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Physical Review A

Scattering of electromagnetic fields of any state of coherence from space-time fluctuations

E. Wolf, John T. Foley

Published Jul 1, 1989• 45 citations
Publisher

Abstract

A theory is developed, valid within the accuracy of the first-order Born approximation, of the scattering of electromagnetic fields that applies under much more general conditions than do the current theories. The incident field is assumed to be statistically homogeneous and stationary, of arbitrary state of coherence, of arbitrary state of polarization, and to have arbitrary spectrum. The medium is assumed to be linear and spatially and temporally random, with the randomness being characterized by an ensemble that is statistically homogeneous, isotropic, and stationary. The analysis is not restricted to scattering from a medium in thermal equilibrium and applies in both off-resonance and near-resonance regions of the medium.

DOI
10.1103/physreva.40.579

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A theory is developed, valid within the accuracy of the first-order Born approximation, of the scattering of electromagnetic fields that applies under much more general conditions than do the current theories. The incident field is assumed to be statistically homogeneous and stationary, of arbitrary state of coherence, of arbitrary state of polarization, and to have arbitrary spectrum. The medium is assumed to be linear and spatially and temporally random, with the randomness being characterized by an ensemble that is statistically homogeneous, isotropic, and stationary. The analysis is not restricted to scattering from a medium in thermal equilibrium and applies in both off-resonance and near-resonance regions of the medium.
Scattering of polychromatic light by a medium whose dielectric susceptibility is a random function of position is considered within the accuracy of the first Born approximation. It is shown, in particular, that if the two-point spatial correlation function of the dielectric susceptibility has Gaussian form and the spectrum of the incident light has a Gaussian profile, the spectrum of the scattered light may be shifted toward the shorter or the longer wavelengths, depending on the angle of scattering. The results are analogous to those derived recently in connection with radiation from partially coherent sources [ Nature (London)326, 363 ( 1987)].
In the light of the perspective of statistical similarity, we examine the maintenance of the second-order coherence of a light wave on weak scattering from a random medium. Some new and nontrivial results relating to properties of the scattered field which remains the second-order coherence of the incident field are presented. By assuming that the scattered field remains the second-order coherence, we can show that all of higher order correlation functions of Fourier component of the scattering potential can reduce to the like-factorization forms with a series of constant coefficients. These coefficients furnish an efficient and direct way to describe the higher order coherence property of the scattered field. We also show that the combination of the maintenance of second- and fourth-order coherence implies the scattered field coherence to all orders. Finally, the structure feature of the random medium is also discussed when the coherence of the incident field is retained up to 2nth order, in particular, in the case of the second-order coherence. Our theory is an important contribution for understanding of spatially fully coherent scattered fields, and also gives a general and new method to discuss the variation of the coherence of the scattered field.
Within the validity of the first-order Born approximation, we study the spectrum changes generated by the scattering of light with a tunable spectral degree of coherence (SDOC) from a spatially deterministic medium. Before interacting with the medium, a scalar plane wave is incident upon Young's pinholes, which are employed to produce the incident wave with a tunable SDOC. By assuming that both the scattering potential of the medium and the spectrum of incident plane waves obey Gaussian profiles, the spectrum of the scattered field is derived as a linear combination of Fourier transforms of the scattering potentials. Changes in the spectrum of the scattered field are strongly dependent on the scattering angle, the effective radius of the medium, and Young's pinhole parameter. Moreover, red shifts occurring in the scattered field can be enhanced by either reducing the scattering angle or increasing the effective radius. The results presented here generalize the previous reports, where the light with prescribed functions of the SDOC scattered from a deterministic medium was considered.
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.
The basic equations for the statistical system of waves in a randomly fluctuating medium are presented in the same form as in quantum mechanics, assuming a Markov process for the temporal change of the medium. A simple model is chosen for the Fokker-Planck equation of the medium which gives rise to a fluctuation of the Gaussian process, and several results are given to illustrate how the methods used in quantum mechanics or quantum field theory can be applied almost without change. Thus, the physical variables are represented by linear operators, and their equations of motion are determined by an equation similar to the Heisenberg equation of motion. The system has two stationary states (corresponding to the vacuum states 〈0∣〈0|〈0∣ and ∣0〉|0〉∣0〉 in quantum field theory), and the rather standard methods in field theory can be used for the evaluation of the Green's functions. When the temporal changes of the medium are sufficiently large compared with those of the waves, an adiabatic approximation is possible, and the isomorphic transformation (corresponding to the unitary transformation) is employed to lead to the result that the statistical system of the waves for this model is in perfect correspondence with the waves of bosons which interact with each other only through a two-body potential.
Introduction We will now study the properties of fluctuating electromagnetic fields, paying attention mainly to the optical region of the electromagnetic spectrum. It seems hardly necessary to stress that every electromagnetic field found in nature has some fluctuations associated with it. Even though these fluctuations are, as a rule, much too rapid to be observed directly, one can deduce their existence from suitable experiments that provide information about correlations between the fluctuations at two or more space-time points. The simplest manifestations of correlations in optical fields are the well-known interference effects that arise when two light beams that originate from the same source are superposed. With the availability of modern light detectors and electronic circuitry of very short resolving time, other types of correlations in optical fields began to be studied in more recent times. These investigations, as well as the development of lasers and other novel types of light sources, led to a systematic classification of optical correlation phenomena and the complete statistical description of optical fields. The area of optics concerned with such questions is now generally known as optical coherence theory . The first investigations of coherence phenomena are due to Verdet (1865, 1869) and von Laue (1907a, b). Some early investigations of Stokes (1852) and Michelson (1890, 1891a, b, c, 1892, 1920) although not explicitly mentioning coherence – because this concept is of a much later origin – have also contributed to the clarification and development of this subject.