Theory of Attenuated Total Reflection Including Effects of Roughness

We have developed a theory of attenuated total reflection applicable to multi-layered samples with roughness at surfaces and/or interfaces. Angle dependence of reflectivity for incident light with any polarization are expressed by a power series of profile functions of roughness, and the terms up to the second order polynomials of the profile functions are presented. Numerical analyses for Au films in the Kretschmann configuration reveal that differences between the predictions taking into account roughness and the ones assuming flat interfaces depend on both lateral and longitudinal sizes of roughness, and that the differences becomes not negligible for roughness greater than a certain level, e.g., roughness whose correlation length $a$ and root-mean-square $\delta$ of the amplitude are 20 and 2 nm, respectively.