Supplementary document

This document provides supplementary information on the theories of SU(2) geometric modes including their formation and parameter explanation. The second part of this document shows the detailed calculation process of the generalized frequency conversion of structured beams, and the third part includes the details of the experimental setup. Moreover, the fourth part explains the asymmetrical effect of the experimental laser mode results. The phase distributions of two types of LG-based SHG SU(2) geometric modes are enlarged in the fifth part to stress the frequency-astigmatism asymmetric nonlinear conversion.