In recent decades, chiral metamaterials with helical structures have gradually attracted extensive attention. Helical chiral structures possess unique optical properties, including optical activity^[1–3], circular dichroism^[4], and negative refractive index^[5]. Based on the unique optical features, helical chiral materials can be used to fabricate polarization conversion devices^[6]. The circular dichroism spectra obtained from their circular dichroism are also the most commonly used method for determining protein molecular structures^[7,8]. In addition, helical chiral structures are also applied in microrobots^[9,10], thermal diffusers^[11], biological medicine^[12,13], and particle transport^[14,15].

The fabrication of helical structures typically involves stereolithographic techniques, such as laser direct writing (DLW), by exposing proper laser light into photosensitive resists to create arbitrary three-dimensional polymerized structures at sub-micrometer scales^[16,17]. However, high-accuracy mechanical scanning trajectories are indispensable in fabricating the three-dimensional helical-shaped structures in the conventional point-by-point DLW procedure. Moreover, it is difficult to apply this time-consuming method in manufacturing large-scale helical arrays. To address this issue, many research groups have attempted to construct a class of structured light fields with helical shapes, thereby achieving the rapid fabrication of the entire helical structure in a single exposure procedure. Several schemes have been proposed to construct helical optical fields, such as the superposition of high-order Laguerre–Gaussian beams^[18] and the coaxial interference between vortex beams and plane waves^[19]. Additionally, researchers have combined Bessel beams with coaxial Gaussian beams to braid single-helix and double-helix optical fields^[20]. Although the above-mentioned methods enable the rapid fabrication of large-area helix arrays, the shape of the single-unit structure is determined by the focal shape of the optical fields. The fabricated structures lack adjustability and only have a limited aspect ratio, greatly restricting the preparation of helical structures of diverse parameters.

In order to meet the demands for high aspect ratio helical structures, Yu’s team introduced a controllable rotation speed formula into the superposition of two high-order Bessel phases, generating a new class of self-accelerating beams (SABs) with a rotating intensity distribution^[21]. This method provides greater flexibility in fabricating high-aspect ratio helical structures. Utilizing this beam, it is possible to rapidly fabricate three-dimensional helical structures with adjustable shapes in large-area regions in a single-exposure procedure. Moreover, by employing the rotation speed formula, it is feasible to control the pitch of the generated helical structures. However, the method relies on changing the orders of the paired Bessel beams to manipulate the radial size of the helix, making it unable to arbitrarily adjust the radial size in this way. A straightforward method to control the radial size of the helical optical fields is to adjust the carrier spatial frequency of the conical waves by tweaking the cone angles of the high-order Bessel beams^[22].

In this Letter, we propose a scheme to generate helical beams with independent control of multiple helical features including the radial size, the lobe number, the chirality, and the pitch. By superimposing two high-order Bessel beams of different topological charges, helical optical fields are generated. The conical angles of the paired beams are carefully investigated to enable the arbitrary control of the radial size, the chirality, and the lobe numbers of the resulting helical fields in cooperation with the selection of the topological charges. Furthermore, a rotation velocity formula is also introduced in the optical fields phase to control the pitch of the helical optical fields. The validity of our comprehensive manipulating scheme is demonstrated in both simulations and experiments. Due to the high flexibility of the multi-freedom control, this scheme is highly suitable for rapid fabrication of large-area, highly adjustable arrays of helical chiral structures.

The coaxial superposition of two high-order Bessel beams can generate a helical-shaped optical field. The structure of the helix is formed by interfering with the main lobes of the two Bessel beams and the overlapped position of the main lobes determines the radial size of the helical structure. In fact, two conical wavefronts with vortex phases can achieve this goal^[23]. Therefore, the phase formulas for the paired Bessel beams with different conical angles and orders l and m can be expressed as {ψl=exp[i(kl,rρ+lθ)]ψm=exp[i(km,rρ+mθ)].(1)Here, ρ and θ are the radial distance and the polar angle theta in the polar coordinate, respectively. kl,r=ksinαl and km,r=ksinαm represent the radial wave numbers of the two Bessel beams with the conical angles αl and αm, respectively. By varying the conical angles of the two Bessel beams, we can alter the radial wave numbers of the Bessel beams, thereby tuning the radii of the main lobes of the two beams. For the lth order Bessel beams of the conical angle αl, the radial size can be readily given by r=Zlλ2πsinαl,(2)where Zl represents the coordinate of its first maximum.

To maximize the intensity contrast of the coherent helical pattern, the main lobes of the two Bessel beams must overlap as much as possible. A matching factor is defined as R=sinαl/sinαm to fulfill the matching condition. By adjusting the conical angles, the maxima of the two paired high-order Bessel beams always coincide. Table 1 lists the matching factors of the paired high-order Bessel beams. In fact, Eq. (2) also gives the radial size of the helical beam generated from a set of two paired high-order Bessel beams with fixed orders. Therefore, the helical radius can be readily achieved by modifying the conical angles. Figure 1 shows the relationship between the helical radius r and the conical angle αl where the helical optical fields are generated from two Bessel beams with (l−m)=1. In Fig. 1, it is also possible to generate the helical beams with identical radial sizes from two paired Bessel beams with different sets of orders. Further discussions on this issue will be given in the next section.

In addition to controlling the radial size, the helical rotation can be also tuned by adding a rotation tuning function v(ρ) in the optical fields phase^[21]. Inspired by the radial-to-axial mapping concept of Bessel beams in the geometric optics^[24], the local rotation speed ω(z) at a given axial distance z can be associated with a circular segment at a certain radial distance ρ in the input phase plane. The local rotation speed ω(z) can be determined from a differential equation in terms of v(ρ)^[21]. The resulting phases of the paired high-order Bessel beams can be expressed as {φ1=[kl,r−l−m2v(ρ)]ρ+lθφ2=[km,r+l−m2v(ρ)]ρ+mθ.(3)

By altering the phase term v(ρ), we can modify the rotation speed along the beam propagation as in Ref. [21].

In practice, a single phase-only spatial light modulator (SLM) is capable of superimposing the paired Bessel beams^[25,26] with the double-phase hologram algorithm. In this scheme, a pair of complementary binary “checkerboard” patterns M1 and M2 as in Ref. [27] would be introduced in each phase of the paired Bessel beams, and we can encode the above phase expressions below into a single hologram as φ=M1φ1+M1φ2. With a Gaussian beam incident on the spatial light modulator with the above phase, the structured beam with the helical intensity distribution is generated in the diffracting beam.

The experimental setup for generating the helical optical fields is shown in Fig. 2(a). The light source is a homemade all-polarization-maintaining ultrafast fiber laser, which can generate Gaussian pulsed beams with central wavelength of λ=1065nm, at a repetition rate of 34 MHz, and a pulse width of 9.92 ps. After the beam passes through a beam expanding collimation system consisting of lenses with a focal length of 35 mm and a focal length of 125 mm, the spot can completely cover the whole SLM active area, which is a reflective phase-only liquid crystal spatial light modulator (Holoeye Pluto, 1920pixel×1080pixel), and the objectives lenses MO1 and MO2 are both 50× microscope objectives (Olympus, NA=0.8).

The phase information of the helical-shaped light fields loaded on the SLM is shown in Fig. 2(b). In order to avoid the undesired interference from the reflected lights from the unloaded phase region as well as the ambient lights, the phase mask is also superimposed with a blazed grating phase so that stray light fields are blocked by the aperture diaphragm. Once the beam in the first-order diffraction passes through a telescope system consisting of a lens L1 (focal length f=1m) and MO1, the target helical optical fields are generated at the back focal plane of MO1. The intensity profiles at each z position along beam propagation are then captured on the CMOS sensor installed in a high-precision motorized imaging system consisting of MO2 and a lens (focal length f=500mm). An exemplar of the helical-shaped fields is shown in the inset of Fig. 2(c).

In Eq. (1), by giving the appropriate conical angles, single-lobe helical light fields with the identical radius are obtained with different topological charges, as shown in Fig. 3. By maintaining the order difference (l−m)=1 and tuning the conical angles αl and αm, single-lobe helical beams with an exemplary radius of 2 µm are generated when the values of m are taken to be 1, 3, 5, and 7.

The simulation and the experimental results in Fig. 3 verify the influence of the conical angles on the radial sizes of the single-lobe helical structure. The radius of the white dotted circle in Fig. 3 is 2 µm, and it is evident that the positions of the maximum light intensity coincide precisely with the white dotted circle. It can be derived from Eq. (2) that, when m=1,3 [Figs. 3(a) and 3(b)], the conical angles of the two lower-order Bessel beams interferences producing a 2-µm-radius single-lobe helix are 13.04° and 22.60,° respectively. When m=5,7 [Figs. 3(c) and 3(d)], the conical angles of the two higher-order Bessel beams interferences producing a 2-µm-radius single-lobe helix are 33.90° and 42.38° respectively. The experimental results demonstrate a significant level of agreement with the simulations. It can be seen that Eq. (2) allows flexible control of the radius of the generated helical light fields. We note that this scheme can also flexibly adjust the width of the helical lobes with the same radius. A thicker lobe can be generated from the pairs of low-order Bessel beams with lower conical angles, and accordingly, a thinner lobe requires the pairs of higher order Bessel beams with higher conical angles.

Besides generating the single-lobe helical optical fields with a constant radius, this method also allows for obtaining multi-lobe helical fields with identical radial features. By varying l and m, the lobe numbers of the helical fields are determined by their difference (l−m). Figure 4 presents the simulation and the experimental results of the helical light fields obtained by taking (l−m)=1,2,3,4. In this work, the lth order of the Bessel beam is set as l=6 with the conical angle αl fixed at 15.41°.

As depicted in Fig. 4, various combinations of the topological charges allow the generation of multi-lobe helical structures with identical radii under the same conical parameters. Once the conical angles of the paired Bessel beams are matched, changing in the order combination has little impact on the radii of the resulting helical structures with the identical lower-order paired Bessel beams. Tuning the higher order of the paired beams only influences the lobe number of the helix, leading to the independent control over the radius and the lobe number of the helix.

In addition, changing the sign of the topological charge difference enables the helical structure to be obtained with different chirality for left- and right-handed rotations, and changing the value of the rotational tuning function in the phase also enables the helical light fields to be obtained with different rotation rates. Figure 5 shows the light fields intensity distributions when the topological charge difference is l−m=1 and the rotational speed is v=0.064rad/μm [Fig. 5(a)] with 0.136 rad/µm [Fig. 5(c)]. When the topological charge number difference is positive, the light fields is rotated clockwise along propagation. Furthermore, with the increased rotation speed, the turning angles of the light fields also increase accordingly along the same transmission distance. When the sign of (l−m) is negative, the rotational directions of the corresponding light fields are counterclockwise, as shown in Figs. 5(b) and 5(d).

It is interesting that light fields with high rotation speeds exceeding ∼1.0rad/μm would transmit in a tube-like fashion, no longer in helical trajectories. On the other hand, light fields with low rotation speeds below 0.01 rad/µm would not generate the full cycle helical structures within the available non-diffractive axial distance. Besides, the tube-like fields transmit at a much shorter axial distance compared with the host-paired Bessel beams, as shown in Figs. 6(a)–6(d). Further investigations show that this is attributed to the overweighted speed tuning term in Eq. (3), which is no longer a perturbation to the radial wave number term in the overspeed case. In contrast, exemplary helical beams with proper rotation speeds have little shortening effects on the axial distance, as shown in Figs. 6(e)–6(h). Therefore, the rotation speed should be carefully evaluated in specific applications.

Figure 7 presents the measured intensity profiles corresponding to the simulated results in Fig. 5. The angle of rotation observed during the transmission of the helical light fields in the simulation aligns well with the expected exemplary rotational speed. A tiny deviation from the ideal case occurs in the intensity distribution along the helical lobes in experiments. These discrepancies are attributed to the high sensitivity of the high-order Bessel beams patterns to several factors in our setup, such as the residual aberrations and the non-ideal radial intensity profiles of the optical fields. However, these deviations remain within the acceptable error range. Above all, the figure again illustrates that altering the helix speed has no impact on the radius of the helix light fields, as indicated by the dashed circles of the identical radius. Moreover, we note that our scheme allows for manipulating each individual feature of the helical structure independently.

In this Letter, we propose a scheme to generate helically structured beams by combining a self-accelerating beam capable of controlling the helical rotational speed with the method of matching the carrier spatial frequencies of two paired high-order Bessel beams. The multi-degree-of-freedom feature control of the helical beam is demonstrated in terms of radial size, rotational speed, chirality, and lobe number. Tuning the conical angles of the paired Bessel waves allows for matching their main lobes, and this would generate helical beams with high-quality helical lobes. Moreover, it is also highly flexible to adjust the radius of the helical beams with this method. Tweaking the orders of the paired Bessel waves can vary the lobe number and the chirality. The pitch of the helix can be controlled by a rotation phase term additionally introduced in the paired Bessel waves. With the above efforts in phase engineering, single- and multi-lobe helical optical fields with fixed radial sizes are produced. Both the simulations and experiments validate the effectiveness of our scheme, and the greatest advantage is also demonstrated in terms of the mutually independent control of the main features of the helical optical fields. Therefore, our flexible solution of multi-degree-of-freedom control can be well adapted to generate helical beams with diverse parameters. Recently, we noted that diverse spiral beams with similar shapes have also been elegantly generated^[28–30], expanding the family of spiral beams with great potential in multi-parameter manipulation. As chiral structures with arbitrarily controllable parameters are demanded in more applications, we expect our scheme to be applied in a variety of fields, such as optical metamaterials, particle transport, and microfluidic mixing.