Optical Communications, 2nd Edition

1.7 RANDOM FIELDS

In an optical system, we are often forced to deal with stochastic or random fields. Such fields arise when dealing with atmospheric or weather effects (which cause random variations to occur in the transmitted field), or in describing background or stray light that may appear in the system. In addition, the laser itself may produce random contributions (self-emission noise) to its own emitted fields. These random variations lead to statistical fluctuations in the optical field, which can only be analyzed after associating proper statistics with the field itself. We wish to develop some basic properties for describing these general random fields.

The complex envelope of a stochastic field must be considered random at each point t and r describing the field over a designated area. As such, random fields are completely described by their probability densities, that is, the probability densities associated with each point, or set of points, of the field. These densities are often difficult to model exactly, and often assumptions must be imposed upon these statistics. For example, a Gaussian random field is one whose probability density at any point (t, r) is taken as a complex Gaussian variable.

Stochastic field analysis is often confined to second-order statistics associated with the field; in particular, its coherence function [12]. In this regard, the time-space (mutual) coherence function of a stochastic field f(t, r) at points (t₁, r₁) and (t₂, r₂) is formally ...