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Journal of the Optical Society of America

Statistical properties of counting distributions for intensity-modulated sources

Paul R. Prucnal, Malvin Carl Teich

Published Apr 1, 1979• 15 citations
Publisher

Abstract

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.

DOI
10.1364/josa.69.000539

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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.
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.
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.
In recent years there has been considerable utilization of photocount statistics for studying theoretical and experimental properties of a quantized electromagnetic field. The electromagnetic field, which has been of greatest interest in these areas, is a superposition of coherent and chaotic radiation. In this paper closed-form recurrence equations are derived for the photocount distribution and the factorial moments for this case. The photocount distribution is then obtained by using a well-known solution for p(0, T), and the factorial moments are generated by using the fact that the zeroth factorial moment is unity.
This paper is concerned with the determination of the quantum statistics of the photoelectric detection, and the E-field detection of multiple-mode superposed chaotic and coherent radiation. The Glauber P-representation is utilized extensively, and results are obtained by a fully quantum-mechanical treatment. The generating function, the mean, and the variance of the photocount statistics are derived for illumination times of arbitrary duration and for modes of arbitrary statistical interdependence. For short illumination times, the complete photocount distribution is obtained. The distribution has the same form as that for a single mode of superposed chaotic and coherent radiation. The multiple-mode distribution, however, is nonstationary. The difference between experimental determinations of the mean and variance which utilize periodic sampling and those which utilize random sampling is derived. A method for determining the P-representation for modulated fields is described, and the detection statistics are derived. The statistics of the E field are found to be Gaussian, and it is found that for multiple-mode classical statistics and quantum statistics differ.
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).
We discuss the validity and accuracy of onefold approximate photocount statistics of mixtures of chaotic and coherent radiation. We determine the relative error for factorial cumulants and moments between approximate and exact solutions for heterodyne detection. It is shown how the accuracy increases as the signal-to-noise ratio grows. Other generalizations of the approximation approach can be made to improve the coincidence between the exact and approximate solutions, but their usefulness is limited. Some numerical results are obtained when the chaotic line shape is Lorentzian.