Fig. 1. Design of the helical beams. (a) Superposition of light rays in blue to form a helical caustic in red. (b) Angular spectrum of the helical beam in the initial plane with the phase distribution in grayscale, and an additional ring-shaped amplitude distribution in red with a spatial frequency width described by δ. (c) The initial field distribution corresponding to different spatial frequency widths δ.

Fig. 2. Cross-sectional intensity distribution of helical beams with spatial frequency width δ set to be (a) 0.03, (b) 0.1, and (c) 1, corresponding to Fig. 1(c). The radius and period of the predesigned helical trajectory are 20 and 2000 μm, respectively.

Fig. 3. Propagation dynamics of helical beams with different numbers of main lobes. The helical beams presented include (a) one-lobe, (b) two-lobe, (c) three-lobe, and (d) four-lobe with slightly different periods of 2018, 1970, 1917, and 2087 μm, respectively.

Fig. 4. Characterization of the helical beams with different numbers of main lobes in Fig. 3. (a) The percentage of power within a circle in the radius of r, which is fitted by 0.033{1+Erf[0.44(x−22.1)]}, 0.034{1+Erf[0.44(x−22.1)]}, 0.034{1+Erf[0.44(x−22.0)]}, and 0.032{1+Erf[0.42(x−22.1)]} for helical beams with one to four main lobes, respectively. Note that the curves for two-lobe, three-lobe, and four-lobe helical beams are shifted upward by 0.01, 0.02, and 0.03 for better visualization. The inset shows the first two flat states in the curve and contains the main lobe. (b) The fluctuation of the peak intensity during the two-rotation helical propagation, which is normalized by the initial peak intensity for a better comparison.