## 4 Theoretical Background and Simulations
### Monte Carlo Simulation  
An alternative to the previously discussed ab initio approach is to use statistical methods [165], Ray tracers (e.g., _LightTools_[166; 167]) allow accurate simulation of light-matter interaction at surfaces. Provided the number of rays is sufficiently large, statistical processes such as the diffuse reflectance can also be approximated satisfactorily [168; 169; 170; 171; 172; 173; 174; 175; 176; 177; 178; 179; 180; 181]. For this, a model of the material under investigation is designed; relevant physical parameters that will be used within the simulation are determined beforehand such as the refractive index, e.g., via the Becke line or Schroeder van der Kolk method [182; 183; 184; 185; 186], or the particle size distribution \(q_{0}\), e.g., via disc centrifugation [159]. Further parameters that enter the simulation result from experimental conditions such as the actual sample thickness or volume filling fraction \(f_{V}\). Lastly, the diffuse reflectance \(R_{\infty}\) is measured as it will be used as a reference to calculate the imaginary part of the refractive index \(m^{\prime\prime}\) within the simulation. In other words, \(m^{\prime\prime}\) (and thereby, the absorption coefficient \(\alpha\) as well, according to Equation (11)) is altered until the simulated diffuse reflectance value coincides with the experimentally determined one. Repeating this procedure for different wavelengths \(\lambda\) thus yields the dispersion of the absorption coefficient \(\alpha(\lambda)\). In order to calculate the diffuse reflectance via Monte Carlo simulation, a ray tracer setup as schematically illustrated in Figure 17 is used. Upon irradiation, individual light beams may interact with particles such that they are either attenuated (or, ultimately, absorbed), scattered according to Mie theory, or remitted. For single light beams, this can be understood as follows: When a light ray impinges upon the sample, it is scattered by a single particle according to Mie theory, i.e., the probability for a light ray to be scattered into a certain direction is defined by the microscopic scattering coefficients. Owing to the statistical nature of this process, there exists a nonzero probability that the ray is scattered backwards, thereby leaving the sample as a remitted beam which is counted by the back half of a hypothetical detector sphere encompassing the entire setup. Alternatively, the ray may be scattered forward where it can interact with another particle. For a non-absorbing sample, the number of scattering events within the pellet is theoretically unlimited. In this case, propagation is only terminated when the light ray leaves the  
Figure 17: Schematic setup for ray tracer simulations. The inset shows a magnified version of the interaction mechanisms inside the sample as calculated within the simulation. sample. As a result, the calculated diffuse reflectance as the number of backscattered rays scaled by the incoming ones approaches unity for thick samples. In case of thin layers, a significant number of beams traverses the sample and are subsequently collected by the front half sphere which counts towards transmission signals. Taking absorption into account provides a specific termination condition: At each scattering event, the incident light ray is slightly attenuated owing to the lossy medium, thereby gradually decreasing the ray's intensity until a lower threshold is attained that halts further propagation due to negligible intensities [187]. In close relation to the Beer-Lambert law, the attenuation rate scales with the product of absorption coefficient and path traversed through individual particles. Hence, both parameters \(\alpha\) and \(q_{0}\) can affect the calculated diffuse reflectance equally; on a per-particle basis, an increase of \(\alpha\) can be compensated by a decreased particle size and vice versa. In order to avoid this ambiguity, \(q_{0}\) is affixed which is reasonable after its experimental determination.