ABSTRACT
The Mueller matrix is a 4×4 real matrix which relates a set of incident Stokes parameters to the exiting Stokes parameters. Despite this simple definition, the mathematical properties of Muller matrices are quite complex due to depolarization. In a depolarizing interaction, polarized light becomes partially polarized; the degree of polarization is reduced.
Retarders are represented by real unitary Mueller matrices, also called orthogonal matrices, which can be related to three retardance components δ H, δ 45, and δ R, which define the Stokes parameters of the retarder fast axis. The basic model of a retarder is an element which splits the light into two orthogonal polarization states (modes) and applies a different phase shift to each mode. This optical path difference is the retardance. Sequences of retarders are analyzed as a series of rotations of the Poincaré sphere.
Ideal diattenuators have two different intensity transmittances for two orthogonal linear eigenpolarizations; thus the name “di” “attenuator”. Similarly, diattenuation has three diattenuation components D H, D 45, and D R. Many commercial polarimeters use the Poincaré sphere to represent the eigenstates associated with the three diattenuation components, three polarizance components, and three retardance components. Ideal diattenuator Mueller matrices are Hermitian matrices with real eigenvalues. The transmitted irradiance of a Mueller matrix and its diattenuation depends only on its first row. Reflection at metal surfaces acts as a diattenuator with retardance. The polarizance is the degree of polarization DoP of the exiting light for unpolarized incident light. The polarizance does not necessarily equal the diattenuation. Weak polarization elements cause only small changes to the polarization state and have Mueller matrices close to the identity matrix times a constant, to account for absorption or transmission losses. Examples include the lens surfaces and mirror surfaces in lenses, microscopes, and telescopes. The Mueller matrix has 16 independent degrees of freedom. One corresponds to loss, three to diattenuation, and three to retardance. The remaining nine degrees of freedom describe depolarization. The depolarization index, the average degree of polarization, and degree of polarization maps and surfaces describe the degree to which a Mueller matrix depolarizes incident states. To be physically realizable, a Mueller matrix must operate on all possible sets of Stokes parameters producing valid exiting Stokes parameters; otherwise the Mueller matrix is not physically realizable. Only a subset of 4×4 matrices are physically realizable Mueller matrices.
