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Journal of Mathematical Physics

Phase Problem in Coherence Theory

H. M. Nussenzveig

Published Mar 1, 1967• 46 citations
Publisher

Abstract

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.

DOI
10.1063/1.1705232

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Proceedings of the Royal Society of Edinburgh Section A Mathematics•Jan 1, 1972•0 citations
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 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.
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 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.
A new class of phase-retrieval methods for 2-D fields is introduced. Phase-space tomography can be used for the experimental determination of the amplitude and phase structure of a quasi-monochromatic wave field in a plane normal to its propagation direction. The wave field ψ(r) may represent either a scalar electromagnetic (EM) field or the quantum-mechanical (QM) wave function of a matter wave. The complex wave field may be coherent or partially coherent, in which case the method reconstructs the two-point spatial correlation function, Γ(r, r′) = ⟨ψ(r)ψ*(r′)⟩. (In the QM case, the analogous quantity is the density matrix.) The experiment uses only intensity measurements and refractive optics (lenses), and the data-analysis algorithm is noniterative and requires no deconvolution.
Synopsis This paper shows that the coherence function associated with a band-limited optical spectrum is expressible as an infinite product involving its zeros. Only a finite number of zeros are physically significant and these can be determined from measurements of the visibility of interference fringes; but an ambiguity remains in the sign of the imaginary part of each zero. If the spectrum is to be recovered using the visibility measurements, an auxiliary experiment is needed to supply the necessary signs, but it need not be especially accurate. If the signs are not known there is only a limited number of different spectra that are compatible with the visibility measurements. Finally it is suggested that wavelength measurements made on an asymmetric spectral line with a Michelson interferometer may yield differing results when used with long and short delays.