A generalized equivalence theorem for polarization theory is formulated and proven. It is shown that anisotropic properties of homogeneous nondepolarizing media can be presented as a combination of four basic mechanisms: linear and circular phase and linear and circular amplitude anisotropy. Expressions for the generalized effect operators of algebraic (or operator) optics are obtained and the inverse problem of crystal optics is solved in terms of physically realizable anisotropy parameters.

The polarization of light when it passes through optical media can change as a result of change in the amplitude (dichroism) or phase shift (birefringence) of the electric vector. The anisotropic properties of media can be determined from these two optical features. We derive the conditions required for polarization elements to be dichroic and birefringent. Our derivation starts from commonly accepted assumptions for dichroism and birefringence. Our main conclusions are that (i) the generalized Jones matrix for dichroic elements has in general nonorthogonal eigenpolarizations and (ii) in the general case, the birefringent and dichroic properties of polarization elements have no direct association with the corresponding phase and dichroic polar forms derived in the polar decomposition of the polarization elements’ Jones matrices.

A theoretical analysis of eigenpolarizations and eigenvalues pertaining to the Jones matrices of dichroic, birefringent, and degenerate polarization elements is presented. The analysis is carried out employing a general model of a polarization element. Expressions for the corresponding polarization elements are derived and analyzed. It is shown that, despite the presence of birefringence, a polarization element can, in a general case, demonstrate a totally dichroic behavior. Moreover, it is proved that birefringence necessarily accompanies dichroic elements with orthogonal eigenpolarizations. A transition between degenerate, dichroic, and birefringent eigenvalues is studied, and examples of synthesis of polarization elements are given.