Effects of rough interfaces in a multilayer stack

The reflected and transmitted fields scattered from a layered structure, build up of N thin films with shallow rough interfaces and bounded by two semi-infinite media are determined to analyze the effects of roughness in the optical response of a given multilayer system. Each layer is assumed to be homogeneous, isotropic, local, linear and characterized by a frequency dependent complex dielectric function. All rough interfaces are modeled by using a stationary random process, and the Rayleigh criteria is used in the mathematical treatment because of the shallowness of the defined roughness. Two independent integral equations relating the reflected and transmitted fields to the incident wave are found. These will allow a recursive calculation of the overall optical responses. The solution involves Fourier coefficients of functions dependent on the roughness profiles. This treatment is shown to be valid for both TM(p) or TE(s) states of polarization. Numerical results are obtained for the case of a two layer system, with two of the interfaces present Gaussian roughnesses and compared to some experimental data.