Conventional optical elements achieve specific optical functions based on the gradual phase changes accumulated along the propagation path, leading to a large form factor that is not compatible with the miniaturized, lightweight, and compact systems [1,2]. Therefore, planar optics components have received extensive attention in recent years benefiting from the compact and ultrathin design and excellent manipulation capability in multi-dimensional physical parameters [3]. The generalized laws of reflection and refraction offer the theoretical explanation for the principle of unique performance in planar optics components, leading to arbitrary wavefront modulation due to the phase discontinuities in light propagation [4]. In general, the phase discontinuities can be implemented by geometric phase, dynamic phase, and so forth. Compared with the dynamic phase arising from the optical path difference in propagation, the geometric phase, also known as Pancharatnam–Berry (PB) phase, originates from the photonic spin–orbit interaction in asymmetric anisotropic structures [5,6]. In contrast with the dynamic phase that is adjusted by the equivalent refractive index of the material, the PB phase is a broadband non-dispersion phase modulation method that is only related to the rotation angle of the anisotropic structure [7]. Therefore, PB phase has been widely used in multitudinous practical applications due to its precise phase control ability and robustness against fabrication tolerances.

Past decades have witnessed extensive efforts on geometric phase. Among these applications, metasurfaces are the most representative planar optical components with promising potential and application value. Metasurfaces, the equivalent artificial two-dimensional (2D) metamaterials, generally comprise plentiful sub-wavelength nanostructures arranged in accordance with specific orders, which can be applied to accurately control multiple physical characteristics of the light field, including polarization, amplitude, and phase [8,9]. Geometric phase metasurfaces have been widely used in optical lenses [10,11], catenary optics [12,13], the spin Hall effect [14], holograms [15,16], vortex beam generators [17,18], and so on. Recently, generalized geometric phase in rotationally symmetric meta-atoms has been demonstrated to enrich the understanding of the geometric phase as well as light–matter interaction in nanophotonics [19]. However, although metasurfaces obtain precise and flexible control capabilities at sub-wavelength scales, the preparation of the metasurfaces comprising plentiful nanopillars or nanoholes is facing huge challenges especially in large-area manufacturing [20]. Currently, numerous nanopatterning techniques have been extensively applied in the nanopatterning, such as focused ion-beam (FIB) milling, electron-beam lithography (EBL), ion-beam lithography (IBL), and atomic layer deposition (ALD). These techniques suffer from high cost and low throughput and are not compatible with the low-cost, high-efficiency, and large-area demands in mass nanofabrication [21].

Actually, the anisotropic liquid crystals (LCs) can also realize a series of impressive functions by utilizing the designed method in geometric phase metasurfaces. To a certain extent, LCs can achieve the arbitrary wavefront manipulation by introducing discontinuous phase gradients at the interface between the two media, which is consistent with the generalized laws of reflection and refraction; thus, the planar LC optical elements can be considered as a kind of generalized metasurface. Here we define the proposed device as a polymer LC metasurface, which is the same as the previous report [22]. On the one hand, although optical lithography has attracted a great deal of interest and shown promising applications in low-cost and large-area nanofabrication, the expensive laser setups and complex processing inevitably restrict their applications. The gradual maturity of LC production lines provides the low-cost and large-scale production for LC metasurfaces, which can be adopted to offer a cost-effective and high-throughput method for the fabrication of metasurfaces. On the other hand, compared to conventional diffraction optical elements, LCs also revealed unparalleled superiority in terms of the work efficiency, dynamic tunability, and processing difficulty. So far, LCs have been widely applied in displays, optical imaging, holography, and so forth. In particular, LC displays and spatial light modulators are the most representative and widespread applications in modern engineering and technology. Recently, geometric phase LCs based on light alignment technology have received tremendous attention and impressive progress in multifarious application scenarios, including near-eye display [23], virtual reality/augmented reality (VR/AR) display [24], integral imaging [25], spiral phase modulation [26,27], beam splitting [28,29], polarization gratings [30], telescope [31], and so on. It is worth mentioning that one of the most widely used applications is LC metalenses, which have received extensive attention due to the planar design with the extreme optical performance and low-cost manufacturing. The planar high-efficiency LC metalenses also have great superiority in advanced optical systems due to the high integration and light weight requirements. Therefore, LC metalenses are of great significance and of great potential in advanced optical imaging systems.

As one of the most representative applications in planar optical elements, polarization-multiplexing metasurfaces have received sufficient attention and research to achieve the multitudinous functions, including lenses [32,33] and holography [34], while these polarization-multiplexing devices cannot balance bandwidth, efficiency, and multifunctionality efficiently. Among these applications, the polarization-multiplexing lenses based on PB phase obtain high error tolerance in nanofabrication, and the ultrabroadband characteristics can greatly broaden the working bandwidth of the designed lenses. In addition, polymerized LC metalenses, as a representative planar optical component based on PB phase, achieve nearly 100% polarization conversion efficiency once the thickness of the LC layer can satisfy the half-wave condition. Previously reported LCs were devoted to the multi-foci functions such as a PB-phase-based polarization-multiplexing bifocal lenses [35] or LC integrated metalenses [36], while the different polarization-switchable focusing behaviors are rarely realized, which greatly limits the application scenarios of the proposed polarization-multiplexing LC lenses.

In this paper, broadband high-efficiency polymerized liquid crystal metasurfaces (BHPLCMs) are proposed for polarization-switchable focusing functions in the visible based on photonic spin–orbit interaction. Our proposed method can achieve excellent focusing performance simultaneously under LCP and RCP incident light based on wavefront engineering and holographic synthesis within the working bandwidth, leading to the enormous progress in the practical application scenarios. The experimental results are consistent with the simulated results, which can indicate that the spin-multiplexed method can be implemented to achieve the polarization-switchable functions. The proposed BHPLCMs are expected to have promising applications in microscopy, VR/AR, and optical interconnections.

The designed BHPLCMs are composed of plentiful anisotropic LC molecules with different rotation directions deposed on a glass substrate. In this work, two different BHPLCMs are designed to achieve different focusing functions: one can realize the diffraction-limited focusing to sub-diffraction focusing with 0.8 times Abbe diffraction limit, and the other can achieve the diffraction-limited focusing to the focusing vortex beam. Therefore, the planar polarization-multiplexing LC-polymer metasurfaces are expected to have flexible imaging functions based on the polarization-multiplexing method. As shown in Fig. 1, the proposed BHPLCMs can achieve different focusing modes under the different incident light with the left-circularly polarized (LCP) and right-circularly polarized (RCP) states.

Anisotropic materials, as an important characteristic of LCs, are an indispensable approach to control the polarization information of the incident light effectively. Theoretically, the design of PB-phase-based LC metasurfaces can be summarized as deriving the corresponding rotation angle to achieve the ideal geometric phase modulation of LC molecules through the Jones matrix, which establishes the relationship between the orientation angle ($\theta$) and phase modulation as [37] $$J_{xy}=R(\theta )J_{uv}R(-\theta )=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{cc}{t}_u& 0\\ 0& t_v\end{array}\right]\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right],$$(1)where the main axis directions in the localized coordinate system of the anisotropic LC molecules are located at $u$ and $v$, the angle between $u$ and the $x$ axis is the orientation angle ($\theta$), and the transmission complex amplitudes along the two main axis directions are $t_u$ and $t_v$, respectively. Subsequently, for the incident circularly polarized electromagnetic wave, the output electric field after passing through the anisotropic LC molecules can be expressed as $$\left[\begin{array}{c}{E}_x\\ E_y\end{array}\right]=\frac{J_{xy}}{\sqrt{2}}\left[\begin{array}{c}1\\ i\delta \end{array}\right]=\frac{1}{2\sqrt{2}}\{(t_u+t_v)\left[\begin{array}{c}1\\ i\delta \end{array}\right]+(t_u-t_v)\exp (2i\delta \theta )\left[\begin{array}{c}1\\ -i\delta \end{array}\right]\},$$(2)where $\delta =\pm 1$, corresponding to the circumstance under LCP and RCP incident light. After the incident circularly polarized electromagnetic waves interact with anisotropic metasurface structure, the electromagnetic field contains two different constituent parts with different polarization states, i.e., the original main polarization electromagnetic wave with complex amplitude $(t_u+t_v)/2\sqrt{2}$ and the orthogonal polarization electromagnetic wave with complex amplitude $(t_u-t_v)\exp (2i\delta \theta )/2\sqrt{2}$. Besides, the cross-polarized electromagnetic wave carries an additional geometric phase of $2\delta \theta$. Accordingly, the phase retardation $\mathrm{\Gamma }$ resulting from the birefringence of LCs can be expressed as [38] $$\mathrm{\Gamma }=\frac{2\pi (n_{\mathrm{eff}}-n_o)d}{\lambda },$$(3)where $d$ is the thickness of the LCs, $\lambda$ is the free-space wavelength of the incident light, $n_o$ is the ordinary refractive index, and $n_{\mathrm{eff}}$ is the extraordinary refractive index. Furthermore, the thickness of the LC layer can be set as a theoretical value to satisfy the half-wave condition, which is $\mathrm{\Gamma }=(2a+1)\pi$ ($a$ is an integer). In general, high-efficiency PB-based LCs can be implemented by specially choosing the refractive index of LCs and optimizing the thickness $d$, and the polarization conversion rate (PCR) can reach 100% in this case. Here, the PCR is defined as $\mathrm{PCR}=T_{\text{cross}}/(T_{\text{cross}}+T_{\mathrm{co}})$, where $T_{\mathrm{cross}}$ and $T_{\mathrm{co}}$ represent the cross-polarization transmittance and co-polarization transmittance.

Hence, the required ideal phase wavefront can be designed based on the geometric $\text{phase}\pm 2\theta$ in the cross-polarized electromagnetic wave, such as diffraction-limited focusing, sub-diffraction focusing, and vortex beams. According to the previous reports, an ideal focusing phase can be expressed as $${\phi }_{\mathrm{focus}}(x,y)=-\frac{2\pi }{\lambda }[\sqrt{f^2+{(x-x_0)}^2+{(y-y_0)}^2}-f],$$(4)where $\lambda$ represents the illumination wavelength, $r=\sqrt{{(x-x_0)}^2+{(y-y_0)}^2}$ represents the distance from arbitrary pixel $(x,y)$ to the center $(x_0,y_0)$ of the designed devices, and ($x_0$, $y_0$, $f$) is the position of the designed foci. In this work, $(x_0,y_0)$ is set as the center of the BHPLCMs that is (0,0). Super-oscillation, as a promising far-field label-free super-resolution method, is a phenomenon where the band-limited function is able to oscillate much faster than the highest Fourier components [39], which originates from delicate interference of a digital metasurface wave [40]. In general, super-oscillatory lenses have been considered as a promising non-invasive universal imaging technique without the luminescence of object [41,42]. To further achieve the sub-diffraction focusing, we further superimpose the binary super-oscillatory phase on the basis of the focusing phase to redefine the light field at the focal plane position. In other words, the fine interference of the light field had been realized according to the principle of super-oscillation. Therefore, according to the Fresnel diffraction integral, the point spread function (PSF) of the system could be approximately expressed as [43] $$I(\rho )\propto {\left(\frac{1}{\lambda f}\right)}^2{\left|{\int }_0^R\exp [i{\phi }_{\text{binary}}(r)]J_0\left(\frac{2\pi r\rho }{\lambda f}\right)r\mathrm{d}r\right|}^2,$$(5)where $J_0$ is the zero-order Bessel function, $R$ is the radius of the super-oscillatory lens, $\rho$ is the radial coordinate on the focal plane, and ${\phi }_{\text{binary}}$ represents the corresponding binary super-oscillatory phase. The required super-oscillatory phase can be optimized by adding some specific constraints, e.g., the constraints on super-resolution capability by limiting the position of the first zero point or the full width at half-maximum (FWHM), and the constraints on sidelobe intensity through restricting the ratio $M$ of the sidelobe intensity to the mainlobe intensity in the corresponding field of view (FOV). Here, this process can be performed by an optimized algorithm, such as the linear programming method (LPM) [44], particle swarm optimization (PSO) [45], and genetic algorithm (GA) [46]. The detailed analysis for the optimization of binary super-oscillatory phase is given in the Appendix A. Consequently, the required phase profile for sub-diffraction focusing can be written as follows: $${\phi }_{\mathrm{SOL}}(x,y)=-\frac{2\pi }{\lambda }(\sqrt{f^2+x^2+y^2}-f)+{\phi }_{\text{binary}}(x,y).$$(6)Here, the binary super-oscillatory phase ${\phi }_{\text{binary}}$ is optimized by the LPM according to the Eqs. (A1) and (A2). Meanwhile, the wavefront phase of the vortex beam generator can be expressed as $${\phi }_{\text{vortex}}(x,y)=-\frac{2\pi }{\lambda }(\sqrt{f^2+x^2+y^2}-f)+l·\arctan \left(\frac{y}{x}\right),$$(7)where $l$ denotes the topological charge of the vortex beam generator and is set as 1 here. In conclusion, the required phase for preset functions can be realized once the relationship between the rotation angle of the LC molecules and the required phase profile is established.

Based on the above design scheme, polarization-multiplexing LC metalenses are proposed to further improve the flexibility in the actual application scenarios, and the key to realizing such polarization-multiplexing LC metasurfaces mainly lies in the capability of a single optical element to simultaneously achieve the different function under different incident light. Theoretically, the required phase distribution for the polarization-multiplexing LC lens is governed by $${\phi }_{\mathrm{BHPLC}\mathrm{M}}(x,y)=\arg \{A_1\exp [i{\phi }_{\mathrm{LCP}}(x,y)]+A_2\exp [i{\phi }_{\mathrm{RCP}}(x,y)]\}.$$(8)

In this work, we designed two BHPLCMs with different focusing modes; for ${\mathrm{BHPLCM}}_1$, ${\phi }_{\mathrm{LCP}}$ and ${\phi }_{\mathrm{RCP}}$ should be satisfied by the following phase profile: $$\{\begin{array}{l}{\phi }_{\mathrm{LCP}}(x,y)={\phi }_{\mathrm{focus}}(x,y)=-\frac{2\pi }{\lambda }(\sqrt{f^2+x^2+y^2}-f)\hfill \\ {\phi }_{\mathrm{RCP}}(x,y)=-{\phi }_{\mathrm{SOL}}(x,y)=-[-\frac{2\pi }{\lambda }(\sqrt{f^2+x^2+y^2}-f)+{\phi }_{\text{binary}}(x,y)]\hfill \end{array}.$$(9)

The binary super-oscillation phase ${\phi }_{\text{binary}}$ is optimized by the LPM to work on a resolvable ability to 0.8 times of Abbe diffraction limit, combining with the constraints of low sidelobe intensity ($M=0.08$) to avoid the huge noise in practical imaging, and $\pi$-phase-jump positions of the ${\mathrm{BHPLCM}}_1$ along the radial direction are $r_1=0.183R$, $r_2=0.308R$, $r_3=0.514R$, and $r_4=0.694R$.

For ${\mathrm{BHPLCM}}_2$, ${\phi }_{\mathrm{LCP}}$ and ${\phi }_{\mathrm{RCP}}$ can be denoted as $$\{\begin{array}{l}{\phi }_{\mathrm{LCP}}(x,y)={\phi }_{\mathrm{focus}}(x,y)=-\frac{2\pi }{\lambda }(\sqrt{f^2+x^2+y^2}-f)\hfill \\ {\phi }_{\mathrm{RCP}}(x,y)=-{\phi }_{\text{vortex}}(x,y)=-[-\frac{2\pi }{\lambda }(\sqrt{f^2+x^2+y^2}-f)+l·\arctan \left(\frac{y}{x}\right)]\hfill \end{array}.$$(10)

Here, the designed wavelength $\lambda$, the focal length $f$, and the entrance pupil diameter $R$ of the proposed BHPLCMs are all fixed as 633 nm, 450 mm, and 5.4 mm, respectively. At the same time, $A_1$ and $A_2$ are adopted as 1 and $\sqrt{2}$ to balance the intensity of the focusing light on the focal plane under the incident light with different polarization. Consequently, the different focusing functions can be changed without changing the other optical elements as long as the quarter-wave plate is rotated to switch the incident light from LCP to RCP.

After clarifying the required wavefront phase of the proposed BHPLCMs, the rotation angle of the LC molecules at the corresponding position can be determined according to the geometric phase theory. Subsequently, the required phase profile is discretized into 18-order phase (i.e., 0, $2\pi /9$, $\pi /3$, $4\pi /9$, $5\pi /9$, $2\pi /3$, $7\pi /9$, $8\pi /9$, $\pi$, $10\pi /9$, $11\pi /9$, $4\pi /3$, $13\pi /9$, $14\pi /9$, $5\pi /3$, $16\pi /9$, $17\pi /9$) to reduce the influence of discontinuous phases on the final result, and the designed BHPLCMs can be fabricated via photoalignment technology. Taking the ${\mathrm{BHPLCM}}_1$ as an example, the optical performance of the ${\mathrm{BHPLCM}}_1$ designed as 18-order phase is basically consistent with the theoretical result without discretization, while there are increasing virtual focal spots as the discrete precision becomes lower, as shown in Fig. 9 in Appendix B; thus, the focusing efficiency will be decreased due to the inevitable loss in the virtual focal spots. Non-invasive optical alignment technology had been widely used in LC manufacturing due to the superiority in avoiding arbitrary mechanical damage, electrostatic charge, and dust contamination. The fabrication process should be carried out in a dust-free environment to produce BHPLCMs with high precision and excellent quality, and the entire process can be summarized into five steps as shown in Fig. 2(a). First, the glass substrate should be cleaned to confirm the high performance of the LCs after being ultrasonic cleaned, sufficiently heated, exposed to ultraviolet (UV), and blown by compressed air. Then, a solution comprising common sulphonic alignment agent azo-dye (SD1, 0.5%) and dimethylformamide (DMF, 99.5%) will be dropped on the glass substrate uniformly to form an alignment layer. Third, a digital micromirror device (DMD) is utilized to achieve the required rotation angles. Fourth, a solution consisting of RM257 (14%), Irgacure184 (1%), and toluene (85%) is implemented to form a uniformly rotating film on the alignment SD1 layer. Finally, the LCs are solidified by exposing in the unpolarized light with a wavelength of 365 nm. Ultimately, part of the fabricated BHPLCM is manifested by a polarized optical microscope (POM), and the POM images under different polarization states are measured as shown in Fig. 2(b). Here, the fabricated BHPLCMs are inserted in between two linear polarizers, and the blue and red arrows correspond to the input and output polarization states, respectively. In this work, the pixel size of DMD adopted in the dynamic mask photopatterning system is 10.8 μm, and the pixel size of the designed LC lenses is 5.4 μm. Actually, a system with higher magnification can be utilized to make the phase modulation more precise with a smaller pixel size.

To further characterize the optical performance of our proposed BHPLCMs, the theoretical simulation results and experimental results are calculated to verify the validity of the polarization-multiplexing method. As shown in Fig. 3, a self-built optical equipment after beam expansion and collimation is utilized to detect the corresponding field distributions in the operating bandwidth. The system, illuminated by a supercontinuum laser source (NKT SuperK Extreme 15) equipped with a tunable filter (NKT Photonics, SuperK SELECT), is through polarizer 1 to achieve the polarization of the incident light. Certainly, polarizer 1 can be regarded as an attenuator to choose a suitable incident light field intensity. The collimator and beam expander are employed to ensure the uniform parallel incident beam on the proposed BHPLCMs. Hence, polarizer 2 and quarter-wave plate 1 (QWP1) are used to yield the LCP and RCP incidence. The QWP2 and polarizer 3 are selected to filter the co-polarization part, although this part is negligible for the final focusing performance benefited from the high PCR of LC metalenses, while it is necessary for the accuracy of efficiency measurement. Finally, the incident light (LCP/RCP) on the BHPLCMs will be focused and collected on the CCD (Daheng MER-310-12UC) mounted on a motorized stage, which is applied to measure light field distributions along the propagation direction.

First of all, the ${\mathrm{BHPLCM}}_1$ is designed to realize the polarization-switchable behavior from diffraction-limited focusing to sub-diffraction focusing. Here, we calculated the theoretical simulation results of ${\mathrm{BHPLCM}}_1$ under different incident light with different polarization states by the vector angular spectrum (VAS) theory and compared them with the actual experimental results. As illustrated in Fig. 3, the supercontinuum laser is utilized to measure the optical performance of ${\mathrm{BHPLCM}}_1$ under a wide spectrum range. Theoretically, the proposed ${\mathrm{BHPLCM}}_1$ based on the non-dispersion geometric phase can achieve preset focusing modes with the wavelength-independent characteristics. However, the propagating wave vector in the free space is different under the incident light with different wavelengths, leading to the inevitable axial chromatic aberration. In principle, the focal plane position of the hyperbolic phase function under the paraxial approximation condition for two different wavelengths should follow these principles: $$\frac{2\pi }{{\lambda }_1}(-\sqrt{f_1^2+r^2}+f_1)+2m\pi =\frac{2\pi }{{\lambda }_2}(-\sqrt{f_2^2+r^2}+f_2)+2n\pi ,$$(11)where $f_1$ and $f_2$ represent the focal length under the incident light with the wavelength ${\lambda }_1$ and ${\lambda }_2$, and $m$ and $n$ are all integers. Thus, the relationship between $f_1$ and $f_2$ can be easy to derive as $${\lambda }_1{f}_1\approx {\lambda }_2{f}_2.$$(12)

The corresponding focal lengths for incident light of 480 nm, 530 nm, 580 nm, 633 nm, and 670 nm can be deduced from Eq. (12) as 593 mm, 537 mm, 491 mm, 450 mm, and 425 mm, respectively. The different focal plane positions at different incident wavelengths are switched by controlling the position of the CCD placed on the electric stage, and the switching focusing function at the single wavelength can be realized by rotating the QWP 90°. The focusing performance located at the left and center of Fig. 4 indicated the broadband characteristics of ${\mathrm{BHPLCM}}_1$ and the polarization-switchable functions from the diffraction-limited focusing to sub-diffraction focusing. In order to accurately calculate the super-resolution performance of the proposed planar element, we extracted the PSF curves of the device on the design focal plane as shown in Fig. 4. The FWHMs of the focal spots under the incidence of 480 nm, 530 nm, 580 nm, 633 nm, and 670 nm LCP incident light are 54.4 μm, 60.8 μm, 54.4 μm, 60.8 μm, and 54.4 μm, respectively, while the FWHMs of the focal spots under RCP incidence are 44.6 μm, 44.6 μm, 43.8 μm, 43.8 μm, and 43.8 μm, respectively, corresponding to 0.83, 0.83, 0.81, 0.81, 0.81 times the Abbe diffraction limit equal to $0.5\lambda /\mathrm{NA}$, where NA is the numerical aperture of the optical system and $\lambda$ is the illumination wavelength.

Simultaneously, such a super-oscillatory wavefront is accompanied with a needle-like light field with a long focal depth, which is crucial for improving the robustness in practical applications. Accordingly, the axial light intensity distributions of the ${\mathrm{BHPLCM}}_1$ along the propagation direction are measured and simulated in Fig. 5. Additionally, to reduce the amount of measurement data efficiently, we selected the light field distributions in the $\pm 50\mathrm{mm}$ range at the design focal length in the diffraction-limited focusing mode (LCP incidence) and $\pm 150\mathrm{mm}$ range in the sub-diffraction focusing case (RCP incidence). There are certain differences between the simulated results and experimental results whether the light distributions are at the focal plane or along the propagation direction, especially for the light field distributions under the sub-diffraction focusing mode. The super-oscillation phenomenon originates from the fine interference of the incident light field; thus, the imperceptible disturbance will destroy the fragile and elaborate super-oscillatory light field. In general, the difference can be mainly speculated from three reasons. One is that the rotation angle of the LC molecules cannot be achieved perfectly in the actual processing especially for the external diameter of ${\mathrm{BHPLCM}}_1$, which originates from the unstable force between the LC molecules so as to cause the foreseeable discrepancy between the processing result and the theoretical design. The other is that the self-built optical system cannot definitely guarantee the uniformity of the expanded incident beam. And finally, the incident light cannot guarantee the perfect collimation and perfect circularly polarized characteristic because of the certain degree ellipticity for the generated circularly polarized incident light, and we have reduced this error as much as possible in the experiment. Therefore, the reasons for the difference between simulated and experimental results mainly lie in the first two factors, and the detailed analysis can be seen in Appendix C.

To further prove the universality of the proposed method, we designed ${\mathrm{BHPLCM}}_2$ to achieve the polarization-switchable behavior from diffraction-limited focusing to the vortex beam generator carrying the orbital angular momentum with a topological charge equal to 1. Similarly, the simulated and experimental light field distributions of the ${\mathrm{BHPLCM}}_2$ are also compared under the LCP and RCP incidence within the working bandwidth. As shown in Fig. 6, the focusing effects of the ${\mathrm{BHPLCM}}_2$ at the preset focal length are measured under the LCP/RCP incident light with the wavelengths 480 nm, 530 nm, 580 nm, 633 nm, and 670 nm, and they are compared with the simulated results. Obviously, the experimental results are basically consistent with the simulation results although there are some negligible differences. Furthermore, the extracted PSF intensity curves also demonstrated the stable broadband response capability of the proposed method.

Subsequently, we also paid attention to the light field distribution along the propagation direction under the incidence of LCP or RCP. Here, the axial light field distributions are simulated and measured around $\pm 50\mathrm{mm}$ at the preset focal length. As illustrated in Fig. 7, the stability of the polarization-switchable function in the broadband range also indicates the superiority of the proposed method, which greatly broadens the bandwidth and flexibility of the conventional optical diffractive element. In principle, the reasons for the existing difference are also the same as the analysis in ${\mathrm{BHPLCM}}_1$. It is worth mentioning that it is hard to give a specific definition for the operating bandwidth of the BHPLCMs due to the ultrabroadband response characteristics for geometric phase. As for the incident light far away from the designed center wavelength, the PCR will become lower and lower, but it is certain that the geometric phase obtains the ultrabroadband phase response theoretically, which had been confirmed in previous reports [7].

One of the most preeminent characteristics of our proposed BHPLCMs is the LC materials with high PCR. Therefore, a powermeter is adopted to measure the PCR and focusing efficiency of BHPLCMs at several discrete wavelengths accurately. Here, the co-polarization part of the light field should be filtered after passing the BHPLCMs, so we add QWP2 and polarizer 3 behind the LC lenses based on the self-built system as shown in Fig. 3 to maximize the accuracy of the measurement for PCR although the intensity of the cross-polarization part is negligible. After the detailed measurement, the PCRs of ${\mathrm{BHPLCM}}_1$ under LCP incident light with wavelengths of 480 nm, 530 nm, 580 nm, 633 nm, and 670 nm are 64.7%, 81%, 92.8%, 99.3%, and 96.41%, respectively. Analogously, the PCRs of ${\mathrm{BHPLCM}}_2$ under LCP incident light with wavelengths of 480 nm, 530 nm, 580 nm, 633 nm, and 670 nm are 70.11%, 81.90%, 92.13%, 96.04, and 93.10%, respectively. Actually, the actual conversion efficiency should be higher because of the existing inevitable energy loss through QWP2 and polarizer 3 in the self-built system, especially under the incident light of 480 nm where the energy of supercontinuum laser is relatively weak. Besides, the focusing efficiency of ${\mathrm{BHPLCM}}_1$ under different incident light is also measured as shown in Fig. 8. Here, the focusing efficiency is defined as the ratio of the energy in the desired region (3 times the corresponding FWHM) at the focal plane to the total energy of the incident light.

Obviously, the proposed ${\mathrm{BHPLCM}}_1$ maintains a high PCR near the designed wavelength benefiting from the satisfaction of the half-wave condition and basically achieves 100% PCR at the designed wavelength 633 nm, which is consistent with the theoretical results as expected. However, the focusing efficiency under LCP/RCP is not very high and originates from two main reasons. One is that the super-oscillatory phenomenon is usually accompanied with relatively low working efficiency, and the other is that certain sacrifices are necessary to realize the polarization-multiplexing switchable focusing function. Undeniably, the high efficiency is attributed to the high transmission coefficient and high PCR in the case of satisfying the half-wave condition, which shows miraculous advantages compared with metallic metasurfaces [47] or transmitted PB-based dielectric metasurfaces [48] in terms of the working efficiency. Besides, the intensity ratio in the sub-diffraction focusing mode is improved in advance to balance the efficiency under the two polarization states LCP/RCP, and the same considerations are also adopted in the design of ${\mathrm{BHPLCM}}_2$. However, in general, the ultra-high PCR also ensures the high efficiency of the device in practical applications scenarios.

Indeed, the super-resolution ability can be further enhanced by adding appropriate constraints for ${\mathrm{BHPLCM}}_1$, while enormous challenges will be generated to balance the relationship among the FOV, sidelobe, mainlobe intensity, and so forth as proved in Appendix D. Moreover, the tolerance in actual processing will also be decreased because the super-oscillatory light field is extremely fragile albeit the slight changes. Meanwhile, the proposed BHPLCMs with a low $f$-number based on the hyperbolic focusing phase can be achieved by using a lens with larger magnification to decrease the pixel size of the DMD-based microlithography system. Such a challenging problem also can be solved by adopting the analogous optimization algorithm in designing traditional phase-type diffractive elements, such as binary optics, which originated from the destructive interference and constructive interference in principle, leading to the depressing efficiency to a certain extent, which is ubiquitous in the design of super-oscillatory lenses [49]. Additionally, the axial chromatic aberration is also worthy of further attention. Although such non-dispersion geometric phase modulation is wavelength-independent, a certain axial chromatic aberration will be generated because of the different propagation wave vectors in the free space under different wavelengths despite that the needle-like field can weaken the negative influence. Accordingly, the elimination of axial chromatic aberration is of great significance to the practical application, and this problem can be handled in accordance with our previous work [50], i.e., combined with traditional commercial achromatic lenses to achieve white light super-resolution imaging, which will also greatly broaden the application scenarios of the proposed method. Another method is utilizing a multi-objective optimization algorithm to control the dispersion efficiently, e.g., the holographic super-resolution method [51] or the Pareto optimal model [52,53] can be employed to achieve the achromatic focusing. Certainly, a fundamentally physical method based on controlling the group delay dispersion [48] or wavelength dispersion engineering [54] can be considered to achieve achromatic focusing from the source combining with dielectric metasurfaces. Ulteriorly, as we know, the optical lenses are crucial to the optical imaging system; thus, the proposed multifunctional focusing lenses can be applied to different imaging modes. For ${\mathrm{BHPLCM}}_1$, diffraction-limited imaging and super-resolution imaging can be achieved by a single element as proved in our previous work [55], and we did not choose a higher resolution because the lower sidelobe and wider FOV need to be prioritized to reduce the adverse noise in the actual imaging system without combining confocal microscopy. For ${\mathrm{BHPLCM}}_2$, the polarization-switchable spiral phase contrast imaging can be realized as proved in previous reports [56]. Finally, LCs, as one of the most promising electro-optical materials, obtain excellent dynamic tunability and huge potential in dynamic display, VR, and AR, leading to unrivalled superiority and miraculous capabilities in tunable and reconfigurable metadevices [57,58]. In general, the proposed method can be widely used in multifunctional imaging modes, and the flexibility of the lenses will be greatly broadened in practical applications.

In this work, broadband high-efficiency polarization-multiplexing LC metalenses are proposed and experimentally verified based on the spin-dependent phase conjugation. It can be found that the efficient polarization-switchable focusing function can be achieved by arranging LC molecules rationally with different rotation directions. The proposed ${\mathrm{BHPLCM}}_1$ can achieve the diffraction-limited focusing under LCP incidence and sub-diffraction focusing with 0.8 times Abbe diffraction limit under RCP incidence, while ${\mathrm{BHPLCM}}_2$ can achieve diffraction-limited focusing under LCP incidence and a focused vortex beam under RCP incidence. The optical performance is characterized by the self-built experimental system within the operating bandwidth, and the experimental results are consistent with the simulated results, which indicates the veracity and reasonability of our proposed method. Above all, the unique non-dispersion based on geometric phase modulation will greatly help to produce constant phase distribution within an ultrabroadband working bandwidth. We believe that the proposed method is expected to have great potential in the fields of super-resolution color imaging, holograms, VR/AR, and so on.