# Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case
## 2 The Physical Model
(the low-contrast material) worsen significantly at very low temperatures. The optimization technique developed in this paper attempts to reduce the worsening of the loss angle due to this cryogenic peak phenomenon. In this connection, we introduce a functional producing the thermal noise of coating as a function of the normalized thicknesses of the layers  
\[\phi_{c}=\sum_{m=1}^{N_{L}}\eta_{m}\bar{z}_{m} \tag{7}\]  
where \(\eta_{m}\) are the specific thermal noise coefficients depending on the material that constitutes the layer. The noise model used here is the one described in [28]; other models are available in the literature [30], but despite their supposed higher accuracy they provide comparable results (see discussion in [31] and supplementary material therein). In the following, we study the dielectric structure made of alternating layers of two materials indicated with the subscripts \(H\) and \(L\), with the refractive index \(n_{H}\) and \(n_{L}\), and specific normalized loss coefficient \(\eta_{H}\) and \(\eta_{L}\). The type \(L\) material is a low refractive index material; thus, we have by definition \(n_{L}<n_{H}\). In view of the linearity of \(\phi_{c}\) functional, see Equation (7), both as a function of \(\bar{z}_{m}\) and \(\eta_{m}\), we can take as a reference the material \(H\) and pose \(\eta_{H}=1\). This normalization takes the lower specific noise coefficient between the two materials as unitary; in the cryogenic case (due to the cryogenic peak of L-type material), the H-type material (supposed to have lower mechanical losses) is taken as a reference (at room temperature the situation is reversed). Considering the above definitions and for a dielectric mirror made of two alternating materials starting with the \(H\) layer, we have the following functional:  
\[\phi_{c}=\sum_{m=0}^{\lfloor N_{L}/2\rfloor}\bar{z}_{2m+1}+\sum_{m=1}^{\lfloor N _{L}/2\rfloor}\eta_{L}\bar{Z}  
The problem in (10) can be alternatively written [32]:  
\[ \begin{array}{ll}\underset{z_{1},\ldots,z_{N_{L}}\in\Omega}{\text{Min}}&\tau_{c} (\tilde{z}_{1},\ldots,\tilde{z}_{N_{L}})\\ \text{subject to}&\phi_{c}(\tilde{z}_{1},\ldots,\tilde{z}_{N_{L}})\leq\phi_{0} \end{array}  \tag{11}\]  
where \(\phi_{0}\) is the maximum allowed loss angle and \(\tau_{0}\) is the constraint on the transmittance. In the paper [33], it has been shown that the constrained optimization problems (10) or (11) can be further formulated in an equivalent multi-objective fashion, as follows:  
\[\underset{z_{1},\ldots,z_{N_{L}}\in\Omega}{\text{Min}}\quad[\phi_{c}(\tilde{z }_{1},\ldots,\tilde{z}_{N_{L}}),\tau_{c}(\tilde{z}_{1},\ldots,\tilde{z}_{N_{L} })]. \tag{12}\]  
The coating optimization using the multi-objective approach reduces to the computation of the Pareto front of the optimization problem (12); for a tutorial on multi-objective optimization, see [34]. In [33], it has been shown that the three formulations of the optimization problem (10)-(12) are equivalent, for the case of coatings designed to operate at room temperature. It must be remembered that the problem (10) can be solved using the Pareto front of (12), by choosing the point on the Pareto boundary with a transmittance component equal to \(\tau_{0}\). Similar considerations can be made for the problem (11), which can be solved by taking the point on the Pareto boundary with a noise component equal to \(\phi_{0}\). Such a result arises from the properties of the Pareto front, which turns out to be continuous and monotonic for the problem (12). In the remainder of this paper, we will show that the above results are also true in the case of the optimization of coatings operating at cryogenic temperatures.