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Journal of the Optical Society of America

Dead-time-corrected photocounting distributions for laser radiation*

Barry I. Cantor, Malvin Carl Teich

Published Jul 1, 1975• 68 citations
Publisher

Abstract

Exact forward photocounting distributions for a source of optical radiation with arbitrary statistics are obtained in the presence of photodetector dead time. In particular, we examine the counting and pulse-interval distributions that arise from amplitude-stabilized radiation, from chaotic radiation, and from a Van der Pol laser with arbitrary excitation. The exact dead-time-corrected Poisson distribution is graphically compared with a previous approximate result and with the uncorrected Poisson. Plots of the Bose–Einstein distribution clearly indicate the dramatic anti-bunching effects of the dead time in overcoming the inherent bunching of this distribution. A simplified approximate solution is also found. for the Van der Pol laser above threshold; this result is similar to light from an amplitude-stabilized source incident on a photodetector with a gaussian-distributed dead time. Information about photodetector dead-time variation can therefore be obtained either by using an amplitude-stabilized source or by properly choosing system parameters such that irradiance fluctuations are averaged out.

DOI
10.1364/josa.65.000786

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Journal of the Optical Society of America•Oct 1, 1978•32 citations
Exact forward photocounting distributions for a source of optical radiation with arbitrary statistics are obtained in the presence of photodetector dead time. In particular, we examine the counting and pulse-interval distributions that arise from amplitude-stabilized radiation, from chaotic radiation, and from a Van der Pol laser with arbitrary excitation. The exact dead-time-corrected Poisson distribution is graphically compared with a previous approximate result and with the uncorrected Poisson. Plots of the Bose–Einstein distribution clearly indicate the dramatic anti-bunching effects of the dead time in overcoming the inherent bunching of this distribution. A simplified approximate solution is also found. for the Van der Pol laser above threshold; this result is similar to light from an amplitude-stabilized source incident on a photodetector with a gaussian-distributed dead time. Information about photodetector dead-time variation can therefore be obtained either by using an amplitude-stabilized source or by properly choosing system parameters such that irradiance fluctuations are averaged out.
A previously constructed laser model with quantum (noncommuting) noise sources was shown to lead near threshold to a quantum rotating-wave Van der Pol oscillator. A full dynamical correspondence between quantum and classical random processes allows one to compute the average of any time-ordered, normal-ordered operator function by averaging the associated function of classical random variables. Numerical calculations for the associated classical Van der Pol oscillator of the steady-state distribution, the total intensity fluctuations, and the linewidth versus operating point were amply confirmed experimentally. Measurements (and calculations) of higher than two-time correlations were sparse and contradictory. Photocount distributions, at times short compared to the intensity correlation time, confirm only the steady state of the laser. Photocount distributions at intermediate and longer times are difficult to compute because they involve multitime correlations of high ($\ensuremath{\infty}$) order. By providing an exact solution for photocount distributions and their moments for all times, we expected to stimulate measurements near threshold which would provide an adequate test of the Van der Pol laser model. Comparison of the results reported here with recent photocount experiments of Meltzer, Davis, and Mandel and of Jakeman, Oliver, and Pike provides gratifying agreement and confirmation of our statistical understanding of laser fluctuations near threshold.
The corrections to the photocount distributions, due to dead-time effects, are evaluated by use of the time-interval probability density of the counts. The normalized cumulative distribution function J k ( T , τ), associated with the detection of fluctuating light beams during a counting time interval T , with a counter having a constant dead-time τ, appear naturally as a convenient quantity to be used in comparing experimental results with theoretical predictions. The quantities J k ( T , τ) obey a simple recurrence relation and are readily derived from the photocount distribution p ( k, T , τ). A general expression is derived for J k ( T , τ) and p ( k, T , τ) associated with the detection of fluctuating light beams. Typical cumulative distribution functions, relating to the detection of thermal light, are evaluated as functions of the degree of polarization of the light, of the average number of photocounts < n 0 > in the absence of dead-time effects and of the ratio τ/ T . The distributions J k ( T , τ) and p ( k, T , τ) relating to detection of laser light are also calculated in a typical case.
Exact photocounting distributions are obtained for a pulse of light whose intensity is exponentially decaying in time, when the underlying photon statistics are Poisson. It is assumed that the starting time for the sampling interval (which is of arbitrary duration) is uniformly distributed. The probability of registering n counts in the fixed time T is given in terms of the incomplete gamma function for n >/= 1 and in terms of the exponential integral for n = 0. Simple closed-form expressions are obtained for the count mean and variance. The results are expected to be of interest in certain studies involving spontaneous emission, radiation damage in solids, and nuclear counting. They will also be useful in neurobiology and psychophysics, since habituation and sensitization processes may sometimes be characterized by the same stochastic model.
A computer simulation method is employed to investigate the dead-time modified photocount detection distribution where the ratio of coherence time of the incident radiation to the counting time τc/T ≦ 1. The principal result is to demonstrate that a dead-time correction formula that was derived for τc/T ≫ 1 may be modified to be applicable to arbitrary coherence times. The simulation technique was verified by comparing simulated results with known results for dead-time modified photocount distribution for radiation with τc/T ≫ 1 and for unmodified photocount distribution for radiation with τc/T ≦ 1.
The problem of obtaining the probability density of the fluctuating light intensity from the knowledge of photocounting statistics, when the detectors have a finite dead time, is considered. The integral relation for the probability of registering nnn photocounts in the time interval TTT in the presence of a dead time $\ensuremath{\tau}$ is inverted, and an expression is obtained for the probability density of the fluctuating intensity of the incident light radiation. The method is illustrated with a few specific examples.
We report a series of optical experiments that verify the theoretical photocounting distributions obtained by Diament and Teich for triangularly and sinusoidally modulated laser radiation. Another series of experiments validates the (nonparalyzable) dead-time-modified versions of these formulas obtained by Cantor and Teich. A new expression is obtained for the counting distribution in the presence of modulation and paralyzable dead time. The results have application in photon counting, nuclear counting, and neural counting.