Fig. 5. Independent control of two momenta of the spatiotemporal beams.

a–c The 2D amplitude and phase profiles at time t = 0, and the envelope structures in (x, y, t) coordinates of the generated rotating-revolving $L{G}_{\overline{\ell },0}$ beams with different rotating $\overline{\ell }$ values but the same revolving speed of 0.2 THz. Scale bar, 1 mm. The phasefronts are exp(iθ) and exp(i2θ) in one circle around the beams’ center intensity nulls in (a) and (b), respectively; panel (c) is a single beam combining an array of four rotating-revolving $L{G}_{\overline{\ell },0}$ beams, where $\overline{\ell }=0,1,2,3$. See Supplementary Fig. 2 for details of the spatial $L{G}_{\ell ,p}$ mode distributions used for superposition. d The corresponding profiles/structures of the generated rotating-revolving $L{G}_{\overline{\ell },0}$ beams with the same $\overline{\ell }$ = 3 value but a different revolving speed. e Examples of an LG_3,0 beam revolving clockwise at different speeds from 0.2 to 0.6 THz; and f an LG_3,0 beam revolving counterclockwise at a speed of 0.2 THz. Except for the varied parameters and the spatial/frequency spectra, all the other parameters are the same as those in Fig. 2.