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Journal of Physics A Mathematical and General

Clip-level crossing rates in photon correlation and analogue signal processing

J C Brown, Eric Jakeman, John G. McWhirter, Edward R. Pike

Published Oct 1, 1977• 1 citation
Publisher

Abstract

An investigation is made of the rate at which the number of counts in consecutive sample times in a photon counting experiment is found to lie on either side of a fixed level. The same method is used to generalise the conventional theory of zero-crossings of an analogue signal to include finite processing bandwidth. In this case the resulting formulae are applicable to certain situations of practical importance, such as the Lorentzian spectrum, which have previously given singular results.

DOI
10.1088/0305-4470/10/10/013

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An investigation is made of the rate at which the number of counts in consecutive sample times in a photon counting experiment is found to lie on either side of a fixed level. The same method is used to generalise the conventional theory of zero-crossings of an analogue signal to include finite processing bandwidth. In this case the resulting formulae are applicable to certain situations of practical importance, such as the Lorentzian spectrum, which have previously given singular results.
Digital autocorrelation of optical signals is discussed with particular reference to the technique of clipping, reported briefly in an earlier publication. A detailed analysis of the relationship between the autocorrelation functions of clipped and unclipped photon-counting fluctuations is given. By evaluating the generating function for the joint distribution of intensity fluctuations, corrections to the earlier results are calculated which arise if the time over which the signal is sampled is not negligible compared with any coherence time of the light. The effect of clipping in heterodyne experiments is also analysed.
The joint probability of finding m2 photons in detection interval 2 and m1 photons in detection interval 1 was previously determined by the author [ J. Mod. Opt.34, 91 ( 1987)] in terms of a series expansion of the time-integrated intensity correlation function with coefficients given by a class of orthogonal polynomials in the discrete variables m2 and m1. This expansion is now used to obtain an expression for the average of two general functions υ(m2) and u(m1), 〈υ(m2)u(m1)〉. These functions are specialized to cover two important situations: singly and doubly clipped covariances for essentially arbitrary counting times. In the doubly clipped case, different clipping levels are permitted in the two channels in order to study the influence of clipping mismatch. Numerical calculations, in the form of graphs, are used to illustrate typical behavior.
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