On the Per-Sample Capacity of Nondispersive Optical Fibers

TL;DR

This paper analyzes the capacity of zero-dispersion optical fibers modeled by the nonlinear Schrödinger equation, deriving bounds and characterizing optimal input distributions, revealing unbounded growth with input power and connections to Rician fading channels.

Contribution

It provides a novel capacity analysis of the nonlinear optical fiber channel in the zero-dispersion regime, including bounds, optimal input distributions, and a simplified algebraic model.

Findings

Per-sample capacity grows unboundedly with input power.

A simple half-Gaussian amplitude and uniform phase distribution is asymptotically capacity-achieving.

The zero-dispersion channel relates to a 2x2 Rician fading MIMO channel.

Abstract

The capacity of the channel defined by the stochastic nonlinear Schr\"odinger equation, which includes the effects of the Kerr nonlinearity and amplified spontaneous emission noise, is considered in the case of zero dispersion. In the absence of dispersion, this channel behaves as a collection of parallel per-sample channels. The conditional probability density function of the nonlinear per-sample channels is derived using both a sum-product and a Fokker-Planck differential equation approach. It is shown that, for a fixed noise power, the per-sample capacity grows unboundedly with input signal. The channel can be partitioned into amplitude and phase subchannels, and it is shown that the contribution to the total capacity of the phase channel declines for large input powers. It is found that a two-dimensional distribution with a half-Gaussian profile on the amplitude and uniform phase provides a lower bound for the zero-dispersion optical fiber channel, which is simple and asymptotically capacity-achieving at high signal-to-noise ratios (SNRs). A lower bound on the capacity is also derived in the medium-SNR region. The exact capacity subject to peak and average power constraints is numerically quantified using dense multiple ring modulation formats. The differential model underlying the zero-dispersion channel is reduced to an algebraic model, which is more tractable for digital communication studies, and in particular it provides a relation between the zero-dispersion optical channel and a $2 \times 2$ multiple-input multiple-output Rician fading channel. It appears that the structure of the capacity-achieving input distribution resembles that of the Rician fading channel, i.e., it is discrete in amplitude with a finite number of mass points, while continuous and uniform in phase.