Generalized matrix method for analysis of coherent and incoherent reflectance and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates

A generalized matrix method is presented for calculating the optical reflectance and transmittance of an arbitrary thin-solid-film multilayer structure on very thick substrates with rough surfaces and interfaces. We show that the effect of roughness and the influence of incoherently reflected light on the back side of a thick layer can be accounted for with a more general transfer matrix that enables the inclusion of modified complex Fresnel coefficients. Coherent, partially coherent, and incoherent multiply reflected light inside the multilayer structure is treated in the same way. We demonstrate the method by applying it to simulated and experimental reflectance spectra of thin epitaxial Si overlayers on very thick SiO(2) substrates and on a separation by ion implantation of oxygen structure with a SiO(2) buried layer exhibiting substantial roughness on both of its interfaces (Si/SiO(2) and SiO(2)/Si).