Fig. 5: Propagation dynamics of intensity and phase distributions of radial carpet beams (RCBs) and beam-like patterns (BLPs) generated by radial gratings and spiral-like structures (SLSs).

a Simulated intensity and phase distributions of an RCB at z = 25 m, generated by a sinusoidal radial grating with m = 10. a1 Intensity distribution; a2 enlarged view of one of the main intensity spots (MISs) and its surrounding region; a3, a4 simulated phase distributions corresponding to the intensity patterns in (a1) and (a2), respectively; a5 phase distribution of the MISs. b Intensity and phase distributions of BLPs generated by an SLS with m = 10 and L = 5 mm⁻¹ under propagation. b1 Intensity distribution; b2 enlarged view of an MIS and its surroundings; b3, b4 corresponding phase distributions; b5 phase distribution of the MISs. i–v illustrate the evolution of these patterns during propagation. c same as (b), but for an SLS with m = 10 and L = 10 mm⁻¹, evaluated at z = 25 m. For dynamic visualizations of phase evolution in the cases with m = 10 and L = 10 mm⁻¹, see Supplementary Movie 4. Wrapped and unwrapped phase profiles along a circular path across the MISs for all cases in (a)-(c), as well as a comparison with an SLS defined by m = 20 and L = 10 mm⁻¹, are provided in Fig. S4.