Communication theory for the free-space optical channel

The current understanding of quantum detectors, the noise mechanisms which limit (are basic to) their operation, and their application to optical communications (theory) is summarized. In this context, we are considering channels in which the electromagnetic field is not subjected to any propagation effects other than a geometric loss. (Such a channel would exist between satellites.) Consequently, we will concentrate on optimum time processing using the tools of statistical communication theory. Fundamental to the study of a detection process is the need to develop a good mathematical model to describe it [1]-[6]. Therefore, approximately one-fifth of the paper is devoted to establishing, in a semi-classical analysis, the quantum detector output electron number as a conditional Poisson process with the conditioning variable being the modulus of the electromagnetic field. Once this has been established, these results are used to derive various limiting probability densities related to actual practice. Although the mathematical details are omitted, these results will be presented from the viewpoint of orthogonal function expansions and interpreted in terms of an eigenspace. The resulting current flow is analyzed next as a shot noise process, and the power density spectrum is calculated. Attention is focused on isolating the signal components from the noise in terms of both the current probability density and the power density spectrum. Examples are given where appropriate. At this point, an understanding of the underlying noise processes will have been presented and attention will shift to analog and digital communications. The analog communication will be presented primarily in terms of the signal-to-noise ratio. The S/N ratio in direct detection will be presented both as a ratio of the integrals of two separate portions of the spectrum and as a ratio of two moments of the probability density describing the current. These calculations will be extended to include heterodyne detection. Digital communications will be discussed in the context of detection theory. It will be shown that the likelihood ratio is often a monotonic function of the random variable representing the number of electrons flowing. Hence optimum processing will consist of a weighted count of electrons from various counting modes. Digital design will be presented in terms of M-ary signaling, error probabilities, and information rates.