1. Introduction

As important information carriers, photons in vector beams (VBs) exhibit two types of angular momentum: spin angular momentum (SAM) associated with polarization and orbital angular momentum (OAM) related to the helical wavefront. Presenting distributions of spatially varying polarizations and helical phases, VBs have facilitated the classical applications such as super-resolution imaging [1], optical trapping [2], optical communication [3], and optical nanofabrication [4]. Fundamentally, the SAM and OAM in VBs are coupled in non-separable way, resembling localized entanglement in quantum systems, leading to the concept of “classical entangled light" [5]. While in quantum information science, VBs involve important applications such as quantum key distributions [6], quantum teleportation [7], and alignment-free communication over long distance using rotationally invariant quantum qubit encoding [8].

Recent investigations on tightly focused vector light fields have revealed intriguing phenomena, including Mobius strips [9,10], Hopf topological links [11], and three-dimensional topological optical fields with vector textures [12,13]. These discoveries have provided new avenues for VB applications and invigorated explorations of novel methods for VB manipulations. To date, numerous techniques have been developed for generating VBs using traditional optical components and intricate optical paths [14–19]. Nevertheless, these methods encounter challenges such as insufficient spatial resolution or suboptimal beam quality for high information capacity, and particularly, the bulky footprint of the used components constrains the miniaturization and integration of the photonic devices [20].

Metasurfaces are ultrathin, multifunctional devices composed of subwavelength meta-atoms that enable customized control of the amplitude, phase, and polarization of light. These devices are extensively applied in fields such as metalenses [21,22], vector holography [23], quantum entanglement [24], high-density optical storage [25], and encryption [26]. When light of a specific wavelength is used as the illuminating beam, the light field transmitting through an arbitrarily shaped meta-atoms acquires propagation and geometric phases, while the former is dependent on the dimension of meta-atom but independent of incident polarization due to the waveguide effect [27], and conversely, the latter is nondispersive and is controlled by the orientation of the meta-atom and incident circular polarizations (CPs) [28]. Nevertheless, the propagation phase is vulnerable to design inaccuracies and manufacturing limitations, which may be unfavorable to the efficacy of the output light field and beam quality. Among the different categories of metasurfaces, geometric metasurface consisting of anisotropic meta-atoms with identical geometry has attracted considerable interests owing to their advantages such as design flexibility, fabrication feasibility and accurate phase control. This metasurface has enabled applications such as metalens with breaking the diffraction limit [29,30], generating multimode vector light fields [31–33], and holographic imaging using Malus’ law [34].

Nevertheless, the intrinsic opposite signs of the geometric phase for the two orthogonal CPs present a challenge. Prior geometric metasurface designs for focusing arbitrarily polarized light necessitate two hyperbolic phase profiles with opposite signs which are configured to match the two CPs [31,35–37], respectively. Yet, the sign-reversed hyperbolic geometric phases corresponding to their individually mismatched incident CPs diverge the light and inevitably introduce background components as noise in the output field. Yao et al. developed a spin-decoupled, intensity-adjustable metalens with four focal points, requiring eight sets of different hyperbolic phases [36]. Zhou et al. utilized numerous hyperbolic phases to simulate multichannel vortex beams in the terahertz band [37]. Wang et al. generated a focused VB array, which also required eight sets of hyperbolic phases [31]. Consequently, integrating more channels for focusing in geometric metasurfaces leads to reduced efficiency and increased inherent crosstalk and background noise [38,39], imposing a fundamental constraint on the information capacity achievable by integrated miniaturized metasurfaces.

In this study, based on Fresnel zone plate [40–42], we generate focused higher-order Poincaré sphere (HOP) beams in multichannel of symmetrical array using a dielectric geometric metasurface composed of two sub-metasurfaces. Each sub-metasurface includes two sets of rings, on which identical meta-atoms of half-wave plate are arranged. The rings in the first sets of the two sub-metasurfaces are on the Fresnel zones of odd and even numbers, and rings in the second sets have an optical path increase by a quarter wavelength. The optical path increment for each set of rings is a wavelength, and with the hyperbolic geometric phase profile configured to meta-atoms on one set of rings, the mirror-symmetrical position-flip of the off-axis focal point is enabled under the chirality switch of the illuminating CP. By the combined design of the hyperbolic and helical geometric phase profiles, the two sets of rings in a sub-metasurface generate vortices of opposite topological charges and orthogonal CPs at each focal point under linearly or elliptically polarized incidence. Owing to superposition of the vortices, two VBs or HOP beams are generated at the two focal points with positional mirror-symmetry. With other two VBs produced by the other sub-metasurface, the generation of VBs at four channels are realized, and their polarization states can be modulated by altering the incident polarizations. Additionally, the optical path difference of a quarter wavelength brought about a propagation phase difference between the two sets of rings within a sub-metasurface, leading to the orthogonal polarization states of VBs at the mirror-focused points. With meta-atoms on four sets of rings, the metasurface of this study generates multichannel focused HOP beams, though the focusing is not so strong as to reach the criteria of tightly focused beam. Initially, we provided theoretical derivations for a sub-metasurface, followed by the simulation of six metasurface samples using finite-difference time-domain (FDTD) simulations. We fabricated and tested four samples, with the results well aligning with theoretical predictions. The proposed design reduces the requisite number of meta-atom sets by half, enhancing metasurface efficiency. Adding to diversity and design flexibility of metasurface, this study holds promise for the miniaturization and integration of optical devices.

2. Theory and simulations

2.1 Design principle

Figure 1(a) illustrates a schematic of the multichannel focused HOP beams generated by our dielectric geometric metasurface under the illumination of arbitrary elliptically polarized light. The light is considered as the linear superposition of left- and right-handed CP (LCP and RCP) lights with σ = ±1 and amplitudes a and b, respectively. Our metasurface comprises two sub-metasurfaces, labeled as j = A and B, where j represents the index of the sub-metasurface. Each sub-metasurface contains two sets of rings on which the meta-atoms are arranged. The four sets of rings in two sub-metasurfaces are alternately distributed in the following way. In sub-metasurface A, the rings in first set are the odd-number rings of Fresnel zones, which have the optical path difference of integer M multiples of wavelength, denoted as ΔD_A= Mλ, relative to the distance of f–λ/2, where the additional –λ/2 is to ensure the first ring of this set to occupy the first ring of Fresnel zone for M = 1; the ring in the second set adjacent to the one in the first set has an increase in the optical path difference by a quarter wavelength, i.e., its path difference is (M + 1/4)λ. While in sub-metasurface B, the rings in first set are the even number rings of Fresnel zones, and accordingly, a ring has an increase in the optical path difference by a half wavelength with respect to its counterpart in sub-metasurface A, and its path difference is (M + 1/2)λ; similarly, the adjacent ring in the second set has the path difference of (M + 3/4)λ. By rotating the meta-atoms versus the azimuthal angle along each ring to control the geometrical phase, the helical and hyperbolic phase profiles are established on each set of rings to yield the vortex wavefronts and to focus them at the corresponding focal points, respectively. We set the spatially varying phase Ψ_j for sub-metasurface j, which is twice the orientation angle of the meta-atom φ_j(x, y), as given by the following expression:

 

Fig. 1. (a) Schematic of the multichannel focused HOP beams generated with the dielectric geometric metasurface under the illumination of elliptically polarized light. (b) The theoretical images of light field intensities at the focal plane and the traditional Poincaré sphere. (c) Figures from left to right are an enlarged view of the meta-atom of a-Si:H with period P_θ = 350 nm, the transmission (T_x), and transmission phase (ϕ_x), respectively. The scale bar is 4 µm and is for all the intensity images.

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When sub-metasurface A is illuminated with linearly polarized light, which contains RCP and LCP components, the first set of rings generates the RCP and LCP vortices situated symmetrically at the focal points (x_FA, y_FA) and (–x_FA, –y_FA), respectively. Meanwhile, the second set of rings generates the RCP and LCP vortices at the opposite points (–x_FA, –y_FA) and (x_FA, y_FA). Moreover, the increase of path difference λ/4 brings about a propagation phase increase of π/2; to match this increase, the designed geometric phase σΨ_j is adjusted, and a phase difference of π in the generated RCP and LCP vortices is brought about owing to the switch of σ from 1 to -1. Resultantly, with the light field of sub-metasurface A taken under consideration as a whole, at points (x_FA, y_FA) and (–x_FA, –y_FA), the RCP and LCP vortices of the first set of rings are superposed with the LCP and RCP vortices of the second set of rings, respectively, forming a VB at each of the two focal points; the phase difference of π sets up the orthogonal polarization states of the two beams. Similarly, under the illumination of linearly polarized light, the sub-metasurface B generates the other two VBs focused at points (x_FB, y_FB) and (–x_FB, –y_FB). More generally, when the metasurface containing the two sub-metasurfaces are illuminated with elliptically polarized light composed of the two unequally weighted CPs, the HOP beams are generated at the four focal points, realizing the manipulations in the multi-channels.

For a schematic illustration of the generated VBs, the theoretical images of light field intensities at the focal plane are given in Fig. 1(b), where the schematic polarization states are overlaid on the images. The horizontal, vertical and curved green arrows indicate the incident polarization states corresponding to points labeled by | H>, | V > and | E_P> on the traditional Poincaré sphere, respectively, and the horizontal red arrows denote the x-component field of the generated VBs. In Fig. 1(c), the left panel schematically shows a meta-atom of a-Si:H with refractive index of n = 3.744 and vanished extinction coefficient of κ = 0.000 at 800 nm wavelength. The meta-atom is designed to function as half-wave plate; it has the rectangular dimension with the parameters of W = 100 nm, L = 180 nm, and H = 480 nm. The dimension is selected from the data maps of the transmission (T_x) and transmission phase (ϕ_x) as shown in the middle and right panels in Fig. 1(c), respectively. The two maps are the simulation results of the two-dimensional parameter sweep conducted over a meta-atom in the size range of 80–330 nm using FDTD method. The white asterisks in the two maps in Fig. 1(c) designate the selected meta-atom, with its dimension and the corresponding transmission and phase retardance satisfying the condition of the half-wave plate [44]. In this study, the geometric metasurfaces were designed and all of them were composed of the meta-atoms of the same dimension.

2.2 Realization of multichannel HOP beams with different polarized light illumination

As depicted in Fig. 2(a), the metasurface is on the object plane oxy, point p(x, y) denotes the position of a meta-atom of the metasurface, and it can be also represented as p(R_j(n), θ) in polar coordinates oRθ, where R_j(n) depicts that the meta-atom is at the n-th ring and belongs to the j-th sub-metasurfaces. Initially, we take sub-metasurface j = A as the example for the mathematical analysis of the VB fields. When a meta-atom in sub-metasurface A is illuminated by CP light with Jones vector [1 iσ]^T=|E^σ>, where T represents the matrix transpose, the light field on the observation plane Ox_Fy_F can be expressed by the Rayleigh-Sommerfeld formula [45]:

 

Fig. 2. Geometric illustration of sub-metasurface A for generating a pair of position-symmetrical (a) optical vortices under illumination of circular polarization and (b) vector beams under illumination of elliptical polarization. The inset is the enlarged view of coordinate geometry for observation point near focal point (x_FA, y_FA). (c) The radial, 45° slant-, azimuthal, and 135° slant-polarized VB arrays of orders 1, 2, and 3 generated by samples S_I-S_III, respectively. (d) The VB arrays in a rhombus lattice pattern generated by samples S_IV–S_VI. The scale bars are 4 µm and are for all the intensity images.

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For the first set of the rings, when the incident light is LCP (σ=1) represented by | L > = [1 i]^T, the light field in Eq. (4) with the “–” sign of the quadratic phase factor ${e^{ik\textrm{[}f\textrm{ + (}1 \mp \sigma \textrm{)}\frac{{\textrm{(}{x^2} + {y^2}\textrm{)}}}{{2f}}\textrm{]}}}$ is focused at (x_FA, y_FA), because this factor is reduced to ${e^{ikf}}$. Conversely, when the incident light is RCP (σ=–1) represented by |R > = [1 –i]^T, the quadratic phase factor also takes “–” in the plus-minus symbol, the factor ${e^{ik\textrm{[}f\textrm{ + (}1 - \sigma \textrm{)}\frac{{\textrm{(}{x^2} + {y^2}\textrm{)}}}{{2f}}\textrm{]}}}$ is reduced to ${e^{ikf}}{e^{i2\pi \textrm{(}2M - 1\textrm{)}}} = {e^{ikf}}$, and the light field is focused at (–x_FA, –y_FA). It is indicated that with the geometric phase profile σΨ_j of the meta-atoms configured according to Eq. (1) and their positions on the rings of Fresnel zone, the first set of the rings can simultaneously focus both the light fields produced under illuminations of both CPs. Here we note that in prior metasurfaces, one set of meta-atoms usually could focus the light field generated under only one CP illumination. Substituting the integral variables (R, θ) for (x, y), and performing the integral over θ [44], we can further calculate the light field $E _{out}^A({{x_F},{y_F}\textrm{;}\sigma } )$ in Eq. (4) and write it as the following two equations for both LCP and RCP illuminations, respectively:

Subsequently, we considered elliptically polarized light as the general illuminating light, as shown in Fig. 2(b), which is expressed in the following form:

2.3 Simulation results

After carrying out the parameter sweep to obtain meta-atom of the half-wave plate, as depicted in the former text, we further performed FDTD simulations to analyze the output optical fields of six samples (S_I-S_VI) under horizontally polarized light illumination, as shown in Figs. 2(c) and (d). In the FDTD simulations, the light fields of the metasurface samples were first calculated in a near-field plane and then projected to the far-field focal plane (z = f = 70µm) to achieve the focused HOP beam. The title bar on the left indicates the topological charges l_A and l_B of the sub-metasurfaces A and B, respectively, in the sample. The images, from the second to fourth rows, display intensities of the total field I_t= |E_x|² + |E_y|², and the x-, and y-component fields I_x= |E_x|² and I_y= |E_y|² with a schematic of the polarization state overlaid on the total field. The purple double arrows in the lower left of each image signify the polarization state of the incident light, whereas the white double arrows in the lower right represent the direction of the component field. Panels (i–iii) in Fig. 2(c) reveal that samples S_I–S_III generate radial, 45° slant-, azimuthal, and 135° slant-polarized VB arrays of orders 1, 2, and 3, respectively, in rectangular lattice patterns; the polarization states of the four VBs in each array are positioned at four equidistant points on the equator of the HOP sphere at longitudes of 0°, 90°, 180°, and 270°, respectively.

It is also observed that with increase of VB orders generated by the different samples, the diameters of the total intensity doughnuts increase. Each intensity image of the component fields comprises 2l_j lobes. Figure 2(d) illustrates the VB arrays in a rhombus lattice pattern generated by samples S_IV–S_VI at the focal plane. Panel (i) in Fig. 2(d) displays VBs identical in polarizations but different in position compared to those in panel (i) in Fig. 2(c). The four VBs of the Bell-like state [46] generated by sample S_V are depicted in panel (ii) in Fig. 2(d), whereas the radial and azimuthal VBs of order 1 and 3 generated by sample S_VI are illustrated in panel (iii) in Fig. 2(d). On the whole, the simulations confirm that the designed metasurface generates a VB array with an arbitrary distribution of the polarization states. Attributing to the annulus doughnut of the intensity distribution I_t, we define the radius of the circle in the annulus with the maximum intensity as the width of the VB, and define the full width at half-maximum (FWHM) of the light intensity profile versus the radius as the thickness of the doughnut annulus. It is clear that the widths of different order VBs on the focal plane are different but at the same scale of magnitude, and the widths of the same order VBs with different polarization states are approximately equal. Since the focused azimuthally polarized VB without the E_z component has a typical doughnut, the azimuthally polarized VBs of orders 1 and 3 produced by the sample S_VI in panel (iii) of Fig. 2(d) are taken as examples for calculating the widths and the thicknesses of the VBs, and the widths of the VBs of orders 1 and 3 are 0.55λ and 1.56λ, and the thicknesses are 0.6λ and 0.7λ, respectively.

From the images in Figs. 2(c) and (d), we may see that the intensity distributions in an HOP beam array are anisotropic. The intensities of four beams are different to some degree. Particularly, for the images in panel (ii) in Fig. 2(d) for sample S_V, the radially and azimuthally polarized VBs in the right and left channels have the weakest and the strongest doughnuts of the total light intensity, respectively. One of the causes for the phenomena might be originated from the anisotropy in the rectangular dimension of meta-atom used in our metasurface. Another reasonable cause might be the existence of the other component E_z, which may take a portion of power of the VB. Among the four images of the VBs of Bell-like states, the E_z component for the radially polarized VB in the right channel is the largest, corresponding to its weakest doughnut and x- and y- component intensities, while the case is opposite for the azimuthally polarized VB in the left channel, with largest doughnut intensity.

3. Experiment

3.1 Experimental setup and sample fabrications

Figure 3(a) outlines the experimental setup. A horizontally polarized light beam of 800 nm wavelength emits from the laser source. The light beam passes through a half-wave plate (HP) and a quarter-wave plate(QWP), and the arbitrary polarization states of the light beam across the traditional Poincaré sphere [40] are achieved to illuminate the metasurface sample. An attenuator (A) then modulates the incident light power. The focused VBs are generated on the focal plane of metasurface at a distance of focal length f = 70 µm from sample. A microscope objective (MO) (Nikon, 0.9, 100x, WD 1.0 mm) images focused VBs, with a scientific complementary metal–oxide semiconductor (sCMOS) image sensor positioned at the image plane of the objective to record the overall field pattern. Introducing an analyzing polarizer (P) before the sCMOS allows to record the intensity patterns of the x-, 45°-, y-, and 135°-component fields of the VBs obtained by rotating its transmitting axis. Figures 3(b) and (c) present the overall metasurface image exported from FDTD simulations and localized magnified scanning electron microscopy (SEM) images of sample S_II, respectively.

 

Fig. 3. (a) Schematic of the experimental setup for generating focused HOP beams with the metasurfaces. (b) The overall metasurface image of sample S_II exported from FDTD simulations. (c) The locally enlarged SEM image of sample S_II.

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3.2 Experimental multichannel HOP beams of different orders and polarization states

Based on the simulation results from Figs. 2(c) and (d), we fabricated samples S_II, S_III, S_V, and S_VI using electron beam lithography (EBL). Employing the experimental setup delineated in Fig. 3(a), we acquired the experimental results presented in Figs. 4 and 5. In Fig. 4(a), the upper and lower portions illustrate doughnut and x-component intensities of the VB arrays generated by samples S_V and S_VI, respectively, under illuminations of light with horizontal, 45°-slant, vertical and 135°-slant linear polarizations. In the x-component intensity image for sample S_V under illumination of horizontal polarization, the four VBs generated at the horizontal and vertical vertex-pairs of the rhombus lattice are of orders 1 and -1, respectively, which are consistent with their counterparts in panel (ii) in Fig. 2(d). The two VBs at a vertex-pair are radially and azimuthally polarized, respectively, indicating their polarization states orthogonal.

 

Fig. 4. Images of the focused HOP beam arrays of different orders experimentally generated from samples S_V and S_VI. (a) Illustrations of doughnut and x-component intensities of the VB arrays generated under illuminations of four linear polarization. (b) and (c) HOP beam arrays generated by sample S_V and sample S_VI, respectively, under the illumination of light with elliptical polarization states on the prime meridian of the traditional Poincaré sphere. The purple and white double-headed arrows denote the linear polarizations of illuminating light and the direction of the VB component fields, respectively, and the curved arrows of changing colors represent elliptical polarization states of the illuminating light. (d) The VQF values and the intensity patterns of the four simulated and experimented VBs of Bell-like states from sample S_V. The scale bars are 4 µm and are for all the intensity images of the same dimension.

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Fig. 5. The experimental results of focused higher-order Poincaré spheres (HOP) beam arrays generated by samples S_II and S_III under the illuminations of different polarizations. (a) HOP sphere of l = 2 and 3. (b) The focused VB arrays generated by samples S_II and S_III under illumination of four linear polarizations. (c) Focused HOP beam arrays generated by sample S_II under different elliptical polarization illuminations. The scale bars are 4 µm and are for all the intensity images.

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Whereas for sample S_VI under illumination of horizontal polarization, the four VBs of orders 1 and 3 are generated at the vertical and horizontal vertex-pairs with radial and azimuthal polarization states, respectively, which are consistent with the counterparts in panel (iii) in Fig. 2(d). From the x-component intensity images of the VB arrays in Fig. 4(a), it is also observed that by changing the linear polarizations of the illumination, the polarization states of the VBs can be modulated.

Under the illumination of light with elliptical polarization states on the prime meridian of the traditional Poincaré sphere, the VB arrays generated from samples S_V and S_VI are shown in Figs. 4(b) and (c), respectively. The polarization states of all VBs are on either the prime or the 180° meridians HOP spheres of different orders. Specifically, the orders of HOP spheres are 1 and -1 for S_V, and 1 and 3 for S_VI, respectively, which are the same as the orders of the aforementioned counterparts of linearly polarized VBs. Points I–IX denote the locations of the VB arrays at nine equidistant points on the meridian of the HOP sphere. The x-component images in Figs. 4(b) and (c) demonstrate that as the incident polarization state transitions along the prime meridian, from the LCP at the south pole to linear polarization at the equator and then to the RCP at the north pole of the traditional Poincaré sphere, the position of the VB array on the HOP spheres correspondingly shifts from point I to IX. Correspondingly, from the south pole to equatorial point, the blurred dark lines between the petal patterns in the component intensity images progressively become distinct, and then from the equatorial point, the dark lines become more and more blurred again.

To quantitatively evaluate the consistency of our result of simulation and experiment, we chose the comprehensive and commonly used quantity of the vector quality factor (VQF) for the evaluation [47,48]. We calculated VQF values for the VBs of Bell-like states (|TE > ₁, |TM > ₁, |HE^e> ₁, and |HE^o> ₁) generated by S_V, as the example for demonstrations. In Fig. 4(d), the VQF values of the four VBs with linearly polarized states were calculated from the data of the corresponding simulated and experimental images in different color rectangles as shown in the Figure, respectively. The upper and lower intensity patterns in each rectangle in Fig. 4(d) are simulated and experimental results of total field I_t of the VB, respectively, where the schematic polarization states are overlaid on the simulated intensity patterns by the black arrows. According to the Refs. [47,48], the theoretical VQF of $\Theta ^{\prime} = {90^ \circ }$ was 1, marked as the black triangle in Fig. 4(d), where Θ’ is independent variable of function for VQF and represents longitude on the HOP sphere. It can be seen from Fig. 4(d), the simulated and experimental VQF values are all close to the theoretical value, demonstrating the consistency between the simulated and experimental results.

Figure 5 displays the experimental results of HOP beam arrays generated by samples S_II and S_III under varying polarizations of incident light illuminations. Within the images, purple double arrows denote polarization direction of linearly polarized incident light, and colored curve arrows signify the polarization state of elliptically polarized incident light. The white double arrows indicate the direction in which the analyzing polarizer permits the transmission of the light field. The polarization states of the generated VBs, as depicted on the HOP sphere in Fig. 5(a), are marked with points A–D and I–IX. Figure 5(b) exhibits the focused VB arrays generated by samples S_II and S_III under illumination of four linear polarizations. In the images for horizontal linear polarization of illuminations in Fig. 5(b), the VBs with doughnuts marked with A–D have the polarization states corresponding to the points A–D on the HOP sphere of orders 2 for S_II and 3 for S_III, respectively. When the incident polarization rotates in the sequence as indicated by the purple arrows from left to right in the Fig. 5(b), the polarization states of the four VBs are also rotated sequentially in the same direction, with fixed relative positions of A–D in the images, and the position of VB with polarization state A is marked in each image. The x-component images under horizontal polarization illumination for S_II and S_III are consistent with their counterparts in panels (ii) and (iii) in Fig. 2(c), respectively. Figure 5(c) reveals that under illumination of light with elliptical polarizations located on the prime meridian of the traditional Poincaré sphere, S_II yields HOP beams situated on the four meridians, echoing the pattern observed in Fig. 4(b) and (c).

4. Discussion and conclusion

To analyze the optical performance of our designed metasurfaces, we used FDTD simulation to calculate the efficiency of sample S_V for generating the focused vector beams of Bell-like states, as an example. It is defined as the ratio of the integral of transmitting power over the monitored area to incident power on the illuminating area, and we calculated the efficiency of VB from sample S_V was 21.23% at wavelength 800 nm. In the calculation, the sample of diameter 84 µm was illuminated by the incident light over an area of 86 µm × 86 µm, and the size of the monitored area is 16µm × 16 µm on the far-field focal plane, which contain four VBs. To analyze the issues of background noise and crosstalk, we first calculated the power proportion of four VBs generated by sample S_V to the total transmitting power. The center of monitored area is set at the center of each focused VB and the size of area is 6µm × 6 µm. The power proportions of four VBs (|TE > ₁, |TM > ₁, |HE^e> ₁, and |HE^o> ₁) on the focal plane were 26.46%, 17.05%, 23.52% and 23.02%, respectively. The power proportion of background noise was defined as the ratio of the power in the 3µm × 3 µm paraxial region, on which the main futile residual portion of power were concentrated, to the total transmitting power, and the calculated power proportion of background noise was 1.31%. Excluding the power flow of both the four VBs and background noise, the remaining power could be considered as crosstalk between the four channels, and the power proportion of the crosstalk averaged to each channel was 2.16%. On the whole, comparing with the prior designs of geometric metasurfaces, the proposed metasurface uses the scheme of arranging meta-atoms on Fresnel zone rings, and it has the advantage of achieving more tightly focused VBs using less sets of meta-atoms. Moreover, based on simulated and experimental results, it is observed that by designing the helical phases with different topological charges and adjusting the linear or elliptical polarizations of the incident light, the generation of focused HOP beam arrays with different orders and polarization states can be achieved. This demonstrates the feasibility and flexibility of the proposed metasurface in generating the diverse VBs, and the quality of the generated VBs is satisfactory according to the qualitative visualization and quantitative VQF analysis.

Theoretically, the metasurface designed with our method can generate focused VBs with a higher number of channels. To generate multi-channel focused VBs of high quality, the diameter of the metasurface should usually be designed to be larger. Because the meta-atoms are arranged on rings related to Fresnel zones with optical path increment of a quarter wavelength, when the diameter of the metasurface is too large, the edge gap of two meta-atoms on the adjacent rings might be smaller than the distance limitation of fabrications. Fortunately, in order to further increase the number of channels of focused VBs, we can set a larger focal length to increase the radius increment between adjacent rings of the metasurface. Overall, the number of channels of focused VBs is limited by the practical fabrication condition.

To conclude, we introduce a design of dielectric geometric metasurfaces capable of generating multichannel focused HOP beams. On the whole, our theoretical analyses, simulations, and experimental findings are in consistency, substantiating the viability of our design. This research may be beneficial to the miniaturization and integration of pertinent optical systems, which have potential applications in fields such as micromanipulations, super-resolution imaging, and optical and quantum communications.