## 3 Numerical Results  
In this section, we compute the Pareto front of the three competing designs, i.e., the proposed design method in Equation (14), the periodic design [20] and the fully (_brute force_) optimized design [18]. Table 1 is used as a reference for the physical parameters of the \(L\) and \(H\) materials. In Figure 2a the Pareto front \(\mathcal{P}_{H}(\bar{\phi}_{c})\) of the problem (14) where \(N_{D}\leq 15\) is shown in red on a log-linear scale as a function of the dimensionless quantity \(\bar{\phi}_{c}\). The continuous black curve represents the Pareto front again for problem (14) but with \(N_{D}=10\) kept fixed. Finally, for fixed \(N_{D}=10\) the dashed curves show the Pareto front of the optimal periodic doublet design. In Figure 2b, a close-up of the central area of the Figure 2a is shown along with several Pareto fronts of [PERSON] (continuous curves) and periodic (dashed curves) designs. From the analysis of Figure 2, it is clear that [PERSON]'s design generates a Pareto front that consists of several bumps (a bumpy curve). This behavior had already been observed in the _brute force_ solution given by the BorgMOEA method used in the paper [18]. This figure reveals that the various bumps of the complete Pareto front are parts of the Pareto curves with fixed \(N_{D}\) that intersect each other (see Figure 2b). Moreover, as will be more evident later, we note that the periodic designs are always worse than [PERSON]'s. Let us take a closer look at this result in Figure 3 to better illustrate it. The Pareto front of the periodic synthesis of the doublet (respectively of the BorgMoea) will be called with \(\mathcal{P}_{D}(\bar{\phi}_{c})\) (respectively with \(\mathcal{P}_{B}(\bar{\phi}_{c})\)). \begin{table}
\begin{tabular}{l l} \hline \hline
**Coating** & **Substrate** \\ \hline \(H\) (amorphous Ti-doped Ta\({}_{2}\)O\({}_{5}\) ) & (bulk crystalline SiO\({}_{2}\)) \\ \(L\) (SiO\({}_{2}\)) & \\ \hline \(n_{H}=2.10\) & \(n_{s}=1.45\) \\ \hline \(n_{L}=1.45\) & \(Y_{s}=72\) GPa \\ \hline \(\kappa_{H}=4.0\times 10^{-8}\) & \(\kappa_{s}=8.4\times 10^{-11}\) \\ \hline \(\kappa_{L}=8.4\times 10^{-11}\) & \(\phi_{s}=7.00\times 10^{-8}\) \\ \hline \(Y_{H}=147\) GPa & \\ \hline \(Y_{L}=72\) GPa & \\ \hline \(\phi_{H}=3.76\times 10^{-4}\) & \\ \hline \(\phi_{L}=5.00\times 10^{-5}\) & \\ \hline \(\gamma=9.5\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of coating and substrate material used in the simulations, we assume that the temperature is \(T=300\) K and free-space wavelength is \(\lambda_{0}=1064\) nm. The two materials described above are those that have the best characteristics, mainly in terms of negligible extinction coefficients, and are used in both the Virgo [1] and LIGO [2] experiments. Figure 2: The figure shows in (**a**), in red, the Pareto front by [PERSON]’s method \(\mathcal{P}_{H}(\tilde{\phi}_{c})\) obtained by keeping \(N_{D}\leq 15\) as a function of dimensionless normalized mechanical loss \(\tilde{\phi}_{c}\). For fixed \(N_{D}=10\) the Herpin Pareto front (solid black) and the periodic Pareto front (dashed black) are displayed for comparison. Below in (**b**) a closeup of the central part of (**a**) where some additional fixed \(N_{D}\) Pareto curves has been displayed. The red dots in (**a**,**b**) are the quarter-wavelength designs for \(N_{D}=6,8,10,12\). In Figure 3 the following normalized differences  
\[\mathcal{D}_{DH} =\mathcal{P}_{H}(\bar{\phi}_{c})^{-1}[\mathcal{P}_{D}(\bar{\phi}_{c })-\mathcal{P}_{H}(\bar{\phi}_{c})],\] \[\mathcal{D}_{HB} =\mathcal{P}_{H}(\bar{\phi}_{c})^{-1}[\mathcal{P}_{H}(\bar{\phi}_{ c})-\mathcal{P}_{B}(\bar{\phi}_{c})], \tag{17}\]  
are displayed. To be precise, in Figure 3 the function \(\mathcal{D}_{DH}\) is the dashed curve while \(\mathcal{D}_{HB}\) is the continuous black one. By inspection of the figure, it is clear that both the two normalized differences are _positive_ and have discontinuity points in the cusps that separate the different bumps in the Pareto front curves.