Fig. 6. The relationship between the quality of the rotating-revolving $L{G}_{\overline{\ell ,0}}$ beams and the frequency spectrum.

a We first calculate the spatiotemporal spectra of the rotating-revolving beam using the approach described in Methods, and then we select a certain number of frequency lines/modes from the calculated spectra for power/mode purity calculation in (b) and (c). b The power distribution on light beams with different rotating $\overline{\ell }$ values for generating a rotating-revolving LG_3,0 beam, when the number of selected frequency comb lines is varied. c The mode purity of a generated rotating-revolving $L{G}_{\overline{\ell },0}$ beam ($\overline{\ell }$ varies from 0 to +3), when the number of selected frequency comb lines is varied. d We calculate the spatiotemporal spectra of different rotating-revolving beams and count the number of frequency lines in the 10-dB bandwidth of the frequency spectra; panels (e) and (f) show the number of frequency comb lines in the 10-dB bandwidth for generating a rotating-revolving $L{G}_{\overline{\ell },0}$ beam with (i) the same beam waist of w₀ = 0.2 mm and a revolving radius of R varied from 0.75 to 1.5 mm; and (ii) the same revolving radius of R = 1.5 mm and a beam waist of w₀ varied from 0.15 to 0.5 mm, respectively. a = 0.3 mm in (e) and (f).