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Proceedings of the Physical Society

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

E. Wolf

Published Feb 1, 1958• 67 citations
Publisher

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.

DOI
10.1088/0370-1328/71/2/315

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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.
Several different definitions of the coherence time Δτ can be found in the literature and it is generally assumed that, in cases of practical interest, the different definitions give values of the same order of magnitude. The same remark applies also to definitions of the effective bandwidth Δν. Moreover it is generally supposed that these two quantities are related by the order of magnitude formula Δτ similar 1/Δν. It is shown from a detailed examination of one particular case of practical interest that the bandwidths as defined by two of the previously proposed formulae are of quite different orders of magnitude, although the correspondingly defined coherence times differ only by a factor of order unity. Similarly it may happen that the two values of Δ r differ greatly, whereas the corresponding values of Δν are nearly equal. This discussion shows that the reciprocity relation Δτ similar 1/Δν must be treated with caution, especially in the case of multiple-peaked distributions, such as arise in optical masers.
This article presents a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum. Analyses based on both the classical and quantum theories are described. After a brief historical introduction, the elementary concepts which are frequently employed in the discussion of interference phenomena are summarized. The measure of second-order coherence is then introduced in connection with the analysis of a simple interference experiment and some of the more important second-order coherence effects are studied. Their uses in stellar interferometry and interference spectroscopy are described. Analysis of partial polarization from the standpoint of correlation theory is also outlined. The general statistical description of the field is discussed in some detail. The recently discovered universal "diagonal" representation of the density operator for free fields is also considered and it is shown how, with the help of the associated generalized phase-space distribution function, the quantum-mechanical correlation functions may be expressed in the same form as the classical ones. The sections which follow deal with the statistical properties of thermal and nonthermal light, and with the temporal and spatial coherence of blackbody radiation. Later sections, dealing with fourth- and higher-order coherence effects include a discussion of the photoelectric detection process. Among the fourth-order effects described in detail are bunching phenomena, the Hanbury Brown-Twiss effect and its application to astronomy. The article concludes with a discussion of various transient superposition effects, such as light beats and interference fringes produced by independent light beams.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
The concept of the Wigner distribution function is applied to stochastic signals, in particular to partially coherent light. Relations between the Wigner distribution function, the power spectrum and related subjects are formulated. Some inequalities for the Wigner distribution function are derived; one of these inequalities leads to the definition of an overall degree of coherence. The overall degree of coherence of a stochastic signal remains invariant when the signal propagates through a lossless linear system.
The concept of the Wigner distribution function is applied to stochastic signals, in particular to partially coherent light. Relations between the Wigner distribution function, the power spectrum and related subjects are formulated. Some inequalities for the Wigner distribution function are derived; one of these inequalities leads to the definition of an overall degree of coherence. The overall degree of coherence of a stochastic signal remains invariant when the signal propagates through a lossless linear system.
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.