Polynomial inversion of the single transparent layer problem in ellipsometry

It is shown that for a uniform transparent layer over a substrate the layer dielectric constant satisfies a fifth-degree polynomial. The problem of extracting the layer index and thickness from the ellipsometric measurement is then reduced to finding the roots of this polynomial. The coefficients of this polynomial are determined by the angle of incidence, the real incident-medium index, the complex substrate index, and the measured complex ellipsometric ratio ρ. This approach to the problem gives directly all the possible physical solutions without the need for initial guesses or ranges. Special cases are examined. Numerical analysis and error analysis are provided for the case of a silicon oxide layer over silicon.