The geometrical optics field at a caustic

The asymptotic expansion of a wave field in powers of
1/k
, where
k
is the wave number, for large k 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 teen becomes 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
\pi/2
in the case of a focal point, but which may differ from
\pi/2
in the case of more general types of caustics. In addition, the geometric 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.