Stokes parameters as a Minkowskian four-vector

Abstract

It is noted that the Jones-matrix formalism for polarization optics is a sixparameter two-by-two representation of the Lorentz group. It is shown that the four independent Stokes parameters form a Minkowskian four-vector, just like the energy-momentum four-vector in special relativity. The optical filters are represented by four-by-four Lorentz-transformation matrices. This fourby-four formalism can deal with partial coherence described by the Stokes parameters. A four-by-four matrix formulation is given for decoherence effects on the Stokes parameters, and a possible experiment is proposed. It is shown also that this Lorentz-group formalism leads to optical filters with a symmetry property corresponding to that of two-dimensional Euclidean transformations.

Symmetries Shared by the Poincaré Group and the Poincaré Sphere

“…This replacement enables us to formulate the Lorentz group for the Stokes parameters [15]. Furthermore, this attenuation matrix makes it possible to make a continuous transformation from one matrix to another by adjusting the attenuation parameters in optical media.…”

“…Let us go back to the four-momentum matrix of Equation (15). Its determinant is m 2 and remains invariant.…”

“…In Section 2, we listed four two-by-two matrices whose repeated applications lead to the most general form of the two-by-two matrix, G. It is known that every G matrix can be translated into a four-by-four Lorentz transformation matrix through [4,9,15]: …”

“…We now have the four-vector, (S 0 , S 3 , S 1 , S 2 ), which is Lorentz-transformed like the space-time four-vector, (t, z, x, y), or the energy-momentum four-vector of Equation (15). This Stokes four-vector has a three-component subspace, (S 3 , S 1 , S 2 ), which is like the three-dimensional Euclidean subspace in the four-dimensional Minkowski space.…”

Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere

“…The Poincaré sphere for polarization optics contains the SL(2, c) symmetry isomorphic to the four-dimensional Lorentz group applicable to the Minkowski space [3][4][5][6][7]. Thus, the Poincaré sphere extends Wigner's picture into the three space-like and one time-like coordinates.…”