Polynomial solution for two thicknesses of a multilayer system from a single ellipsometric measurement

It is shown that the unknown thicknesses of any two transparent layers in an arbitrary multilayer system from a single ellipsometric measurement can be found by the solution of an eighth-degree real polynomial. The method gives directly all the possible physical solutions, which are computed from the real roots of the polynomial. The coefficients of the polynomial are determined by the angle of incidence, the refractive indices of all the phases, and the thicknesses of the other layers. The method is used on a simulated system of air/silicon nitride/silicon oxide/silicon.