## 3 Simulation and Experimental Results
### The Reflectance Curve of the Edge Filter  
To accurately measure the reflectance curve of the edge filter, we demonstrated a high-precision measurement system (Figure 3), including a broad-linewidth (1030-1110 nm)-amplified spontaneous emission (ASE) light source, a fiber collimator (FC), a polarization rotator, a convex lens with a 100 mm focal length, and an optical spectrum analyzer. Firstly, we measure the spectrum of the ASE light source in the absence of an edge filter and polarizer (as shown in the inset (a) of Figure 3). By inserting and rotating the polarization rotator, the collimated broad-linewidth laser is injected into a linear polarizer (\(p\) or \(s\)). The laser with a pure polarization state is incident onto the surface of the edge filter, and then it is focused by a convex lens. The focus beam is collected by the probe of the spectrometer. Comparing the spectrum before and after the edge filter, the transmittance curve of the filter can be calculated accurately. With a low absorption coefficient, the absorption of the edge filter is negligible, and thus, the reflectance curve of the filter can be derived from the expression of the transmittance curve. On the other hand, by rotating the edge filter, the reflectance curves at various angles can also be measured (as shown in the inset (b) of Figure 3), which is of benefit for studying the relationship between the transmittance curves and the angle of incidence. In general, there are several typical factors that influence the filter's optical characteristics, such as the surface roughness, the thickness error, and the angle of incidence. According to the uncorrelated surface roughness model (Equation (16)) and the structure of the edge filter, we calculated the reflectance curve of the edge filter under different surface roughness values. The simulation results for the normal  
Figure 3: Test setup for measuring the reflectance curve of the edge filter. FC, fiber collimator; ASE light source, amplified spontaneous emission light source. Inset (**a**) shows the measured spectra; Inset (**b**) shows the experimental results of the reflectance curves under different incident angles (\(0^{\circ}\), \(2.5^{\circ}\), and \(5^{\circ}\)). Figure 2: Experimental setup for a spectral beam formed from the combination of two high-power, broad-linewidth single beams. DL, diode laser; HR FBG, highly reflective fiber Bragg grating; OC FBG, output coupler fiber Bragg grating; YDCF, ytterbium-doped double-clad fiber; CPS, cladding power stripper; HR, high reflector [11; 24]. incidence are shown in Figure 4a. With an increase in the surface roughness, a shift in the reflectance curve position is noticeable, and the rising edge tends to be gentler. The experimental result, as measured by the spectrometer, is indicated by the red line, and it shows a good agreement with the simulated results. Moreover, we have compared the model with the Thin Film Design Software TFCalc, and considered a multilayer coating with smooth surface and the thickness error not taken into account. The simulated reflectance curves for the normal incidence are shown in Figure 4b,c. The results indicate that the simulation derived from the proposed model agrees well with those obtained with the TFCalc, which proves the reliability of the model. Except for the surface roughness, we also calculated the thickness error of the multilayer. A system of 268 dielectric layers was considered, each having the same variations from \(-0.5\) nm to \(0.5\) nm, and the surface roughness was set to \(0.85\) nm. To treat the case of normal incidence, the reflectance curves of the edge filter with respect to the thickness error are shown in Figure 4d, in which only a shift in the position of the reflectance curve occurs, and where the thickness error does not affect the steepness of the rising edge. It should be noted that the thickness errors on each layer always occur randomly during the coating process.