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A multiple mode rate equations model of the dual-cavity solid-state vortex laser has been established and used to investigate the dynamic process of the Laguerre-Gaussian (LG) modes competition. We calculated the dynamic processes of the modes LG01, LG02 and LG03. The results show that the laser exhibits a complex cross-spiking and cross-relaxation characteristic during the early stage of mode competition. The later start of a mode would cause the cross-spiking and cross-relaxation process, and ultimately the mode started firstly may even not be the one that can be sustained at steady state. To ensure the successful mode selection, the reflectivity of the secondary cavity should be larger than that of the primary cavity, but a too large one would decrease the stable output power of the mode LG01, even to its suppression. The pumping beam distribution has a great influence on the dynamic process and the stable output power of the modes, so the radius and the order of the pumping beam should be optimized. In our case, the optimized beam radius is 0.4 mm, slightly larger than the beam radius of the mode LG00, i.e.0.3 mm, and the optimized order is 4. Moreover, if the laser do not have proper reflectivities of output couplers and pumping beam distribution, the mode selection may not be demonstrated only by optimization of the aperture radius, which would only delay the crossspiking.
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