Uniform asymptotic expansions for wave propagators via fractional transformations

By use of the fractional Fourier transformation, a new representation is developed for the wave propagator of scalar diffraction theory, and its asymptotic form is then derived by using the fractional Legendre transformation. The resulting expression remains valid in the presence of caustics and automatically accounts for the associated and problematic phase shifts. Although the derivation is long, the final result is simple and takes a form that is strikingly similar to the conventional expression. The new asymptotic method that is established here has both extended validity and significant advantages.