Abstract

Longitudinal cusp caustics (which unfold along the direction of propagation) are of interest when modeling the focusing produced by the passage of light through irregular media¹ or reflected by curved surfaces. In the present research I model diffraction patterns characteristic of transverse cusp caustics which unfold perpendicular to the propagation direction of the most strongly focused ray. It is assumed that the amplitude exp [ik(g — ct)] in an x-y plane results from propagation (from an unfocused source) through irregularities or from reflection. If this wave is to propagate beyond the x-y plane in a homogeneous medium to produce a shear-free transverse cusp, it is sufficient to consider g = a₁x² + a₂y²x + a₃y², a₂ ≠ 0, with real a_j. The Fresnel approximation of the resulting 2-D diffraction integral gives a wave field proportional to the 1-D Pearcy integral P(X, Y) or to P^*(X, Y) depending on the sign of a₁ + (2z)⁻¹ where z is the distance beyond the x-y plane. The real parameters X, Y depend on the a_j, z, k, and the transverse coordinates in the observation plane. Stationary-phase points of the diffraction integral locate the rays from the x-y plane to the observation point.

© 1986 Optical Society of America