Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike

The paraxial propagation of a beam incident along an optic axis of a biaxial crystal slab is studied in detail. Analytical descriptions are given for the Poggendorff bright and dark rings (associated with the conical singularity of the dispersion surface), and the axial spike (associated with the toroidal ring in the dispersion surface). The rings and spike depend on distance from the crystal. In sharpest focus, the rings are close and asymmetrical, and the spike is faint. Further away, the rings separate, they develop weak diffraction oscillations, and the spike grows in intensity. Eventually the oscillations disappear and the rings become symmetrical, and the axial spike dominates. The images depend on the profile of the incident beam; explicit formulae are given for a Gaussian beam and a coherently illuminated pinhole. Geometrical optics (extended to complex rays for the Gaussian beam) can describe some aspects of the images, in particular the Poggendorf dark ring, which arises from antifocusing and for which an explicit description is given.