Correlation Functions for Coherent Fields

A factorization condition which must be satisfied by the first $n$ correlation functions for the electromagnetic field operators has been used to define nth-order coherence. The first-order coherence condition has been shown to imply maximum fringe contrast in interference patterns. In the present paper we investigate the mathematical consequences of assuming the condition for maximum fringe contrast. By considering the correlation functions as scalar products and formulating rigorous inequalities for them we are able to show that the assumed condition in turn implies factorization of the first-order correlation function. By extending the same methods we are able to show that all of the higher order correlation functions factorize into forms similar to those required for full coherence, but differing from them through the inclusion of a sequence of constant numerical factors. These coefficients are shown to furnish a convenient description of the higher-order coherence properties of the field. Their values are presented for some typical examples. We derive a number of inequalities satisfied by the coefficients for the case of fields which possess positive-definite weight functions in the $P$ representation. Some inequalities obeyed by the correlation functions for such fields are derived as well.