Treatment of optical propagation in crystals using projection dyadics

The propagation of light in crystals has many features that suggest an eigenvalue problem involving a real symmetric dyadic, such as the occurrence of two distinct phase velocities vo and ve for the ordinary and the extraordinary waves and the orthogonality of their polarizations. The usual treatment by the use of Fresnel’s formulas, however, does not reveal an eigenvalue problem. It is shown in this paper that there is indeed an eigenvalue problem but that it relates, not directly to the permittivity dyadic ?, but to the dyadic P↘⋅?−1⋅P↘, where P↘ is the projection dyadic onto a plane orthogonal to the propagation vector k↘. The displacement vectors ?o and ?e for the ordinary and extraordinary waves act as eigenvectors of this dyadic with eigenvalues 1/εo and 1/εe. These effective permittivities εo and εe determine the phase velocities vo= (μεo)−1/2 and ve= (μεe)−1/2 of the ordinary and extraordinary waves, respectively.