Anisotropic extension of the Parratt formalism

Neutron and X-ray reflectometry are widely used for investigating thin multilayer structures. Numerical modeling of such systems is commonly performed using the Parratt formalism or the method of characteristic (transfer) matrices. The conventional Parratt method was originally formulated for isotropic media, whereas the transfer-matrix approach can treat anisotropic (magnetic) systems but is prone to numerical instabilities, particularly for multilayers with a large number of periods, where the total thickness becomes large under grazing-incidence conditions. In this work, a generalized Parratt formalism applicable to anisotropic layered systems is derived. Explicit expressions are obtained for both reflectivity and transmissivity. A simulation and data-analysis program implementing the new formalism has been developed and used to perform numerical studies of representative multilayer structures. Direct comparisons with transfer-matrix calculations demonstrate the improved numerical stability of the generalized Parratt approach. A mathematical analysis clarifies the origin of the instabilities inherent in the transfer-matrix formulation. The treatment of rough interfaces is also discussed, and several commonly used approximations are compared.