Numerical analysis of LG_3,3 second harmonic generation in comparison to the LG_0,0 case

For coating Brownian thermal noise reduction in future gravitational wave detectors, it is proposed to use light in the helical Laguerre-Gaussian LG_3,3 mode instead of the currently used LG_0,0 mode. However, the simultaneous reduction of quantum noise would then require the efficient generation of squeezed vacuum states in the LG_3,3 mode. Current squeezed light generation techniques employ continuous-wave second harmonic generation (SHG). Here, we simulate the SHG for both modes numerically to derive first insights into the transferability of standard squeezed light generation techniques to the LG_3,3 mode. In the first part of this paper, we therefore theoretically discuss SHG in the case of a single undepleted pump mode, which, in general, excites a superposition of harmonic modes. Based on the differential equation for the harmonic field, we derive individual phase matching conditions and hence conversion efficiencies for the excited harmonic modes. In the second part, we analyse the numerical simulations of the LG_0,0 and LG_3,3 SHG in a single-pass, double-pass and cavity-enhanced configuration under the influence of the focusing, the different pump intensity distributions and the individual phase matching conditions. Our results predict that the LG_3,3 mode requires about 14 times the pump power of the LG_0,0 mode to achieve the same SHG conversion efficiency in an ideal, realistic cavity design and mainly generates the harmonic LG_6,6 mode.