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Physical Review

Coherence Properties of Blackbody Radiation. III. Cross-Spectral Tensors

C. L. Mehta, E. Wolf

Published Sep 25, 1967• 63 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.161.1328

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Journal of the Physical Society of Japan•Apr 1, 1968•2 citations
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 \ensuremathλ\ensuremath{\lambda}\ensuremathλ is itself of the order of \ensuremathλ\ensuremath{\lambda}\ensuremathλ and is therefore independent of the temperature of the radiation.
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 Paper I of this investigation, dynamical laws which describe the space-time development of second-order coherence tensors of a quantized electromagnetic field in vacuo, and the associated conservation laws, were derived. In the first part of the present paper these results are specialized to stationary fields. Second-order cross-spectral tensors for such fields are then introduced, and some of their properties are discussed. A relation analogous to the Wiener-Khintchine theorem of the theory of stationary random processes is derived. Various non-negative-definiteness conditions obeyed by the cross-spectral tensors are established, and equations which govern the spatial variation of these tensors are deduced. Certain analytic properties of the correlation tensors are derived, and some of their consequences are examined. It is also shown that in the limiting case when the two space-time arguments of the coherence tensors coincide, two of our conservation laws reduce to the averaged form of the energy and the momentum conservation laws of the electromagnetic field.
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.
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).
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.