Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh-Rice theory

摘要： An efficient and numerically stable method for calculating the optical quantities of multi-layer systems with slightly rough boundaries using the second order Rayleigh–Rice theory is developed. It is assumed that the mean planes of the boundaries are parallel and all the media forming the system are nonmagnetic, isotropic and homogeneous. The perturbation series is formulated using the four-dimensional formalism inspired by the Yeh matrix formalism, but the final result is written using the two-dimensional formalism which is more efficient for the numerical calculations. The final formulae, which are expressed using an arbitrary power spectral density function, include the mixing between the p and s polarizations occurring for anisotropic roughness. Although in the general case the calculation of optical quantities requires evaluation of double integrals, it is shown that for the power spectral density function given by the isotropic Gaussian function some integrals can be calculated analytically and only single integrals have to be evaluated numerically. The random roughness of boundaries is a defect that occurs frequently in practice, and it must be taken into account in the optical characterization and synthesis of thin film systems exhibiting this defect. The presented method is suitable for these purposes, since both of the mentioned applications require methods that are very fast.