A probabilistic model for optical fiber channels with zero dispersion

Signal evolution in optical fibers with zero dispersion and distributed Raman amplification is modeled by a stochastic nonlinear ordinary differential equation (ODE) which includes the effects of the Kerr nonlinearity and amplified spontaneous emission noise. Such a mathematical model in the form of a stochastic nonlinear ODE is not initially suitable for information-theoretic analysis. In this paper we provide a simple framework to probabilistically model signal propagation in optical fibers. The analysis is based on discretizing the fiber as a cascade of an infinite number of infinitesimal pieces of fiber in the distance dimension, while at the same time quantizing the signal into a large number of small bins in the complex plane. This can be understood in the context of the sum-product algorithm, known in coding theory. Though the method can be also applied to fibers with dispersion, in this paper it is illustrated for the special case of zero dispersion. In particular, for this case we find the conditional probability density function of the output signal given the input signal. We further show that the capacity of the dispersion-free optical channel as a function of signal-to-noise ratio (SNR) goes to infinity with SNR → ∞.