Representation of birefringent filters by directly composed 4 × 4 matrices

The interaction of an obliquely incident plane wave with an arbitrary birefringent plane structure can be described exactly by a so-called 4 × 4 layer matrix. However, its formation is rather complex, and therefore a simplified method to set up this matrix is proposed. It is deduced from the boundary conditions for the tangential components of the E and the H fields at the interfaces of the birefringent layer and the phase shifts of the four waves within the layer between both interfaces. The refractive indices and the directions of polarization of the four waves are calculated from the well-known dispersion relation in the principal coordinate system and a coordinate transformation. Transmission curves of Lyot filters made of uniaxial and biaxial materials are calculated in the cases of plane waves of finite or infinite lateral extension that are incident under Brewster's angle upon the filter positioned outside or within a laser resonator. The special examples given for a quartz filter and a YAlO(3) filter are characterized by a high selectivity within a broad bandwidth if Fabry-Perot effects can be neglected. Transmission curves of a quartz filter were verified by measurements.