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Journal of the Optical Society of America A

Dead-time modified photocount distributions for chaotic radiation with arbitrary coherence times

Gerard Lachs, Lin Wen-Tsung

Published Oct 1, 1985• 1 citation
Publisher

Abstract

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.

DOI
10.1364/josaa.2.001687

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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 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 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.
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.
There has been considerable interest in recent years in the statistical properties of optical frequency electromagnetic fields. Of particular interest is the study of photon bunching and the probability density for waiting time between adjacent photocounts. In this paper we derive an expression for the probability density of the time interval distribution as well as some other quantities of interest in photon bunching for the case where the incident radiation is comprised of superposed coherent and chaotic radiation of arbitrary spectral shapes. Calculations show that the key parameters are the average number of chaotic counts per coherence time and the ratio of the strengths of coherent and chaotic parts (signal-to-noise ratio).
The paper deals with the modifications in photocount distributions due to dead-time following every registered count. In order to study the effects due to dead-time, the distribution of the interval between successive photocounts is studied. For the case of a constant dead-time, the mean count rate is derived using the dead-time neglected two-fold counting generating function. The result is also generalised to cover the case when the dead-time has a distribution with the different dead-times independently and identically distributed. In view of the complexity of the problem, a lower bound is obtained for the mean square number of photocounts in an arbitrary interval.
A simulation technique is described and utilized to generate superposed coherent and chaotic (thermal) radiation of arbitrary spectral shape. The statistical properties of a simulated radiation field with a Lorentzian spectral density is investigated with a photoelectron counting experiment. The experimental photocount distribution and normalized mth-order factorial moments are compared to theory and verify that the simulated radiation field, an appropriately modulated laser beam, has the expected statistical properties. The concept described indicates that one may, in principle, generate a thermal source of comparable intensity to that of a laser, arbitrary spectral shape, and bandwidth.