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Journal of the Optical Society of America

Photoelectron-Counting Distributions for Irradiance-Modulated Radiation*

Paul Diament, Malvin Carl Teich

Published May 1, 1970• 30 citations
Publisher

Abstract

Photoelectron-counting distributions are obtained for modulated radiation sources, with arbitrary modulation waveforms and depths. Cases treated in detail include chaotic, amplitude-stabilized, and Risken sources, with square-wave, triangular, and sinusoidal modulation. Modulation is shown to broaden the counting distributions, an effect interpretable as accentuated photon bunching. The broadening is not marked for the chaotic source, but the shape of the distribution changes drastically for the amplitude-stabilized source. Varying the radiation statistics and the modulation waveforms can produce a variety of counting distributions, from double peaked to extremely flat. Modulation may serve to accentuate the distinctions between counting distributions associated with sources of different radiation statistics.

DOI
10.1364/josa.60.000682

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Photoelectron-counting distributions are obtained for modulated radiation sources, with arbitrary modulation waveforms and depths. Cases treated in detail include chaotic, amplitude-stabilized, and Risken sources, with square-wave, triangular, and sinusoidal modulation. Modulation is shown to broaden the counting distributions, an effect interpretable as accentuated photon bunching. The broadening is not marked for the chaotic source, but the shape of the distribution changes drastically for the amplitude-stabilized source. Varying the radiation statistics and the modulation waveforms can produce a variety of counting distributions, from double peaked to extremely flat. Modulation may serve to accentuate the distinctions between counting distributions associated with sources of different radiation statistics.
Statistical properties such as the cumulants, and the central, factorial, and ordinary moments are obtained for photocounting distributions when the incident radiation is intensity-modulated with arbitrary statistics. The results are applied in some detail to the cases of triangular, sinusoidal, and square-wave modulation of multimode superposed coherent and chaotic radiation. The coefficient of variation, skewness, and kurtosis are obtained as a function of modulation depth. Comparison is made with experimental data in the cases of triangular and sinusoidal modulation of a laser source.
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 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.
This paper is concerned with the study of statistical properties of various optical fields obtained by several types of laser intensity modulation from the stationary random point process point of view. We essentially consider intensity correlations and time intervals distributions (TID). A generalization of the already known TID is given: It is the probability distribution (MTID) of the time interval between one photoelectron and the n th following photoelectron. Some theoretical and experimental results are presented in the general case (the time intervals considered are not necessarily small compared to the coherence time of the field). The modulation effect can be characterized by the photoelectron bunching effect. The TID and the MTID are significantly dependent upon the spectral profile of the field, if the used time scale is of the order of the coherence time. On the other hand, the MTID can be interpreted as the probability of the sum over time intervals between two successive photoelectrons. We treat two cases: If the variables of this sum are uncorrelated, we verify the central-limit theorem; if the variables are correlated we study the influence of the correlations when the number of summed variables becomes large.
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.