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Journal of Applied Physics

Effects of Spectral Truncation on Photocount Statistics

Gerard Lachs, Neil F. Ruggieri

Published Mar 1, 1972• 3 citations
Publisher

Abstract

For many photocounting experiments, it is desirable to know the theoretical photocount distribution for radiation whose spectral shape may be altered by an optical filter or truncated in some manner. The effect of spectral truncation (band limiting) is investigated by computing the photocount distribution by a method that makes direct use of the power spectral density. Some data are presented for truncated Lorentzian spectra.

DOI
10.1063/1.1661263

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For many photocounting experiments, it is desirable to know the theoretical photocount distribution for radiation whose spectral shape may be altered by an optical filter or truncated in some manner. The effect of spectral truncation (band limiting) is investigated by computing the photocount distribution by a method that makes direct use of the power spectral density. Some data are presented for truncated Lorentzian spectra.
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.
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.
It is well known that the photocount distribution associated with the photoelectric detection of fluctuating light beams carries information about the probability density of the light intensity. In this paper, we present an exact solution to a problem of particular interest which arises in connection with the photoelectric detection of narrow-band Gaussian (thermal or pseudothermal) light, namely, the problem of determining the photocount distribution when the counting-time interval is not necessarily short compared to the coherence time of the light. The method developed in this paper is applied to the case where the spectral profile of the light is Lorentzian; it leads to an exact expression for the photocount generating function and it provides simple recurrence relations for the photocount distribution and for its factorial moments.
The evaluation of a photon-pair source employs characteristic metrics like the photon-pair generation rate, heralding efficiency, and second-order correlation function, all of which are determined by the photon number distribution of the source. The photon number distribution, however, can be altered due to spectral or spatial filtering and optical losses, leading to changes in the above characteristics. This paper theoretically describes the effects of different filterings, losses, and noise counts on the photon number distribution and related characteristics. From the theoretical description, an analytic expression for the effective mode number of the joint spectral density is also derived. Compared with previous methods for estimating the photon number distribution and characteristics, an improved methodology is introduced along with a suitable metric of accuracy for estimating the photon number distribution, focusing on photon-pair sources. We discuss the accuracy of the calculated characteristics from the estimated (or reconstructed) photon number distribution through repeated simulations and bootstrapped experimental data.