The operator of refractive index N, which characterizes velocities and polarizations of natural waves in their co- and counterpropagation, plays an important role in the dynamics of an electromagnetic field. In an isotropic medium, the N operators form an infinite family of isometries that represent generators of Lie involute algebras, which describe propagation of a polarized light beam. Representations of N in terms of second-rank tensors [the SO(3) group] and traceless 2 2 matrices [the SU(2) group], obtained as a result of extracting the square root from the unit matrix, are considered. These representations are related to the continuous group of rotations about the beam and the continuous group of reflections with respect to surfaces containing the beam. Topological properties of mirrors are characterized by two complex normal vectors S and C, which satisfy the conditions SC = 1 or –1 and characterize inherent properties of wave fronts. Relations of these vectors to the Pauli matrices and the vectors of energy flows are considered for various light involutions.
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