Fresnel Equations Solver

Fresnel Reflection Coefficients:

\\[ r_s = \frac{n_1 \cos \theta_i – n_2 \cos \theta_t}{n_1 \cos \theta_i + n_2 \cos \theta_t}, \quad r_p = \frac{n_2 \cos \theta_i – n_1 \cos \theta_t}{n_2 \cos \theta_i + n_1 \cos \theta_t} \\]

Fresnel Transmission Coefficients:

\\[ t_s = \frac{2 n_1 \cos \theta_i}{n_1 \cos \theta_i + n_2 \cos \theta_t}, \quad t_p = \frac{2 n_1 \cos \theta_i}{n_2 \cos \theta_i + n_1 \cos \theta_t} \\]

Snell’s Law:

\\[ n_1 \sin \theta_i = n_2 \sin \theta_t \\]

Reflectance and Transmittance:

\\[ R_s = |r_s|^2, \quad R_p = |r_p|^2 \\] \\[ T_s = \frac{n_2 \cos \theta_t}{n_1 \cos \theta_i} |t_s|^2, \quad T_p = \frac{n_2 \cos \theta_t}{n_1 \cos \theta_i} |t_p|^2 \\]

Where:

• \\( n_1 \\): Refractive index of incident medium
• \\( n_2 \\): Refractive index of refracted medium
• \\( \theta_i \\): Angle of incidence (degrees)
• \\( \theta_t \\): Angle of transmission (degrees)
• \\( r_s, r_p \\): Reflection coefficients for s- and p-polarized light
• \\( t_s, t_p \\): Transmission coefficients for s- and p-polarized light
• \\( R_s, R_p \\): Reflectance for s- and p-polarized light
• \\( T_s, T_p \\): Transmittance for s- and p-polarized light

\\[ \theta_t = \arcsin\left(\frac{1.0 \sin 45^\circ}{1.5}\right) \approx 28.13^\circ \\] \\[ r_s = \frac{1.0 \cos 45^\circ – 1.5 \cos 28.13^\circ}{1.0 \cos 45^\circ + 1.5 \cos 28.13^\circ} \approx -0.170 \\] \\[ r_p = \frac{1.5 \cos 45^\circ – 1.0 \cos 28.13^\circ}{1.5 \cos 45^\circ + 1.0 \cos 28.13^\circ} \approx 0.106 \\] \\[ t_s = \frac{2 \cdot 1.0 \cos 45^\circ}{1.0 \cos 45^\circ + 1.5 \cos 28.13^\circ} \approx 0.830 \\] \\[ t_p = \frac{2 \cdot 1.0 \cos 45^\circ}{1.5 \cos 45^\circ + 1.0 \cos 28.13^\circ} \approx 0.895 \\] \\[ R_s = (-0.170)^2 \approx 0.029, \quad R_p = (0.106)^2 \approx 0.011 \\] \\[ T_s = \frac{1.5 \cos 28.13^\circ}{1.0 \cos 45^\circ} \cdot (0.830)^2 \approx 0.971 \\] \\[ T_p = \frac{1.5 \cos 28.13^\circ}{1.0 \cos 45^\circ} \cdot (0.895)^2 \approx 1.048 \\]