Bessel beams, which are a set of solutions of the free space Helmholtz equation and were first suggested by Durnin in 1987 , have attracted intensive attention due to their unique nondiffractive  and superior self-healing capabilities . Higher-order Bessel beams (HOBBs), which carry orbital angular momentum with transport information encoded [4,5], are in principle excellent tools for stable, long-range optical quantum communications and optical manipulation [6,7]. The typical device to generate Bessel beams is an axicon  or an annular aperture . Furthermore, the generation of HOBBs requires a Laguerre–Gaussian mode beam to illuminate the axicon . However, these traditional methods suffer from low efficiency, bulky size, and a complicated manufacturing procedure. The emergence of metasurface provides a new approach to realize ultrathin, compact Bessel beam generators. Metasurfaces are the 2D equivalent of metamaterials, which are constructed from subwavelength planar microstructures with carefully tailored geometrics [10–12]. Metasurfaces exhibit extraordinary capabilities to control electromagnetic (EM) waves. Numerous ultrathin devices such as lenses based on metasurfaces [13,14], complex light field generation , anomalous reflection of light [16,17], and metasurface holograms [18,19] have been developed.
Many Bessel beam generators based on metallic metasurfaces also have been proposed [20–25]. However, most are limited by low efficiency due to the high loss in metals. Although a few high-efficiency Bessel beam generators have been demonstrated [26–29], all these elements unfortunately lack tunability and mostly focus on the generation of zeroth-order Bessel beams. The tunable metasurfaces integrated with functional materials, controlled by thermal , electrical , optical , and chemical  trigger modes have been proposed. These active devices can overcome the limitations of static metasurfaces. For example, an electronically engineered reflective metasurface was designed to flexibly manipulate the nondiffraction region of Bessel beams in the gigahertz (GHz) band , in which a varactor diode is embedded in the gap between two strips in the metasurface. Nevertheless, the implementation of this method faces a great challenge in the terahertz (THz) and higher frequencies range because active elements such as PIN diodes and varactors are difficult to integrate with the reduced unit cell structures for shortened wavelengths. Very recently, an emerging moiré metasurface cascaded by two metasurfaces with mutual movement offers an alternative approach to dynamically control the wavefront of EM waves, which can be operated in a very wide frequency band from microwave to visible light [35,36]. Compared to a commonly tunable metasurface integrated with active materials, the moiré metasurfaces offer advantages that include continuous control of the wavefront, a large tuning range, more stable property, and easier operation [37–39].
Here, a moiré meta-device integrated with the phase distributions of a spiral plate and an axicon to flexibly control the order and nondiffraction length of a Bessel beam is proposed. As a proof of concept, an all-dielectric moiré meta-device to actively control the order of Bessel beams is designed and fabricated. Experimental results demonstrate that the proposed moiré meta-device exhibits the ability to generate not only the zeroth-order but also order-variable HOBBs without using additional vortex phase elements. The order of the HOBBs can be changed continuously by rotating one metasurface relative to the other. In addition, it is also numerically proven that the nondiffraction length of a Bessel beam can be continuously modulated by designing a moiré meta-device.
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2. PRINCIPLE AND DESIGN METHOD
The schematic diagram of the designed moiré meta-device to generate and manipulate Bessel beams is illustrated in Fig. 1. A pair of planar all dielectric metasurfaces with different predesigned phase distributions and placed face to face was used to construct the moiré meta-device. The terahertz (THz) Bessel beams can be generated after a plane THz beam passes through the cascaded metasurfaces. Both the order and the nondiffraction length of the Bessel beams can be adjusted flexibly by the mutual rotation of the two metasurfaces.
Figure 1.Schematic of the proposed moiré meta-device for order-variable Bessel beams generation. The moiré meta-device is cascaded by two metasurfaces with different phase profiles. Bessel beams can be obtained after a THz beam passes through the moiré meta-device. The order of the Bessel beam can be changed by mutual rotating with a step of 20° between the two metasurfaces.
The basic principle used to construct a moiré meta-device is described in this way. Suppose two components and to express the phase profiles of the two metasurfaces, respectively, where and are the coordinate parameters in the polar coordinates. If a metasurface has a shift in one of the coordinates, for example, metasurface 1 has a shift of and metasurface 2 has a shift of along the axis, the joint phase after two cascading metasurfaces has the form 
The field distribution of a Bessel beam propagating along the axis in the cylindrical coordinates (, , ) can be written as 
Figure 2.Design principle of the proposed moiré meta-device to generate order-variable Bessel beams. (a) Schematic of the combination of a flat spiral and a flat axicon. (b) Design principle of the proposed moiré meta-device to generate order-variable Bessel beams.
The design schematic diagram of the proposed moiré meta-device is shown in Fig. 3. The basic element constructing the metasurfaces is high-resistivity silicon () cylindrical pillars with uniform height and variable diameters , which are prepared on the same silicon substrate with a height of and period of , as shown in Fig. 3(a). To achieve full-phase modulation of at the target frequency of 0.6 THz, we performed full-wave numerical simulations using commercial software with the finite difference time domain (FDTD) method to sweep the parameters , , , and . The values of , , and are optimized and determined to be 200 μm, 700 μm, and 300 μm, respectively. Under the x-polarized illumination, the transmission amplitude and phase shifts of the co-polarization components were abstracted. Figure 3(b) gives the dependence of the transmission amplitude and phase on the diameter of the silicon pillar at the operation frequency. The amplitude is maintained around 0.75 and the phase increases from 0 to as increases from 30 to 180 μm. Then, the pillars with different diameters were arranged according to the predesigned phase modulation profiles to construct corresponding metasurfaces. To verify the feasibility of the devices in the experiment, a moiré meta-device with a diameter of 12 mm was fabricated using standard ultraviolet lithography and inductively coupled plasma (ICP) etching technology. Details of the sample fabrication can be found in Appendix B. Figures 3(c) and 3(d) illustrate the structure arrangements and SEM images of a part of the fabricated metasurface 1 and metasurface 2, respectively.
Figure 3.Schematic of designed processes of the metasurfaces. (a) Basic elements of the metasurface are high-resistivity silicon pillars with a height
3. RESULTS AND DISCUSSION
A THz focal-plane imaging system shown in Fig. 4(a) is used to characterize the function of the proposed device. The details about the system can be found in Appendix B. To verify the function validity of the proposed moiré meta-device as a Bessel beam generator, we measured the transmitted field distribution of the moiré meta-device with a mutual rotation of 0° in serial image planes by moving the sample along the z direction with a 28 mm scan range and a 0.5 mm scan step, and then mapped the longitudinal cross-section of the transmitted field. The position of the sample closest to the probe crystal was set as , and the center of the beam spot in each x–y plane was set as the origin. It should be mentioned that the alignment between the two metasurfaces can significantly affect the performance of the moiré meta-device. Thus, a mask method for the alignment is designed in our experiment (see Appendix B). Figure 4(b1) illustrates the longitudinal amplitude profile of the generated optical field along the propagation direction at the operating frequency of 0.6 THz, whereas Fig. 4(b2) gives the transverse amplitude profile at . The longitudinal amplitude shows a propagation invariance in the process of propagation and the transverse amplitude profile presents an axially symmetric circular central peak as well as two concentric ring side lobes. In addition, the corresponding longitudinal (on the plane) and transverse (on the plane) phase profiles are shown in Figs. 4(b3) and 4(b4), respectively. On the plane, the longitudinal phase distribution manifests the interference effect of the THz beams and the phase always keeps a flat plane around the optical axis in the diffraction-free region, which suggests that the THz field consistently maintains the constructive interference at different propagation distances. This is the reason causing the nondiffractive feature of the Bessel beam. On the plane, the transverse phase distribution presents several concentric circles, and the phase value almost keeps constant in each circular region. Besides, a phase jump of occurs on the interface between the adjacent circles. The features described above are consistent with the typical properties of a zeroth-order Bessel beam ; that is, the zeroth-order Bessel beam was generated by the proposed moiré meta-device when the mutual rotation between two metasurfaces is 0°. These are also observed in corresponding simulation results shown in Figs. 4(c1)–4(c4). Detailed simulation execution can be found in Appendix B. It is noted that the experimental results are in good agreement with the simulation results. To more intuitively compare the properties of the Bessel beam obtained in the experiment and simulation, profiles along the horizontal cut lines across the centers of Figs. 4(b1)–4(b4) and 4(c1)–4(c4) are extracted and plotted together. The nondiffraction range is defined as the FWHM of the amplitude profile along the propagation direction . As shown in Fig. 4(d1), the of the Bessel beam along the propagation direction is evaluated as 10.5 mm in experiment, and 15.5 mm in the simulation. The FWHM of the transverse field illustrated in Fig. 4(d2) is 0.83 mm in the experiment, and 0.80 mm in the simulation. The phase variation is almost linear in the nondiffractive region, as shown in Fig. 4(d3), and the transverse phase jumps periodically between 0 and in the experiment and simulation, as shown in Fig. 4(d4). As such, the moiré meta-device can generate a Bessel beam quite well, and the experimental results agree well with the simulation results, except for a slight deviation in the FWHM along the propagation direction due to the fabricating error, misalignment, and inherent errors in the system.
Figure 4.Experimental results. (a) Schematic of experimental setup. PM, parabolic mirror; HWP, half-wave plate; P, polarizer; BS, beam splitter; QWP, quarter-wave plate; WP, Wollaston prism; and CCD, charge-coupled device. (b) Experimental and (c) simulated results for the zeroth-order Bessel beam when the mutual rotation of two metasurfaces is 0°. Two columns on the left are the normalized amplitudes on the
Usually, by adding a spiral phase to the axicon phase, an HOBB will be formed. In addition, the order of Bessel beam could be changed by using a spiral phase with different helical orders. In our experiment, the helical orders of the spiral phase can be flexibly controlled by mutually rotating two metasurfaces with different angles in the proposed moiré meta-device. To assess the capacity of our moiré meta-device in changing the order of the Bessel beams, the transmitted fields of the moiré meta-device under different mutual rotations between two metasurfaces of 0°, 20°, 40°, 60°, and 80° were measured. Both amplitudes and phase profiles in simulations and experiments were obtained at 0.6 THz and the results are displayed in Fig. 5. When the mutual rotation angle equals 0°, the amplitude distribution shown in Fig. 5(a1) presents an axially symmetric circular central peak as well as two concentric ring side lobes, whereas the phase profile shown in Fig. 5(b1) is also a concentric ring with a phase jump of , which means that a zeroth-order Bessel beam is generated. The circular central peak in the amplitude distribution was transformed into the doughnut shape with a central dark core and an annular side lobe when the mutual rotation was changed. The size of central dark core was gradually increased when the mutual rotation varied from 20° to 80° with a step of 20°, as shown in Figs. 5(a2)–5(a5). Going around the circle, the phase presents spiral modalities with multiple periods, and the periodicity increased from 1 to 4, as shown in Figs. 5(b2)–5(b5). This means that the HOBBs are generated by the proposed moiré meta-device and the orders of the Bessel beam were changed linearly from 0 to 4 by rotating one metasurface relative to another metasurface. Figures 5(c) and 5(d) give the simulation amplitude and phase profile, respectively. Good agreement with experimental data is found. This example demonstrates the power of the proposed moiré meta-device in active control of the Bessel beam generation.
Figure 5.Experimental and simulated results for order-variable Bessel beam generation based on the moiré meta-device with the mutual rotation of two metasurfaces changing from 0° to 80° with a step of 20°. (a)–(b) Normalized amplitude and phase profiles measured in the experiment. (c)–(d) Normalized amplitude and phase profiles obtained in the simulation.
Usually, the nondiffraction length of a Bessel beam can be adjusted by changing the base angle of an axicon. As shown in Eq. (7), using our designed moiré meta-device we can flexibly control the base angle of a flat axicon by mutually rotating two metasurfaces with different angles. This means that nondiffraction length of the Bessel beam can be flexibly controlled by the proposed moiré meta-device. To demonstrate the ability of this flexible control, we designed another moiré meta-device that consisted of two other metasurfaces with different phase profiles and simulated the transmission field distribution of the moiré meta-device along the direction of propagation in the Fresnel region. According to Eqs. (4), (5), and (7), the phase profiles of two metasurfaces, and , can be obtained by setting , , , and . Again, the undesired sectoring effect exists in the joint phase of the devices. Such a sector can be avoided by adopting a rounding operation , and the phase profiles of two metasurfaces can be written as
Figure 6.Simulated results for nondiffraction region variable Bessel beam generation based on the moiré meta-device with the mutual rotation of two metasurfaces changing from 15° to 60° with a step of 15°. (a) Near-field phase distribution of entire 12 mm diameter cascaded metasurfaces on the plane that is 1 mm away from the device obtained using the FDTD solver and field distribution of the device along the direction of propagation on the
In summary, we have proposed a platform for a moiré meta-device that can dynamically manipulate the wavefront of EM waves without additional active elements. Based on this platform, both the order and nondiffraction range of the Bessel beam can be actively controlled. An all-dielectric moiré meta-device to generate order-variable Bessel beams was designed and experimentally demonstrated. By a simple mutual rotation of two cascading metasurfaces, the order of the Bessel beam can be continuously changed. The experimental results are in good agreement with the simulations. In addition, the numerical results proved that the length of the nondiffraction region in the Bessel beam can also be flexibly controlled by the designed moiré meta-device. We believe this moiré meta-device platform provides a simple, effective strategy to flexibly manipulate the properties of a Bessel beam, both the order and nondiffraction region, which could find applications in stable, long-range optical quantum communications, particle manipulation, and high-resolution imaging.
APPENDIX A: DERIVATION OF PHASE AMBIGUITY CORRECTION
A sectoring effect caused by the ambiguity of rotation is inevitable in the moiré meta-device because part of the region shifted out of the defined region [0, ] when the metasurface is rotated, and the overall emergent phase profile is split into two sectors. For example, the phase profiles of two metasurfaces used to control the order of the Bessel beams are given as
Considering the rotation angle of , the total phase of the device is
APPENDIX B: SIMULATION AND EXPERIMENTAL SECTION
For the simulation of the cascaded metasurfaces, we obtained the complex amplitude distribution of the entire 12 mm diameter cascaded metasurfaces on the plane, which is 1 mm away from the device using the FDTD solver, where the distance between two metasurfaces is selected as 1 mm. Then, we calculated the field distribution in Fresnel region of the near-field complex amplitude using the Fresnel integral diffraction algorithm.
Standard ultraviolet lithography and inductively coupled plasma (ICP) etching technology were used to fabricate the samples. First, a high-resistivity silicon wafer (, 2000 ) that was 1000 μm thick with a diameter of 4 inches was cleaned in an ultrasonic bath. Then, a positive photoresist (AZP4620) film was put onto the silicon wafer by spin-coating and then baking. An aligned mask was used for the exposure of the photoresist film. After the development process, the image on the mask was transferred to the photoresist film. Next, the silicon wafer was etched using ICP etching with SF6 etch gas. Finally, the silicon pillars were obtained after the remaining photoresist was removed with acetone, and the wafer was cleaned and dried.
A THz focal-plane imaging system [
A key point of the moiré meta-device is the alignment of two metasurfaces. Although the device will not lose its intended function if the two metasurfaces are not aligned perfectly, the performance of device will be significantly affected. Thus, an alignment method is proposed in our experiment. As illustrated in Fig.
Figure 7.Diagram of alignment method for two metasurfaces.
APPENDIX C: MANIPULATION OF NONDIFFRACTION REGION IN HIGHER-ORDER BESSEL BEAMS
To characterize the change in the nondiffraction region with the rotation angle in higher-order Bessel beams, we simulate the far-field diffraction of the higher-order Bessel beams where the mutual rotation angles between two metasurface are equal to 15°, 30°, 45°, and 60°, respectively. Figure
Figure 8.Simulation results for nondiffraction region variable higher-order Bessel beams generation based on the moiré meta-device with the mutual rotation of two metasurfaces changing from 15° to 60° with a step of 15°.
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