Asymptotic Evaluation of the Field at a Caustic

TLDR

In high‑frequency wave theory, the leading‑order asymptotic term is the geometrical optics field, whose caustics—points where this term diverges—can occur even when the exact field remains regular. The study applies the general asymptotic method to compute the lowest‑order field for a plane wave incident on a parabolic cylinder and for a circular‑cylinder segment reflector. They derived the geometrical optics field for a plane wave on a parabolic cylinder, evaluating its lowest‑order term at the focus and surrounding region, and performed a similar analysis for a circular‑cylinder segment reflector. Reflection from cylindrical walls shows that caustic points alter the asymptotic expansion so that the lowest‑order term depends on k with a positive power, and a π/2 phase jump occurs along rays crossing a caustic.

Abstract

The asymptotic expansion of wave fields in powers of 1/k for large k, where k is the wave number, has as its lowest-order term what is commonly known as the geometrical optics field. The caustics of geometrical optics are those point sets on which the zero-order terms become infinite. It is well known that caustics may exist even where the exact wave field is perfectly regular. An investigation of reflection from cylindrical walls of arbitrary cross section shows that the occurrence of caustic points means a change in character of the asymptotic expansion of the true field such that the lowest-order term is no longer independent of k but actually contains a factor k raised to a positive power. There also occurs a jump in phase along a ray passing through a caustic which, as is well known, equals π/2. As an application of the general method, the geometrical optics field is worked out in detail for the case of a plane wave incident on a parabolic cylinder, and the field is obtained in its lowest order at the focus and in the neighborhood of the focus. A similar problem for a reflector consisting of a segment of a circular cylinder is also considered in detail.