## 3 Model Violations
### Geometry
#### 3.2.4 Coherence  
In thin film optics, a loss in coherence between multiply internally reflected waves is usually caused by either a coating thickness that increases the coherence length of the incident light, or by the loss of phase information of the wave when passing rough interfaces (see Section 3.2.3). In this section, our focus will be on the first mechanism. In optical multilayer systems with smooth interfaces, a distinction is usually made in the description between the extreme cases of incoherent (e.g., substrate with consideration of the back side) and coherent light propagation [11]. In the case of incoherent light propagation, no phase information is included, because the coherence length of the light is assumed to be smaller than the layer thickness. Therefore, no interference effects occur and only the wave amplitudes are relevant and geometrical optics can be used for modeling. In the case of coherent light propagation, the coherence length is assumed to be significantly larger than the layer thickness. For this reason, the phase information remains important and interference effects between multiply reflected waves will occur. Therefore, the wave optic is the corresponding theoretical model here. Optical coatings containing both types of layers can be modeled with the general transfer matrix method outlined in many textbooks (e.g., [92]). In the case of anisotropic layers, a generalized method is proposed in [93]. In the intermediate regime, when the layer thickness is in the range of the coherence length, the two mentioned extreme cases are of no relevance, and an adapted theoretical approach based on partial coherence is required. Bearing in mind that the coherence length is affected by spectral resolution and wavelength of the light source, a given sample may appear either \"coherent\" or \"incoherent\", depending on the illumination conditions [10]. When an optical coating is measured in a sufficiently broad spectral range, the transition from \"coherent\" to \"incoherent\" may even be observed within a single transmission or reflection spectrum. As a result, the loss in coherence may be observed by a depletion of the amplitude of the interference fringes usually observed in both transmission and reflection spectra. Let us finish this section with a short literature review. Initially, the theoretical description of partial coherence was limited to single-layer coatings at normal incidence [94], but later these limitations could be eliminated in a more general theory [95]. In this approach, a complex degree of coherence (cdc) has been introduced. The limiting cases of the absolute values are \(|\cdc|=0\) for incoherent and \(|\cdc|=1\) for coherent light propagation. In [77], a simpler coherence function is proposed and spectral averaging over an interval is used. In [96], partially coherent light propagation is also modeled with the transfer matrix method, but the transition from coherent to incoherent propagation is achieved by introducing a random phase of increasing intensity. In [97], the coherence length is considered using a Fourier transform of a randomly generated partially coherent wave. Thereby, the statistical field distribution of partially coherent light is modeled using a rigorous coupled wave analysis. In [72; 75], a generalized matrix method is presented, where interface roughness has been addressed as a special case of partially coherent light propagation. The phase-shift integration method in [98] provides an alternative mathematical approach for the modeling and is used to derive an analytical expression for irradiance at an arbitrary depth of the multilayer stack. The method is advantageous for gradient optimization methods because analytical layer thickness derivatives are provided. In [99], the net-radiation method is adapted for modeling the reflectance, transmittance, and absorption depth profile of thin film multilayer structures such as solar cells.