Fig. 1. Structure of the spatiotemporal vortex generating device. (a), (b), (c), and (d) Schematic structure of the photonic crystal plate, with the following structural parameters: a=210  μm; b1=130  μm; b2=150  μm; w=60  μm; t1=52.5  μm; and t2=37.5  μm. Additionally, the aperture tilt angle of the photonic crystal plate is denoted as θ.

Fig. 2. Reflection working mode. (a) When the state of VO2 is in a fully metallic state, the system’s reflection channel is activated, allowing the structure to operate in reflection mode. At this stage, the system is responsive to left- and right-circularly polarized light, enabling the generation of spatiotemporal vortices with distinct orbital angular momentum (OAM) directions depending on the observation angle. (b) The diagram includes four ports labeled 1, 2, 3, and 4 for the photonic crystal plate structure. (c) Complete polarization conversion at the resonant frequency follows a trajectory from one pole to the other on the Poincaré sphere, as indicated by the black arrow in the figure. (d) A 2D resonance band diagram of the photonic crystal plate structure is presented, showcasing vanadium dioxide in various states.

Fig. 3. Transmission working mode. (a) When the temperature of VO2 is below 68°C, the system operates in transmission mode, with the transmission channel open. This configuration adapts to the generation of spatiotemporal vortices caused by the incidence of linearly polarized light along the y direction. (b) The four ports labeled 1, 2, 3, and 4 are designated for the photonic crystal plate structure.

Fig. 4. Plots of the copolarized reflection phase and copolarized reflectivity in the plane of the wave vector at different θ. (a) and (c) Copolarized reflection phase diagram and copolarized reflectivity diagram, respectively, for θ=8° and an incident frequency of 1.108 THz. In these plots, the red points denote the intersections of the topologically dark lines within the wave vector plane in the 3D frequency-momentum space. (b) and (d) Copolarized reflection phase and copolarized reflectivity plots at the same frequency, with the angle θ increased to 15°. As θ increases, the red singularity shifts outward; however, the vortex phase pattern remains unchanged.

Fig. 5. Topological dark line diagram in frequency-momentum space and at frequency-transverse wave vector plane section and frequency-longitudinal wave vector plane section. The illustration at the lower-left corner of the diagram is the schematic illustration where the two parallel oblique topological dark lines are located in the section of the wave vector plane in the 3D frequency-momentum space, as shown in the polarization reflection amplitude diagram and phase diagram. Four phase vortices appear in the diagram with higher frequency. The phase vortices (±1) with opposite topological charges in the diagram all annihilate at the frequency of 1.1045 THz. The section diagrams of the topological dark line and the frequence-x wave vector plane when kya/(2π)=0.02 and kya/(2π)=−0.02, and the section diagrams of the topological dark line and the frequence-y wave vector plane when kxa/(2π)=−0.02 are, respectively, drawn.

Fig. 6. Graphical representation of the copolarized reflection phase and copolarized reflectivity obtained at different cuts by changing the vanadium dioxide state. (a), (b), (e), and (f) Copolarized reflection phase diagrams for a fixed observation cut at kxa/(2π)=−0.02, with VO2 conductivities of 200,000, 150,000, 100,000, and 70,000 S/m, respectively. (i) and (j) Copolarized reflection phase diagrams in the wave vector plane at an incident frequency of 1.105 THz, specifically for VO2 conductivities of 200,000 and 100,000 S/m. The red dots in these figures indicate the intersection points between the topological dark line and the wave vector plane. (c), (d), (g), and (h) Copolarized reflection amplitude spectra corresponding to (a), (b), (e), and (f), highlighting how variations in VO2 conductivity affect the maximum reflectivity and the position of singularities, with maximum reflectivity fluctuating within a range of 0.1. (k) and (l) Copolarized reflection amplitude related to (i) and (j).

Fig. 7. Schematic of the direction of movement of the topological dark line with temperature in 3D frequency-momentum space. The black curve in the figure is the topological dark line, and the red markers are the intersections of the topological dark line with the wave vector plane at a fixed frequency.

Fig. 8. Demonstration plots of singularity displacements due to different inclination angles θ. (a)–(d) Evolution of the transmission amplitude in the frequency-transverse wave vector plane at θ=15°, θ=22°, θ=28°, and θ=32°. (e)–(h) Phase evolution of the singularity in the frequency-transverse wave vector plane for angles θ=15°, θ=22°, θ=28°, and θ=32°, respectively. The black circle in the figure indicates the position of the phase singularity. As θ increases, the singularity’s location progressively approaches the Γ point. (i) Evolution of the topological singularity in frequency-momentum space as the inclination varies from 8° to 54°.

Fig. 9. Conductivity of vanadium dioxide with temperature fluctuations.

Fig. 10. The time reversal symmetry of the system is broken after the addition of an external magnetic field, resulting in changes in the position of the topological singularity. (a) Phase of topological singularities in frequency-momentum plane without an external magnetic field. (a) Phase of the frequency-momentum plane topological singularities when an external magnetic field is added.

Fig. 11. The metasurface machining process of temporal and spatial vortices in this paper.