Fresnel equations specify the amplitude coefficients for transmission and reflection at the interface between two transparent homogeneous media – for two different polarization directions:
- s polarization: electric field vector perpendicular to the plane of incidence (the plane containing all three beams, which is the drawing plane of Fig. 1)
- p polarization: electric field vector in the plane of incidence
The equations are:
For example, ts is the amplitude transmission coefficient for s polarization; the transmitted amplitude is that factor times the incident amplitude in that case (disregarding any phase changes for transmission in the media). n1 and n2 are the refractive indices of the two media. The corresponding propagation angles (measured against the normal direction) are θ1 and θ2 (see Figure 1).
For example, the amplitude transmission coefficient is ts for s polarization, i.e., if the electric field vector is perpendicular to the plane of incidence.
The power reflection coefficients (reflectivity or reflectance values) are obtained simply by taking the modulus squared of the corresponding amplitude coefficients. For the transmissivity, one must add a factor (n2 cos θ2) / (n1 cos θ1) in order to take into account the different propagation angles.
Figure 2 shows in an example case how the reflectivity of the interface depends on the angle of incidence and the polarization. The reflection coefficient vanishes for p polarization if the angle of incidence is Brewster's angle (here: ≈55.4°).
For the simplest case with normal incidence on the interface, the power reflectivity (which is the modulus squared of the amplitude reflectivity) can be calculated with the following equation:
Application to Absorbing Media
Fresnel equations can also be applied to absorbing media, including metals. In that case, a complex refractive index needs to be used; the imaginary part is related to the absorbance. The reflection and transmission coefficients are then generally no more real numbers.
A difficulty occurs concerning the calculation of power transmission for absorbing media in cases with non-normal incidence. The above mentioned correction factor for taking into account the different propagation angles will be complex, but the power transmission factor would of course have to be a real number. The problem is related to the fact that the optical intensity is no longer constant along the wavefronts.
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