Theory of Gyrotropic Birefringence

Abstract

A theoretical treatment of the optical effect known as gyrotropic or nonreciprocal birefringence is presented. By suitably renormalizing the electric dipole moment tensor, it is shown, for the case of lossless media, that 10 of the 18 independent quantities in the gyrotropic-birefringence tensor have their origin in electric quadrupole effects. The other eight are shown to be related to the magnetoelectric effect. The general results are applied to the materials ${\mathrm{Cr}}_2$ ${\mathrm{O}}_3$ and MnTi${\mathrm{O}}_3$. The propagation of a plane wave along one of the crystalline axes of ${\mathrm{Cr}}_2$ ${\mathrm{O}}_3$ is then considered. It is shown that the gyrotropic birefringence exhibits itself as a rotation of the principal optic axes, together with a change in the velocity of propagation of the wave in the medium. Next, the modified boundary conditions corresponding to the renormalized field vectors are given, and the case of a plane wave normally incident on a gyrotropically birefringent medium is discussed. It is noted that the field relations at a boundary will be modified even when the quadrupole contribution vanishes and the magnetoelectric tensor is isotropic, a case in which there is no gyrotropic birefringence in the medium itself. Foinally, a quantum-mechanical calculation of the gyrotropic-birefringence tensor at 0°K is given. The expression obtained is applied to the case of ${\mathrm{Cr}}_2$ ${\mathrm{O}}_3$, and the electric quadrupole and magnetoelectric contributions are separated. It is roughly estimated that, at optical frequencies, the electric-quadrupole-induced rotation of the principal optic axes of ${\mathrm{Cr}}_2$ ${\mathrm{O}}_3$ is of the order of ${10}^{-6\mathrm{}}$ rad, and the magnetoelectric-induced shift is two orders of magnitude less.