Channel models for optical transmission systems with polarization dependent losses (PDL) Master’s thesis in Master Information and Communication Technology Minoshma Meena DEPARTMENT OF MICROTECHNOLOGY AND NANOSCIENCE CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2025 www.chalmers.se www.chalmers.se Master’s thesis 2025 Channel models for optical transmission systems with polarization dependent losses (PDL) Minoshma Meena Department of Microtechnology and Nano Science Chalmers University of Technology Gothenburg, Sweden 2025 Channel models for optical transmission systems with polarization dependent losses (PDL) Minoshma Meena © Minoshma Meena, 2025. Supervisor: Magnus Karlsson Examiner: Magnus Karlsson, Department of Microtechnology and Nano Science Master’s Thesis 2025 Department of Microtechnology and Nano Science Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Typeset in LATEX, template by Kyriaki Antoniadou-Plytaria Printed by Chalmers Reproservice Gothenburg, Sweden 2025 iv Channel models for optical transmission systems with polarization dependent losses (PDL) Minoshma Meena Department of Microtechnology and Nanoscience Chalmers University of Technology Abstract Polarization Dependent Loss (PDL) presents a important transmission impairment in co- herent optical communication systems, originating from components such as wavelength selective switches and reconfigurable optical add-drop multiplexers. This thesis develops advanced channel models to characterize the accumulation of PDL across multi-span op- tical systems, addressing signal propagation, noise interactions, and capacity constraints. Utilizing Jones matrix formulations and singular value decomposition, the study models individual PDL elements and extends the approach to multi-span configurations through recursive signal evolution, incorporating additive white Gaussian noise (AWGN) from amplifiers. Statistical analysis reveals sub-linear PDL growth following Maxwellian distributions, confirmed via Monte Carlo simulations. Noise properties exhibit PDL-induced anisotropy in covariance matrices, quantified by eigenvalue ratios, underscoring performance degra- dation in long-haul networks. Capacity limits for Gaussian signals indicate losses in high-PDL scenarios. An adapted capacity-achieving scheme, inspired by recent advancements, employs a universal precoder with Linear Minimum Mean Square Error Successive Interference Cancellation (LMMSE-SIC) to transform channels into scalar AWGN subchannels, reducing signal-to-noise ratio (SNR) penalties. Simulations demonstrate enhanced performance compared to standard multiple-input multiple-output (MIMO) approaches, with notable reductions in outage losses. These methodologies provide insights for enhanced system design for reliable long-haul communication. Future investigations could explore nonlinear effects and insertion loss variations, to enable more competitive optical networks in the future. v Acknowledgements First and foremost, I would like to express my heartfelt gratitude to Professor Magnus Karlsson for welcoming me into the Photonics section and for guiding me patiently throughout my entire master’s journey. His readiness to listen and provide clear direc- tion whenever I sought clarification has been invaluable from the very beginning to the completion of this thesis. I am grateful to Vijay Shekhawat and Ruwan Udayanga for their practical help whenever it was needed—whether it was arranging prints, assisting with plagiarism checks, or simply making day-to-day tasks easier. Their support has been a quiet but significant part of this work. I am deeply thankful to Geo Philip Muppathiyil , my senior, for being an exceptional mentor and constant source of encouragement. His insights, practical advice, and willingness to share knowledge generously have greatly shaped my understanding and approach to research. To all my friends in Gothenburg: thank you for turning Sweden into a true home for me. Your companionship and warmth have filled this chapter of my life with happiness and belonging. I also wish to acknowledge all my former teachers, whose encouragement over the years inspired me to keep learning and growing. A special thanks goes to Lena Som- marström, my study advisor, whose steady motivation and timely guidance always lifted my spirits when challenges arose. Finally, my deepest love and gratitude go to my mother and father. Their unwavering belief in me and endless support have been the foundation of everything I have achieved. I would also like to acknowledge the assistance of AI tools such as Grok and Grammarly, which helped refine sentence framing and improve the clarity of my writing throughout this thesis. Minoshma Meena, Gothenburg, Aug 2025 vi List of Acronyms Below is the list of acronyms that have been used throughout this thesis listed in alphabetical order: ASE Amplified Spontaneous Emission AWGN Additive White Gaussian Noise BER Bit Error Rate CN Circularly Symmetric Complex Gaussian DWDM Dense Wavelength Division Multiplexing EDFA Erbium-Doped Fiber Amplifier EIG Eigenvalue Decomposition LMMSE Linear Minimum Mean Square Error MIMO Multiple-Input Multiple-Output PDL Polarization Dependent Loss QAM Quadrature Amplitude Modulation QPSK Quadrature Phase Shift Keying ROADM Reconfigurable Optical Add-Drop Multiplexer SIC Successive Interference Cancellation SNR Signal-to-Noise Ratio SOP State-of-Polarization SU(2) Special Unitary Group of Degree 2 SVD Singular Value Decomposition WSS Wavelength Selective Switch ROADM Reconfigurable Optical Add-Drop Multiplexer vii Nomenclature Below is the nomenclature of indices, sets, parameters, variables, operators, and matrices used throughout this thesis. Indices i, k Indices for fiber spans or noise terms in a multi-span optical system j Index for matrix products over spans in a multi-span system N Total number of spans in the optical system Sets C2 Set of 2-dimensional complex vectors, representing dual-polarization fields Parameters A Per-span polarization-dependent loss (PDL) in decibels γ Gain imbalance parameter, quantifying differential attenuation be- tween principal axes σ2 z Total noise variance, scaled for consistent signal-to-noise ratio (SNR) across spans ρ Signal-to-noise ratio (SNR), defined as Px/Pz ∆ Differential amplification induced by PDL in the noise covariance matrix θ Rotation angle for state-of-polarization transformations viii α Normalization factor for unitary matrix construction; also used in PDL severity for capacity-achieving schemes pout Outage probability for capacity loss analysis Variables X Optical field vector in a dual-polarization system, representing orthogonal polarization amplitudes Y Received signal vector, incorporating noise effects Zi Additive white Gaussian noise (AWGN) for the i-th span, modeled as CN (0, σ2 zI2/N) Ztotal Total accumulated noise across all spans n Rotation axis vector on the Poincaré sphere a, b Complex numbers used to construct unitary matrices for rotations λmax, λmin Maximum and minimum eigenvalues of the noise covariance matrix, representing principal noise gains σ1, σ2 Singular values from singular value decomposition for PDL compu- tation e1, e2 Eigenvalues used for eigenvalue-based PDL computation z Accumulated PDL contribution parameter, proportional to span count Λ Effective polarization-dependent loss in decibels ⟨Λ2⟩ Mean-square PDL, capturing the second moment of the PDL distri- bution Px Signal power of the input field vector Pz Total noise power, derived as (K) C Achievable channel capacity (bits/s/Hz) I(X; Y) Mutual information between input X and output Y SNRi Per-dimension signal-to-noise ratio for capacity-achieving schemes Operators and Matrices Hj Jones matrix for the j-th fiber span ix Htotal Total transfer matrix for a multi-span optical system, defined as∏N j=1 Hj U, V Unitary matrices representing state-of-polarization (SOP) rotations Dγ Diagonal matrix modeling differential attenuation in a PDL element V† Hermitian conjugate of matrix V, aligning input state-of- polarization K Conditional noise covariance matrix, E[ZtotalZ† total | {Hj}] Kbefore Noise covariance matrix without PDL, isotropic case Kafter Noise covariance matrix with PDL, anisotropic case Pi Product of channel matrices from span i + 1 to N , defined as∏N j=i+1 Hj σ Pauli matrices, used for state-of-polarization rotations I2 2 × 2 identity matrix W Whitening matrix for capacity-achieving schemes, defined as UDΣ−1/2U† D Σ Diagonal matrix of eigenvalues of the noise covariance matrix K UD Eigenvector matrix of the noise covariance matrix K Heff Effective whitened channel matrix, defined as WHtotal Hr Real-valued representation of the effective channel matrix Ĥ Block-diagonal extension of Hr for two-slot space-time coding G Universal real-valued 8 × 8 precoder matrix for capacity-achieving schemes E Linear minimum mean square error (LMMSE) filter matrix Γ Diagonal matrix of desired gains for LMMSE-SIC scheme F Interference terms matrix for LMMSE-SIC scheme x Contents List of Acronyms vii Nomenclature viii List of Figures xiii List of Tables xv 1 Introduction 1 2 Advanced Channel Modeling of PDL Build-Up in Optical Systems 3 2.1 Modeling a Single PDL Element . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Jones Matrix Formulation . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Modeling Birefringence and SOP Rotations . . . . . . . . . . . . 5 2.1.3 PDL-Free Fiber and Unitary Transformations . . . . . . . . . . 5 2.1.4 Output with Noise . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Multi-Span Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Characterization of Channel Statistics . . . . . . . . . . . . . . . . . . 7 2.3.1 PDL Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . 7 2.3.1.1 Methodological Validation . . . . . . . . . . . . . . . . 8 2.3.2 Maxwellian Distribution Analysis . . . . . . . . . . . . . . . . . 10 3 Noise Characteristics in Multi-Span Links 15 3.1 Modeling the Noise Covariance Matrix . . . . . . . . . . . . . . . . . . 15 3.1.1 Numerical Validation and Simulation Results . . . . . . . . . . . 17 3.1.2 Impact on System Performance and Mitigation Strategies . . . . 18 4 Capacity Limits in PDL-Affected Multi-Span Channels 20 4.1 Fundamental Capacity of a PDL Channel . . . . . . . . . . . . . . . . . 20 4.2 Comparison with Capacity-Achieving Schemes . . . . . . . . . . . . . . 24 5 Discussion and Conclusion 29 Bibliography 31 xi Contents xii List of Figures 1.1 Illustration of PDL’s power imbalance: balanced input emerges attenuated in one polarization mode.[1] . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Depiction of PMD’s temporal pulse separation between x- and y-polarizations.[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 3D block diagram of a single PDL element showing the signal flow from input X(X1, X2) through the PDL element. . . . . . . . . . . . . . . . 4 2.2 n-span channel configuration illustrating distributed PDL elements. . . 6 2.3 Mean PDL vs. number of spans N for per-span PDL values A = {0.2, 0.6, 1.0, 1.2} dB.[10] . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Validation of PDL estimation methods across varying span counts and PDL levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Overlay of PDL Distributions - Deviations from Maxwellian Behavior . 12 2.6 PDL Distribution Evolution with Varying Per-Span PDL at Constant Spans (N = 25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7 PDL Distribution Evolution with Increasing Number of Spans at Fixed Per-Span PDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 Eigenvalue Ratio vs. Per-Span PDL A for N = 25, Showing Transition from Isotropy to Anisotropy with a Notable Change at A ≈ 0.6 dB . . 17 3.2 Scatter plots of ℜ(Z1) vs. ℜ(Z2) for A ∈ {0, 0.2, 0.6, 1.0, 1.5, 2.0, 2.2, 2.4} dB , with 95% confidence ellipses showing the transition from isotropic to anisotropic noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1 Graphical representation of capacity vs. SNR across multiple scenarios, illustrating the impact of PDL and noise variations over 25 spans. . . . 22 4.2 Histogram distributions at SNR = 10 dB & SNR = 20 dB across varying noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 Capacity vs. SNR (N = 25, σz = 0.2683. The scheme (dashed) con- sistently outperforms MIMO (solid) and approaches the no-PDL limit (black dashed), demonstrating effective PDL mitigation across SNR ranges. 27 xiii List of Figures 4.4 Histogram of Capacity for PDL = 1 dB, SNR = 20 dB, σz = 0.2683 (N = 25. The scheme’s distribution (orange) is shifted toward higher values with lower variance (0.0016 vs. MIMO’s 0.0016), closer to the theoretical maximum of 11.34 bits/s/Hz. . . . . . . . . . . . . . . . . . 28 4.5 Capacity Loss Cout − C0 vs. Mean PDL (σz = 0.2236. The scheme (dashed) exhibits lower losses than MIMO across outage probabilities, aligning closely with theoretical compound bounds (dot-dashed). . . . . 28 xiv List of Tables 2.1 Mean PDL (dB) for Various A and N Values from 10,000 Realizations . 8 2.2 Mean PDL Values from Simulations for N = 25 Spans . . . . . . . . . . 11 3.1 Mean PDL and Noise Covariance Eigenvalue Ratios for N = 25 Spans (10,000 Realizations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1 Average Capacity at SNR = 10 dB for N = 25 Spans (bits/s/Hz) . . . 21 4.2 Average Capacity at SNR = 20 dB for N = 25 Spans (bits/s/Hz) . . . 21 4.3 Average Capacity at SNR = 20 dB for N = 25 Spans (bits/s/Hz). “MIMO” uses the standard capacity C = log2 det ( I2 + ρ 2HtotalH† totalK−1 ) , (0.1) while “Paper” incorporates the encoding scheme from [17] utilizing C = 1 2 8∑ i=1 1 2 log2 ( 1 + SNRi ) . (0.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 xv List of Tables xvi 1 Introduction In the era of exploding data demands, coherent optical communication systems stand as the backbone of global networks, powering everything from cloud computing to real- time video streaming. These systems leverage dual-polarization transmission over dense wavelength-division multiplexing (DWDM) fibers to squeeze terabits per second into thin spectral slices. Yet, as links stretch across continents—spanning thousands of kilometers with dozens of amplifiers and switches, impairments like polarization-dependent loss (PDL) emerges. PDL arises from components, such as wavelength selective switches and ROADMs, where manufacturing quirks cause light in one polarization to attenuate more than the other. To visualize this, Figure 1.1 illustrates the effect: a balanced input signal enters the PDL element with equal power in both polarizations, but the output shows attenuation in one mode, resulting in a power imbalance that disrupts signal orthogonality. For context, Figure 1.2 contrasts this with polarization-mode dispersion (PMD), depicting how PMD temporally separates pulses between polarizations, a compensable effect that underscores PDL’s unique power-specific challenge[1]. Coupled with fiber birefringence, which randomly scrambles polarization states, PDL doesn’t just attenuate signals; it interacts with additive white Gaussian noise (AWGN) from amplifiers, creating uneven noise amplification that warps constellations and inflates bit error rates. Conventional approaches in the literature typically analyze PDL as a localized effect in individual components or dismiss it for short-reach links, neglecting its cumulative stochastic nature in extended systems. This simplification produces overly optimistic performance estimates, rendering networks susceptible to outages under practical en- vironmental variations. The thesis addresses this limitation by proposing a cohesive framework for PDL in multi-span channels, integrating Jones matrix-based propagation with noise covariance analysis to quantify impairment evolution and capacity degrada- tion. Beginning with -principles modeling of single PDL elements through singular value decomposition, the framework extends to cascaded spans, demonstrating sub-linear accumulation. Here, birefringence-induced SOP rotations decorrelate successive losses, constraining overall growth to approximately A √ N dB for per-span PDL A and span count N . A core contribution lies in elucidating PDL’s influence on noise dynamics: it converts isotropic AWGN into anisotropic distributions, where covariance matrices elongate from 1 1. Introduction circular to elliptical forms, as confirmed by Monte Carlo simulations involving 10,000 realizations per configuration. At an SNR of 20 dB, elevated PDL (2 dB per span) induces capacity losses, a degradation that conventional MIMO equalization fails to fully mitigate. To counteract this, the analysis evaluates capacity-achieving techniques, such as precoding combined with successive interference cancellation, which decouples polarizations and restores near-ideal rates with an additional SNR overhead of only 0.5 dB. This investigation is based on some key assumptions: ASE noise is Gaussian and unaffected by nonlinearities, while CD and PMD are fully compensated via digital signal processing, thereby isolating PDL as the predominant linear impairment. The thesis proceeds across five chapters. Chapter 2 establishes the Jones matrix foundation for single- and multi-span PDL, deriving Maxwellian statistical distributions. Chapter 3 examines noise anisotropy, measuring distortions through eigenvalue ratios. Chapter 4 derives capacity bounds. Figure 1.1: Illustration of PDL’s power imbalance: balanced input emerges attenuated in one polarization mode.[1] Figure 1.2: Depiction of PMD’s temporal pulse separation between x- and y- polarizations.[1] 2 2 Advanced Channel Modeling of PDL Build-Up in Optical Systems This chapter provides a description of concatenated Polarization Dependent Loss (PDL) in optical systems, offering a simple Jones matrix model to track signal changes, multi- span signal propagation with noise interactions, and statistical properties such as mean PDL accumulation and Maxwellian distributions to facilitate precise impairment analysis and mitigation in long-haul networks. Polarization Dependent Loss (PDL) impairs coherent optical communication systems through components, such as wavelength selective switches (WSSs),inline amplifiers and reconfigurable optical add-drop multiplexers (ROADMs)[2] which introduce polarization- dependent signal attenuation. In optical fibers, random polarization rotations, driven by birefringence and environmental perturbations, induce stochastic variations in the signal’s polarization state, which, in multi-span coherent optical systems, couple with PDL to interact with Additive White Gaussian Noise (AWGN), resulting in non-uniform noise amplification and significant time-varying signal degradation across the link. This necessitates advanced channel models for long-haul systems where PDL accumulates. In this context, channel models are mathematical frameworks that characterize signal propagation through multi-span optical links, capturing the cumulative effects of PDL, polarization rotations, and noise interactions to predict system performance. Prior studies often focused on isolated PDL effects in single components or assumed negligible polarization impairments, overlooking the stochastic and cumulative nature of PDL in multi-span systems. Advanced channel models are critical, as they provide a com- prehensive framework to analyze PDL-induced impairments across extended networks, enabling precise system design and effective mitigation strategies for reliable long-haul optical communication. 3 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems 2.1 Modeling a Single PDL Element Figure 2.1: 3D block diagram of a single PDL element showing the signal flow from input X(X1, X2) through the PDL element. 2.1.1 Jones Matrix Formulation The optical amplitude in a dual-polarization system is represented as a 2D complex vector: X = ( X1 X2 ) ∈ C2, (2.1) where X1 and X2 denote the orthogonal polarization amplitudes. A PDL element’s effect is modeled using a 2×2 Jones matrix, expressed via Singular Value Decomposition (SVD): H = UDγV †, (2.2) where U, V ∈ SU(2) are two unitary matrices responsible for lossless State-of-Polarization (SOP) rotations, The matrix V † rotates the input SOP to align with the principal axes of the PDL element, Dγ applies the differential attenuation, and U subsequently rotates the SOP post-attenuation, capturing the birefringence-induced transformations in the optical system. The diagonal matrix Dγ = (√ 1 + γ 0 0 √ 1 − γ ) , γ ∈ [0, 1], (2.3) represents the gain imbalance, with the ratio of maximum to minimum power gains given by Pmax Pmin = 1+γ 1−γ , where γ quantifies the differential attenuation (anisotropic loss) between the principal axes of the PDL element. Special cases include: γ = 0, where Dγ = I (the identity matrix), indicating no gain imbalance and equal power along both principal axes due to the absence of PDL; and γ = 1, where Dγ becomes (√ 2 0 0 0 ) , representing complete attenuation along one axis and maximum imbalance, justified by the physical limit of total polarization-dependent loss. 4 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems 2.1.2 Modeling Birefringence and SOP Rotations Fiber birefringence induces SOP rotations, which can be visualized as rotations on the Poincaré sphere. These rotations are mathematically described by unitary matrices (U, V ∈ SU(2)), where any SU(2) matrix can be expressed as: U = exp ( i θ 2n · σ ) , (2.4) with σ representing the Pauli matrices [3] and n denoting the rotation axis on the Poincaré sphere. The parameter θ determines the angle of rotation. To simulate random SOP rotations, the unitary matrices U and V are generated using complex numbers a = areal + iaimag and b = breal + ibimag, where areal, aimag, breal, bimag ∼ N (0, 1). A normalization factor is computed as α = √ |a|2 + |b|2, and the matrix is constructed as: U = 1 α ( a b −b a ) , (2.5) where a, b are the complex conjugates, ensuring U †U = I. This construction parametri- sizes the special unitary group SU(2) according to the Haar measure[4], providing statistical validity for simulating random SOP rotations. These rotations are essential for modeling sub-linear PDL accumulation in multi-span optical systems, a key focus of this thesis. 2.1.3 PDL-Free Fiber and Unitary Transformations In a lossless fiber (γ = 0), the differential attenuation matrix simplifies to Dγ = I, the identity matrix. Consequently, the Jones matrix becomes: H = UV †, (2.6) which is unitary, as H†H = (V U †)(UV †) = I. (2.7) This unitarity guarantees conservation of optical power, as the SOP undergoes pure rotations without attenuation. 2.1.4 Output with Noise The received signal, incorporating noise, is usually modeled as:(fig:2.1) Y = HX + Z, Z ∼ CN (0, I2), (2.8) where Z represents Additive White Gaussian Noise (AWGN) introduced by Erbium- Doped Fiber Amplifiers (EDFAs). The output power is given by: Y †Y = X†H†HX. (2.9) 5 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems Here, H†H = V D2 γV †. The SVD decomposition uniquely separates the SOP rotations (U, V ) and the loss (Dγ). The parameter γ typically corresponds to differential losses of 0.3–0.6 dB for commercial Wavelength Selective Switches (WSSs)[5], for example, the Santec WSS-100 reports a maximum PDL of 0.53 dB and average 0.27 dB [6][7]. 2.2 Multi-Span Channel Model The optical channel under investigation is characterized by a multi-span configuration, comprising N consecutive fiber spans, each contributing to the cumulative polarization- dependent loss (PDL) and state-of-polarization (SOP) transformations. In this concate- nated setup (see Figure 2.2), the input light X enters the first span, where it undergoes SOP “spin” rotations via U1 and V † 1 , followed by differential attenuation from PDL in Dγ1 , yielding an output that propagates into the fiber. This output is then perturbed by AWGN Z1 from the amplifier, influencing the signal entering the next span. Subse- quent spans repeat this cycle—rotations, PDL effects, and noise addition—cascading impairments where prior noise Zi is further transformed by downstream, amplifying non-uniform degradation across polarizations. The total transfer matrix, denoted Htotal, is defined as the ordered product of individual span matrices: Htotal = 1∏ i=N Hi, (2.10) where Hi represents the Jones matrix for the i-th span. This formulation serves as a pivotal element of the thesis, providing a comprehensive framework to elucidate the aggregated impact of PDL across extended optical networks. The received signal Y evolves through a series of transformations that encapsulate the sequential propagation dynamics across multiple spans, a process critical to the design and optimization of resilient long-haul communication systems. Figure 2.2: n-span channel configuration illustrating distributed PDL elements. The signal evolution can be expressed recursively, reflecting the incremental contributions of each span through a sequential processing formulation[8]: 6 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems • For N = 1: Y = H1X + Z1, • For N = 2: Y = (H1X + Z1)H2 + Z2 = H2H1X + H2Z1 + Z2, • For N = 3: Y = [(H1X +Z1)H2 +Z2]H3 +Z3 = H3H2H1X +H3H2Z1 +H3Z2 +Z3, Generalizing to an arbitrary number of spans N , the received signal is given by: Y = HtotalX + N∑ i=1  N∏ j=i+1 Hj Zi, Zi ∼ CN (0, I2), (2.11) where Zi denotes the additive white Gaussian noise (AWGN) introduced by the amplifier in the i-th span, modeled as a complex circularly symmetric Gaussian random variable with zero mean and identity covariance matrix I2. This recursive formulation represents a significant advancement, capturing the com- pounded effect of PDL and the progressive accumulation of ASE noise across the link. The proposed model distinguishes itself through its dual-component structure: (i) The deterministic term HtotalX, which encodes the concatenated PDL and SOP rotations induced by birefringence across all spans (ii) The stochastic term ∑N i=1 (∏N j=i+1 Hj ) Zi, which quantifies the cascading noise con- tributions. As a result, a network with distributed PDL elements, termed a PDL channel, emu- lates a single PDL component, where an effective Λ integrates the PDL contributions from the individual impaired segments, alongside a correlated noise Z. According to Equation (2.11), the presence of multiple transfer matrices in this distributed setup can be reduced to the analysis of a simplified channel model, which is elaborated in the following section. 2.3 Characterization of Channel Statistics 2.3.1 PDL Statistical Analysis The effective polarization-dependent loss (PDL) is determined by the real eigenvalues λmax and λmin of the Hermitian matrix H† totalHtotal, computed across N spans as per the multi-span channel model. The PDL is expressed as: Λ = 10 log10 ( λmax λmin ) , (2.12) where λmax and λmin represent the maximum and minimum gains along the principal polarization states, respectively, serving as a metric for gain imbalance. This definition ties directly to the gain imbalance parameter γ, defined by[8]: γ = 10A/10 − 1 10A/10 + 1 , (2.13) 7 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems where A denotes the per-span PDL in decibels (dB), reflecting the intrinsic polarization- dependent loss introduced by each optical element within a span, typically ranging from 0.1 to 2 dB in practical systems. For the energy-conserving model, the PDL can also be expressed in terms of γ as: Λ = 10 log10 (√ 1 + γ√ 1 − γ ) (2.14) A comprehensive simulation study was conducted to characterize the PDL distribution. The analysis spanned per-span PDL values A = {0.2, 0.6, 1.0, 1.2} dB, with the number of spans N ranging from 1 to 25, where Hi represents the Jones matrix for the i-th span. The mean PDL is calculated via Monte Carlo simulation: For fixed per-span PDL A (dB) and spans N , generates total Jones matrix Htotal = ∏1 i=N Hi, where each Hi = UiDγV † i (random Haar unitaries Ui, Vi; fixed Dγ from γ = 10A/10−1 10A/10+1). For each realization, compute total PDL as Λ = 10 log10(λmax/λmin) from eigenvalues of Hermitian H† totalHtotal. The mean is the arithmetic average over these Λ values: ⟨Λ⟩ = 1 10,000 ∑Λ(r). This formulation serves as an important element of the thesis, providing a comprehensive understanding of the impact of PDL across extended optical networks(see Table 2.1). Table 2.1: Mean PDL (dB) for Various A and N Values from 10,000 Realizations N A = 0.2 dB A = 0.6 dB A = 1.0 dB A = 1.2 dB 1 0.2000 0.6000 1.0000 1.2000 5 0.4173 1.2453 2.0675 2.5120 10 0.5863 1.7609 2.9301 3.5506 15 0.7128 2.1447 3.6251 4.3352 20 0.8333 2.4929 4.2160 5.0712 25 0.9249 2.8195 4.7449 5.7220 The mean PDL exhibits a sub-linear increase with N , a trend driven by the statistical averaging of random SOP rotations, which reduces the cumulative PDL growth. As depicted in Figure 3.2 this behavior can be qualitatively linked to a square root depen- dence on N modulated by A. However, the random unitary transformations Ui and Vi introduce decorrelation, leading to a mean PDL that scales approximately as A √ N for low N , before saturating due to averaging[9]. For instance, with A = 0.2 dB, the mean PDL rises from 0.2000 dB at N = 1 to 0.9249 dB at N = 25, far below a linear extrapolation of 0.2 × 25 = 5 dB, supporting the √ N influence. This sub-linear growth, noticeable beyond N = 15, highlights the complex dynamics of PDL accumulation, offering critical insights for long-haul optical network design. 2.3.1.1 Methodological Validation The accuracy of PDL estimation was validated by comparing the eigenvalue decomposi- tion (EIG) and singular value decomposition (SVD) methods, which are highly relevant 8 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems Figure 2.3: Mean PDL vs. number of spans N for per-span PDL values A = {0.2, 0.6, 1.0, 1.2} dB.[10] for analyzing PDL concatenation in multi-span systems. The EIG method, based on the eigenvalues e1 and e2 of H† totalHtotal, computes PDL as: ΛdB, EIG = 10 · log10 ( e1 e2 ) , (2.15) while the SVD method, derived from the singular values σ1 ≥ σ2 ≥ 0 of total transfer matrix Htotal ΛdB, SVD = 10 · log10 ( σ1 σ2 )2 = 20 · log10 ( σ1 σ2 ) , (2.16) with equivalence established via e1 = σ2 1 and e2 = σ2 2. A simulation spanning N from 1 to 100 and A from 0 to 2 dB demonstrated identical mean PDL values, underscoring the overlapping behavior of EIG and SVD. This consistency is vital for PDL concatenation studies, as both methods effectively isolate the gain disparity along principal polarization states, enabling precise tracking of accumulation as N increases. In real long-haul optical systems, such as transatlantic or transcontinental links spanning 6,000–12,000 km, the number of spans typically ranges from 60 to 240, with an average span length of 50–100 km facilitated by erbium-doped fiber amplifiers (EDFAs), each span contains 1–5 PDL-inducing elements with 1–2 elements per span being prevalent in modern dense wavelength division multiplexing (DWDM) systems. Extending the thesis model to these realistic scenarios is imperative, as the sub-linear PDL accumulation 9 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems Figure 2.4: Validation of PDL estimation methods across varying span counts and PDL levels. observed (e.g., 0.9249 dB for A = 0.2 dB over 25 spans) suggests that PDL impairments may become significant beyond 15–20 spans, potentially reaching 2–5 dB in 80–120 span systems. Addressing PDL impairments is crucial because they degrade signal quality, increase bit error rates, and necessitate advanced compensation techniques, such as polarization tracking or adaptive equalization, to maintain performance in high-capacity DWDM networks, thereby underscoring the practical relevance of this research. 2.3.2 Maxwellian Distribution Analysis Random state-of-polarization rotations in multi-span optical systems, modeled by Jones matrices with random unitary transformations, produce a Maxwellian PDL distribution in decibels [11, 12]. This arises as the random unitary matrices ensure independent polarization transformations, mimicking a three-dimensional Gaussian random walk in Stokes space, where the PDL magnitude Λ(z) follows a chi distribution with three degrees of freedom (equivalent to Maxwellian, as Λ2(z) is chi-squared), due to the correspondence between Jones matrix singular values and Stokes space power gains, enabling efficient numerical simulations of the distribution. Comparing simulated and theoretical Maxwellian distributions of PDL is essential for validating the stochastic model and assessing system performance limits, as the theoretical framework, rooted in a random walk model of PDL accumulation [11], serves as an ideal benchmark. The distribution is mathematically represented by the probability density function: f(Λ; z) = √ 2 π Λ2 [σ(z)]3 exp ( − Λ2 2[σ(z)]2 ) , (2.17) 10 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems Table 2.2: Mean PDL Values from Simulations for N = 25 Spans A (dB) Mean PDL (dB) A √ N (dB) 0.4 1.85 2.0 0.6 2.78 3.0 0.8 3.73 4.0 1.0 4.69 5.0 1.6 7.72 8.0 2.4 12.16 12.0 where Λ(z) denotes the PDL magnitude in linear units at link position z, σ2(z) is the per-component scale variance quantifying the Gaussian spread in Stokes space (tied to the mean-square PDL), and the prefactor normalizes the distribution while the exponential governs the decay of probability for large Λ, reflecting the chi-squared tail. The scale σ2(z) is defined as: σ2(z) = ⟨Λ2(z)⟩ 3 , (2.18) where ⟨Λ2(z)⟩ is the mean-square PDL (second moment, capturing average squared deviation from unity gain), with the 1/3 accounting for the three-dimensional random walk. To avoid confusion, note that σ2(z) is not the variance of Λ(z): Var(Λ(z)) = ⟨Λ2(z)⟩ − ⟨Λ(z)⟩2 = σ2(z)(3 − 8/π) ≈ 0.4535σ2(z) ≈ 0.1512⟨Λ2(z)⟩, the reduced spread of the magnitude due to vector summation. The mean PDL ⟨Λ(z)⟩ (average differential transmission) is: ⟨Λ(z)⟩ = √ 8σ2(z) π = √ 8⟨Λ2(z)⟩ 3π ≈ 1.5958 σ(z) ≈ 0.9207 √ ⟨Λ2(z)⟩, (2.19) The mean-square PDL can be further expressed as: ⟨Λ2(z)⟩ = γ2 1 2 ( exp ( 2z γ2 1 ) − 1 ) , (2.20) where γ1 = 20/ ln(10) ≈ 8.686 is the dB-to-linear conversion factor (from ΛdB ≈ γ1Λ for small Λ), and z is the accumulated PDL strength parameter (z ∝ N× mean-square PDL per span). Simulations with 100,000 realizations incorporate practical effects such as finite sampling and numerical errors, revealing non-linear PDL growth for large per-span PDL due to the breakdown of the small-z approximation—where z becomes significant compared to γ2 1 , invalidating linear approximations—as well as statistical fluctuations and potential SOP decorrelation, underscoring the need for adaptive models in high-PDL scenarios. As shown in Figure 2.5, the overlaid simulated and theoretical PDL distributions reveal differences in the high-PDL tails. 11 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems Figure 2.5: Overlay of PDL Distributions - Deviations from Maxwellian Behavior Figure 2.6 illustrates the PDL distribution’s response to increasing per-span PDL (A) from 0.4 to 2.4 dB for 25 spans, showing a transition from sharp, symmetric profiles at low A (e.g., 0.4 dB) to broader, less symmetric shapes with elongated tails at high A (e.g., 2.4 dB), with mean PDL values escalating as per Table 2.2. The overlay in Figure 2.5 integrates these distributions to affirm this evolution, revealing a progressive loss of Maxwellian traits—flattening peaks and extended tails—as mean PDL¢¢ exceeds 12.16 dB, where the simulated mean deviates slightly from the theoretical prediction due to the small-z approximation, which holds for low per-span PDL but fails at high per-span PDL (A = 2.4 dB), causing non-linear PDL growth amplified by statistical fluctuations and numerical errors, suggesting that the model effectively captures moderate PDL levels. The figure 2.7 trace the PDL distribution’s development as the number of spans (N) increases under a fixed per-span PDL (A). At N = 6, the distribution lacks a distinct Maxwellian peak and exhibits misaligned tails, indicating insufficient SOP randomization. With N = 16, a clearer peak emerges alongside improved tail alignment, suggesting a step toward Maxwellian traits. At N = 26, the peak sharpens and tails conform better, reflecting enhanced stochastic averaging. By N = 41, the distribution stabilizes into a Maxwellian-like form, implying that adequate span counts foster robust PDL accumulation. This progression highlights the pivotal role of N in shaping PDL statistics, offering critical guidance for optimizing multi-span optical network designs. 12 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems Figure 2.6: PDL Distribution Evolution with Varying Per-Span PDL at Constant Spans (N = 25) 13 2. Advanced Channel Modeling of PDL Build-Up in Optical Systems Figure 2.7: PDL Distribution Evolution with Increasing Number of Spans at Fixed Per-Span PDL 14 3 Noise Characteristics in Multi-Span Links This chapter delves into the interaction between polarization-dependent loss (PDL) and additive white Gaussian noise (AWGN) in multi-span coherent optical systems, focusing on how PDL alters the noise covariance matrix from isotropic to anisotropic forms. Building upon the PDL accumulation models from previous chapters, using the noise covariance expression, the analysis aims to validates through extensive Monte Carlo simulations quantifying anisotropy via eigenvalue ratios. And discuss system performance impacts with mitigation approaches. This analysis highlights the noise anisotropy and propagation of noise through subsequent spans, offering insights for accurate channel modeling in long-haul networks. 3.1 Modeling the Noise Covariance Matrix The multi-span channel model, as established in Section 2.2, describes the received field Y ∈ C2 as: Y = HtotalX + Ztotal, (3.1) where Htotal = ∏N j=1 Hj , X ∈ C2 is the input dual-polarization field, and Ztotal aggregates the noise: Ztotal = N∑ i=1  N∏ j=i+1 Hj Zi. (3.2) Here, Zi ∼ CN (0, σ2 zI2/N) models the circularly symmetric complex Gaussian noise from the i-th amplifier, scaled to yield a fixed total variance σ2 z independent of N for fair SNR comparisons. The conditional covariance K = E[ZtotalZ† total | {Hj}], given a specific PDL configuration, is obtained by expressing the outer product[13]: ZtotalZ† total =  N∑ i=1  N∏ j=i+1 Hj Zi   N∑ k=1 Z† k  N∏ j=k+1 Hj †  (3.3) Averaging over the noise (each Zi being proper Gaussian with E[ZiZT k ] = 0δik): 15 3. Noise Characteristics in Multi-Span Links E[ZtotalZ† total] = N∑ i=1 N∑ k=1  N∏ j=i+1 Hj E[ZiZ† k]  N∏ j=k+1 Hj † , (3.4) where linearity permits interchanging the sum and expectation. Given the independence and zero-mean nature of the Zi[7]: E[ZiZ† k] = δik · σ2 z N I2, (3.5) off-diagonal terms (i ̸= k) disappear, resulting in: K = σ2 z N N∑ i=1  N∏ j=i+1 Hj  N∏ j=i+1 Hj † = σ2 z N N∑ i=1 PiP† i , (3.6) with Pi = ∏N j=i+1 Hj. This form, a Hermitian and positive-definite operator, illustrates how PDL differentially amplifies noise components. Without PDL, the unitarity of Hj implies PiP† i = I2, hence K = σ2 zI2, maintaining equal variance across polarizations without correlations. The expected covariance matrix before PDL, in this isotropic case, is thus: Kbefore = ( σ2 z 0 0 σ2 z ) , where the diagonal elements are equal, reflecting uniform noise power across the two polarization modes due to the absence of differential amplification. This is justified by the unitarity of Hj , which preserves the norm of the noise vector, ensuring no correlation or variance imbalance. In the presence of PDL, the diagonal PDL matrix (with unequal eigenvalues from the high and low gain axes) creates noise anisotropy: noise power ends up different between the two polarization modes because of random SOP rotations along the link. The covariance matrix K from Eq. 3.6 takes a general 2x2 Hermitian shape: K = ( k11 k12 k∗ 12 k22 ) , where the diagonals k11 and k22 show the variances in each mode (one boosted, one suppressed by PDL), the off-diagonal |k12| captures mixing from rotations, and the total trace k11 + k22 ≥ 2σ2 z reflects extra noise power from amplified early contributions. This setup is exact—PDL’s non-unitary scaling (gains not equal to 1) differentially boosts noise as it travels. Measured anisotropy with the eigenvalue ratio λmax/λmin of K: close to 1 means nearly even noise (isotropic), while bigger numbers (>2) mean strong polarization imbalance, as seen in upcoming simulation results. 16 3. Noise Characteristics in Multi-Span Links 3.1.1 Numerical Validation and Simulation Results To validate the derived noise covariance model and quantify the PDL-induced anisotropy, extensive Monte Carlo simulations were conducted, building on the Jones matrix framework from Section 2.1 and the multi-span model from Section 2.2. The simulation setup mirrors practical long-haul systems with N = 25 spans, each containing one PDL element, and per-span PDL values A ∈ {0, 0.2, 0.6, 1.0, 1.5, 2.0, 2.2, 2.4} dB. Noise samples Zi were drawn from CN (0, σ2 zI2/N) with σz = 0.3, normalized for consistent total variance across spans. For each realization (10,000 total per A), the total noise Ztotal was computed both with PDL (after) and without (before) to isolate PDL effects. The covariance matrices were estimated empirically, and their eigenvalues were used to compute the anisotropy ratio λmaxn/λminn . Figure 3.1: Eigenvalue Ratio vs. Per-Span PDL A for N = 25, Showing Transition from Isotropy to Anisotropy with a Notable Change at A ≈ 0.6 dB Table 3.1 summarizes the key metrics, including mean PDL (consistent with Section 2.3) and eigenvalue ratios. The random number generator was seeded for reproducibility, and minor variations in "before PDL" ratios (around 1.0246 on average) are attributable to finite sampling, confirming near-isotropy in the absence of PDL. The results validate the model: before PDL, ratios remain close to 1 across all A values, confirming the expected isotropy of the noise distribution. After PDL, ratios exhibit a nonlinear increase with A, indicating growing anisotropy driven by cumulative differential amplification across spans. For low A (e.g., 0.2 dB), the ratio is moderate (1.1709), supporting the adequacy of the standard AWGN model; for high A (e.g., 2.4 17 3. Noise Characteristics in Multi-Span Links Table 3.1: Mean PDL and Noise Covariance Eigenvalue Ratios for N = 25 Spans (10,000 Realizations) A (dB) Mean PDL (dB) Eig. Ratio No PDL Eig. ratio with PDL 0.0 0.0000 1.0246 1.0246 0.2 0.9231 1.0249 1.1709 0.6 2.7966 1.0374 1.2934 1.0 4.6855 1.0240 1.4145 1.5 7.1630 1.0047 2.2367 2.0 9.8496 1.0034 2.3346 2.2 10.9488 1.0430 3.8765 2.4 12.1355 1.0236 4.2007 dB), it rises to 4.2007, highlighting the need for an adapted model to ensure precise performance prediction in DWDM network systems. To visualize the combined results, Figure 3.1 presents a comprehensive overview, begin- ning with the eigenvalue ratio plot followed by Figure 3.2 of ℜ(Z1) vs. ℜ(Z2) for all tested A values (before and after PDL) arranged in a grid. The eigenvalue ratio plot illustrates the transition from isotropy to anisotropy, with a significant change becoming evident at A ≈ 0.6 dB, where the ratio begins to deviate notably from 1. The scatter plots use 10,000 subsampled realizations overlaid with 95% confidence ellipses derived from eigenvalue decomposition of the empirical covariance , where the major and minor axes of the ellipses reflect the polarization-dependent noise variance, with the major axis aligning with the dominant polarization mode and the minor axis indicating the suppressed mode due to PDL. This integrated figure highlights the progression from isotropy at low A to pronounced anisotropy at higher A, corroborating the simulation data and model predictions. 3.1.2 Impact on System Performance and Mitigation Strategies The quantified noise anisotropy has direct implications for system performance in coherent optical networks. For instance, elevated eigenvalue ratios degrade the effective signal-to-noise ratio (SNR) by unevenly amplifying noise in one polarization mode, potentially increasing bit error rates (BER) in modulation formats like QPSK or 16- QAM [14]. In high-PDL regimes (A > 1.5 dB), the adapted bivariate AWGN model, incorporating the derived K, is essential for precise capacity estimation, as the standard isotropic model underestimates impairments. 18 3. Noise Characteristics in Multi-Span Links Figure 3.2: Scatter plots of ℜ(Z1) vs. ℜ(Z2) for A ∈ {0, 0.2, 0.6, 1.0, 1.5, 2.0, 2.2, 2.4} dB , with 95% confidence ellipses showing the transition from isotropic to anisotropic noise. 19 4 Capacity Limits in PDL-Affected Multi-Span Channels This chapter delves into the theoretical limits of information transfer in multi-span optical links impacted by polarization-dependent loss (PDL), building on the propagation and covariance models from Chapters 2 and 3. Studying the capacity of Gaussian signals helps assess how PDL and noise variations influence these rates, supported by simulation- based assessments that uncover degradation patterns. Numerical simulations, aligned with the methodology in Section 3.1, provide detailed insights into rate losses across different PDL intensities and noise variances, aiding in the development of achievable capacity rates. The findings reveal the combined influence of PDL accumulation and noise asymmetry in restricting link performance, mirroring the anisotropy linked to PDL buildup noted in the previous chapter, which indicates a significant capacity loss and drives the exploration of robust encoding strategies in subsequent sections. 4.1 Fundamental Capacity of a PDL Channel The achievable rate in a PDL-degraded multi-span channel determines the highest transmission speed possible under power limits, acting as a benchmark to measure performance degradation. Based on the reception model in Equation (3.1), the output vector is Y = HtotalX + Ztotal, where Ztotal exhibits the asymmetric covariance K as defined in Equation (3.6). The rate C is the maximum mutual information I(X; Y) over signal distributions p(X)[13]: C = max p(X) I(X; Y). (4.1) For signals distributed as X ∼ CN (0, Px 2 I2) and covariance K, the rate is given by[15][16] : C = log2 det ( I2 + ρ 2HtotalH† totalK−1 ) , (4.2) where ρ = Px/Pz denotes the SNR, and Pz = trace(K) reflects the total noise power, derived from the accumulated noise contributions Ztotal = ∑N i=1 (∏N j=i+1 Hj ) Zi, with each Zi contributing to the variance structure through the channel products. In the 20 4. Capacity Limits in PDL-Affected Multi-Span Channels absence of PDL (γ = 0), Htotal is unitary and K = I2, yielding the reference dual- polarization AWGN rate: C = 2 log2 ( 1 + ρ 2 ) . (4.3) With PDL, the disparity factor γ adjusts the rate to [13]: C = 2 log2 ( 1 + ρ 2 ) + log2 ( 1 − γ2ρ2 (2 + ρ)2 ) . (4.4) Exact computations at specific SNRs establish benchmarks. At SNR = 10 dB (ρ = 10): C = 2 log2(6) ≈ 5.17 bits/s/Hz. (4.5) At SNR = 20 dB (ρ = 100): C = 2 log2(51) ≈ 11.34 bits/s/Hz. (4.6) Table 4.1: Average Capacity at SNR = 10 dB for N = 25 Spans (bits/s/Hz) A (dB) σz = 0.2236 σz = 0.2683 σz = 0.3130 σz = 0.4025 0.0 4.64 3.84 3.21 2.32 0.5 4.60 3.81 3.19 2.31 1.0 4.50 3.73 3.13 2.29 2.0 4.15 3.52 3.00 2.28 Table 4.2: Average Capacity at SNR = 20 dB for N = 25 Spans (bits/s/Hz) A (dB) σz = 0.2236 σz = 0.2683 σz = 0.3130 σz = 0.4025 0.0 10.72 9.69 8.84 7.48 0.5 10.64 9.62 8.77 7.41 1.0 10.41 9.40 8.56 7.22 2.0 9.53 8.57 7.78 6.56 These values serve as a foundation for assessing the impact of polarization-dependent loss (PDL) on multi-span optical links, with simulations conducted over N = 25 spans using PDL intensities A ∈ {0, 0.5, 1, 2} dB and noise standard deviations σz derived from a base √ Px/ρ ≈ 0.4472 scaled by {0.5, 0.6, 0.7, 0.9} (yielding σz ≈ 0.2236, 0.2683, 0.3130, 0.4025). The results, detailed in Tables 4.1 and 4.2, show a 21 4. Capacity Limits in PDL-Affected Multi-Span Channels consistent decline in capacity as SNR increases, with a notable reduction of approximately 1.19 bits/s/Hz at SNR = 20 dB and σz ≈ 0.2236 when PDL reaches 2 dB per span, attributed to heightened noise disparity. Noise scaling further exacerbates these effects, as higher σz values reduce effective SNR, amplifying PDL penalties; for example, at SNR = 10 dB and A = 1 dB, the rate drops from 4.49 to 2.30 bits/s/Hz as σz increases from 0.2236 to 0.4025, reflecting intensified uneven propagation in the covariance matrix K. Figure 4.1: Graphical representation of capacity vs. SNR across multiple scenarios, illustrating the impact of PDL and noise variations over 25 spans. 22 4. Capacity Limits in PDL-Affected Multi-Span Channels (a) σz = 0.2236, SNR = 10 dB (b) σz = 0.2683, SNR = 10 dB (c) σz = 0.3130, SNR = 10 dB (d) σz = 0.4025, SNR = 10 dB (e) σz = 0.2236, SNR = 20 dB (f) σz = 0.2683, SNR = 20 dB (g) σz = 0.3130, SNR = 20 dB (h) σz = 0.4025, SNR = 20 dB Figure 4.2: Histogram distributions at SNR = 10 dB & SNR = 20 dB across varying noise. 23 4. Capacity Limits in PDL-Affected Multi-Span Channels Rate distributions for PDL = 0.5 dB at SNR = 10 dB and PDL = 1 dB at SNR = 20 dB display expanded ranges with growing σz, diverging from ideal maxima owing to noise asymmetry. With σz ≈ 0.4025, the 20 dB distribution covers 6–12 bits/s/Hz, indicating heightened failure risks in unmitigated setups. Figure 4.1 illustrates the capacity versus SNR relationship, where increasing PDL levels A induce progressive downward shifts in the curves such as a 1.19 bits/s/Hz reduction at 20 dB SNR for σz ≈ 0.2236 and A = 2 dB emphasizing the pronounced rate compression under low-noise conditions due to accumulated polarization imbalances. Complementing this Figure 4.2 shows simulated capacities deviating from theoretical dashed benchmarks (∼5.17 bits/s/Hz at 10 dB SNR, ∼11.34 bits/s/Hz at 20 dB SNR). At 10 dB (PDL,A=1dB), subfigures (a)–(d) undershoot from near-ideal in (a) (σz = 0.2236, peak ∼4.6 bits/s/Hz) to ∼55% loss in (d) (σz = 0.4025, mean ∼2.3 bits/s/Hz. At 20 dB (PDL=1 dB) in (e)–(h),mirrors the 10 dB pattern with a leftward shift confirming PDL’s steady drops and extra uncertainty beyond basic noise channels in tough conditions. These insights align with [13], showing PDL restricts rates through asymmetry and noise interactions. In lengthy networks, such constraints (1–3 bits/s/Hz) endorse advanced encoding for enhanced stability. 4.2 Comparison with Capacity-Achieving Schemes Building upon the mitigation strategies discussed, recent advancements have introduced capacity-achieving schemes specifically tailored for PDL-impaired channels. The study by Shehadeh and Kschischang [17] introduces an efficient and optimal method utilizing a universal precoder alongside linear minimum mean square error estimation followed by successive interference cancellation (LMMSE-SIC). This method effectively converts the PDL-affected channel into independent scalar AWGN subchannels. It attains the lowest possible SNR penalty given by 10 log10 ( 1√ 1 − α2 ) dB for the most severe PDL scenario of 10 log10 (1 + α 1 − α ) dB, 0 ≤ α < 1. This methodology resonates with prior investigations into PDL-induced capacity con- straints, as examined by Nafta et al. [18], who quantified outage-capacity reductions and required SNR reserves in systems exhibiting typical PDL levels. Their conclusions indi- cate that systems optimized for performance can endure greater PDL with reduced degra- dation relative to metrics based on bit error rates. To assess these schemes in the context of our multi-span framework, comprehensive Monte Carlo simulations were performed. The setup included N = 25 spans, per-span PDL levels A ∈ {0, 0.3, 1.0, 1.8, 2.4} dB, and noise standard deviations σz ∈ {0.1789, 0.2012, 0.2236, 0.2683}. The outcomes illustrate 24 4. Capacity Limits in PDL-Affected Multi-Span Channels notable enhancements. In implementing the scheme from Shehadeh and Kschischang [17], this thesis adhered to the procedure for complex-valued channels outlined in Section V of their work. The key steps executed in our simulations are as follows: 1. Noise covariance: K = N∑ i=1  N∏ j=i+1 Hj  σ2 z N  N∏ j=i+1 Hj † . (4.7) Justification: Maintains constant total noise variance across spans, aligning with [17]’s noise model and supporting realistic noise in long-haul systems. 2. Whitening matrix: W = UDΣ−1/2U† D, (4.8) where Σ is the diagonal matrix of eigenvalues of K, UD is the eigenvector matrix, and Σ−1/2 inverts the square root of eigenvalues. Justification: UD rotates to the eigenbasis, Σ−1/2 scales variances to unity, and U† D reverts the basis, per noise normalization. 3. Whitened (effective) channel: Heff = WHtotal. (4.9) 4. Real-valued representation: Hr = [ ℜ(Heff) −ℑ(Heff) ℑ(Heff) ℜ(Heff) ] . (4.10) Converts complex signals to a real-valued form for precoder, preserving phase/am- plitude integrity. 5. Block-diagonal extension (two time slots): Ĥ = blkdiag(Hr, Hr). (4.11) Implements two-slot space-time coding, enhancing capacity. 6. Apply precoder G: H = ĤG, (4.12) where G is the universal real-valued 8 × 8 precoder derived. 7. LMMSE filter: E = H† ( HH† + (1/SNR)I )−1 . (4.13) 8. Decision statistics: Compute EH, let Γ = diag(EH) be the desired gains and F = EH − Γ the interference terms, isolates signal and interference enhancing analysis. 9. Per-dimension SNRs: Compute SNR for each of the first four real dimensions (signal vs. interference + noise variances). 25 4. Capacity Limits in PDL-Affected Multi-Span Channels 10. SIC on remaining block: Repeat LMMSE on columns 5-8 after cancelling previously-decoded components. 11. Achievable rate aggregation: Sum real-dimension rates and divide by two (two time slots): C = 1 2 8∑ i=1 1 2 log2 ( 1 + SNRi ) . (4.14) This implementation enables the use of conventional codes designed for scalar AWGN channels on the decoupled subchannels, thereby attaining the channel capacity in the information-theoretic sense. The data in Table 4.3 reveal that as noise level (σz) in- creases, simulating lower effective SNR conditions, the standard MIMO capacities decline significantly, particularly at higher PDL. In contrast, the implemented scheme sustains capacities near the no-PDL theoretical maximum of approximately 11.34 bits/s/Hz for low to moderate PDL, with only gradual degradation at higher PDL. This resilience stems from the scheme’s ability to orthogonalize the channel and cancel interference effectively, minimizing the PDL penalty. The theoretical compound capacities, computed using the worst-case α from simulations, serve as a lower bound, and the scheme closely approximates or exceeds this in many cases, validating its optimality. Table 4.3: Average Capacity at SNR = 20 dB for N = 25 Spans (bits/s/Hz). “MIMO” uses the standard capacity C = log2 det ( I2 + ρ 2HtotalH† totalK−1 ) , (4.15) while “Paper” incorporates the encoding scheme from [17] utilizing C = 1 2 8∑ i=1 1 2 log2 ( 1 + SNRi ) . (4.16) σz Method A = 0.0 A = 0.3 A = 1.0 A = 1.8 A = 2.4 0.1789 MIMO 11.34 11.34 11.34 10.97 10.20 Paper 11.34 11.34 11.34 11.34 11.33 0.2012 MIMO 11.31 11.28 11.00 10.32 9.60 Paper 11.34 11.34 11.34 11.34 10.99 0.2236 MIMO 10.72 10.69 10.41 9.75 9.06 Paper 11.34 11.34 11.34 11.16 10.44 0.2683 MIMO 9.69 9.67 9.40 8.77 8.14 Paper 11.11 11.08 10.81 10.17 9.51 26 4. Capacity Limits in PDL-Affected Multi-Span Channels Figure 4.3: Capacity vs. SNR (N = 25, σz = 0.2683. The scheme (dashed) consis- tently outperforms MIMO (solid) and approaches the no-PDL limit (black dashed), demonstrating effective PDL mitigation across SNR ranges. Figures 4.3–4.5 collectively substantiate performance and robustness of the proposed capacity-achieving scheme, highlighting its superior SNR scaling (Figure 4.3), tighter capacity distribution under PDL (Figure 4.4), and reduced outage-induced losses across mean PDL values (Figure 4.5). The results justify the superior performance: at SNR = 20 dB and A = 2.4 dB, the scheme achieves 9.51 bits/s/Hz . The plot also confirms that for pout = 0.01 and mean PDL = 10 dB, the scheme reduces loss by approximately 1 dB compared to MIMO, justifying its efficacy in outage-limited scenarios by better handling worst-case PDL instances. This histogram underscores the scheme’s consistency, with capacities clustered around 10.81 bits/s/Hz versus MIMO’s 9.40 bits/s/Hz, indicating reduced sensitivity to random SOP rotations and PDL accumulations. Incorporating this scheme into the concatenated PDL model for N = 25 spans and A = 2.0 dB (α ≈ 0.33), the SNR penalty drops to about 0.5 dB, reclaiming up to 1.5 bits/s/Hz at SNR = 20 dB relative to baseline cases. The precoder proposed by Shehadeh and Kschischang [17] operates independently of specific channel states, rendering it ideal for dynamic long-haul networks subject to fluctuating PDL. Paired with established AWGN coding techniques, it facilitates seamless integration. Prospective developments might encompass handling insertion loss variations, as highlighted by Nafta et al. [18], to additionally curb outage risks. 27 4. Capacity Limits in PDL-Affected Multi-Span Channels Figure 4.4: Histogram of Capacity for PDL = 1 dB, SNR = 20 dB, σz = 0.2683 (N = 25. The scheme’s distribution (orange) is shifted toward higher values with lower variance (0.0016 vs. MIMO’s 0.0016), closer to the theoretical maximum of 11.34 bits/s/Hz. Figure 4.5: Capacity Loss Cout − C0 vs. Mean PDL (σz = 0.2236. The scheme (dashed) exhibits lower losses than MIMO across outage probabilities, aligning closely with theoretical compound bounds (dot-dashed). 28 5 Discussion and Conclusion The models and analyses developed in this thesis illuminate the critical yet often underestimated influence of polarization-dependent loss (PDL) on the performance of multi-span coherent optical links. The results reveal that random state-of-polarization (SOP) rotations constrain cumulative impairment to a sub-linear A √ N scaling, while simultaneously inducing anisotropic noise that can reduce capacity by up to 3 bits/s/Hz at typical signal-to-noise ratios (SNRs). By integrating Jones matrix propagation with covariance-based formulations and extensive Monte Carlo validation, this work quantifies how PDL transforms isotropic additive white Gaussian noise (AWGN) into non-uniform distributions, with eigenvalue ratios exceeding 1 under high per-span losses. The thesis further evaluates mitigation strategies, demonstrating that precoding com- bined with successive interference cancellation can restore near-ideal throughput with only a modest SNR penalty, outperforming traditional MIMO techniques in skewed chan- nel conditions. These findings bridge theoretical modeling and practical design, offering engineers actionable tools to refine DWDM link margins—particularly for transoceanic systems where 60–240 spans amplify PDL’s cumulative impact. The assumptions of isolated Gaussian amplified spontaneous emission (ASE) noise and fully compensated chromatic dispersion (CD) and polarization-mode dispersion (PMD) were verified to remain valid, reinforcing PDL’s role as the dominant linear impairment in coherent transmission. In conclusion, this thesis repositions PDL from a secondary concern to a principal determi- nant of link robustness, connecting high-fidelity simulations with practical methodologies to preserve terabit-scale capacities in the presence of real-world birefringence. Future research could incorporate nonlinear effects such as Kerr nonlinearity for extended C+L band operation or employ machine learning for real-time covariance estimation, potentially decreasing DSP complexity in modular systems. Experimental validation in recirculating loops may refine outage probability models, while advanced coding schemes like the Dual-Alamouti code—achieving a 0.13 bits/sym GMI gain at 10.5 dB OSNR and 8 dB PDL—offer promising means to counteract PDL-induced penalties. 29 5. Discussion and Conclusion 30 Bibliography [1] Mohammad Farsi. Phase Noise and Polarization Effects in Fiber-Optic Commu- nication Systems: Modeling, Compensation, Capacity, and Sensing. Phd thesis, Chalmers Tekniska Högskola, 2024. Available from PQDT-Global. [2] Xiaotian Zhu, Xiang Wang, Yanlu Huang, Liyan Wu, Chunfei Zhao, Mingzhu Xiao, Luyi Wang, Roy Davidson, Yanni Ou, Brent E. Little, and Sai T. Chu. Low-Loss and Polarization Insensitive 32 × 4 Optical Switch for ROADM Applications. Light: Science & Applications, 13(1):94, December 2024. [3] Tiberiu Tudor and Gabriel Voitcu. Revisiting Poincaré Sphere and Pauli Algebra in Polarization Optics. Photonics, 11(4), 2024. [4] Uwe Edeke and R. Ariyo. 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Optics Letters, 34(23):3613–3615, December 2009. 32 DEPARTMENT OF MICROTECHNOLOGY AND NANOSCIENCE CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden www.chalmers.se www.chalmers.se List of Acronyms Nomenclature List of Figures List of Tables Introduction Advanced Channel Modeling of PDL Build-Up in Optical Systems Modeling a Single PDL Element Jones Matrix Formulation Modeling Birefringence and SOP Rotations PDL-Free Fiber and Unitary Transformations Output with Noise Multi-Span Channel Model Characterization of Channel Statistics PDL Statistical Analysis Methodological Validation Maxwellian Distribution Analysis Noise Characteristics in Multi-Span Links Modeling the Noise Covariance Matrix Numerical Validation and Simulation Results Impact on System Performance and Mitigation Strategies Capacity Limits in PDL-Affected Multi-Span Channels Fundamental Capacity of a PDL Channel Comparison with Capacity-Achieving Schemes Discussion and Conclusion Bibliography