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Abstract
The Bessel beam has become significant in optical research due to its properties such as a long focal depth, self-healing, and non-diffraction. However, conventional methods for generating Bessel beams have drawbacks such as limited flexibility and tunability and the use of bulky optics. These factors lead to the complexity of the optical systems. This paper presents what we believe is a novel approach to generating Bessel beams by utilizing a fully phase-modulated all-dielectric metasurface. The proposed method enables the arbitrary and independent manipulation of cross-polarized and co-polarized components, allowing the creation of Bessel beams featuring multiple polarization conversions when subjected to left-handed circularly polarized (LCP) incidence. To demonstrate the versatility and effectiveness of the method, three metasurfaces with distinct characteristics are designed. The simulated generated Bessel beams exhibit qualities including long focal depth, non-diffraction behavior, self-healing capabilities, and polarization conversion, which align with the theoretical predictions. This work presents novel possibilities for effectively generating and multi-functional application of Bessel beams.
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1. Introduction
Bessel beams are a specific solution to the scalar wave equation in free space, initially discovered by Durning in 1987 [1]. They possess a transverse intensity distribution characterized by a Bessel function of the first kind. These beams demonstrate remarkable properties, including long depth of focus, self-healing, and non-diffraction. Consequently, they find extensive applications in tractor beams [2], laser processing [3,4], particle trapping [5], and microscopic imaging [6]. Conventional Bessel beam generation methods are generated using axicons [7], holograms [8], and spatial light modulators [9,10]. However, these methods have some drawbacks, including fixed functionality and large size, factors that are gradually failing to keep up with the trend towards integration, limiting the further application of Bessel beams.
Metasurfaces with an extraordinary ability to manipulate electromagnetic waves have attracted great interest in recent years [11–16]. This two-dimensional material, consisting of artificially designed arrays of sub-wavelength unit structures, overcomes the limitations of conventional optical elements by its excellent optical field control capability, high resolution, and ultrathin thickness. The ability of the metasurface to flexibly control polarization, amplitude, and phase of the optical field at the sub-wavelength scale has been widely used to realize many new phenomena and functions, such as metalens [17–21], vortex beam generators [22–26], non-diffractive beam generators [27–31], holograms [32–35], polarization converters [36–40], and so on.
Recently, the use of metasurfaces to generate non-diffractive Bessel beams has become a research hotspot. Chen and colleagues demonstrated a method based on geometrical phase for generating Bessel beams using titanium dioxide (TiO2) metasurfaces [41]. They experimentally verified that the resulting Bessel beams exhibit wavelength-independent transverse intensity distributions in the visible spectral range. Christina et al. introduced a technique relying on a supercellular TiO2 metasurface [42]. In this approach, incident laser light is diffracted simultaneously into a Gaussian beam, a helical beam, and a Bessel beam over a wide range of angles. However, this method has certain limitations, specifically regarding the polarization angle of the generated beams, which is constrained by the angle of incidence and lacks flexible tuning. Lin et al. have also demonstrated the use of a Huygens metasurface to generate Bessel beam arrays, which integrate the functionality of Dammann grating and meta-axicons [43]. Chen et al. built on their work by presenting a geometrical-phase based method for generating Bessel beam arrays, which provides high-uniformity and high-resolution characteristics by optimizing the supercells of the Dammann gratings [27]. Nevertheless, the metasurfaces mentioned earlier exhibit limited capability in generating Bessel beams, and the main focus is on the quality of the generated Bessel beams, while the study of the ability to manipulate the polarization of the Bessel beams is neglected, which is not able to meet the multifunctional requirements of practical applications. In addition, Li et al. demonstrated an all-dielectric metasurface to generate double Bessel beams with independent orders and different numerical apertures through the interaction of geometric and propagation phases [44]. The essence of this method lies in the utilization of cross-polarization, namely L-R (RHCP output under LHCP input) and R-L (LHCP output under RHCP input). Therefore, in the above methods for generating Bessel beams from metasurfaces, either the geometric phase, the propagation phase, or the combination of geometric and propagation phases, only the cross-polarized or co-polarized components are considered individually, and it cannot control the co-polarized and cross-polarized output components independently at the same time, which limits the flexibility and versatility of the application to a certain extent.
In this paper, a fully phase-modulated all-dielectric metasurface for generating Bessel beams is proposed. It allows arbitrary independent manipulation of the output cross-polarized and co-polarized components for generating Bessel beams with multiple polarization conversions at LCP incidence. To demonstrate the flexibility and efficiency of the method, three metasurfaces with different functions are designed: the first generates dual Bessel beams with orthogonal circular polarization, the second generates dual Bessel beams with orthogonal linear polarization, and the third generates arrays of orthogonal linearly polarized Bessel beams, as shown in Fig. 1. We verify the excellent polarization conversion properties by analyzing the electric field distribution as well as the Stokes distribution of the three metasurfaces. Additionally, we demonstrate the long focal depth and non-diffraction properties by analyzing the results of the focal depth and half-height width of the generated Bessel beams. To verify its self-healing property, we positioned obstacles of varying sizes in the propagation path of the Bessel beam and obtained simulation results that agree with the theoretical results. This approach overcomes the constraint that earlier metasurfaces can generate Bessel beams with only a single polarized component, thus providing novel perspectives and chances for the effective production and multifaceted implementation of Bessel beams. In addition, the method could be expanded to cover other ranges of operating frequency.
Fig. 1. Schematic diagrams of three all-dielectric metasurfaces. (a) Schematic of the metasurface generating LCP and right-hand circularly polarized (RCP) dual Bessel beams. (b) Schematic of the metasurface generating x-polarized (X-P) and y-polarized (Y-P) dual Bessel beams. (c) Schematic diagram of a metasurface device generating an array of X-P and Y-P Bessel beams.
2. Theory and design
Bessel beams, as solutions to the Helmholtz wave equation in free space, can be characterized in the scalar form when propagating along the z-axis in cylindrical coordinates $(r,\phi ,\textrm{ }z)$ as follows [41]:
where A represents the amplitude, ${J_n}$ represents the n-order Bessel function of the first class, and $k = 2\pi /\lambda$ denotes the total wave number. Here, ${k_z}$ and ${k_r}$ indicate the number of longitudinal and transverse waves that comply with the relation $\sqrt {{k_z}^2 + {k_r}^2} = k$, respectively. Additionally, $\phi = arctan({y/x} )$ denotes the azimuthal angle, which characterizes the higher-order Bessel beam provided by the optical vortex.To align with the phase requirements of a fully phased metasurface, the desired phase profile for the Bessel beam can be expressed as:
Then, we describe how to use a fully phase-modulated metasurface to realize the proposed multifunctional polarization-converting metasurface. We begin our derivation with meta-atoms with doubly symmetric structures, where the linear transmittance properties of the meta-atom rotation angle $\theta $ can be characterized by the Jones matrix [23,39]:
Here, ${\phi _2} = \left( {\sum \varphi } \right)/2$ represents the phase distribution of a co-polarized wave, implying that the LCP output wavefront can be directly phase-tuned by adjusting the sum of propagation phase delays along the x- and y-axes. Simultaneously, the phase distribution of the cross-polarized wave can be expressed as ${\phi _1} = \left( {\sum \varphi - \pi } \right)/2 + 2\theta$, indicating that, within the original sum of phase delays along the x- and y-axes, introducing a specific rotation angle of the meta-atom allows independent manipulation of the phase distribution of the cross-polarized output wave to achieve the desired RCP function. That is to say, the geometric phase offered by the rotation angle is $\theta = ({{\phi_1} - {\phi_2} + \pi /2} )/2$. Where the amplitude of the co-polarized component is ${T_{co}} = \cos ({\Delta \varphi /2} )$, the amplitude of the cross-polarized component is ${T_{cross}} = \sin ({\Delta \varphi /2} )$, and the power ratio of the two terms is defined as $\eta = {\tan ^2}({\varDelta \varphi /2} )$. Subsequently, to ensure that the output power distributions between the co-polarized and cross-polarized channels are equal ($\eta = 1$), the phase difference $\varDelta \varphi = \pi /2$ is set.
The basic units of the metasurface are titanium dioxide nanopillars featuring dimensions including width (W), length (L), height (H), rotation angle ($\theta $), and lattice constant (P), positioned on a fused silica substrate, as depicted in Fig. 2(a). TiO2 is known for its high refractive index and low loss in the visible region, which effectively controls the propagation and refraction of light. Optimizations are performed by finite-difference time-domain (FDTD) simulations using commercial software FDTD Solutions (Lumerical Solutions). Periodic boundary conditions are employed along both the x and y directions, while a perfectly matched layer (PML) is implemented in the z-direction to mitigate the impact of wave reflections. The operational wavelength is configured as $\lambda = 532\textrm{ nm}$, and the chosen nanopillar maintains a constant height of $H = 600\textrm{ nm}$, along with a lattice constant of $P = 350\textrm{ nm}$. Here, the corresponding refractive indices of TiO2 and substrate are set to ${n_{\textrm{Ti}{\textrm{O}_\textrm{2}}}} = 2.44488$ and ${n_{\textrm{sub}}} = 1.5$, respectively [41,45]. The nanopillar L and W range from 50 to 250 nm, with an increase of 2 nm in each geometrical variable. The results for the transmission amplitudes ${T_x}$ and ${T_y}$ of the meta-atoms under X-P and Y-P incident light and the phase delays ${\varphi _x}$ and ${\varphi _y}$ along the x and y axes are shown in Fig. 2(b). ${T_x}$ and ${T_y}$ show the high transmittance of the designed structure, while ${\varphi _x}$ and ${\varphi _y}$ cover the full $2\pi $ phase shift. We carefully designed eight subwavelength meta-atoms for the formation of metasurfaces, as shown in Fig. 2(c), with the L and W from left to right as (213 nm, 57 nm), (170 nm, 95 nm), (177 nm, 118 nm), (191 nm, 129 nm), (123 nm, 179 nm), (130 nm, 190 nm), (143 nm, 223 nm), and (148 nm, 240 nm), respectively. Since the eight nanopillars have different sizes and rotation angles, the phase information of the metasurface can be fully satisfied. These structures are capable of providing the necessary phase difference $\varDelta \varphi = \pi /2$ to ensure equal amplitudes of the transmitted co-polarized and cross-polarized components. Moreover, they can yield the phase sum $\sum \varphi$ required to cover the entire $2\pi $ range, while both ${T_x}$ and ${T_y}$ exceed 0.9. Therefore, the designed eight meta-atoms can independently modulate the phases of output co-polarized and cross-polarized components to form the designed three kinds of metasurfaces.
Fig. 2. (a) Perspective and top view of a meta-atom consisting of a rectangular TiO2 nanopillar and a square silica substrate. (b) Transmittances (Tx and Ty) and phase delays (${\varphi _x}$ and ${\varphi _y}$) as a function of the nanopillar size parameters L and W. (c) Corresponding transmittances and phase delays for the selected eight nanopillars.
3. Results and discussion
Based on the above theory, three all-dielectric metasurfaces consisting of TiO2 nanopillars with a diameter of 48${\mathrm{\mu} \mathrm{m}}$ are constructed to produce Bessel beams with controlled order and polarization properties. Simulations for three types of metasurfaces are being performed using the FDTD method on a workstation with 512GB of RAM. The operating wavelength is set to 532 nm, with an NA of 0.2. The value of NA directly affects the beam characteristics; the larger the numerical aperture, the more side lobes, the smaller the beam half-height width, and the shorter the propagation distance. Therefore, we determine this value according to the specific application requirements and design objectives.
3.1 Metasurfaces generating orthogonally circularly polarized dual Bessel beams
Firstly, we design the metasurface capable of generating orthogonal circularly polarized dual Bessel beams upon incidence of LCP waves. Based on the full phase modulation approach, we need to encode the phase information into LCP and RCP components, which can be described as:
Figure 3 shows a phase flow diagram for the generation of an orthogonal circularly polarized dual Bessel beam metasurface. The RCP state, illustrated in the top half of the figure, is comprised of a zeroth-order ${J_0}$ Bessel beam phase ${\varphi _{{J_0}}}$ and offset component ${\varphi _G}$, which shifts the beam in a negative direction along the x-axis. The lower portion of Fig. 3 shows phase distribution of the LCP state, which consists of a first-order ${J_1}$ Bessel beam phase ${\varphi _{{J_1}}}$ and offset component $- {\varphi _G}$ that moves the beam in the positive direction of the x-axis. The geometric phase required to achieve cross-polarization, i.e., the additional meta-atom rotation angle $\theta $, is obtained from the LCP and RCP component phases, illustrated on the right side of Fig. 3. Finally, by combining these rotational angle phases with the propagation phases provided by the eight meta-atoms, a metasurface capable of generating orthogonally polarized dual Bessel beams is constructed.
Fig. 3. Phase flow diagram for generating orthogonal circularly polarized dual Bessel beams metasurface.
We conducted FDTD simulations on the metasurface generating orthogonally circularly polarized dual Bessel beams. Figure 4(a1) demonstrates the longitudinal electric field distributions of orthogonal circular-polarized dual Bessel beams of different orders under LCP incidence. Where the incident LCP wave interacts with the metasurface, it generates dual Bessel beams with different offset directions and orders. Notably, these generated beams maintain relatively small spot diameters and exhibit uniform and clear spot energies within a certain range. Furthermore, the produced beams maintain stability along the z-axis, resulting in relatively long focal depths. The focal depths of ${J_0}$ and ${J_1}$ Bessel beams are 117 and 115${\mathrm{\mu} \mathrm{m}}$, respectively, which closely match the theoretical focal depths of ${J_0}$ and ${J_1}$ Bessel beams, $D/2\textrm{ }tan({asin({NA} )} )= 118.3{\mathrm{\mu} \mathrm{m}}$(${\sim} 222\mathrm{\lambda }$) [41]. Since the polarization properties of the two Bessel beams differ, the longitudinal cross-polarized part and the co-polarized part can be distinguished from the overall intensity distribution map through theoretical derivation and programming. Figures 4(a2) and 4(a3) display separated intensities of two parts, respectively, showing that the two Bessel beams are independent and complete. Figure 4(a4) illustrates the distribution of Stokes parameter S3, revealing a value close to 1 for the RCP component and a value close to -1 for the LCP component. The left half of the figure shows the cross-polarized RCP component and the right half shows the co-polarized LCP component, demonstrating the polarization conversion characteristics of the designed device from the longitudinal direction.
Fig. 4. Simulation results of the metasurface for the generation of orthogonally circularly polarized dual Bessel beams at LCP incidence. (a1) Longitudinal intensity distribution of the orthogonal circularly polarized double Bessel beam. (a2, a3) Longitudinal intensity distributions of the RCP component and the LCP component. (a4) Longitudinal Stokes distribution of an orthogonal circularly polarized dual Bessel beam. (b1-b3) Electric field strength distributions on different focal planes. (c1-c3) Stokes distributions at different focal planes. (d1-d3) Intensity distribution of LCP and RCP components at different propagation distances.
Figure 4(b1-b3) shows the simulated light intensity distributions at focal planes 70, 90, and 110${\mathrm{\mu} \mathrm{m}}$ from the metasurface. The center points of the ${J_0}$ and ${J_1}$ Bessel beams can be observed and appear as concentric rings spreading outward, caused by the interference of the Bessel beams. The distribution of the transverse Stokes parameter S3 is shown in Fig. 4(c1-c3). This reveals the transverse polarization distributions located at different focal planes, thus further validating the co-polarization and cross-polarization distributions in the generated Bessel beam. It is noteworthy that the full width at half maximums (FWHM) of the ${J_0}$ Bessel beam in the RCP component of the three focal planes are 0.928, 0.945, and 0.951${\mathrm{\mu} \mathrm{m}}$, as shown in Fig. 4(d1-d3), which are in good agreement with their theoretical values, $\textrm{FWH}{\textrm{M}_{{J_0}}} = \textrm{ }0.358\lambda \textrm{/}NA = 0.952{\mathrm{\mu} \mathrm{m}}$[41]. Meanwhile, the FWHMs of the ${J_1}$ Bessel beam in the LCP component are 0.774, 0.772, and 0.776${\mathrm{\mu} \mathrm{m}}$, very nearly the theoretical value, i.e., $\textrm{FWH}{\textrm{M}_{{J_1}}} = \textrm{ }0.292\lambda \textrm{/}NA = 0.776{\mathrm{\mu} \mathrm{m}}$. From the above discussion, it can be observed that the intensity distribution and polarization performance of the beam focal planes at different positions are in line with the design expectation. To facilitate the comparison of the three metasurfaces, the 90${\mathrm{\mu} \mathrm{m}}$ focal plane is uniformly selected for observation in the following. To demonstrate the polarization conversion function of the metasurface more accurately, the polarization conversion efficiency (PCE) is defined as the ratio of the light intensity of the output target polarization state to the total light intensity of the incident light wave, and the conversion efficiency of the RCP component can be expressed as: $PC{E_{\textrm{RL}}} = {{{t_{\textrm{RL}}}^2} / {{t_{\textrm{in}}}^2}} = 43.76\%$.
3.2 Metasurfaces generating orthogonal linearly polarized dual Bessel beams
Next, we constructed a second entirely dielectric metasurface to produce equal-intensity dual Bessel beams with orthogonal linear polarization. The LCP and RCP components are superimposed with equal amplitude and phase to eliminate the circularly polarized components, leaving only the linearly polarized components along the direction of superposition. Therefore, LCP light passing through the merged phase metasurface will exhibit as X-P and Y-P light. With the above method, to generate orthogonal line-polarized dual Bessel beams, we can encode the phase information into co-polarized and cross-polarized components:
To demonstrate the feasibility of the above scheme, the metasurfaces generating orthogonally linearly polarized ${J_0}$ and ${J_1}$ dual Bessel beams are simulated here. Figures 5(a) and 5(b) show the longitudinal intensity distributions of the LCP light incident on the two metasurfaces. The co-polarized and cross-polarized components of beams are symmetrically distributed along the z-axis and have nearly identical intensity distributions. Besides, the generated focal depths of 116 and 115${\mathrm{\mu} \mathrm{m}}$ align closely with the expected theoretical values. Figures 5(c) and 5(d) show intensity distribution and the distribution of Stokes parameter S1 at the focal plane 90 ${\mathrm{\mu} \mathrm{m}}$ from the two metasurfaces. Complete spot formations with discernible concentric rings are observed. Meanwhile, the polarization nature of the generated beam is visible in the distribution of the generated Stokes parameter S1. The red area of the value close to 1 corresponds to X-P, while the blue area of the value close to -1 represents Y-P, which is in agreement with the designed one. The orthogonal polarization intensity distribution curves in the focal plane of the ${J_0}$ dual Bessel beam are presented in Fig. 5(e). The calculated FWHMs of X-P and Y-P are 0.932 and 0.937${\mathrm{\mu} \mathrm{m}}$, respectively, closely approximating the theoretical value of 0.952${\mathrm{\mu} \mathrm{m}}$. Similarly, the orthogonal line polarization intensity distribution curves in the focal plane of the ${J_1}$ dual Bessel beam are shown in Fig. 5(f). The calculated FWHMs for X-P and Y-P are 0.743 and 0.736${\mathrm{\mu} \mathrm{m}}$, respectively, close to the theoretical value of 0.776${\mathrm{\mu} \mathrm{m}}$. In addition, when the LCP is incident on the metasurface, the PCE of the X-P component is defined as $PC{E_{\textrm{XL}}} = {{{t_{\textrm{XL}}}^2} / {{t_{\textrm{in}}}^2}}$, and the PCE of the Y-P component is defined as $PC{E_{\textrm{YL}}} = {{{t_{\textrm{YL}}}^2} / {{t_{\textrm{in}}}^2}}$. The PECs are 42.79% and 48.9% for the ${J_0}$ dual Bessel beam X-P and Y-P, and 42.7% and 48.91% for the ${J_1}$ dual Bessel beam X-P and Y-P, respectively.
Fig. 5. Results of the metasurface simulation for the generation of orthogonally linearly polarized dual Bessel beams at the incidence of LCP light. (a, b) Longitudinal intensity distributions of orthogonal linearly polarized ${J_0}$ and ${J_1}$ dual Bessel beams. (c, d) Electric field intensity distributions and Stokes distributions at the focal plane for orthogonal linearly polarized ${J_0}$ and ${J_1}$ dual Bessel beams at z = 90 µm propagation distance. (e, f) Intensity distribution curves of X-P and Y-P at z = 90 µm for orthogonal linearly polarized ${J_0}$ and ${J_1}$ dual Bessel beams.
Self-healing behavior is one of the properties of Bessel beams, meaning that the propagation of a Bessel beam is not affected by obstacles in its path. To verify the self-healing property of the generated orthogonal linearly polarized Bessel beams, we place perfect electrical conductor (PEC) spheres with different radii and positions on the propagation path of the ${J_0}$ dual Bessel beams, respectively. First, we place the PEC sphere with $r\mathrm{\ =\ 1\mu m}$ at ($x\textrm{ ={-} 1}\mathrm{.33\mu m}$, $z\mathrm{\ =\ 25\mu m}$), and then we place the ball at a higher intensity at a longer distance ($x\textrm{ ={-} 3}\mathrm{.724\mu m}$, $z\mathrm{\ =\ 70\mu m}$), as shown in Figs. 6(a) and 6(b). Upon careful observation, it is observed that the main lobe intensity is only minimally affected upon encountering the obstacle. The intensity of the main lobe weakened suddenly for a short distance and then gradually returned to the size of the light intensity it would have been if there had been no obstruction. Then, we increased the radius of the obstacle to 5 ${\mathrm{\mu} \mathrm{m}}$ and localized it at ($x\textrm{ = 0}$, $z\mathrm{\ =\ 25\mu m}$), i.e., in the propagation paths of the main lobes of the two Bessel beams, as shown in Fig. 6(c). We observed this time that an increase in obstacle size significantly affects the beam's ability to self-heal. As the obstacle size increases, the distance required for the Bessel beam to self-heal increases. After passing the obstacle, the light intensity of the main lobe diminishes significantly, nearing zero, over an extended distance. However, at $z\mathrm{\ =\ 45\mu m}$, the light intensity of the main lobe gradually reverts to its normal level. Similarly, we place the obstacle with a diameter of 5 ${\mathrm{\mu} \mathrm{m}}$ at $z\mathrm{\ =\ 70\mu m}$. Compared with Fig. 6(b), the beam is refocused at a greater distance, but the energy after refocusing is relatively weaker. It is important to note that the self-healing nature of the Bessel beam does not mean that it is completely self-healing at every location in the beam. In particular, self-healing may be reduced in areas of the beam that are far from the center of the beam or in cases where the light travels a long distance.
Fig. 6. Self-healing characteristics of orthogonal linearly polarized dual Bessel beams under main lobe obstruction. (a, b) Longitudinal field diagrams of the generated Bessel beams when the obstacles of radius r = 1 µm are placed at z = 25 µm and z = 70 µm of the propagation path of the left main lobe, respectively. (c) Longitudinal field map of the Bessel beams when the obstacle of radius r = 5 µm is placed at z = 25 µm in the propagation paths of main lobes. (d) Longitudinal field diagram of the Bessel beam generated when an obstacle of radius r = 5 µm is placed at z = 70 µm in the propagation path of the left main lobe.
3.3 Metasurfaces generating arrays of orthogonal linearly polarized Bessel beams
In this section, we further construct metasurfaces that can realize 2 × 4 orthogonal linearly polarized Bessel beam arrays at LCP incidence based on Section 3.2. To realize the above function, the arrangement of the meta-atoms constituting the metasurface needs to satisfy the following conditions:
Here, we introduce the phase ${\phi _{DG}}$ of a 1D Dammann grating. By setting the appropriate phase transition point, it becomes possible to generate a 2 × 4 orthogonal linearly polarized Bessel beam array from the original dual Bessel beams. The phase transition points are set to 0.22057, 0.44563, 0.5, 0.72057, and 0.94563 [27,28].
The simulation results of orthogonal linearly polarized Bessel beam array metasurfaces at $z = 90{\mathrm{\mu} \mathrm{m}}$ focal plane under LCP light incidence are shown in Fig. 7. Figures 7(a) and 7(c) depict the 2 × 4 ${J_0}$ and ${J_1}$ Bessel beam arrays with transverse light intensity distributions. The spot intensity uniformities of the ${J_0}$ and ${J_1}$ Bessel beam arrays are calculated as $Q = 1 - ({{I_{\max }} - {I_{\min }}} )/({{I_{\max }} + {I_{\min }}} )$[27], which are as high as 83.7% and 71.1%, respectively. The uniformity of the ${J_0}$ Bessel beam array decreases because the main lobe of the beam changes from a focal spot to a hollow ring, and the energy disperses from being concentrated to scattered. Simultaneously, it is also influenced by neighboring beams, resulting in non-uniformity when splitting into a Bessel beam array. In order to optimize the inhomogeneity of the intensity distribution, the uniformity can be improved by increasing the grid fineness in the FDTD simulation, as well as optimizing the cell parameters of the Dammann grating and increasing the distance between adjacent beams. In addition, the orthogonal line polarization properties of the metasurface generated beam arrays continue to be investigated, and the intensity distributions of the X-P and Y-P components of the ${J_0}$ and ${J_1}$ Bessel beam arrays are shown in Figs. 7(b) and 7(d). By observing the intensity distributions of the X-P and Y-P components as well as the 2D distribution plots, the Bessel beam arrays after the superposition of cross-polarized and co-polarized components and after the phase action of the Dammann grating still have good polarization conversion properties. The PECs are 42.71% and 48.92% for the ${J_0}$ dual Bessel beam array X-P and Y-P, and 42.85% and 48.49% for the ${J_1}$ dual Bessel beam array X-P and Y-P, respectively. Figures 7(e) and 7(f) show the intensity distribution curves of the X-P and Y-P components of the ${J_0}$ and ${J_1}$ Bessel beam arrays, with FWHMs of 0.93 and 0.91${\mathrm{\mu} \mathrm{m}}$ for the X-P and Y-P components of the ${J_0}$ Bessel beam array, and FWHMs of 0.73 and 0.74 ${\mathrm{\mu} \mathrm{m}}$ for the X-P and Y-P components of the ${J_1}$ Bessel beam array.
Fig. 7. Simulation results of orthogonal linearly polarized Bessel beam array at z = 90 µm focal plane. (a) Intensity distribution of orthogonally linearly polarized ${J_0}$ Bessel beam array. (b) Intensity distribution of the X-P and Y-P components of the orthogonal linearly polarized ${J_0}$ Bessel beam array. (c) Intensity distribution of an orthogonal linearly polarized ${J_1}$ Bessel beam array. (d) Intensity distribution of the X-P and Y-P components of the orthogonal linearly polarized ${J_1}$ Bessel beam array. (e, f) Intensity distribution curves of X-P and Y-P components of the orthogonal linearly polarized ${J_0}$ and ${J_1}$ Bessel beam arrays.
4. Conclusion
In summary, we propose a fully phase-modulated method for generating Bessel beams from all-dielectric metasurfaces. The method achieves phase modulation of co-polarized and cross-polarized states by simultaneously and independently modulating the transmission phase and geometrical phase, which can efficiently generate Bessel beams with orthogonal circular polarization and orthogonal linear polarization of different orders. Subsequently, the Dammann grating phase is introduced to realize an array of orthogonal linear polarization Bessel beams with high-uniformity. Simulation results show that the proposed metasurface successfully achieves the polarization conversion function set as expected and produces Bessel beams with the characteristics of depth of focus, non-diffraction, and self-healing in agreement with the theoretical values. This work provides new ideas for the efficient generation and versatile application of Bessel beams and has great significance for several disciplines, such as laser fabrication, particle trapping, and optical communications.
Funding
Ministry of Science and Technology of the People's Republic of China (2022YFC2808203); National Natural Science Foundation of China (62375245).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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