18 to +4 dBm. The plotted curves depict /S opt N1 1 in logarithmic scale. The second receive branch (orthogonal RX polarization) delivers the same results; therefore, the index denoting the state of polarization is dropped in the sequel. There are two distinctive optical power regions. For low power levels, the

−6 12 spans −8 8 spans 4 spans −10

→ −12 −14 [dB] −16

/S −18 N −20 10 −22

10 log −24 −26 −28 −30 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 P [dBm] opt → Figure 5.6: Estimated relative noise power versus optical signal power for various distances; 12.5 GHz OFDM system. 5.2. Interrelation of Optical Signal Power and Equivalent Noise Power 67

−6 6 spans −8 4 spans −10 2 spans

→ −12 −14

[dB] −16

/S −18

N −20 10 −22 −24 10 log −26 −28 −30 −12 −10 −8 −6 −4 −2 0 2 4 6 8 P [dBm] opt → Figure 5.7: Estimated relative noise power versus optical signal power for various distances; 50 GHz OFDM system. estimated /S decreases by one dB per one dB signal power increment. Obviously, ASE N noise dominates the signal distortion. The fiber channel behaves like a linear transmission system. At a certain power level, the decay of the relative noise power turns to an increment of two dB, when the optical launch power is raised by one dB. In this region the Kerr effect induced non-linear signal distortion dominates the observed noise power. As this kind of signal distortion is caused by non-linear interaction between the single optical signal and the waveguide, it is referred to as self phase modulation (SPM)1. The three curves are associated with transmission distances of 4, 8, and 12 spans, respectively. When the number of spans is increased, the curves are shifted to higher /S values. Both, N more ASE noise and larger noise power due to SPM accumulate when the distances are increased. The curves’ slopes stay constant within both mentioned launch power regions. The optical launch power corresponding to the minimum relative noise power decreases with longer distances. Respective simulation results were obtained for “system 2” (48.7 GHz signal bandwidth; c.f. Table 4.2) and are shown in Figure 5.7. The transmission distances are set to 2, 4, and 6 fiber spans. Analog observations like with the first scenario can be made, especially with respect to the /S slopes within the mentioned signal power regions. As a differing N aspect, the launch power for minimum /S is larger than for system 1 of Figure 5.6. N 1In the context of OFDM, sometimes the term “intra-channel four-wave-mixing” is used. Moreover, for dual polarization signals, the non-linear interaction between orthogonally polarized signal contributions is often accounted for separately – then called cross-polarization modulation. 68 5. Impact of the Kerr Effect

Obviously, the signal power density is relevant for the SPM induced signal distortion.

5.2.2 Multiple OFDM Channels

For further investigations, the simulation setup was extended by surrounding wavelength division multiplexing channels. In order to evaluate the influence of the WDM channel power on the Kerr effect induced distortion, an odd number of WDM channels on a 12.5 GHz grid is simulated. The observed channel is placed in the center of the used WDM spectrum. Its power is varied independently from the neighboring channels’ power. Their launch powers in turn are kept equal in the following. Figure 5.8 depicts the resulting /S estimates versus the optical powers as a contour plot. N When the powers of the neighboring WDM channels are low (low ordinate values), their influence on the observed channel is marginal and thus the results are similar to Figure 5.6. The relative noise power is affected for WDM channel powers exceeding -6 dBm. For this case, the Kerr effect causes signal distortion within the observed channel. The respective mechanism can be understood as non-linear crosstalk, usually referred to as cross phase modulation (XPM). Moreover, in the literature the effect of four wave mixing (FWM) is described. Signals at three distinct frequencies generate a (fourth) distorting signal contribution within the observed frequency band. In the sequel, this effect is not investigated separately. Figure 5.9 shows /S results for equal powers in N all WDM channels. Hence, the curves correspond to a cross-section through the contour plot in Figure 5.8 for equal powers. Besides four fiber spans, distances up to 12 spans are

−6 −10 −12 2 −8 −14 0 −16 −12 −16 −18 −14 −20

→ −2 −18 −22 −4 [dBm]

−6 −20

−24

opt,2-5 −8 P

−22

−10 −20 −18

−12 −24 −22 −12 −10 −8 −6 −4 −2 0 2 P [dBm] opt,1 → Figure 5.8: Estimated /S versus optical power of observed channel and launch powers N of neighboring WDM channels; 5 WDM channels on 12.5 GHz grid; 4 spans. 5.3. Separation of Noise Power Contributions 69

−1 12 spans −3 8 spans 4 spans −5

→ −7 −9 [dB] −11

/S −13 N −15 10 −17

10 log −19 −21 −23 −25 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 P [dBm] opt,1-5 → Figure 5.9: Estimated /S versus WDM channel powers; 5 WDM channels on 12.5 GHz N grid. considered. The curves exhibit a similar behavior like in Figure 5.6 in terms of slopes. For low signal powers, the ﬁber channel is linear and thus the distortion is ASE noise dominated. Therefore, the relative noise powers are the same. At high signal powers, the non-linear signal distortion is larger in the WDM case as SPM and XPM induced distortion add up. However, still the slope is 2 dB increment per 1 dB launch power raise.

5.3 Separation of Noise Power Contributions

The observed interrelations between optical signal powers and the relative noise powers lead to a set of equations which allow for determining separate noise power contributions. Assuming that signal distortion due to ASE noise, SPM, and non-linear crosstalk are statistically independent, their individual powers add up

N + N + N N = ASE SPM XPM . (5.20) S S Based on the slopes of the inverse SNR curves, a measurement technique for the evaluation of the summands can be formulated. In order to extract three power contributions, three /S estimates at distinct signal power conﬁgurations are required. In a ﬁrst step, the N relative noise power at the operating point of interest is estimated

N + N + N m = N = ASE SPM XPM . (5.21) 0 S S 70 5. Impact of the Kerr Eﬀect

Then the optical input power of the observed WDM channel is increased by 10 log10(km) dB, which results in a second estimate

N k2 N N m = N = ASE + m SPM + XPM (5.22) 1 k S k S S S m · m according to above mentioned observations. Noteworthy, the 2 dB /S increment per one 2 N dB signal power raise translates to the factor km for the SPM induced signal distortion. For the separation of the distorting powers due to XPM and SPM a third measurement has to be carried out. The power of channel 1 is reduced to its original value while the power levels of the co-propagating channels are (equally) increased by 10 log10(km) dB. This procedure yields

N + N k2 N m = N = ASE SPM + m XPM . (5.23) 2 S S S

From m2 and m0, the ASE noise and SPM contributions are eliminated leading to

N m m XPM = 2 − 0 . (5.24) S k2 1 m −

This result is inserted in (5.22) and (5.23). NASE/S is then eliminated by evaluating

NSPM 3 km (m2 m0) m2 km m1 = 1 km + · − , (5.25) − · S − km + 1 yielding N k m + k (k + 1)m m SPM = − m 0 m m 1 − 2 . (5.26) S (k + 1)(k3 1) m m −

Finally, (5.24) and (5.26) are subtracted from m0 in order to obtain the ASE noise power contribution N N N ASE = m SPM XPM . (5.27) S 0 − S − S The separation technique is applied to the simulation results of Figure 5.8. Five WDM channels on a 12.5 GHz grid propagate over four spans. The power variation parameter

10 log10 km is set to 1 dB. The lower contour plot in Figure 5.10 shows the estimated ASE noise contribution. Its values decrease with increasing Popt,1. At the same time the results are independent from the adjacent channels’ power. The dashed lines denote region borders where valid results are obtained. Outside this region the accuracy becomes bad as the dominating sources of distortion are several decibels larger for these points of operation. 5.3. Separation of Noise Power Contributions 71

2 −15 10 log NXPM/S −10 10 −15 0 −20 −15 −20 −2 → −20 −25 −4 −25

−25 −30 [dBm] −6 −30

−8 −30 −35 opt,2-5 P −35 −10

−12

2 10 log10 NSPM/S 0

−20 −2 −22

→ −24

−26

−28 −4 −30 −32

[dBm] −6

−36

−34

−8 −20 opt,2-5 P −24 −10 −22 −26

−38

−36

−28 −12 −30

−32

2 10 log10 NASE/S 0

−2

−26 −28 −28 → −20 −22 −24

−4 −30

[dBm] −6

−18

−8 opt,2-5

−24 P −10

−28

−30

−26 −12

−22 −20

−12 −10 −8 −6 −4 −2 0 2 P [dBm] opt,1 → Figure 5.10: Estimated separated noise power contributions versus optical power of observed channel and launch powers of neighboring WDM channels; 5 WDM channels on 12.5 GHz grid; 4 spans. 72 5. Impact of the Kerr Eﬀect

0 10 log10 NASE/S −5 10 log10 NSPM/S 5 WDM ch. 9 WDM ch.

→ −10 15 WDM ch. −15 [dB]

/S −20 N

10 −25

−30 10 log

−35

−40 −10 −8 −6 −4 −2 0 2 4 P [dBm] opt,1-5 → Figure 5.11: Estimated separated noise power contributions versus optical signal powers; various WDM channel counts; transmission over 12 spans.

The second contour plot corresponds to the SPM contribution, which is present in a trapezoidal region. Especially, valid results are obtained when the signal power of the observed channel is not larger than the neighboring channels’ powers. For the latter case, XPM becomes the dominating source of distortion. The corresponding estimated values after separation are shown in the first diagram of Figure 5.10. Further simulations have been carried out for a 12 span scenario with equal WDM channel powers. Here, the number of WDM channels on the 12.5 GHz grid is varied. Again, the observed channel is located in the center of the used frequency spectrum. For all WDM settings, the separated contribution of ASE noise and SPM is the same. The ASE noise summand cannot be accurately estimated, when the non-linear signal distortion dominates distinctly. Therefore, the dashed line in Figure 5.11 depicts an extrapolation from valid measures. The results of the separation procedure reveal for the 5 channel scenario distortion due to non-linear crosstalk being larger than SPM. Of course, the XPM contribution becomes even larger when the number of WDM channels is increased. As a benefit, the /S estimation technique allows for finding optimum optical power N settings in a WDM transmission scenario. With the help of separating the total observed noise power into three contributions, one can find out, which signal degrading effect dominates the overall distortion. This result can then serve as a basis for a decision on optimizing optical signal powers. The presented formulation is restricted to the characterization of one observed WDM channel, while – at the same time – all the neighboring WDM channels are treated as 5.4. Channel Capacity 73 an entity. A possible extension of the approach can be achieved by splitting the XPM contribution into W 1 summands corresponding to the XPM induced distortion, which − is generated through the presence of each individual WDM channel. This step leads to a larger set of equations requiring more measurements in order to determine all the contributions. As a result, this technique provides information about ASE noise, distortion due to SPM and – most importantly – non-linear crosstalk from each neighboring WDM channel (c.f. Appendix B). The method can serve as a monitoring tool within meshed transparent optical networks for optimization of individual signal power levels [MH09a].

5.4 Channel Capacity

There are various publications on determining the capacity – in an information theoretic sense – of the non-linear fiber channel. Usually perturbation approaches are utilized in order to describe the influence of the non-linearity as distortion. For example, a Volterra approach can be used for modeling non-linear signal distortion as an additive contribution [Lou06], [Tan06]. Alternatively, Mitra proposes to describe the non-linear signal distortion as a multiplicative perturbation [MS01], leading to an adaptation of Shannon’s capacity equation. A further approach is based on a frequency domain analysis of four-wave mixing products along with respective powers [EFWK09], [EKW+10], [Göb10]. In this work, the capacity investigations shall be carried out on the basis of numerical results, i.e., by analysis of corresponding OFDM symbol pairs at the transmitter and – after simulated propagation over the fiber channel – at the receiver.

5.4.1 Mutual Information

The measure of interest is the mutual information, which describes the dependence between random variables. Assuming discrete, memory-less, time-invariant sources

X = x , x , . . . , x ,Y = y , y , . . . , y , (5.28) ∈ X { 1 2 Mx } ∈ Y { 1 2 My } the mutual information is deﬁned as

M Mx y Pr(Y = y X = x ) I(X,Y ) = Pr(X = x ,Y = y ) log j| i . (5.29) i j 2 Pr(Y = y ) i=1 j=1 j X X For the scenario of transmitting discrete symbols over a distorting channel, we model the interrelation of X and Y as a discrete variable being distorted by a continuous noise sample N

Y = X + N. (5.30) 74 5. Impact of the Kerr Eﬀect

16 Capacity 128QAM 14

→ 64QAM 32QAM 12 16QAM 8QAM 10 4QAM

mutual information [bit] 4

2 −15 −10 −5 0 P [dBm] opt,1-5 → Figure 5.12: Estimated mutual information versus optical launch power; application of various symbol alphabets.

Then the mutual information equation reads

Mx p (y X = x ) I(X,Y ) = Pr(X = x ) p (y X = x ) log Y | i dy. (5.31) i · Y | i 2 p (y) i=1 Y X ZY

The numerical evaluation of (5.31) was carried out along with simulated TX and RX data. The parameters of “system 1” are used, with a fiber link consisting of six spans. The OFDM sub-carriers are modulated utilizing QAM symbols. The size of the constellations is varied from 4QAM up to 128QAM. At the same time, the signal bandwidth is kept constant. Hence, by increasing the constellation size the overall bitrate is raised. For each OFDM system setting variant, the optical signal power is swept from -19 to +4 dBm. The resulting values of the estimated mutual information are shown in Figure 5.12. When the optical launch power is small, ASE noise limits the mutual information. Respective values increase with larger signal powers – up to the maximum value, which is given by the size of the symbol alphabet. For the simulated transmission scenario, 14 bit/channel-use can be achieved utilizing 128QAM modulation at a launch power of -6 dBm. When the input power exceeds this value, the influence of the Kerr effect appears. The effective noise power increases and therefore the mutual information is reduced. 5.5. Achievable Spectral Efficiency 75

5.4.2 Gaussian Signaling

The mutual information becomes maximum when the source signal is Gaussian distributed. Describing the inﬂuence of the channel’s non-linearity through additive noise-like distortion makes Shannon’s capacity equation applicable. Therefore, an estimate for the upper bound can be obtained with the help of (5.19) by, at ﬁrst, evaluating the sub-carrier capacity

2 E Hˆi1[q]X1[q] + Hˆi2[q]X2[q]

Γi[q] = log2 1 +  . (5.32) Φˆ nn,ii[q]       Through averaging over the OFDM sub-carriers along with summing up the contributions of the orthogonal states of polarization one obtains

Q 1 Q 1 1 − − Γ = Γ [q] + Γ [q] . (5.33) Q 1 2 q=0 q=0 ! X X Evaluating this equation using previously discussed simulation results yields the topmost curve in Figure 5.12. As an upper bound, the channel capacity of the investigated scenario amounts to approximately 14.7 bits/channel-use at an optical signal power of -6 dBm.

5.5 Achievable Spectral Eﬃciency

When applying the aforementioned method for estimating an upper bound for the maximum achievable spectral eﬃciency for WDM transmission systems, a “WDM ﬁll factor”

ηW shall be considered. This measure corresponds to the ratio of the used OFDM signal bandwidth over the WDM grid. Hence, in the sequel the results of (5.33) shall be weighted with ηW = 12.2/12.5 and ηW = 48.7/50 for system 1 and 2 according to Table 4.1 and Table 4.2, respectively. Resulting Γ values versus optical powers are summarized in Figure 5.13 for different channel settings. The diagram corresponds to the 12.5 GHz scenario, where the number of co-propagating WDM channels is varied from one to 15. Moreover, two sets of curves are shown corresponding to 6 spans (bright color) and 12 spans (dark color). The simulation results confirm that non-linear crosstalk increasingly delimits the maximum achievable spectral efficiency for higher WDM channel counts. For example, the values range from 13.6 down to 10.2 bit/s/Hz for transmission over 12 spans. At the same time the maximum is reached for different optical signal powers, i.e., the optimum launch power decreases, when the number of WDM channels is raised – from -4 to -9 dBm in the aforementioned scenario. An analog observation is made for the shorter transmission setup, 76 5. Impact of the Kerr Effect

20 1 channel 18 5 channels 9 channels 16 15 channels → 14

[bit/s/Hz] 10 Γ 8

4 −12 −10 −8 −6 −4 −2 0 2 4 P [dBm] opt → Figure 5.13: Achievable spectral efficiency versus optical signal power for various WDM channel numbers. where 12.8 to 15.9 bit/s/Hz can be reached (15 WDM channels to single WDM channel). Correspondingly, the optical signal powers have to be set to -8 dBm to -4 dBm. The equivalent analysis was carried out for the transmitted OFDM symbols of the 50 GHz setup, which was simulated for 1, 3, and 5 WDM channels. The results are summarized in Figure 5.14. Two sets of curves correspond to the transmission over 2 spans (bright color) and 6 spans (dark color). The 6 span setup leads to 13 and 14bit/s/Hz for WDM configurations with 3 and 5 channels, respectively. These values are in the same region as for the previous 12.5 GHz setup, when looking at 9 and 15 WDM channels. A further striking aspect is the position of the maximum Γ value. For the 50 GHz scenario -2 and 0 dBm are the optimum channel powers for the transmission over 6 spans. This is 6 dB higher than for system 1, which occupies a quarter of the frequency band per WDM channel compared to system 2. Apparently, the signal power density is the critical parameter for the maximum achievable spectral efficiency. This observation agrees with the results presented in [EZC10], where analytical expressions are developed for the non- linear Shannon limit based on [MS01].

Comparison of Capacity Results

From the data basis of the diagram in Figure 5.13, the maximum Γ values for different transmission distances have been extracted. In the following, these results are compared with further approaches for the evaluation of the fiber channel’s capacity. 5.5. Achievable Spectral Efficiency 77

22 1 channel 20 3 channels 5 channels 18

→ 16

[bit/s/Hz] 10 Γ 8

4 −8 −6 −4 −2 0 2 4 6 8 P [dBm] opt → Figure 5.14: Achievable spectral eﬃciency versus optical signal power for various WDM channel numbers; 50 GHz WDM setup.

Theoretical upper bound The theoretical bound for the spectral efficiency on SSMF has been addressed in the literature, e.g. [EFW+08], [EFWK09]. For distances of 500 km, 1000 km, and 2000 km maximum values have been extracted from the respective publications (denoted with stars in Figure 5.15). For example, those investigations predict a channel capacity of approximately 14 bit/channel-use for a transmission distance of 1000 km. These results indicate an upper bound for SSMF channels assuming an ideal configuration, which includes distributed Raman amplification and pulse pre-distortion for SPM compensation. In order to approximate Gaussian signaling, constellations in concentrical circles were applied. Linear AWGN channel An alternative upper bound can be obtained by modeling the concatenation of SSMF spans with EDFAs in-between as a linear AWGN channel. For various distances, the optimum launch powers are taken from the simulation results of Figures 5.13 and 5.14. Solely ASE noise accumulation is taken into account yielding SNR values at the receiving end of the fiber chain. Inserting these measures into Shannon’s equation provides the values shown as the dashed curve in Figure 5.15. It lies slightly below the aforementioned values, as the EDFA based transmission chain provides worse SNR performance than the Raman setup. However, ignoring non-linear signal distortion does overestimate the achievable spectral efficiency significantly. Equivalent noise power The estimation error for the assumption of a linear AWGN channel becomes evident 78 5. Impact of the Kerr Effect

20 [EFWK09] AWGN channel 18 weakly nl. sys.

→ 14

[bit/s/Hz] 10 Γ

4 3 10 2 10 3 7⋅10 distance [km] → Figure 5.15: Maximum achievable spectral efficiency versus transmission distance. when comparing the results with those based on the concept of the equivalent noise power estimation. The solid curve in Figure 5.15 summarizes the values for the maximum achievable spectral efficiency extracted from the simulation data of Figure 5.13 for various distances (15 WDM channels). This method yields around 10.5 bit/s/Hz at 1000 km. Thus, the analysis predicts an achievable datarate of slightly above 500Gbit/s on a 50GHz WDM grid. Distinctly larger rates are possible for shorter distances. The estimation technique for the presented upper bounds is based on a weakly non- linear system approach. At this point, the model incorporates noise of the optical amplifiers as well as noise-like distortion, which is caused by the interaction of signals and the waveguide through the Kerr effect. For approaching the channel capacity, ideally implemented transmitters and receivers would be necessary as well as ideal forward error correction schemes. The utilized approach allows for incorporating sources of distortion which have their origin in implementation related limitations. The next chapter is dedicated to this topic in order to obtain estimates for the maximum achievable spectral efficiency accounting for impairments in real systems. 79

6 Impairments through TX and RX Implementation

The preceding investigations assumed ideal transmitters and receivers in order to focus on the effects of the optical transmission channel. With the help of the weakly non-linear system approach an estimate for the maximum achievable spectral efficiency has been obtained – which is delimited by linear and non-linear fiber effects as well as noise. In this chapter impairments through non-ideal implementation of transmitters and receivers shall be included in the presented framework. Several idealizing assumptions shall be dropped for investigating their risk of limiting the system performance.

6.1 Laser Phase Noise

So far, laser sources have been modeled as ideal oscillators, i.e., as an optical device which generates a single harmonic signal. In that case the power spectral density equals the Dirac delta function. Practically, measured power spectral densities deviate from the ideal characteristic due to several eﬀects adding intensity noise and phase noise. The spectrum around the carrier frequency is broadened – the two-sided 3 dB width is usually referred to as the “linewidth” of the laser. In the context of coherent optical systems, laser phase noise may be critical and shall therefore be considered in further simulations. The eﬀect is accounted for by multipli- cation of the optical signal with exp(jφ(t)). The random variable φ(t) is modeled as a Wiener process [KBW96] – also known as “random walk process” – and obtained through 80 6. Impairments through TX and RX Implementation

0 → [rad] ) t (

φ −5

−10 0 5 10 15 20 t [µs] → Figure 6.1: Exemplary phase progression; linewidth: 1 MHz. the integral

φ(t) = φ′(τ)dτ. (6.1)

The term φ′(τ) denotes zero-mean white Gaussian noise with power spectral density 2π∆ν. This expression comprises the parameter ∆ν, which corresponds to the laser linewidth. Figure 6.1 shows an exemplary realization of φ(t), when the linewidth equals 1 MHz. Within the plotted snapshot of 20 µs duration, one can observe rapid phase ﬂuctuations but also distinctly slower variations. The corresponding power spectral density of the phase-noise impaired laser signal is shown in Figure 6.2. The shape equals a Lorentzian distribution [KBW96]. The impact of phase noise on the performance of OFDM transmission has been investigated by many authors, e.g., [PVBM95], [AC98], [Tom98], [WBN04]. The common starting point is to extend the time-domain OFDM signal model of Chapter 3 by the above mentioned random phase modulation term

y[k] = [x[k] h[k]] ejφ[k] + n[k]. (6.2) ⊗ · Obviously, through the DFT phase noise has an impact on all OFDM sub-carriers, which has two aspects. On one hand the useful signal power is reduced. On the other hand inter- carrier interference occurs. Through phase noise, the orthogonality of the sub-carriers is lost, which then causes noise-like interference from any sub-carrier to all the others. 6.1. Laser Phase Noise 81

−5 → −10

−15 [dBc/Hz] )

f −20 ( P

10 −25

10 log −30

−35 −20 −10 0 10 20 f [MHz] → Figure 6.2: Power spectral density of oscillator with 1 MHz linewidth.

For the purpose of quantifying the phase noise impact, in [PVBM95] an estimate for the SNR degradation is derived in closed form. It is valid for small phase noise and a large number of sub-carriers1

10 11 ∆ν E 10 log η 2πQ S [dB]. (6.3) 10 p ≈ log 10 60 B · N 0 A more general signal-to-interference-plus-noise ratio (SINR) derivation is carried out in [WBN04]; from their results an approximation for small linewidths, i.e., 2π∆νQ/B 1, ≪ can be derived S 1 π∆νQ − 3B . (6.4) N ≈ π∆νQ + N0 3B ES Above mentioned condition for small phase noise gives rise for neglecting the reduction of the useful signal power in the numerator; in that case a handy expression for the inverse SNR is obtained N N π∆νQ 0 + . (6.5) S ≈ ES 3B By this approximation the inﬂuence of phase noise is modeled as an additive noise power contribution, which ﬁts well to the distortion model elaborated in Chapter 5. The validity

1In contrast to [PVBM95] we deﬁne the laser linewidth as two-sided 3-dB bandwidth, which requires to divide the term by two. 82 6. Impairments through TX and RX Implementation

−1 10 numerical res. (6.3) (6.4) −2 10 (6.5) AWGN ref.

−3

→ 10

BER −4 10

−5 10

−6 10 2 4 6 8 10 12 14 10 log E /N [dB] 10 b 0 → Figure 6.3: BER performance at the presence of phase noise with the laser linewidth equaling one percent of the sub-carrier bandwidth. of these approaches for the description of the phase noise induced signal distortion shall be examined in the following. For this purpose the OFDM system simulation of Chapter 4 is set up in back-to-back configuration, i.e., no fiber channel shall be considered. AWGN is added at the receiver in order to investigate arbitrary SNR settings. The OFDM parameters have been selected according to Table 4.2. The laser linewidth ∆ν is accounted for by setting a fixed ratio

∆νQ r = , (6.6) LW B where Q and B denote the number of sub-carriers and the overall signal bandwidth, respectively. Hence, rLW expresses the ratio of laser linewidth and sub-carrier bandwidth. This term is included in (6.3) to (6.5) and is the crucial measure for the phase noise impairment.

When we set rLW = 0.01 for the system with B = 48.7 GHz and Q = 1616 sub- carriers, the corresponding linewidth ∆ν amounts to approximately 300 kHz. Figure 6.3 summarizes results of BER values versus Eb/N0, when the sub-carriers are modulated with symbols taken from a 4QAM constellation. Two curves with markers show the AWGN channel reference and results of Monte-Carlo simulations. The curves in between depict the results, which are obtained analytically – the SNR values are adapted on the basis of the introduced approaches. For example, the modiﬁed SNR of (6.5) leads to the following 6.1. Laser Phase Noise 83 estimate for the bit error probability for 4QAM modulation at the presence of phase noise

1 1 Pb erfc   . (6.7) ≈ 2 · N0 π v2 + rLW u ES 3 u ·  t  3 The diagram reveals that for BER values larger than 10− there is almost no diﬀerence between the performance estimation techniques. For lower BER values, i.e., for larger

Eb/N0 there is an increasing gap between the simulated BER curve and the predicted performance according to (6.3) to (6.5). The Monte-Carlo simulations exhibit an error ﬂoor which is not covered by the SNR approximation equations. A further observation is, that there are hardly any diﬀerences between the analytic curves. An alternative presentation of the results considers the SNR penalty for distinct BER values with the ratio of linewidth and sub-carrier bandwidth rLW as the free parameter. 3 4 Figures 6.4 and 6.5 show respective curves for bit error ratios equaling 10− and 10− . The penalty according to [PVBM95] is given directly in (6.3). The respective computation for (6.4) and (6.5) is done as follows: For the AWGN scenario, we consider an operating point at an arbitrary ES/N0; in the phase noise impaired case, the (initially higher) SNR shall be denoted by γS, but phase noise then leads to a reduced SINR. The penalty shall be expressed as γ η = S . (6.8) p ES N0 We postulate equal BER values for the AWGN and phase noise impaired situation, i.e., for (6.5) we write

ES ! 1 = 1 π . (6.9) N0 + r γS 3 LW

With the help of this equation γS in (6.8) can be eliminated, resulting in

1 E π − η = 1 S r . (6.10) p,(6.5) − N · 3 LW 0 The situation for (6.4) is treated analogously:

π ES ! 1 3 rLW = 1 − π (6.11) N0 + r γS 3 LW leads to 1 π ES π − ηp,(6.4) = 1 rLW rLW . (6.12) − 3 − N0 · 3 84 6. Impairments through TX and RX Implementation

3 The penalty curves for a target BER of 10− in Figure 6.4 show less than 0.2 dB deviation between penalty estimates and numerical results for relative linewidths up to 1%. There are just marginal diﬀerences among the curves corresponding to estimates based on (6.3) to (6.5) in the whole range of linewidth values. The actual SNR penalties are un- derestimated, e.g., for rLW = 2% the penalty estimates are approximately 0.4 dB too low. 4 Respective results for a BER of 10− in Figure 6.5 reveal that the gap between estimates and actual penalties becomes larger. This observation is attributed to the fact that an error ﬂoor is not covered by the penalty estimation techniques. Therefore, the model is solely valid in SNR regions where phase noise distortion is not the performance limiting impairment. In other words, the maximum relative linewidth, where the approximations are valid becomes smaller when the SNR is increased. As a conclusion we can state that distortion due to phase noise can be regarded as additive noise-like distortion. The crucial parameter is the ratio of linewidth and sub- carrier bandwidth rLW. The respective noise power contribution can be approximated by

N π pn = r , (6.13) S 3 LW as long as it leads to a small penalty, i.e., when phase noise is not dominating other sources of distortion. The model is valid for small values of rLW. Given the relative laser linewidth is in the range of 0.1% to 0.5%, the associated noise powers amount to -29.8 to -22.8 dB.

1.4 numerical res. (6.3) 1.2 (6.4) (6.5) 1 →

0.8 [dB] p η

10 0.6

0.4 10 log

0.2

0 0 0.005 0.01 0.015 0.02 r LW → 3 Figure 6.4: SNR penalty versus linewidth to sub-carrier bandwidth ratio at BER = 10− . 6.2. Clipping 85

3.5 numerical res. (6.3) 3 (6.4) (6.5) 2.5 →

2 [dB] p η

10 1.5

1 10 log

0.5

0 0 0.005 0.01 0.015 0.02 r LW → 4 Figure 6.5: SNR penalty versus linewidth to sub-carrier bandwidth ratio at BER = 10− .

6.2 Clipping

A further cause of performance limitation is signal clipping. Since the OFDM source signal consists of the superposition of independently modulated sub-carriers, the transmit signal is nearly Gaussian distributed. Therefore, it exhibits a very high peak-to-average power ratio. Due to this property, limiting the transmit signals’ amplitudes (clipping) and setting the driving levels for the modulators become important aspects. The upconversion of the complex valued baseband signal shall be carried out by nested Mach-Zehnder structures as introduced in Chapter 3. Separate modulators are used for inphase and quadrature component; both are driven in push-pull operation and therefore act as pure amplitude modulators. The following considerations are based on real valued signals and thus we restrict them on the inphase component sI(t) of the baseband signal. The ﬁrst step of the generation of the optical transmit signal is the limitation of the amplitude by a soft-limiter (c.f. Figure 6.6)

s for s (t) < s − I,max I − I,max scl(t) = s (t) for s s (t) s . (6.14) I  I − I,max ≤ I ≤ I,max  +sI,max for sI(t) > sI,max

As an important parameter, the clipping ratio is deﬁned as

2 sI,max 1 rcl = 2 = 2 . (6.15) σsI σsI 86 6. Impairments through TX and RX Implementation

u(t)

pre- sI(t) cl dist. sI (t) vI(t)

Figure 6.6: Limitation of source signal amplitudes and compensation of MZM characteristic by pre-distortion.

The second equality holds for setting the maximum signal amplitude sI,max = 1. The denominator refers to the variance of the signal. A further building block has the task to linearize the characteristic of the Mach- Zehnder modulator by pre-distorting the signal by

2v v (t) = s arccos scl(t) . (6.16) I π · I The resulting signal drives the modulator in such a way that there is a linear relation between source signal and the optical signal u(t), which becomes obvious by inserting vI(t) into (3.10)

u(t) = u scl(t). (6.17) cw · I

In the case of ucw = 1 the signal distortion, which is introduced by the clipping operation is determined by

n (t) = u(t) s (t) = scl(t) s (t). (6.18) cl − I I − I In a similar fashion like the characterization of the fiber channel effects, we define a relative noise power due to clipping as 2 Ncl σncl = 2 . (6.19) S σsI Respective values are depicted in Figure 6.7 for Gaussian distributed input signals sI(t). As the free parameter, the clipping ratio is varied from 5 to 12 dB. Apparently, the noise power decays with increasing clipping ratio, e.g. below -40dB for rcl exceeding 10.5 dB. For actual system designs, the clipping ratio shall not be set to very high values as rcl is directly related with additional attenuation. Therefore, the choice of its value has 6.3. Quantization Noise 87

−15

−20

−25 →

] −30 dB [ cl

S −35 N

10 −40

10 log −45

−50

−55 5 6 7 8 9 10 11 12 10 log r [dB] 10 cl → Figure 6.7: Relative noise power due to clipping of a Gaussian input signal versus clipping ratio. to be done considering the overall TX power budget in order to guarantee a certain TX OSNR2.

6.3 Quantization Noise

The preceding considerations on signal distortion through clipping are based on a continuous time model. In order to incorporate the discrete time nature of digital signal processing and the ﬁnite number of amplitude values at the DAC, the model is extended as shown in Figure 6.8. The digital signal xI is limited to a certain range of values. The linearization of the MZM characteristic is carried out in the digital domain. The resulting p amplitude values xI [k] is converted to an analog signal. The behavior of the DAC is considered by dividing the allowed amplitude region

v v < v (6.20) min ≤ I max into MQ intervals Ii, i = 1 ...MQ. The signal amplitudes in the digital domain address reconstructions values, which are associated with the respective intervals Ii. In the sequel, uniform quantization shall be assumed, i.e., the intervals Ii shall have equal widths. The limited number of amplitudes causes quantization noise. Its variance depends on

2In the context of wireless OFDM systems much research has been done on the peak-to-average power ratio (PAR) problem, especially, numerous PAR reduction techniques have been proposed [Sie10]. 88 6. Impairments through TX and RX Implementation

u(t)

pre- xI[k] DAC cl dist. p xI [k] xI [k] vI(t)

Figure 6.8: Limitation of source signal amplitudes and compensation of MZM characteristic by pre-distortion in the digital domain; digital-to-analog conversion causes quantization errors. the number of quantization levels as well as the probability density function of the signal to be converted. The MZM pre-distortion operation has an inﬂuence on the amplitude distribution. As an example, Figure 6.9 shows the pre-distorted signal’s pdf of an associated Gaussian input at a clipping ratio of 8 dB revealing that it is concentrated to relatively small amplitudes compared to the original Gaussian distributed source signal (dashed curve). By means of numerical simulations the power of the error signal which is caused by clipping and quantization has been evaluated. The results are plotted versus the clipping

1.6 p p xI [k] E xI [k] cl − { } 1.4 xI [k]

1.2

→ 0.8

pdf 0.6

0.4

0.2

−1 −0.5 0 0.5 1 signal amplitude → Figure 6.9: Pdf of clipped Gaussian signal (dashed line) and of output of MZM pre- distortion; clipping ratio: 8 dB. 6.4. Transmission Simulations 89 ratio in Figure 6.10 – diﬀerent curves correspond to diﬀerent numbers of quantization levels. Small rcl values make signal clipping the dominating source of distortion (comparable with results given in Figure 6.7). Larger clipping ratios reduce the amount of clipped signal energy but increase distortion due to quantization. One additional bit DAC resolution reduces quantization noise by 6 dB [Hub08]. Minimum signal distortion amounts to -22 to -43 dB at 9 to 12 dB clipping ratio for DAC resolution values in the range from 5 to 9 bit.

6.4 Transmission Simulations

The simulation model used in Chapter 5 has been extended by incorporating laser phase- noise, as well as clipping and quantization at the transmitter and receiver. By this means, a more realistic model of relevant sources of signal distortion is investigated, which finally leads to an estimate for the maximum achievable spectral efficiency incorporating implementation impairments. The actual simulation model regards phase noise as a single source of distortion at the receiver while the laser at the transmit side is assumed to be ideal. As the linewidths of the transmit and local laser add up [PHN09], in the following, all mentioned linewidth values have to be understood as the sum of both linewidths. At first, system 1 (c.f. Table 4.1) occupying a 12.5 GHz WDM grid is investigated. The parameters of the first column are chosen – except for doubling the number of sub-carriers in order to allow for transmission over 1000 km SSMF. The DAC and ADC resolution is varied from 5 to 9 bit. Respective clipping ratio values are chosen in such a way to

−15 log2 MQ = 5 log2 MQ = 6

→ −20 log2 MQ = 7

] log2 MQ = 8

dB log2 MQ = 9 [ −25 q S N −30 + cl S N

−35 10

−40 10 log

−45 6 8 10 12 14 16 10 log r [dB] 10 cl → Figure 6.10: Relative noise power due to clipping and quantization versus clipping ratio. 90 6. Impairments through TX and RX Implementation achieve minimum distortion according to Figure 6.10. In order to keep the parameter space manageable, the same clipping and quantization parameters are set up for the transmitter and the receiver. The propagation of 15 WDM channels is simulated, while the observed signal is located in the center of the used wavelength band. The assumption of a weakly non-linear system is kept up. Thus, the same method as in Chapter 5 is utilized for the characterization of noise powers. Estimated inverse SNR values are summarized in the diagrams of Figure 6.11. The plots depict /S measures versus transmission distance for different DAC and ADC N resolution values. Three diagrams correspond to simulated laser linewidths of 0.1, 0.2, and 0.5%. The OFDM sub-carrier bandwidth amounts to 60 MHz. Therefore, the mentioned linewidth sub-carrier ratios correspond to absolute sums of TX and RX laser linewidths of 60, 120, and 300 kHz, respectively. The diagrams reveal, that for long distances, there are similar noise power values of approximately -15 dB (1000 km) – independent from the laser linewidths. The estimated noise powers stay constant when the DAC/ADC resolution is increased from 7 to 9 bit. There is a slight degradation of 0.2 dB when solely 6 bit quantization is utilized. A further reduction to 5 bit results in approximately 0.7 dB SNR loss compared to 7 to 9 bit quantization. The picture changes significantly when shorter distances are examined. For example, at a link length of 160 km the observed noise powers amount to approximately -19.7 to -23.7 dB for relative laser linewidths of 0.5 to 0.1%. In order to investigate very short distances while still considering non-linear fiber effects, the transmission is also simulated for a single span of 3 km SSMF. The data points for 3km in Figure 6.11 are repeated in Figure 6.12. Here, the estimated noise power values are plotted versus the number of bits of the quantizers. The noise powers can become as low as -21.3 to -26 dB for relative laser linewidths of 0.5 to 0.1% – given the resolution of the quantizers is larger than 6 bit. The plot also shows that there are solely several tens of dBs improvement when the DAC/ADC resolution is increased from 7 to 9 bit. As already stated above, significant deterioration is observed for lower resolution values. The OFDM system parameters have been chosen to comply with the chromatic dispersion of 1000 km SSMF. For shorter distances, a reduced cyclic extension length is sufficient. When the relative cyclic extension ratio is kept up, a larger sub-carrier bandwidth is obtained (c.f. design rule (4.10)). By this means a larger tolerance to (the absolute) laser linewidth can be achieved. For example, when the number of sub-carriers of the investigated OFDM setup is reduced to 50, the sub-carrier bandwidth is increased by a factor of approximately four. Consequently, now rLW = 0.1 to 0.5% correspond to linewidths of 240 kHz to 1.2 MHz. Estimated /S values for this modified system are shown as N bright curves in Figure 6.12, which approximate the results of the original configuration quite well. In the region of 7 to 9 bit resolution the total distortion is approximately 1 dB 6.4. Transmission Simulations 91

−14 rLW = 0.1 % −16 → −18 ] dB [ −20 /S

N −22 10 −24 10 log −26

−28

−14 rLW = 0.2 %

−16 →

] −18 dB [

/S −20 N

10 −22

10 log −24

−26

−14 rLW = 0.5 % −15

→ −16 ]

dB −17 [

/S −18 N

10 −19

log MQ = 5 −20 2 log MQ = 6 10 log 2 log MQ = 7 −21 2 log2 MQ = 8 log MQ = 9 −22 2 0 200 400 600 800 1000 distance [km] → Figure 6.11: Estimated /S versus distance for various DAC/ADC resolution values. The N diagrams correspond to relative laser linewidths of 0.1, 0.2, and 0.5%. 92 6. Impairments through TX and RX Implementation

−19 rLW = 0.5 % −20 rLW = 0.2 % rLW = 0.1 % −21 →

] −22 dB [ −23 /S

N −24 10 −25

10 log −26

−27

−28 5 6 7 8 9 log M 2 Q → Figure 6.12: Estimated /S versus DAC/ADC resolution for various laser linewidth val- N ues; ﬁber channel: 3 km SSMF; bright curves: system parameters adapted to 50 sub-carriers.

lowered. This deviation is caused by the fact that (6.5) approximates (6.4) by neglecting the reduction of the useful signal power caused by phase noise induced inter-carrier interference. This equation in turn is an estimate for the resulting SNR. Thus, modification of the system parameters has some influence on the model error, but still (6.13) is a useful design rule for finding the tolerable laser linewidths.

Accordingly, the 50 GHz system (parameters of system 2, Table 4.2, first column) is simulated for OFDM transmission over 2 to 6 fiber spans. The resulting effective noise variances are given in Figure 6.13 with rLW set to 0.1%. The simulation quite accurately reproduces the results shown in the upper diagram in Figure 6.11 for respective distances. The sub-carrier bandwidth for this set of system parameters amounts to 30.1 MHz. There- fore, 30 kHz linewidth is necessary for this scenario.

The main findings of this section are, at first, that there is hardly a SNR gain when the DAC/ADC resolution is increased beyond 7 bit. Furthermore, laser linewidth turns out to be the crucial system parameter. The definition of a maximum allowable phase noise power contribution along with (6.13) and with the sub-carrier bandwidth determines the absolute value of the sum of the TX and RX laser linewidths. The required maximum value is in the region of tens to few hundreds of kHz. Especially, this requirement becomes rather strict for large signal bandwidths along with large tolerable chromatic dispersion. 6.5. Optical Frontend 93

−17 log2 MQ = 5 −18 log2 MQ = 6 log2 MQ = 7 −19 log2 MQ = 8 → log2 MQ = 9 ] −20 dB [

/S −21 N

10 −22

−23 10 log

−24

−25 100 200 300 400 500 distance [km] → Figure 6.13: Estimated /S versus distance for various DAC resolution values; investi- N gated system: 50 GHz signal bandwidth with 0.1% relative laser linewidth.

6.5 Optical Frontend

The preceding sections have illuminated the performance limitation due to laser phase noise, clipping and quantization. However, even if costly implementation would help to reduce these impairments, the SNR of the received signal cannot be increased to arbitrarily high values because of fundamental noise contributions at the optical receiver frontend. These noise sources consist mainly of shot noise and thermal noise of the first electrical amplifier stage. Shot noise describes the observation that the energy of the received signal is carried by a finite number of energy quanta (“photons”), eventually leading to fluctuations in the photo detection process. The relative power of shot noise for homodyne receiver architectures is given by [KBW96],[Ho05]

N q B shot = el · , (6.21) S 2RPr where B denotes the two-sided signal bandwidth and qel abbreviates the charge per elec- 19 tron (1.6 10− As). The photo detector’s responsivity is denoted by R. Finally, the · expression is dependent from the power Pr of the received signal. The equation is valid for a large local laser power – compared to the power of the received signal. Furthermore the photo diode’s dark current has been neglected. When, for example, the receiver bandwidth is B = 50 GHz with R = 0.1 W/A, the in- 94 6. Impairments through TX and RX Implementation verse SNR due to shot noise amounts to -34 dB for a received signal power P = 10 dBm. r − Further values for the shot noise limit versus the received optical power are shown in Fig- ure 6.14. This noise power contribution is distinctly smaller than those caused by phase noise, clipping and quantization. A further source of distortion is thermal noise of the transimpedance amplifier (TIA), which is utilized as the first stage of electrical amplification after photo detection. Usually, the noise property of electrical circuits is expressed by a noise figure. In the TIA case usage of the input-referred noise current is advantageous as it is (virtually) located at the same point of the circuit as the shot noise contribution [Säc05]. The corresponding inverse SNR reads N σ2 B TIA n,TIA · = 2 . (6.22) S 4R PrPLO In contrast to the shot noise contribution this term depends from the power of the local laser. Hence, the impact of this noise source can be controlled in the receiver design process and may become sufficiently low for adequately high LO laser power levels. In Figure 6.14 the sum powers of shot noise and TIA thermal noise are shown for two values for the input-referred noise current, i.e., 10 and 20 pA/√Hz. The power of the local laser was chosen to PLO = 10 mW. The diagram makes obvious that the noise power contribution due to TIA thermal noise can be made small and thus the coherent receiver is able to operate quite close to the shot noise limit. Altogether the sum power of noise occurring in the coherent optical receiver’s frontend

−25 shot noise limit σn,TIA = 10 pA/√Hz → σn,TIA = 20 pA/√Hz ]

dB −30 [ S TIA N −35 + S shot N −40 10 10 log −45

−15 −10 −5 0 P [dBm] r → Figure 6.14: Shot noise and TIA thermal noise versus received optical signal power for 10 mW local laser power. 6.6. Achievable Spectral Eﬃciency 95 is distinctly smaller than the contributions of phase noise, clipping and quantization. Therefore, shot noise and thermal noise of transimpedance ampliﬁers shall be neglected in the sequel.

6.6 Achievable Spectral Eﬃciency

The diagram in Figure 6.15 extends the results for achievable spectral efficiencies versus transmission distance of Figure 5.15. It repeats the upperbound values found in [EFWK09] and the maximum spectral efficiency for the linear AWGN fiber channel. As pointed out in Chapter 5, in that scenario the channel is assumed as a concatenation of 80 km SSMF spans. EDFAs with low noise figure compensate for attenuation. The optical powers of the transmit signals are set up to the values found as optimum in simulations which incorporate non-linear fiber effects. The weakly non-linear system approach delivers achievable spectral efficiencies versus transmission distance as depicted by the solid line. The results are obtained by simulation of optical wave propagation with the help of the split-step Fourier method. The influence of clipping, quantization and phase noise on the achievable spectral efficiency becomes evident by translating the results of Figure 6.11 to capacities. The curves for 8 bit resolution (11 dB clipping ratio) are chosen. The laser linewidths are

20 [EFWK09] 18 AWGN channel weakly nl. sys. 16 rLW = 0.1% rLW = 0.2% 14 rLW = 0.5% (6.23) → 12 experiments

8 [bit/s/Hz] Γ 6

0 3 10 2 10 3 7⋅10 distance [km] → Figure 6.15: Maximum achievable spectral eﬃciency versus transmission distance. 96 6. Impairments through TX and RX Implementation

publication Γ distance modulation remarks [bit/s/Hz] [km] [JMST08] 0.8 4160 QPSK OFDM, SSMF, Raman+EDFA, ∆ν 100 kHz ≈ [CRM+08] 2.0 2550 QPSK single carrier, ocean fiber, Raman, DFB lasers [JAAT+09] 4.0 1300 8QAM OFDM, SSMF, Raman+EDFA, ∆ν 100 kHz ≈ [GW09] 4.1 1022 16QAM single carrier, SSMF, Raman+EDFA, ∆ν 100 kHz ≈ [LYCS10] 6.3 990 32QAM OFDM, SSMF, EDFA, ∆ν 100 kHz, TCM ≈ [TAAJ+08] 5.6 640 16QAM OFDM, SSMF, Raman+EDFA, ∆ν 100 kHz ≈ [GWDB09] 6.2 630 16QAM single carrier, SSMF, Raman+EDFA, ∆ν 100 kHz ≈ [ZYH+11] 8.0 320 36QAM single carrier, ULAF, Raman+EDFA, ∆ν 100 kHz ≈ [TAAJ+09] 7.0 240 32QAM OFDM, SSMF, Raman+EDFA, ∆ν: 10 and 100 kHz [NMGF10] 7.2 80 16 and loading of OFDM sub-carriers, 64QAM SSMF, EDFA, ∆ν 100 kHz ≈ Table 6.1: Publications on transmission experiments at high spectral efficiency. set to 0.1, 0.2 and 0.5%. At 1000 km approximately 0.8 bit/s/Hz are lost compared to the ideal TX/RX implementation case. For short distances however, there is a distinct performance limitation as implementation impairments become the dominating sources of distortion. The case of 0.5% linewidth for example, leads to 4.2 bit/s/Hz loss at 100 km with respect to the solid curve, i.e., this loss is attributed to implementation impairments. Compared to the linear AWGN fiber channel, there is even 9 bit/s/Hz degradation. For comparison, the diagram depicts published results on transmission experiments at high spectral efficiency by circles; distance and spectral efficiency data along with the utilized modulation format and some remarks are listed in Table 6.1. The experimental setups are based on quite different prerequisites. Thus, the results are difficult to compare. The authors have used single carrier as well as OFDM systems. Narrow band lasers have been applied, with linewidths not exceeding 100 kHz. However, exact values are not always reported. Partly, Raman pumping is implemented in order to realize low effective noise 6.6. Achievable Spectral Efficiency 97

ﬁgures. Despite of these diﬀerences, the common strategy for evaluating the data points is as follows: For all experiments, the maximum transmission distance is determined, 3 which still guarantees a BER of less than approximately 10− . At this BER, a standard

FEC coding scheme operating at a code rate rc = 0.93 is assumed to provide error-free decoded data [Int03]. This code rate value has to be taken into consideration when the achieved spectral eﬃciency is computed. The experimental results are located in a band from 8bit/s/Hz at 300km to 2bit/s/Hz at 2500km. There is a gap of 4 to 5bit/s/Hz to the determined upper bound which considers implementation impairments. This method is also applicable to the simulated data, which is the basis for Figure 6.11. For that purpose, those distances are determined, which lead to SNR values corresponding 3 to BER = 10− for 4, 8, 16, and 32QAM. The maximum achievable spectral eﬃciency is then evaluated through

Γ = 2 rc ηW log2 M , (6.23) · · · A with ηW denoting the WDM fill factor, i.e., the ratio of signal bandwidth and WDM frequency grid. M abbreviates the symbol constellation size. A The resulting Γ versus distance pairs are plotted as diamonds in Figure 6.15, which show slightly larger capacities than the experiments. This observation is attributed to signaling overhead, e.g., pilot symbols for channel estimation, which are necessary in the actual implementation, but which are not considered in the simulations. In the literature, often a value between 5 to 10 percent is reported [JSM+09]. However, an exact measure required for a given setup is difficult to determine. It depends on many parameters like channel dynamics, the chosen estimation/equalization algorithms, as well as their actual implementation as performance–complexity trade-offs are mandatory. In [JMST08], for example, 6% of the transmitted symbols are known to the receiver and can thus be used for channel estimation purposes. A reduction of the estimated achievable spectral efficiencies by 5 to 10 percent still leaves a gap to the experimental results. In order to reduce this gap, advanced forward error correction techniques have to be applied. A brief outlook on this topic is given in the following section. Experiment [LYCS10] listed in Table 6.1 makes use of trellis coded modulation (TCM) as an inner code. Together with a standard block code, a larger net coding gain can be achieved – at the same time the overall code rate is reduced below 93%. An alternative way to increase the spectral efficiency is given by modifying the channel characteristic, e.g., through the usage of ultra large area fibers (ULAF) as done in [ZYH+11] (c.f. Table 6.1). These fibers exhibit less Kerr effect induced signal distortion and thus, higher optical signal powers can be applied. The overall fiber channel capacity is increased in ULAFs. 98 6. Impairments through TX and RX Implementation

6.7 Outlook: Forward Error Correction

The reported experiments for transmission at high spectral efficiency make use of dual polarization quadrature amplitude modulation utilizing symbol alphabets with 4 to 64 constellation points. The deduced spectral efficiency may be increased through more powerful coding schemes, which can be achieved through larger coding overhead and application of soft-decision decoding. A respective scheme for optical transmission has been proposed in [MSM+10]: It consists of the concatenation of a low-density parity check (LDPC) code (12.94% redundancy) with a standard algebraic block code [Int03]. For the inner LDPC code, soft decision decoding is done with the help of the belief propagation algorithm. It shows very good 2 performance at relatively high pre-FEC BER values (10− ) at the cost of an error floor at higher SNR values. These remaining bit errors are corrected by the outer code. Al- together, at 20.5% redundancy this scheme exhibits a net coding gain of 10.8 dB at a 15 decoded BER = 10− . However, this advanced forward error correction scheme as well as the ones standardized for digital optical transmission work on binary symbols. Their application to non-binary modulation symbol sets requires to map binary code words to symbol sequences. In this context, the question arises what is the best way to carry out this mapping and what are the consequences for the decoder. Different approaches for coded transmission using large signal constellations have been proposed. The idea of trellis coded modulation [Ung82] is to use a convolutional encoder in order to map k˜ binary information symbols to k˜ + 1 code symbols, which then address one constellation point. Thus, the cardinality of the modulation alphabet M has to be A 2k˜+1. The actual mapping of the k˜ + 1 code symbols to the constellation points is found through “set-partitioning”. At the receiver, the Viterbi algorithm is utilized for decoding. The concept of multilevel coding makes use of several binary codes – one for each bit of the binary label of the constellation symbols. At the receiver, at first the lowest bitlevel is decoded. Each subsequent code is decoded individually – for each level, the decisions of the previous stages are taken into account (multi-stage decoder) [HWF98]. An alternative approach does not treat coding and modulation as an entity. Instead, it separates coding from modulation through an interleaving permutation which gave this technique the name bit-interleaved coded modulation (BICM) [iFMC08]. For error protection, a single binary code is used, e.g., a Turbo code or LDPC code, for which there are powerful decoding algorithms at manageable complexity. The encoded binary symbols are fed to an interleaver an then mapped to constellation symbols. At the receiver side, the demapper determines soft-information – either as a-posteriori probabilities or log-likelihood ratios – for each bit of the binary label of the received symbol. After de-interleaving they are given to the decoder. BICM is a promising technique for high bitrate implementation and therefore well 6.7. Outlook: Forward Error Correction 99

Outer zi Encoder Encoder Mapping X[q] Q

Figure 6.16: BICM at the transmitter.

iterative decoding

˜ 1 Outer Y [q] Demapper − Decoder Decoder zˆi Q

Figure 6.17: Receiver: BICM along with iterative decoding. suited for long haul communications close to the channel capacity. The authors of [LASGU11], [LSGU11] apply the distortion model elaborated in this work for optimizing a BICM based coding scheme for optical OFDM transmission. The architecture of the transmitter and receiver is sketched in Figures 6.16 and 6.17. In [LSGU11] for the inner code a LDPC code is designed at a rate of 0.59. Several degrees of freedom are considered for optimization, especially the binary labeling of the complex valued modulation symbols. The constellation of 64 points is designed in such a way that it better reproduces a complex Gaussian source than a square QAM alphabet. Importantly, iterative decoding is applied. For that purpose the output of the LDPC decoder is interleaved and fed back to the demapper. With the help of this information the demapper results can be improved. The simulated OFDM system achieved an overall spectral eﬃciency of 5.9 bit/s/Hz at 2000 km transmission distance. The result considers an outer algebraic code of rate 0.93. It is expected that next generation commercial systems will utilize these advanced FEC coding techniques [Miz08]. Especially soft decision decoding will be applied. Furthermore increased signal processing resources allow for larger coding overhead; accordingly future systems will operate at lower code rates [DS11] in order to reduce the gap to the capacity limit. Such an improved forward error correction performance is already assumed in [LCW+10] in order to evaluate the performance of the investigated system in terms of 100 6. Impairments through TX and RX Implementation

BER, net data rate and net spectral eﬃciency. Results on actual LDPC coded optical OFDM transmission can be found in [YSD11]. 101

7 Compensation of Non-linear Signal Distortion

In the previous chapters, the Kerr effect induced signal distortion is modeled as additive noise-like distortion. The weakly non-linear system approach assumes that there is no correlation between this non-linear distortion and the signal itself. As a consequence, the capacity of the fiber channel is limited. It exhibits a maximum value at a certain optical signal power where the powers of all the investigated sources of distortion add up to a minimum. However, the contribution of self phase modulation is generated through non-linear interaction of the propagating signal with the fiber. This effect is not a random process; there is a well-defined mapping between fiber input and output signals. It consists of a distributed non-linearity with memory, which – in principle – can be inverted. The aim of this chapter is to sketch several approaches for SPM compensation and to point out the interrelation among them. Moreover, the performance in single channel and WDM scenarios is investigated.

7.1 Backpropagation

Backpropagation is a versatile technique for the compensation of non-linear signal distortion. First works considered the method for application at the transmitter side in order to pre-compensate signal distortion in direct detection systems. Since coherent detection schemes provide linear conversion of the optical signal into the electrical domain, backpropagation can also be utilized at the receiver side – for multi-carrier as well as 102 7. Compensation of Non-linear Signal Distortion single-carrier transmission. An elaborate presentation of this compensation method can be found in [Ip10]. The basic idea is, that the description of the propagation of optical signals in silica fibers through the non-linear Schrödinger equation can be inverted. In other words, when the optical signal at the end of a fiber link is known, the corresponding input can be determined by “backpropagating” it through the transmission channel, i.e., by solving the NLSE for negative steps ∂z. Equivalently, the usual NLSE can be solved along with the RX signal as the input – however, after inverting the fiber parameters α, β, and γ. Hence, the modified non-linear Schrödinger equation to be solved then reads

∂u α ∂u j ∂2u 1 ∂3u 2 1 u + β 1 + β 1 + β 1 = jγ u 2 + u 2 u ∂z − 2 1 1,1 ∂t 2 2 ∂t2 6 3 ∂t3 | 1| 3| 2| 1 (7.1) ∂u α ∂u j ∂2u 1 ∂3u 2 2 u + β 2 + β 2 + β 2 = jγ u 2 + u 2 u ∂z − 2 2 1,2 ∂t 2 2 ∂t2 6 3 ∂t3 | 2| 3| 1| 2 for the dual polarization case. Obviously, there are restrictions for this method for realtime applications. For optimum compensation performance the received signal has to be sampled at a high frequency, i.e., several times the original signal bandwidth as non-linear signal distortion causes inter- modulation contributions. Moreover, as with the simulation of signal propagation in silica ﬁbers, the stepsize for the numerical solution of the NLSE is a critical parameter. These limitations exclude backpropagation from implementation in (current) realtime receivers. However, the method shall serve as an estimate for best achievable compensation performance in the following. In practice, less complex – and hence sub-optimum – methods are required.

7.2 Non-linear Phase Rotation

This compensation technique is based on a signal power dependent phase modulation, which is implemented at the transmitter and/or receiver [Low07]. The method is based on solving a simpliﬁed version of the NLSE.

7.2.1 Approach

As a starting point one assumes a dispersion free ﬁber channel. In this case, the NLSE (single polarization version, c.f. (2.34)) reduces to

∂u(t, z) α + u(t, z) = jγ u(t, z) 2u(t, z). (7.2) ∂z 2 − · | | 7.2. Non-linear Phase Rotation 103

For this simpliﬁed propagation equation an analytic solution can be given:

α z jΦNL(t,z) u(t, z) = u(t, 0) e− 2 e (7.3) · · with

αz 1 e− Φ (t, z) = u(t, 0) 2 γ − . (7.4) NL −| | · · α

Leﬀ

Apparently, the inﬂuence of the Kerr eﬀect shows up| as{z power} dependent phase modulation. It can be compensated by phase-shifting the signal by an angle ΦSPMC(t), which is linearly proportional to the instantaneous signal power

Φ (t) P (t). (7.5) SPMC ∝

The constant of proportionality contains the Kerr non-linearity coeﬃcient γ

Φ (t) = γ n L P (t). (7.6) SPMC · s eﬀ ·

Moreover, the fraction in (7.4) is interpreted as an “eﬀective length” and abbreviated by

Leff. If the fiber link consists of multiple spans together with optical amplifiers, ΦSPMC(t) is scaled by the number of fiber spans ns. Non-linear phase rotation is an operation in the time domain and can either be carried out at the OFDM transmitter right after the inverse Fourier transform, or at the receiver before the DFT (Figure 7.1). For this purpose the corresponding discrete time description is required

Popt 2 ΦSPMC[k] = γ ns Leﬀ 2 y[k] . (7.7) · · σy · | |

Practically, the assumption of a dispersion free channel does not hold. The optical signal – disturbed by non-linear phase modulation through the Kerr effect – experiences chromatic dispersion all along the transmission path. The phase modulation according to (7.6) has a compensating effect at the beginning of the link, when it is applied at the transmitter. In the case of post-compensation, one can reduce non-linear signal distortion which originates from signal–waveguide interaction at the final stages of the path [Low07]. The method cannot ideally compensate for SPM, but it can improve the signal quality. For an actual implementation, the constant of proportionality Leff is subject to optimization, while at the same time the parameters γ and Popt have to be known. 104 7. Compensation of Non-linear Signal Distortion

y[k] CP DFT Y [q]

ejφSPMC[k]

Figure 7.1: Non-linear phase rotation at the receiver.

Dual Polarization Description

The respective dual polarization description of the dispersion free, but Kerr eﬀect induced wave propagation is given by two coupled diﬀerential equations

∂u (t, z) α 2 1 + u (t, z) = jγ u (t, z) 2 + u (t, z) 2 u (t, z) ∂z 2 1 − · | 1 | 3| 2 | 1 (7.8) ∂u (t, z) α 2 2 + u (t, z) = jγ u (t, z) 2 + u (t, z) 2 u (t, z). ∂z 2 2 − · | 2 | 3| 1 | 2 In a similar fashion as pointed out above this set of equation is solved by

α z jΦ1,NL(t,z) u1(t, z) = u1(t, 0) e− 2 e · α (7.9) z jΦ ,NL(t,z) u (t, z) = u (t, 0) e− 2 e 2 . 2 2 ·

Now, there are two different phase modulation terms Φ1,NL(t, z), Φ2,NL(t, z) affecting both orthogonally polarized signal contributions. Accordingly, the compensation method based on non-linear phase rotation has to consider two different angles

P 2 Φ [k] = γ n L opt y [k] 2 + y [k] 2 1,SPMC · s eﬀ · σ2 · | 1 | 3 | 2 | y (7.10) P 2 Φ [k] = γ n L opt y [k] 2 + y [k] 2 . 2,SPMC · s eﬀ · σ2 · | 2 | 3 | 1 | y Both depend on the complex amplitude of the signal contributions of both states of polarization [LBT09]. 7.2. Non-linear Phase Rotation 105

7.2.2 Phase Rotation per Span

A better compensation performance is expected from multiple phase rotation stages. In- stead of a single phase correction term, which is proportional to n L , the phase rotation s · eff is carried out considering a single effective length. Next, the linear dispersive effects of a single fiber span is compensated for. Both steps are repeated ns times. The phase rotation operation takes place in the time domain whereas the equalization of the channel is advantageously done in the frequency domain. Obviously, this technique is related with the backpropagation method: One split step operation is implemented per fiber span [Ip10].

7.2.3 Simulation Results

In the following, the effectiveness of the presented SPM compensation methods shall be investigated through simulations. For this purpose the simulation environment used in Chapter 5 is extended by implementations of the backpropagation and phase rotation algorithms. The analysis of the signal distortion is carried out as in Chapter 5. Especially, the assumption of a weakly non-linear system is kept up. At first, system 1 of Table 4.1 is examined. A single OFDM transmitter generates signals of 12.2 GHz bandwidth (no WDM); two to 12 spans (80 km SSMF each) are considered as transmission distances. As pointed out previously, for the non-linear phase rotation method the optimum effective length (c.f. (7.7)) has to be determined. Figure 7.2 summarizes estimated SNR values versus transmission distance. The parameter Leff is varied in the range of 3 to

7 km. The application of non-linear phase rotation yields a SNR gain for Leff values up to 6 km compared to uncompensated transmission. The optimum SNR is reached for Leff = 4 km. For this compensation parameter setting, the SNR gain amounts to approximately 1.9 dB at 1000 km. Shorter distances allow for reduced compensation gains, e.g., 1.5 dB at 300 km. Despite of smaller gains for lower span counts, these results confirm that the proportionality coefficient for ΦSPMC[k] in (7.7) shall comprise the factor ns, i.e., the number of spans.

Moreover, the diagram shows that an inappropriate choice for Leﬀ (larger than 6 km) leads to a signiﬁcant performance deterioration – below the SNR of uncompensated transmission. Analogously, imprecise knowledge of γ or Popt would lead to the same observation. In a similar fashion, parameter optimization is necessary for non-linear phase rotation based on span-wise computation. Corresponding results in terms of estimated SNR values are given in Figure 7.3. The investigated transmission system still operates with a single optical channel of 12.2 GHz bandwidth. For comparison purposes, the SNR measures for the uncompensated case as well as the optimum performance for the previously investigated phase rotation approach are given. 106 7. Compensation of Non-linear Signal Distortion

uncompensated 32 Leff = 3 km L = 4 km 30 eff Leff = 5 km → Leff = 6 km 28 Leff = 7 km [dB] 26 N S/ 24 10

22 10 log

18 100 300 500 700 900 distance [km] → Figure 7.2: Estimated S/ versus transmission distance for various settings of the eﬀective N length parameter; SPM compensation through non-linear phase rotation.

uncompensated 32 phase rotation L = 3 km 30 eff Leff = 5 km → Leff = 7 km 28 Leff = 8 km [dB] 26 N S/ 24 10

22 10 log

18 100 300 500 700 900 distance [km] → Figure 7.3: Estimated S/ versus transmission distance for various settings of the eﬀective N length parameter; SPM compensation through phase rotation stages per span. 7.2. Non-linear Phase Rotation 107

As with the single stage phase rotation method, appropriate values for Leff are in the region of a few kilometers. Now the optimum performance is achieved at effective lengths of 5 to 7 km. For these settings there is a SNR gain of approximately 2.3 dB with respect to the uncompensated case at 1000 km. This measure corresponds to a 0.4 dB improvement compared to the optimum result in Figure 7.2. Like with the previous findings the compensation gains are lower at shorter transmission links. In order to obtain the presented SNR results versus transmission distance, the optimum launch power has to be determined for each link length. Estimated SNR values versus the optical signal power are given in Figure 7.4 for a transmission link length of 12 spans. For uncompensated transmission the maximum SNR is obtained at -4 dBm launch power and amounts to 20.8 dB. As mentioned above non-linear phase rotation allows for 1.9 dB SNR improvement. When phase rotation and linear channel equalization are carried out per span, 2.3 dB SNR gain is observed. The diagram shows that the new optimum operating points are given at -3 dBm. Moreover we find that backpropagation is able to compensate for SPM induced signal distortion to distinctly higher launch powers. The maximum SNR exceeds 30 dB which is reached at +4.5 dBm optical signal power. Thus, backpropagation attains a SNR gain of more than 9 dB compared to uncompensated transmission. There is quite a large gap between the backpropagation performance and the compensation techniques based on non-linear phase rotation. As mentioned previously, the backpropagation results shall be understood as upper bound. There is a large parameter space for complexity reduction, e.g., stepsize of the split-step Fourier method, oversampling factor, and quantization. Therefore, the remaining SNR region is subject to complexity performance trade-offs [Ip10]. Same investigations have been carried out for system 2 (parameters Table 4.2), which operates at 48.7 GHz signal bandwidth. The transmission link length is now reduced to 6 spans, i.e., 480 km. Respective SNR results versus optical signal powers are summarized in Figure 7.5. The presented SPM compensation techniques deliver similar SNR gains like with the previous scenario. 108 7. Compensation of Non-linear Signal Distortion

uncompensated 30 backpropagation rot. per span 28 phase rotation →

26 [dB]

N 24 S/

10 22

10 log 20

−10 −8 −6 −4 −2 0 2 4 6 P [dBm] opt → Figure 7.4: Estimated S/ versus optical signal power for diﬀerent SPM compensation N techniques; 12.5 GHz system, 12 spans.

32 uncompensated 30 backpropagation rot. per span 28 phase rotation → 26

[dB] 24 N

S/ 22 10 20

10 log 18

14 −4 −2 0 2 4 6 8 10 12 P [dBm] opt → Figure 7.5: Estimated S/ versus optical signal power for diﬀerent SPM compensation N techniques; 50 GHz system, 6 spans. 7.2. Non-linear Phase Rotation 109

7.2.4 Frequency Domain Description

The SPM compensation technique based on a single stage of non-linear phase rotation at the receiver (section 7.2.1) makes use of a non-linear operation in the time domain. The respective discrete time implementation in the OFDM receiver reads (c.f. Figure 7.1)

y [k] = y[k] ejΦSPMC[k] y[k] (1 + jΦ [k]) . (7.11) SPMC · ≈ · SPMC

The approximation in the second step is valid for small phase values ΦSPMC[k]. This expression can be transformed to the frequency domain

Y [q] = Y [q] + jY [q] DFT Φ [k] . (7.12) SPMC ⊗ Q{ SPMC }

The discrete Fourier transform of ΦSPMC[k] can be given using (7.7) and symmetry properties of the DFT

Popt ∗ DFTQ ΦSPMC[k] = γ nsLeﬀ 2 (Y [q] Y [ q]) . (7.13) { } · · σy · ⊗ −

As assumed throughout this work, the Kerr non-linearity is modeled as an additive noise- like distortion (weakly non-linear system assumption). Thus, in Chapter 5 we formulated the input-output relation for OFDM sub-carrier symbol as

Y [q] = H[q] X[q] + N[q]. (7.14) · The noise sample N[q] is assumed to comprise a SPM induced contribution. Comparing this expression with (7.12) an estimate for this portion can be given

Popt ∗ NSPM,est[q] = jγ nsLeﬀ 2 Y [q] (Y [q] Y [ q]) . (7.15) − · · σy · ⊗ ⊗ −

In the dual polarization case, these considerations are applied to (7.10) and one obtains

Popt 2 N [q] = jγ n L Y [q] Y [q] Y ∗[ q] + Y [q] Y ∗[ q] 1,SPM,est − · s eﬀ · σ2 · 1 ⊗ 1 ⊗ 1 − 3 2 ⊗ 2 − y (7.16) Popt 2 N [q] = jγ n L Y [q] Y [q] Y ∗[ q] + Y [q] Y ∗[ q] . 2,SPM,est − · s eﬀ · σ2 · 2 ⊗ 2 ⊗ 2 − 3 1 ⊗ 1 − y Using these equations, an estimate for the SPM induced signal distortion can be determined in the frequency domain. The compensation operation itself is then carried out by subtracting these samples from the received sub-carrier symbols Y [q]. Altogether, this method is equivalent with non-linear phase rotation in the time domain for small 110 7. Compensation of Non-linear Signal Distortion

phase values ΦSPMC[k] – simulation results confirmed this assumption. As the convolution operation exhibits high computational complexity, non-linear phase rotation is preferably carried out in the time domain. However, the frequency domain description is useful during the pilot symbol based channel estimation procedure. When transmitted symbols are known at the receiver, (7.14) can be solved for the noise sample N[q]. The correlation with NSPM,est[q] then yields an estimate for Leff, which – up to this point – had to be found by an adaptation process. Finally, the frequency domain description elaborated in this section gives a hint to further potential for optimization, which is offered by decision directed methods.

7.3 Decision Directed Compensation

The basic idea of an enhanced SPM compensation technique is to generate estimates for the SPM induced signal distortion on the basis of received symbols after decision. A block diagram for this approach is given in Figure 7.6. The resulting estimated noise samples have to be fed back before the decision stage and subtracted from the received sub-carrier symbols. By this procedure, the signal quality can be improved in an iterative fashion – given an appropriately low symbol error ratio. This method has the advantage that it potentially delivers more accurate estimates NSPM,est[q] compared to (7.15) where other sources of distortion, especially ASE noise are present and therefore limit the accuracy of the estimate. Based on the actual noise-free symbols one may determine the pure SPM contribution.

The question arises how samples NSPM,est[q] shall be generated out of Xˆ[q]. One possible approach is provided in [WNNS09], [WNNS10], where Volterra ﬁltering is utilized for this step. The authors apply non-linear phase rotation as presented in section 7.2.1 in the initial stage when the symbols Xˆ[q] are detected for the ﬁrst time. After two further loop runs a performance gain of 4 dB with respect to uncompensated

Y [q] y[k] DFT Equ. Decision Xˆ[q]

NSPM,est[q] Estimate SPM Distortion

Figure 7.6: Decision directed SPM compensation. 7.4. SPM Compensation in WDM Scenarios 111 transmission is observed in [WNNS10]. Thus, the compensation performance is improved by 2 dB compared to non-linear phase rotation. As a disadvantage one has to mention the large computational complexity of Volterra filtering. In [WNNS10] several simplifying assumptions on the Volterra kernels are introduced. Moreover, the number of calculation operations is reduced through partially carrying out the computation in the time domain. The difficulty of parameter extraction is another drawback. Altogether one can state that gains in the region between non-linear phase rotation and backpropagation can be achieved by sophisticated SPM compensation methods. However, a more detailed investigation on this topic shall not be pursued in the sequel. Instead more relevant transmission scenarios are to be addressed. Since a high spectral efficiency is desirable over a large wavelength range, it is important to examine the performance of the presented compensation techniques along with wavelength division multiplexing.

7.4 SPM Compensation in WDM Scenarios

System 1 is now extended to 5, 9, and 15 WDM channels operating on a 12.5GHz grid. As assumed in previous chapters, the observed WDM channel is located in the center of the used frequency band. The non-linear phase rotation algorithm is set to Leff = 4 km which provided optimum compensation performance in the single channel scenario as pointed out above (Leff = 5 km for phase rotation per span). Corresponding estimated SNR values versus transmission distance are shown in Figure 7.7. For the cases of 9 and 15 WDM channels, the non-linear phase rotation based compensation methods have no effect. When there are just 5 channels, slight SNR gains remain – approximately 0.1 and 0.2 dB for both compensation techniques. When the same investigations are carried out for system 2 operating with 3 and 5 WDM channels on a 50 GHz grid, we can observe a larger compensation gain. Phase rotation per span delivers approximately 0.8 and 0.3 dB for the 3 and 5 channel case at 1000 km distance (c.f. Figure 7.8). When non-linear phase rotation is applied in a single stage, these gains reduce to about their halves. In order to compare these values with the maximum achievable gains, backpropagation has also been applied for these WDM scenarios. Respective estimated SNR measures are provided in Figure 7.9 and Figure 7.10. Now the results are given in terms of the optical signal power at distances of 12 and 6 fiber spans, respectively. Even backpropagation is not able to enhance the signal quality significantly. There is just about 0.35 dB gain for 5 channels and 0.15 dB along with 9 channels for the 12.5 GHz system. The 50 GHz system performance can be improved by approximately 0.9 and 0.4 dB at 500 km when 3 and 5 WDM channels are present. These results reveal that the compensation of Kerr effect induced signal distortion benefits from large signal bandwidths. For larger bandwidths a larger portion of non-linear signal distortion can be attributed to SPM which then can be effectively reduced. 112 7. Compensation of Non-linear Signal Distortion

28 5 channels 9 channels 26 15 channels rot. per span → 24 phase rotation

[dB] 22 N

S/ 20 10

18 10 log

14 100 300 500 700 900 distance [km] → Figure 7.7: Estimated S/ versus transmission distance for 5, 9, and 15 WDM channels N on a 12.5 GHz grid. Solid lines correspond to uncompensated transmission; dashed and dash dotted lines denote non-linear phase rotation in a single stage and per span.

28 3 channels 5 channels rot. per span 26 phase rotation →

24 [dB] N S/ 22 10

10 log 20

18 100 200 300 400 500 distance [km] → Figure 7.8: Estimated S/ versus transmission distance for 3 and 5 WDM channels on N a 50 GHz grid. Solid lines correspond to uncompensated transmission; dashed and dash dotted lines denote non-linear phase rotation in a single stage and per span. 7.4. SPM Compensation in WDM Scenarios 113

28 1 channel 26 5 channels 9 channels 24 backpropagation → uncompensated 22

[dB] 20 N

S/ 18 10 16

10 log 14

10 −8 −6 −4 −2 0 P [dBm] opt → Figure 7.9: Estimated S/ versus optical signal power for 1, 5, and 9 WDM channels on N a 12.5 GHz grid over 12 spans; uncompensated versus backpropagation transmission performance.

28 1 channel 3 channels 26 5 channels backpropagation → 24 uncompensated

[dB] 22 N

S/ 20 10

18 10 log

14 −4 −2 0 2 4 6 P [dBm] opt → Figure 7.10: Estimated S/ versus optical signal power for 1, 3, and 5 WDM channels N on a 50 GHz grid over 6 spans; uncompensated versus backpropagation transmission performance. 114 7. Compensation of Non-linear Signal Distortion

Separation of Noise Power Contributions

The effectiveness of the SPM compensation techniques can be illustrated with the help of the noise power separation approach presented in Chapter 5. This is done by estimating the relative noise powers at three different channel power settings (c.f. section 5.3). From these measurements one can deduce the individual contributions of ASE-noise, SPM and non-linear crosstalk from adjacent WDM channels. The results summarized in Figure 7.11 and Figure 7.12 correspond to the 12.5 GHz system with 5 and 9 channels. The optical signal power was set to -6 dBm which is the optimum launch power setting for both scenarios. The diagrams indicate that non-linear phase rotation is actually able to reduce the SPM induced signal distortion. At the same time the ASE-noise and the XPM contribution are not affected. Therefore, the overall performance is hardly improved as the total signal distortion is dominated by ASE-noise and non-linear crosstalk.

Discussion

As a summary, one can state that for relevant transmission scenarios, i.e., for dense wavelength division multiplexing, distortion contributions which are caused by SPM play a minor role. Thus, the discussed compensation techniques provide just negligible SNR gains and therefore cannot increase achievable spectral efficiencies for these scenarios. Hence, in this context Figure 6.15 of Chapter 6 is still relevant with respect to capacity limitations. As already pointed out, optimization of FEC coding schemes may provide some reduction of the gap to the channel capacity. But the spectral efficiency cannot be increased beyond that point for the investigated fiber channel setups. If higher spectral efficiencies are desired, new technologies are necessary, which primarily requires to migrate from standard single mode fibers along with lumped optical amplification to new optical waveguide techniques. One potential fiber type is mentioned in [EZC10]. Hollow-core photonic crystal fiber exhibit a very low non-linear coefficient γ and therefore allow for distinctly higher optical signal powers for the optimum operating point where optical amplifier noise and non-linear signal distortion are balanced. Therefore, these fibers are supposed to facilitate spectral efficiency measures greater than 20bit/s/Hz for 2000km transmission distance [EZC10]. An alternative approach consists of introducing spatial dimensions by utilizing multi- core, multi-mode, or few-mode fibers [WF11],[BAHS10],[RRG+11]. MIMO signal processing at the transmitter and receiver side handles the unavoidable crosstalk between coupled signals. In this context, one may learn from wireless transmission where MIMO signal processing is utilized for quite some time. However, the techniques have to be adapted to the special properties of fiber optical transmission scenarios. Most importantly, these include distributed noise generation and non-linearities as well as implementation constraints at the desired bitrates. 7.4. SPM Compensation in WDM Scenarios 115

−20 NASE/S NXPM/S NSPM/S −25 uncompensated → rot. per span [dB]

/S −30 N 10

−35 10 log

−40 300 500 700 900 distance [km] → Figure 7.11: Estimated noise power contributions versus distance for 5 WDM channels on a 12.5 GHz grid; uncompensated transmission versus phase rotation per span.

−15 NASE/S NXPM/S NSPM/S −20 uncompensated → rot. per span

[dB] −25 /S N −30 10

10 log −35

−40 300 500 700 900 distance [km] → Figure 7.12: Estimated noise power contributions versus distance for 9 WDM channels on a 12.5 GHz grid; uncompensated transmission versus phase rotation per span. 116 7. Compensation of Non-linear Signal Distortion 117

8 Summary and Conclusions

The latest evolution in high bitrate optical data transmission brought coherent detection along with powerful signal processing. There are many concepts known from wireline and wireless communications becoming applicable for these setups. The aim of this work is to investigate the suitability of orthogonal frequency-division multiplexing for optical transmission scenarios. In this context, dual polarization transmission at high spectral efficiency is of special interest. The fiber channel exhibits properties which differ from wireless and wireline scenarios and thus have to be investigated thoroughly for OFDM transmission: Dispersive effects, polarization dependent loss, and the Kerr non-linearity. Due to the required high bitrates there are restrictions with respect to implementation impairments. As a major result this work elaborates a method for estimating the maximum achievable spectral efficiency while considering these fiber channel effects and also impairments imposed by non-ideal transmitter and receiver implementation. From another perspec- tive, it helps to deduce requirements for crucial components in order to not severely limit the achievable transmission performance. The most relevant results which have emerged during the investigation of optical OFDM transmission in this work can be summarized as follows:

Chromatic dispersion is the dominant source of linear signal distortion in long un- • compensated links. In order to choose appropriate OFDM system parameters the related impulse broadening has to be evaluated. The duration of the cyclic extension must not be smaller than this value and – at the same time – should not exceed a certain percentage of the overall symbol duration in order to keep the overhead 118 8. Summary and Conclusions

low. A design rule is developed which shows the link between the required number of sub-carriers, the signal bandwidth and the dispersion parameter. A proper system design then allows for penalty-free compensation of chromatic dispersion and PMD. In contrast, PDL may deteriorate the transmission performance. The impairment of single PDL elements and multiple concatenated PDL elements is investigated through a statistical analysis of SNR penalties with respect to BER and channel capacity. Lower penalties are observed in the latter case which gives rise for the investigation of polarization-time coding schemes in order to mitigate the BER penalties.

As a very important aspect, the influence of the Kerr effect on the signal quality • is examined. Throughout the work the fiber channel is assumed to be a weakly non-linear system. Thus, the non-linear signal distortion is treated as a noise-like additive contribution. In this context, extensive investigations by means of Monte Carlo simulations are conducted for the characterization of the noise variances for different OFDM system configurations. The main finding can be summarized in the slopes of the noise powers versus the optical signal powers. For low signal powers, i.e., in the linear regime, the observed relative noise power decays by 1 dB per 1 dB signal power raise whereas at high power levels it increases by 2 dB per 1 dB launch power raise. From this result we deduce a separation method yielding estimates for noise power contributions of ASE noise, distortion due to SPM and non-linear crosstalk between WDM channels. This method may be of interest with respect to performance monitoring as it facilitates the control of operating conditions of reconfigurable transparent networks. Through coordinated launch power variation one can find out which kind of signal distortion limits the performance of each WDM channel.

Moreover, the presented characterization of the Kerr effect induced signal distortion • allows for estimating the capacity of the fiber channel. There are distinct operating points where the maximum capacity is achieved. A reduction of the signal power leads to a stronger influence of ASE noise whereas at higher launch powers non-linear signal distortion reduces the capacity. By this method a value of 10.5 bit/s/Hz is found for dual polarization transmission over 1000 km standard single-mode fiber.

The introduced system model is extended by incorporating implementation impair- • ments like laser phase noise and clipping/quantization at DACs and ADCs. When characterizing the simulated system, their inﬂuence show up as additional noise variance summands. Laser phase noise is identiﬁed as the most relevant source of performance limitation. The ratio of the laser linewidth and sub-carrier bandwidth is the crucial parameter 8. Summary and Conclusions 119

in a further design rule. It delivers a strict requirement for OFDM systems operating at large signal bandwidths in scenarios with large amounts of chromatic dispersion. We ﬁnd that the laser linewidth has to be in the range of a few tens of kilohertz in order to keep the phase noise induced distortion suﬃciently low.

The determined noise variances translate to estimated achievable spectral eﬃciencies • for realistic setups, i.e., including major limiting eﬀects which stem from non-ideal implementation. When this upper bound is compared with published experimental results, a gap of approximately 4 bit/s/Hz at 1000 km is observed which can be reduced by enhanced FEC coding schemes, especially with the help of soft decision decoding.

Finally, the performance of SPM compensation for optical OFDM transmission is • investigated. For this purpose several published methods are reviewed and their interrelation is elaborated. The presented techniques exhibit diﬀerent compensation performance in terms of SNR gains in single channel scenarios. However, these gains vanish for WDM setups as the overall signal distortion is dominated by non-linear crosstalk and ASE noise for optimum power settings in the multi-channel case. This result is conﬁrmed by the application of the presented noise variance separation method.

As the SPM compensation techniques are not able to increase the overall achievable • spectral efficiency, the above mentioned bound is still valid. When higher values for given distances are desired, fundamental changes in the infrastructure are required, especially, enhanced optical waveguide technology has to be introduced. Accord- ingly, as the main requirement for new fibers low Kerr effect induced non-linear distortion has to be named. For such a fiber channel, higher signal powers can be chosen leading to a higher overall SNR and finally to a larger achievable spectral efficiency. 120 8. Summary and Conclusions 121

A Abbreviations and Symbols

A.1 Abbreviations

Acronym Description ADC analog-to-digital converter ASE amplified spontaneous emission AWGN additive white Gaussian noise BER bit error ratio BICM bit-interleaved coded modulation CD chromatic dispersion CP cyclic prefix DAC digital-to-analog converter DFB distributed feedback (laser) DFT discrete Fourier transform DGD differential group delay DSL digital subscriber line DVB digital video broadcasting E/O electro-optical conversion ECL external cavity laser EDFA erbium doped fiber amplifier FEC forwarderrorcorrection FWM four wave mixing GVD group velocity dispersion 122 A. Abbreviations and Symbols

Acronym Description IDFT inverse discrete Fourier transform ITU-T Internation Telecommunication Union – Telecommunication Stan- dardization Sector LO local oscillator MIMO multiple-input/multiple-output ML maximum likelihood MZM Mach-Zehnder modulator NLSE non-linear Schrödinger equation OFDM orthogonal frequency-division multiplexing P/S parallel-to-serial conversion PAM pulse amplitude modulation PDL polarization dependent loss PDM polarization-division multiplexing PMD polarization mode dispersion PSK phase shift keying PSP principal state of polarization QAM quadrature amplitude modulation RF radio frequency RX receiver S/P serial-to-parallel conversion SINR signal-to-interference-plus-noise power ratio SNR signal-to-noise power ratio SOA semi-conductor optical amplifier SOP stateofpolarization SPM self phase modulation SSMF standard single mode fibre TCM trellis coded modulation TIA transimpedance amplifier TX transmitter ULAF ultra large area fiber VSB vestigial side-band WDM wavelength-division multiplexing WiMAX worldwide interoperability for microwave access XPM cross phase modulation A.3. Symbols and Functions 123

A.2 Mathematical Operations

Operation Description x 2 squared Euclidian vector norm || || AH conjugate transpose of matrix A arg max . argument maximizing subsequent expression { } arg min . argument minimizing subsequent expression { } AT transpose of matrix A

ccdfx(xth) complementary cumulative distribution function of x with respect to

threshold xth

x∗ conjugate of complex number x ∂/∂t partial derivative d/dz derivative DFT . Q-point discrete Fourier transform Q{ } E 2 2 identity matrix 2 × erfc(.) complementary error function exp(.) exponential function E . expectation { } . Fourier transform F{ } f ′(x) derivative of function f log natural logarithm

log10 logarithm with respect to base 10

log2 logarithm with respect to base 2 pdf(x) probability density function Pr a > b probability for event a > b { } cyclic convolution ⊗ A.3 Symbols and Functions

Symbol Description ﬁnite set of symbols A α attenuation coeﬃcient (in Neper per distance)

amax, amin maximum and minimum attenuation

αdB attenuation coeﬃcient (in dB per distance)

Aeﬀ eﬀective area

atot,1, atot,2 total attenuation in polarization 1 and 2 β phase characteristic of ﬁber B signal bandwidth 124 A. Abbreviations and Symbols

Symbol Description

β1, β2,... Taylor series coeﬃcients of ﬁber’s phase characteristic

β1,1, β1,2 phase shift coeﬃcient β1 for both states of polarization

Bref reference bandwidth

Bsc sub-carrier bandwidth c speed of light in vacuum C capacity Golden code Cgolden Silver code Csilver D dispersion coeﬃcient

∆fLO frequency oﬀset between signal carrier and local laser ∆λ wavelength diﬀerence ∆ν laser linewidth

∆Φ1(t), ∆Φ2(t) phase shift in MZM arms ∆τ diﬀerential group delay ∆t impulse broadening time e Euler’s constant e Jones vector

Eb/N0 energy per bit over noise power density

ηC SNR penalty with respect to capacity

ηC,th SNR penalty threshold with respect to capacity

ηp SNR penalty due to phase noise

ηS SNR penalty

ηS,th SNR penalty threshold

ηW WDM ﬁll factor f frequency

fc carrier frequency

FN noise figure γ non-linearity coefficient (Kerr effect) G gain g(t) pulse shaping filter Γ spectral efficiency

Γi[q] spectral eﬃciency per polarization and sub-carrier

γS SNR (without considering phase noise contribution) h Planck’s constant H[q] channel coeﬃcient of sub-carrier q

Hβ2 (f, z) transfer function associated with β2

hβ2 (t, z) impulse response associated with β2

HCD,PMD(f) transfer function associated with CD and PMD A.3. Symbols and Functions 125

Symbol Description

HDGD(f) transfer function of DGD element

H∆z(f) transfer function of ﬁber piece of length ∆z Hˆ [q] estimated sub-carrier coeﬃcient

Hpdl transfer function of PDL element

HZF matrix for linear zero-forcing equalization

Ii quantization interval i I(X,Y ) mutual information j imaginary unit k discrete time variable

km power variation factor λ wavelength L ﬁber length

Leﬀ eﬀective length

λ1, λ2,... wavelengths of WDM carriers

m0, m1, m2 estimated inverse SNR as result of measurement 0, 1, 2

µPDL expectation of PDL

MQ number of quantization intervals

Mx number of symbols N[q] noise sample on sub-carrier q

n2 non-linear refractive index

NASE ASE noise power

nASE(t) ASE noise signal representation

ncl(t) signal distortion due to clipping

Ncl power of clipping noise power of noise and non-linear signal distortion Ni nP number of pilot symbols

Npn power of phase noise induced distortion

Nq power of quantization noise

ns number of spans

Nshot power of shot noise

nsp spontaneous emission factor

NSPM power of signal distortion caused by SPM

NSPM,est[q] estimated SPM distortion on sub-carrier q n(t) noise signal representation

NTIA transimpedance ampliﬁer noise power

ntot total noise in multi-span PDL link

NXPM power of signal distortion caused by XPM OSNR optical signal to noise power ratio 126 A. Abbreviations and Symbols

Symbol Description p proportionality constant p(y x) probability for receiving y given x | Pb bit error probability

Pch channel power φ(t) phase

pi polynomial coeﬃcient

PLO power of local laser signal

Popt optical signal power

Pr power of received signal

ΦNN ASE noise power spectral density per polarization

ΦNL(t, z) Kerr induced phase modulation

ΦSPMC[k] SPM compensating phase modulation

ΦSPMC(t) SPM compensating phase modulation Ψ scaled noise covariance matrix

Ψˆ nn[q] estimated noise power density

Ψnn[q] power spectral density of noise-like signal Q number of sub-carriers q sub-carrier number

Qcp number of cyclic preﬁx samples

qel elementary charge R responsivity R SOP rotation matrix ρ PDL value r(t) receive signal

Rb bitrate

rc code rate

rcl clipping ratio

rcp cyclic preﬁx duration ratio

ros oversampling factor

ρdB polarization dependent loss in dB

ρdB,span PDL per span in dB

rL(t) linear system output signal

rLW laser linewidth to sub-carrier ratio s(t) continous time transmit signal

Si signal power

sI(t) inphase component cl sI (t) clipped inphase component 2 σn noise variance A.3. Symbols and Functions 127

Symbol Description 2 σncl variance of ncl(t) 2 σnTIA variance of input-referred TIA noise current 2 σsI inphase signal variance 2 σx variance of signal x sI,max maximum amplitude

SN non-linear system

sQ(t) quadrature component T OFDM symbol duration

T0 overall transfer matrix of PDL aﬀected ﬁber channel τ time delay

Tcp cyclic preﬁx duration U(f, z) optical signal in frequency domain u(t, z) optical signal

ucw continous wave laser signal amplitude

uLO(t) local laser signal

uˆLO local laser signal amplitude

uVSB(t) signal after vestigial side-band ﬁltering

uWDM(t, z) WDM signal in time domain

v1(t), v2(t) voltages applied to MZM electrodes

vI(t) inphase component applied to MZM

vmax maximum voltage

vmin minimum voltage

vs MZM switching voltage ω angular frequency x[k] transmit signal in discrete time representation X[q] symbol on sub-carrier q xcp[k] transmit signal after cyclic extension xˆ estimated symbol cl xI [k] transmit samples after clipping p xI [k] transmit samples after MZM pre-distortion x(os)[k] oversampled signal in discrete time representation

XP[q] pilot symbol ﬁnite set of symbols X xξ[k] realizations of discrete time signals y[k] receive signal in discrete time representation Y [q] received symbol on sub-carrier q ﬁnite set of symbols Y y˜[k] receive signal in discrete time representation 128 A. Abbreviations and Symbols

Symbol Description Y˜ [q] received symbol after equalization

ySPMC[k] received sample after SPM compensation z distance

z1, z2, z3, z4 symbols before polarization time encoding 129

B Separation of Equivalent Noise Power

In this chapter, the WDM scenario of Chapter 5 is examined one more time. The setup shall consist of W channels; without loss of generality, the observed channel shall be channel w = 1. Within its frequency band, distortion due to non-linear crosstalk is denoted by NXPM. In the following, we assume that this noise power is the result of the sum

NXPM = NXPM,w. (B.1) w=2 X Each summand corresponds to the noise power contribution which is caused by channel w. With the help of this assumption, the separation approach presented in Chapter 5 can be extended to observing non-linear signal distortion which is generated by each singular WDM channel. In total, there are W +1 noise power contributions – accordingly W +1 measurements are necessary. Initially, the total relative noise power at the operating point of interest is estimated N + N + W N m = ASE SPM w=2 XPM,w . (B.2) 0 S P After increasing the input power of the observed channel by 10log10km dB, the measured noise power becomes

N k2 N W N m = ASE + m SPM + w=2 XPM,w . (B.3) 1 k S S S m P 130 B. Separation of Equivalent Noise Power

Now the co-propagating WDM-channels also have to vary their powers – one after the other W NXPM,w w=2 2 NASE NSPM w=6 v kmNXPM,v mv = + + P + , v = 2 . . . W. (B.4) S S S S The underlying assumption of a 2 dB distortion increase per 1 dB signal power raise is kept up. Then, from measurements m0 and mv, one can compute the distortion due to XPM which is caused by channel v

N m m XPM,v = v − 0 , v = 2 . . . W. (B.5) S k2 1 m − The result of (B.5) is inserted in (B.2)

N N W m (W 1)m m = ASE + SPM + w − 0 . (B.6) 0 S S k2 1 − k2 1 w=2 m m X − − Similarly (B.5) along with (B.3) yields

N k2 N W m (W 1)m m = ASE + m SPM + w − 0 . (B.7) 1 k S S k2 1 − k2 1 m w=2 m m X − −

(B.6) - km times (B.7) then reads

N 1 k W (W 1)(k 1)m m k m = 1 k3 SPM + − m m + − m − 0 , (B.8) 0 − m · 1 − m S k2 1 w k2 1 m − w=2 m − X which is the basis for evaluating the SPM contribution

N (W 2 k )m + k (k + 1)m W m SPM = − − m 0 m m 1 − w=2 w . (B.9) S (k + 1)(k3 1) m m − P This expression has a very similar structure like (5.26) in Chapter 5, where distortion due to XPM is treated as one entity.

Finally, with the help of m0 and the results of (B.5) and (B.9) the relative ASE-noise power reads

N N W N ASE = m SPM XPM,w . (B.10) S 0 − S − S w=2 X As a result, the proposed technique allows for determining the dominating source of distortion by coordinated noise power variation. Noteworthy, after completing one cycle B. Separation of Equivalent Noise Power 131 of power variation for all W channels, all noise power contributions can be computed for all channels. The numbering of the observed and co-propagating channels have to be adapted in the presented equations accordingly. This monitoring technique may be useful for controlling distortion in transparent meshed optical networks ([MH09a]). One may think of diﬀerent optimization criteria, e.g., minimizing the SNR of certain links or maximizing the total capacity of the network. In order to introduce this functionality in a meshed network, a synchronization mechanism has to be implemented which guarantees the required power variation taking place in a coordinated fashion. 132 B. Separation of Equivalent Noise Power 133

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