Analytic Models for the Capacity Distribution in MDG-impaired Optical SDM Transmission

TL;DR

This paper tackles the stochastic capacity problem in strongly-coupled SDM optical links afflicted by MDG/MDL. It develops analytic capacity expressions for arbitrary mode counts by invoking Gaussian approximations for per-mode and total capacities, deriving per-mode distributions from random-matrix theory (GUE with fixed trace) and, for large D, from the Wigner semicircular law. A data-driven correlation model connects per-mode capacities, enabling a joint Gaussian description of the total capacity and its variance. The framework is validated against simulations and extended to frequency diversity, yielding practical outage-capacity estimates and demonstrating accurate performance predictions for SDM systems under MDG/MDL impairments.

Abstract

In coupled space-division multiplexing (SDM) transmission systems, imperfections in optical amplifiers and passive devices introduce mode-dependent loss (MDL) and gain (MDG). These effects render the channel capacity stochastic and result in a decrease in average capacity. Several previous studies employ multi-section simulations to model the capacity of these systems. Additionally, relevant works derive analytically the capacity distribution for a single-mode system with polarization-dependent gain and loss (mode count D = 2). However, to the best of our knowledge, analytic expressions of the capacity distribution for systems with D > 2 have not been presented. In this paper, we provide analytic expressions for the capacity of optical systems with arbitrary mode counts. The expressions rely on Gaussian approximations for the per-mode capacity distributions and for the overall capacity distribution, as well as on fitting parameters for the capacity cross-correlation among different modes. Compared to simulations, the derived analytical expressions exhibit a suitable level of accuracy across a wide range of practical scenarios.

Figures (11)

• Figure 1: Representation of a coupled sdm system with $N_{c}$ spectral channels and $D$ spatial channels. The spectral and spatial channels are first modulated. Subsequently, a spectral multiplexer generates a WDM signal, and a spatial multiplexer generates an SDM signal. The transmitted symbol sequences $s_{i}[n]$, for $i\in[1,D]$, are recovered independently for each spectral channel by a $D\times D$ MIMO equalization process, generating estimates $\widehat{s}_{i}[n]$ of the transmitted symbol sequence. The optical carrier modulator and the receiver front-end perform electrical-to-optical and optical-to-electrical conversion, respectively. The channel transfer matrix $\mathbf{H}$ and modal gains $\lambda_{\mathrm{dB}}$ are frequency dependent, but we omit the frequency dependent variable for simplicity, assuming a narrowband channel. The insets illustrate the evolution of the logarithmic scale modal gains pdf ($f_{\lambda_{\text{dB}}}(\lambda_{\text{dB}})$).
• Figure 2: (a) Logarithmic gain $\lambda_{\mathrm{dB}}$ distribution of each mode and ensemble, considering the sum of the linear gains equals to $D$. The ensemble distribution is normalized by $D$ in order to facilitate visual analysis. The per-mode distributions are obtained via simulation for ${D=6}$ and ${\sigma_{\mathrm{mdg}}=5~\mathrm{dB}}$. (b) Instantaneous capacity distribution of each mode and total. The distributions are obtained via simulation for ${D=6}$, ${\mathrm{SNR}=10~\mathrm{dB}}$, and ${\sigma_{\mathrm{mdg}}=5~\mathrm{dB}}$. Dotted lines connect the simulated histogram values for visualization purposes.
• Figure 3: Analytical approximation of per-mode capacity mean (a) and standard deviation (b) for ${\mathrm{SNR}=10~\mathrm{dB}}$ and ${\sigma_{\mathrm{mdg}}=5~\mathrm{dB}}$. For ${D\leq 8}$ the gue distribution method is considered. Higher mode counts use the Wigner semicircular cdf method.
• Figure 4: Comparison between the analytically obtained curves, assuming a Gaussian distribution, and simulation values for the per-mode capacity pdfs for ${D=6}$, ${\mathrm{SNR}=10~\mathrm{dB}}$, and ${\sigma_{\mathrm{mdg}}=5~\mathrm{dB}}$.
• Figure 5: Capacity cdf and ${\mu_{C_{i}}}$ estimation for ${D=6}$, ${\mathrm{SNR}=10~\mathrm{dB}}$, and ${\sigma_{\mathrm{mdg}}=5~\mathrm{dB}}$.