A quantitative analysis of internal conical refraction in biaxial crystals, for which an extensive knowledge of ray geometry is not required, is presented. Except for the wave incident along an optic axis, the various waves in the incident wave bundle are shown to couple to modes of definite wave velocity and polarization in the crystal, and the dependence of these quantities on wave direction is found, using a first-order expansion in the angle between an optic axis and the wave vector. The Poynting vector for each mode is determined. From the dependence of the Poynting vector on the wave direction, the shape of the cone is determined. Our mathematical expansion permits this to be done in a simple manner. In particular, this analysis shows that rays with the same polarization do not lie along radii of the cone, a result not previously given. Pecularities associated with the wave incident along the optic axis, and reasons for not observing them, are discussed. Use of such an explicit analysis aids in clarifying the nature of internal conical refraction and its consequences.
D. L. PORTIGAL and E. BURSTEIN, "Internal Conical Refraction," J. Opt. Soc. Am. 59, 1567-1573 (1969)