Significance   Driven by the recent gradual maturity of digital holography and flat optics with geometric phase, great advances have been made in shaping and application of structured light in the linear optics. In comparison, relevant study based on nonlinear optics, although enabling many crucial functions, such as information exchange between light fields or photons, is still in its infancy. Focusing on this frontier topic of structured nonlinear optics-i.e., nonlinear generation, transformation and interface of classical/quantum states encoded by complex spatial modes-some theoretical and technical bottlenecks to parametrically control all the spatial dimensions of light have been broken. These results lay a solid foundation for future relevant studies on high-dimensional quantum optics experiments.

  Progress  Advances in parametrically controlling all the spatial dimensions of light can be divided into theoretical and applied aspects. On the theoretical side, a parameter transformation theory for the full-field selection rule of spatial modes in cylindrical coordinates during small signal three-wave mixing (Fig.1(a)) and a theoretical model of spin-orbit-coupling-mediated nonlinear polarizations (Fig.1(b)) were first proposed. These two theoretical tools can be used together to describe and predict the nonlinear propagation of the vector spatial structure of light fields (amplitude, phase, and polarization) in any paraxial second-order parameter process. Guided by theoretical tools, a nonlinear astigmatism frequency interface has been proposed (Fig.1(c)). In this parametric astigmatism system, an unexpected new physical effect called anomalous orbital angular momentum conservation has been uncovered. This discovery renewed the perception of the nonlinear orbital angular momentum conservation. On the applied side, a frequency conversion technique based on a Sagnac nonlinear interferometer pumped by a super-Gauss mode was first proposed to achieve a spatial polarization independent conformal frequency interface (Fig.2(a)). On this basis, the apparatus can also act as a spatial-amplitude independent frequency interface for orbital angular momentum, which enables a simultaneous conversion of frequency and orbital angular momentum without impacting on the radial mode of signals, with the introduction of vortex super-Gaussian modes (Fig.2(b)). What's more, the new theory has also inspired a new application called spatially-resolved autocorrelation technique, which is based on spatially multimodal nonlinear optical effects. This technique allows for the characterization of the temporal envelope and spatial modes of ultrafast light simultaneously (Fig.2(c)).

  Conclusions and Prospects   These systematic research results fill the key theoretical gaps in the field of nonlinear control of structured light in all spatial degrees of freedom. They also provide inspiration for new ideas in the field of light field shaping and have significant implications for related studies, such as structured laser technology, modulation of high-dimensional quantum states and polarization-resolved up-conversion imaging.