# Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case
## 1 Introduction  
The experimental research and the construction of large interferometric ground-based antennas [1, 2, 3, 4, 5, 6, 7] aimed at the direct detection of gravitational waves have been very relevant in recent years (see [8, 9] for an introduction to the topics). During this period, we have witnessed the first detection of a gravitational wave, as well as the first direct evidence of a black hole mergers [10]; the joint detection achieved with three detectors, which allowed for the great sky localization precision [11] (gravitational astronomy was born); and the the first detection of a multi-messenger event [12, 13], the publication of which has ended with the third catalog [14]. To achieve these far-reaching results, there are many problems and technological challenges that researchers have had to overcome, and it is precisely one of these problems that we will discuss below.  
In this work, we address the problem of identifying the optimal design of a totally reflective dielectric mirror. The solution to the problem is of particular interest for the realization of gravitational wave detectors [1, 5, 15, 16], which must have the highest possible reflectance and at the same time reduced thermal noise [17, 18]. In fact, the sensitivity of gravitational waves detectors is ultimately limited by the various noise sources determining random fluctuations in the physical optical path length, in particular suspension thermal noise and mirror thermal noise.  
Therefore, the problem of optimal design naturally presents itself as a multi-objective optimization problem. The highly non-linear and non-convex nature of the reflectance function due to the variation in the length of the layers makes it necessary to use optimizationtechniques that do not fall within the canonical methods of mathematical programming [19]. Furthermore, an analytical solution to this problem is not known in the current literature, although solutions very close to the analytical one can be found in the literature [20] for the case of detectors operating at room temperature.  
Of course, there are other strategies for noise reduction in gravitational wave coatings: those based on the use of three or more materials [21,22], and the production of nano-engineered materials [23]. In this area, the study of the crystallization of materials as a function of deposited thickness becomes crucial [24].  
In this paper, we address the problem using a meta-heuristic approach. In particular, we ask whether there is a single criterion (i.e., independent of the optimization method used) that is representative of the set of solutions that make up the Pareto front.  
In other words, the objective of the paper, rather than focusing on a specific optimization technique, is to answer the question of whether there is also a unique criterion of optimal design in the case of cryogenic detectors, and of what its physical interpretation is. To achieve this goal, we compare the solutions offered by a multiplicity of multi-objective meta-heuristic optimization methods with constraints. We will select among multiple optimization methods those that have demonstrated well-established validity in applications. Furthermore, the solution found offers some non-trivial physical interpretation cues and original design criteria, showing a performance limit.  
The remainder of this paper is organized as follows: Section 2 illustrates the physical model of the dielectric mirror in the context of gravitational wave detection; Section 3 illustrates the mathematical formulation of the optimization problem; Section 4 outlines the methods employed for solving the optimization problem; Section 5 reports the experimental findings, describing the physical meaning of results; and the conclusions and perspective are discussed in Section 6.