ABSTRACT
Polarization elements comprise polarizers, retarders, and depolarizers. Mirrors, lenses, thin films, and nearly all optical elements alter polarization to some extent, but are not considered as polarization elements because this is not their primary role. Polarization properties are classified into three categories: diattenuation or polarization dependent amplitude change, retardance or polarization dependent phase change, and depolarization, a random reduction in the degree of polarization. Dichroic material have a polarization dependent absorption coefficient. The diattenuation is a metric for the strength of a polarizer or partial polarizer. Birefringent materials have a polarization dependent refractive index. Birefringent materials are used for constructing retarders. Retardance is the optical path difference between the two eigenpolarizations. Retardance is commonly specified in four different units: radians, waves, degrees, or nanometers.
The Jones matrix relates an arbitrary incident monochromatic polarization state to the corresponding exiting polarization state by matrix-vector multiplication. The Jones matrix provides a powerful method to describe sequences of polarization elements and the intrinsic polarization properties of ray paths through optical systems. Homogeneous polarization element have orthogonal eigenpolarizations. Otherwise the element is inhomogeneous. Polarizers and partial polarizers have hermitian Jones matrices with real eigenvalues. Retarders have unitary Jones matrices with unimodular eigenvalues. Unitary transformations change the basis of a Jones matrix. Arbitrary Jones matrices can be decomposed into products of a hermitian and unitary matrix by the polar decomposition. The polar decomposition is calculated using the singular valued decomposition or by an algorithm involving matrix square roots. Two identical polarization elements with nilpotent Jones matrices in series pass no light.
Vortex retarders are analyzed as a Jones matrix example. Vortex retarders are spatially varying half wave retarders where the fast axis of the retarder rotates as a function of the angle from the x-axis. Vortex retarders can generate an optical vortex, a phase singularity about a zero of an electromagnetic field where the phase is continuous around the zero, but discontinuous crossing the zero, a screw dislocation.
