## Abstract

Characterizing the amplitude, phase profile, and polarization of optical beams is critical in modern optics. With a series of cascaded optical components, one can accurately resolve the optical singularity and polarization state in traditional polarimetry systems. However, complicated optical setups and bulky configurations inevitably hinder future applications for integration. Here, we demonstrate a metadevice that fully resolves arbitrary beams on a higher-order Poincaré sphere (HOPS) via a single-layer all-silicon metasurface. The device is compact and capable of detecting optical singularities and higher-order Stokes parameters simultaneously through a single intensity measurement. To verify the validity of the proposed metadevice, different beams on ${\mathrm{HOPS}}_{0,0}$ and ${\mathrm{HOPS}}_{1,-1}$ are illuminated on the metadevices. The beams are fully resolved, and the reconstructed higher-order Stokes parameters show good agreement with the original ones. Taking the signal-to-noise ratio into account, the numerical simulations indicate that the design strategy can be extended to fully resolve arbitrary beams on HOPS with order up to 4. Because of the advantages of compact configuration and compatibility with current semiconductor technology, the metadevice will facilitate potential applications in information processing and optical communications.

© 2021 Chinese Laser Press

## 1. INTRODUCTION

Polarization and phase are two intrinsic characteristics of electromagnetic waves, in which polarization indicates the vectorial nature of an oscillating electric field [1]. In 1892, researchers proposed a geometric representation of polarization states termed a “Poincaré sphere” (PS), which unifies all the fundamental polarizations (circular, linear, and elliptical) that exhibit spatially homogeneous distributions [2]. In such a geometric representation, the polarization states are mapped to the PS’ surface through the Stokes parameters in the sphere’s Cartesian coordinates. This geometric representation is a general tool for polarization-pertinent issues in numerous fields. However, the polarization states represented by a conventional PS are limited to electromagnetic waves with homogeneous polarization. For representing waves with spatially inhomogeneous polarizations (such as vector vortex beams), researchers proposed an extended generalized geometric representation known as a higher-order Poincaré sphere (HOPS) [3–8]. The HOPS incorporates spin angular momentum (SAM) and orbital angular momentum (OAM), describes the evolution of both phase and polarization, and provides a new degree of freedom for manipulating light beams. Recently, because of their intriguing properties, beams on HOPS have attracted extensive attention in optical microscopy [9,10], optical trapping [11,12], quantum information technology [13], and mode division multiplexing in optical communication [14,15].

With the development of complex optical beams and their applications, it is important to simultaneously measure the higher-order Stokes parameters and optical singularities in a compact and reliable manner. However, direct polarization measurement is unavailable because of the lack of phase information in conventional intensity probing schemes. As such, to accurately determine the polarization states in conventional methods, one must implement a series of intensity measurements via dividing the input electromagnetic waves temporally or spatially [16,17]. In particular, for characterizing the inhomogeneous polarization state, the optical singularity associated with angular momentum must be measured as well [3]. In such measurements, complicated optical setups (such as a wave plate, polarizer, and diffractive fork gratings) and bulky configurations are inevitably introduced, hindering the development of integrated optical devices.

Metasurface, ultrathin optical components, which are capable of either replacing traditional bulky optical components or achieving exceptional functionalities, have been considered a major advancement in nano-optics [18]. The components are composed of 2D discrete meta-atoms that can locally modify the light’s property and show unprecedent superiority in wavefront manipulation, thus providing an excellent platform for planar and ultracompact optical devices [19,20]. To date, researchers have demonstrated a variety of ultrathin devices based on metasurfaces, such as metalenses [21–27], vortex beam generators [6,28–30], and holograms [31–34]. Likewise, metasurface-enabled polarimeters, which accurately measure Stokes parameters in a compact configuration with straightforward setups compared with conventional counterparts have also attracted much attention [35–42]. For these devices, researchers accurately resolve the polarization states by retrieving the Stokes parameters via detecting the polarization-dependent intensity response of the metasurfaces. Despite their powerful functionalities and compact sizes, these devices are mostly limited to resolving the polarization states on standard PS. For beams on HOPSs, researchers have simultaneously detected the spin and angular momenta of these beams [43–46]. However, these devices cannot fully resolve the incident beams because they are unable to retrieve the Stokes parameters. Therefore, a compact setup capable of fully resolving beams on HOPSs is highly desirable and to date remains unexplored, to our best knowledge.

In this work, to overcome the aforementioned limitation of current polarimetry systems, we demonstrate a compact all-silicon metadevice that completely resolves arbitrary beams on HOPSs. The metadevice is composed of two main parts that detect the angular momentum and higher-order Stokes parameters, respectively. The former can be accomplished by recognizing solid spots in the helicity-decoupled focusing vortex generator, whereas the latter can be achieved by analyzing intensity contrasts in the four predesigned polarization-dependent metalenses. As a result, researchers can completely resolve incident beams on an arbitrary HOPS via a single measurement, without the need for any additional optical components.

We selected two beams of ${\mathrm{HOPS}}_{0,0}$ and ${\mathrm{HOPS}}_{1,-1}$ to verify the metadevice, and the polarization states are well reconstructed by numerical results. Our design strategy is general and can be extended to fully resolve arbitrary beams on any HOPS.

## 2. PRINCIPLES OF METASURFACE DESIGN

For a paraxial beam on a HOPS, the Jones vector can be expressed in a form of superposition of two circular polarization bases as

where representing two circularly polarized vortexes with topological charges $m$ and $n$, respectively. The vectors $\widehat{x}$ and $\widehat{y}$ represent the unit vectors along the $x$ and $y$ axes, respectively, and $\phi =\mathrm{arctan}(y/x)$ is the azimuthal angle. Therefore, by adopting a proper set of coefficients, any point on a HOPS can be expressed as a superposition of these two circularly polarized vortexes.For a vector vortex beam on a HOPS, the high-order Stokes parameters in the sphere’s Cartesian coordinates can be derived as

where ${|{A}_{R}^{m}|}^{2}$ and ${|{A}_{L}^{m}|}^{2}$ are the intensity of $|{R}_{m}\u27e9$ and $|{L}_{n}\u27e9$, respectively. $\phi =\mathrm{arg}({A}_{R}^{m})-\mathrm{arg}({A}_{L}^{n})$ is the phase difference between the two circularly polarized (CP) vortices. In particular, for the specific case $m=n=0$, a HOPS reduces to the standard plane wave PS, and the higher-order Stokes parameters in Eqs. (4)–(7) reduce to the standard plane wave Stokes parameters. As such, to obtain a generalized description of all the beams, we treat the standard PS as a special case of HOPS in the remainder of this paper.To fully resolve a light beam on a HOPS, three key parameters—spin of the photons $\sigma $, topological charges $m$ and $n$, and higher-order Stokes parameters in Eqs. (4)–(7)—should all be resolved. The former two parameters pertain to the optical singularities, and the last parameter corresponds to the polarization states of the light beams. Figure 1(a) shows a schematic of the metasurface-based device that fully resolves the beams on a HOPS. Figure 1(a) shows an all-silicon meta-atom, from which one can see the period along the $x$- and $y$-axes $P$, and the height of the meta-atom $h$ (see detailed parameters in Fig. 6 in Appendix A). To resolve the first two key parameters, we designed a CP-decoupled focusing vortex generator located in the center of the metadevice [47,48]. With the capacity to generate an independent focusing vortex for the two orthogonal circular polarizations [right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP)], it discriminates the spin of the photons *σ* along with the topological charges $m$ and $n$ of the incident beam. To resolve the last key parameter, we added four elaborately designed distinctive metalenses in the four corners, with the ability to efficiently focus four predesigned polarizations [RCP, LCP, $x$-linearly polarized (XLP), and diagonal linearly polarized (DLP)]. By measuring the focusing efficiencies of the four metalenses and with a matrix derive calculation, one can obtain the higher-order Stokes parameters. With the three key parameters, the designed metadevice fully resolves the beams on an arbitrary HOPS. Here, we choose two common cases for $m=n=0$ and $m=-n=1$, which correspond to ${\mathrm{HOPS}}_{0,0}$ (the standard plane wave PS) and ${\mathrm{HOPS}}_{1,-1}$, respectively. Figures 1(b) and 1(c) show typical examples for light beam modes on two selected HOPSs. Compared with the standard PS, where the poles indicate two orthogonal CP beams (with opposite SAM), the poles of ${\mathrm{HOPS}}_{1,-1}$ indicate two CP vortex beams (with different orbital angular momentum values: $m=1$ and $n=-1$).

## 3. RESULTS AND DISCUSSIONS

#### A. Discriminating the Spin and Topological Charge of Incident Beams

To resolve the spin of photon $\sigma $ as well as the topological charges $m$ and $n$ of the incident beam, we designed a CP-decoupled focusing vortex generator. For such a focusing vortex generator, the key issue is to obtain independent phase control under the two CPs. By accommodating both the propagation and geometric phase in the birefringent meta-atoms, one can achieve independent phase control for the two CP light beams (see detailed process in Appendix A). The phase profile $\phi (x,y)$ of the CP decoupled focusing vortex generator can be expressed as

By using Eqs. (8)–(10), we arranged the selected meta-atoms to form the CP-decoupled focusing vortex generator. The operating wavelength is $\lambda =1.55\text{\hspace{0.17em}}\mathrm{\mu m}$, and the focal length is set as $f=20\text{\hspace{0.17em}}\mathrm{\mu m}$. Here, we designated the CP-decoupled focusing vortex generator to generate six focusing vortexes. Three of them are for the LCP state with topological charge $l=-1,0$, and $+1$, respectively, located at ($-8\text{\hspace{0.17em}}\mathrm{\mu m}$, 8 μm), (0 μm, 8 μm), and (8 μm, 8 μm) in the focal plane. The other three are for the RCP state with topological charge $l=+1$, 0, and $-1$, respectively, located at ($-8\text{\hspace{0.17em}}\mathrm{\mu m}$, $-8\text{\hspace{0.17em}}\mathrm{\mu m}$), (0 μm, $-8\text{\hspace{0.17em}}\mathrm{\mu m}$), and (8 μm, $-8\text{\hspace{0.17em}}\mathrm{\mu m}$), respectively, in the focal plane. One can decompose an arbitrary vector vortex beam on the HOPS into two orthogonal circular beams in the two poles [Eqs. (1)–(3)]. The superposition effect of the angular momentum gives a visualized method for resolving the angular momentum information because annihilation of the angular momentum (${l}_{\mathrm{tot}}=0$) results in a solid intensity spot. Therefore, one can use the CP-decoupled focusing vortex generator to fully discriminate the spin and topological charge of the incident beam: focusing spots (when ${l}_{\mathrm{tot}}=0$) manifest at certain positions.

To demonstrate its functionality, Fig. 2 shows simulated intensity profiles of the CP decoupled focusing vortex generator under various incident light beams on ${\mathrm{HOPS}}_{0,0}$. The solid spot emerges at the specific locations where total angular momentum ${l}_{\mathrm{tot}}=0$, which resolves the topological charge of the incident beams. Details of the resolving process are as follows. For convenience, we adopted a result matrix, ${L}_{ij}=\left[\begin{array}{ccc}-1& 0& +1\\ +1& 0& -1\end{array}\right]$, to express the topological charges of the generated optical vortex, in which the first and second lines represent the topological charges of the focusing vortices generated by the LCP and RCP incident light, respectively. For the case that a solid intensity spot manifests at ${L}_{22}$ [Fig. 2(a)], one can conclude that the incident beam is an RCP beam ($\sigma =-1$) with topological charge $m=0$ (using the relationship ${l}_{\mathrm{tot}\text{\hspace{0.22em}}}={L}_{22}+m=0$). Similarly, for the case that a solid intensity spot manifests at ${L}_{12}$ [Fig. 2(b)], the incident beam can be recognized as an LCP beam ($\sigma =+1$) with topological charge $n=0$. Otherwise, for the case that two solid intensity spots manifest at both ${L}_{12}$ and ${L}_{22}$ [Figs. 2(c) and 2(d)], the incident beam can be expressed as a combination of the two CP beams [Eqs. (1)–(3)], with topological charges $m=n=0$. Similarly, for the beams on ${\mathrm{HOPS}}_{1,-1}$, the topological charge can be resolved as well (see details in Appendix B). Moreover, with the same strategy, discriminating the spin and topological charge on HOPS with order up to 4 is also demonstrated (see details in Appendix B).

#### B. Fully Resolved Beams on ${\mathrm{HOPS}}_{0,0}$ by the Designed Metadevice

To confirm the functionality of the proposed metadevice, Figs. 3(a)–3(h) show simulated intensity (${\left|E\right|}^{2}$) profiles of the metadevice under different incident beams on ${\mathrm{HOPS}}_{0,0}$. Figures 3(a)–3(d) illustrate the simulated intensity profiles of the metadevice for incident light with four polarization states on ${\mathrm{HOPS}}_{0,0}\text{:}|{R}_{0,0}\u27e9$, $|{L}_{0,0}\u27e9$, $|{X}_{0,0}\u27e9$, and $|{D}_{0,0}\u27e9$, respectively. The simulated intensity profiles for incident light with other polarization states on ${\mathrm{HOPS}}_{0,0}$ are demonstrated in Fig. 10 in Appendix B. In accordance with the previous subsection, one can resolve the topological charges of the incident beams on ${\mathrm{HOPS}}_{m,n}$ as $m=n=0$. Figures 3(e)–3(h) show the normalized intensity profiles along the *x* axis at $y=\pm 8\text{\hspace{0.17em}}\mathrm{\mu m}$ in Figs. 3(a)–3(d), further confirming the correctly measured positions of the solid intensity spots. We measured the focusing efficiencies of the predesigned four distinctive metalenses (located in the four corners) illuminated with different light on ${\mathrm{HOPS}}_{0,0}$ (see detailed configurations in Appendix C). One can calculate the Stokes parameters of the selected incident beams by the focusing efficiency contrasts. Table 1 shows the corresponding focusing efficiencies and reconstructed Stokes parameters (see Appendix D for the detailed reconstructing process). The reconstructed Stokes parameters show good agreement with the original parameters, verifying the veracity of our metadevice. Therefore, via a single measurement, the metadevice fully resolves an arbitrary beam on ${\mathrm{HOPS}}_{0,0}$.

#### C. Fully Resolved Beams on ${\mathrm{HOPS}}_{1,-1}$ by the Designed Metadevice

Figures 4(a)–4(d) illustrate the simulated intensity profiles of the metadevice for incident light with four polarization states on ${\mathrm{HOPS}}_{1,-1}$: $|{R}_{1,-1}\u27e9$, $|{L}_{1,-1}\u27e9$, $|{X}_{1,-1}\u27e9$, and $|{D}_{1,-1}\u27e9$, respectively. The simulated intensity profiles for incident light with other polarization states on ${\mathrm{HOPS}}_{1,-1}$ are demonstrated in Fig. 11 in Appendix B. Similarly, by recognizing the solid intensity spots exhibited at certain positions, we successfully find the topological charges of the incident beams on ${\mathrm{HOPS}}_{m,n}$ as $m=-n=1$. Although there is a small scattered intensity field around the solid spots, one can accurately confirm the correctly measured positions of the solid spots [see the normalized intensity profiles in Figs. 4(e)–4(h)]. Moreover, Table 2 shows the corresponding focusing efficiencies and reconstructed Stokes parameters. One can reconstruct the Stokes parameters of the selected incident beams on ${\mathrm{HOPS}}_{1,-1}$ by the focusing efficiency contrasts. Therefore, via a single measurement, such a metadevice also fully resolves an arbitrary beam on ${\mathrm{HOPS}}_{1,-1}$.

Figure 5 shows the original (red circles) and reconstructed (blue asterisks) higher-order Stokes parameters in Tables 1 and 2. The reconstructed polarization states show good agreement with the original states for both cases, verifying the veracity of our proposed metadevice. Here, we selected incident beams on the 0- and 1-order HOPSs as examples. For our designed metadevice composed of a CP-decoupled focusing vortex generator and four elaborately designed distinctive metalenses, the simulated results indicate that the CP-decoupled focusing vortex generator can be extended to fully resolve arbitrary beams on HOPS with order up to 4 (see details in Appendix B). The four metalenses would slightly worsen the performance of the designed CP-decoupled focusing vortex generator. Considering the reported experimental results [49], the HOPS with order up to 4 is not the limitation of our proposed metadevice. Hence, it is safe to say that the design strategy can be extended to fully resolve arbitrary beams on HOPS with order up to 4.

We also considered the experimental feasibility and potential applications of the proposed all-silicon metadevice. One can fabricate the single-layer all-silicon metadevice with one-step lithography and dry etching. Here, we choose an all-silicon configuration for its simpler fabrication procedure than that of the a-Si on silica substrate configuration (see Appendix E for the detailed comparison of these two configurations). The metadevice has potential applications in polarization-related information processing and optical communications. For example, orthogonally structured light beams such as optical vortex beams and cylindrical vector beams are recent candidates for high-capacity multiplexing optical communication. The metadevice would be important to this process because of its capacity to simultaneously measure the polarization states and optical singularities in a compact and reliable manner.

## 4. CONCLUSION

In conclusion, we proposed a compact metadevice composed of an all-silicon metasurface, which is capable of completely resolving arbitrary beams on HOPSs. Via a single measurement, one can simultaneously detect the OAM and higher-order Stokes parameters of the incident beams, without the need for additional optical components. To verify the validity of the metadevice, we selected ${\mathrm{HOPS}}_{0,0}$ and ${\mathrm{HOPS}}_{1,-1}$ as examples. Different beams on these two are illuminated on the metadevices, and the reconstructed polarization states show good agreement with the preset ones. Here, these two HOPSs are chosen as proof of concept, and the design strategy can be extended to fully resolve arbitrary beams on HOPS with order up to 4. Diverse applications such as polarization-related information processing and optical communications will be facilitated by the compact configuration of the metadevice and its compatibility with current semiconductor technology.

## APPENDIX A: INDEPENDENT PHASE CONTROL FOR THE TWO CIRCULAR POLARIZATIONS

As demonstrated in previous works, birefringent meta-atoms with both propagation and geometric phase could achieve independent phase control of arbitrary orthogonal states of polarization [31]. To obtain independent phase control of circular polarizations, the phase relationships between the two orthogonal linear and the two orthogonal circular polarizations can be expressed as

where ${\phi}_{x}$, ${\phi}_{y}$, ${\phi}_{L}$, and ${\phi}_{R}$ are the phase shifts of $x$-linearly polarized (XLP), $y$-linearly polarized (YLP), LCP, and RCP light, respectively. Therefore, for an arbitrary set of ${\phi}_{L}$ and ${\phi}_{R}$, one just needs to design the ${\phi}_{x}$, ${\phi}_{y}$, and $\theta $ according to the above equations. In other words, we can arbitrarily and independently manipulate the LCP and RCP phases by designing three parameters: the $x$ polarized phase ${\phi}_{x}$, the $y$ polarized ${\phi}_{y}$, and the rotation angle $\theta $.The meta-atom capable of providing independent phase control at two orthogonal linearly polarized lights (XLP and YLP) is shown in Figs. 6(a) and 6(b). The birefringent meta-atom consists of an all-silicon elliptical nanoblock (with refractive index $n=3.47$). The detailed parameters of the meta-atom are height $h$, orientation angle $\theta $, length along $x$-axis $b$, length along $y$-axis $a$, and lattice constant $P$. Figures 6(c)–6(f) show the transmission properties of the meta-atoms, with the simulated phase and transmission amplitude of the meta-atoms as functions of $a$ and $b$ for the two linear polarizations.

To achieve independent phase modulation at two orthogonal linearly polarized lights, the polarization multiplexed technology is adopted. The selected geometrical parameters of the meta-atoms that independently cover the phase coverage ($-\pi $ to $\pi $) are shown in Fig. 6. In order to obtain these optimized meta-atoms, the errors between the required and realized phase shifts as well as amplitudes are taken into consideration. The optimized geometrical parameter set ($a$ and $b$) is achieved by taking minimum error value

## APPENDIX B: DISCRIMINATING THE SPIN AND TOPOLOGICAL CHARGE ON ${\mathrm{HOPS}}_{m,-n}$

The simulated intensity profiles of the CP decoupled focusing vortex generator under different incident lights on ${\mathrm{HOPS}}_{1,-1}$ are depicted in Fig. 8. As also depicted in Fig. 8, for the beams on ${\mathrm{HOPS}}_{1,-1}$, the solid spot emerges at the specific locations where total angular momentum ${l}_{\mathrm{tot}}=0$, which is capable of resolving the topological charge of the incident beams. The detail-resolving process is as follows. Here, a result matrix ${L}_{ij}$ is adopted to express the topological charges of the generated vortex beams in which the first and second lines represent the topological charges of OAMs generated by LCP and RCP incident lights, respectively. For the case that a solid intensity spot exhibits at ${L}_{23}$, one can conclude that the incident beam is an RCP beam with topological charge $m=+1$ (using the relationship ${l}_{\mathrm{tot}}={L}_{22}+m=0$). Similarly, for the case that a solid intensity spot exhibits at ${L}_{13}$, the incident beam is an LCP beam with topological charge $n=-1$. Otherwise, for the case that two solid intensity spots exhibit at both ${L}_{13}$ and ${L}_{23}$, the incident beam can be expressed as a combination of the two circularly polarized beams as expressed in Eqs. (1)–(3) in the main text, with topological charges $m=-n=1$. Therefore, the designed CP decoupled focusing vortex generator is able to simultaneously resolve the spin of photon and topological charges of the incident beam on ${\mathrm{HOPS}}_{1,-1}$.

Moreover, with the same design strategy, discriminating the spin and topological charge on HOPS with order up to 4 is also demonstrated. By using Eqs. (8)–(10), we arranged the selected meta-atoms to form the required CP-decoupled focusing vortex generator. The operating wavelength is $\lambda =1.55\text{\hspace{0.17em}}\mathrm{\mu m}$, and the focal length is set as $f=40\text{\hspace{0.17em}}\mathrm{\mu m}$. Here, we designated the CP-decoupled focusing vortex generator to generate 18 focusing vortexes. Nine of them for the LCP state with topological charge $l=-4$ to $+4$, respectively, are located at (6 μm, $-20\text{\hspace{0.17em}}\mathrm{\mu m}$), (6 μm, $-10\text{\hspace{0.17em}}\mathrm{\mu m}$), (6 μm, 0 μm), (6 μm, 10 μm), (6 μm, 20 μm), (18 μm, 10 μm), (18 μm, 0 μm), (18 μm, $-10\text{\hspace{0.17em}}\mathrm{\mu m}$), and (18 μm, $-20\text{\hspace{0.17em}}\mathrm{\mu m}$) in the focal plane. The other nine are for the RCP state with topological charge $\mathit{l}=+4$ to $-4$, respectively, located at ($-6\text{\hspace{0.17em}}\mathrm{\mu m}$, $-20\text{\hspace{0.17em}}\mathrm{\mu m}$), ($-6\text{\hspace{0.17em}}\mathrm{\mu m}$, $-10\text{\hspace{0.17em}}\mathrm{\mu m}$), ($-6\text{\hspace{0.17em}}\mathrm{\mu m}$, 0 μm), ($-6\text{\hspace{0.17em}}\mathrm{\mu m}$, 10 μm), ($-6\text{\hspace{0.17em}}\mathrm{\mu m}$, 20 μm), ($-18\text{\hspace{0.17em}}\mathrm{\mu m}$, 10 μm), ($-18\text{\hspace{0.17em}}\mathrm{\mu m}$, 0 μm), ($-18\text{\hspace{0.17em}}\mathrm{\mu m}$, $-10\text{\hspace{0.17em}}\mathrm{\mu m}$), and ($-18\text{\hspace{0.17em}}\mathrm{\mu m}$, $-20\text{\hspace{0.17em}}\mathrm{\mu m}$) in the focal plane. For convenience, a matrix $P$ is adopted to express the appointed topological charges of the CP-decoupled focusing vortex generator as

## APPENDIX C: DETAILED CONFIGURATION OF THE FOUR PREDESIGNED METALENSES

The P-B phase is introduced to the design of the two metalenses for focusing the LCP and RCP incident lights. To fulfill the P-B phase condition, the meta-atom is optimized to function as a half-wave plate, and the polarization conversion rate is also optimized as unity at the working wavelength. The corresponding parameters of the all-silicon meta-atom are $a=200\text{\hspace{0.17em}}\mathrm{nm}$, $b=480\text{\hspace{0.17em}}\mathrm{nm}$, and $h=1000\text{\hspace{0.17em}}\mathrm{nm}$. The polarization conversion rate of the meta-atom as a function of the incidence wavelength is depicted in Fig. 12(a). The polarization conversion rate over 99% at 1550 nm indicates the half-wave plate functionality of the meta-atom. In this condition, the meta-atom is able to flip the helicity of the input circularly polarized light and to impart an additional phase shift of $2\theta $ on the output light. The phase shift of the meta-atom as a function of rotation angles $\theta $ is plotted in Fig. 12(b). It is obvious that the phase shift and the rotation angle satisfy the P–B phase condition ($\phi =2\theta $).

For the metalens designed for XLP incident light, the meta-atoms are selected from Figs. 7(c) and 7(d) with minimum error value, where ${\phi}_{x}$ and ${\phi}_{y}$ are the desired and object phase shifts for XLP incident light. Moreover, rotating all the meta-atoms of the metalens for XLP incident light with an angle of 45°, the metalens designed for DLP incident light is achieved.

## APPENDIX D: CALCULATING THE HIGHER-ORDER STOKES PARAMETERS

For the vector vortex beam on HOPS, the higher-order Stokes parameters in the sphere’s Cartesian coordinates can be derived as

From Eqs. (D1)–(D9), there is a linear relationship among the four intensity components (${I}_{R}^{m},{I}_{L}^{n},{I}_{X}^{m,n},{I}_{D}^{m,n}$) and the higher-order Stokes parameters, which can be described by a $4\times 4$ device matrix ${M}^{m,n}$ as

However, this is not the final device matrix since the measured intensities are composed of the four polarization components ${I}_{0}^{m,n}={({I}_{R}^{m},{I}_{L}^{n},{I}_{X}^{m,n},{I}_{D}^{m,n})}^{T}$ and the background intensities ${B}_{0}^{m,n}={({B}_{R}^{m},{B}_{L}^{n},{B}_{X}^{m,n},{B}_{D}^{m,n})}^{T}$, which consist of transmitted light not modulated by the metasurfaces as well as cross-talk between the different metasurfaces structures. Let us assume there is also a linear relationship among the four polarization components and the background intensities:

Then, the designed metadevice is capable of accurately resolving the polarization states by calculating the higher-order Stokes parameters from Eq. (D11).

## APPENDIX E: METADEVICE WITH A-SI ON SILICA SUBSTRATE CONFIGURATION

In the main text, we choose an all-silicon configuration and demonstrate a metadevice for fully resolving arbitrary beams on a HOPS. The all-silicon configuration is selected for its simpler fabrication procedure (requires no additional film deposition during the fabrication procedure) compared with the a-Si on silica substrate configuration. Actually, identical functionality could be achieved for our proposed metadevice with a-Si on silica substrate configuration. To show that, the all-silicon metadevice is replaced with a-Si on silica substrate configuration (the refractive indexes of a-Si and silica are set as 3.7 and 1.46, respectively). The unit cell consists of an a-Si nanoblock placed on a silica substrate. The lattice constant is $P=600\text{\hspace{0.17em}}\mathrm{nm}$, and the height of the nanoblock is $h=800\text{\hspace{0.17em}}\mathrm{nm}$. With the same strategy, we repeated all the optimization processes and accomplished the same functionalities (resolving different beams on ${\mathrm{HOPS}}_{0,0}$ and ${\mathrm{HOPS}}_{1,-1}$). The intensity profiles for the metadevice with a-Si on the silica substrate configuration are identical with those in the main text and hence are not shown here. The corresponding simulated focusing efficiencies and reconstructed Stokes parameters are shown in Tables 3 and 4, respectively. For both cases, the reconstructed Stokes parameters show good agreement with the original ones. From the four tables, one can find that the total intensity of the a-Si on silica substrate configuration is slightly larger than that of the all-silicon configuration. However, the metadevice based on these two configurations shows approximate performance. Therefore, we choose the all-silicon configuration for its simpler fabrication procedure.

## Funding

Guangdong Major Project of Basic and Applied Basic Research (2020B0301030009); National Key Research and Development Program of China (2018YFB1801801); National Natural Science Foundation of China (U1701661, 61935013, 61975133, 11604218, 11774240, 11947017); Natural Science Foundation of Guangdong Province (2016A030312010, 2020A1515011185); Leadership of Guangdong Province Program (00201505); Science and Technology Innovation Commission of Shenzhen grants Shenzhen Peacock Plan (JCYJ20180507182035270, KQJSCX20170727100838364, KQTD20170330110444030, ZDSYS201703031605029, JCYJ20200109114018750, JCYJ20180305125418079); Shenzhen University (2019075).

## Acknowledgment

H. Wang acknowledges financial support from the Ministry of Education of Singapore (MOE2016-T2-1-052).

## Disclosures

The authors declare no conflicts of interest.

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