Characterized by the pivotal attributes of self-healing properties and constant transverse intensity profiles over extended distances, nondiffracting beams (NDBs) have found wide applications across multiple fields [1]. The origin of NDBs dates back to 1987, when Durnin derived a Bessel-type solution based on the cylindrical coordinate Helmholtz equation and demonstrated the experimental generation of the zeroth-order Bessel beam [2,3]. The long-distance propagation property makes the zeroth-order Bessel beam a powerful tool in imaging [4,5] and micro/nanofabrication [6]. Subsequently, higher-order Bessel beams carrying orbital angular momentum were demonstrated, enabling potential applications in fields such as optical tweezers [7,8] and quantum communication [9]. By solving wave equations in different coordinate systems, various families of NDBs have been reported, including Mathieu beams [10], Airy beams [11], parabolic beams [12], etc. These NDBs have different transverse intensity profiles, and some even propagate along curved paths, extending the applications of NDBs [13–15]. To date, NDBs with tunable transverse intensity patterns and customizable propagation trajectories have drawn increasing interest due to their distinctive properties.

NDBs can be considered as the interference of plane waves with propagation vectors forming a conical surface [16]. Based on this principle, specialized optical components have been developed to produce NDBs. Durnin et al. initially used a circular slit to generate the zeroth-order Bessel beams experimentally [3], which is straightforward but comes with the drawback of low efficiency. Axicons are preferred for their simplicity and high conversion efficiency, but limiting the nondiffraction propagation length to a few centimeters due to a relatively large base angle [17]. Optimizing the holographic axicon with a programmable spatial light modulator (SLM) instead of a physical axicon can produce tunable axicons without fabrication errors [18]. Furthermore, a range of customized optical components were designed for generating high-quality NDBs, such as meta-surfaces [19,20], Fresnel zone plates [21], enclosed cylindrical lenses [22], and integrated silicon photonic chips [23]. However, it brings the problem that the light field structure of the NDBs produced by these elements is limited to a few modes. To solve this problem, the SLM is used to generate NDBs with customized transverse intensity patterns by shaping complex angular spectrum distributions before the operation of Fourier transforming [15,24–26]. Nevertheless, shaping the complex amplitude of these beams with complex coding algorithms leads to significant power loss [27]. Therefore, a high-freedom and high-efficiency method for generating customized long-propagation-distance NDBs is still desirable.

In this paper, we report a method for generating NDBs based on free lens modulation by using a phase-only SLM. We can directly shape the angular spectra of NDBs by designing the free lens phase imprinted onto the SLM. Specifically, the shape, position, size, and transverse phase gradient [i.e., the topological charge (TC) for a closed vortex] can be controlled by the elaborate design of the digital lenses. Thus we readily modulate the pattern, propagation direction, propagation distance, and orbital angular momentum distribution of the produced NDBs. Using the reported NDBs generator, we have realized the generation of long-distance, high-quality Bessel beams, polymorphic generalized NDBs, tilted NDBs, and asymmetric NDBs. Furthermore, the ingenious combination of double-free lenses can be applied to the superposition of co-propagating NDBs to produce specially structured light beams, such as circular spot arrays, helical beams, and optical conveyor beams. The reported method provides a significant high power efficiency advantage due to the pure phase modulation and high flexibility because of the free shaping of the digital lens. Experimental measurements show that for generating the same diffraction-free modes, the efficiency of our method is approximately seven times higher than the complex modulation methods, which is significant for applications concerned with high power.

The nondiffracting optical fields without acceleration in the transverse plane can be described by the Whittaker-type integral [16]: Ψ(x,y,z)=exp(ikzz)∫02πA(φ)⁢exp[ikt(xcosφ+ysinφ)]dφ,(1)where (x, y, z) are the spatial coordinates. A(φ) refers to the angular spectrum with φ denoting the azimuthal angle in the frequency domain. kz and kt are the longitudinal and transverse components of the wave vector, respectively. Various intensity profiles of nondiffracting fields can be produced by manipulating the angular spectrum. Here, the following complex function is used as the angular spectrum of polymorphic beams [28]: ϕ(x,y)=1L∫0Texp{ip2[yx0(t)−xy0(t)]}⁢exp[i2πmS(T)S(t)]|c′(t)|dt.(2)

In Eq. (2), t∈[0,T], c(t)=(x0(t),y0(t)) represents a two-dimensional curve, which determines the shape of the polymorphic beam. L=∫0T|c′(t)|dt represents the length of the curve and S(t)=∫0t[x0(τ)y0′(τ)−y0(τ)x0′(τ)]dτ. σ is a constant. m denotes the topological charge. To avoid the optical power loss caused by the complex amplitude modulation, in this paper we propose to create the corresponding angular spectra by the digital free lenses. Specifically, the digital free lens phase can be expressed as [29] ϕFreelens(r,φ)=−π[r−ρ0(φ)]2λf0,(3)where (r, φ) denote the polar coordinates, λ is the wavelength of the laser, ρ0(φ) controls the shape of the digital free lens, and f0 is the focal distance. Taking the Bessel beam as an example, the angular spectrum of the Bessel function forms a thin ring. Therefore, the key point for generating NDBs is to produce a bright and sharp ring by using an annular free lens [Fig. 1(a)]. A spherical lens performs a Fourier transform on the annular light field to produce a Bessel beam. In this case, the predesigned radius ρ0(φ) is a constant, denoted as ρ0. When the annular beam possesses a flat phase profile, a zeroth-order Bessel beam is obtained behind the spherical lens. High-order Bessel beams carrying orbital angular momentum are generated by directly superimposing a vortex phase ϕVortex=mφ on the annular lens [Fig. 1(b)]. Consequently, the free lens phase can be written as ϕFreelens(r,φ)=−π(r−ρ0)2λf0+mφ.(4)

Due to the high flexibility, the digital lenses’ shape can be customized to generate polychromatic beams with transverse intensity patterns such as triangle, square, pentagon, and oval [Fig. 1(c)], thereby creating novel nondiffracting fields with diversified transverse intensity distributions.

The schematic diagram of the experimental setup for generating NDBs is shown in Fig. 1(d). A laser beam featuring a collimated light with wavelength of 633 nm passes through a half-wave plate (HWP) and a polarizing beam splitter (PBS), after which the beam’s polarization state is converted to horizontal polarization. Next, the linearly polarized light illuminates a phase-only SLM (HDSLM80R Plus, UPOLabs, China), which possesses an aperture with dimensions measuring 15.42mm×9.66mm and a resolution of 1920×1200 pixels. We project the annular lens phase onto the SLM with f0=100mm, ρ0=0.45mm, and m=1 so that the laser beam is modulated and focused at the focal plane of the free lens to form a thin annular beam [Fig. 1(e)]. Here, a ring-actuated iris diaphragm serves as a spatial filter to block unmodulated light beams. After passing through a spherical lens with a focal length of f1=300mm, the NDB is generated. A camera is mounted on a movable rail with a length of 110 cm to translate along the optical axis and retrieve the corresponding longitudinal intensity evolutions of the NDBs. Figure 1(f) shows the transverse intensity distribution of the first-order Bessel beam recorded at z=10cm behind the spherical lens and the corresponding radial intensity profile.

Compared with the axicon and circular slit, the proposed method does not require a trade-off between performance and flexibility. This is attributed to the formidable capabilities of the SLM and the adaptability of the free lens modulation, allowing us to directly create annular patterns with varying radii in experiments by phase modulation and investigate the propagation characteristics of the Bessel beams. Experimentally, annular free lenses [Fig. 2(a)] with ρ0 set to 0.315 mm, 0.45 mm, 0.9 mm, 1.35 mm, and m=0, are used to generate annular light fields of different radii [Fig. 2(b)]. The corresponding transverse intensity patterns of the zeroth-order Bessel beams at z=10cm distinctly reveal a trend that the central peak width of the beams narrows as ρ0 increases [Fig. 2(c)]. To further quantify this trend, we measured the changes in the central peak’s full width at half maximum (FWHM) of the zeroth-order Bessel beams as ρ0 increased from 0.3 to 1.4 mm, showing that the FWHM decreased nonlinearly with the increasing ρ0 [Fig. 2(d)]. The results indicate that the spot sizes of the generated Bessel beams can be conveniently and effectively tuned.

Figure 2(e) portrays the normalized peak intensity of the zeroth-order Bessel beams dependent on the propagation distance. The intensity values exhibit a gradual decrement with the extension of the propagation distance. We recorded the evolution of the Bessel beams over a normalized intensity range from 1 to 0.13. The Bessel beams’ FWHM remains almost constant over this propagation range [Fig. 2(f)]. To characterize the nondiffracting propagation properties of the produced zeroth-order Bessel beams, we delineate the nondiffracting propagation distance as the distance from the beginning to the position where the beam intensity decreases to 13% of the maximal intensity. For ρ0 set to 0.315 mm, 0.45 mm, 0.9 mm, and 1.35 mm, the corresponding nondiffracting distances are measured as approximately 102 cm, 95 cm, 77 cm, and 61 cm. The trend of the propagation distance decreasing with the ring size ρ0 arises from the finite aperture for shaping the input beam. Consequently, a larger value of ρ0 leads to a larger cone angle and a smaller interference volume and thus a shorter nondiffracting propagation distance.

Introducing spiral phase distributions to the annular lenses enables the generation of high-quality, topological charge-controllable higher-order Bessel beams. Figures 3(a)–3(c) show the longitudinal cross sections of intensity of the experimentally generated zeroth-order, first-order, and tenth-order Bessel beams in the range of z=[0,95]cm after the spherical lens, respectively, and Figs. 3(d)–3(f) show the transverse cross sections of intensity of these beams at z=10cm. Here, the parameter ρ0 of annular free lenses is set to 0.45 mm. Three-dimensional light field reconstructions of these beams are shown in Visualization 1. It is evident that for the zeroth-order and first-order Bessel beams, both the central peak/ring and the outer-lying satellite rings effectively resist diffraction spreading, achieving nearly nondiffractive propagation. For the tenth-order Bessel beam, the diameter of all Bessel rings decreases slowly due to the influence of high topological charges. Further analysis for the experimentally measured variation of the central ring’s diameter and the propagation distance of high-order Bessel beams is shown in Appendix A. The experimental radial intensity profiles (red dashed lines) and the corresponding numerically fitted curves (blue solid lines) demonstrate that the produced Bessel beams closely conform to the Bessel function distribution [Figs. 3(g)–3(i)].

An advantage of free lens modulation is the flexibility to adapt the lenses’ shape and enable the generation of polymorphic fields. Consequently, not limited to Bessel beams, the NDB generator can be customized to produce a variety of complex diffracting-free beams. In this case, the parameter ρ0(φ) in Eq. (1) is no longer a constant but rather follows the following expression: ρ0(φ)=1−1pcos(qφ).(5)Here, p determines the level of smoothness of the polygon, while q determines the geometric shape of the modulated light field. As the value of q increases, p should be gradually increased to avoid the quality degradation of the NDBs caused by the large curvature of the focused polymorphic field. By setting the appropriate parameters (p, q), we get the free lenses with different shapes [Fig. 4(a)] and high-quality polymorphic beams at the focal plane [Fig. 4(b)], including triangular lenses (p=10, q=3) with m=0, 1, and 10, square lenses (p=15, q=4) with m=10, pentagonal lenses (p=20, q=5) with m=10, and oval lenses (p=7, q=2) with m=10. Here, the resolution of the displayed holograms is 600×600 to visualize the phase distribution clearly. Figure 4(c) shows the transverse intensity patterns at different distances behind the spherical lens. The three-dimensionally reconstructed 10th-order polymorphic NDBs are shown in Fig. 4(d) (see Visualization 2). Slices S1–S4 display the intensity patterns in some transverse planes during the propagation. The zeroth-order and high-order beams exhibit concentric geometric shapes and long-range propagation properties, similar to the abovementioned Bessel beams. Since the noncircle polymorphic beams generated by the free lenses are not the exact solutions of the Helmholtz wave equation, the intensity profiles of these beams fail to remain invariant and gradually experience distortion and rotation with different angles during long-distance propagation. However, it should be noted that these beams still have anti-diffraction properties, and we therefore refer to such beams as polymorphic generalized NDBs. Taking the oval beam as an example, at z=5cm, the angle (α1) between the long axis of the oval beam and the x direction is about 137.6 deg, while at z=95cm, the angle (α2) turns to about 77.3 deg. The long axis of the oval is rotated about 60.3 deg in the clockwise direction [Fig. 4(c)]. The relationship between the topological charge value m and the rotation angle α1−α2 of the oval generalized NDBs within the propagation range from z=5cm to z=95cm is analyzed [Fig. 4(e)]. It is shown that the rotation angles nonlinearly decrease when m increases from 0 to 35. Owing to the intriguing properties, we believe that polymorphic generalized NDBs hold potential applications in imaging, high-capacity optical communication, and various other fields [30–33]. Note that since the free lens modulation technique can only produce geometrically shaped beams with parametric expressions, our method cannot customize nondiffracting beams with arbitrary transverse shapes. In recent decades, freeform optics has attracted significant attention as an effective optical shaping tool with a high degree of design freedom for efficient control of the beam wavefront [34,35]. Given the robustness of the technique, we expect to generate highly efficient, arbitrarily customizable NDBs with freeform optics techniques following the works reported in this manuscript.

Tilted nondiffracting beams provide an opportunity to realize various intriguing applications. Song et al. introduced a volumetric two-photon imaging method of neurons with an elongated, stereoscopic V-shaped point spread function configuration, which allows for an increase in the number of neurons recorded while maintaining a high frame rate [36]. Yang et al. designed two-photon laser scanning stereomicroscopy based on stereoscopically scanning two tilted Bessel beams in the excitation pathway. The need for only two scanning images from different angles has significantly enhanced the speed of volumetric imaging [37]. In these applications, precise and flexible control of the tilt angle of Bessel beams is crucial. One technique for deviating Bessel beams from the optical axis involves illuminating the axicon with incident light at an inclined angle, but precise correction of aberrations introduced by oblique incidence necessitates using a second SLM [38]. Based on the metasurface, it is possible to generate Bessel beams with any inclination angle but at the expense of flexibility [39].

Effortlessly, our method employs the decentered free lens technique to generate tilted NDBs with easily controlled tilt angles. Specifically, the resolution of the phase hologram is 1200×1200. We establish a coordinate system with the central pixel as the origin (the yellow dot o) on the hologram [Fig. 5(a)]. For the Bessel beam propagating along the optical axis, the center of the free lens is located at the origin. To generate tilted Bessel beams, altering the center position (ul, vl) of the free lens is required. For example, we maintain vl=0 while setting ul=180 pixels. The predesigned radius ρ0 of the free lens is 0.45 and m=0. In this case, the intensity pattern of the annular beam on the focal plane of the free lens exhibits nonuniformity along the azimuthal direction [Fig. 5(b)]. After the Fourier transform, the tilted zeroth-order Bessel beam is generated behind the spherical lens. We recorded the evolution of the tilted beam along the z direction from 0 to 95 cm and obtained the three-dimensional optical field distribution [Fig. 5(c)]. Slices S1 to S3 illustrate some transverse intensity distributions during propagation, displaying distinct concentric ring patterns. The FWHM of the central peak exhibits remarkable stability during propagation, validating that the generated tilted beam maintains high-quality nondiffracting characteristics [Fig. 5(d)]. Figure 5(e) represents schematic diagram of the tilted zeroth-order Bessel beams on the y−z plane with vl=0 but various lateral displacements ul of 0, 60, 120, 180, and 240 pixels. The corresponding tilt angles θ are approximately 0, 0.08, 0.16, 0.24, and 0.32 deg, respectively, indicating a linear increase in the tilt angle θ with the parameter ul [Fig. 5(f)]. Therefore, we can flexibly control the tilt angle θ relative to the z axis by simply adjusting the center pixels (ul, vl) of the free lenses. This approach is also universally applicable to both higher-order Bessel beams and polymorphic generalized NDBs (Appendix B).

The asymmetric Bessel modes were theoretically proposed by Kotlyar et al. [24]. The transverse profile intensity of this beam has azimuthal variant intensity, resembling the shape of a crescent. Owing to its unique optical properties, the asymmetric Bessel beam offers potential applications in fields such as optical manipulation and microlithography. Our method demonstrates a new scheme for generating asymmetric Bessel beams and extends these beams into a wide family of asymmetric polymorphic generalized NDBs with various intensity profiles. The method is based on the Fourier transform of the three-dimensional tilted focal plane fields. Spatially tilted angular spectrum results in the intensity distribution of the generated NDBs becoming nonuniform. In this case, the focal length of the free lens in Eq. (1) should be f(φ)=f0a−sinφa,(6)where a controls the slope of the focal field. The tilt angle of the focal field decreases nonlinearly with the increase of a [29], as demonstrated in Figs. 6(a) and 6(b), which show two light field models with respective slope parameters of 2 and 4. The corresponding simulation results of the intensity and phase of the first-order asymmetric Bessel beams are shown in Figs. 6(c)–6(f), illustrating a smooth and continuous nonuniform distribution of intensity along the central ring of beams. The dependence of the transverse intensity profiles on the slope parameter a was revealed by the normalized transverse intensity [Fig. 6(g)]. As a increases from 2 to 6, the intensity of the left side of the center ring of the beams increases nonlinearly. Figures 6(h)–6(j), respectively, show the experimentally measured intensity patterns at different propagation distances for the first-order asymmetric Bessel beam, the fifth-order asymmetric triangular generalized NDB, and the tenth-order asymmetric square generalized NDB, with a=2. These beams provide distinct nonuniform intensity distribution features, confirming the high feasibility of our method in producing polymorphic asymmetric generalized NDBs.

The superposition of co-propagating Bessel beams can form some distinctive structured light beams, such as circular spots array [40], helical beams [41], and optical conveyor beams [42]. The circular spots array consists of a controlled number of diffraction-free petal-like spots arranged on a circular intensity pattern with radius associated with the m-th-order Bessel function. Helical beams, a subset of radially self-accelerating beams, manifest a continuous helical trajectory throughout their propagation. Optical conveyor beams display a sequence of focal spots aligned along the propagation vector that act as efficient optical traps for particles. Experimentally, the superposition of Bessel beams can be achieved by superimposing two concentric ring free lens phases in a phase hologram. This arrangement yields interfering ring or concentric double-ring light patterns on the focal plane (first row of Fig. 7). Experimentally measured transverse intensity patterns of the circular spot array, helical beams, and conveyor beams at z=10cm, as well as the three-dimensional volume reconstructions, are respectively shown in the second and third rows of Fig. 7. See Visualization 4 for a more visual impression. Concretely, the circular light spot array is generated by two spatially overlapping high-order Bessel beams (ρ0 set to 0.45 mm) with topological charges of the same magnitude but opposite signs. The number of diffraction-free spots and the diameter of the dot array are related to the magnitude of the topological charges. When the absolute value of m is 3, the number of spots in the array is 6 [Fig. 7(a)], and it increases to 10 when the absolute value of m is 5 [Fig. 7(b)]. Figure 7(c) shows two free lenses with ρ0 set to 0.45 mm and 0.9 mm and corresponding phase orders m of −1 and +2. The generated helical beam is featured with threefold rotational symmetry. By modulating the m of the outer ring to +3 and keeping the m of the inner ring to −1, a four-component helical beam can be easily realized [Fig. 7(d)]. Additionally, the optical conveyor beam is generated based on two free lenses with flat phase (m=0). Here, ρ0 is set at 0.45 mm and 1.35 mm, respectively, and within a range of 53 cm, eight high-intensity focus-like spots are measured [Fig. 7(e)]. The spacing between adjacent spots along the propagation direction progressively increases while the spots gradually decrease in size. By resetting the ρ0 corresponding to the outer ring at 1.2 mm and fixing the center coordinates (ul, vl) of the free lens at (0, −120), a tilted optical conveyor beam with focal spots of about seven is obtained [Fig. 7(f)]. The tilt angle θ relative to the optical axis is about 0.16 deg.

In summary, a highly efficient method for generating NDBs is proposed based on the free lens modulation. The generator employs a phase-only SLM without the requirement for additional specialized optical components, thereby providing enhanced simplicity and cost-effectiveness. The excellent tunability of free lenses offers the advantage of high flexibility. It is shown that the propagation length and FWHM of the produced NDBs can be controlled via tailoring the parameters of free lenses. High-quality NDBs with negligible transverse evolution over the propagation distance of 95 cm have been experimentally produced. By precisely varying the parameters of the free lens, we achieved customized polygonal transverse profiles and propagation directions. Experimental results have demonstrated the self-healing capabilities of NDBs. Placing an obstacle in the propagation path, the produced NDBs were found restoring their transverse profiles with very good fidelity. The details are discussed in Appendix C. Due to the significant advantages of high flexibility and great versatility, predictably, the reported method holds potential applications in fields such as optical trapping, high-capacity optical communication, and three-dimensional microscopy [43–45].

Along with high flexibility, high power utilization is one of the outstanding merits of the reported method. Take the generation of Bessel beams as an example; we have experimentally compared the power utilization rate for various methods (Appendix D). The power efficiency is defined as the ratio between the power of the produced Bessel beam and that of the laser beam impinging on the SLM panel. The experimental results show that the power efficiency of our method remains roughly constant at about 58% when the order of Bessel beams changes. In contrast, for the same angular spectra produced by using the complex amplitude modulation technique proposed by Arrizòn et al. [46], Bolduc et al. [47], and Davis et al. [48], the maximum efficiency is about 8.5%. Note that the power efficiency of some methods is related to the order of Bessel beams. Taking Bolduc’s method as an example, the efficiency of the 30th-order Bessel beam is around 3.45%, while that of the zeroth-order Bessel beam is only about 0.33%. Moreover, the generation of more complex NDBs using the complex amplitude method is less efficient than that of Bessel beams. This means that to generate a complex light field with the desired power, it is necessary to illuminate the SLM with laser energy tens or even hundreds of times greater for modulation, which would cause damage to the SLM. Nevertheless, high-power NDBs are essential for many applications. Diffraction-free beams have been proven effective tools in improving penetration depth through scattering media. Sufficiently high energy ensures that the beams can successfully self-reconstruct after scattering, allowing dielectric particles to be optically trapped simultaneously on multiple planes [49]. In the field of biomedical imaging, NDBs with high power and long axial extent would be more suitable for high-contrast and high-resolution volumetric imaging through deep tissue [50]. Considering that the axial intensity profile of the practically generated NDBs decreases gradually during propagation [51], it might be requisite to compensate for the energy depletion by increasing the laser power for a more stable optical trap or deeper imaging. Therefore, the high power usage of the reported method makes it particularly suitable for applications such as deep-tissue microscopy that requires not only high mode quality but also high penetration depth.