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Journal of the Optical Society of America

Moment estimator approach to the retrieval problem in coherence theory

Richard G. Barakat

Published Jun 1, 1980• 3 citations
Publisher

Abstract

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.

DOI
10.1364/josa.70.000688

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•Jan 21, 2013•0 citations
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.
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.
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 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.
A solution to the Hubbard model is presented which invokes the single assumption that the low-energy peak in the spectral weight function can be well approximated by a function which duplicates its zeroth, first, and second moments to a given order in the bandwidth. The solution takes advantage of the rigorous moment relations of Harris and Lange. A simple, exact, and unambiguous technique to evaluate the correlation functions, required for the moment relations, in the atomic limit is developed. These functions are used to calculate the moments which in turn generate a spectral weight function with no restriction on the electron density. The ground-state energy for the paramagnetic and ferromagnetic configurations are calculated and compared. The model is found to be ferromagnetic for moderate densities.
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.
This thesis is concerned with two generalized moment problems arising in the estimation of stochastic models. Firstly, we consider the THREE approach, introduced by Byrnes Georgiou and Lindquist, for estimating spectral densities. Here, the output covariance matrix of a known bank of filters is used to extract information on the input spectral density which needs to be estimated. The parametrization of the family of spectral densities matching the output covariance is a generalized moment problem. An estimate of the input spectral density is then chosen from this family. The choice criterium is based on the minimization of a suitable divergence index among spectral densities. After the introduction of the THREE-like paradigm, we present a multivariate extension of the Beta divergence for solving the problem. Afterward, we deal with the estimation of the output covariance of the filters bank given a finite-length data generated by the unknown input spectral density. Secondly, we deal with the quantum process tomography. This problem consists in the estimation of a quantum channel which can be thought as the quantum equivalent of the Markov transition matrix in the classical setting. Here, a quantum system prepared in a known pure state is fed to the unknown channel. A measurement of an observable is performed on the output state. The set of the employed pure states and observables represents the experimental setting. Again, the parametrization of the family of quantum channels matching the measurements is a generalized moment problem. The choice criterium for the best estimate in this family is based on the maximization of maximum likelihood functionals. The corresponding estimate, however, may not be unique since the experimental setting is not "rich" enough in many cases of interest. We characterize the minimal experimental setting which guarantees the uniqueness of the estimate. Numerical simulation evidences that experimental settings richer than the minimal one do not lead to better performances