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Journal of the Physical Society of Japan

Coherent Properties of the Superposed Radiation. I. Temporal Coherence

Yutaka Kano

Published Apr 1, 1976• 2 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.40.1122

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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.
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.
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 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.
The profound distinctions between photon and Fermion optics appear in the second-order coherence properties of the particle field.1 Particularly significant are blackbody radiation2 and thermal electron fields; the correlation functions are uniquely specified by temperature θ and chemical potential u. Differences in the second-order coherence arise principally from considerations of (a) particle conservation (photons are not conserved; electron conservation is rigorous and tantamount to charge conservation); (b) tensorial character of the fields (electromagnetic fields are vector-valued; electron fields are spinor-valued); (c) quantum statistics (photons and electrons are subject to Bose-Einstein and Fermi-Dirac statistics, respectively).
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.