Deriving the Directional Dependence of Surface Properties for Dielectric Windows
A thin dielectric layer may reflect, absorb, or transmit any incoming radiation depending on the angle of incidence, the refractive indices of the layer and the surrounding media, and the absorption coefficient of the layer. Instead of modeling the exact path of each ray through the layer, the global reflectance, transmittance and absorptance can be derived. The Figure 4-15 shows how an incoming ray is reflected and refracted through a dielectric layer.
Figure 4-15: Ray path through a dielectric layer.
For the following derivation, it is supposed that the imaginary part of the refractive index is negligible.
Windows are optically flat (specular) surfaces. The reflection and transmission at each dielectric interface is calculated starting with Snell’s law. We consider three media: upside (u subscript, where the ray comes from), layer (l subscript), downside (d subscript, where the ray goes if transmitted).
Where nu and nd are the refractive indices of the adjacent domains, and nl is the refractive index of the layer. The angle of incidence θu is measured with respect to the surface normal so it corresponds to the polar angle with the semitransparent surface. The angle of the ray within the material is noted θl, and the angle of the ray in the downside domain is noted θd.
The Fresnel coefficients give the reflection coefficients of each s- and p-polarization.
For a ray from the upside domain and going to the layer but reflected back at the interface:
For a ray from the layer and going to the downside domain but reflected back at the interface:
For a ray from the layer and going to the upside domain but reflected back at the interface:
The reflected power of each polarization at the interface is given by
Thermal radiation is randomly polarized because it is a result of multiple incoherent combinations between variously polarized wave trains. Therefore, one can take the average of the s- and p-polarization to compute reflection and transmission at each interface:
The reflection, and therefore the transmission, is identical on the incident and transmitted sides for each interface, for example, Ru → l = Rl → u.
From the Beer–Lambert law, the transmission of a beam of light as it travels through the interior of the layer, is:
Light will experience multiple reflections between the two parallel interfaces, so the total reflection, transmission, and absorption has to be computed through an infinite series of the form: for x < 1.
The total transmissivity through the glass window is given by
The directional specular reflectivity is:
And the absorptivity is equal to the emissivity. The following relation should be true on both sides: . It implies that incident radiation coming from one side should contribute to the blackbody emission on that same side. The directional emissivity function is: