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Efficiency

wiki for efficiency calculations

Snell's law

A fraction of the light is reflected and another transmitted:

refraction_fresnel
θi= (normal) incidence angle
θr= (normal) reflection angle (same as θi)
θt= (normal) transmittance angle
N1= refraction index of material from which the ray is coming (left in image)
N2= refraction index of material into which the ray is going (right in image)

all parameters are potentially complex numbers. The refractive indices are retrieved from files (Palik, Henke, Cromer..)

Snell's law: N1sinθi=N2sinθtsinθt=N1N2sinθi

θi, N1, N2 are known, we are looking for θt.
We do not calculate the angle specifically but only the cosinus, which is sufficient for further calculations and more efficient/precise than calculating the angle itself because we do not need to use more trigonometric functions. We can calculate the incidence angle θi of each ray from its direction and the surface normal. Then we calculate cos(θi) and from that we can derive cos(θt) with snell's law:

(sinθi)2=1(cosθi)2(sinθt)2=(N1N2)2(sinθi)2cosθt=1(sinθt)2=1(N1N2sinθi)2

The cosine of both angles is then used in the Fresnel equations to calculate the s- and p-polarization

Fresnel equation

Any polarization state can be described by two components: one vertical and one horizontal. Or - relative to the plane of incidence - s- and p-polarization. p-polarization (parallel, left image) lies parallel in the plane of incidence and s-polarization (senkrecht, right image) is orthogonal to the plane of incidence.

ppol spol

the reflectance of both polarizations is calculated with the fresnel equations:

rs=N1cosθiN2cosθtN1cosθi+N2cosθt rp=N2cosθiN1cosθtN2cosθi+N1cosθt

(The transmitted power is then "the rest": ts=1rs and tp=1rp)