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Journal of Applied Physics

Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation

S. Laxpati, Gerard Lachs

Published Nov 1, 1972• 18 citations
Publisher

Abstract

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.

DOI
10.1063/1.1661006

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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.
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).
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.
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.
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.
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]
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.