  Nonlinear optics of structured light has recently delivered intriguing
fundamental physical phenomena in light-matter interactions and advanced
applications from classical imaging to quantum informatics. The mutual
interaction between spin, orbital angular momentum (OAM) and wavelength is
extensively studied in such cases. In this work, we go beyond only considering
OAM and wavelength by taking the nonlinear frequency conversion and transverse
mode astigmatism conversion as two building blocks and investigating how single
modes and complicated multiplexed modes evolve after them. In particular, We
found a generalized law of nonlinear conversion structured light from
experiments and theories, that the converted modes are highly related to the
sequence of these two blocks, obeying an inherent (non)commutative rule in
which. This effect not only creates extended structured laser modes but serve
as new rules in nonlinear structured light manipulation.

Optics