Design and Analysis of Digital Direct-Detection Fiber-Optic Communication Systems Using Volterra Series Approach

Transcription

1 Design and Analysis of Digital Direct-Detection Fiber-Optic Communication Systems Using Volterra Series Approach A Dissertation Presented to the Faculty of the School of Engineering and Applied Science University of Virginia In Partial Fulfillment of the requirements for the Degree of Doctor of Philosophy (Electrical Engineering) by Kumar V. Peddanarappagari October, 1997

2 Abstract Optical fiber communication systems are the most efficient means of handling the heavy data traffic in this information age. Efforts are being made to increase the already phenomenal capacity of these high bandwidth communication systems. Highly coherent optical sources that can generate high power narrow pulses are being developed and fiber amplifiers are making repeaterless transmission over long distances possible. Considerable attention is being paid to the limitations placed on these systems by linear dispersion, fiber nonlinearities, and amplified spontaneous emission (ASE) noise from fiber amplifiers. However, most of the analysis is based on single pulse-propagation experiments and simplified analytical expressions; pulse-to-pulse interactions are generally ignored. The design of these systems is still empirical due to lack of analytical expressions for the output field in terms of input, fiber, and amplifier parameters. This study develops analytical tools to analyze and design systems to reduce the influence of linear dispersion, fiber nonlinearities, and ASE noise in single-user and multi-user systems. The generalized nonlinear Schroedinger (NLS) wave equation may be used to explain the effects of linear dispersion and fiber nonlinearities on the evolution of the complex envelope of the optical field in an optical fiber. The NLS equation is typically solved using numerical (recursive) methods. In this work, a Volterra series based nonlinear transfer function of an optical fiber is derived based on solving the NLS equation in the frequency-domain and retaining only the most significant terms (Volterra kernels) in the resulting transfer function. Single pulse-propagation in single-mode optical fibers is studied and the results are compared to available literature which uses numerical solutions. The linear portion of the above model is then used for the theoretical study of the effects of phase noise on linear dispersion in single-mode fiber-optic communication systems using a directdetection receiver. The effect of coherence time on dispersion, nonlinear interference (pulse-topulse interactions), and intensity noise on the performance of a single-user communication system is studied. A study of the effects of phase uncertainty in the received pulses due to timing jitter in ii

3 modern lasers is also presented. Using the Volterra series transfer function (VSTF), the effects of linear dispersion and fiber nonlinearities on the performance of single-user fiber-optic communication systems are studied; the criterion used is the signal-to-interference ratio (SIR) of intensity of the optical field and an upper bound on the probability of error at the receiver. The model helps us choose input pulse parameters to maximize the SIR (minimize probability of error) and design the complete optical link or design lumped nonlinear equalizers at the receiver to compensate for the effects of linear dispersion and fiber nonlinearities. Using the analytical expressions provided by the VSTF and the upper bound on the probability of error, optimal dispersion parameters for the fiber segments in a fiber amplifier based optical communication system are obtained to minimize the linear dispersion, fiber nonlinearities and ASE noise. The effect of input power levels and amplifier gains on the system performance in several different possible configurations of a point-to-point optical communication system is studied. Finally, useful mathematical expressions for studying the nonlinear effects in WDM systems are derived and possible ways of optimally extracting the information transmitted by different users are discussed. iii

4 Contents 1 Introduction Fiber-optic Communication Systems Optical Transmitter Optical Fiber Linear Attenuation Linear (Chromatic) Dispersion Fiber Nonlinearities Optical Amplifiers Optical Receiver System Analysis and Design Dissertation Organization Signal Degradations within Fiber-optic Systems Degradations Caused by Laser Source Imperfections Degradations Caused by the Optical Fiber Nonlinear Schroedinger Wave Equation Linear Dispersion Nonlinear Phenomenon Optical Amplifiers Photo-detector and Receiver iv

5 2.4 Performance Evaluation Volterra Series Transfer Function (VSTF) Derivation of Volterra Series Transfer Function Numerical Results Potential Applications Two-signal Analysis Nonlinear Equalizer Optimal Input Parameters Summarized Observations Linear Dispersion in Fiber-optic Communications Derivation of SIR for Arbitrary Light Source Completely Coherent Light Completely Incoherent Light Numerical Results Communication System Performance Summarized Observations System Design Transfer Function of the System Receiver Statistics Modified Chernoff Bound (MCB) Design Example Concluding Remarks Conclusions and Future Work Summary v

6 6.2 Future Work A Analysis of WDM Systems Using VSTF method 116 vi

7 List of Figures 1.1 A block diagram of a typical fiber-optic communication system Normalized square deviation of the output field for no Raman effect and no dispersion for different input peak powers. The SSF method from the true solution is shown with lines, the third-order VSTF method from the true solution is shown with, and the fifth-order VSTF method from the true solution is shown with Normalized square deviation of the output field for no Raman effect for different input peak powers the third-order VSTF method from the SSF method, shown with lines, and the fifth-order VSTF method from the SSF method, shown with Magnitude squared of the Fourier transform of output field Normalized square deviation of the output field of third- (shown with lines) and fifthorder VSTF method (shown with ) from SSF method, showing the dependence of NSD on input RMS pulse-width for a length of km Interference-to-signal ratio due to the presence of a pump pulse at different frequencies Plots of output intensity of completely coherent light for different power levels Plot of RMS widths of output pulses, showing the effect of peak input power on dispersion of the input pulse Output SIR as a function of symbol period for different power levels Output SIR as a function of peak pulse power for different symbol periods vii

8 4.1 Output pulse shapes for different levels of coherence of light for input pulses of psec RMS width: (a) GVD dominant case (b) operation at zero-dispersion wavelength, Plot of RMS pulse-widths of output pulses, showing the effect of coherence time on dispersion: (a) GVD dominant case (b) operation at zero-dispersion wavelength, Plots of different waveforms of expected received signal (signal and mean nonlinear interference) and showing the effect of pulse separation on the nonlinear interference: (a) GVD dominant case (b) operation at zero-dispersion wavelength Plot of SIR, SIR, SIR, SIR, and SIR showing the effect of coherence time and timing jitter for a fixed symbol period of s indicate the asymptotic values, psec and a pulse-width of psec, and! #", calculated from formulae in Sections and 4.1.2: (a) GVD dominant case, (b) operation at zero-dispersion wavelength, Plot of SIR (shown with o s), and SIR, showing the effect of coherence time for an input RMS pulse-width of psec and different symbol periods for (a) GVD dominant case (b) operation at zero-dispersion wavelength Plot of the (a) SIR (shown with o s) and SIR, and (b) SIR$ showing the effect of coherence time for a symbol period of psec for different pulse-widths for a GVD dominant case Plot of the (a) optimal input pulse-widths and (b) SIR$ s for optimal pulse-width for different symbol periods for a GVD dominant case Typical communication systems used to demonstrate the design procedure Block Diagram of the cascade of % th fiber amplifier and % th fiber segment Input pulse shape used. The input pulse corresponding to &' is shown with a solid line to stress that this is the bit of interest viii

9 5.4 Upper bound on the probability of error using the optimal dispersion map for different configurations shown in Figure 5.1 as a function of amplifier gain for different input powers The optimal total accumulated dispersion parameter determined by minimizing the MCB for different configurations shown in Figure 5.1 and for different input powers. The lines indicate dispersion parameters for amplifier gains close to the optimal amplifier gains and the indicate dispersion parameters for amplifier gains higher by db per amplifier than the optimal amplifier gains ix

10 List of Tables 3.1 Various components of the output signal due to fiber nonlinearities and the presence of pump pulse x

11 Chapter 1 Introduction The main objectives of this study are to (i) derive analytical expressions for the nonlinear (Volterra series) transfer function of an optical fiber, (ii) analyze the effect of phase noise (or source coherence) on linear dispersion in direct-detection fiber-optic communication systems, (iii) analyze the effects of linear dispersion and fiber nonlinearities in fiber-optic communication systems that use directdetection receiver, (iv) develop a mechanism to choose input pulse parameters to maximize the signal-to-interference ratio (SIR) of light incident on the photo-detector, and (v) design the entire fiber-optic communication link to minimize the effects of linear dispersion, fiber nonlinearities, and amplified spontaneous emission (ASE) noise from the fiber amplifiers. This chapter briefly describes how an optical fiber communication system works and how linear dispersion, fiber nonlinearities, and ASE noise affect its performance. 1.1 Fiber-optic Communication Systems The last decade has witnessed an unprecedented increase in data traffic and one main strategy has emerged to handle this increase. Fiber-optic networks are used to transfer data between remote but fixed switches at very high data rates. Data is then transferred to the nearby destinations by either wireless (possibly mobile) or by wired copper links. The huge bandwidth of the optical fiber is exploited to obtain high data rates between these switches, which are separated by large dis- 1

12 tances. Optical communication systems are becoming more and more complex due to attempts to utilize the enormous bandwidth provided by the optical fiber; optimal system design is of great interest. In spite of the possibility of communicating at rates of Terabits per second (Tbps) using optical fibers, the linear dispersion in the fibers, fiber nonlinearities, ASE noise from optical amplifiers, and the low-speed electronic interfaces at the destinations limit data rates to Gigabits per seconds (Gbps). To exploit this large bandwidth, a large number of channels are multiplexed with an electronic interface for each channel on the same fiber. These channels are either multiplexed in time (time-division multiplexed, TDM), wavelength (wavelength-division multiplexed, WDM), or in signature-sequence (code-division multiplexed, CDM). Since optical links use guided medium, they are expensive to install and maintain; multiplexing numerous channels is an effective approach to minimize the cost. Figure 1.1 shows a block diagram of a typical fiber-optic communication system using erbium doped fiber amplifiers (EDFAs). A (possible) booster amplifier used to increase power transmitted by a low power laser is shown, a possible receiver pre-amplifier used to increase the sensitivity of the receiver, and a few in-line amplifiers used to compensate for the power losses in the optical fiber are also shown. Current commercial fiber-optic communication systems (e.g., transpacific fiber-cable system (TPC-5), fiber-optic loop around the globe (FLAG) [1] system) use semiconductor lasers operating at a wavelength of instantaneous power of m, that can generate a series of - psec pulses of maximum mw. This stream of pulses is on-off modulated by the data to be transmitted (i.e., each bit is transmitted by either sending or not sending a pulse). In WDM system, each user s pulse stream is modulated on a carrier with a different wavelength, and these pulse streams are multiplexed to increase the overall data transfer possible on a single fiber. At the receiver, the pulse stream from each user are separated by optical filtering and individual stream of pulses is photo-detected, (sometimes integrated over the symbol period) and compared with a threshold to determine whether a pulse is present in a given slot or not. 2

13 Laser... EDFA EDFA EDFA Receiver Booster Amplifier Receiver Pre-amplifier Figure 1.1: A block diagram of a typical fiber-optic communication system. Long-distance fiber-optic systems can broadly be categorized into terrestrial long-haul systems and undersea systems. Networking is an important issue for terrestrial systems whereas undersea systems are currently point-to-point links. Terrestrial systems are typically of km in length, operating with optical amplifiers spaced at km. The undersea systems are typically of in length, with the amplifier spacing of about km km. The amplifier spacing is small to keep the accumulated ASE noise from the large number of amplifiers to a minimum. Unfortunately, the information transfer is not perfect. Laser imperfections like phase noise, timing jitter, modal distribution of light from a laser, and nonlinearities introduced by the modulation process affect the performance of a digital communication system. Linear attenuation, dispersion, nonlinearities from the fiber, and ASE noise from optical amplifiers limit the data rate that can be achieved and the number of users that can be placed on a given optical fiber. In addition, detector response time, the statistical nature of the photo-detection process, which introduces shot noise, and thermal noise from the electronic circuits increase the probability of error considerably. 1.2 Optical Transmitter Most of the study of lasers in the recent past has concentrated on the development of highly coherent lasers that can generate pulses as short as psec with coherence times as large as sec. It is hoped that because of the availability of such short pulses, very high data rates can be achieved and using low attenuation and low dispersion fibers (or using fiber amplifiers with dispersion compensation), the performance of the optical fiber communication systems can be increased enormously. However, such short pulses suffer more from linear dispersion and fiber nonlinearities, even when we use these 3

14 low attenuation and low dispersion fibers. For a given fiber length, it has been shown that because of chromatic (linear) dispersion, the received pulses are wider if a short pulse is sent than if a wider pulse is transmitted (as long as the input RMS pulse-width is smaller than psec) [2, 3]. The stream of pulses from the laser is modulated with the data stream to be transmitted. This modulation can be done either internally or externally. An internal modulator is driven by the electric current representing the data stream and the output of the laser is the modulated output. External modulation involves modulating the stream of pulses from the laser with the data stream. Either method of modulation introduces mild nonlinearities (produces chirp), which affects the performance of the system [4, 5]. Typically the laser and the modulator are packaged together to reduce coupling losses and other stray effects. When a laser is on-off modulated with electric pulses representing the data stream, there is a delay between the actual generation of the optical pulse from the time the electric pulse is applied. This delay depends on the optical power build-up in the laser, which depends upon the bias level of the semiconductor laser diode. When the electrical pulse exceeds the bias level of the laser, the optical build-up starts [6]. The quantum component of the intensity noise (is an important consideration in analog communication systems, where it is called relative intensity noise (RIN) [7]) adds to this signal making the actual pulse generation time random, giving rise to random timing jitter. In mode-locked lasers, the timing jitter has a periodic autocorrelation properties owing to the saturation of the laser gain medium. Timing jitter is typically on the order of psec for pulse-widths of about psec [8]. In most studies of optical communications, it is assumed that the lasers generate monochromatic light; however, because of finite cavity length in the laser oscillator, the light produced by a laser consists of numerous distinct modes, each having very small line-width. These spectral lines (modes) are broadened by various mechanisms in the laser as well as the fiber. Homogeneous broadening is caused by the linear dispersion of the laser gain medium. Inhomogeneous broadening is mainly due to (i) Doppler broadening, due to different atoms in the laser medium having different kinetic 4

15 energies and hence having different apparent resonance frequencies as seen by the applied signal, and (ii) lattice broadening, in which various atoms see different frequencies due to their positions in the lattice (local surrounding). These broadening mechanisms increase the line-width of each mode generated, producing phase noise. Modulation with the pulse shape and fiber nonlinearities also broaden the already broadened spectrum, increasing the severity of the linear dispersion and fiber nonlinearities. It is well understood in the literature that the broader the line-width (i.e., the more the phase noise), the worse the dispersion in direct-detection receivers [9, 10]. Alternatively, the phase noise at the laser can be mathematically modeled as randomly occurring spontaneous emission events [6], which cause random changes (in magnitude and sign) in the phase of the electro-magnetic field generated by the laser. Therefore, as time evolves, the phase executes a random walk away from the value it would have had in the absence of spontaneous emission. The phase noise spectrum has two components, one low frequency component that has or characteristic up to around MHz, and a white component (quantum noise) that is associated with quantum fluctuations and is the principal cause of line broadening. Analysis of the much smaller low-frequency component is quite tedious and attention can be restricted to quantum noise only. It has been recognized that phase noise can severely affect the performance of coherent detection of the information bearing signal [11, 12, 13, 14, 15, 16, 17, 18]. Considerable attention has been paid to characterize phase noise in different kinds of lasers [8] and attempts have been made to reduce phase noise in lasers [19]. The effect of coherence time on nonlinear processes like self-phase modulation has been studied in [20]. Saleh [21] considered the effect of phase noise on completely coherent and completely incoherent light when direct-detection receivers are used. Incoherent light (intensity) gets filtered linearly [9], yet with an impulse response that is magnitude squared of the impulse response for the coherent light (field). When coherent light is incident on the photo-detector, the electric current is the sum of the intensity of different pulses taken individually and other crossterms due to pulse-to-pulse interactions. The cross-terms introduce a nonlinear interference term which needs to be included in the error analysis at the receiver. 5

16 1.3 Optical Fiber There are three major challenges faced in long-distance optical communication systems: attenuation, linear dispersion, and fiber nonlinearities. Linear attenuation reduces the power level of the signal below the thermal noise threshold at the receiver increasing the probability of error. Attenuation is typically on the order of db/km, and limits the transmission distance to about km. Fiber amplifiers are an attractive means of compensating for attenuation, since they typically have low noise, high bandwidth, and low cost. Amplifiers gains are in the range of distance of km, db, so for a transmission we require about 10 amplifiers. However, amplified spontaneous emission (ASE) noise (proportional to the amplifier gain) adds to the amplified signal, which accumulates with each amplification stage in the link, thereby degrading the signal-to-noise ratio and increasing the probability of error at the receiver. Linear dispersion spreads the pulses, while fiber nonlinearities introduce phase effects that accumulate along the fiber due to the increased power levels (due to amplification) and large transmission distances. Both these phenomena severely affect the performance of the communication system by introducing inter-symbol interference (ISI) and inter-channel interference (ICI). The typical limits on transmission distances set by linear dispersion and fiber nonlinearities are km and km, respectively. These two deleterious effects can be reduced by three methods: input pulse shaping (chirping, phase encoding, polarization scrambling, etc.), dispersion management (using dispersion compensating fiber), or using fiber gratings. The optical fiber supports various number of modes of the optical wave. These different modes travel at different speeds owing to the difference in their path-lengths in the fiber. This gives rise to what is known as modal dispersion. By adjusting the fiber core radius and core-cladding index difference, we can allow only one mode to propagate. For high-speed long-haul communication systems, consideration can be restricted to these single-mode fibers (SMF), and the modal dispersion can assumed to be zero. There are other random nonlinear problems like polarization mode dispersion 6

17 (PMD) [22, 23] that affect these long-distance transmission systems. Physically, PMD has its origin in the birefringence that is present in any optical fiber. While this birefringence is small in absolute terms in fibers, the corresponding beat length is only about m, far smaller than the dispersive or nonlinear scale lengths which are typically hundreds of kilometers. This large birefringence would be devastating in communication systems but for the fact that the orientation of the birefringence is randomly varying on a length scale that is on the order of m. The rapid variation of the birefringence orientation tends to make the effect of the birefringence average out to zero. The residual effect leads to pulse spreading, referred to as polarization mode dispersion (PMD). PMD is significant only in very long-haul systems (lengths work, since we concentrate on links varying from km); therefore, we ignore this effect in this km. The remaining signal degradations caused by the fiber are attenuation, chromatic (linear) dispersion and deterministic nonlinearities Linear Attenuation Linear attenuation and dispersion are widely studied in fiber-optic systems, as they limit the rate and reliability of information transfer [21, 24, 25, 26]. Attenuation reduces the power levels, thus scaling the signal as the exponent of the length of the fiber. Absorption of an optical wave occur mainly because of material absorption and Rayleigh scattering. Rayleigh scattering is generally described by the absorption coefficient, which is known to obey the relation [20], where - db m /km for silica (depending on the constituents of the fiber core) and is the optical wavelength. Modern optical systems are expected to operate in the low-loss region of the spectrum, m- m, where Rayleigh scattering is less than db/km for silica. Absorption reduces the intensity of light in the fiber, which in turn reduces the magnitude of fiber nonlinearities thus limiting the length over which nonlinear interactions are effective. Fiber amplifiers can be used to restore the signal to its original power levels; however, that increases the effect of fiber nonlinearities considerably. We introduce optical amplifiers in Section

18 1.3.2 Linear (Chromatic) Dispersion Linear dispersion has been recognized as the primary limiting factor on the maximum data rate for single-user optical fiber communication systems [24, 25, 26, 27] that use using optical amplifiers to remove the effects of attenuation. The linear dispersion introduces linear inter-symbol interference (ISI) in single-user and multi-user systems, and fiber nonlinearities introduce nonlinear ISI and inter-channel interference (ICI) in multi-user systems. The second-order dispersion, commonly known as group velocity dispersion (GVD), widens the pulses, thus introducing inter-symbol interference (ISI). GVD is the primary limiting factor on the maximum data rate achievable, as it makes the information retrieval at the receiver difficult for long lengths of fibers. Thus, recent fiber-optic systems operate at the zero-dispersion wavelength ( m for silica), at which GVD is zero, thus avoiding its effects. GVD consists of two distinct components, one due to material dispersion (depends on the material used to fabricate the fiber) and the other due to waveguide dispersion (depends upon core radius and the core-cladding index difference). By controlling the waveguide dispersion, can be shifted to the vicinity of m where the fiber loss is minimum, giving us what are commonly known as dispersion-shifted fibers [20]. Using multiple claddings, waveguide dispersion and material dispersion can be carefully controlled to give dispersion-flattened fibers; these fibers have low dispersion ( large wavelength range - m for use in WDM or CDM application. ps/km nm) over a relatively For wavelengths, GVD is positive and the fiber is said to exhibit normal dispersion, i.e., higher frequency components of an optical pulse travels slower than the lower frequency terms. For, GVD is negative and the fiber is said to exhibit anomalous-dispersion. The residual dispersion in long haul applications is typically compensated by inserting a fiber having GVD with an opposite sign that of the original fiber, so that the phase changes induced by the original fiber are cancelled by the phase changes introduced by the inserted fiber. This methodology is generally called dispersion mapping. 8

19 We have to include higher-order dispersion terms for wavelengths closer than a nanometer to. The third-order dispersion, which is much smaller than GVD, is the primary concern for the present fiber-optic links operating at. The third-order dispersion introduces an oscillatory structure near one of the pulse edges (intensity), i.e., the pulse shape is asymmetric [20, 28]. For positive (negative) third-order dispersion, oscillations appear near the trailing (leading) edge of the pulse. For polarization maintaining single-mode fibers operating at, polarization mode dispersion, modal dispersion, and group velocity dispersion (GVD) can be ignored; third-order dispersion is the major limitation on the data rate. Using single pulse propagation results, Marcuse [28] evaluated the effect of input spectral width (line-width) and the input pulse-width on the output pulse-width (approximating the output pulse shape with a Gaussian shaped pulse). It has been shown that data rates of be achieved over fiber lengths of km when limited only by third-order dispersion [20]. Gbps can Research has been conducted into choosing system parameters like input pulse-width, input chirp, etc. to minimize output RMS pulse-width [3, 28], inter-symbol interference in the electronic domain [24], and probability of error in the electronic domain [29, 30]. All these studies model dispersion properly but does not take into account the effect of laser phase noise or fiber nonlinearities on the performance of the communication system. The electronics and the impulse response of the photo-detector at the receiver limit the data rate at each link (on each channel in the fiber) to about Gbps (pulse separations of psec), and typically voice channels are multiplexed on such a link. These voice channels are typically multiplexed in time, i.e., TDM. Such wide pulses do not suffer much from dispersion and together with the use of fiber amplifiers, allow us to have long spans of the fiber without any significant dispersion or attenuation. Future systems are expected to use - psec input pulses, generated by highly coherent semiconductor lasers for single-user systems and even smaller pulse-widths for multi-user systems like TDM and CDM. For CDM systems, optical preprocessing is required before photo-detection to separate the multiple signals so that the total throughput is not limited by the photo-detector response. Assuming a minimum signature-sequence length of, pulse-widths of 9

20 psec are required to obtain the same data rate in CDM systems as the single-user system; dispersion effects are more severe, which combined with fiber nonlinearities make long fibers impractical. Thus, a realistic study of the effect of linear dispersion and fiber nonlinearities on the system performance is required to harness the potential of these high data rate fiber-optic systems Fiber Nonlinearities It is well recognized that fiber nonlinearities limit the performance of current optical communication systems [20, 31, 32, 33, 34]. Various nonlinear effects such as self-phase modulation (SPM) [35], cross-phase modulation (CPM), stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), and four-wave mixing (FWM) can cause significant cross-talk in multi-user systems such as WDM [31, 33] and CDM [34] systems. Nonlinear effects increase with increase in the power level of the laser and the length of the fiber. In previous single-user optical fiber communication systems using power levels below mw (in a typical single-mode fiber of cross-sectional area m, this power gives an intensity of MW/m, which increases the refractive index of the fiber core by ) and link lengths of km, fiber nonlinearities did not play a significant role in degrading the system performance. Current systems use peak input power levels of less than mw over link lengths exceeding km. WDM systems using solitons as basic pulse shapes anticipate peak power levels of mw per user at each frequency and CDM systems are expected to transmit - times that power at a single carrier frequency. Due to the presence of multiple users in the system, power levels are much higher in multi-user systems; fiber nonlinearities introduce inter-symbol interference (ISI) and interchannel interference (ICI) [36], leading to deterioration in the performance of these systems. The availability of low attenuation (using fiber amplifiers) and low dispersion fibers makes it possible to transmit data over long lengths without requiring reconstruction of the signal. These increased lengths together with higher power levels make the study of fiber nonlinearities useful at this juncture of time, especially for future WDM and CDM systems [33, 34, 37]. 10

21 Fiber nonlinearities, especially self-phase modulation spreads the spectrum of the signals by creating new frequencies. The frequency spread of the signal depends on the rate at which the signal changes; the narrower the pulses, the steeper the edges and the more the time spreading due to higher-order dispersion and higher-order fiber nonlinearities. Higher-order dispersion and fiber nonlinearities have to be included in an accurate model when pulse-widths are smaller than psec in long-distance systems. Current research in optical communications uses root mean-square (RMS) pulse-width, optical and electrical signal-to-interference (SIR), power loss and crosstalk from other users (estimated from single pulse propagation experiments) [31, 33] as measures of degradation. Other effects on pulseshape (e.g., spreading and steepening of the pulse) and the resulting effect on the probability of error of a communication system have not been studied. Due to the high coherence of light from modern lasers, the nonlinear interaction between pulses is significant. Fiber nonlinearities introduce additional pulse-to-pulse interactions that are generally ignored in the current design process. Analytical expressions provide a powerful tool for the study and accurate modeling of these nonlinear (individually and cumulatively) effects. In this dissertation, we present a closed-form nonlinear transfer function that can describe linear dispersion, fiber nonlinearities and the pulse-to-pulse interactions in single-user and multi-user fiber-optic communication system. The generalized nonlinear Schroedinger (NLS) wave equation is commonly used to describe the slowly varying complex envelope of the optical field (valid for pulses with widths as short as fsec) in the fiber. The NLS equation is derived from Maxwell s equations either completely in the time-domain [20, 34] or completely in the frequency-domain [38, 39]. NLS equation has also been derived including the waveguide properties of the fiber [40]. It can explain most of the linear and nonlinear phenomena in an optical fiber. Previous methods of solving the NLS equation, either in the time domain or in the frequency domain, have been recursive and numerical. It is not practical to optimize the system performance using such methods when many variables are considered since the optimization must be performed numerically. The availability of an analytical model of the NLS 11 -

22 equation allows the design of high-performance optical amplifier based long-haul systems, which requires the minimization of linear dispersion, fiber nonlinearities, and ASE noise from amplifiers simultaneously. The NLS equation is generally solved using (recursive) numerical methods such as the Split-step Fourier (SSF) method, finite-difference methods [20] or the Runge-Kutta method [38]. The splitstep Fourier method divides the fiber into small segments and the output of each segment is found numerically using the output of the previous segment as the input. Very small segment lengths are required to get accurate results; therefore, the computational cost becomes prohibitively high for long lengths of fibers and short pulse-widths, which are important for future systems. If we increase the segment lengths to reduce these effects, the accuracy in representing the fiber nonlinearities decreases. Furthermore, the discretization errors can accumulate along the length of the fiber, thus generating erroneous results. Taha et. al. [41] have investigated various recursive solutions for the time-domain NLS equation, and have shown that the SSF method is the most accurate and computationally cheap algorithm available (when the nonlinear portion of the NLS equation is implemented in the time domain). Several variations of the SSF method have been proposed recently, one based on an orthonormal expansion of the output field [42] and another based on a wavelet expansion [43]; however, both of these methods are still recursive, and may suffer from the same problems as the original SSF method. These recursive methods of solving the NLS equation do not give any indication of how to remove the nonlinear effects, especially when dealing with communication applications, where a series of pulses are usually transmitted and nonlinear interaction between the pulses should be considered. A closed-form analytical description of the linear and nonlinear effects is required to optimize communication system performance. An analytical method based on a Volterra series transfer function (VSTF) is presented in this dissertation. 12

23 1.4 Optical Amplifiers Linear attenuation reduces the signal levels as an exponential function of the length of the fiber. Previous optical systems had repeaters every km, with the optical signals photo-detected, converted to electronic signals, and then converted to optical signals using optoelectronic components. With the advent of all optical fiber amplifiers, the need for the slow interface between optics and electronics is bypassed and higher data rates are now possible. The fiber amplifiers have a typical spacing of about - km that restore the signal powers by optical means to the original values. Using fiber amplifiers, link lengths of the order of km are in operation without ever requiring reconstruction of the signal [27]. The invention of the erbium-doped fiber amplifier (EDFA) paved way for the development of high bit rate, all-optical ultra long-distance communication systems [27, 44, 45, 46]. There are three major types of optical amplifiers available for use in optical communication systems: fiber amplifiers (mostly erbium doped, EDFA), semiconductor optical amplifiers (SOA) [47], and Raman amplifiers. Raman amplifiers require pump powers of the order of - W which can not be obtained from current semiconductor lasers and therefore these are not used. SOAs are polarization sensitive, suffer more coupling losses, and introduce more inter-channel interference than the fiber amplifiers. Fiber amplifiers have low insertion loss, high gain, large bandwidth, low noise, and low crosstalk; therefore, they are most commonly used for long-haul applications. We consider only the fiber amplifiers in this work. Although fiber amplifiers provide a good means of compensating for attenuation, they add considerable amount of ASE noise, that accumulates as the number of amplifiers increases. The ASE noise together with the interaction of linear dispersion and fiber nonlinearities (modulation instability) gets amplified thus affecting the receiver statistics. So while reducing the problems introduced by linear attenuation, optical amplifiers introduce (of course less significant) ASE noise problems. Fiber amplifiers use the energy provided by the laser pump to amplify the optical signals pro- 13

24 portional to the pump power and length of the doped fiber used in the amplifier. Power conversion is most efficient for EDFAs when the pump wavelength is close to nm, although recently proposed distributed amplifiers use pump wavelength of nm to keep the pump power loss due to fiber attenuation to a minimum. Amplification efficiencies of db/mw of pump power have been achieved at a pump of power of about mw. The pump power is fed to the amplifiers either in the same direction as the signal (forward-pumping), or opposite direction to the signal (backwardpumping), when higher gains are desired, pump power is fed from both directions using two pump lasers. This has created considerable interest in bi-directional systems, where the huge bandwidth of the fiber can exploited to a fuller extent. The intrinsic gain spectrum (the wavelength range over which the signal experiences significant gain) of pure silica is about nm (full-width at half maximum, FWHM). The gain of alumino-silicate glasses with homogeneous and inhomogeneous broadening increases the amplifier bandwidth to about nm [20]. After the pump and signal are coupled into the erbium doped fiber, there is power conversion from pump to signal. The electrons in the doped fiber transition to a higher energy level after stimulated absorption of pump energy, which return to their original energy states by either stimulated emission or spontaneous emission. Stimulated emission due to the signal power provides the basic amplification mechanism in the EDFA. The spontaneous emission causes the electrons to return to the ground state randomly, thereby adding (incoherent) spontaneous noise to the amplified signal, which gets amplified further by the amplifier gain mechanism, producing what is called amplified spontaneous emission (ASE) noise. Therefore, both the noise power and the amplifier gain are proportional to the pump power and doped fiber length, i.e., the larger the gain, the more noise power is added to the amplified signal. Optimum pump powers and doped fiber lengths can be found to improve the performance, which is quantified by the noise figure that is defined as the the ratio of the signal-to-noise ratios before and after the amplifier. Typical noise figures are on the order of db, and typical doped fiber lengths are m. For input power levels below dbm, the amplifier acts like a linear amplifier; however, for higher signal powers the excited carrier concentration de- 14

25 creases, deteriorating the gain mechanism, thereby reducing the gain. The fiber amplifier is said to be in saturation and acts like a nonlinear device for higher signal powers. 1.5 Optical Receiver The photo-detector is a square-law device that detects the magnitude-squared of the real envelope of the optical field. The photo-detector produces a number of electrons following a Poisson process with rate proportional to the intensity of the incident optical field. These electrons thus generate an electric current proportional to the number of photons incident on the photo-detector. In a more accurate model the impulse response of the photo-detector (due to diode capacitance and carrier transit times) can also be included in the Poisson model, giving what is known as a filtered Poisson model [48]. The current from ASE noise component and the beat terms between the signal and the ASE noise add to the electronic/thermal noise from the electronic circuitry, increasing the noise content of the output current from the photo-detector. There are two methods of extracting the information bearing signal from the received optical field, coherent detection and direct-detection. Coherent detection involves translating the incoming optical wave to an intermediate frequency (IF) by mixing with an optical wave from a local oscillator. The local oscillator output field is added to the incoming signal and then the photo-detector detects the magnitude-squared of the real envelope of the sum signal and then low-pass filters it, thus providing a scaled version of the incoming optical field at the IF (i.e., phase information is retained). Due to the availability of highly coherent short pulse-width sources, considerable attention is being paid to the possibility of coherent receivers [11, 12, 49]. For multi-user systems (called optical frequency division multiplexed (OFDM) or dense WDM) with coherent receivers, the channel spacing can be small of the order of GHz, as isolating the channels at the receiver is relatively simple using a local oscillator. However, the transmitting lasers and local oscillator are required to have a wavelength stability of the order of nm, which although has been achieved in commercial 15

26 systems, is still not very popular. Systems using coherent detection are likely to be more sensitive to the effects of fiber nonlinearities and laser phase noise than the present direct-detection systems. This has created considerable interest in finding ways of reducing the line-width of lasers [19]. Modern lasers generate highly coherent light (i.e., very little phase noise), thus reducing the effect of phase noise on the received signal. Although coherent systems offer many interesting challenges in the area of nonlinear modeling and system design, we have focused our work on direct detection. Direct-detection uses the intensity of the real envelope of the incoming signal to make decisions about the transmitted bits, i.e., the phase information in the optical field is not utilized. Directdetection receivers are very popular because of their low cost and their insensitivity to the state of polarization of the received signal. Since the intensity of the information bearing signal only can be observed and the phase is lost at the detector, it is generally assumed that the phase noise does not affect the performance of a communication system using direct-detection [3, 24, 25, 29, 30]. We show that phase noise does affect the performance of a direct-detection receiver. For multi-user systems using direct-detection systems (called WDM systems), the channel spacing is required to be larger than GHz, so large bandwidth optical amplifiers are required. 1.6 System Analysis and Design Several phenomena limit the transmission performance of long-haul optical transmission systems including noise, dispersion and nonlinearities [50]. There are various methods of reducing the effects of these fiber imperfections. Given the statistical properties of the input signals and the channel description, an optimal receiver can be designed to minimize the deleterious effects of the fiber; however, such a method is not popular or practical for optical communication systems due to limited optical processing capabilities. Unlike the wireless or satellite communication channels, the fiber-optic communication channel itself can be tailored to provide the required performance; the fiber parameters can be determined that provide the optimal performance (of course, within few 16

27 practical limitations). Therefore, the problem reduces to either choosing the input parameters to suit a given channel or choose the channel to suit given input parameters. In this work, we design the system by either choosing the input peak power and input pulse-width to maximize the optical signal-to-interference ratio (SIR) at the receiver or choosing the channel parameters for a given input parameters to minimize the probability of error. We do not consider the optimization of receiver processing and leave that for future work. In the initial long-haul systems using optical amplifiers, it was believed that if we maintain the total losses to be zero and use dispersion shifted fiber (DSF), we can achieve the best performance. However, when the system is operated at the fiber s zero dispersion wavelength, the signals and the amplifier noise (with the wavelengths close to the signal) travel at same velocities. Under these conditions, the signal and the noise waves have long interaction lengths and can mix together. Linear dispersion causes different wavelengths to travel at different group velocities in the SMF. Linear dispersion thus reduces phase matching, or the propagation distance over which closely spaced wavelengths interact. Therefore, we can use dispersion parameters intelligently to reduce the amount of nonlinear interaction in the fiber. Thus, in systems operating over long distances, the nonlinear interaction can be reduced by tailoring the accumulated dispersion so that the phase-matching lengths are short, and the end-to-end dispersion is small. This technique of dispersion mapping has been used in both single channel as well as WDM systems to reduce the nonlinear interaction between signal and noise and different frequencies in WDM systems. Current WDM systems use non dispersion shifted fiber (NDSF) for most of the length, and rely on using short lengths of dispersion compensating fiber (DCF) to get the total dispersion in the link close to zero. In the equi-modular dispersion compensation scheme, there is a segment each of NDSF and DCF between the pairs of amplifiers, thus keeping the total dispersion of each fiber segment between the amplifiers close to zero. When dispersion compensating fiber segments with negative dispersion are used in the optical link, the interaction between dispersion and fiber nonlinearities introduces modulation instability 17

28 (MI), which is a parametric gain process. This parametric process amplifies the ASE noise over a major portion of the spectrum, increasing the already accumulated noise at the receiver. Therefore, the choice of dispersion parameters play a significant role in determining the performance of an optical communication system. Even though there has been increasing interest in developing better dispersion management schemes that minimize modulation instability [51, 52], the optimal dispersion management scheme has not been found, and there are no measures of optimality available. Analytical results for describing MI are available only for a single fiber segment [20], and the effect of having fiber amplifiers on modulation instability is not clear. Volterra series model includes the effect of fiber amplifier parameters on modulation instability and the effects of modulation instability on ASE noise, with statistical description of the output current at the receiver. This is an example of a situation where the analysis is not possible with the SSF method due to lack of analytical expressions for the output field including all these effects; the VSTF method can excel as an excellent design tool in this case. For systems that have already been installed, fiber gratings are a very efficient means of compensating for dispersion [53]. They are compact, passive and relatively simple to fabricate. With the commercially available cm long phase masks, the bandwidth over which these gratings can compensate for dispersion is increasing rapidly. Although it is possible to model and design the optimal fiber gratings using the VSTF approach, we do not investigate this topic in this dissertation. The most logical performance measure in the design of digital optical communication systems is the probability of error. Unfortunately, analytical expressions for the probability of error are intractable. Most of the performance evaluation methods in optical communication systems rely on simulations, eye-diagrams and receiver. A majority of researchers rely on the receiver (often determined from eye-diagrams), which is proportional to the signal-to-noise ratio of the photodetector output current when a Gaussian distribution is used for the Poisson counting process (using the central-limit theorem), i.e., it depends only upon the first two moments of output current of the photo-detector. The probability of error predicted using receiver is very conservative [54], and 18