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Journal of Modern Optics

Autocorrelation Central Hole in Clipped Photon-counting Detection Estimation from the Zero-count Probability Density

Manuel P. Cagigal, Pedro M. Prieto

Published Jul 1, 1995• 0 citations
Publisher

Abstract

In this paper we develop a procedure to avoid the autocorrelation central hole that appears in clipped photon-counting detection. This method can be understood as a higher order approximation than the Hofmann procedure. It can be applied in combination with any of the alternative methods to evaluate the autocorrelation function in clipped conditions proposed up until the present. We especially analyse that based on the zero-count probability measurement. Fairly good autocorrelation estimates are obtained for low intensity signals.

DOI
10.1080/09500349514551281

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In this paper we develop a procedure to avoid the autocorrelation central hole that appears in clipped photon-counting detection. This method can be understood as a higher order approximation than the Hofmann procedure. It can be applied in combination with any of the alternative methods to evaluate the autocorrelation function in clipped conditions proposed up until the present. We especially analyse that based on the zero-count probability measurement. Fairly good autocorrelation estimates are obtained for low intensity signals.
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.
The theoretical treatment of clipped photocount autocorrelation of Gaussian light, introduced by Jakeman and Pike, has been extended to include the effects of finite-duration sampling intervals and incomplete spatial coherence. A system of ``interval partitioning'' is used to derive theoretical expressions for 〈Nq(0)N(τ)〉 and 〈N0(0) N0(τ)〉, the single- and double-clipped photocount autocorrelation functions. Here N(t) is the number of photons detected in a sampling interval centered at t and Nq(t) is the clipped count at level q. Each original interval is systematically partitioned in space and time. Multidimensional generating functions are applied to the sets of subintervals with the object of expressing g(1) (t), the first-order field correlation function, in terms of measurable quantities. It is shown that under certain well-defined conditions, 〈Nq(0) N(τ)〉−〈Nq〉〈N〉=γq|g(1)(τ)|2 and 〈1−N0〉−2−[1−2〈N0〉+〈N0(0)N0(τ)〉]−1=φ|g(1)(τ)|2, where, for a given set of experimental parameters, γq and φ can be taken as simple multiplicative factors independent of the delay time τ. For comparative purposes, a similar analysis is performed for the unclipped correlation function 〈N(0)N(τ)〉. Finally, computer calculated values of various quantities of interest are presented for several representative cases.