Optimizing precision of rotating-analyzer and rotating-compensator ellipsometers.

I investigate the dependence of shot-noise-limited uncertainties of the ellipsometric parameters psi, and delta for the rotating-analyzer ellipsometer (RAE) and the rotating-compensator ellipsometer (RCE) of the polarizer-sample-compensator-analyzer type. The development is general and takes into account correlations among the Fourier coefficients of the transmitted intensity, in particular the average intensity, which is necessarily correlated with all other coefficients through normalization. The results are expressed in terms of the traditional uncertainties delta(psi) and delta(delta) of the ellipsometric parameters psi and delta, respectively, although a more appropriate measure of uncertainty is the differential area 2delta(psi) x sin psi(delta)delta on the unit-radius Poincaré sphere. Numerical results for broadband operation from 1.5 to 6.0 eV with a Si sample show that the optimum measurement conditions for both configurations occur when the intensity of light reflected from the sample is approximately balanced between the TE and the TM modes, and, for the RCE, when the analyzer azimuth is essentially equal to that of the polarizer. Under typical broadband operating conditions in which components cannot be optimized on a wavelength-by-wavelength basis, the RCE is better at determining delta, whereas the RAE is better at determining psi. The approach is easily generalized to other configurations and other types of experimental uncertainty, both random and systematic.