Frequency-astigmatism asymmetric nonlinear conversion of structured light lasers

Nonlinear optics of structured light carrying orbital angular momentum (OAM) has recently led to intriguing fundamental physical effects in light–matter interactions and advanced applications from classical imaging to quantum informatics. However, the mutual interaction between spatial mode and frequency conversions in nonlinear processes is still elusive. In this work, we go beyond only considering OAM and frequency in nonlinear structured light conversion and investigate the complex mode evolution in cascaded nonlinear frequency conversion and spatial astigmatic mode conversion. In particular, we unveil a generalized law of nonlinear structured light from experiments and theories, that the converted modes are highly related to the sequence of frequency and spatial mode conversions, obeying an inherent (non)commutative rule in which. This effect not only creates extended structured laser modes but serves as regular rules in nonlinear optics.