Conical diffraction complexified: dichroism and the transition to double refraction

With dichroism (anisotropic absorption), the dielectric tensor of a crystal becomes non-Hermitian, and each optic axis (diabolical point) splits into two singular axes (branch-point degeneracies). For a Gaussian beam incident on a dichroic crystal in a direction near the singular axes, the polarized light beyond the crystal is given by the same diffraction integrals as for a transparent crystal illuminated along the optic axis, but with the variables complexified. Unexpectedly, the effects of absorption and beam direction can be described with a single parameter. The theory predicts several new types of interference, visible in logarithmic intensity plots: between the nonorthogonally polarized geometrical rays, and between these rays and waves diffracted by the singular axes. The phenomena are analysed in terms of saddle-point, end-point and uniform asymptotics of the diffraction integrals.