Analytical solutions for layer and substrate parameters determination using amplitude reflection coefficients

This study provides analytical solutions for determining layer and substrate parameters using amplitude reflection coefficients, which can be experimentally obtained through immersion ellipsometry or by applying Kramers-Kronig relations to power reflection data. Four cases are considered: an absorbing layer on an arbitrary or homogeneous substrate, and a non-absorbing layer on an arbitrary or homogeneous substrate. For each case, explicit formulas are provided to determine the layer’s refractive index and thickness. In the case of a non-absorbing layer on a non-absorbing homogeneous substrate, all system parameters, including the layer’s refractive index, thickness, and the substrate’s refractive index, are determined. Error propagation for different methods is analyzed. Experimental examples using immersion ellipsometry are given, demonstrating its effectiveness, particularly in characterizing very thin layers. Software tools for solving these inverse problems have been developed and are available in the Supplementary Material.