## 2 Theory
\begin{array}{cc}e^{i\varphi_{j}}&0\\ 0&e^{-i\varphi_{j}}\end{array} \right] \tag{4}\]  
where \(r_{ij}\) and \(t_{ij}\) are the Fresnel reflection coefficient and the Fresnel transmission coefficient, respectively. Under the condition of a small incidence angle \(\theta_{i}\), they can be expressed as:  
\[r_{\mathrm{s}}=\frac{n_{i}\cos\theta_{i}-n_{j}\cos\theta_{j}}{n_{i}\cos \theta_{i}+n_{j}\cos\theta_{j}} \tag{5}\]  
\[r_{p}=\frac{n_{i}\cos\theta_{j}-n_{j}\cos\theta_{i}}{n_{i}\cos\theta_{j}+n_{j} \cos\theta_{i}} \tag{6}\]  
\[t_{s}=\frac{2n_{i}\cos\theta_{i}}{n_{i}\cos\theta_{i}+n_{j}\cos\theta_{j}} \tag{7}\]  
\[t_{p}=\frac{2n_{i}\cos\theta_{i}}{n_{i}\cos\theta_{j}+n_{j}\cos\theta_{i}} \tag{8}\]  
where \(\varphi_{j}\) is the optical phase thickness of layer \(j\), and it is given as:  
\[\varphi_{j}=\frac{2\pi n_{j}d_{j}\cos\theta_{j}}{\lambda} \tag{9}\]  
Figure 1: (**a**) Schematic diagrams of the steep-edge filters; (**b**) the profile of the rough surface [21; 24]. The matrix \(S\) represents the effect of the surface roughness, and it can be expressed as:  
\[S_{ij}=\frac{1}{\left(1-r_{ij}^{2}\right)}\left[ \begin{array}{cc}a-r_{ij}^{2} b&r_{ij}(b-a)\\ r_{ij}\left(b^{-1}-a^{-1}\right)&a^{-1}-r_{ij}^{2}b^{-1}\end{array} \right] \tag{10}\]  
\[a=\exp\left\{i\left[\frac{2\pi f(x,y)(n_{i}\cos\theta_{i}-n_{j}\cos\theta_{j}) }{\lambda}\right]\right\} \tag{11}\]  
\[b=\exp\left\{i\left[\frac{2\pi f(x,y)(n_{i}\cos\theta_{i}+n_{j}\cos\theta_{j}) }{\lambda}\right]\right\} \tag{12}\]  
where \(z=f(x,y)\) is the function of the surface profile, and its associated probability density \(\omega(z)\) can be expressed as:  
\[w(z)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{z^{2}}{2\sigma^{2}}\right) \tag{13}\]  
where \(\sigma\) is the RMS surface roughness. Thus, the amplitude reflectance \(r(\lambda)\) and the transmittance \(t(\lambda)\) of the edge filter can be given as:  
\[r(\lambda)=p_{2}/p_{1} \tag{14}\]  
\[t(\lambda)=1/p_{1} \tag{15}\]  
The radiant reflectance \(R(\lambda)\) is the absolute square of the amplitude reflectance:  
\[R(\lambda)=r\cdot r^{\star} \tag{16}\]  
If the angle of incidence is \(\theta_{0}\) and the angle of refraction is \(\theta_{s}\), the transmittance curve is deduced as follows:  
\[T(\lambda)=(n_{s}\cos\theta_{s}/n_{0}\cos\theta_{0})\cdot t\cdot t^{\star} \tag{17}\]  
For the case of the spectral beam-combining system, the filter as the combining element needs to overlap two beams at different wavelengths in both the near and far fields, and the optical reflectance curve \(R(\lambda)\) of the filter will have a marked impact on the combining efficiency. Therefore, a combining efficiency model that is calculated from the laser spectrum is introduced into the uncorrelated surface roughness model, and the combining efficiency can be expressed as [11]:  
\[\eta=\frac{\sum\limits_{\lambda_{1}}^{\lambda_{2}}[F_{2}(\lambda)\star T( \lambda)]+\sum\limits_{\lambda_{1}}^{\lambda_{2}}[F_{1}(\lambda)\star R( \lambda)]}{\sum\limits_{\lambda_{1}}^{\lambda_{2}}F_{1}(\lambda)+\sum\limits_ {\lambda_{1}}^{\lambda_{2}}F_{2}(\lambda)} \tag{18}\]  
where \(F_{1}(\lambda)\) and \(F_{2}(\lambda)\) represent the spectral curves of the incident laser beams, with \(R(\lambda)\) being the reflectance curve of the edge filter and (\(\lambda_{1}\), \(\lambda_{2}\)) being the calculation interval.