Reciprocity Inequalities, Coherence Time and Bandwidth in Signal Analysis and Optics

Abstract

This paper is concerned with establishing a rigorous reciprocity relation of the Heisenberg type between the coherence time and the bandwidth of polychromatic radiation. Following upon an observation of Gabor (1946) that the usual formulation of the reciprocity inequality which relates the effective duration of a signal and its effective bandwidth is unsatisfactory for signals that are intrinsically real (e.g., an electric signal) a modified version is here presented. Our analysis is closely related to that of Gabor and supplements it in several respects. In particular we show that his formulation, like our own, is restricted to signals with zero mean value. Contrary to general belief, there is no evidence that the inequality holds when this restriction is removed. A definition of the coherence time of light is then proposed in terms of the (measurable) correlation function that plays a central part in the theory of partial coherence. The coherence time obeys rigorously a reciprocity inequality of the required type and is shown to have a simple interpretation in terms of averages involving the correlations in fluctuations of the instantaneous intensity. It is also shown that the coherence time may be determined experimentally from Michelson's visibility curve of interference fringes.