# Fresnel reflection and transmission calculator

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Output

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This calculator computes the power reflectivity and transmission of a plan wave at a dielectric interface using the Fresnel equations. These depend on the refractive indexes of the two materials as well as the incidence angle and polarization of the wave in material 1.

Therefore, the results are different for S polarization (E-field vector sticking out of the paper if you draw the interface) and P polarization (E-field vector within the incidence plane). For convenience, the calculator also computes an average result corresponding to unpolarized light.

Additionally, this calculator computes the refraction angle, the angle for total internal reflection (TIR) and Brewster's angle. TIR can occur only if material 2 has a lower refractive index than material 1 and even then the incidence angle must be larger than a so called critical angle. Brewster's angle is a special incidence angle which results in P polarized being completely transmitted (i.e. the reflected beam is purely S polarized). It, like the TIR critical angle, depends only on the materials' refractive indexes.

The refraction angle $\theta_2$ can be computed from Snell's law as

$\theta_2 = \arcsin\left(\dfrac{n_1}{n_2} \sin{\theta_1}\right)$

where $n_1$, $n_2$ are the refractive indexes of materials 1 and 2, respectively and $\theta_1$is the angle of incidence.

The TIR critical angle corresponds to $\theta_2 = 90^{\circ}$ in the previous equation. This is true when

$\dfrac{n_1}{n_2} \sin{\theta_1} = 1$
$\theta_1 = \arcsin\left(\dfrac{n_2}{n_1}\right)$

The Fresnel equations giving the amplitude reflection and transmission coefficients for S and P polarized light are

$r_s = \dfrac{n_1 \cos \theta_1 - n_2 \cos \theta_2}{n_1 \cos \theta_1 + n_2 \cos \theta_2}$
$t_s = \dfrac{2 n_1 \cos \theta_1}{n_1 \cos \theta_1 + n_2 \cos \theta_2}$
$r_p = \dfrac{n_1 \cos \theta_2 - n_2 \cos \theta_1}{n_1 \cos \theta_2 + n_2 \cos \theta_1}$
$t_p = \dfrac{2 n_1 \cos \theta_1}{n_1 \cos \theta_2 + n_2 \cos \theta_1}$

Power reflectivity and transmission is then calculated as

$R_{s,p} = \left|r_{s,p}\right|^2$
$T_{s,p} = \dfrac{n_2 \cos \theta_2}{n_1 \cos \theta_1} \left|t_{s,p}\right|^2$

where transmission requires an extra scaling factor. The unpolarized case is computed simply as an average between S and P polarizations. The reflected and transmitted powers are computed by multiplying the incident power by $R_{s,p}$ and $T_{s,p}$.

The condition for Brewster's angle $\theta_B$ can be derived by setting $r_p = 0$. This results in

$\theta_B = \arctan \left( \dfrac{n_2}{n_1} \right)$