High harmonic spin-orbit angular momentum generation in crystalline solids preserving multiscale dynamical symmetry

Symmetries essentially provide conservation rules in nonlinear light-matter interactions, that facilitate control and understanding of photon conversion processes or electron dynamics. Since anisotropic solids have rich symmetries, they are strong candidate to control both optical micro- and macroscale structures, namely spin (circular polarization) and orbital angular momentum (spiral wavefront), respectively. Here, we show structured high harmonic generation linked to the anisotropic symmetry of a solid. By strategically preserving a dynamical symmetry arising from the spin-orbit interaction of light, we generate multiple orbital angular momentum states in high-order harmonics. The experimental results exhibit the total angular momentum conservation rule of light even in the extreme nonlinear region, which is evidence that the mechanism originates from a dynamical symmetry. Our study provides a deeper understanding of multiscale nonlinear optical phenomena and a general guideline for using electronic structure to control structured light, such as through Floquet engineering.