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Determination of the characteristic directions of lossless linear optical elements

We show that the problem of finding the primary and secondary characteristic directions of a linear lossless optical element can be formulated in terms of an eigenvalue problem related to the unimodular factor of the transfer matrix of the optical device. This formulation makes any actual computation of the characteristic directions amenable to pre-implemented numerical routines, thereby facilitating the decomposition of the transfer matrix into equivalent linear retarders and rotators according to the related Poincaré equivalence theorem. We explain in detail how this issue arises in the context of stress analysis based on integrated photoelasticity or hybrid methods combining photoelastic measurements with analytical stress models and/or numerical Finite-Element computations for the stress tensor field. Furthermore we show how our results can be applied when algorithms for the reconstruction of the dielectric tensor in the interior of a photoelastic model (dielectric tensor imaging) are tested for their stability against noise in the measurement data. For the sake of completeness we provide a brief derivation of the basic equations governing integrated photoelasticity.