Fresnel reflection and transmission calculator

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About this calculator

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 θ2\theta_2 can be computed from Snell's law as

θ2=arcsin(n1n2sinθ1)\theta_2 = \arcsin\left(\dfrac{n_1}{n_2} \sin{\theta_1}\right)

where n1n_1, n2n_2 are the refractive indexes of materials 1 and 2, respectively and θ1\theta_1is the angle of incidence.

The TIR critical angle corresponds to θ2=90\theta_2 = 90^{\circ} in the previous equation. This is true when

n1n2sinθ1=1\dfrac{n_1}{n_2} \sin{\theta_1} = 1
θ1=arcsin(n2n1)\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

rs=n1cosθ1n2cosθ2n1cosθ1+n2cosθ2r_s = \dfrac{n_1 \cos \theta_1 - n_2 \cos \theta_2}{n_1 \cos \theta_1 + n_2 \cos \theta_2}
ts=2n1cosθ1n1cosθ1+n2cosθ2t_s = \dfrac{2 n_1 \cos \theta_1}{n_1 \cos \theta_1 + n_2 \cos \theta_2}
rp=n1cosθ2n2cosθ1n1cosθ2+n2cosθ1r_p = \dfrac{n_1 \cos \theta_2 - n_2 \cos \theta_1}{n_1 \cos \theta_2 + n_2 \cos \theta_1}
tp=2n1cosθ1n1cosθ2+n2cosθ1t_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

Rs,p=rs,p2R_{s,p} = \left|r_{s,p}\right|^2
Ts,p=n2cosθ2n1cosθ1ts,p2T_{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 Rs,pR_{s,p} and Ts,pT_{s,p}.

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

θB=arctan(n2n1)\theta_B = \arctan \left( \dfrac{n_2}{n_1} \right)