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

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.