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Physical Review A

Correlation-induced spectral changes and energy conservation

G. S. Agarwal, E. Wolf

Published Nov 1, 1996• 12 citations
Publisher

Abstract

An energy conservation law is derived for fields generated by random, statistically stationary, scalar sources of any state of coherence. It is shown that correlation-induced spectral changes are in strict agreement with this law and that, basic to the understanding of such changes, is a distinction that must be made between the spectrum of a source and the spectrum of the field that the source generates. This distinction, which is obviously relevant for spectroscopy, does not appear to have been previously recognized. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.4424

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An energy conservation law is derived for fields generated by random, statistically stationary, scalar sources of any state of coherence. It is shown that correlation-induced spectral changes are in strict agreement with this law and that, basic to the understanding of such changes, is a distinction that must be made between the spectrum of a source and the spectrum of the field that the source generates. This distinction, which is obviously relevant for spectroscopy, does not appear to have been previously recognized. \textcopyright{} 1996 The American Physical Society.
Recent developments in coherence theory have shown that the spectrum of the field produced by a partially coherent source depends not only on the source spectrum but also on the spatial correlations of the source fluctuations.1 In particular, it has been found that some correlations may give rise to frequency shifts of spectral lines.2 Questions have been raised concerning energy conservation in such situations. In this paper we first discuss energy conservation laws pertaining to fields generated by sources of any state of coherence, and we use these fields to demonstrate that conservation laws are not violated when source correlations produce spectral changes in the emitted radiation. We also show that spectral shifts may be generated by isotropic sources, without violating energy conservation.
This paper presents a review of research, both theoretical and experimental, concerning the influence of coherence properties of fluctuating light sources and of correlation properties of scattering media on the spectra of radiated and scattered fields. Much of this research followed a discovery made in 1986, that the spectrum of light may change on propagation, even in free space. More than 100 papers on this topic have been published to date and many of them are reviewed, or at least mentioned, in this article. After an introduction and a summary of some of the main mathematical results relating to second-order coherence theory of statistically stationary optical fields, spectral changes that may take place on superposing fields produced by two partially correlated sources are discussed. Spectral effects in fields produced by two-dimensional secondary sources and by three-dimensional primary sources are then considered. The section which follows describes spectral changes that may arise when polychromatic light is scattered on media whose physical properties vary randomly either in space and/or in time. A review is also presented of recent research, which has revealed that under certain circumstances the changes in the spectrum of light scattered on random media may imitate the Doppler effect, even though the source, the medium and the observer are all at rest with respect to one another. In the final section a brief review is given of a new emerging technique sometimes called spatial-coherence spectroscopy. It is based on the discovery that it is possible, under certain circumstances, to determine field correlations from spectral measurements.
Abstract : It is generally taken for granted that the spectrum of light does not change on propagation. This belief is implicit in all spectroscopy. In the last few years it was discovered that while such spectral invariance holds for radiation from commonly used sources it does not hold, in general, because the spectrum of emitted radiation depends not only on the source spectrum but also on the coherence properties of the source. It was found that in some cases the induced changes may be appreciable, possibly resulting in frequency shifts of spectral lines. It has also been demonstrated that similar spectral changes may be produced by scattering on random media whose dielectric susceptibility fluctuations are appropriately correlated. Very recently it was shown that under certain circumstances the frequency shifts generated by dynamic scattering may imitate the Doppler effect, even though the source, the scatterer and the observer are all at rest relative to each other. (emk)
It is generally taken for granted that the spectrum of light does not change on propagation. This belief is implicit in all spectroscopy. In the last few years it was discovered that while such spectral invariance holds for radiation from commonly used sources it does not hold, in general, because the spectrum of emitted radiation depends not only on the source spectrum but also on the coherence properties of the source. It was found that in some cases the induced changes may be appreciable, possibly resulting in frequency shifts of spectral lines. It has also been demonstrated that similar spectral changes may be produced by scattering on media whose dielectric susceptibility is appropriately correlated.
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 correlations of the light emitted by a partially coherent source can produce frequency shifts in the spectrum observed in the far field if the correlation function of the emitted radiation does not satisfy a certain scaling law. A Fourier achromat is used to generate a secondary source in which the degree of spectral coherence is independent of wavelength; i.e., it violates the scaling law. The spectrum detected in the far zone of the secondary source is, in general, found to be displaced in frequency and distorted relative to the spectrum measured at the secondary source. The displacement can be toward the higher frequencies or the lower frequencies depending on the direction of observation.