  For coating Brownian thermal noise reduction in future gravitational wave
detectors, it is proposed to use light in the helical Laguerre-Gaussian LG33
mode instead of the currently used LG00 mode. However, the simultaneous
reduction of quantum noise would then require the efficient generation of
squeezed vacuum states in the LG33 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 LG33
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 LG00 and LG33 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 LG33 mode requires about 14
times the pump power of the LG00 mode to achieve the same SHG conversion
efficiency in an ideal, realistic cavity design and mainly generates the
harmonic LG66 mode.
