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Journal of the Optical Society of America

Propagation of Higher-Order Coherence Functions in Optically Lossy Media

Joel L. Zuckerman, John B. DeVelis

Published Oct 1, 1967• 2 citations
Publisher

Abstract

In this paper we extend the calculation of the propagation of the mutual power spectrum, Γˆ(x1,x2,ν), for nonlinear dielectrics to the case of optically lossy media. In particular, in the quasi-monochromatic limit, the dependence of the mutual coherence function of the second harmonic wave on the full fourth-order coherence of the incoming radiant flux is discussed. An experiment utilizing this phenomenon is suggested.

DOI
10.1364/josa.57.001176

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In this paper we extend the calculation of the propagation of the mutual power spectrum, Γˆ(x1,x2,ν), for nonlinear dielectrics to the case of optically lossy media. In particular, in the quasi-monochromatic limit, the dependence of the mutual coherence function of the second harmonic wave on the full fourth-order coherence of the incoming radiant flux is discussed. An experiment utilizing this phenomenon is suggested.
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 this paper we extend the formulation of the propagation of higher-order coherence functions in nonlinear media to include optical parametric amplification. It is shown that, in the strictly monochromatic limit the coherence formulation leads exactly to the solutions previously reported. This is demonstrated for complete phase matching. In addition, the quasi-monochromatic case is also considered and suggests that measurements of the higher-order coherence functions can give information about the nonlinear medium.
In this paper we define the normalized coherence function of arbitrary order (m, n), in a manner which seems to be a natural generalization of that defined for the second-order coherence function. Both classical and quantized optical fields are considered and the results are compared. It is shown that for classical fields and also for quantized optical fields having nonnegative definite diagonal coherent state representations of the density operator, the modulus of these normalized coherence functions is bounded by the values 0 and 1. This definition differs from the one recently given by Glauber for quantized optical fields, where the normalized coherence functions may take arbitrarily large values even for fields having nonnegative definite diagonal representations of the density operator. Conditions for ``complete coherence,'' i.e., those under which the modulus of the normalized coherence function attains the limiting value 1, are discussed. Some consequences of stationarity and quasi-monochromaticity are also discussed.
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, 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 higher-order coherence functions of optical fields involving phase fluctuations have not yet been studied extensively. We discuss here an experiment which can in principle yield the fourth-order coherence function. This experiment is a study of the correlation function of the irradiance fluctuations obtained in some interference experiments. Two kinds of delay are then used: optical delays in the interferometer and electrical delays with suitable variations in the correlator.The results of the experiment are calculated theoretically with a particular statistical model of laser light. The possibility of using a spectrum analyzer instead of a correlator is discussed.