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Physical Review A

Photon Counting Statistics of Harmonic Signal Mixed with Thermal Light. II. N -Fold Photocounting

C. L. Mehta, A. K. Jaiswal

Published Dec 1, 1970• 6 citations
Publisher

Abstract

The one-detector photoelectron counting statistics of a harmonic signal mixed with thermal light was considered in paper I. In the present paper, we consider the extension to the counting statistics with several detectors. An expression for the factorial moment generating function of the number of counts is obtained in terms of the eigenvalues of an associated matrix integral equation. This expression is used to evaluate the factorial cumulants of the number of counts.

DOI
10.1103/physreva.2.2570

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The one-detector photoelectron counting statistics of a harmonic signal mixed with thermal light was considered in paper I. In the present paper, we consider the extension to the counting statistics with several detectors. An expression for the factorial moment generating function of the number of counts is obtained in terms of the eigenvalues of an associated matrix integral equation. This expression is used to evaluate the factorial cumulants of the number of counts.
In this paper, we consider the photoelectron counting distribution when the wave field incident on the detector is a mixture of harmonic signal and thermal noise. An expression for the factorial moment-generating function of the number of counts is obtained in terms of the eigenvalues of an associated integral equation. This expression is then used to evaluate the factorial cumulants of the number of counts. The results are valid for an arbitrary counting interval. The extension to photoelectron counting with several detectors is considered in paper II of the present investigation.
We consider onefold photoelectron-counting statistics, for arbitrary counting times, when the wave field on the detector is a mixture of thermal light of a specified spectral profile (incoherent component) and a harmonic signal (coherent component). The probability density function of the time-integrated intensity W(Ω) is obtained from the inverse Laplace transform of the moment-generating function by the Dubner–Abate algorithm. The corresponding probability distribution of the detected photoelectrons, P(m), is then evaluated directly by numerical integration of W(Ω). Numerical results for W(Ω) and P(m) have been obtained when the thermal line shape is Lorentzian for both the homodyne situation (coherent line coincides with the mean frequency of the thermal spectral profile) and the heterodyne situation (coherent line is now off center with respect to the mean frequency of the thermal spectral profile).
General formulas for the factorial cumulants of the photocounting distribution and the cumulants of the integrated intensity are obtained for chaotic light superposed with coherent light, for experiments involving NNN different counting intervals. The chaotic light may be partially polarized, need not be time stationary or single mode, may differ in frequency from the coherent light, and may have any spectral distribution.
Analytic aspects of the photon-counting distributions in quantum optics are investigated for the case where the underlying field includes a mixture of harmonic signal and thermal noise. A Lorentzian spectral profile for the thermal light is used to find simple iterative relations for the factorial cumulant moments. Explicit analytic expressions for the moments are then given to eighth order. Approximate interpolation formulas, which display proper limiting behavior at small and large correlation length, are also investigated.
The conventional model of the photodetection process and the short-time approximation are used to obtain the factorial moment generating function for the NNN-fold detection of mixed thermal and coherent radiation. This generating function is then used to obtain the photocount distribution for twofold joint photodetection. The photocount distribution is expressed as a threefold summation over Laguerre polynomials. The distribution is examined for the limiting cases of purely thermal radiation and uncorrelated fields.
For a mixture of an arbitrary number of sinusoidal coherent components and chaotic light with a rational spectral density having simple poles, the probability-generating function h(z) of the number of photoelectrons ejected by the light in an interval is expressed in terms of certain finite determinants. It figures in the integrands of contour integrals in the complex plane for the cumulative and complementary cumulative distributions of the number of photoelectron counts. When these are evaluated by numerical integration, h(z) can be computed at each point on the contour by standard computer routines for solving algebraic equations and evaluating determinants, permitting precise and efficient computation of the photocount distributions.