## 3 Model Violations
### Materials
#### 3.3.4 Non-Linear Response  
According to model assumption VIII, T and R are independent of the incident light intensity. However, T and R may become intensity-dependent, provided that Equation (1) is replaced by its non-linear counterpart according to [119]:  
\[ \begin{array}{l}P(t)=P^{(1)}+P^{(2)}+P^{(3)}+\ldots=\epsilon_{0}\int_{0}^{ \infty}\kappa^{(1)}(\xi)E(t-\xi)d\xi+\\ \epsilon_{0}\int_{0}^{\infty}\int_{0}^{\infty}\kappa^{(2)}(\xi_{1},\xi_{2})E( t-\xi_{1})E(t-\xi_{1}-\xi_{2})d\xi_{1}d\xi_{2}+\\ \epsilon_{0}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\kappa^{(3)}( \xi_{1},\xi_{2},\xi_{3})E(t-\xi_{1})E(t-\xi_{1}-\xi_{2})E(t-\xi_{1}-\xi_{2}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\xi_{3})d\xi_{1}d\xi_ {2}d\xi_{3}+\ldots\end{array}  \tag{24}\]  
In the more familiar frequency domain, Equation (24) may be written in the following simplified manner:  
\[P=P^{(1)}+P^{(2)}+P^{(3)}+\ldots=\epsilon_{0}\chi^{(1)}E+\epsilon_{0}\chi^{(2 )}E^{2}+\epsilon_{0}\chi^{(3)}E^{3}+\ldots \tag{25}\]  
As is evident from Equation (24) or Equation (25), non-linear optical effects become significant only at rather large light intensities. Relevant with respect to thin film optics examples include harmonic generation as well as the observation of intensity-dependent transmitances or reflectances [120; 121; 122; 123; 124]. In ultrashort light pulses, peak intensities may become rather large [125]. In such a constellation, the reflectance of a thin film reflector may be significantly reduced when compared to what is expected from the standard (linear) theory. The physical process behind this is called non-linear absorption. Let us mention for completeness that the second harmonic generation is related to second-order terms in Equations (24) and (25), while the third harmonic generation as well as non-linear absorption result from the third-order terms. As an example, Figure 5 shows the simulated normal incidence reflectances and transmitances of three fictive antireflection coatings depending on the intensity of the incident light. For details of the simulation algorithm, see [126; 127; 128], but let us shortly mention that the calculation of T and R is based on the solution of a system of equations:  
\[ \begin{array}{l}\frac{dE}{dz}=-i\mu_{0}\omega H\\ \frac{dH}{dz}=-i\varepsilon_{0}\varepsilon\omega E-\frac{3}{4}i\varepsilon_{0 }\omega\chi^{(3)}|E|^{2}E\end{array}  \tag{26}\]  
Here, for simplicity, normal incidence is assumed. \(E\) stands for the electric, and \(H\) for the magnetic field strength. This is a non-linear system of equations which needs to be solved numerically for the assumed \(z\)-dependence of \(\varepsilon\) and \(\chi^{(3)}\). Note that in the special case of \(\chi^{(3)}=0\), Equation (26) becomes linear and coincides with the system of equations relevant for calculating the linear response of any stratified medium (compare [9; 10; 26]). In particular, the well-known matrix formalism may be directly derived from the linear version of Equation (23). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(n\) & \(K\) & \begin{tabular}{c} **[PERSON](3)** \\ **/m\({}^{2}\)V\({}^{-2}\)** \\ \end{tabular} & \begin{tabular}{c} **[PERSON](3)** \\ **/m\({}^{2}\)V\({}^{-2}\)** \\ \end{tabular} & \begin{tabular}{c} **V-Coating** \\ \end{tabular} & \
\begin{tabular}{c} **R-Optimized** \\ **Design** \\ \end{tabular} &
\begin{tabular}{c} **T-Optimized** \\ **Design** \\ \end{tabular} \\ \hline Layer 1 & 2.249 & 0 & 1.86\(\cdot 10^{-20}\) & 2.74\(\cdot 10^{-21}\) & 17.5 & 11.7 & 8.4 \\ Layer 2 & 1.477 & 0 & 2.05\(\cdot 10^{-22}\) & 7.32\(\cdot 10^{-24}\) & 121.5 & 122.4 & 125.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic input parameters for the calculations provided in Figure 5. Substrate is fused silica (\(n_{Siu}\)@532 nm = 1.4607), ambient air, normal incidence. Layer 1 is next to substrate, layer 2 next to ambient.