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

##### Milad Sefidgaran, Mansoor Yousefi

The capacity of a discrete-time model of optical fiber described by the split-step Fourier method (SSFM) as a function of the average input power $\mathcal P$ and the number of segments in distance $K$ is considered. It is shown that if $K\geq \mathcal{P}^{2/3}$ and $\mathcal P\rightarrow \infty$, 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 the signal-to-noise ratio $\text{SNR}$. As $K\rightarrow \infty$, the intra-channel signal-noise interactions average 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 $K{=}\mathcal{P}^{1/\delta}$ is generally characterized in terms of $\delta$. We consider the SSFM model where the dispersion matrix does not depend on $K$, e.g., when the step size in distance is fixed. It is shown that the capacity of this model when $K\geq \mathcal{P}^3$ and $\mathcal P \rightarrow \infty$ is $\frac{1}{2n}\log_2(1+\text{SNR})+ o(1)$. Thus, there is only one DoF in this model. Finally, it is shown that if the nonlinearity parameter $\gamma\rightarrow\infty$, the capacity of the continuous-space model is $\frac{1}{2}\log_2(1+\text{SNR})+ o(1)$ for any $\text{SNR}$.

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