## 3 Model Violations
### Geometry
#### 3.2.3 Rough Surfaces  
Rough surfaces, as well as any laterally structured coatings, violate the model assumption I defined in Section 2.1 because they introduce an additional \(x\)- and/or \(y\)-dependence of the dielectric function. Indeed, any surface profile is usually defined in terms of a surface height function \(h(x,y)\). The rms surface roughness \(\sigma\) is then defined through the following:  
\[\sigma=\frac{1}{L}\sqrt{\int_{0}^{L}\int_{0}^{L}\left[h(x,y)-\langle h\rangle \right]^{2}dxdy};\langle h\rangle=\frac{1}{L^{2}}\int_{0}^{L}\int_{0}^{L}h(x, y)dxdy \tag{16}\]  
where \(L\) is a characteristic length scale. Usually, in a practical situation, the surface profile is not exactly known. Let us for simplicity consider the situation of a one-dimensional surface height profile \(h(x)\) that is assumed to be known on a certain length interval \(L\) in a number of \(N\) points. That interval may then periodically continue, so that it can be approximated by the following [56]:  
\[h(x)=\sum_{l=-N}^{N}h_{l^{\prime}}e^{\frac{2\pi lx}{L}} \tag{17}\]  
In this case, the rms surface roughness can be written as follows:  
\[\sigma_{rms}=\sum_{l=-N}^{N}|h_{l}|^{2} \tag{18}\]From here, we obtain the expressions for the so-called small-scale roughness \(\sigma_{small}\) and large-scale roughness \(\sigma_{large}\) via the following:  
\[\sigma_{small} =\sum_{[l>>\frac{k}{\lambda}]}|h_{l}|^{2} \tag{19}\] \[\sigma_{large} =\sum_{[l<<\frac{k}{\lambda}]}|h_{l}|^{2} \tag{20}\]  
Small- and large-scale roughnesses have a different impact on the optical properties of a coating. Large-scale roughness gives rise to diffuse light scattering and thus provides an optical loss mechanism. Small-scale roughness, on the contrary, rather acts like an antireflection coating. A small-scale roughness is usually modeled by introducing a fictive mixture layer on the surface, making use of a mixing model like the Lorentz-Lorenz or the Maxwell Garnett approach. This calculation strategy is well established and incorporated today in commercially available film characterization software, particularly for ellipsometry purposes [57]. The introduction of an (optically homogeneous) mixture model eliminates the \(x\)- and \(y\)-dependence of the optical properties from the system, so that the calculation of the optical properties may again be performed in terms of the model assumptions formulated in Section 2.1. The situation is different in the case of a large-scale roughness. Different approximation formulas have been proposed to incorporate the scatter losses into the optical model. A crucial point is that the effect of the roughness on the thin film spectra is now twofold: The surface roughness may not only change the propagation direction of the incident light through elastic scatter, but has also an impact on the coherence of light trains superimposing as a result of multiple internal reflections at the interfaces of the film system. Therefore, at the moment, no unique generally accepted manageable approach is available, but there exists a large diversity of different approaches (as examples, we cite [56; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 89; 90; 91]), all having their individual strength and shortcomings. As seen from the references, surface roughness in optical coatings is a hot topic from the sixtieth of the last century of the previous millennium until today. A detailed discussion of the available literature is beyond the frames of this review. Clearly, a rigorous electrodynamic treatment of wave propagation through a system with a specified surface geometry will provide the necessary information, but in thin film practice, any practitioner will seek for a couple of manageable formulas that allow for a quick-and-dirty estimation of the effect of roughness on a spectrum that is otherwise calculated in the usual way.