Characterization of heterogeneous media using nonlinear Stokes–Mueller polarimetry

Abstract

The nonlinear optical properties of heterogeneous media are described by an algebraic decomposition of the correlation matrix X(n) of the nth-order nonlinear susceptibility tensor values, obtained from the generalized nonlinear Stokes–Mueller polarimetry measurements. The correlation matrix X(n) forms a square Hermitian semidefinite positive matrix, which can be further decomposed into separate components using projection matrices and calculating eigenvalues and the corresponding eigenvectors for pure states of the heterogeneous media. Up to eight pure states can be deduced for the sum-frequency generation process and up to sixteen states can be presented for a four-wave mixing process. The obtained eigenvalues are used to define the entropy characterizing a heterogeneous media. Filtering and maximum-likelihood estimation approaches are presented to determine the physically realizable values of the nonlinear susceptibility from the X(n) matrix that is obtained by the Stokes–Mueller polarimetry experiments. The Cholesky decomposition is employed for the maximum-likelihood estimation of the correlation matrix X(n). The structural characterization can be used, for example, in nonlinear microscopy for determining the ultrastructure organization of heterogeneous biological samples.