# Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case
## 6 Conclusions  
This work conclusively addresses a problem of great importance in the construction of gravitational wave antennas. The problem of optimizing the dielectric mirrors of the Fabry-Perot constitutes one of the arms of the Michelson-Morley interferometer. The aforementioned problem, in our opinion, also has relevance for the optics since this research field is new and full of possible applications. Normally, in the theory and practice of multilayer coatings in optics, the focus is on optimizing reflectance alone. The coating-design optimization problem for dielectric mirrors of gravitational wave antennas is an inherently multi-objective problem that we propose to address with appropriate global optimization heuristics. In this connection, we propose to simultaneously minimize the transmittance and the thermal noise of the mirrors.  
In a previous paper [33], we analyzed the case of global multi-objective optimization of a coating operating at room temperature, where the relation \(\eta_{H}>\eta_{L}\) holds. In [33],  
Figure 8: The panel (**a**) of the figure shows the common Pareto front in the case \(\eta_{H}=\eta_{L}=1\); the dielectric parameters are the same as in Table 1. The red dots represent the \(\lambda/4\) design. For the case \(\tau_{c}=0.05\) in the right panel (**c**), the corresponding design is shown, and in the above panel (**b**), the reduced design is shown. In the panels (**b,c**), the blue bar (resp. red) corresponds to \(H\)-type material (resp. \(L\)-type). The common Pareto front is computed by implementing the above-mentioned meta-heuristics, with each running for \(10^{5}\) s.  
however, a thorough convergence analysis has never been done, using only an optimization method (i.e., BorgMOEA). Unfortunately, when using heuristic global optimization methods, for nonlinear and nonconvex problems, there is always the possibility that they will be trapped in local minima by not converging to the sought solution.  
In this work, we extend the study to the cryogenic case \(\eta_{H}<\eta_{L}\), and for completeness to the case of purely mathematical interest \(\eta_{H}=\eta_{L}\) (coating length minimization). We find that in all physically relevant cases, the global multi-objective optimization heuristics (GDE3, NSGA2, NSGA3, MOEA/D, Borg MOEA, and MOPSO) converge to the same trade-off curves, and all features of the Pareto fronts and the optimal designs shown in [20,33] hold regardless of the global optimization method used. This is the first strong evidence of the correctness and uniqueness of the optimal designs previously found and is also the main result of this paper. The optimal design found, i.e., the sequence of the layers thicknesses of the dielectric mirrors, falls in all cases in the class of tweaked periodic structures. These structures improve the performance of the periodic designs analyzed in Refs. [32,52], which are therefore sub-optimal. Furthermore, it is definitively excluded that other types of designs, i.e., fractal or aperiodic, can improve the performance of the dielectric mirror. In fact, we have verified that the heuristics used in the present work to compute Pareto fronts and optimal designs, when applied to the room-temperature case, reproduce the results reported in Ref. [33]. In conclusion, we believe that methods developed in this work will enable us to understand the goodness of convergence of the global optimization heuristics, even in cases of multi-material and possibly nano-layered coating design.  
Formal analysis and Supervision, V.P.; Writing--review and editing, V.G., L.T. and V.P. All authors have read and agreed to the published version of the manuscript. This research received no external funding.  
Not applicable.  
The authors are grateful for the discussion and suggestions received from the colleagues of the INFN Virgo Sannio/Salerno Group, the Virgo Coating R&D Group, and the Optics Working Group of the LIGO Scientific Collaboration.  
The authors declare no conflict of interest.