Lorentz transformations in beam polarization optics

Summary form only given. In beam optics, polarization properties of beams can be alternatively described by a vector of the four stokes parameters, by the two-by-two polarization matrix, or by the two, in general, Jones vectors in two-dimensional polarization space. The evolution of the beam along an optical system is then given, in paraxial approximation, with the aid of the Mueller four-by-four matrices or the Jones two-by-two matrices. On the other hand, in the special theory of relativity, we deal with the space-time four-vectors, second-rank spinor matrices or with the two-dimensional spinor vectors. All these entities are transformed under four-dimensional and two-dimensional representations of the restricted six-parameter Lorentz group, respectively. Although these transformations are of different origin in both cases, the Lorentz invariance of a space-time interval in the coordinate, Minkowski space corresponds to the invariance of a polarization degree of the transforming beam. Basic features of the above correspondence are reviewed and a few examples of the Lorentz transformations of the coherent beam polarization are given. Some aspects of this correspondence, not considered in this context in the literature until very recently, are outlined. It is shown that the invariance of the beam polarization transformations specifies a definition of beam amplitude, with transformation properties independent of those related to the beam polarization. A suitable reference frame for the treatment of the beam amplitude and the beam polarization is proposed. A case of the beam transformations at a dielectric interface is discussed in more detail. This leads to a complete identification between scattering coefficients of a multilayer optical structure and the parameters of the corresponding Lorentz transformation.