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Physical Review

Measurement of Fourth-Order Coherence Functions

Mark J. Beran, John B. DeVelis, George B. Parrent

Published Feb 25, 1967• 8 citations
Publisher

Abstract

In this paper we outline a method which allows one, in principle, to measure $〈{E}_{i}({\mathrm{x}}_{1}, t){E}_{j}({\mathrm{x}}_{2}, t){{E}_{k}}^{*}({\mathrm{x}}_{3}, t)\ifmmode\times\else\texttimes\fi{}{{E}_{l}}^{*}({\mathrm{x}}_{4}, t)〉$ (in general, ${\mathbf{x}}_{1}$\ensuremath{\ne}${\mathbf{x}}_{2}$\ensuremath{\ne}${\mathbf{x}}_{3}$\ensuremath{\ne}${\mathbf{x}}_{4}$), for stationary quasimonochromatic light propagating principally in the forward direction. The procedure makes use of the properties of nonlinear dielectrics in a sequence of interference experiments. No correlation of photoelectron currents is used, and as a special case the contracted moment $〈{|{E}_{i}({\mathrm{x}}_{1}, t)|}^{2}{|{E}_{j}({\mathrm{x}}_{2}, t)|}^{2}〉$ measured by Hanbury Brown and Twiss could be obtained in this manner. It is shown that it should be possible to measure the full fourth-order coherence function in the laboratory for laser light, using presently obtainable intensity levels.

DOI
10.1103/physrev.154.1224

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In this paper we outline a method which allows one, in principle, to measure 〈Ei(x1,t)Ej(x2,t)Ek∗(x3,t)\ifmmode×\else\texttimes\fiEl∗(x4,t)〉〈{E}_{i}({\mathrm{x}}_{1}, t){E}_{j}({\mathrm{x}}_{2}, t){{E}_{k}}^{*}({\mathrm{x}}_{3}, t)\ifmmode\times\else\texttimes\fi{}{{E}_{l}}^{*}({\mathrm{x}}_{4}, t)〉〈Ei​(x1​,t)Ej​(x2​,t)Ek​∗(x3​,t)\ifmmode×\else\texttimes\fiEl​∗(x4​,t)〉 (in general, x1{\mathbf{x}}_{1}x1​\ensuremath{\ne}x2{\mathbf{x}}_{2}x2​\ensuremath{\ne}x3{\mathbf{x}}_{3}x3​\ensuremath{\ne}x4{\mathbf{x}}_{4}x4​), for stationary quasimonochromatic light propagating principally in the forward direction. The procedure makes use of the properties of nonlinear dielectrics in a sequence of interference experiments. No correlation of photoelectron currents is used, and as a special case the contracted moment 〈∣Ei(x1,t)∣2∣Ej(x2,t)∣2〉〈{|{E}_{i}({\mathrm{x}}_{1}, t)|}^{2}{|{E}_{j}({\mathrm{x}}_{2}, t)|}^{2}〉〈∣Ei​(x1​,t)∣2∣Ej​(x2​,t)∣2〉 measured by Hanbury Brown and Twiss could be obtained in this manner. It is shown that it should be possible to measure the full fourth-order coherence function in the laboratory for laser light, using presently obtainable intensity levels.
The second-order autocorrelation function, g(2)(0){{g}^{{(2)}}}(0)g(2)(0) , measured in the Hanbury Brown–Twiss scheme is the strong indicator separating the thermal (incoherent) from the coherent state of light emitted by lasers or photon or polariton condensates. Consequently, g(2)(0){{g}^{{(2)}}}(0)g(2)(0) can be used to determine if the Bose–Einstein condensate forms in the system. The fast dynamics of non-equilibrium light–matter Bose–Einstein condensate make implementing the Hanbury Brown–Twiss scheme challenging and expensive. A good time resolution requires single-photon detectors operating at cryogenic temperatures. An alternative approach to measuring g(2)(0){{g}^{{(2)}}}(0)g(2)(0) , taken in many experiments, is to rely on the scheme with a single detector and measure the intensity fluctuations to directly obtain g(2)(0){{g}^{{(2)}}}(0)g(2)(0) . In this paper, we show that the Hanbury Brown–Twiss scheme and the single detector scheme are not equivalent in the general case because we cannot directly use the intensity fluctuations measured in the single detector scheme to obtain g(2)(0){{g}^{{(2)}}}(0)g(2)(0) of light emitted by a multi-mode system. Nevertheless, we develop a method allowing for the reconstruction of g(2)(0){{g}^{{(2)}}}(0)g(2)(0) from the measurements in the single detector scheme for a multi-mode system with fast thermalization and illustrate it for a special case of non-equilibrium light–matter Bose–Einstein condensate.
This article presents a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum. Analyses based on both the classical and quantum theories are described. After a brief historical introduction, the elementary concepts which are frequently employed in the discussion of interference phenomena are summarized. The measure of second-order coherence is then introduced in connection with the analysis of a simple interference experiment and some of the more important second-order coherence effects are studied. Their uses in stellar interferometry and interference spectroscopy are described. Analysis of partial polarization from the standpoint of correlation theory is also outlined. The general statistical description of the field is discussed in some detail. The recently discovered universal "diagonal" representation of the density operator for free fields is also considered and it is shown how, with the help of the associated generalized phase-space distribution function, the quantum-mechanical correlation functions may be expressed in the same form as the classical ones. The sections which follow deal with the statistical properties of thermal and nonthermal light, and with the temporal and spatial coherence of blackbody radiation. Later sections, dealing with fourth- and higher-order coherence effects include a discussion of the photoelectric detection process. Among the fourth-order effects described in detail are bunching phenomena, the Hanbury Brown-Twiss effect and its application to astronomy. The article concludes with a discussion of various transient superposition effects, such as light beats and interference fringes produced by independent light beams.
An intensity-correlation experiment is proposed to test the second-order coherence of laser light from a cw-operated laser based on the Glauber formalism. The calculation utilizes the known form of the density matrix for radiation from a randomly excited source such as a discharge tube which is equivalent to filtered black-body radiation, and represents the laser radiation field by a coherent state. The intensity correlation calculated exhibits the standard Hanbury Brown-Twiss term and an extra term due to intensity interference between the Fourier components of the thermal field and the laser mode.
We investigate the propagation of Γˆ(x1,x2,ν) the Fourier transform of the mutual coherence function Γ(x1,x2,τ) in nonlinear dielectrics. We assume that the classical phenomenological equations govern the electric field within the dielectric, and we restrict our discussion to the propagation of fields with narrow angular spread. The indeterminate character of the propagation equations is emphasized, and it is shown that a determinate equation results only if the nonlinear term is small. In the quasimonochromatic limit, we investigate for this latter case the dependence of the two-point coherence of the second harmonic on the full fourth-order coherence of the incident beam, a result implied by Ducing and Bloembergen (1964), and Akhmanov et al. (1966). We also show under what circumstances the Hanbury-Brown and Twiss experiment may be performed using a single nonlinear dielectric.
In the history of optical science, we were satisfied with the measurement of mean intensity of light, namely the measurement of first-order coherence of light, for optical studies and observations, until the middle of 20th century when two astrophysicists Hanbury Brown and Twiss (HBT) introduced their “intensity interferometer”. “Correlation” emphasizes its statistical nature, while “coherence” emphasizes its interference nature. The chapter extends the concept of second-order coherence to Nth-order coherence (N > 2) and analyzes a few simple measurements on the third-order temporal and spatial coherence of thermal field. It also calculates the second-order temporal coherence function and the second-order spatial coherence function of thermal field by assuming idealized photodetectors and correlation measurement electronics.
A new experimental procedure for determining the (second-order) coherence time of a light beam is described that is based on a combination of conventional interferometry with an intensity-correlation technique. It permits measurements of coherence times that are several orders of magnitude shorter than the resolving times of the detectors, which ordinarily limit correlation measurements. The validity of the method is demonstrated by an experiment in which the transverse fluorescent light from a dye laser is measured.