## 1 Introduction
The refractive index of the \(m\)-th layer is in general \(n^{(m)}\), for binary coating the sequence of refractive index is alternating with values \(n_{H}\) (high refractive index) and \(n_{L}\) (low refractive index). The substrate refractive index is \(n_{s}\), the leftmost half space has, refractive index \(n^{(0)}\) (here it is supposed to be the vacuum). These multilayer structures are made of \(N_{L}\) alternating high \(n_{H}\) and low \(n_{L}\) refractive indexes deposited on a substrate of refractive index \(n_{s}\). The coefficients \(\phi_{H}\) and \(\phi_{L}\) are the mechanical losses and \(Y_{H}\) and \(Y_{L}\) are the Young moduli of the two materials. Let us also introduce the specific noise coefficients that will be used in the following:  
\[\eta_{L}=\frac{1}{\sqrt{\pi w}}\phi_{L}\bigg{(}\frac{Y_{L}}{Y_{s}}+\frac{Y_{s} }{Y_{L}}\bigg{)}\quad\eta_{H}=\frac{1}{\sqrt{\pi w}}\phi_{H}\bigg{(}\frac{Y_{ H}}{Y_{s}}+\frac{Y_{s}}{Y_{H}}\bigg{)}, \tag{6}\]  
where \(w\) is the (assumed Gaussian) laser-beam waist. Thermal noise in gravitational detectors is the most important limitation for their operation. We refer the reader to the works in [13; 14; 15] for an exhaustive description of the problem and the proposed solutions [16; 17] in the operating observatory Virgo [1] and LIGO [2]. A straightforward formulation of the coating optimization problem for the design of low noise dielectric mirror can be:  
\[ \begin{split}\underset{z_{1},\ldots,z_{N_{L}}\in\Omega}{\text{ Minimize}}&\tilde{\phi}_{c}\\ \text{subject to}&\tau_{c}\leq\tau_{0}\end{split}  \tag{7}\]  
where the constraint transmittance \(\tau_{0}\) should be a few parts per million (ppm), typically \(\tau_{0}=6\) ppm. Defining the normalized loss angle \(\tilde{\phi}_{c}=\phi_{c}/(\lambda_{0}\eta_{L})\) and introducing the normalized physical length \(z_{m}=d_{m}/\lambda_{0}\), where \(\lambda_{0}\) is the free-space wavelength of the laser, we have for binary coatings:  
\[\tilde{\phi}_{c}=\,^{\circ}\Sigma_{m=1}^{N_{L}}\gamma_{2m}+\,^{\circ}\Sigma_{m =1}^{N_{L}}z_{m}, \tag{8}\]  
where summation \({}^{\circ}\Sigma\) (resp. \({}^{\circ}\Sigma\)) is on the even (resp. odd) integer such that \(1\leq m\leq N_{L}\). The noise ratio coefficient in Equation (8), i.e., \(\gamma=\eta_{H}/\eta_{L}\), can be explicitly written for binary coating as:  
\[\gamma=\frac{\phi_{H}}{\phi_{L}}\bigg{(}\frac{Y_{H}}{Y_{s}}+\frac{Y_{s}}{Y_{H} }\bigg{)}\bigg{(}\frac{Y_{L}}{Y_{s}}+\frac{Y_{s}}{Y_{L}}\bigg{)}^{-1}. \tag{9}\]  
In the case where the refractive index \(n_{L}\) is the same as that \(n_{s}\) of the substrate material (as in current gravitational wave detectors) \(N_{L}\) is an odd number. The choice of an even \(N_{L}\) would lead to a configuration with the rightmost layer (near the substrate) made of low refractive index material (the same as the substrate) that would increase noise without having any effect on reflectivity. The search space \(\Omega\) is defined by the inequalities \(0\leq z_{m}\leq 0.25/n_{H}\) for odd \(m\) and \(0\leq z_{m}\leq 0.5/n_{L}\) for even \(m\). An alternative and equivalent way to formulate the optimization problem (as shown in [18]) is  
\[ \begin{split}\underset{z_{1},\ldots,z_{N_{L}}\in\Omega}{\text{ Minimize}}&\tau_{c}\\ \text{subject to}&\tilde{\phi}_{c}\leq\tilde{\phi}_{0} \end{split}  \tag{10}\]  
where \(\tilde{\phi}_{0}\) is a prescribed maximum allowed loss angle.