Characteristic Parameters in Integrated Photoelasticity: An Application of Poincaré’s Equivalence Theorem

The Poincaré Equivalence Theorem states that any optical element which contains no absorbing components can be replaced by an equivalent optical model which consists of one linear retarder and one rotator only, both of which are uniquely determined. This has many useful applications in the field of Optics of Polarized Light. In particular, it arises naturally in attempts to reconstruct spatially varying refractive tensors or dielectric tensors from measurements of the change of state of polarization of light beams passing through the medium, a field which is known as Tensor Tomography. A special case is Photoelasticity, where the internal stress of a transparent material may be reconstructed from knowledge of the local optical tensors by using the stress-optical laws. – We present a rigorous approach to the Poincaré Equivalence Theorem by explicitly proving a matrix decomposition theorem, from which the Poincaré Equivalence Theorem follows as a corollary. To make the paper self-contained we supplement a brief account of the Jones matrix formalism, at least as far as linear retarders and rotators are concerned. We point out the connection between the parameters of the Poincaré-equivalent model to previously introduced notions of the Characteristic Parameters of an optical model in the engineering literature. Finally, we briefly illustrate how characteristic parameters and Poincaré-equivalent models naturally arise in Photoelasticity.