## 2 Theory  
The formula for calculating the reflectivity of multilayer elements is commonly derived from the Fresnel formula and the iteration algorithm [14]. However, the experimental reflectivity is usually lower than the theoretical value, which can be attributed to three aspects: firstly, the film stack is usually considered to be homogeneous and smooth in simulation, while the roughness of each interface can result in significant losses in the reflectivity of the filters during the coating process [15]; secondly, when limited by the coating devices and manufacturing techniques, the thickness of the film is inaccurate, and the accumulated thickness errors have an obvious effect on the reflectivity. The last important factor is the deviation of the angle from the normal incidence, which causes a change in the optical path difference between the reflected lights. In this case, several optical models and modeling approaches to accurately calculating the reflectivity of multilayer optical coatings have been presented, such as uncorrelated surface roughness model, additive surface roughness model, and uncorrelated bulk inhomogeneity model [16; 17; 18; 19; 20; 21]. After repeated supplementation and correction [22; 23], the uncorrelated surface roughness model is now considered to be an effective and practical approach for academic research into the relationship between the surface roughness and reflectivity values of multilayer elements. In this model, each interface is treated as being completely uncorrelated with the other interfaces in the multilayer coating, and some assumptions should be emphasized: (I) the root mean square (RMS) surface roughness, \(\sigma\), is typically much less than a wavelength; (II) for the case of a small incidence angle, the polarization effect is not considered; (III) the function of the surface profile \(f(x,y)\) is a random function, and its associated probability density is a Gaussian distribution; (IV) the absorption of the edge filter is not included in this model; (V) the substrate is considered to be semi-infinite, so that its reflectance can be ignored [21]; (VI) the roughness of each interface is treated as the same as the filter's. The schematic diagrams of the edge filter used in the 10-kW-class FSBC system are shown in Figure 1a [11; 24]. The filter consists of \(m\) (\(m=268\)) dielectric layers, each having an index \(n\) and a thickness \(d\). With regard to considerations for increasing the threshold, SiO\({}_{2}\) (\(n_{1}=1.433\)) and ZrO\({}_{2}\) (\(n_{2}=1.97\)) films are deposited in an alternating fashion on the fused silica substrate (\(n_{s}=1.5\)). In this case, the filter has a high damage threshold (\(>\)20 MW/cm\({}^{2}\)) and a steep rising edge. The surface roughness \(\sigma\) of each interface here is considered as the same as the filter's, which could be accurately measured by the Veeco white light profiler. Dozens of positions are measured and the average value is 0.85 nm [24]. Figure 1b illustrates the profile of the rough surface, and for the case of a small incidence angle \(\theta_{i}\) (\(\theta_{i}<6^{\circ}\)), the amplitudes of the incident wave \(E_{0}^{+}\), the reflected wave \(E_{0}^{-}\), and the transmitted wave \(E_{\mathrm{s}}^{+}\) can be related through the following [21]:  
\[\left[ \begin{array}{c}E_{0}^{+}\\ E_{0}^{-}\end{array} \right]\ \ =\ \ \ P\ \ \ \left[ \begin{array}{c}E_{s}^{+}\\ 0\end{array} \right] \tag{1}\]  
where \(P\) is the characteristic matrix of the multilayered optical coatings and described as:  
\[P=S_{01}I_{01}T_{1}S_{12}I_{12}\cdots T_{m}S_{ms}I_{ms}=\left[ \begin{array}[] {cc}p_{1}&p_{3}\\ p_{2}&p_{4}\end{array} \right] \tag{2}\]  
where \(I_{ij}\) represents the propagation of the field from layer \(i\) to layer \(j\), and \(T_{j}\) represents the propagation of the field across the \(j_{th}\) layer. The expressions of \(I_{ij}\) and \(T_{j}\) are given by:  
\[I_{ij}=\frac{1}{t_{ij}}\left[ \begin{array}{cc}1&r_{ij}\\ r_{ij}&1\end{array} \right] \tag{3}\]  
\[T_{j}=\left[