Closed-form inversion of the reflection ellipsometric function of a film-substrate system: absorbing-substrate optical constant

A closed-form inversion expression for obtaining the optical constant (complex refractive index) of the substrate of an absorbing-film-absorbing-substrate system from one reflection ellipsometry measurement is derived. If, in addition, the film thickness is to be determined, a second measurement at another angle of incidence may or may not be used. The derived formula does not introduce errors itself, and tolerates errors in input variables very well. Random and systematic errors in the measured ellipsometric parameters do not affect the value obtained for the optical constant of the substrate: it is always the true value to two decimal places. Two examples in ellipsometry and in the design of reflection-type optical devices, one each, are presented and discussed. In addition, experimental results for a commercially available wafer are also presented. Two other closed-form inversion expressions for obtaining the optical constant of the substrate from two and three measurements are also presented and briefly discussed.