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Physical Review

Coherence Properties of Blackbody Radiation. II. Correlation Tensors of the Quantized Field

C. L. Mehta, E. Wolf

Published Jun 1, 1964• 63 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.134.a1149

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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.
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.
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.
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.
Statistical properties of blackbody radiation are discussed by the techniques of second quantization. Various probability distribution functions are derived, in particular the Bose-Einstein distribution function and its generalization for an arbitrary number of momentum-spin states. The Sudarshan phase space distribution function for blackbody radiation is obtained. Although in general the Sudarshan distribution function appears to be a generalized function, it is found that for the radiation field considered here, it becomes an ordinary function and is in agreement with expressions derived recently somewhat heuristically by other authors. A certain new probability distribution function for a statistcal radiation field is introduced and is expressed in terms of the eigenvalues of the annihilation operator. Unlike the phase space distribution function of Sudarshan, this function is non-negative. Finally an expression is derived for this probability distribution function appropriate to blackbody radiation.
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.
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.