Analytic solutions for calculating the surface inclination of isotropic media and bare substrates by using reflection-based generalized ellipsometry

Using ellipsometry for curved-surface characterization requires the knowledge of the surfacenormal vector in order to determine material-related surface parameters like refractiveindex, layer thickness, or birefringence of the surface material at the incidence point, becausethe recorded signal depends on both the (unknown) surface normal vector n(x) andmaterial-related surface characteristics. It is convenient in ellipsometry to parametrize thesurface normal vector by the angle of incidence q and the azimuthal rotation angle f . Dependingon the design of the ellipsometer, there may be two angles f1 and f2 necessary,which describe the azimuthal rotation before and after the light is reflected off the sample,respectively. The authors present analytic formulas to determine f1 and f2 for opticallyisotropic samples using generalized ellipsometry. The resulting measurement uncertaintyis lower than that of previously known methods. Furthermore, the authors provide an analyticformula to calculate q from the ellipsometric angles Y and D for bare substrateswith known refractive index N1 = n1−ik1. The formulas have been evaluated with experimentaldata acquired with a conventional and an imaging retroreflection-based return-pathellipsometer.