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Journal of Physics A Mathematical Nuclear and General

Joint probability of instants of occurrence of photon events and estimation of optical parameters

Bahaa E. A. Saleh

Published Jul 21, 1974• 2 citations
Publisher

Abstract

Using a Karhunen-Loeve modal expansion of a stationary and gaussian optical field an expression is derived for the joint probability of instants of occurrence of photon events. When the observation time is much longer than the coherence time, this expression takes a simple form. In this case, and for lorentzian light, it is shown that measurement of the total number of counts is sufficient to determine the maximum likelihood estimate (MLE) of the average count rate. A nonlinear equation is found whose solution gives the MLE of the bandwidth.

DOI
10.1088/0305-4470/7/11/015

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Using a Karhunen-Loeve modal expansion of a stationary and gaussian optical field an expression is derived for the joint probability of instants of occurrence of photon events. When the observation time is much longer than the coherence time, this expression takes a simple form. In this case, and for lorentzian light, it is shown that measurement of the total number of counts is sufficient to determine the maximum likelihood estimate (MLE) of the average count rate. A nonlinear equation is found whose solution gives the MLE of the bandwidth.
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It is well known that the photocount distribution associated with the photoelectric detection of fluctuating light beams carries information about the probability density of the light intensity. In this paper, we present an exact solution to a problem of particular interest which arises in connection with the photoelectric detection of narrow-band Gaussian (thermal or pseudothermal) light, namely, the problem of determining the photocount distribution when the counting-time interval is not necessarily short compared to the coherence time of the light. The method developed in this paper is applied to the case where the spectral profile of the light is Lorentzian; it leads to an exact expression for the photocount generating function and it provides simple recurrence relations for the photocount distribution and for its factorial moments.
A statistical model linking the probability of detecting photons in a photon counting experiment to the average number of input photons is used to analytically derive an error distribution of the estimates of the average number of photons in coherent input pulses. The estimates distribution determines the required number of experimental count runs required for an acceptable measurement error, for any value of the average number of input photons. An expression for the minimum detectable power by photon counting is obtained, fully capturing the effect of the photon counting statistics thereby completing an earlier analysis.
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