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Journal of the Optical Society of America A

Estimation of the squared modulus of the mutual intensity from high-light-level intensity measurements

Timothy J. Schulz

Published Jun 1, 1995• 1 citation
Publisher

Abstract

The problem of estimating the squared modulus of the mutual intensity (or the complex coherence factor) from high-light-level intensity measurements is addressed for the situation in which the fluctuations of the complex-valued amplitude are governed by circular-Gaussian statistics and the light level is high enough that all nonclassical fluctuations of the measurements can be ignored. A lower bound on the variance of any unbiased estimator is presented, and this bound is compared with the variance of Ebstein’s polynomial estimators [ J. Opt. Soc. Am. A8, 1450 ( 1991)] along with the variance of the maximum-likelihood estimator.

DOI
10.1364/josaa.12.001331

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The problem of estimating the squared modulus of the mutual intensity (or the complex coherence factor) from high-light-level intensity measurements is addressed for the situation in which the fluctuations of the complex-valued amplitude are governed by circular-Gaussian statistics and the light level is high enough that all nonclassical fluctuations of the measurements can be ignored. A lower bound on the variance of any unbiased estimator is presented, and this bound is compared with the variance of Ebstein’s polynomial estimators [ J. Opt. Soc. Am. A8, 1450 ( 1991)] along with the variance of the maximum-likelihood estimator.
Amplitude interferometry and intensity interferometry are methods for processing the complex amplitude and the intensity, respectively, of an electromagnetic field to estimate the mutual intensity of the field at two spatial locations. Whereas complex amplitude measurements allow for the direct estimation of the mutual-intensity phase, intensity measurements do not. We consider applications for which the estimation of the magnitude or the squared magnitude of the mutual intensity is adequate, and we provide fundamental limits on the estimation accuracy of any unbiased estimator of the squared magnitude of the mutual intensity from coherent (complex amplitude) or incoherent (intensity) measurements. Our analysis is performed for the high-light-level (classical-noise-limit) case and quantifies the advantages of making the more difficult coherent measurements.
In many applications using light for practical purposes, the accuracy of optical experiment is limited most fundamentally by the finite amount of light utilized in the measurements. Special emphasis should be put on interferometry aimed at measuring the parameters of simple fringe patterns due to the following reasons. First, fringe parameter measurement provides a relatively well defined and tractable example of application of the theory. The desired parameters are easily defined, and methods for their measurement are readily devised based on common sense. Second, fringe parameter measurement is central to all problems involving coherence. The fundamental descriptors of light waves utilized in coherence theory are in fact measurable parameters of fringes. By examining the limitations to fringe parameter measurement, we are actually examining the limitations to the measurability of coherence itself. In this paper, fundamental limitations of estimating the amplitudes and phases of polar-interferograms recorded at low light levels are investigated. By modeling the receiver as a spatial array of photon-counting detectors, results are obtained that permit specification of the minimum number of photoevents required for estimation of fringe parameters to a given accuracy. Both a discrete Fourier-transform estimator and an optimum joint maximum-likelihood estimator are considered to specify the limiting performance of all unbiased estimators in terms of the collected light flux.
We derive the full statistics of the product events in homodyne correlation measurements, involving a single mode signal, a local oscillator, a linear optical network, and two linear photodetectors. This is performed for the regime of high intensities impinging on the detectors. Our description incorporates earlier proposed homodyne correlation measurement schemes, such as the homodyne cross-correlation and homodyne intensity-correlation measurements. This analysis extends the amount of information retrieved from such types of measurements, since previously attention was paid only to the expectation value of the correlation statistics. As an example, we consider the correlation statistics of coherent, Gaussian, and Fock states. Moreover, nonclassical light is certified on the basis of the variance of the measurement outcome.
The signal-to-noise ratio (S/N) of the Brown–Twiss stellar intensity interferometer, in which the resolving time (τr) of the photodetectors is much longer than the coherence time of incident radiation (τc), is derived, especially for incident radiation with a very high effective temperature. The statistical properties of the wave amplitude of the incident radiation is assumed to be described by a gaussian distribution function. In order to evaluate the mean square error in the intensity-correlation measurement, the eighth-order moment of complex wave amplitudes is obtained by using the moment theorem for a gaussian random process. The mean square error of the high-intensity correlation experiment during a finite observation time (τ0) is determined chiefly by the random correlation of noise currents due to intensity fluctuations of the incident radiation. The S/N in the case of extremely high-intensity radiation, thus, is given by (τ0/τc)12∣γ12∣2, where γ12 is the complex degree of coherence between two observation points independent of the intensity of incident radiation. In the case of low-intensity radiation the mean square error of an intensity-correlation measurement is determined by the random correlation of shot noise in photoelectric currents. The S/N for low-intensity radiation, as originally discussed by Hanbury Brown and Twiss, is given by (τ0/τr)12∣γ12∣2ηnτc where n is the number of incident photons per second and η is the quantum efficiency of a photodetector. The S/N for the intermediate case is also discussed.
Single photon count distributions at the outputs of a Mach-Zehnder interferometer are highly dependent on the phase difference between the two arms of the apparatus, so photocount records can be used to estimate an unknown optical phase. Experimental results have shown that phase information can be inferred from the data even at low mean input intensities of ⪅10 photons. Here we consider the optimal Bayesian parameter estimate for the model used by a previous frequentist analysis. In the limit of high input intensity the observation is well-modelled as an intensity measurement of a classical wave in Gaussian measurement noise, however in the low intensity regime this simple model does not apply. It seems that no generally valid closed-form estimator for this problem exists without making the Gaussian approximation. Moreover the 2π radian ambiguity in phase estimates presents a problem in defining cost functions to compare estimator performance, and we consider two reasonable alternatives. We present a numerical study of the experiment along with a novel numerical approach to this problem which out-performs the existing estimators and with a squared error that approaches the Cramer-Rao lower bound previously derived in the literature, in the limit where that bound is valid.
We address several estimation problems in quantum optics by means of the maximum-likelihood principle. We consider Gaussian state estimation and the determination of the coupling parameters of quadratic Hamiltonians. Moreover, we analyze different schemes of phase-shift estimation. Finally, the absolute estimation of the quantum efficiency of both linear and avalanche photodetectors is studied. In all the considered applications, the Gaussian bound on statistical errors is attained with a few thousand data.