Gaussian effective medium theory of dielectric properties of inhomogeneous systems

Inhomogeneous systems, where optical properties change continually, can be analyzed in the ellipsometrical modeling software of Woollam by the use of option GRADED LAYERS, where a main layer is divided into several sublayers, and optical properties of these sublayers lead to effective properties. From a mathematical point of view, this modeling involves evaluations of appropriate integrals, which are assessed so far by only view data points. Precise assessment of integrals by the simplest trapezoidal rule would require, however, hundreds of data points. In this presentation, the authors show a theoretical approach which can improve such modeling in which still several points are needed but the integration outcome is much more precise. The approach uses a well known formula for a chain of capacitors and the principles of Gaussian integration leading to the model of the Gaussian effective medium approximation. As an example, an ultrathin layer of liquid crystal has been used with inhomogeneity induced by the walls. The properties of the order parameter tensor are obtained here from the Onsager theory. The connection of the order parameter tensor with the dielectric tensor components with the rule of chain capacitors allows for obtaining effective dielectric constants for the whole sample of the inhomogeneous layer. Changes in the values of such effective dielectric constants induced by the existence of the walls with respect to uniform bulk values for examined liquid crystal are shown to be so strong that they almost canceled anisotropy. Such a result can be a good motivation for further considering this model for future applications. The proposed effective medium model has been applied then to thin films of the poly vinylo carbazol polymer for which ellipsometric measurements and modeling provide effective values of the optical properties. The agreement between the experimental results and the outcome of the effective medium theory Gaussian effective medium approximation is found to be very satisfactory. Application of this method to porous materials has been also discussed. The method is general and can be applied to any system with inhomogeneous and continually changing optical properties. The proposition how to improve the option Graded Layer in completeease software of Woollam has been also given.