Some Theorems on the Unimodular Complex Degree of Optical Coherence

Abstract

The complex degree of coherence γ(r₁, r₂, t₁ − t₂) of a stationary optical field is defined as the normalized cross‐correlation function (|γ| ≤ 1) of the light disturbances at two space‐time points (r₁, t₁), (r₂, t₂), the disturbances being represented by means of Gabor's analytic signals. In the present paper the general form of γ is examined under the condition that |γ| takes on the extreme value unity for all possible time differences τ = t₁ − t₂ (− ∞ < τ < ∞). Several cases are distinguished, depending on whether this condition is satisfied for some points or for all points in some fixed domain of space. It is pointed out, that the previously published derivations of the relevant theorems contain serious errors. The methods employed in the present paper make use of the property of nonnegative definiteness which the complex degree of coherence is shown to obey. The results have a bearing on the important but as yet unsolved ``phase problem'' of optical coherence theory.