Vector representation of behavior of polarized light in a weakly inhomogeneous medium with birefringence and dichroism

TLDR

Maxwell’s equations in a weakly inhomogeneous, absorbing, anisotropic, optically active medium reduce to a differential equation whose coefficients (Ω and T) encode linear and circular birefringence and dichroism along the propagation axis. The study investigates a vectorial and geometrical representation of light polarization in such a medium. By transforming the differential equation into Stokes and normalized polarization vectors, the authors derive a compact vector equation ∂s/∂x3=Ω^×s+(T^×s)×s that explicitly incorporates birefringence and dichroism. This vector equation clearly describes the geometrical evolution of polarization on the Poincaré sphere for the inhomogeneous absorbing medium.

Abstract

An investigation is made of the vectorial and geometrical representation of the polarization of light propagating through a weakly inhomogeneous absorbing anisotropic and optically active medium. When the approximations of geometrical optics are used, Maxwell's equations lead to the equation ∂G/∂x3 = (i/2)(Ω + iT)G, governing the behavior of polarized light propagating along the x3 axis in the medium, where x3 is the propagation distance along a light path, G is the complex amplitude of the electric vector, the vectors Ω = (0, Ω1, Ω2, Ω3) and T = (T0, T1, T2, T3), whose basis vectors are the unit matrix and the Pauli spin matrices, represent the optical properties of the medium. The two successive transformations of the resulting equation by the Stokes vector and the normalized polarization vector s yield a simple vector equation ∂s/∂x3=Ω^×s+(T^×s)×s, where Ω^ = (Ω1, Ω2, Ω3) and T^ = (T1, T2, T3) are defined as the birefringent vector and the dichroic vector, respectively, representing the birefringence and the dichroism of the absorbing medium. The component Ω1 (or T1) shows the linear birefringence (or the dichroism) along the x1 and x2 axes, Ω2 (or T2) shows the linear birefringence (or dichroism) along the bisectors of the x1 and x2 axes, and Ω3 (or T3) shows the circular birefringence (or dichroism). The vector equation can represent clearly the geometrical behavior of the polarization of light in the inhomogeneous absorbing medium with the help of the Poincaré sphere.