Prism: Ellipsometry Curve-Fitting

Inverse Regression & Joint Optimization

1. Analytical Formulations & Objective Function

Variable Angle Ellipsometry (VAE) measures the complex reflectance ratio (ρ) between the p-polarized (r_p) and s-polarized (r_s) components of light. This ratio is parameterized by the amplitude ratio (Ψ) and phase difference (Δ):

For a 3-layer system (ambient 0, film 1, substrate 2), the total reflection coefficients incorporate the Fresnel interface coefficients (r₀₁, r₁₂) and the film phase thickness (β):

where β = 2π (d / λ) n_film cos θ_f. Here, d is the contamination thickness, λ is the wavelength, n_film is the film index, and θ_f is the refracted angle within the film.

The 3-Pass Adaptive Grid Search algorithm iteratively evaluates combinations of n_glass ∈ [1.50, 1.90] and d ∈ [0, 20] nm to minimize the joint Mean Squared Error (MSE) across N discrete measurement points:

2. Model Mismatch

In Variable Angle Ellipsometry (VAE), the dominant error is Model Mismatch rather than random noise. If experimental data is generated containing a thin physical contamination layer (e.g., adventitious hydrocarbons or native oxide, modeled here with a static index of n_film = 1.4500), but the statistical solver is instructed to fit it using an air-to-bulk assumption (a 2-layer model), the solver is constrained. (Note: While all materials exhibit optical dispersion, the spectral variation of the ultra-thin film's index induces a negligible phase shift compared to the bulk substrate, and is thus held constant in this algorithm). The solver will mathematically force a fit by artificially skewing the extracted bulk refractive index (n_glass) to absorb the physical discrepancy.

3. Phase (Δ) Residuals

By plotting the regression residuals for both Ψ and Δ, the experimenter can visually diagnose this mismatch. While the Ψ residuals might appear acceptable when a 2-layer model is forced onto 3-layer data, the Δ residuals will exhibit a highly structured systematic error signature exactly at the Brewster angle. This alerts the experimenter that their mathematical model is incomplete. Switching to the Rigorous 3-Layer solver restores the flat residual line and decouples the bulk material properties from the surface chemistry.

4. Hardware Leakage, Geometric Misalignment & Solver Constraints

When adjusting the Extinction Ratio (ε) or Analyzer Misalignment (γ) sliders, the extracted parameters (n_glass and d) and the theoretical fit lines remain highly stable, whereas they are highly sensitive to actual surface contamination (d). While hardware leakage increases the amplitude residuals (Ψ), the regression solver remains anchored. This illustrates four key principles:

• Parameter Covariance: The Extinction Ratio (ε) and Analyzer Misalignment (γ) induce mathematically degenerate effects on the measured amplitude ratio (Ψ). Both variables leak the orthogonal polarization state into the detector, symmetrically washing out the intensity null. Because their residual error signatures are nearly identical, these parameters are highly covariant. A statistical regression algorithm cannot uniquely extract both variables simultaneously from the Ψ curve without additional external constraints.
• Phase Immunity: Hardware leakage washes out the Ψ intensity null, but the phase difference (Δ) is a relational property of the boundary. The physical phase shift remains intact regardless of polarization leakage. Therefore, the experimental Δ curve retains its step-function profile.
• Solver's Constraint: The non-linear regression solver minimizes the joint MSE of both Ψ and Δ. If the solver attempted to fit the corrupted, shallow Ψ null, it would have to artificially increase the modeled film thickness. However, doing so would relax the theoretical Δ curve into a sigmoid, which would violate the experimental Δ data and incur a large increase in the MSE.
• Zero-Offset Vulnerability (θ_off): In contrast to amplitude-focused hardware leakage, a mechanical misalignment of the goniometer's baseline shifts the apparent measurement coordinates relative to the physical interface. Because the ellipsometric phase (Δ) transitions sharply at the Brewster angle, an uncorrected angular offset displaces the theoretical step-function away from the measured data. This generates large, highly structured residuals in both Ψ and Δ, forcing the solver to extract an inaccurate n_glass to compensate for the geometric misalignment.

Result: The solver is mathematically anchored by the immune phase data. To minimize the joint MSE and preserve the fit on the Δ curve, the algorithm must maintain d ≈ 0. It mathematically tolerates increased residuals at the minimum of the Ψ curve to prevent large error penalties in the phase domain. This demonstrates why VAE provides more robust constraints than simple intensity-based extraction, provided the system's geometric alignment (θ_off) is strictly calibrated.