Polarization and phase are two fundamental characteristics of electromagnetic (EM) waves, whose spatial distributions can be engineered to manipulate the properties of EM waves, leading to a variety of special beams, i.e., vortex beams [1] and vector vortex beams [2,3]. EM waves can possess spin angular momentum (SAM) and orbital angular momentum (OAM) states. SAM is associated with the circular polarization of EM waves, where the SAM of each photon is $\pm \hslash$. EM waves can also carry OAM with a helical phase defined as $\exp (il\theta )$, where $l$ is the topological charge of the vortex beam. Since the OAM of EM waves has an infinite number of eigen modes, i.e., $l=\pm 1,\pm 2,\pm 3,\dots$, and the different OAM modes are orthogonal, they possess unprecedented quantities of data for high-capacity data transmission in optical and quantum communication systems [4,5]. Currently, a variety of approaches, i.e., spiral phase plates [6], Q-plates [7], cylindrical mode converters [8], and spatial light modulators [9], have been developed to generate and manipulate vortex beams. However, all of these conventional methods for generating and manipulating multiple OAMs (e.g., superposition of OAM states) require many bulky components (functional devices) that need to be well aligned with each other. In addition, the aberrations induced by the mismatch or incompatibility between these individual components will inevitably degrade the quality of the generated vortex beams.

Geometric metasurfaces, which are composed of predesigned anisotropic meta-atoms with identical shapes but different in-plane orientations, provide a flexible platform to accurately tailor the amplitude, phase, and polarization of EM waves at subwavelength resolution. The manipulation of EM waves based on geometric metasurfaces depends on the spatially varying geometric phase response (Pancharatnam–Berry phase) rather than the phase (propagation phase) accumulation along optical paths. The main advantage of geometric metasurfaces is the abrupt phase shifts that can be realized by anisotropic meta-atoms with thicknesses less than the wavelengths of EM waves, making geometric metasurfaces ideal candidates for device miniaturization and system integration. Benefiting from the ease of fabrication, ultrathin nature, and multifunctional integration characteristics, a plethora of applications, such as generalized Snell’s law of refraction/reflection [10,11], metalenses [12–18], polarization manipulation [19–28], spin Hall effects [29–32], holograms [33–39], and nonlinear photonics [40–43], have been demonstrated. Moreover, by judiciously designing metasurfaces that introduce spatially varying geometric phase response, a variety of structured beams with various OAMs or topological charges have been demonstrated using circularly/linearly polarized incident beams, such as vortex beams [44–48], vector vortex beams [49], perfect vortex beams [50–54], and the multiplexing of multiple vortex beams [55–59]. However, previous studies of the multiplexing of vortices mainly focus on the transverse direction. The multiplexing of multiple vortex beams with polarization-controllable functionality in the longitudinal direction (or both transverse and longitudinal directions) offers further flexibilities to the manipulation of EM waves. Indeed, longitudinal multiplexing of multiple vortices can be realized by focusing spin-dependent vortices in the propagation direction, but the inherent conjugation in traditional geometric phase response between the two helical components hinders the simultaneous focusing and independent modulation of topological charges between left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) components.

In this paper, we propose an approach to design a geometric metasurface that can realize the polarization synthesis and multidimensional multiplexing of terahertz converged vortices, based only on the geometric phase response. The polarization-independent converged vortices are first demonstrated by the superposition of two converged vortices with orthogonal circularly polarized (CP) states and identical topological charges. By further integrating the multiple focusing phases and polarization-rotatable functionality into a single geometric metasurface, the multiplexing of polarization-rotatable multiple vortices in the longitudinal direction or multiple dimensions (both transverse and longitudinal directions) has been realized. Furthermore, a converged vortex beam with an extended focal depth is realized through a partial overlap of two orthogonal linearly polarized (LP) vortices. This robust approach for superposition and multiplexing of THz converged vortices enables a far-reaching impact on designing flat devices with multiple functionalities and applications in polarization-dependent sensing or imaging, detection, and quantum information.

The operating principle for superposition and multiplexing of THz converged vortices is schematically shown in Fig. 1. As depicted in Fig. 1(a), a metasurface is designed to focus the incident THz waves with arbitrary polarization states into a converged vortex with identical topological charges. The incident THz waves can be divided into two orthogonal circular components (LCP and RCP). To simultaneously focus the LCP and RCP components into a converged vortex with the same topological charge, the required phase profile for the geometric metasurface is governed by [55,60] $${\mathrm{\Phi }}_1=\arg [\exp (i{\phi }_{\mathrm{LCP}})+\exp (i{\phi }_{\mathrm{RCP}})],$$(1)where ${\phi }_{\mathrm{LCP}}=\frac{2\pi }{\lambda }(\sqrt{x^2+y^2+f_L^2}-f_L)+l_L\phi$ and ${\phi }_{\mathrm{RCP}}=-\frac{2\pi }{\lambda }(\sqrt{x^2+y^2+f_R^2}-f_R)-l_R\phi$. “Arg” returns the phase angles of the total field of “$\exp (i{\phi }_{\mathrm{LCP}})+\exp (i{\phi }_{\mathrm{RCP}})$”. $f_L$ and $f_R$ are the focal lengths of the converged RCP and LCP vortices, respectively, and $l_L$ and $l_R$ are topological charges. $\lambda$ is the working wavelength. When $f_l=f_R$ and $l_L=l_R=l$, a converged vortex with a topological charge of $l$ can be observed after the metasurface under illumination of a THz wave with an arbitrary polarization state. It should be noticed that the phase profile in Eq. (1) contains a convex lens and a concave lens relative to both incident LCP and RCP components, and thus the spin-independent converged vortex can be generated by the designed metasurface. For the incidence of LCP/RCP THz waves, it is modulated by both ${\phi }_{\mathrm{LCP}}$ (for focusing the incident LCP/RCP THz waves into an RCP/LCP converged vortex) and ${\phi }_{\mathrm{RCP}}$ (for defocusing it into background). Only half of the transmitted waves will be converted to the RCP/LCP converged vortex. Therefore, the maximal focusing efficiency of the designed metasurfaces cannot exceed 50% in theory.

Since the incident LCP and RCP components with the same amplitude can be combined (by the metasurface) into an LP converged vortex, our approach can be extended to generate a polarization-rotatable converged vortex by embedding polarization modulation into the geometric metasurfaces. The phase requirement can be expressed as $${\mathrm{\Phi }}_2=\arg \left\{\exp \left\{i\left[\frac{2\pi }{\lambda }\right(\sqrt{x^2+y^2+f_L^2}-f_L)+l_L\phi ]+\varphi \right\}+\exp \{-i[\frac{2\pi }{\lambda }(\sqrt{x^2+y^2+f_R^2}-f_R)+l_R\phi ]-\varphi \}\right\}.$$(2)

According to Eq. (2), the converged vortex after the metasurface will be generated with the corresponding polarization state (LP state) rotated with an angle $\varphi$ in comparison with the incident THz waves. Furthermore, we can integrate a series of converged vortices with different orientations of polarization into a single metasurface to realize the multiplexing of polarization-rotatable multiple vortices on multiple spatial dimensions [see Fig. 1(b)]. The phase profile of the geometric metasurface can be written as $${\mathrm{\Phi }}_3=\arg \left\{\sum _{i=1}^N[\exp (i{\phi }_{i-\mathrm{LCP}})+\exp (i{\phi }_{i-\mathrm{RCP}})]\right\},$$(3)where ${\phi }_{i-\mathrm{LCP}}=\left\{\frac{2\pi }{\lambda }[\sqrt{{(x-x_{i-L})}^2+{(y-y_{i-L})}^2+f_{i-L}^2}-f_{i-L}]+l_{i-L}\phi \right\}+{\varphi }_i$ (${\phi }_{i-\mathrm{LCP}}$ is the phase profile for converting the LCP incident beam into the $i$th RCP converged vortex), and ${\phi }_{i-\mathrm{RCP}}=-\left\{\frac{2\pi }{\lambda }[\sqrt{{(x-x_{i-R})}^2+{(y-y_{i-R})}^2+f_{i-R}^2}-f_{i-R}]+l_{i-R}\phi \right\}+{\varphi }_i$ (${\phi }_{i-\mathrm{RCP}}$ is the required phase profile for converting the RCP incident beam into the $i$th LCP converged vortex). $i=1,2,3,\dots$, and $f_{i-L}=f_{i-R}$ (defined as the focal length of the $i$th LCP and RCP converged vortex). A detailed discussion of the metasurfaces for generating multiple converged vortices with polarization-rotatable functionality is given in Appendix A.

We design a series of silicon-based geometric metasurfaces to demonstrate the aforementioned functionalities. As shown in Fig. 2(a), the silicon-based dielectric metasurface consists of anisotropic meta-atoms with identical shapes but different orientations. Each meta-atom should be optimized as a quasi-half-wave plate that can switch the incident LCP (RCP) THz waves into RCP (LCP) beams with an abrupt phase change (Pancharatnam–Berry phase). The optimized structure parameters of the meta-atom are $L=80\mathrm{\mu m}$, $W=42\mathrm{\mu m}$, $h_1=400\mathrm{\mu m}$, and $h_2=600\mathrm{\mu m}$. When the long axis of the meta-atom is placed along the $x$ axis, the transmittance between transverse magnetic (TM) and transverse electric (TE) waves is the same at $f=0.8\mathrm{THz}$ (working frequency), and the phase difference between the transmitted TE and TM waves is about $\pi$, which demonstrates that our designed meta-atom works as a quasi-half-wave plate [see the dark and green lines in Fig. 2(b)]. Under illumination of an LCP THz beam, the transmittance of RCP THz waves at 0.8 THz is nearly 90%, which can be defined as conversion efficiency (the ratio between the power of transmitted cross-polarized THz waves and incident power). The oscillations in transmission [see Fig. 2(b)] can be qualitatively explained as Fabry–Perot resonance, as in Ref. [44]. Figures 2(c)–2(f) show fabricated samples for generating the spin-independent vortex [Fig. 2(c)], the multiplexing of polarization-rotatable vortices in the longitudinal direction [Fig. 2(d)], multiple spatial dimensions [Fig. 2(e)], and the vortex with an extended focal depth [Fig. 2(f)].

As a guiding study, the generation of a polarization-independent THz vortex based on the designed metasurface [see Fig. 2(c)] is initially demonstrated. As depicted in Eq. (1), the phase profile induced by each meta-atom contains a spiral phase and focusing functionality for both LCP and RCP components. The corresponding focal length is theoretically designed as $f=8\mathrm{mm}$, and the topological charge of the polarization-independent THz vortex is $l_L=l_R=1$. Figure 3 shows numerical simulations and experimental results of our designed metasurface for generating the polarization-independent THz vortex. Under illumination of LCP THz waves, we calculate the corresponding electric-field intensity and phase distributions, as shown in Figs. 3(a₁) and 3(b₁). A converged vortex (at $f=0.8\mathrm{THz}$) can be observed after the metasurface, and the focal length is about $z=7.5\mathrm{mm}$. There is a slight discrepancy in focal length between the calculated result and theoretical case that can be attributed to the finite meta-atoms in numerical simulations (100 meta-atoms in the radial direction). The phase distribution illustrated in Fig. 3(b₁) shows a $2\pi$ phase change around point (0, 0, 7.5 mm), which demonstrates that the converged vortex possesses a topological charge of $l=1$. Figures 3(c₁) and 3(d₁) show the measured results of the fabricated sample [see Fig. 2(c)]. A converged vortex is generated after the sample with the focal length of $z=7.5\mathrm{mm}$ [see Fig. 3(c₁)]. The measured phase distribution in the focal plane ($z=7.5\mathrm{mm}$) shows that the phase change around point (0, 0, 7.5 mm) is also $2\pi$. Therefore, a converged vortex with a topological charge of one is experimentally demonstrated. In comparison with Figs. 3(a₁) and 3(b₁) and Figs. 3(c₁) and 3(d₁), the numerical simulations agree well with the measured results, except for a slight mismatch in field and phase distributions, which can be ascribed to sample imperfection. Figures 3(a₂)–3(d₅) show the numerical calculations and measured results of the electric-field intensity and phase distributions under the illumination of THz waves with polarization switched from left-handed elliptically polarized (LECP) state to RCP state. One converged vortex with a focal length of $z=7.5\mathrm{mm}$ and the corresponding topological charge of $l=1$ is observed after the geometric metasurface for the incidence of THz waves with an arbitrary polarization state. The designed geometric metasurface can focus both LCP and RCP components into a converged vortex (located at the same position) with the same topological charge, which demonstrates that the proposed approach can simultaneously focus and independently modulate the topological charges of the two helical components. The electric-field intensity distributions of the polarization-independent converged vortex in the $x–z$ plane are given in Appendix B.

The polarization-independent characteristic demonstrated above can indeed be realized by designing a metasurface with isotropic meta-atoms, i.e., symmetric cylindrical or square-shaped meta-atoms with different sizes, but that method loses a degree-of-freedom in the design space, due to the limited tailoring functionality (on the wavefront of EM waves) of isotropic meta-atoms. The proposed method based on geometric metasurfaces consisting of anisotropic meta-atoms cannot only independently focus two orthogonal helical components into a converged vortex, but also enable the capability to rotate the orientation of polarization, resulting in polarization-rotatable converged vortex/vortices. As depicted in Eq. (2), an angle ($\varphi$) of polarization rotation is introduced into the phase profile to rotate the polarization of the converged vortex. When $\varphi =90°$ and under illumination of $x$-polarized THz waves, a $y$-polarized converged vortex can be observed behind the designed geometric metasurface. The electric-field intensity and phase distributions of such a polarization-rotatable converged vortex are shown in Appendix C.1. By integrating the multiple focusing phases and spiral phases, and the polarization-rotation modulations into a geometric metasurface, the multiplexing of multiple converged vortices with polarization-rotatable functionality can be realized [see Eq. (3)]. Figure 4 shows the demonstration of two longitudinally multiplexed converged vortices with two orthogonal LP states. The designed phase parameters are $x_{1-L}=x_{1-R}=y_{1-L}=y_{1-R}=0$, $x_{2-L}=x_{2-R}=y_{2-L}=y_{2-R}=0$, $f_{1-L}=f_{1-R}=4.5\mathrm{mm}$, $f_{2-L}=f_{2-R}=8.5\mathrm{mm}$, ${\varphi }_1=0°$, and ${\varphi }_2=90°$. For the incidence of $x$-polarized THz waves, an $x$-polarized converged THz vortex is observed with a focal plane at $z=4.3\mathrm{mm}$, as shown in Fig. 4(a₁). In addition, another converged vortex is coaxially located at the focal plane of $z=7.9\mathrm{mm}$, but the corresponding polarization state of such a converged vortex is rotated counterclockwise with an angle of 90° in comparison to that of the incident THz waves, resulting in a $y$-polarized (converged) vortex [see Fig. 4(a₂)]. The simulated phase distributions in Figs. 4(a₃) and 4(a₄) show that both multiplexed converged vortices in the longitudinal direction have the same topological charge of one. The electric-field intensity distributions in the $x–z$ plane of these two converged vortices are illustrated in Figs. 4(a₅) and 4(a₆). The main intensity distribution of the left converged vortex ranges from $z=3.8\mathrm{mm}$ to $z=5\mathrm{mm}$, while it is located between $z=7\mathrm{mm}$ and $z=9\mathrm{mm}$ for the right one, which demonstrates two separated vortices in the longitudinal direction. Figures 4(b₁)–4(b₆) show the measured electric-field intensity and phase distributions. Under the illumination of an $x$-polarized THz beam, there is one $x$-polarized converged vortex and one $y$-polarized (converged) vortex after the fabricated sample. The focal lengths of both converged vortices corresponding to two orthogonal polarization states are $z=4.3\mathrm{mm}$ and $z=7.9\mathrm{mm}$, respectively [see Figs. 4(b₁) and 4(b₂)]. The topological charge of each vortex is one [see Figs. 4(b₃) and 4(b₄)]. The main electric-field intensity distribution of the vortex on the left (or right) side is located from 3.5 mm (6.0 mm) to 5.8 mm (9.5 mm), which also demonstrates that two separated vortices in the longitudinal direction are observed in experiment [see Figs. 4(b₅) and 4(b₆)]. The multiplexing of an $x$-polarized vortex and a $y$-polarized vortex in the transverse direction is supplied in Appendix C.2. In addition, the multiplexing of two vortices with two orthogonal helical states (LCP and RCP) in the longitudinal direction (or transverse direction) is given in Appendix C.3 and Appendix C.4.

To further demonstrate the unique and robust characteristic of our proposed approach, the multiplexing of vortex superposition in both transverse and longitudinal directions is calculated and experimentally investigated in Fig. 5. The structure parameters in this case are designed as $x_{1-L}=x_{1-R}=y_{1-L}=y_{1-R}=0$, $x_{2-L}=x_{2-R}=y_{2-L}=y_{2-R}=0$, $f_{1-L}=f_{1-R}=4.5\mathrm{mm}$, $f_{2-L}=f_{2-R}=8.5\mathrm{mm}$, ${\varphi }_1={\varphi }_2=90°$, $x_{3-L}=x_{3-R}=1.5\mathrm{mm}$, $x_{4-L}=x_{4-R}=-1.5\mathrm{mm}$$y_{3-L}=y_{3-R}=y_{4-L}=y_{4-R}=0$, $f_{3-L}=f_{4-R}=6.5\mathrm{mm}$, and ${\varphi }_3={\varphi }_4=0°$. Under illumination of $x$-polarized THz waves, two $x$-polarized converged vortices are symmetrically distributed along the $x$ axis, as shown in Figs. 5(a₁) and 5(a₂). One vortex is located at (1.5 mm, 0, 6.3 mm), and the other vortex is placed at ($-1.5\mathrm{mm}$, 0, 6.3 mm). The topological charge of each vortex is one [see Figs. 5(d₁) and 5(d₂)]. Since these two converged vortices possess the same focal plane ($z=6.3\mathrm{mm}$), the multiplexing of vortices in the transverse direction is realized. In contrast, when the $y$-polarized electric-field intensity is calculated and measured, two coaxially distributed converged vortices can be observed in the longitudinal direction [see Figs. 5(b₁), 5(b₂), 5(c₁), and 5(c₂)]. One vortex is focused at the focal plane of $z=4.3\mathrm{mm}$, while the other one is coaxially located at the focal plane of $z=7.9\mathrm{mm}$. The polarizations of these two longitudinal multiplexed converged vortices are orthogonal to those of the incident THz waves. Both $x$-polarized and $y$-polarized electric-field intensities in the $x–z$ plane [see Figs. 5(g₁), 5(g₂), 5(h₁), and 5(h₂)] show that two separated transversely distributed and longitudinally distributed converged vortices are generated by combining the modulations of focusing phases, spiral phases, and polarization-rotatable modulations. The multiplexing of multiple vortices with LP and CP states is also demonstrated and given in Appendix D. In addition, the multiplexing of multiple (converged) vortices with CP or LP states and different topological charges is supplied in Appendix E.

The converged vortex/vortices realized by combining the functionalities of a lens and a vortex plate into geometric metasurfaces can be potentially used in imaging, in particular, for edge detection and edge imaging, which will boost the efficiency of computer vision applications. However, traditional focal points or converged vortex/vortices are limited to the chromatic aberration due to the finite focal depth. A converged vortex with an extended focal depth can further tolerate focal shifts, and thus, it can be applied in high-tolerance edge imaging. As a proof of concept, a converged vortex with an extended focal depth is demonstrated, as shown in Fig. 6. The designed geometric metasurface can focus LP THz waves into two coaxially distributed converged vortices, and hence, one can generate two converged vortices with partial overlapping to each other, leading to the converged vortex with an extended focal length. The structure parameters are $x_{1-L}=x_{1-R}=y_{1-L}=y_{1-R}=0$, $x_{2-L}=x_{2-R}=y_{2-L}=y_{2-R}=0$, $f_{1-L}=f_{1-R}=7.0\mathrm{mm}$, $f_{2-L}=f_{2-R}=8.5\mathrm{mm}$, ${\varphi }_1=0°$, and ${\varphi }_2=90°$. As depicted in Fig. 6(a₁), an $x$-polarized vortex with the main electric-field intensity distribution (${|E_x|}^2$) ranging from 5.8 mm to 7.6 mm is observed. In addition, a $y$-polarized converged vortex is also generated after the designed geometric metasurface. The main electric-field intensity distribution (${|E_y|}^2$) is located between $z=7\mathrm{mm}$ and $z=9.1\mathrm{mm}$ [see Fig. 6(b₁)]. In comparison with Figs. 6(a₁) and 6(b₁), both the $x$-polarized and $y$-polarized converged vortices are partially overlapped with each other. As a result, a converged vortex with an extended focal length can be realized with the main electric-field intensity distribution (${\left|E_x\right|}^{2+}{|E_y|}^2$) ranging from 5.8 mm to 9.0 mm [see the total electric-field intensity distribution in Fig. 6(c₁)]. The electric-field intensity distributions in the $x–y$ plane for ${|E_x|}^2$, ${|E_y|}^2$, and ${|E_x|}^2+{|E_y|}^2$ are shown in Figs. 6(d₁)–6(h₃). Three $x$-polarized converged vortices and four $y$-polarized converged vortices can be observed in different $x–y$ planes, while seven converged vortices are revealed in the total electric-field intensity distributions. The measured results are shown in Figs. 6(a₂)–6(i₆). The $x$-polarized converged vortex is mainly distributed from 4.5 mm to 7.6 mm [see Fig. 6(a₂)], while the $y$-polarized converged vortex is mainly located between 6.2 mm and 10.4 mm [see Fig. 6(b₂)]. The total electric-field intensity distribution in experiment ranges from 4.5 mm to 10.6 mm, demonstrating a converged vortex with an extended focal length [see Fig. 6(c₂)]. The measured electric-field intensity distributions in the $x–y$ plane for ${|E_x|}^2$, ${|E_y|}^2$, and ${|E_x|}^2+{|E_y|}^2$ are shown in Figs. 6(e₁)–6(i₆), and they are also confirmed as a converged vortex with an extended focal length. The calculated and measured phase distributions in the $x–y$ plane are given in Appendix F.

In fact, a vortex enables the functionality in manipulating micro-particles. Our proposed geometric metasurfaces for generating converged vortices with well-defined focal planes provide a compact platform and an essential ingredient for preferably trapping and rotating micro-particles, especially for polarization-dependent micro-particles. Since a vortex can be applied to edge imaging/detection [61] based on a Fourier transform setup incorporating metasurfaces (that can generate a vortex), our proposed geometric metasurfaces for generating multiple converged vortices can be extended to realize multifocal edge imaging/detection [62]. Furthermore, the proposed geometric metasurface that can generate a vortex with an extended focal length can further successively mitigate aberrations, leading to applications in high-tolerance edge imaging/detection [63] in the longitudinal direction.

In summary, we have proposed and experimentally demonstrated an approach to the independent manipulation of two orthogonal helical components of THz waves. By integrating the focusing functionality and spiral phases into a geometric metasurface, the polarization-independent vortex was realized by the superposition of two orthogonal helical vortices with identical topological charges. Although the combination of geometric phase and propagation phase can be applied to independently manipulate these two orthogonal helical components, it suffers from the disadvantage of scanning a large number of structural parameters of meta-atoms to achieve locally customized objectives of spatial phase sampling. In addition, the proposed geometric metasurface enables the polarization-rotatable functionality by unitarily rotating each meta-atom. The multiplexing of multiple polarization-controllable vortices in the longitudinal/transverse direction or both longitudinal and transverse directions has been realized by introducing multiple focusing and spiral phases and polarization-rotatable functionality into the geometric metasurface. Finally, a vortex with an extended focal length was further demonstrated. This unique and robust approach provides a platform for integrating multiple functionalities into a single metasurface, leading to the straightforward design and miniaturization of multifunctional optical devices and systems.