Abstract

© 2002 Optical Society of America

Introduction

Before optical interference phenomena were first observed, ray optics was widely believed to give an exact description of light propagation. Rays were thought of as infinitesimal conduits of luminous flux or as trajectories of “light particles”. As Hamilton noted, rays are mathematically analogous to particle trajectories in mechanics, which, as the quantum revolution showed, also present a wave character. The discovery of the wave nature of light, however, did not bring an end to the use of rays. In many practical situations, a rigorous wave description of a light field is unattainable, and the simpler ray model can give useful information. In fact, accurate estimates of the wave field can be extracted from the corresponding rays by means of a range of different methods. The purpose of this Focus Issue is precisely to give an overview of some of the latest developments in this area. It must be noted that these methods apply not only to optical fields but are also useful in the description of any wave phenomenon. In fact, many of these tools were originally developed within the fields of acoustics, seismology, and quantum mechanics.

Formally, ray optics can be deduced from wave optics by taking either (or both) of two limits. The first is that of incoherence. The coherence of a field is a measure of its ability to interfere with itself (in a statistical sense). When a field is considerably incoherent, it can be accurately described in a radiometric-like fashion, where each ray carries a certain amount of optical power, and the light intensity at a given point results from the sum of the contributions of all the rays through that point.

The second (and better known) limit that links rays to waves is that of small wavelength. By letting the wavelength tend to zero, the laws of ray optics can be deduced from Maxwell’s equations. In fact, by using asymptotic procedures, ray-based estimates of the wave field can be obtained which are valid for sufficiently small but finite wavelengths. Different alternatives result from applying asymptotic techniques to different representations of the wave field (i.e. the field itself, its spatial Fourier transform, a phase space representation, etc.). The interpretation of what a ray is depends on the chosen alternative.

One of the approaches represented in this Focus Issue is that of sums of Gaussian beamlets. The wave field that is perhaps closest to a ray is a Gaussian beam, since it presents (at least at its waist) maximum localization both spatially and directionally. Due to an optical analogue of Ehrenfest’s theorem, the core line of these beamlets corresponds asymptotically to a ray. A general coherent field can be expressed as a sum of these beamlets, which propagate following simple rules. The paper by Arnold proposes the use of a new discrete orthogonal basis of beamlets. Another alternative, discussed in the paper by Alonso and Forbes, is based on a continuous sum of Gaussian contributions, which are not restricted to propagate like beamlets. Diffraction effects resulting from opaque obstacles can also be modeled in terms of rays. This formalism, introduced by Joseph Keller, is known as the geometrical theory of diffraction. The article by Stamnes compares two strategies for estimating diffracted wave fields, each using a different set of rays.

Finally, this Focus Issue includes two articles concerning the application of ray-based methods to specific problems. The article by Tureci et al. presents a new method, based on the refraction and reflection of Gaussian and Hermite-Gaussian beams centered at stable periodic rays, for estimating quasi-bound modes of dielectric micro-cavities. Flatté’s article gives an overview of the theory and experimental results for wave propagation in random media like the atmosphere and the ocean, and compares the accuracy in this context of ray-based and parabolic wave equation approximations.

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