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Recovery of the Matrix Operators in the Similarity and Congruency Transformations: Applications in Polarimetry

L. November
Journal / Anthology

Journal of the Optical Society of America
Year: 1993
Volume: 10
Issue: 4
Page range: 719-739

Formulas are presented for the recovery of the matrix operators in arbitrary-order similarity and congruency transformations. Two independent input and output matrix pairs exactly determine the similarity-transformation matrix operator, while three independent Hermitian-matrix pairs are required for the congruency-transformation operator. The congruency transformation is the natural form for the quantum observables of a multiple-element wave function, e.g., for polarized-light transfer: the recovery of the Jones matrix for a nondepolarizing device is demonstrated, given any three linearly independent partially polarized input Stokes states. The recovery formula gives a good solution even with large added noise in the test matrices. Combined with numerical least-squares methods, the formula can give an optimized solution for measures of observation error. A more general operator, which includes the effect of isotropic depolarization, is defined, and its recovery is demonstrated also. The recovery formulas have a three-dimensional geometric interpretation in the second-order case, e.g., in the Poincaré sphere. It is pointed out that the geometric property is a purely mathematical property of quantum observables that arises without referring to spatial characteristics for the underlying wave function.

*Science > Physics > Optics
*Science > Physics > Quantum Physics