Simulates a numerical inverse-problem solver to extract the refractive index of a glass prism from the measured ellipsometric intensity ratio (I135 / I45). In Total Internal Reflection (TIR) ellipsometry, the relative phase shift between s- and p-polarized light alters the polarization state of the totally reflected beam. This simulation models the complete optical path, including off-normal Fresnel transmission at the entrance and exit boundaries, to calculate the theoretical intensity ratio. The solver iteratively adjusts the hypothesized refractive index until the theoretical ratio matches the user-defined experimental intensities. The algorithm then recalculates and outputs the deterministic internal beam geometry based on the converged material index.
Algorithmic Note: In physical experiments, the prism index is an immutable property of the material while the transmitted intensities are empirical variables. Computationally, manually adjusting the intensity sliders alters the objective target, forcing the solver to extract a hypothetical refractive index to match the offset or corrupted intensity data.
The determination of a material's refractive index (nglass) via Total Internal Reflection (TIR) relies on measuring the relative phase shift (ΔΦ) between the s-polarized (perpendicular) and p-polarized (parallel) components of light. Recent methodologies propose determining the refractive index by measuring the polarization ellipticity resulting from this TIR phase shift. These idealized models utilize a semi-cylindrical geometry to ensure normal incidence and assume antireflection coatings to avoid boundary effects. The ellipticity is then directly related to the phase shift via ε = tan(ΔΦ / 2). While these simplified models assume normal incidence at the entrance and exit boundaries, utilizing a triangular prism introduces off-normal refractions. These planar boundaries exhibit polarization-dependent amplitude transmission (diattenuation), which alters the polarization state independent of the TIR phase shift.
To extract an accurate refractive index, the system must be modeled using sequential Jones matrix calculus to account for both the phase accumulation and the Fresnel transmission coefficients.
By applying Snell's Law at the entrance interface, the internal refracted angle (θin) is:
Using the geometric properties of the prism, the internal angle of incidence upon the TIR face (θint) is defined as:
For a beam entering and exiting the prism, the sequential amplitude transmission coefficients for s- and p-polarizations (Ts and Tp) deviate from unity. Assuming a symmetric beam path (where exit refraction mirrors entrance refraction), the total amplitude transmissions are governed by the Fresnel equations:
At the TIR boundary, the evanescent wave interaction induces a phase delay of the p-polarization relative to the s-polarization. The theoretical phase shift (ΔΦ) is:
The incident optical field is represented by the Jones vector Ein = [cos θ1, sin θ1]T. Propagating this vector through the entrance transmission matrix, the TIR phase matrix, and the exit transmission matrix yields the final polarization state before the analyzer.
Projecting this final state onto an analyzer oriented at an arbitrary angle θ2 provides the output intensity function I(θ2). The ratio of the minor axis intensity (I135) to the major axis intensity (I45) is analytically derived as:
The intensity ratio equation operates as the objective constraint. Because Ts, Tp, and ΔΦ are interdependent functions of nglass, a closed-form algebraic inversion is computationally intractable. The JavaScript engine evaluates this constraint using a hybrid numerical root-finding algorithm. It first performs a linear scan across the refractive index domain [1.4, 2.5] using 1100 discrete steps to bracket the zero-crossing of the objective function (where the theoretical intensity ratio equals the empirical ratio). Once bracketed, a bisection search isolates the exact root, converging to a prescribed tolerance limit of 10−8. If the requested intensities correspond to a non-physical state (e.g., violating the TIR critical angle threshold), the algorithm terminates the search and flags the evaluation matrix as invalid.
Chang, J.-P.; Tsai, C.-M.; Weng, J.-H.; Han, P. Refractive Index and Dispersion Measurement Principle with Polarization Change in Total Internal Reflection. Photonics 2024, 11, 505.