# Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case
## 5 Numerical Experiments
### Physical Properties of the Optimal Design
The physical parameters are those of Table 1, and the common Pareto fronts are computed by implementing the above-mentioned meta-heuristics each running for \(10^{5}\) s.  
curve that varies with the number of doublets, i.e., it varies with the number of \(H\)-type layers. It is necessary to take a small precaution for the design that, at the first interface (near to the vacuum), has an \(H\)-type material layer; the last \(L\)-type layer should not be taken into account in the noise computation, being of the same material as the substrate (in principle it can be removed without affecting the coating transmittance). With these considerations in mind, we comment on Figure 6, where the common Pareto front for \(N_{H}=12\) (i.e., \(N_{L}=23\)) is shown together with the Pareto fronts of the periodic design; the numbers of \(H\)-type layers change accordingly \(N_{H}=3(1)8\). The envelope of the periodic case Pareto fronts always lies above the common Pareto front. Furthermore, in Figure 6 it is noted that each bump of the common Pareto front is close to a periodic design Pareto front with given \(N_{H}\). This is further elucidated in Figure 7; in the upper right b) panel we display the closeup of two common Pareto fronts with \(N_{H}=12\), \(14\) (they join together, as noted before). In the same panel, the periodic design Pareto fronts for \(N_{H}=3\), \(4\), \(5\), are displayed and all the curves of the Pareto fronts are intersected with a horizontal straight line, indicating a transmittance of \(\tau_{c}=0.2\). The intersection identifies a point on (both) the common Pareto fronts that is in the fourth bump (counting from above). Furthermore, the line \(\tau_{c}=0.2\) intersects the periodic design Pareto curve with \(N_{H}=4\) i.e., the curve with four \(H\)-type layers. In the upper left panel (a) this periodic design is shown, and in the same column in the panels (c) and (e), the design relative to the common Pareto fronts with \(N_{H}=12\), \(14\) is shown. The periodic design for \(\tau_{c}=0.2\) in panel (a) does not seem to be similar to those obtained from the common Pareto fronts in panels (c) and (e), but if we manipulate the thicknesses sequence by eliminating layers with thicknesses much smaller than \(\lambda/4\) and joining two contiguous layers of the same material, then the situation changes. This reduction was performed on the sequences displayed in (c) and (e) by providing panels (d) and (f); it is clear that the two solutions are very similar to panel a), and all three correspond to a quasi-periodic design with initial and final tweaking. Figure 6: The common Pareto fronts for \(N_{H}=12\) are shown in the figure; furthermore, in the figure we show the Pareto fronts for periodic designs with the variable \(N_{H}=3(1)8\). The envelope of the periodic Pareto fronts always lies above the common Pareto front. The physical parameters are those of Table 1, and the common Pareto front is computed by implementing the above-mentioned meta-heuristics, with each running for \(10^{5}\) s. The Pareto tradeoff curves of the periodic design are computed in \(\sim\)\(10^{2}\) s.