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Reflection and Refraction#

Reflection and Refraction are two wave phenomena that cause a change in direction of a wavefront at an interface between two different media. In general, a light wave is partially refracted and partially reflected when it passes from one medium to another at any angle other than 0° from the normal.

Reflection: The angle at which the wave is incident on the surface equals the angle at which it is reflected.

\[\alpha_1 = - \alpha_2\]

with the angle of incidence \(\alpha_1\) and the angle of reflection \(\alpha_2\).

Refraction: The direction of the propagating wave changes according to the refractive indices.


with the angle of incidence \(\alpha\), the angle of refraction \(\beta\), the refractive indices \(n_1\) and \(n_2\).

image/svg+xml n₁ = 1.0 n₂ = 1.5 α = 60

Reflection angle \(\alpha_2 = - \alpha_1 =\) -30 \({}^\circ\)
Refraction angle \(\beta = \arcsin\left( \frac{n_1}{n_2} \sin(\alpha_1) \right) =\) -30 \({}^\circ\)

The refractive index \(n\) of a material

\[n = \sqrt{\mu_r \epsilon_r}\]

with \(\epsilon_r\) is the material's relative permittivity, and \(\mu_r\) is its relative permeability.

Total Internal Reflection#

The total reflection will happen at the critical angle

\[\alpha_C = \arcsin\left( \frac{n_2}{n_1} \right)\]

where \(\alpha_C\) is the critical angle, \(n_1\) and \(n_2\) are the refractive index.


  • If \(\alpha \le \alpha_C\), the ray will split; some of the ray will reflect
  • If \(\alpha \gt \alpha_C\), the entire ray reflects from the boundary.

Polarization Angle#

Light that is reflected at Brewster's angle

\[\alpha_G = \arctan\left( \frac{n_2}{n_1} \right)\]