Limits of applicability of the method of geometric optics and related problems

We present on a heuristic basis the universal sufficient conditions for applicability of the method of geometric optics. In formulating the criteria, we make essential use of the concept of the “Fresnel volume” of a ray, whose boundary links the first Fresnel zones “threaded” on the ray. The fundamental criterion of applicability requires that the parameters of the medium and of the wave should vary little over the transverse section of the Fresnel volume. The second criterion, which stems from the first, requires that rays incident on the same given point should lie mostly outside the Fresnel volume of an adjacent ray. The effectiveness of these criteria has been demonstrated in many problems of electrodynamics and acoustics that allow a solution more precise than the ray solution. On the basis of the presented criteria, one can reveal the regions of inapplicability of the ray method (focal and caustic regions, penumbra regions in diffraction by screens and convex bodies, regions where lateral waves arise, etc.). If we know the dimensions of the regions of inapplicability, we can also solve a number of related problems. The most important of these problems are: the problem of determining the field in the neighborhood of caustics and foci and the problem of analyzing the wave pattern as a whole. The proposed criteria also allow a generalization to three-dimensional quantummechanical problems, while outlining the limits of applicability of the quasiclassical approximation.