🔍 Wave Optics & Polarization Physics
Fresnel Reflection & Transmission Calculator
Compute power reflectance ($R_s, R_p, R_{\text{avg}}$) and transmittance ($T = 1 - R$) for polarized light at dielectric optical boundaries and determine Brewster's polarizing angle ($\theta_B$).
🪟 Air ➔ Glass ($0^\circ$, 4.0% Reflectance)
🕶️ Air ➔ Glass at Brewster ($56.3^\circ$)
💧 Air ➔ Water at Brewster ($53.1^\circ$)
💎 Air ➔ Diamond (45°)
Optical Boundary Inputs
degrees (°)
Average Unpolarized Reflectance (R_avg)
7.40% Reflected
s-Polarized ($R_s$)
14.79%
Perpendicular
p-Polarized ($R_p$)
0.00%
Parallel to plane
Normal Refl ($R_0$)
4.00%
At θ = 0°
💡 Optical Application Insight:
At 56.31° (Brewster's Angle), parallel p-polarized light passes through with 0% reflection, producing 100% linearly polarized reflected light (how polarized sunglasses eliminate glare).
The Fresnel Equations for Reflection & Transmission
Augustin-Jean Fresnel derived the amplitude reflection coefficients for planar electromagnetic waves:
- s-Polarized (TE): $$R_s = \left|\frac{n_1 \cos\theta_i - n_2 \cos\theta_t}{n_1 \cos\theta_i + n_2 \cos\theta_t}\right|^2$$
- p-Polarized (TM): $$R_p = \left|\frac{n_1 \cos\theta_t - n_2 \cos\theta_i}{n_1 \cos\theta_t + n_2 \cos\theta_i}\right|^2$$
- Brewster's Polarizing Angle: $$\tan(\theta_B) = \frac{n_2}{n_1} \implies \theta_B = \arctan\left(\frac{n_2}{n_1}\right)$$
- Normal Incidence ($0^\circ$): $$R_0 = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2$$