Plotting Amplitude Reflectance and Transmittance for p and s Polarizations vs Incident Angle using MATLAB

Fresnel formulas describe the amplitude changes of light waves during reflection and transmission at interfaces between different media, where p-polarization (parallel to the incident plane) and s-polarization (perpendicular to the incident plane) exhibit distinct reflection and transmission characteristics.

### Core Theory Snell's Law: The incident angle θ₁ and refraction angle θ₂ satisfy n₁sinθ₁ = n₂sinθ₂, where n₁=1 (air) and n₂=1.45 (quartz glass). Amplitude Reflectance (r) and Transmittance (t): s-polarization: Reflectance rₛ = (n₁cosθ₁ - n₂cosθ₂) / (n₁cosθ₁ + n₂cosθ₂) Transmittance tₛ = 2n₁cosθ₁ / (n₁cosθ₁ + n₂cosθ₂) p-polarization: Reflectance rₚ = (n₂cosθ₁ - n₁cosθ₂) / (n₂cosθ₁ + n₁cosθ₂) Transmittance tₚ = 2n₁cosθ₁ / (n₂cosθ₁ + n₁cosθ₂)

### MATLAB Implementation Approach Parameter Definition: Define incident angle range from 0° to 90°, convert to radians using deg2rad function. Calculate corresponding refraction angles θ₂ using Snell's law with asin function. Reflectance and Transmittance Calculation: Apply Fresnel formulas separately for s and p polarizations using element-wise operations. Handle total internal reflection cases (when θ₁ exceeds critical angle, reflectance magnitude becomes 1) using conditional statements. Plotting Strategy: Subplot 1: Plot real and imaginary parts of rₛ, rₚ, tₛ, tₚ versus incident angle using plot function with different line styles and colors. Subplot 2: Plot absolute values |rₛ|, |rₚ|, |tₛ|, |tₚ| versus incident angle, highlighting total reflection effects beyond critical angle using semilogx or specialized scaling.

### Extended Analysis Brewster's Angle: p-polarization reflectance drops to zero at specific angle (θ_B = arctan(n₂/n₁)), where only s-polarization is reflected - implement using atan function and zero-crossing detection. Energy Conservation: Verify |r|² + (n₂cosθ₂/n₁cosθ₁)|t|² = 1 using array operations and error calculation to ensure computational accuracy through validation checks.

This methodology clearly demonstrates polarization-dependent behavior at dielectric interfaces, providing visual foundation for optical design applications through systematic MATLAB implementation.