ABSTRACT
A rigorous mathematical justification of the principle of stationary phase for double integrals may be found in Focke (1954), Braun (1956), Jones and Kline (1958) and Chako (1965). Four of the elementary diffraction catastrophes are associated with two-dimensional diffraction problems, or one-dimensional diffraction integrals. They are called cuspoid diffraction catastrophes. In three-dimensional diffraction problems we must deal with double integrals. Even though catastrophe theory combined with uniform asymptotic approximations suffices to describe a number of caustic phenomena, it is inadequate for studying the focusing of a wavefront whose aberration function is of a general form. Other problems that are beyond the scope of catastrophe theory occur when several nearby stationary points approach the boundary of the integration domain, or when one or several critical points of the first or second kind approach a boundary corner.
