Equations Of Geometrical Optics Research Articles

An asymptotic theory of guided waves

A procedure is developed for finding asymptotic expansions at high frequencies of the solutions of Helmholtz's equation subject to boundary conditions on certain guiding surfaces. This includes surface waves along surfaces of rather general shapes, and wave-guide modes in a class of non-uniform waveguides. Guided waves have some features of both eigenvalue (mode) and radiation problems. The method of this paper combines the two techniques, finding “modes” that propagate along rays in the general waveguide region and whose amplitudes vary along the paths of propagation. The phases of these modes are found from two coupled equations, one analogous to the eiconal equation of geometrical optics, and the other analogous to the eigenvalue or “transverse resonance” equation of waveguides. The amplitudes are asymptotic series in inverse powers of the wavenumber, and the coefficients satisfy a set of ordinary differential equations that can be solved recursively. It is found that the ray paths are not only functions of the refractive index (as in “pure” radiation problems), but depend also on the local geometrical properties of the guiding surface.

Ray-optical techniques for guided waves

While ray-optical methods have been employed for the analysis of high-frequency radiation and scattering problems and also for the study of optical resonators, their use for description of guided modes has been less evident. This paper presents a systematic procedure whereby the asymptotic functional dependences and eigenvalues of modes in uniform and nonuniform guiding structures are deduced by purely ray-optical means, utilizing either the eiconal and transport equations of geometrical optics or a modal consistency argument. The method is illustrated first on the simple example of a homogeneous plane slab, and is then applied to various nonuniform waveguides including an angular sector, a more general taper of either parabolic or hyperbolic shape, and a guide filled with an axially inhomogeneous dielectric. The role of caustics in the latter configurations is emphasized, and stress is placed throughout on a physical interpretation of relevant concepts which retain their validity even under conditions more general than those considered at present.

Completely coherent radiation

Some aspects of the scalar coherence theory of radiation employing two-point correlation functions are considered. In particular, it is shown that “real” waveforms in polychromatic fields need not appear similar to be considered coherent under the present definition. The decomposition of the mutual coherence function for such fields is discussed. Under certain assumptions a general form for coherent fields is derived and these fields are shown to satisfy the equations of geometric optics. For homogeneous media the surfaces of constant phase are the so-called cyclides of Dupin. The significance of these results in relating coherent sources and fields is mentioned as is the decomposition of incoherent sources.