## 1 Introduction  
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]. Indeed, the coating Brownian noise is the most relevant source of noise in the band of interest for astrophysical observation. To reduce this source of noise, researchers can act by using different materials with the best properties for the mirrors, optimizing the interferometer's laser beam, lowering the temperature to cryogenic values, and finally acting on the coating design. Unfortunately, there are not many glassy materials that satisfy the optical requirements necessary for gravitational wave detectors, the improvement of the laser beam would require a rethinking of the whole interferometer cavity as well as make this cavity cryogenic (see the papers [3, 4, 5] for a review on the subject). In this work, we explore the possibility of optimizing the design of the mirrors by acting on the thicknesses of the layers that form the coating, with the materials currently available. Although 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 in Figure 1, can be described in a semi-analytic way with the method of the characteristic matrices of the layers (also called transmission matrices [10]). The characteristic matrix of the \(m\)-th layer can be written [11, 12]:  
\[\mathbf{T}_{m}= \begin{bmatrix}\cos(\psi_{m})&t(n^{(m)})^{-1}\sin(\psi_{m})\\ m^{(m)}\sin(\psi_{m})&\cos(\psi_{m})\end{bmatrix} , \tag{1}\]where \(\psi_{m}=\frac{2\pi}{\lambda_{0}}n^{(m)}d_{m}\) is the electric phase, \(\lambda_{0}\) and \(d_{m}\) are the free space wavelength and the layer thickness, respectively, and \(n^{(m)}=n_{r}^{(m)}-\imath K^{(m)}\) is the complex refractive index. Here \(\kappa^{(m)}\) is the extinction coefficient, which for the considered materials will be negligible (i.e., \(\kappa^{(m)}\sim 10^{-8}\)). The optical response of the whole coating (i.e., the transmittance) can be computed from the multilayer characteristic matrix,  
\[\mathbf{T}=\mathbf{T}_{1}\cdot\mathbf{T}_{1}\cdot...\cdot\mathbf{T}_{N_{L}} \tag{2}\]  
where \(N_{L}\) is the total number of layers numbered from the vacuum to the substrate as illustrated in Figure 1. To write Equation (2) we use the property that the characteristic matrix of a sequence of layers is the product of the characteristic matrices of the individual layers. The transmittance calculation is done in two steps, first the equivalent reflection index \(n_{c}\) of the multi-layer structure, and then the reflection coefficient \(\Gamma_{c}\) at the vacuum interface are computed. The complex reflection coefficient \(\Gamma_{c}\) at the vacuum/coating interface is given by:  
\[\Gamma_{c}=\frac{n^{(0)}-n_{c}}{n^{(0)}+n_{c}} \tag{3}\]  
where \(n_{c}\) is the effective refractive index of the whole multilayer structure,  
\[n_{c}=\frac{T_{21}+n_{s}T_{22}}{T_{11}+n_{s}T_{12}}. \tag{4}\]  
The power transmittance at the vacuum/coating interface is \(\tau_{c}=1-|\Gamma_{c}|^{2}\). In the case of binary coating we have:  
\[\left\{ \begin{array}{ll}n^{(m)}=n_{H}-\imath K_{H},\ m\ \text{odd},&n^{(m)}=n_{L}-\imath K_{L},\ m\ \text{even};\\ \phi_{m}=\phi_{H},m\ \text{odd}\,&\phi_{m}=\phi_{L},\ m\ \text{even};\\ Y_{m}=Y_{H},\ m\ \text{odd},&Y_{m}=Y_{L},m\ \text{even}\.\end{array} \right. \tag{5}\]  
Figure 1: In the figure an example of binary coating made of alternating \(L\) and \(H\) type materials is displayed. The laser light is normally incident form the left (i.e., a monochromatic wave, an exp \(\imath\omega t\) dependence on time is supposed), \(\Gamma_{c}\) is the reflection coefficient of the stratified medium.