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Journal of Applied Physics

Approximate Photocount Statistics for Coherent and Chaotic Radiation of Arbitrary Spectral Shape

Gerard Lachs

Published Feb 1, 1971• 34 citations
Publisher

Abstract

A method for computing the approximate photocount statistics for Gaussian light is presented. This method may be used for the superposition of coherent radiation with chaotic radiation of arbitrary spectral shape. Data is presented for Gaussian-, triangular-, and square-shaped spectra as well as for Lorentzian-shaped spectra. The results show that the photocount statistics are significantly dependent upon spectral shape for intermediate time-bandwidth products. The results also show that the Bedard, Chang, and Mandel approximation gives a better fit to a Gaussian-shaped spectrum then to a Lorentzian-shaped spectrum. Significant differences between the photocount statistics for a time-bandwidth product of 10 and Poisson statistics were also obtained; even for a signal-to-noise ratio of 40:1.

DOI
10.1063/1.1660069

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A method for computing the approximate photocount statistics for Gaussian light is presented. This method may be used for the superposition of coherent radiation with chaotic radiation of arbitrary spectral shape. Data is presented for Gaussian-, triangular-, and square-shaped spectra as well as for Lorentzian-shaped spectra. The results show that the photocount statistics are significantly dependent upon spectral shape for intermediate time-bandwidth products. The results also show that the Bedard, Chang, and Mandel approximation gives a better fit to a Gaussian-shaped spectrum then to a Lorentzian-shaped spectrum. Significant differences between the photocount statistics for a time-bandwidth product of 10 and Poisson statistics were also obtained; even for a signal-to-noise ratio of 40:1.
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.
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).
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.
Recently derived formulae for the integrated intensity distribution, the photon-counting distribution and its factorial moments in the statistics of the superposition of coherent and chaotic multimode fields are proposed as approximate formulae for light of arbitrary spectrum. It is shown by explicit calculations of the third factorial moment for the superposition of a one-mode coherent field with a Gaussian-Lorentzian field that the proposed formulae hold with good accuracy over a wide range of conditions. An application to the determination of spectral parameters of light is given.
Results have been presented by Bedard (see abstr. A33736 of 1967) on the derivation of the N-fold joint photocount distribution of a Gaussian (thermal) radiation field of arbitrary spectral profile, when the counting-time intervals are short compared with the coherence time of the light. In this paper these results are extended for arbitrary counting-time intervals. The authors assume that the fluctuating radiation field has a fixed direction of propagation and that all the photodetector surfaces lie within one coherence area on a plane normal to this direction of propagation. The evaluation of the -fold joint photocount distribution and the photocount moments can be reduced to the solution of either a homogeneous or an inhomogeneous (Fredholm) integral equation of the second kind.