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.
Highlights
The development of optimized coatings for the end test-masses of the gravitational wave interferometers is one of the important goals to be achieved for improving the sensitivities of gravitational wave detectors [1,2]
Herpin’s theorem allows obtaining an equivalent stratified material, consisting of three layers arranged in a palindrome sequence, which mimic exactly a quarter-wave layer. These quarterwave equivalent layers are used in conjunction with normal layers made of low refractive index material, to produce optimized designs of coatings
The method reduces to an optimization problem with two independent parameters, namely the number of equivalent layers ND and the normalized thickness of one of the materials defining the equivalent layer
Summary
The development of optimized coatings for the end test-masses of the gravitational wave interferometers is one of the important goals to be achieved for improving the sensitivities of gravitational wave detectors [1,2]. Coatings made of multiple materials [6,7,8] (obtained, in some cases, from cascades of binary designs) have been recently proposed, further study of binary coating theory provides the theoretical tools to understand more complex approaches It is well known [9] that the electrodynamics of multilayers structures, like those depicted, can be described in a semi-analytic way with the method of the characteristic matrices of the layers ( called transmission matrices [10]). The transmittance calculation is done in two steps, first the equivalent reflection index nc of the multi-layer structure, and the reflection coefficient Γc at the vacuum interface are computed. In the case where the refractive index nL is the same as that ns of the substrate material (as in current gravitational wave detectors) NL is an odd number. Subject to φc ≤ φ0 where φ0 is a prescribed maximum allowed loss angle
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