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Journal of Mathematical Physics

Some Theorems Concerning the Phase Problem of Coherence Theory

Demosthenes Dialetis

Published Aug 1, 1967• 9 citations
Publisher

Abstract

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.

DOI
10.1063/1.1705403

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Journal of Mathematical Physics•Nov 1, 1972•4 citations
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.
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.
For a class of quasi-monochromatic spectra, including Lorentzian and Gaussian line shapes, it is shown that knowledge of the modulus of the complex degree of temporal coherence γ(τ) does not suffice to reconstruct the spectrum. This is due to the existence of zeros of γ(τ) in the complex τ plane, giving rise to a significant contribution to the phase of γ(τ). The position of the zeros and their physical interpretation are investigated. The case of band-limited spectra is also treated, and some general properties of the distribution of zeros in this case are given.
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.
Many years ago, a paper by Emil Wolf inspired one of us (MAF) to look more closely at the analytic properties of propagating and scattered optical fields. In 1962, Wolf discussed how Michelson’s interferometer could provide information about the energy distribution in the spectrum of a light beam from measurements of the visibility of interference fringes. A simple relationship exists between the visibility function V(τ) and the modulus of the complex degree of coherence, γ(τ). The methods of Hanbury Brown, and Twiss showed that |γ(τ)| could be determined in various ways, and the important question arose as to how one might recover spectral information, g(ν) from |γ(τ)|. Since V(τ) is proportional to the Fourier transform of g(ν), one might think that only the autocorrelation function of g(ν) might be found, without explicitly knowing the phase of γ(τ). How does one determine the missing phase? The deep insight used to address this question was to take into account the analytic properties from which a complete recovery of g(ν) became possible. It is the underlying analytic properties of the functions that we so routinely employ in modeling optical processes that can provide a deeper understanding of relationships between field parameters and information; this is what we address here.
The purpose of this paper is to suggest a possible approach to the recovery of the spectral density function g(ω) through a knowledge of the first few measured complex zeros of the complex degree of coherence γ(τ). The assumption that γ(τ) is band-limited allows us to express the sums of inverse powers of the complex zeros of γ(τ) in terms of the moments of g(ω). Only the lowest-order moments can be evaluated in this manner with any accuracy for reasons discussed in the text. We use two estimation-type solutions that utilize lower-order moments: beta distribution model and the Shannon maximum entropy model to estimate g(CD). Representative numerical calculations are discussed.
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.