## 2 The Herpin Equivalent Layer Optimization Problem  
According to [PERSON]'s equivalent layer theorem [19] a symmetrical multilayer stack (i.e., a _palindrome_ sequence of dielectric thin films) is equivalent to a single layer. This theorem is based on the fact that the two elements on the main diagonal of the characteristic matrix of any palindromic sequence of materials are equal. In this paper, we consider an equivalent Herpin layer that consists of three physical layers arranged in an \(LHL\)-type sequence. According to the general theorem, this sequence must be dielectrically and geometrically symmetric, so both materials and layer thicknesses must be palindromic. Thus denoting by \(p\) and \(q\) the normalized lengths of the layers \(L\) and \(H\) respectively, below are shown, using a simple computation, the relevant elements of the transmission matrix \(\mathbf{T}^{(E)}\) of the considered _virtual layer_\(E=LHL\):\[T_{11}^{(E)}=\cos(2\pi qn_{H})\cos(4\pi pn_{L})\ -\ \frac{\left(n_{H}^{2}+n_{L}^{2} \right)\sin(2\pi qn_{H})\sin(4\pi pn_{L})}{2n_{H}n_{L}}, \tag{11}\]  
\[T_{12}^{(E)}=i\left(-\frac{n_{H}\sin(2\pi qn_{H})\sin^{2}(2\pi pn_{L})}{n_{L}^{ 2}}+\frac{\sin(2\pi qn_{H})\cos^{2}(2\pi pn_{L})}{n_{H}}+\frac{\cos(2\pi qn_{H })\sin(4\pi pn_{L})}{n_{L}}\right). \tag{12}\]  
It can be verified by inspection, and this is also the main result of [PERSON]'s theorem, that \(T_{11}^{(E)}=T_{22}^{(E)}\), and because of the unitary constraint on the determinant of the characteristic matrix \(T\), the last element is uniquely determined by solving the following equation w.r.t. \(T_{21}^{(E)}\):  
\[(T_{11}^{(E)})^{2}-T_{12}^{(E)}T_{21}^{(E)}=1. \tag{13}\]  
To understand the reasons that lead to formulating the present analytical solution, we summarize the results of the papers [18; 20]. In the paper [18], a multi-objective optimization (with the BorgMOEA algorithm [21]) without _a priori_ hypothesis was applied to the problem of general optimization of binary coatings for gravitational wave detectors mirrors. It has been shown in [18] that the Pareto front remains the same whether one sets up a code that solves problem (7) or implements problem (10), so the two formulations considered are equivalent. Moreover, in the same paper we show that the optimal design is made by the following sequences of layers \(L(HL)^{N_{D}}H\). The sequence of thicknesses associated with this solution is given by \(z_{L,i}(z_{H}z_{L})^{N_{D}}z_{H,f}\), where in general \(z_{L,i}\
eq z_{L}\), \(z_{H,f}\
eq z_{H}\). This solution is periodic except for the first and last layers, and that is why the solution has been called periodic with initial and final tweaking. Furthermore \(z_{L},\ z_{H}\) satisfy an approximate Bragg condition \(n_{L}z_{L}+n_{H}z_{H}\sim 0.5\). In the paper [20], a very simple periodic solution \((HL)^{N_{D}}H\) was studied and experimentally validated [22]. In these articles the tweaking procedure has been considered but only as a possible second step of improvement of the periodic design, considering the thicknesses of the innermost layers fixed to the values calculated in the first step. This solution approximates that which would be found by optimizing on all four layers simultaneously, that is implemented in [18]. Finally, in the paper [18] (see Equation (20) therein) it has been shown that the Pareto front of the optimized solutions is placed close to the transmittance versus thermal noise line relative to a suitable (virtual) quarter-wave design. So far, we have mentioned almost exhaustively all the existing literature on the coating optimization problem. For the sake of completeness, although not completely relevant, let us mention [23] which proposes a physical-mathematical approach to the computation of the best periodic design, we emphasize that the method is not based on solving an optimization problem.