Cylindrical vector beams (CVBs) with axially symmetric polarizations also called first-order vector beams [1] are generally classified into radially polarized vector beams (RPVBs), azimuthally polarized vector beams (APVBs), and hybridly polarized vector beams [2,3]. On account of the characteristic of inhomogeneous polarization, CVBs can be applied to multifarious fields, such as 3D polarization control [4], optical trapping [5,6], microscopic imaging [7–10], metrology [11–13], and optical communication [14–18]. Moreover, CVBs of the terahertz band, a wave band with plentiful applications of cutting-edge technology [19,20], can also be used for electron acceleration [21] and improve the coupling efficiency of the waveguide [22]. Radially and azimuthally polarized beams as two typical CVBs have been paid more and more attention. RPVBs have significant longitudinally polarized spots when tightly focused [23], making it more efficient for axial optical manipulation of nanoparticles as compared with linearly polarized light [24]. In addition, RPVBs can break the diffraction limit [25], promising for high-resolution lithography and optical sensing [26]. On the other hand, it was demonstrated that APVBs had stronger ability in transverse trapping, in contrast to RPVBs [27–29]. Several technical methods for generating such CVBs were developed, for example, retardation phase plates [23,30,31], subwavelength space-variant metallic gratings [32], and dielectric gratings [33]. Nevertheless, these methods are often plagued by problems, such as bulky size and fabrication difficulty.

Metamaterials as a kind of artificial materials different from natural materials provide an innovative method of optical manipulation [34,35]. Metasurfaces, two-dimensional forms of metamaterials, have also been widely applied in various aspects, such as ultrathin polarizer [36–38], metalens [39,40], and holographic imaging [41,42] for their superior manipulations of amplitude, polarization, and phase on the subwavelength scale. Moreover, emerging all-dielectric metasurfaces can solve the problem of ohmic losses in plasmonic metasurfaces [39,40,43–45]. Therefore, metasurfaces are promising for highly-compact low-loss CVB nanodevices, and, in fact, there have been works on CVB generation [46,47] or focusing [48,49] by metasurfaces. However, there are few works on simultaneously generating RPVBs and APVBs, meanwhile focusing them in multidimensional positions, although recently, the hybrid dual-focusing effect has been demonstrated (i.e., one focusing is a CVB spot, whereas the other is a homogeneously polarized scalar-beam spot). Recently, there have been some works on multifocus metalens, but the vast majority of them concentrated on multifocusing of the phase-vortex beam with a helical wavefront, which is different from our vector beams with inhomogeneous polarization rather than phase [50,51]. In our previous work, we have already experimentally demonstrated multidimensional and multifunctional metalens based on the photonic spin Hall effect [52] as well as spin-dependent dual-wavelength multiplexing metalens [53]. In this paper, we use the design criterion, so-called spin-decoupled phase control previously adopted for the generation of a single CVB [54] to realize multidimensional simultaneous focusing of RPVBs and APVBs based on an all-silicon metalens, indicating that this design criterion is sophisticated for complex vector-light metalenses. Attributed to the anisotropic bar structure, both polarization and wavefront of transmitted light are controlled by the combined modulation of geometric and dynamic phases. The polarization dependence of the vector beam on incident light determines that the focal positions of both vector beams can be easily exchanged by switching the polarization of incident light, which provides much convenience for practical optical operations. Optical forces of the focused RPVBs and APVBs on a glass sphere were calculated, confirming the contribution of the focusing RPVB (or APVB) to longitudinal (or transverse) optical force. The advantage of this paper for multidimensional simultaneous focusing of RPVBs and APVBs is that multichannel simultaneous manipulation (transverse and longitudinal manipulations) in a multidimensional position can be realized, that is, two kinds of CVBs can be focused or two different types of manipulation (the transverse and longitudinal trappings) can be realized at the same time. Meanwhile, these two kinds of manipulation will not be confined to the same point in space, which is different from a single focused beam of RPVB or APVB that can perform only one manipulation (transverse manipulation or longitudinal manipulation) at a time, and all of these controls will be stuck at the same point in space. In this way, the system of a multidimensional polarization-dependent dual-focusing CVB will enrich the optical control effect [55–57], and more possibilities will be brought to the integration of optical systems.

Figure 1(a) shows the elementary bar on a silicon substrate, a simple anisotropic structure usually adopted in literature [36,39,43–45,54]. The silicon bar has a height $h=200\mathrm{\mu m}$, parametric lengths $a$, $b$, and period $p=150\mathrm{\mu m}$. Silicon substrate plays a supporting role and does not affect the polarization distribution of transmitted light. Due to the dielectric characteristic of our metalens design, it is free of ohmic loss on interacting with incident light.

In this paper, we use the spin-decoupled phase control method [54] based on dynamic and geometric phases to generate multidimensional polarization-dependent dual-focusing CVBs. Excitation light is incident normally from the substrate end of the structure. The superposition of the dynamic and geometric phases is used to represent the phase shift due to the interaction between the structure and the incident light, $${\varphi }_{\mathrm{LR}}={\phi }_d-2\theta ,$$(1)$${\varphi }_{\mathrm{RL}}={\phi }_d+2\theta .$$(2)

${\varphi }_{\mathrm{LR}}$ (${\varphi }_{\mathrm{RL}}$) represents the phase shift attached to the original incident light where the incident component is right-handed circularly polarized (RCP), left-handed circularly polarized (LCP), and the transmitted component is LCP (RCP). ${\phi }_d$ represents the dynamic phase determined by the size of the bar, and $\theta$ is the rotation angle of the bar [Fig. 1(b)]. As shown in Fig. 1(c), the linearly polarized incident light with amplitude $E$ and arbitrary polarization orientation $\beta$ (the angle with respect to the $x$ axis) is divided into RCP and LCP components with same amplitude of $\frac{E}{2}$ but opposite initial phases of $\beta$ and $-\beta$, respectively. Upon transmission through the metasurface, RCP and LCP components are converted into cross-polarized LCP and RCP, respectively. As a consequence, these two converted circularly polarized beams may be reconstituted into a linearly polarized beam on the condition that we design the unit cell of the structure to be a perfect half-wave plate. In this ideal circumstance, both circularly polarized components of the incidence are completely converted to orthogonal circular polarizations upon transmission as shown in Fig. 1(c).

For RCP incident component $[E_R^{\mathrm{in}}=\frac{E}{2}{e}^{i\beta }(\begin{array}{c}1\\ -i\end{array}\left)\right]$, it is transmitted to be LCP with $$E_{\mathrm{LR}}=A\frac{E}{2}{e}^{i\beta }{e}^{i{\varphi }_{\mathrm{LR}}}\left(\begin{array}{c}1\\ i\end{array}\right),$$(3)and for the LCP incident component $[E_L^{\mathrm{in}}=\frac{E}{2}{e}^{-i\beta }(\begin{array}{c}1\\ i\end{array}\left)\right]$, it is transmitted to be RCP with $$E_{\mathrm{RL}}=A\frac{E}{2}{e}^{-i\beta }{e}^{i{\varphi }_{\mathrm{RL}}}\left(\begin{array}{c}1\\ -i\end{array}\right),$$(4)where $A$ in Eqs. (3) and (4) represents a constant, ${\varphi }_{\mathrm{LR}}$ and ${\varphi }_{\mathrm{RL}}$ represent the additional phase shifts after the incident light passes through the metasurface. Consequently, the total transmission can be written as $$\begin{gathered} E_t=E_{\mathrm{LR}}+E_{\mathrm{RL}}=A\frac{E}{2}{e}^{i\beta }{e}^{i({\phi }_d-2\theta )}\left(\begin{array}{c}1\\ i\end{array}\right)+A\frac{E}{2}{e}^{-i\beta }{e}^{i({\phi }_d+2\theta )}\left(\begin{array}{c}1\\ -i\end{array}\right) \\ =A\frac{E}{2}{e}^{i{\phi }_d}\left[e^{i(\beta -2\theta )}\right(\begin{array}{c}1\\ i\end{array})+e^{i(2\theta -\beta )}(\begin{array}{c}1\\ -i\end{array}\left)\right] \\ =A\frac{E}{2}{e}^{i{\phi }_d}\left(\begin{array}{c}{e}^{i(\beta -2\theta )}+e^{i(2\theta -\beta )}\\ i{e}^{i(\beta -2\theta )}-i{e}^{i(2\theta -\beta )}\end{array}\right) \\ =2A\frac{E}{2}{e}^{i{\phi }_d}\left(\begin{array}{c}\cos (\beta -2\theta )\\ -\sin (\beta -2\theta )\end{array}\right) \\ =AE{e}^{i{\phi }_d}\left(\begin{array}{c}\cos (2\theta -\beta )\\ \sin (2\theta -\beta )\end{array}\right). \end{gathered}$$(5)

Equation (5) indicates that the transmitted light is linearly polarized along a direction angle ($2\theta -\beta$) with respect to the $x$ axis.

In order to achieve a transmission beam with radial polarization ($2{\theta }_r-\beta =\alpha$), where $\alpha =a\tan 2(y,x)\in (-\pi ,\pi ]$ is the polar angle, and ${\theta }_r$ represents the rotation angle of the bar. Then, for this RPVB case, we get $${\theta }_r=\frac{\alpha +\beta }{2}.$$(6)

To achieve azimuthally polarized transmission, i.e., $2{\theta }_a-\beta =\alpha -90°$, ${\theta }_a$ represents the rotation angle of the bar for the APVB case, which, thus, satisfies $${\theta }_a=\frac{\alpha -90°+\beta }{2}.$$(7)

It is noted that for the respective bar with determined rotation direction ${\theta }_r$ (or ${\theta }_a$), changing the incident polarization angle $\beta$ in Fig. 1(c) to its orthogonal direction, the transmitted light will be switched from a radially (or azimuthally) polarized to an azimuthally (or radially) polarized CVB.

In addition, as the dynamic phase, ${\phi }_d$ in Eq. (5) plays the role of controlling the phase wavefront of transmitted light. In order to realize highly-compact focusing CVB, ${\phi }_d$ needs to meet the following equation: $${\phi }_d=\frac{360°}{\lambda }[\sqrt{{(x-x_1)}^2+{(y-y_1)}^2+f^2}-f],$$(8)where $\lambda$ represents the wavelength of incident light and $f$ represents the focal length, that is, the distance between the metasurface and the focal point ($x_1$, $y_1$, $f$). According to the above design method, we can realize multidimensional polarization-dependent dual-focusing CVB.

Before assembling metalens with the corresponding properties, elementary bars with suitable sizes are optimized not only to satisfy the half-wave plate condition (phase difference $\pi$ and equal amplitude for the orthogonal polarization components of transmission) for CVB polarization modulation, but also to satisfy the evolution of the dynamic phase in the range of $0–2\pi$ for CVB focusing. As shown in Fig. 2, the database of transmission amplitude and phase shift ($t_{xx}$, $t_{yy}$, ${\phi }_{xx}$, ${\phi }_{yy}$) in dependence of bar size ($a$ and $b$) at 1-THz incidence is established, where $t_{xx}$ ($t_{\mathit{yy}}$) and ${\phi }_{xx}$ (${\phi }_{yy}$) represent the transmission amplitude and the additional phase shift of transmitted $x$-polarized ($y$-polarized) light at the incidence of $x$-polarized ($y$-polarized) light, respectively. By choosing bar sizes to meet $t_{yy}=t_{xx}$ and $|{\phi }_{yy}-{\phi }_{xx}|=\pi$, we pick eight bars out with dimensions of $a=87.35$, 49.6, 49.52, 45.31, 30.9, 103.8, 79.45, 74.74 μm and, correspondingly, $b=30.9$, 103.8, 79.45, 74.74, 87.35, 49.6, 49.52, 45.31 μm.

In order to verify whether these eight bars meet the conditions of half-wave plates and explore the working bandwidth of the bars, amplitudes ($t_{xx}$, $t_{yy}$) and phase shifts (${\phi }_{xx}$, ${\phi }_{yy}$) together with their respective differences as a function of the incident frequency are shown in Figs. 3(a)–3(f). When the incident frequency is 1 THz, obviously, for each of the eight bars, the difference between phase shifts of transmission is always about $\pi$. Meanwhile, the amplitude difference between $t_{xx}$ and $t_{yy}$ remains around zero. In addition, it can be found from Fig. 3(b) that at 1 THz, the dynamic phases ${\phi }_d={\phi }_{xx}$ do satisfy a gradient variation with a step of $\frac{\pi }{4}$ and cover the full $2\pi$-modulation preference. Unfortunately, these bars cannot maintain superior half-wave plate characteristics over a wide spectral range, that is, the working bandwidth is only about 0.1 THz.

Next, with the amplitude and phase databases for eight bars, we design the arrangement of elements based on the aforesaid design principle, which is feasible to realize multidimensional dual-focusing CVB.

Figure 4 illustrates the schematic of a transverse dual-focusing CVB system. For simplicity, the incident light is assumed to be $x$ polarized, which is split into double CVB focal points at the coordinates of ($x_1$, 0, $f$) and ($x_2$, 0, $f$). The former one is APVB, whereas the latter is RPVB, and we set $x_1=-500\mathrm{\mu m}$, $x_2=500\mathrm{\mu m}$, and focal length $f=1200\mathrm{\mu m}$. The enlarged view of the metalens (Fig. 4 inset) is divided into blue and pink regions, corresponding to focal spots with azimuthal and radial polarizations at $x_1$ and $x_2$, respectively. The rotation angles of the bar and dynamic phase in these two regions are ${\theta }_a$ and ${\phi }_{d1}$ for the APVB spot at $x_1$ (or ${\theta }_r$ and ${\phi }_{d2}$ for the RPVB at $x_2$), which satisfy Equations (6)–(8), respectively.

Figure 5 shows the finite-difference time-domain simulation results of the transverse dual-focusing CVB system. It can be clearly found from Fig. 5(a) that there are two CVB focal spots separated perpendicular to the propagation direction of incident light, namely, the transverse $x$ direction where polarization distributions are azimuthally polarized (focus $x_1$) and radially polarized (focus $x_2$), respectively. In order to make the polarization distribution at the focal points clearer, Fig. 5(a) has been appropriately enlarged compared with Figs. 5(c)–5(f). The radial polarization distribution at focus $x_2$ is slightly distorted because there is, in fact, an angle $\gamma$ between the intercepted $x–y$ plane [black dashed line in Fig. 5(b)] and the section where the true radial polarization is located [pink solid line in Fig. 5(b)], and the polarization distribution at focus $x_1$ can also be understood in this way. Figure 5(b) shows the $E$-field intensity distribution on the $x–z$ plane with $y=0\mathrm{\mu m}$. Figures 5(c)–5(f) present the $E$-field intensity map on the $x–y$ plane with $z=1200\mathrm{\mu m}$. The doughnut shape in Fig. 5(c) confirms the focusing characteristic of APVB. Figures 5(d) and 5(e) illustrate that, for the APVB focusing at $x_1$, ${|E_x|}^2$ (or ${|E_y|}^2$) is mainly distributed along the $y$ (or $x$) direction, i.e., perpendicular to the polarization direction, verifying the focus at $x_1$ is azimuthally polarized. Inversely, the ${|E_x|}^2$ (or ${|E_y|}^2$) component for focus $x_2$ is mainly distributed along the $x$ (or $y$) direction, i.e., parallel to the polarization direction, indicating the radially polarized characteristic of focus $x_2$. Thus, the transverse dual-focusing CVB effect is achieved. The phenomenon that the longitudinal component of the focal spot of radially polarized light [Fig. 5(f)] does not dominate the intensity distribution of the focal spot [Fig. 5(c)] as expected is due to the fact that the radially polarized beam is not focused tightly enough [1,23].

The longitudinal dual-focusing CVB system is illustrated in Fig. 6. Similarly, $x$-polarized light is incident normally onto the silicon substrate, it transmits through the two regions (pink and blue) of the metalens, and then longitudinally generates two CVB focusing spots with radial and azimuthal polarizations at positions (0, 0, $f_1$) and (0, 0, $f_2$), respectively, where we set $f_1=1200\mathrm{\mu m}$ and $f_2=3300\mathrm{\mu m}$. The focal lengths of the system have been predetermined in the theoretical design of the structure. As long as the incident condition (i.e., the incident light frequency is 1 THz, and the incident light is normally incident from the substrate end of structure) is satisfied in the experiment, the desired focal spots will be generated at specific focal positions. Therefore, no additional manipulation is required to control the focal lengths experimentally.

Figures 7(a) and 7(b) illustrate the $E$-field distribution of the longitudinal dual-focusing CVB system, verifying the radial and azimuthal distributions, respectively. Figures 7(c)–7(e) illustrate the $E$-field intensity map on the $x–z$ plane with $y=0\mathrm{\mu m}$. In Fig. 7(c), two focusing spots are separately generated at $z=1200$ and 3300 μm. The ${|E_y|}^2$ map on the $x–z$ plane in Fig. 7(d) indicates the APVB characteristic for the focal spot at $z=3300\mathrm{\mu m}$, whereas, the ${|E_z|}^2$ map in Fig. 7(e) shows the RPVB tight-focusing characteristic at $z=1200\mathrm{\mu m}$.

As can be found from the principle described in Section 2, the polarization distribution at the focal position can be switched from radial/azimuthal polarization to azimuthal/radial polarization by converting the polarization direction of the incident light orthogonally. Taking the longitudinal dual-focusing metalens as an example, converting the incident light from $x$ to $y$ polarization will lead to an exchange between the azimuthally and the radially polarized focal spots.

Figures 8(a) and 8(b) show simulation results to confirm the polarization dependence of our longitudinal dual-focusing CVB metalens. Obviously, the original focal spot with radial/azimuthal polarization does interconvert into a focal spot with orthogonal azimuthal/radial polarization. Figures 8(c)–8(h) can verify the azimuthal and radial polarization characteristics at spots $f_1$ and $f_2$, respectively, as discussed earlier.

Taking the longitudinal dual-focusing CVB system as an example, based on the ray tracing method [24,27,58], optical forces, corresponding to the tight-focusing fields of APVB and RPVB, were calculated to investigate the characteristics of tiny particle manipulation by such dual-focusing CVBs, which should contribute to multidimensional vector-light tweezers.

For simplicity, we consider the optical forces stressed by the focused RPVB (or APVB) at focal point $f_1$ in the longitudinal dual-focusing system under the $x$- (or $y$-) polarized incidence. A glass sphere with refractive index of 1.56 and radius of 300 μm was used as the stressed object as shown in Fig. 9(a).

Figure 9(b) illustrates the calculated longitudinal optical force $F_z$ of RPVB focusing at $f_1$ under $x$-polarized incident light. Figures 9(c) and 9(d) illustrate the calculated transverse optical forces $F_x$ and $F_y$ of APVB focusing at $f_1$ under $y$-polarized incident light. The insets illustrate the planar $E$-field distributions, where $\mathrm{\Delta }z$ and $\mathrm{\Delta }x$ (or $\mathrm{\Delta }y$) represent the displacement between the sphere center and the focal point $f_1$ along the longitudinal and transverse directions, respectively. It can be clearly found that the focused RPVB and APVB contribute to the longitudinal and transverse trapping characteristics on the glass sphere, respectively. Taking Fig. 9(c) as an example, when the sphere is deviated from the focal point ($\mathrm{\Delta }x\ne 0$), $F_x$ shows nonzero values. Depending on the offset in different directions, the forces are always toward the focus. That is, the sphere will always be subjected to a force pulling it back to the equilibrium position until it finally stabilizes there, so-called optical trapping. The absorption of glass spheres may have a certain influence on the simulation results, but it will not affect the trend of the optical force, which acts as the trapping force. What needs to be paid attention to is that for the small longitudinal component as discussed in Fig. 5(f) in Section 3.B, the longitudinal gradient force $F_z$ that we calculated in Fig. 9(b) may not be able to overcome the gravity force of the glass sphere and trap the glass sphere longitudinally in the experiment, which has nothing to do with the size of the glass sphere because both optical and gravitational forces are dependent on the particle radius as $r^3$ ($r$ represents the radius of the glass sphere). Therefore, further research and improvement are still needed to experimentally realize longitudinal trapping. Nevertheless, such a tiny trapping force is still meaningful for situations, such as bonding rather than levitation. Moreover, the trapping effect is not limited to the considered silica particle, and it is, in principle, also applicable for other absorptive objects enduring the focusing vector fields. It is worth discussing the special case that the incident light is neither $x$ polarized nor $y$ polarized; instead, the incident light is the superposition of $x$-polarized incidence and $y$-polarized incidence in which the transmitted light will exert both longitudinal and transverse optical forces on the glass sphere at the same time if the interference of optical forces can be ignored.

In this paper, the multidimensional dual-focusing CVB metalens with an all-silicon configuration was proposed under the design principle of spin-decoupled phase control from which a pair of focused vector beams with radial and azimuthal polarizations can be generated in the transverse as well as the longitudinal directions. Furthermore, we calculated the optical forces of such APVB and RPVB focusing spots exerted on a glass sphere and, consequently, revealed the transverse and longitudinal trapping characteristics, attributed to the hollow singularity of APVB and to the convergence singularity of RPVB in the case of focusing, respectively. Moreover, the polarization distributions of the dual-focusing CVB metalens can be interchanged by simply switching the polarization direction of incident light, providing much convenience in multidimensional multichannel optical trapping by focused vector beams. What needs to be pointed out is that our paper only considered forces of APVB and RPVB acting on the glass sphere and numerically proved the transverse and longitudinal trapping forces of these two focused vector beams on the glass sphere. Further research is needed to trap the sphere experimentally. For example, for the focused RPVB in this paper, its longitudinal trapping force is not large enough to fight against other forces, if any, such as the gravity force of the glass sphere or air resistance, and considering the difficulty of providing incident power strong enough in the experiment if we really want to experimentally achieve the longitudinal trapping of the sphere against its gravity with the focused RPVB, Some effort, such as adjusting the operating band to the visible/near-infrared band is necessary to improve the trapping force. Lastly, it is noted that the design strategy, so-called spin-decoupled phase control, can be extended to high-order vector beams, promising for the high integration of the optical system with potential applications in polarization-dependent optical communication, information encryption, imaging, and so on.