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

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 SNR and the number of segments in distance K is considered. It is shown that if K ≥ SNR2/3 and SNR → ∞, the capacity of the resulting continuous-space lossless model is lower bounded by 1/2 log2(1 + SNR) - 1/2 + o(1), where o(1) tends to zero with SNR. As K → ∞, 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=δ√SNR is generally characterized in terms of δ. It is shown that if the nonlinearity parameter γ → ∞, the capacity of the continuous-space model is 1/2 log2(1 + 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=δ√SNR, δ > 3, and SNR → ∞ is 1/2n log2(1 + 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. The AIR characteristically increases with SNR, reaching a peak at a certain optimal power, and then decreases as SNR is further increased. The peak is attributed to a balance between noise and stochastic ISI. However, if the power is further increased, the AIR will increase again, approaching the lower bound 1/2 log(1 + SNR) - 1/2 + o(1). The second ascent is because the ISI averages out to zero with K → ∞ sufficiently fast.