Lower Bound on the Capacity of the Continuous-Space SSFM Model of Optical Fiber

TL;DR

This paper establishes a lower bound on the capacity of the continuous-space SSFM model for optical fibers, showing it retains at least half the input degrees of freedom at high SNR, contrasting with the discrete model.

Contribution

It provides the first rigorous capacity lower bounds for the continuous-space SSFM model, revealing the asymptotic behavior and degrees of freedom as SNR and segments increase.

Findings

Capacity lower bounded by (1/2)log2(1+SNR) - 1/2 + o(1) at high SNR.

Number of degrees of freedom in the continuous model is at least half of the input dimension.

Numerical simulations show the maximum achievable information rates follow a double-ascent curve.

Abstract

The capacity of a discrete-time model of optical fiber described by the split-step Fourier method (SSFM) as a function of the signal-to-noise ratio $\text{SNR}$ and the number of segments in distance $K$ is considered. It is shown that if $K\geq \text{SNR}^{2/3}$ and $\text{SNR} \rightarrow \infty$, the capacity of the resulting continuous-space lossless model is lower bounded by $\frac{1}{2}\log_2(1+\text{SNR}) - \frac{1}{2}+ o(1)$, where $o(1)$ tends to zero with $\text{SNR}$. As $K\rightarrow \infty$, the inter-symbol interference (ISI) averages out to zero due to the law of large numbers and the SSFM model tends to a diagonal phase noise model. It follows that, in contrast to the discrete-space model where there is only one signal degree-of-freedom (DoF) at high powers, the number of DoFs in the continuous-space model is at least half of the input dimension $n$. Intensity-modulation and direct detection achieves this rate. The pre-log in the lower bound when $K= \sqrt[\delta]{\text{SNR}}$ is generally characterized in terms of $\delta$. It is shown that if the nonlinearity parameter $\gamma\rightarrow \infty$, the capacity of the continuous-space model is $\frac{1}{2}\log_2(1+\text{SNR})+ o(1)$. The SSFM model when the dispersion matrix does not depend on $K$ is considered. It is shown that the capacity of this model when $K= \sqrt[\delta]{\text{SNR}}$, $\delta>3$, and $\text{SNR} \rightarrow \infty$ is $\frac{1}{2n}\log_2(1+\text{SNR})+ O(1)$. Thus, there is only one DoF in this model. Finally, it is found that the maximum achievable information rates (AIRs) of the SSFM model with back-propagation equalization obtained using numerical simulation follows a double-ascent curve.