Lorentz Group in Classical Optics

In this chapter, Jones vector formalism is considered for studying the polarization properties of the two-component electric field vector. A general form for the attenuator is introduced, and thereof attenuations, phase-shifts, and the rotation of the polarization axes are reformulated. Their combined effects are investigated and it is shown that they amount to a two-by-two representation of the six parameter SL(2, c) group. In connection with these combined effects, the Wigner rotation is also reviewed. Ray transfer matrices are introduced and their equi-diagonalization process is explained. It is shown how the results of this process facilitate computations when a large number of cycles is required to account for the behaviour of a beam subjected to the effects of a series of optical elements. It is also noted that the Wigner and Bargmann decompositions can be explained in terms of the decomposition properties of the ABCD matrix. This matrix as a general element of the Sp(2) group can be composed by assembling some specific combinations of lens and translation matrices. Concrete physical examples of ABCD matrices governing the propagation of rays in a camera, in a laser cavity, in multi-layers and in an optically active medium are presented. It is observed that they serve as analog computers for the essential features of Wigner’s little groups dictating the internal space– time symmetries of relativistic particles.