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Journal of Mathematical Physics

Some Theorems on the Unimodular Complex Degree of Optical Coherence

C. L. Mehta, E. Wolf, A. P. Balachandran

Published Jan 1, 1966• 34 citations
Publisher

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.

DOI
10.1063/1.1704799

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Journal of Mathematical Physics•Nov 1, 1972•4 citations
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.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
It is shown that the analytic properties of the temporal complex degree of coherence γ(τ) in the complex time plane (τ complex) impose a relationship between|γ(τ)| and arg γ(τ) on the real time axis. This relationship involves, in general, the location of the zeros of the degree of coherence in the lower half of the complex τ plane. It is suggested that the analytic continuation of the temporal degree of coherence of many spectral distributions has no zeros at all in this half plane. The spectral profiles of such distributions could be uniquely determined from measurements of|γ(τ)| alone. This possibility is of interest in connection with Michelson's well-known method of visibility curves. It is also of interest in connection with the recently proposed correlation and coincidence techniques (employing square-law detection) for determining narrow spectral profiles, such as those found in the output from an optical maser.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
In this paper we define the normalized coherence function of arbitrary order (m, n), in a manner which seems to be a natural generalization of that defined for the second-order coherence function. Both classical and quantized optical fields are considered and the results are compared. It is shown that for classical fields and also for quantized optical fields having nonnegative definite diagonal coherent state representations of the density operator, the modulus of these normalized coherence functions is bounded by the values 0 and 1. This definition differs from the one recently given by Glauber for quantized optical fields, where the normalized coherence functions may take arbitrarily large values even for fields having nonnegative definite diagonal representations of the density operator. Conditions for ``complete coherence,'' i.e., those under which the modulus of the normalized coherence function attains the limiting value 1, are discussed. Some consequences of stationarity and quasi-monochromaticity are also discussed.
The spectral density of a fluctuating light beam may be determined from the knowledge of both the modulus and the phase of the complex degree of self-coherence γ(τ) of the beam. The phase itself may be determined from the modulus and from the location of the zeros of the analytic continuation of γ(τ) in the lower half of the complex τ plane. In the present paper results of an investigation are presented which show that the determination of the zeros is equivalent to the solution of a certain inhomogeneous eigenvalue problem of the Sturm-Liouville type on a semi-infinite frequency range. This eigenvalue problem is found to be equivalent to a certain stability problem in mechanics. Although no general technique for the solution of this type of an eigenvalue problem appears to be known, the new formulation may be used to determine spectral profiles for which the associated degree of self-coherence has zeros at prescribed points in the complex τ plane. Some illustrative examples are given.
In the general theory of optical coherence, the following problem discussed in the present paper, arises: to determine the statistical properties of a field represented by an analytic signal from the knowledge of the statistical properties of the corresponding real field. It is shown by the use of the characteristic functionals that in order to determine the joint probability distributions of the complex field at N space-time points, the knowledge of the complete statistical description of the real field is required; on the other hand, the moments of the complex field up to that order can be determined from the knowledge of the moments of the real field up to the same order. The results are illustrated by explicit calculations relating to the Gaussian random process, which, as is well known, characterizes the fluctuations of thermal light. A converse of a well-known theorem of Kac and Siegert relating to a Gaussian random process is derived as an immediate consequence of our analysis.