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Physical Review

Correlation in the Fluctuating Outputs from Two Square-Law Detectors Illuminated by Light of Any State of Coherence and Polarization

L. Mandeļ, E. Wolf

Published Dec 15, 1961• 68 citations
Publisher

Abstract

By treating light fluctuations as a random process (stationary and Gaussian to second order), an expression is derived for the correlation in the output fluctuations of two square-law detectors which are illuminated by a plane light wave of any state of coherence and polarization. The expression takes a particularly simple form when, as is usually the case, the light is spectrally pure in the sense of a definition introduced elsewhere. The solution yields, as a special case, the basic formula relating to the Hanbury Brown-Twiss effect. The generalization discussed here is of particular interest for correlation experiments performed with light beams from optical maser sources. Moreover, the present analysis appears to be simpler than other treatments previously given in connection with more restricted cases.

DOI
10.1103/physrev.124.1696

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By treating light fluctuations as a random process (stationary and Gaussian to second order), an expression is derived for the correlation in the output fluctuations of two square-law detectors which are illuminated by a plane light wave of any state of coherence and polarization. The expression takes a particularly simple form when, as is usually the case, the light is spectrally pure in the sense of a definition introduced elsewhere. The solution yields, as a special case, the basic formula relating to the Hanbury Brown-Twiss effect. The generalization discussed here is of particular interest for correlation experiments performed with light beams from optical maser sources. Moreover, the present analysis appears to be simpler than other treatments previously given in connection with more restricted cases.
The basic formulae governing the fluctuations of counts registered by photoelectric detectors in an optical field are derived. The treatment, which has its origin in Purcell's explanation of the Hanbury Brown-Twiss effect, is shown to apply to any quasi-monochromatic light, whether stationary or not, and whether of thermal origin or not. The representation of the classical wave amplitude, of the light by Gabor's complex analytic signal appears naturally in this treatment. It is shown that the correlation of counts registered by N separate photodetectors at N points in space is determined by a 2 N th order correlation function of the complex classical field. The variance of the individual counts is shown to be expressible as the sum of terms representing the effects of classical particles and classical waves, in analogy to a well-known result of Einstein relating to black-body radiation. Since the theory applies to correlation effects obtained with any type of light it applies, in particular, to the output of an optical maser, although, for a maser operating on one mode, correlation effects are likely to be very small.
The distribution of counts from a photoelectric detector illuminated by light of bandwidth Δν 0 is analysed by associating the photons with Gaussian random waves. This is shown to lead to a full statistical description of the counts. It is shown that the number n T in a time interval T << 1/Δν 0 obeys pure Bose-Einstein statistics, and that the fluctuations in longer intervals T >> 1/Δν 0 are simply the density fluctuations of a boson assembly in a phase space of ~ Δν 0 T cells. The correlation coefficient ρ of the fluctuations of counts from two detectors illuminated by partially coherent beams is found to be proportional to the local time average of the square of the coherence function ⟨γ 12 2 ⟩. The correlation is shown to depend on the degeneracy of the beams in such a way that ρ → 2⟨γ 12 2 ⟩ for highly degenerate beams. The results are all consistent with those obtained by Hanbury Brown and Twiss in 1957.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
We discuss general polarization-sensitive correlation measurements of light fluctuations based on the Hanbury Brown and Twiss interference detection scheme. Such measurements are considered as a basis for correlation spectroscopy, which allows us to observe the correlations of the fluctuations of all polarization components (Stokes parameters) of the probe light. The interferometer scheme is discussed in application to the study of the polarization-dependent interaction of light with an atomic ensemble with polarized angular momentum and put into an external magnetic field. We describe the general procedure for evaluating the photocurrent correlation function in the case of an optically thin atomic medium. Our calculations are based on the Keldysh diagram technique which is usually used for the study of non-equilibrium systems by means of the second quantized formalism. It allows us to express the Raman field-correlation functions and parametric scattering processes describing the output light fluctuations in terms of a different type of time-ordered electric field and atomic Green functions. We show that with definite conditions the photocurrent correlation function can be written as the sum of two terms proportional to the correlation functions of atomic density matrix fluctuations in symmetric and antisymmetric forms. This expression of the photocurrent correlation function can be used in various different applications.