Fresnel Interface Lab

Analyze electromagnetic wave refraction, reflection, and transmission at a single planar interface between two media with complex refractive indices.

Boundary Configurations

nm
Source: Birch & Downs (Edlén)
Source: Schott Datasheet
deg

Reflectance & Transmittance Angle Sweep

0.000.250.500.751.000°15°30°45°60°75°90°θ_B (56.6°)
── R_s- - R_p── T_s- - T_p

Reflection & Transmission Results

Refracted Angle (θ_2)27.783°
Power Reflectance (R_s)0.096025
Power Reflectance (R_p)0.0092209
Power Transmittance (T_s)0.90397
Power Transmittance (T_p)0.99078
Unpolarized Reflectance0.052623
Brewster Angle56.607°
Critical AngleN/A

Complex Amplitude Coefficients

s-Polarization (TE)

Reflection Coefficient (r_s):

-0.30988 + i(0)

Transmission Coefficient (t_s):

0.69012 + i(0)

p-Polarization (TM)

Reflection Coefficient (r_p):

0.09603 + i(0)

Transmission Coefficient (t_p):

0.72250 + i(0)

Conservation Residual: |1 - R_s - T_s| = 0.000e+0 | |1 - R_p - T_p| = 3.331e-16

Fresnel Coefficients & Snell's Law Theory

Complex Snell's Law

n~1sin⁡θ1=n~2sin⁡θ2\tilde{n}_1 \sin\theta_1 = \tilde{n}_2 \sin\theta_2

Solves complex refraction angles when materials are absorbing or index matched above TIR threshold.

s-Polarization (TE) Coefficients

rs=n~1cos⁡θ1−n~2cos⁡θ2n~1cos⁡θ1+n~2cos⁡θ2r_s = \frac{\tilde{n}_1 \cos\theta_1 - \tilde{n}_2 \cos\theta_2}{\tilde{n}_1 \cos\theta_1 + \tilde{n}_2 \cos\theta_2}ts=2n~1cos⁡θ1n~1cos⁡θ1+n~2cos⁡θ2t_s = \frac{2 \tilde{n}_1 \cos\theta_1}{\tilde{n}_1 \cos\theta_1 + \tilde{n}_2 \cos\theta_2}

p-Polarization (TM) Coefficients

rp=n~2cos⁡θ1−n~1cos⁡θ2n~2cos⁡θ1+n~1cos⁡θ2r_p = \frac{\tilde{n}_2 \cos\theta_1 - \tilde{n}_1 \cos\theta_2}{\tilde{n}_2 \cos\theta_1 + \tilde{n}_1 \cos\theta_2}tp=2n~1cos⁡θ1n~2cos⁡θ1+n~1cos⁡θ2t_p = \frac{2 \tilde{n}_1 \cos\theta_1}{\tilde{n}_2 \cos\theta_1 + \tilde{n}_1 \cos\theta_2}

Poynting Vector Normal Transmittance

Ts=Re(n~2cos⁡θ2n~1cos⁡θ1)∣ts∣2T_s = \mathrm{Re}\left( \frac{\tilde{n}_2 \cos\theta_2}{\tilde{n}_1 \cos\theta_1} \right) |t_s|^2

Calculates power transmittance fluxes across boundaries exactly, preserving conservation bounds.