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Journal of the Optical Society of America A

New theory of partial coherence in the space-frequency domain Part II: Steady-state fields and higher-order correlations

E. Wolf

Published Jan 1, 1986• 140 citations
Publisher

Abstract

In Part I of this investigation [ WolfE., J. Opt. Soc. Am.72. 343 ( 1982)] new representations were introduced for the cross-spectral density of a steady-state source of any state of coherence. The central concept in that formulation was the notion of a coherent source mode (a natural mode of oscillation). In the present paper the theory is developed further and new representations are obtained for the cross-spectral densities of all orders, both of the source and of the field that the source generates. These representations involve only the previously introduced coherent source modes and the moments of certain random coefficients that characterize the statistical properties of the source. The results provide a new mathematical framework for analyzing coherence properties of all orders of stationary sources and of stationary fields. Some potential applications of the theory are mentioned.

DOI
10.1364/josaa.3.000076

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In Part I of this investigation [ WolfE., J. Opt. Soc. Am.72. 343 ( 1982)] new representations were introduced for the cross-spectral density of a steady-state source of any state of coherence. The central concept in that formulation was the notion of a coherent source mode (a natural mode of oscillation). In the present paper the theory is developed further and new representations are obtained for the cross-spectral densities of all orders, both of the source and of the field that the source generates. These representations involve only the previously introduced coherent source modes and the moments of certain random coefficients that characterize the statistical properties of the source. The results provide a new mathematical framework for analyzing coherence properties of all orders of stationary sources and of stationary fields. Some potential applications of the theory are mentioned.
It is shown that, under very general conditions, the cross-spectral density of a steady-state source of any state of coherence may be expressed in terms of certain new modes of oscillations, each of which represents a completely spatially coherent elementary excitation. Making use of this result, a statistical ensemble of strictly monochromatic oscillations, all of the same temporal frequency, is then introduced that yields the cross-spectral density as a correlation function in the space–frequency domain. From these results two new expressions for the Wiener–Khintchine spectrum of the source and also a new mode representation of the cross-correlation function of the source follow at once.
A new representation is obtained for the cross‐spectral density function of stationary optical wavefields. We show that under very general conditions it is possible to construct an ensemble of monochromatic wave functions, all of the same frequency ω, with the property that the cross‐spectral density function W(r−1,r−2,ω) of the field is expressible as a correlation between the values of the wave functions at the points r−1 and r−2. The representation is intimately related to a new mode expansion of the field and has a number of potential applications to various problems that involve wavefields of any state of coherence.
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.
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.