Optimal Design of Coatings for Mirrors of Gravitational Wave Detectors: Analytic Turbo Solution via Herpin Equivalent Layers

Abstract

In this paper, an analytical solution to the problem of optimal dielectric coating design of mirrors for gravitational wave detectors is found. The technique used to solve this problem is based on Herpin’s equivalent layers, which provide a simple, constructive, and analytical solution. The performance of the Herpin-type design exceeds that of the periodic design and is almost equal to the performance of the numerical, non-constructive optimized design obtained by brute force. Note that the existence of explicit analytic constructive solutions of a constrained optimization problem is not guaranteed in general, when such a solution is found, we speak of turbo optimal solutions.

Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case

“…It has been verified that for the realisitc values of N H (i.e., ∼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 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…”

“…In this work, we extend the study to the cryogenic case η H < η L , and for completeness to the case of purely mathematical interest η H = η 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 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 optimization techniques 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.…”