Generalized Fresnel coefficients

4.7.1. Background and theory

The derivations and equations presented below are provided in different forms in various publications. A comprehensive derivation that starts with the Maxwell equations in vector form can be found in the book Thin-film Optical filters (Macleod, 2001). For the practical implementation, the publications by Craig (1987) and Keller (2001) were helpful.
The commonly known Fresnel’s equations are used as a starting point for reference. They can be derived by solving Maxwell’s equations for non-absorbing media. They are commonly stated as:

The ratio of the incidence angle 𝜃1 and transmittance angle 𝜃2 is determined by Snell’s law, see eq. (18). The Fresnel coefficients 𝑟𝑠,𝑝 and 𝑡𝑠,𝑝 determine the amplitudes of the reflected and transmitted radiation for each polarisation state. The coefficients for the reflected and transmitted power, referred to as reflectance 𝑅𝑠,𝑝 and transmittance 𝑇𝑠,𝑝 are given by:

This takes into account that the power is proportional to the square of the amplitude and that the angle as well as the impedance of the medium will change for the transmitted beam. For normal incidence (𝜃1 ≡ 𝜃2 ≡ 0) the amplitude coefficients (𝑟,𝑡) and power coefficients (𝑅, 𝑇) simplify to the elegant form:

By using complex-valued refraction indices and introducing the concept of effective refractive indices, more general expressions valid for absorbing media and oblique (=non-normal) incidence angles will be derived now. The finally obtained relations show a striking similarity to equations (43).
The derivation is carried out by solving Maxwell’s equation for interfaces of materials and demanding continuity of the electric and magnetic field at the boundary. While this can be done by applying vector calculus in three dimensions, a more straightforward and more efficient approach is possible when the energy flow normal to the interface is considered. In this case, the relevant electric field vector 𝐸⃗ and the magnetic field vector 𝐻⃗ are oriented parallel (tangential) to the boundary. It is, therefore, sufficient to consider the amplitudes only, denoted as 𝐸̃ and 𝐻̃. In order to include phase information, the amplitudes are represented by complex-valued numbers. The tilde in superscript is further used to indicate all quantities that are complex numbers. s and p subscripts are used in the following to indicate the direction of polarisation for which the relation is valid. If the letters for Light, sun and optics – applied principles, models and methods RadiCal, D. Rüdisser 63 both directions are indicated in an equation (separated by a comma), it will mean that the equation is separately valid for either polarisation state (using only s or only p polarised quantities throughout the equation).
Following the approach of considering the energy flows perpendicular to the boundary, it is helpful to introduce an effective index of refraction, which links the two tangential electromagnetic field components 𝐸̃ and 𝐻̃:

In the applied approach considering the flow of energy perpendicular to the boundary, it can be shown that for the s-polarised and p-polarised components, the effective indices of refraction can be derived of the complex-valued index of refraction 𝑛̃ and the propagation angle 𝜃̃ by:

It is important to note that the general propagation angle 𝜃̃ is now a complex quantity. Consequently, the angle of refraction 𝜃̃ 2 for the transmitted beam has to be derived from the angle of incidence 𝜃̃ 1 by solving Snell’s law in the complex-valued form:

Unlike its real-valued counterpart, the complex-valued effective index of refraction ̃ contains angular information, as well as information on the attenuation (through the complex-valued index of refraction 𝑛̃). The amplitude transmission coefficients for reflection (𝑟̃) and transmission (𝑡̃) and the resulting reflectance (𝑅) and transmittance (𝑇), at the boundary of two absorbing or non-absorbing media 1 and 2, can for any incidence angle be stated as:

It can be seen how this very elegant and simple form of the generalized form of Fresnel equations results only through the introduction of the complex-valued effective index of refraction. The similarity to the earlier, non-complex-valued form is remarkable, see equation (43). However, taking the generalization a step further, a different definition for the amplitude coefficients is chosen to additionally consider the effect of potentially present thin-films in between the two media. It can be shown (Macleod, 2001) that they can be calculated by the tangential electric and magnetic field amplitudes in the incident medium:

If the incident field amplitudes 𝐸̃ 1 𝑠,𝑝 and 𝐻̃ 1 𝑠,𝑝 are determined by:

Where 𝐶̅𝑠,𝑝 is the characteristic matrix of a thin-film assembly consisting of N thin-films.

The specific propagation angles 𝜃̃ 𝑗 and the corresponding effective refractive indices for each layer j can again be found by applying Snell’s law, eq. (46).
Finally, the reflectance coefficient 𝑅 and transmittance coefficient 𝑇 to calculate the power of the reflected and the transmitted beam for each polarisation state can again be found by:

The additional factor required to calculate the transmittance coefficient considers that the impedance, as well as the propagation direction, is generally different in the two media. In the further text, these coefficients will be referred to as generalized Fresnel coefficients. In the following application, it is essential to additionally consider the phase change 𝜖 of the reflected and transmitted beam. They are simply represented by the arguments of the complex coefficients 𝑟̃and 𝑡̃:

The corresponding coefficient for absorptance 𝐴 can be found by applying the law of energy conservation:

Note that, again, all relations in (53),(54) and (55) are valid for each polarisation state separately; for the purpose of clarity, the indices s and p were omitted here.

4.7.2. Implementation

The implementation of above formulae is performed in the module TFresnel of the library rc_fresnel:

The module contains, on the one hand, private functions which perform the actual calculation in line with the relations given above. The operator overloading features of the Pascal language allows a convenient and transparent implementation of the relevant equations, as the complex-number algebra can be converted into programming code straightforwardly. On the other hand, the public functions of the module allow the initialisation of the materials and the direct calculation of the Müller matrices for reflection and transmission (see section 4.9). Alternatively, only the parameters required to derive the Müller matrices can be generated. These essential parameters are contained in the type definition TTRpolRecord and reflect the results of equations (48) and (53). In the actual implementation of the raytracing (see chapter 7), the function getRTandThetaS is called to perform the Fresnel calculations, while the function getRandTMuMat is called to establish the Müller matrices of the calculated parameters.

4.7.3. Testing and validation

The full validation is implicitly included in the Müller matrix section (4.9.3), as the Müller matrix elements rely on the Fresnel calculations. During the implementation, the Fresnel module was additionally tested based on commonly known material behaviours.
Exemplary, Figure 34 illustrates the angular reflectivity for three different materials, resolved for three different wavelengths and the main polarisation states. As can be seen, the dielectric (i.e. nonconducting) glass exhibits limited wavelength dependence and a pronounced Brewster angle at 57° where only s-polarised light is reflected off the surface. Gold shows high reflectivity in the visual and infrared spectral range; however, less than half of the incident radiation is reflected in the UV range. Aluminium generally exhibits a high reflectivity for all wavelengths considered.