## 2 Materials and Methods  
The schematic diagram of VSBC is shown in Figure 1. The VSBC setup includes a diode laser array (anti-reflection coated in the front facet), a fast-axis collimation lens (FAC), a beam transform system (BTS), a slow-axis collimation lens (SAC), a Fourier lens (TFL), a half-wave plate (HWP), a reflection grating, and an HR mirror. The beams from the diode array are collimated in the fast axis, rotated 90\({}^{\circ}\), and collimated in the slow axis by the SAC. The position error of the emitters due to the smile effect in the horizontal direction is converted to that in the vertical direction. The effect of smile, crosstalk of each emitter, and the divergence of the slow axis are reduced. The BTS increases the stability of the cavity significantly [36]. The HWP is inserted to correct the polarization between lasers from the diode array and the grating. The diode array and the reflection grating are respectively placed on the front and rear focal planes of the transform lens with the focal length \(f_{T}\). The position of each array element was transformed into the angle of incidence on the grating. The HR mirror supplies partial feedback with an off-axis reflecting angle \(\alpha\) shown in Figure 1b. Thus, all emitters of the array are forced to work at a specific wavelength in the external cavity structure formed between the back facet of the laser array and the HR mirror. All beams from the diode array are spatially overlapped on the grating at different incident angles with different wavelengths; they diffract at the same angle, and are perpendicular to the HR mirror in the direction of the laser. In principle, the beam quality of VSBC goes beyond the single emitter and traditional SBC. The wavelength of each emitter can be calculated by \(\lambda_{i}=\Lambda(\sin\theta_{i}+\sin\theta_{d})\). \(\lambda_{i}\) is the wavelength of each emitter, the \(\theta_{i}\) and \(\theta_{d}\) are the incident angle in the grating and the diffraction angle, respectively. The linewidth of VSBC \(\Delta\lambda\) can be calculated by \(\Delta\lambda\approx d^{*}\,\Lambda^{*}\cos\theta_{L}/f_{T}\). The \(d\) is the lateral dimension of the array, \(\Lambda\) is the grating period, and the \(\theta_{L}\) is the Littrow angle. A standard centimeter BAL bar with 19 emitters working at 940 nm is employed in our experimental setup. All the emitters are coated with a reflectivity of R < 5% on the front facet. The parameters of the diode laser array are shown in Table 1. The BTS and the FAC with focal lengths of 0.286 mm used are commercial products from LIMO. The SAC is a cylindrical lens with a focal length of 7 mm. The focal length of the transform lens is 350 mm. All optical components mentioned above have an anti-reflection coating of R < 0.5%. The diffraction grating is 1800 lines/mm with a first-order diffraction efficiency of 91% at 940 nm. From the measured result, we found that the grating brings a significant fraction of the loss about 10%. The HR mirror is a K9 mirror coated with Au. To improve the beam quality further, a BSPC was employed after the VSBC. The schematic of the experimental setup of BSPC is shown in Figure 2. BSPC includes the laser combined VSBC, a precisely cut prism, three HR mirrors, an HWP, and a beam combiner (PBC). The laser from the array combined by VSBC was split into two loops equally by a precisely cut prism along the beam waist in the slow axis. The polarization of one loop was changed by the HWP. The two loops with different polarization were combined by the (PBC) using three HR mirrors. The beam quality will be improved by 50% in principle because that the beam waist of the combined laser by BSPC is reduced to half of that original. BSPC is suitable for the laser from VSBC.