Theory of light scattering from self-affine surfaces: Relationship between surface morphology and effective medium roughness

Using Rayleigh-Rice scattering theory we have studied the influence of surface morphology on the optical response of self-affine surfaces. We have established a mathematical relationship between the surface roughness (d) as determined by spectroscopic ellipsometry (SE) using the effective medium approximation (EMA) and the parameters controlling the morphology of the surface: root-mean-square roughness (w), correlation length (ξ), and roughness (Hurst) exponent (α). These three parameters affect the roughness value measured by ellipsometry. However, when the correlation length is smaller than the wavelength, the dependence is contained in a single parameter wδ that is proportional to the product of the surface roughness and the local slope δ=w/wξαξα. The fact that the local slope of a surface increases only very slowly during growth explains the linear dependence experimentally found between w as measured by scanning-probe microscopy and the vertical roughness determined by the effective medium approach.