# Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case
## 2 The Physical Model  
In this section, we start describing the electromagnetic model of a coating made of dielectric layers. This device is designed to reproduce the behaviour of a low-absorption perfect mirror, i.e., of a mirror with a reflection coefficient near to the unity. The electromagnetic behaviour of a coating can be obtained by a simple matrix method [25] (for a comprehensive introduction to the topic of layered structures, see [26,27]). The optical response of a coating can be computed from the multi-layer characteristic matrix,  
\[\mathcal{T}=\mathcal{T}_{1}\cdot\dots\cdot\mathcal{T}_{m}\cdot\dots\cdot \mathcal{T}_{N_{L}} \tag{1}\]  
where \(N_{L}\) is the total number of layers numbered from the vacuum to the substrate, as illustrated in Figure 1. The matrix \(\mathcal{T}_{m}\) is associated with the geometrical and material properties of the layer  
\[\mathcal{T}_{m}= \begin{bmatrix}\cos(\xi_{m})&t(n^{(m)})^{-1}\sin(\xi_{m})\\ n^{(m)}\sin(\xi_{m})&\cos(\xi_{m})\end{bmatrix} , \tag{2}\]  
where in the case of normal incidence  
\[\xi_{m}=\frac{2\pi}{\lambda_{0}}n^{(m)}z_{m}. \tag{3}\]We have defined \(\lambda_{0}\) and \(z_{m}\) as the free space wavelength (i.e., the laser frequency) and the layer thickness, respectively, and \(n^{(m)}\) is the refractive index of the \(m\)-th layer. The refractive index of the whole coating can be obtained by using \(n_{l}\), i.e., the refractive index of the layered structure:  
\[n_{l}=\frac{\mathcal{T}_{21}+n_{s}\mathcal{T}_{22}}{\mathcal{T}_{11}+n_{s} \mathcal{T}_{12}}, \tag{4}\]  
where \(n_{s}\) is the refractive index of substrate (see Figure 1) and \(\mathcal{T}_{ij}\) represents the components of characteristic Matrix (2). The Formula (4) can be used to compute the power transmittance \(\tau_{c}\) at the vacuum/coating interface :  
\[\tau_{c}=1-\left|\frac{n^{(0)}-n_{l}}{n^{(0)}+n_{l}}\right|^{2}. \tag{5}\]  
where \(n^{(0)}=1\) is the refractive index of the vacuum. In the following, we should minimize the transmittance \(\tau_{c}\) as a function of \(z_{m}=z_{m}/\lambda_{0}\), the normalized layer thicknesses. The reduction of transmittance is a general problem of the optical design of dielectric layered structures, and the peculiarity of the gravitational wave coating design problem consists of obtaining such a minimum constraint on the thermal noise. Thermal noise in the coatings of the test-masses is presently the dominant noise source in interferometric gravitatinal wave detectors [17], their ultimate visibility distance is set in the (40-300) Hz band. The power spectral density of coating thermal noise is given, under suitable simplifying assumptions, by [28]  
\[S_{coat}^{B}=\frac{2k_{B}T}{f\pi^{3/2}}\frac{1-\sigma_{s}^{2}}{wY_{s}}\phi_{c} \tag{6}\]  
where \(f\) is the frequency; \(T\) is the (absolute) temperature; \(w\) is the (assumed Gaussian) laser-beam waist; and \(\sigma_{s}\) and \(Y_{s}\) are the Poisson and Young modulus of the substrate, and the coating loss angle \(\phi_{c}\). It is clear from Equation (6) that a viable method of reducing thermal noise is to lower the temperature down to cryogenic values (i.e., \(T\sim 20\) K). The use of cryogenic techniques to reduce the thermal noise of the coating, unfortunately, is limited by the phenomenon of cryogenic peak of silica [29]. In particular, the mechanical losses of silica  
Figure 1: The figure shows the sketch of a multilayer structure with an electromagnetic wave coming from the left, i.e., a monochromatic plane wave; an exp \(\iota\omega t\) dependence on time is supposed, and \(\lambda_{0}\) is the free space wavelength. The symbol \(\Gamma_{c}\) denotes the complex reflection coefficients, while \(n_{H}\) and \(n_{L}\) are the refractive indices of the alternating layers. The parameters of the densest material have the subscript \(H\), while those of the least dense material have the subscript \(L\). The indices \(n^{(0)}\) and \(n_{s}\) are the refractive indices of the vacuum and substrate, respectively.