Chapter 2 Conical diffraction: Hamilton's diabolical point at the heart of crystal optics

This chapter describes how the current understanding of conical refraction has been acquired after nearly two centuries of theoretical and experimental studies. A systematic use of the simplifying approximation of paraxiality is made in the chapter. This is justified by the small angles involved in conical diffraction. Conical diffraction exemplifies a fundamental feature of crystal optics––namely, the diabolical point. Belsky and Khapalyuk's exact paraxial theory of conical diffraction is explored in the chapter with an emphasis on pinhole and Gaussian beams. The chapter focuses on chiral and biaxially birefringent materials. Chirality has been found to destroy the diabolical point by separating the two sheets of the wave surface, reflecting the change of the dielectric matrix from real symmetric to complex Hermitian. The chapter demonstrates that each diabolical point splits into two branch-points, reflecting the fact that the dielectric matrix is non-Hermitian. The effects of absorption on the pattern of emerging light, as well as the combined effects of dichroism and chirality, are described.