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Journal of the Physical Society of Japan

An Application of Green's Function in the Theory of Optical Coherence

Yutaka Kano

Published Apr 1, 1968• 2 citations
Publisher

Abstract

The relation between the retarded Green's function and second-order correlation tensor of electromagnetic potential for a pure electromagnetic radiation in thermal equilibrium is derived. This relation holds also for a long wavelength electromagnetic radiation in media, so long as the radiation is in thermal equilibrium. The electromagnetic second-order correlation tensors for blackbody radiation is obtained, using the relation mentioned above. Upon following Abrikosov, Gorkov and Dzyaloshinski the retarded Green's function for a electromagnetic radiation interacting with matter is expressed in terms of the complex dielectric constant which is necessarily introduced for carrying out a macroscopic formulation. With the assumption of spatial homogeneity of media the electromagnetic second-order correlation tensors for an optical field are obtained. These correlation tensors are also expressed in tems of the complex dielectric constant. This retarded Green's function is simply related to the temperature Green's function, so that the diagram analysis will possibly be applied in the theory of optical coherence for an interacting radiation with matter.

DOI
10.1143/jpsj.24.869

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The relation between the retarded Green's function and second-order correlation tensor of electromagnetic potential for a pure electromagnetic radiation in thermal equilibrium is derived. This relation holds also for a long wavelength electromagnetic radiation in media, so long as the radiation is in thermal equilibrium. The electromagnetic second-order correlation tensors for blackbody radiation is obtained, using the relation mentioned above. Upon following Abrikosov, Gorkov and Dzyaloshinski the retarded Green's function for a electromagnetic radiation interacting with matter is expressed in terms of the complex dielectric constant which is necessarily introduced for carrying out a macroscopic formulation. With the assumption of spatial homogeneity of media the electromagnetic second-order correlation tensors for an optical field are obtained. These correlation tensors are also expressed in tems of the complex dielectric constant. This retarded Green's function is simply related to the temperature Green's function, so that the diagram analysis will possibly be applied in the theory of optical coherence for an interacting radiation with matter.
Expressions are derived for the electromagnetic correlation tensors of blackbody radiation defined on the basis of the theory of the quantized field. Correlation functions of all order are considered, but second-order ones are discussed in detail; it is found that these are identical with those obtained on the basis of semi-classical theory in part I of this investigation. This result illustrates a recent theorem of E.C.G. Sudarshan relating to the equivalence between semiclassical and quantum mechanical description of statistical light beams.
In a recent paper, cross-spectra tensors of quantized stationary electromagnetic fields were introduced and some of their properties were discussed. In the present paper, closed expressions for the cross-spectral tensors of blackbody radiation are derived and their behavior is illustrated by a number of diagrams. It is found that the correlation distance of a spectral component of wavelength $\ensuremath{\lambda}$ is itself of the order of $\ensuremath{\lambda}$ and is therefore independent of the temperature of the radiation.
The electromagnetic correlation tensors of second order is calculated for a superposition of a single coherent mode and blackbody radiation, using Glauber's formula for the density operator of superimposed multi-mode fields. In order to investigate the temporal coherence of this particular radiation, upon restricting to the electric correlation tensors, the complex degree of coherence is evaluated at various temperatures of blackbody radiation. The results are shown in several figures. For comparison the complex degree of coherence of blackbody radiation is also shown in the figure.
An investigation is made of the coherence of the electromagnetic field in a plane dielectric waveguide excited by an external field with specific coherence properties. The quantum theory of an electromagnetic field in one-dimensionally inhomogeneous media is employed to derive an expression which relates the correlation functions of the field inside the waveguide to the external field. This expression is used to study the variation of the coherence of the radiation at various distances z from the area where the radiation is coupled into the waveguide. If the external field has a finite line width, the spatial coherence is a periodic function of z when z ≪ z cr and tends to a constant value when z ≫ z cr . When z ≪ z cr , the temporal coherence is the same as for the external source, whilst when z ≫ z cr , the temporal coherence as a function of τ has maxima which move away from the point τ = 0 with increasing z . It is found that the coherence tunes τ 1 and τ 2 determined following Wolf and Mandel behave differently when z ≫ z cr : τ 1 2 increases quadratically with increasing z , whereas τ 2 tends to a constant value. The form of the radiation modulation spectrum is deduced. Curves are plotted for the spatial and temporal coherence by way of a specific example.
This paper is concerned with the extension of some recently reported results, especially those of Bourret (1960), relating to coherence properties of blackbody radiation. An explicit expression for the complex electric correlation tensor ${\mathcal{E}}_{\mathrm{ij}}(\mathrm{r}, \ensuremath{\tau})$ of blackbody radiation is derived, and on the basis of it spatial coherence ($\ensuremath{\tau}=0$) is discussed in detail. The behavior of diagonal as well as nondiagonal components is illustrated by contour diagrams. In particular, it is found that the nondiagonal components of the correlation tensor, even though being zero for zero-space separation (r=0), acquire, in general, nonvanishing values when r \ensuremath{\ne} 0. The magnetic and mixed correlation tensors are also discussed.
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.