From the anisotropic T wave propagator, a numerical method for the film-substrate interface exit-matrices is derived. The principal issues: are emphasized such as the use of the 4x4 Berreman matrices, the Cayley-Hamilton theorem for the calculation of the exponential operator, and then the choice of the eigenvectors implied in the exit-matrices This code is well adapted to simulate and compare to transmission or reflection data measured with either a Generalized Ellipsometer or a Mueller polarimeter.
For a large number of anisotropic substrates such as flexible displays and very thin films, the optical properties can be measured by Mueller polarimeters as well as standard generalized ellipsometers. The experimentalist has then to deal with a large quantity of acquired data such as with these 4x4 Mueller matrices Mij. The aim is to revise the existing formalism and give a global view in order to build a general algorithm and calculate the Mij’s and the associated Jones matrix elements measured in Generalized Ellipsometry (GE). The explicit equations given hereafter treat the transmitted and reflected amplitude of the electromagnetic waves when propagating in anisotropic multilayer media deposited on an anisotropic substrate.
The problem of the propagation of electromagnetic waves has been established and treated since the beginning of the sixties, following the formalism of F. Abelés and J. Billard , which has been adapted by D. W. Berreman  and P. Yeh . A pioneering introduction to the ellipsometric measurements of anisotropic materials has been presented in R.M.A. Azzam’s book . Many authors have succeeded in implementing different computational codes but using restrictive approaches. A remarkable synthesis is given by M. Schubert and B. Rheinländer,   (the same annotations as in , are used here for sake of clarity). Since 1996, the topic is still being under consideration. . Current research about the propagation of light in anisotropic material is in fact driven by the growing interest of new synthesized materials for semiconductor nanotechnologies and liquid crystal display industry and the new class of metamaterials. An attempt is made here, to describe a consistent approach for numerically solving the problem where some highlights of the reverse problem are given....
see the article...Mueller matrix formalism I : how to implement an efficient numerical code when considering the Analysis of Generalized Ellipsometry data : the propagation of light in stratified anisotropic media
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