Simulates the inverse curve-fitting problem for absolute intensity data. Contrasts a Naive 2-DOF model, a Rigorous 2-DOF model, and a Covariant 5-DOF model to demonstrate the mathematical instability of over-parameterization when analyzing data corrupted by hardware polarization leakage.
Unlike Variable Angle Ellipsometry (which is ratiometric), reflectometry relies on measuring absolute intensity. If the statistical solver assumes perfect hardware (a Naive Model), it interprets the elevated signal floor near the Brewster minimum as a fundamental change in the material's optical properties. Because the global amplitude parameter (A0) is strictly multiplicative, it cannot mathematically absorb the additive leakage caused by polarization errors (ε and γ). Consequently, the solver is forced to artificially skew the refractive index (nglass) to minimize the mean squared error, resulting in severe extraction inaccuracies and a highly bowed residual signature.
In proper metrological practice, hardware constraints (ε, γ, θoff) are characterized independently prior to the sample measurement. By passing these fixed, pre-calibrated values into the theoretical model, the solver successfully decouples the background leakage from the interfacial reflectance. This Rigorous Model reliably extracts the true index, limited only by standard Gaussian noise variance.
When encountering a poor fit under the Naive assumption, a common analytical mistake is to simply "unconstrain" the hardware parameters and allow the solver to fit them simultaneously alongside nglass and A0. This 5-DOF Covariant Model introduces massive mathematical degeneracy. The parameters ε and γ contribute near-identical additive constants to the intensity minimum, creating a perfectly flat "valley" in the objective function. The gradient descent algorithm falls into this trough and stops prematurely, achieving a visually flat residual line by hallucinating wildly incorrect physical parameters. This demonstrates why over-parameterization destabilizes experimental extraction.