For future generations of gravitational wave detectors, it is proposed to use the helical Laguerre-Gaussian LG3,3 mode to reduce thermal noise, which limits the detector sensitivity. At the same time, this requires the efficient generation of squeezed vacuum states in the LG3,3 mode for quantum noise reduction. Since this technique includes the process of second harmonic generation (SHG), we experimentally compare the conversion efficiency and harmonic output field of the LG0,0 and LG3,3 modes in a cavity-enhanced SHG using the same 7% doped MgO:LiNbO3 crystal. Conversion efficiencies of 96% and 45% are achieved, respectively. The influence of mode mismatches and astigmatism is analyzed to estimate the ratio of the pump mode-dependent effective nonlinearities to be d0,0/d3,3∼5. Furthermore, we show that absorption loss in the crystal is more relevant for the LG3,3 mode.