Fresnel Equations

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When light intersects the interface between two mediums, the Fresnel equations give us the proportion of light that is reflected and the proportion of light that is refracted into a medium.

Given an angle of incidence of \(\theta_i\), and an angle of refraction of \(\theta_t\) (calculated via Snell’s law), the reflectance of S-polarized light is given by:

\[R_s = \left\lvert \frac{n_1\cos{\theta_i} - n_2\cos{\theta_t}}{n_1\cos{\theta_i} + n_2\cos{\theta_t}} \right\rvert^2\]

while the reflectance of the P-polarized light is given by:

\[R_p = \left\lvert \frac{n_1\cos{\theta_t} - n_2\cos{\theta_i}}{n_1\cos{\theta_t} + n_2\cos{\theta_i}} \right\rvert^2\]

Because of the conservation of energy, the transmitted power in each case can simply be found with:

\[\begin{split}\begin{align} T_s &= 1 - R_s\\ T_p &= 1 - R_p \end{align}\end{split}\]

Schlick’s approximation

Because evaluating the full Fresnel equations can be expensive we can use Schlick’s approximation. Schlick’s approximation is much faster to evaluate:

\[R(\theta_i) = R_0 + (1-R_0)(1-\cos{\theta_i})^5\]

where \(R_0\) is the base reflectivity, given by:

\[R_0 = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2\]

Note

Schlick’s approximation does not take into account separate polarization components.