Reflection and Transmission

System Parameters
θi
n1
n2
Fresnel Data
Incident Angle
0°
Transmitted Angle
0°
Brewster Angle
0°
Critical Angle
N/A
Reflectance (S)
0.000
Reflectance (P)
0.000
Transmittance (S)
0.000
Transmittance (P)
0.000

Variable and Data Information

Incident Angle (θi) : Angle between incoming ray and normal to the surface.

Transmitted Angle (θt) : Angle between the transmitted ray and the normal to the interface.

Brewster Angle (θB) : Incident angle at which no p-polarized light is reflected (\(\theta_i + \theta_t = 90°\)). $$\tan\theta_B = \frac{n_2}{n_1}$$ Critical Angle (θc) : Incident angle beyond which total internal reflection occurs (\(\theta_t = 90°\)) $$\sin\theta_c = \frac{n_2}{n_1}$$ S-Reflectance (Rs) : Fraction of incident s-polarized optical power that is reflected. $$ R_s = |r_s|^2 = \left|\frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}\right|^2 $$ P-Reflectance (Rp) : Fraction of incident p-polarized optical power that is reflected. $$ R_p = |r_p|^2 = \left|\frac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}\right|^2 $$ S-Transmittance (Ts) : Fraction of incident s-polarized optical power that is transmitted. $$ T_s = |t_s|^2 = \left|\frac{2n_1\cos\theta_i}{n_1\cos\theta_i + n_2\cos\theta_t}\right|^2 $$ P-Transmittance (Tp) : Fraction of incident p-polarized optical power that is transmitted. $$ T_p = |t_p|^2 = \left|\frac{2n_1\cos\theta_i}{n_2\cos\theta_i + n_1\cos\theta_t}\right|^2 $$

Exit

Fresnel Equations: Reflection and Transmission

The Fresnel equations describe how light partially reflects and transmits when it encounters a boundary between two materials with different refractive indices. The amount of reflected and transmitted light depends on the angle of incidence, refractive indices, and polarization. Light polarized along different axes exhibits different behaviors at the interface. These equations are fundamental to many optical phenomenon including thin-film interference, anti-reflection coatings, and fiber optics.

The two polarization cases (S and P), each have distinct reflection and transmission equations:

  • S-polarization: the electric field is perpendicular to the plane of incidence
    • $$ r_s =\frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t} $$
    $$ t_s =\frac{2n_1\cos\theta_i}{n_1\cos\theta_i + n_2\cos\theta_t} $$
  • P-polarization: the electric field is parallel to the plane of incidence
    • $$ r_p = \frac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t} $$
    $$ t_p = \frac{2n_1\cos\theta_i}{n_2\cos\theta_i + n_1\cos\theta_t} $$
At the Brewster angle ( \( \theta_B\) ), the reflected and transmitted rays become perpendicular to each other. For P-polarized light, the reflected wave vanishes (\(r_p = 0 \)) and transmission is maximized.

At the critical angle ( \( \theta_c\) ), the transmitted wave fails to propagate into the second medium, resulting in total internal reflection (TIR). This only occurs when light travels from a medium of higher refractive index into one of lower refractive index. Although the transmitted ray disappears during TIR, the electromagnetic field still exists beyond the boundary as an evanescent wave that decays exponentially with distance.

How to use: Adjust the angle of incidence ( \( \theta_i\) ) of the incoming wave and refractive indices of the first (\(~n_1~\)) and second ( \( n_2\) ) medium, and observe changes to the power coefficients and angles. Toggle between s- and p-polarization to observe their different field orientations.