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Journal of Applied Physics

Photocount time interval distribution for superposed coherent and chaotic radiation

Gerard Lachs, David R. Voltmer

Published Jan 1, 1976• 9 citations
Publisher

Abstract

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).

DOI
10.1063/1.322323

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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).
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.
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 consider the photocount statistics of a field formed by superposition of a coherent signal of an arbitrary form, and the Brownian noise (Ornstein--Uhlenbeck process). We obtain an analytical expression for the generating function of photocounts. In the case of a harmonic coherent signal, we analyze the count statistics of the superposed field as a function of the shift of the carrier frequency of the harmonic signal relative to the maximum of the Lorentz spectral contour, as a function of the partial intensity components of the superposed radiation, and also as a function of the product of the duration of photodetection and the spectral line width of the noise. I. The aim of the present work is to derive and study a formula for the probability of counts under the assumption that the photodetector registers an optical field formed by the superposition of a coherent signal and Brownian noise. This problem is of interest in the study of the count distribution in a detector which absorbs a mixture of coherent and incoherent radiation [i]. In the derivation of the formula for the probability density of the number of counts, one encounters specific mathematical difficulties associated with the fact that the photodetection process is extended in time. The probability P(m, T) that, during a time interval T, m photoelectrons are detected when the superposition of a coherent signal B(t) and noise ~(t) falls on the detector, is given by the Mandel formula [2]
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.
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.
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.