# Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case
## 5 Numerical Experiments
### Physical Properties of the Optimal Design  
In this section, we try to understand what happens to the common Pareto fronts when the number of layers of the two materials used is increased. This comparison is made in Figure 5, where using the same parameters of Table 1, different Pareto fronts, obtained by changing the number of layers, are shown. The analysis of Figure 5 reveals that common Pareto fronts tend to collapse on a single asymptotic (bumpy) curve. It has been verified that for the realisitc values of \(N_{H}\) (i.e., \(\sim\)20), this asymptotic Pareto front is near a decreasing exponential, i.e., the gray dashed line shown in Figure 5. This behavior is the analog of  
Figure 4: The figure shows the common Pareto Front (yellow) and the two curves computed with and without the constraint on the transmittance, for the two optimization procedures MOEA/D and GDE3. A fortiori, it can be seen that without constraint on transmittance, the two heuristics poorly reconstruct the low-transmittance part of the trade-off curve (the green curve). The red curve shows the Pareto curve with the constraint to look for solutions below the transmittance values of \(\tau_{c}<0.01\). The computation time is \(10^{5}\) s, and the physical parameters are reported in Table 1.  
that found in [33] and further elucidated in [20]; the gray dashed line in Figure 5 is the perfomance limit of the optimization method. The equation of the decreasing exponential in the cryogenic case becomes  
\[\log(\tau_{c}) =\log\biggl{(}\frac{4}{n_{L}}\biggr{)}-\frac{2n_{H}\eta_{e}}{\eta_ {e}n_{H}+n_{L}}\log\biggl{(}\frac{n_{H}}{n_{L}}\biggr{)}+\] \[\qquad-\tilde{\varphi}_{c}\frac{8n_{L}n_{H}}{\eta_{e}n_{H}+n_{L} }\log\biggl{(}\frac{n_{H}}{n_{L}}\biggr{)} \tag{13}\]  
where \(\tau_{c}\) is the trasmittance, \(\phi_{c}\) is the normalized thermal noise, and \(\eta_{e}\sim 6\) is the effective noise ratio (the value is relevant for the materials considered in Table 1). The result (13) is obtained by specializing the reasoning made in the Appendix of paper [33] to the cryogenic case. The asymptotic curve (13) is dominated by the red straight line in Figure 5, which represents the curve on which the quarter-wave designs are located. Figures 3 and 5 suggest that the number of bumps present on the Pareto Front is equal to the number \(N_{H}\) of layers made of the \(H\)-type material. Let us recall that for designs starting with an \(H\)-type material at the interface with the vacuum, the following relationship applies \(N_{L}=2N_{H}-1\). In the previous section, we have seen that the common Pareto front has bumps. In this section, we will try to explain where they come from. To this end, recall that the trade-off curves shown above were obtained by considering that the thicknesses of all the layers may vary without a priori assumptions. Let us now introduce the trade-off curves for the periodic structures, i.e., the periodic designs in the thicknesses and in the materials. The periodicity assumption reduces the optimization problem to the maximization of a function of two variables with a linear constraint (see Refs. [32; 52] for the development of this idea). The thicknesses of the two alternating materials are the unknowns to be optimized; the number of periodic doublets is to be considered as a parameter. Additionally, in this simple case, we can identify a Pareto  
Figure 5: Various common Pareto fronts for \(N_{H}=8\left(1\right)14\) are shown in the figure. It can be seen that the common Pareto fronts join together in a single curve in the region with high transmittances and low noise. The bump-like structure is also present here; the number of bumps is equal to the number of layers with a high refractive index \(N_{H}\). The gray dashed line in the figure represents the curve in Equation (13) to which the Pareto fronts approach when \(N_{H}\) increases.