Prism: TIR Ellipsometry (Polarization Ratio)

Theoretical Derivation: Fresnel-Corrected TIR Ellipsometry

The determination of a material's refractive index (n_glass) 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 anti-reflection 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.

1. Experimental Parameters

• Apex Angle (A): The physical vertex angle of the prism. Together with the incidence angle, it deterministically constrains the internal geometry and the TIR angle.
• Ambient Index (n_air): The refractive index of the surrounding medium, dependent on wavelength and local thermodynamic state.
• External Incident Angle (θ_i): The angle of the incoming beam relative to the surface normal of the prism's entrance face.
• Input Polarization (θ₁): The linear polarization angle of the incident beam relative to the s-polarization axis (perpendicular to the plane of incidence).
• Measured Intensities (I₁₃₅ & I₄₅): The optical power recorded after the beam passes through an analyzer oriented at 135° (minor axis) and 45° (major axis), respectively.

2. Internal Optical Geometry

By applying Snell's Law at the entrance interface, the internal refracted angle (θ_in) is:

Using the geometric properties of the prism, the internal angle of incidence upon the TIR face (θ_int) is defined as:

3. Fresnel Diattenuation and TIR Phase Shift

For a beam entering and exiting the prism, the sequential amplitude transmission coefficients for s- and p-polarizations (T_s and T_p) 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:

4. Intensity Ratio Formulation

The incident optical field is represented by the Jones vector E_in = [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:

This equation operates as the objective constraint. Because T_s, T_p, and ΔΦ are all interdependent functions of n_glass, a numerical root-finding algorithm (such as bisection) is employed to converge on the specific value of n_glass that forces the theoretical intensity ratio to equal the experimentally measured ratio.

Reference

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.