Morphology of wave fronts and caustics associated with a circular wave refracted by an arbitrary curve: The two-dimensional axicon

In this work we use geometrical optics to study the global behavior of the morphology of the wave fronts and their respective caustic associated with the refraction of a circular wave propagating between two given optical media with constant refraction indexes, which are separated by an arbitrary curve. For this purpose, we construct an expression for Stavroudis's general solution to the eikonal equation and obtain analytical equations for the wave fronts and caustic associated with a circular wave refracted by an arbitrary curve after being emitted by a point light source located at an arbitrary position immersed in a two-dimensional homogeneous optical medium. The caustic is computed by using the definitions of the critical and caustic sets of a two-dimensional (2D) map that describes the evolution of an arbitrary wave front associated with Stavroudis's general solution. We find that the caustics have only one branch. Our general results are applied to a conical refracting singular curve of the v-wedge type, which is called the 2D axicon. We find that when the point light source is lying on the optical axis of the axicon curve and near the apex, the virtual caustic locally has singularities of the cusp type, while the singularities associated with the reversion of the refracted wave fronts in the virtual caustic region are of the cusp and double-cusp with self-intersections types. We conclude that the caustic is associated with a local property of nonparallel light rays when the refracting curve is smooth locally and that the multiple-image formation process is associated with a global behavior between nonlocal light rays emanating from disjoint regions of the refracting curve, such that they intersect in the disjuncaustic region. In particular, we remark that caustic fields of the needle type do not exist in a 2D geometry and are limited to surfaces with a symmetry of infinitesimal rotations about the same axis in three dimensions.