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Definition

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.

$$\frac{\sin(\alpha)}{\sin(\beta)}=\frac{n_2}{n_1}=\frac{c_1}{c_2}=\frac{\lambda_1}{\lambda_2}$$

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

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.

Explanation

  • 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

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

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