Fig. 1. Principle of generating NDBs based on free lens modulation. (a) Abridged general view of the method. (b) Superposition of the annular lens phase and the vortex phase to generate the free lens phase hologram. (c) Polymorphic free lenses and the corresponding focal plane field profiles. (d) Schematic of the experimental setup. HWP, half-wave plate; PBS, polarization beam splitter; SLM, spatial light modulator. (e) Experimental intensity pattern of an annular beam (m=1) at the focal plane of the free lens. (f) Transverse intensity distribution and the corresponding radial intensity profile of the experimentally produced first-order Bessel beam.

Fig. 2. Experimentally generated zeroth-order Bessel beams by using annular free lenses with different ρ0. (a) Free lens phase holograms with m=0, ρ0=0.315  mm, 0.45 mm, 0.9 mm, and 1.35 mm, respectively. (b) Experimental intensity patterns of the annular beams at the focal plane of the free lenses. (c) Transverse intensity patterns of Bessel beams generated at z=10  cm. (d) Central peak FWHM of the Bessel beams against ρ0. (e) Variation of the normalized peak intensity of the corresponding Bessel beams with propagation distance. (f) Central peak FWHM of Bessel beams against the propagation distance.

Fig. 3. Experimentally generated zeroth-order, first-order, and tenth-order Bessel beams by using free lenses with ρ0=0.45  mm. (a)–(c) Longitudinal progression of the transverse cross sections of Bessel beams at distances ranging from z=0  cm to z=95  cm behind the spherical lens (see Visualization 1). (d)–(f) Transverse intensity profile of these Bessel beams at z=10  cm. (g)–(i) Comparisons between the experimental transverse intensity distributions of the Bessel beams (shown as red dashed lines) and the theoretical transverse intensity distributions (depicted as blue solid lines).

Fig. 4. Experimental generation of polymorphic generalized NDBs. (a) Phase holograms of free lenses (note that to visualize the phase distribution clearly, here, the resolution of the displayed holograms is 600×600). (b) Light field intensity patterns at the focal plane of the free lenses. (c) Transverse strength distribution of polymorphic generalized NDBs at different positions in the z direction. (d) Three-dimensional light field reconstructions of 10th-order polymorphic generalized NDBs (see Visualization 2). (e) Topological charge value m against the rotation angle α1−α2.

Fig. 5. Experimental generation of tilted NDBs. (a) Off-axis free lens phase hologram for generating a tilted zeroth-order Bessel beam. The center of the free lens is located at coordinates (ul, vl), with the yellow dot o serving as the origin of the coordinate system (u, v). (b) Intensity distribution of the annular beams at the focal plane of the free lens. (c) 3D intensity profile of the tilted zeroth-order Bessel beam (see Visualization 3). The slices S1–S3 show the transverse intensity distribution at various axial locations. (d) Peak width FWHM of tilted zeroth-order Bessel beams against the propagation distance. (e) Schematic of the propagation of tilted zeroth-order Bessel beams corresponding to different ul (0, 60, 120, 180, and 240 pixels, respectively) in the y−z plane. (f) Tilt angle θ of the zeroth-order Bessel beam as a function of the off-axis displacement ul.

Fig. 6. Generation of asymmetric NDBs. (a), (b) Three-dimensional tilted annular ring light field models. (c), (d) Simulated intensity patterns of the asymmetric first-order Bessel beams with a of 2 and 4, respectively. (e), (f) Corresponding phase. (g) Intensity profiles of a=2, 3, 4, 5, and 6. (h)–(j) Transverse intensity profiles of the experimentally generated first-order asymmetric Bessel beam, fifth-order asymmetric triangular generalized NDB, and tenth-order asymmetric square generalized NDB at different propagation positions.

Fig. 7. Experimental generation of (a) circular six-spot array, (b) circular ten-spot array, (c) three-component helicon beam, (d) four-component helicon beam, (e) optical conveyor beam, and (f) tilted optical conveyor beam. The first row depicts the light fields generated at the focal plane of the two concentric free lenses. The transverse intensity patterns measured at a distance of z=10  cm are shown in the second row. The third row presents the volumetric reconstructions of the beams (see Visualization 4).

Fig. 8. High-order Bessel beams. (a) Experimentally measured central ring diameters of high-order Bessel beams against the propagation distance. (b) Longitudinal progression of the transverse cross sections of 40th-order Bessel beams.

Fig. 9. Volumetric reconstruction of (a) tilted fifth-order Bessel beam and (b) tilted 10th-order square generalized NDB. See Visualization 3 for more details.

Fig. 10. Self-healing capability of the NDBs with the topological charges m of 0 and 10. (a) Bessel beams. (b) Triangular generalized NDBs. (c) Tilted Bessel beams. (d) Asymmetric Bessel beams. The beam obstacle is placed at z=3  cm. The beams heal is at about z=35  cm.