Polarization and phase are inherent characteristics of light^[1–3]. By flexibly manipulating these properties, a diverse range of structured light beams has been created, including vortex beams (VBs) and vector vortex beams (VVBs) that carry both spin angular momentum (SAM) and orbital angular momentum (OAM)^[4–7]. The interplay between SAM (represented by the circular polarization chirality) and OAM (defined by the beams’ topological charge) constitutes the total angular momentum (TAM) of a light beam^[8,9]. Optical VBs feature helical phase fronts and distinctive doughnut-shaped intensity patterns arising from phase singularities^[10], while VVBs showcase spatially varying polarization distributions resulting from the intricate interplay between optical vortices with opposing SAM and OAM^[11,12].

A convenient framework for characterizing the polarization and spatial degrees of freedom of optical beams spin–orbit interaction^[4,5,13] is the hybrid-order Poincaré sphere (HyOPS), which generalizes the Poincaré sphere (PS)^[14] and higher-order Poincaré sphere (HOPS)^[15]. In the HyOPS, the poles represent two orthogonal circularly polarized (CP) VBs with arbitrary topological charge (TC), indicating that the beams carry opposite SAM values and differing OAM values. Each point on the HyOPS characterizes the space-variant polarization field of a light beam with OAM, constructed through a linear superposition of the two poles, known as Poincaré beams (PBs)^[16]. Combining features of VVBs and VBs, PBs have been successfully demonstrated and utilized in various applications, such as nanoparticle manipulation^[17,18], optical communication^[19,20], and optical encryption^[21,22]. However, conventional PBs often exhibit varying radii of annular intensities tied to their TCs, hindering their utility in scenarios like coupling multiple PBs with different TCs into a single fiber for mode division multiplexing^[23,24]. Furthermore, generating a stable and high-quality PB by combining two orthogonally CP optical vortices with significantly different TCs poses challenges, as the beam may collapse upon propagation^[25].

To address this challenge, the innovative concept of perfect vortex beams (PVBs) has emerged^[26], aiming to maintain consistent ring radii and uniform divergences for all vortex orders, while allowing for the flexibility of adjusting TC. Consequently, perfect Poincaré beams (PPBs), achieved through the linear superposition of PVBs with opposing chirality, have been successfully demonstrated^[27–29]. However, currently, the traditional feat involves the integration of multiple bulky and intricate optical components with diverse functionalities, including axicons, spatial light modulators, q-plates, and Fourier transform lenses, which lack flexibility in regulation and present integration challenges in compact, flat, and miniaturized optical devices. Furthermore, optical aberrations resulting from misalignments among components pose risks to the quality and performance of PPBs.

Metasurfaces, composed of nanoscale optical scatterers, provide a powerful tool for manipulating light in terms of amplitude, phase, and polarization at subwavelength scales, unlocking unprecedented opportunities in the field of highly integrated and miniaturized planar optics, such as imaging metalenses^[30–33], metaholograms^[34–38], reconfigurable optics^[39,40], and beyond. Moreover, metasurfaces offer a versatile approach for crafting diverse spatially structured light beams, with a particular emphasis on PVBs and PPBs. For instance, Liu et al. in 2017 demonstrated the generation of PVBs using three geometric metasurfaces as substitutes for a spiral plate, an axion lens, and a Fourier transform lens, albeit with some complexity in design^[41]. In 2018, Zhang et al. merged phase profiles of these three bulky optical elements into one plasmonic metasurface to achieve focused three-dimensional (3D) PVBs, albeit with reduced efficiency due to high ohmic absorption^[42]. The advent of transmission-type all-dielectric metasurfaces has opened up new possibilities for efficient PVB and PPB generation. Notable recent advancements include the experimental demonstration in 2021 by Liu et al. of broadband generation of PPBs using dielectric spin-multiplexed metasurfaces^[43] and the numerical demonstration by Tian et al. in 2022 of PVB generation with polarization-rotated functionality via a single-layer geometric-phase metasurface^[44]. Additionally, Andrea et al. employed a single metasurface platform to successfully generate compact perfect vector beams^[45] and double-ring perfect vector beams^[46]. Ongoing research continues to flourish in the field of PVBs and PPBs, with notable contributions such as the generation and manipulation of perfect high-order PBs by Cheng et al. using an all-dielectric geometric metasurface^[47], the focused higher-order PB achievement by Gu et al. through a dielectric supercell metasurface^[48], and so forth. However, current metasurfaces can flexibly manipulate the polarization states and OAM of PVBs or PPBs in the x–y plane, i.e., the polarization distributions and OAM still remain constant in every output plane along the propagation path, delving into the exploration of 3D spatial-variation polarization and varying OAM for PVBs and PPBs, which holds immense promise in the realm of structured light beams.

In this study, we present a novel methodology for creating and manipulating multiple multidimensional PPBs utilizing a single-layer all-dielectric geometric metasurface platform. By harnessing a spin-multiplexed scheme propelled by pure geometric phase modulation, the metasurface designs can generate two distinct phase profiles that produce two sets of orthogonal PVBs when subjected to two perpendicular CP incidences. These orthogonal PVB components are superposed to form the desired PPBs. Through precise manipulation of phase and amplitude differences between the two perpendicular PVB components, all possible polarization states of PPBs can be achieved. By adjusting the TCs allocated to two perpendicular PVB components, various polarization orders of PPBs can also be realized. Furthermore, by tailoring the propagation path, the resulting PPBs enable the evolution of polarization states and polarization orders of PPBs in a 3D space. Notably, the tripartite modulation operates independently. To illustrate the efficacy of this recipe, two metasurfaces are meticulously designed: one is for generating a longitudinally propagating PPB array with tailored polarization states along spatially cylindrical helical trajectories, and the other for creating a dual array of PPBs with personalized OAM and polarization eigenstates distributed across two misaligned focal planes. As a proof-of-concept illustration, we showcase an innovative optical information encryption scheme through a single metasurface to encode the personalized polarization state and OAM in parallel channels embedded within multiple PPBs. This work endeavors to establish an ultra-compact and robust platform for the generation of multiple PPBs in the mid-infrared range, potentially advancing their applications in optical encryption, particle manipulation, and quantum optics.

We utilize a design outline for geometric metasurfaces to generate generalized PPBs. Figure 1(a) depicts the operation principles of our presented metasurface. When illuminated by linearly polarized (LP) light, this metasurface facilitates the generation of multiple PPBs that exhibit diverse polarization states and OAM transmitted along customized 3D spatial paths, such as the cylindrical helical trajectories shown in the figure. According to Ref. [43], a PPB can be actualized by integrating two orthogonal polarized PVB components with the same ellipticity yet different TCs in an orthogonal circular polarization basis. The representation for the PPB can be articulated as $$\begin{gathered} |Dm,n⟩=cos(χ/2)exp(iψ/2)|PVBR,lm⟩ \\ +sin(χ/2)exp(−iψ/2)|PVBL,ln⟩, \end{gathered}$$(1)where |PVBR,lm⟩ and |PVBL,ln⟩ denote the right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) PVB components with identical ellipticity but distinct TC numbers of lm and ln, respectively. The terms cos(χ/2) and sin(χ/2) embody the amplitudes of RCP and LCP PVBs, while ψ signifies the relative phase difference between them, where χ∈[0,π] and ψ∈[0,2π]. |Dm,n⟩ represents an arbitrary point on the surface of HyOPS with spherical coordinates (χ,ψ), characterizing a PPB with a unique polarization state.

Given that the PPBs are composed of two orthogonal CP PVBs, our initial consideration involves utilizing geometric metasurfaces to generate a single PPB. Previous studies have shown that PVBs can be achieved in the far field of a Gaussian beam incident on a spiral axicon, essentially as the Fourier transform of a high-order Bessel-Gaussian beam^[12,13,46]. Subsequently, the sequential integration of a spiral phase plate and an axicon forms a conventional optical setup, followed by a lens in a Fourier–Fourier (f−f) configuration. On the back-focal plane of the lens, the electric field of the Fourier lens can be expressed as^[49]E(r,θ)=il−1wgw0exp(ilϑ)exp(−r2+R2w02)Il(2Rrw02),(2)where wg represents the waist of the incident Gaussian beam and w0 corresponds to the Gaussian beam waist at the focal point. R symbolizes the ring radius of the PVBs and equals df/k, where d denotes the axicon parameter and f signifies the focal length of the lens. Il is the modified Bessel function of the first kind. Equation (2) reveals that the intensity profile of PVB is influenced by both the Gaussian function and the modified Bessel function. This outcome suggests that a Gaussian beam traversing through an axicon, a spiral phase plate, and a Fourier lens can generate a PVB carrying TC that aligns with the TC imparted by the spiral phase plate, and its profile can be manipulated by adjusting the parameters of the axicon. Here, we implement the functionalities of these aforementioned elements by amalgamating their phase profiles through a geometric metasurface, as depicted in the bottom panel of Fig. 1(a).

To focus the LCP incidence into a converged PVB at position (xL,yL,f), the requisite phase profile for the spin-multiplexed metasurface is determined by^[49]$$\begin{gathered} φLCP=φlens+φLl+φaxicon+Δδ \\ =2πλ(f−(x−xL)2+(y−yL)2+f2) \\ +lmϑ+2πd(x2+y2)+Δδ, \end{gathered}$$(3)where f represents the focal length, λ is the working wavelength, lm denotes the TC of the generated RCP PVB, ϑ=arctan(y/x) symbolizes the azimuth angle, and d is the parameter governing the diameters of annular intensity of PVBs (PPBs), respectively. Δδ denotes the initial phase difference between LCP and RCP PVB components, which can tailor the polarization states of the merged PPBs. Accordingly, to converge the RCP incident waves into a focused PVB at position (xR,yR,f), the phase criterion of the spin-multiplexed metasurface can be expressed as $$\begin{gathered} φRCP=φlens+φRl+φaxicon \\ =−[2πλ(f−(x−xR)2+(y−yR)2+f2)+lnϑ+2πd(x2+y2)]. \end{gathered}$$(4)

In this expression, ln represents the TC of the produced LCP PVB. Therefore, the total phase profile φtotal of the spin-multiplexed metasurface based on pure geometric phase can be articulated as^[50]φtotal=arg[Mexp(i*φLCP)+Nexp(i*φRCP)],(5)where M and N correspond to the amplitude coefficient of two orthogonal polarization eigenstates, and xL=xR=Δx, yL=yR=Δy.

To produce multiple PPBs with varying polarization states and OAM by utilizing the spin-multiplexed metasurfaces, the overall phase profile φtotal can be delineated as^[50]φtotal=arg{∑sCs[Msexp(i*φLCPs)+Nsexp(i*φRCPs)]},(6)where s signifies the spin-multiplexed order and Cs is the weight factor allotted to each PPB to ensure uniform annular intensity. The amplitude ratio and phase difference between two orthogonal polarization eigenstates determine the polarization states of PPBs on the propagation. Hence, the phase profiles of two orthogonal polarization eigenstates can be represented as $$\begin{gathered} φLCPs=2πλ(fLs−(x−xLs)2+(y−yLs)2+(fLs)2) \\ +lLsϑ+2πds(x2+y2)+Δδs, \end{gathered}$$(7)$$\begin{gathered} φRCPs=−[2πλ(fRs−(x−xRs)2+(y−yRs)2+(fRs)2) \\ +lRsϑ+2πds(x2+y2)]. \end{gathered}$$(8)

In this scenario, the focal positions of multiple PPBs are determined by (Δxs, Δys, fs), where xLs=xRs=Δxs, yLs=yRs=Δys. It should be noted that Δxs and Δys remain constant and are independent of the focal length fs (or the longitudinal propagation position vector z).

In order to facilitate the propagation of multiple PPBs with distinct polarizations and OAM along arbitrary 3D spatial paths, the metasurface design should integrate two critical components: polarization and OAM control, and trajectory shaping. Equations (6)–(8) reveal that, by controlling the amplitude ratio and phase differences between two orthogonal polarization states, the metasurface can possibly generate all polarization states of PPBs carrying tailored OAM, effectively spanning the full HyOPS. On the other hand, to design the transmission path for multiple PPBs, the spin-multiplexed metasurface must skillfully manipulate the phase distribution of the incident wavefront, directing light along specified trajectories. For this purpose, the necessary phase profiles of the metasurface for the LCP and RCP PVB components can be delineated as $$\begin{gathered} φLCPs=2πλ{f(t)Ls−[x−x(t)Ls]2+[y−y(t)Ls]2+[f(t)Ls]2} \\ +lLsϑ+2πds(x2+y2)+Δδs, \end{gathered}$$(9)$$\begin{gathered} φRCPs=−{2πλ{f(t)Rs−[x−x(t)Rs]2+[y−y(t)Rs]2+[f(t)Rs]2} \\ +lRsϑ+2πds(x2+y2)}. \end{gathered}$$(10)

In this scenario, the functions x(t), y(t), and f(t) are dependent on the parameter t and jointly determine the 3D transmission path of the PPBs. By substituting Eqs. (9) and (10) into Eq. (6), one can deduce the phase pattern of the spin-multiplexed metasurface aimed at generating multiple PPBs following personalized 3D transmission routes.

Figure 1(b) depicts the schematic of the proposed geometric metasurface engineered for the production of perfect vector beams. When LCP and RCP beams interact with the metasurface, two PVB eigenstates with opposite chirality, labeled as |PVBR,lm⟩ and |PVBL,ln⟩, are generated. These two eigenstates are superposed to create focused PPBs on the observation plane. The basic unit cell of the metasurface is composed of a Ge2Sb2Se4Te1 (GSST) nanopillar arrayed on a CaF2 square substrate with a subwavelength square lattice constant Px=Py=3μm. GSST is selected as the nanopillar material for its high refractive index (ni=3.19+0.001i) and minimal optical losses at the operational wavelength of 4.4 µm. The nanopillar has a height (h) of 4.8 µm, with its length (L) and width (W) as variable parameters. Geometric phase control of the incident wave is attained by rotating the nanopillars at an angle θ(x,y)=φtotal/2. The unit cell behaves akin to subwavelength half-wave plates (HWPs), with the phase difference between the fast and slow axes set at π to achieve optimal geometric phase manipulation of the incident wave. Following optimization via finite-element method (FEM) simulations, the dimensions of L=1.1μm and W=0.65μm are determined. These dimensions ensure that the phase difference Δϕ between the fast and slow axes is close to π, while maintaining high transmittance (Txx≈Tyy≈1) for the unit cell under both s- and p-polarized waves at the 4.4 µm incident wavelength, as illustrated in Fig. 1(c).

It is worth noting that the feasibility of fabricating GSST-patterned metasurfaces has been successfully demonstrated^[51]. A prototype of the proposed metasurfaces can be experimentally realized by patterning thermally co-evaporated GSST films onto a CaF2 substrate using electron beam lithography (EBL), followed by standard reactive ion etching (RIE). The stoichiometry of the GSST film can be precisely controlled by managing the ratio of evaporation rates of two separate targets: Ge2Sb2Te5 and Ge2Sb2Se5.

Drawing from the aforementioned principles, four metasurface samples (referred to as MF1, MF2, MF3, and MF4) that span an area of π×1202μm2 are meticulously crafted to engender four distinct mode PPBs (denoted as mode 1, mode 2, mode 3, and mode 4). The four metasurfaces share uniform device parameters: M=N=1, d=30μm, f=200μm, xL=xR=Δx=0, and yL=yR=Δy=0 at the design wavelength of λ=4.4μm. The four-mode PPBs produced by these metasurfaces are the superposition of two orthogonal PVBs distinguished by their TCs and phase differences. Specifically, mode 1 PPB features lm,1=1, ln,1=−1, and Δδ=0; mode 2 PPB showcases lm,2=5, ln,2=−1, and Δδ=π/2; mode 3 PPB presents lm,3=−4, ln,3=2, and Δδ=3π/2; and mode 4 PPB encompasses lm,4=8, ln,4=−8, and Δδ=π, respectively. The rotation angle θ of the birefringent GSST nanopillars as a function of spatial coordinates in the four metasurface planes can be deduced from the spatial phase profiles defined by Eqs. (3)–(5).

To illustrate the characteristics of the PPBs generated by these metasurfaces, numerical simulations utilizing the finite-difference time-domain (FDTD) method are meticulously carried out. Perfectly matching layers (PMLs) are strategically incorporated in x, y, and z directions, with incident light gracefully impinging on the metasurface vertically from the substrate side, and the transmitted light being elegantly collected by the field monitor on the opposite side to scrutinize field distributions. Figure 2(a) showcases the simulated electric field intensities in the x–z plane for the four-mode PPBs at the operation wavelength of λo = 4.4 µm, revealing strikingly similar spatial patterns along the propagation direction, irrespective of polarization order and topological Pancharatnam charge.

The polarization order p=(lm−ln)/2, a crucial parameter defining the number of polarization rotations per round trip, as well as the lobe numbers of the PPB pattern, is deftly derived from the anisotropic polarization profile using a linear polarizer. Simultaneously, the topological Pancharatnam charge lp=(lm+ln)/2 can be ascertained. For a rigorous quantitative assessment of the excellence of the generated PPBs, horizontal cross-sections of the annular intensity at the designed focal position z=200μm are eloquently portrayed in Fig. 2(b). The simulated radii are indicated as R1=27μm for mode 1 PPB, R2=28.4μm for mode 2 PPB, R3=29μm for mode 3 PPB, and R4=31.2μm for mode 4 PPB, closely aligning with the theoretical values of R1–4=fNA=fλ/d=29.3μm that are accentuated in Fig. 2(c). Furthermore, Fig. 2(d) meticulously highlights the consistency of the annular intensity radii in two supplementary observation planes at z=140 and z=260μm for the four-mode PPBs, accentuating the uniformity in radii along the transmission direction. Another pivotal metric for evaluating the quality of PPBs is the thickness T of the annular intensity, which is defined as the distance when the maximum intensity value in a specific direction drops to half. Figure 2(e) artfully portrays the thickness of the annular intensity for the four-mode PPBs precisely at the designated focal position z=200μm, effectively showcasing minor fluctuations and a consistent trend. Collectively, these results underscore the resilience of the annular intensity profiles of the PPBs generated by these metasurfaces to variations in polarization order and topological Pancharatnam charge, thereby solidifying the synthetic beams as truly exemplary PPBs.

To comprehensively evaluate the distinctive polarizations of each mode PPB generated by the metasurfaces, six specific points on the HyOPS, encompassing two poles and four equatorial points, are strategically chosen. With a keen focus on the TCs associated with the constituent spin-multiplexed PVBs, it is revealed that mode 1 to mode 4 PPBs, produced by MF1 to MF4, exhibit unique polarization orders (p) and topological Pancharatnam charges (lp). Specifically, mode 1 PPB exhibits a polarization order of p1=1 and a topological Pancharatnam charge of lp,1=0; mode 2 PPB showcases p2=3 and lp,2=2; mode 3 PPB features p3=−3 and lp,3=−1; and mode 4 PPB presents p4=8 with lp,4=0, respectively. The intensity patterns at the focal plane (z=200μm) corresponding to each mode PPB at a free-space wavelength of 4.4 µm are skillfully illustrated in Fig. 2(g) for the selected six points on the HyOPS. Notably, the annular intensity patterns captured via a horizontal linear polarizer when LCP or RCP light interacts with the specified metasurfaces remain consistent, regardless of the polarization order and topological Pancharatnam charge, affirming the nature of the synthetic beams as genuine exemplars of PVBs. Similarly, for the four metasurfaces under LP illuminations, the intensity patterns within the focal planes captured through a polarizer orientated horizontally (x-axis), vertically (y-axis), diagonally (1/4π), and anti-diagonally (3/4π) exhibit invariant contours, independent of polarization order and topological Pancharatnam charge. However, these patterns depict petal-shaped field distributions with distinct lobe numbers, validating the synthetic beams as genuine manifestations of PPBs. Through the anisotropic polarization distribution analysis using a linear polarizer, it can be determined that the number of lobes within PPB patterns is 2 multiplied by the absolute value of the polarization order |p|. As a result, the four-mode PPBs are split into two, six, six, and sixteen lobes, respectively, aligning seamlessly with theoretical expectations.

Additionally, the validation of the anisotropic polarization distribution is further supported by analyzing the Stokes parameters (S0, S1, S2, and S3) and the polarization orientations (Ω) of the four-mode PPBs at specified focal planes, as depicted in Fig. 2(h). The Stokes parameters are defined as follows: S0=IL+IR, S1=Ix−Iy, S2=Iπ/4−I3π/4, and S3=IL−IR, where IL, IR, Ix, Iy, Iπ/4, and I3π/4 represent the intensities of the LCP, RCP, x, y, π/4, and 3π/4 components of the PPBs, respectively. With equal weight factors assigned to two orthogonal polarization eigenstates (|PVBR,lm⟩ and |PVBL,ln⟩) constituting the PPBs, the S3 patterns for the four-mode PPBs approach approximately 0, consistent with the simulation results shown in the fourth column of Fig. 2(h). Furthermore, S1 and S2 patterns for the four-mode PPBs display an even number of lobes, with the maximum and minimum intensity lobes alternating sequentially along the azimuthal direction, and the number of maximum or minimum intensity lobes corresponding precisely to the polarization order 2|p|. Importantly, for PPBs with opposing polarization orders—such as the instances of lm=5, ln=−1 for p=3, and lm=−4, ln=2 for p=−3—the positions of maximum intensity in S1 and S2 are opposite to those of the minimum intensity, underscoring the intricate polarization dynamics within the PPB structures. Based on Ref. [13], the spatial distribution of the polarization orientation angle Ω for the PPBs represented by any point on the HyOPS can be calculated as follows: Ω=arctan(S2/S1)/2. It is evident that the polarization orientations undergo rotations of 4π, 12π, −12π, and 32π per round trip for mode 1, mode 2, mode 3, and mode 4 PPBs, corresponding to the polarization orders of 1, 3, −3, and 8, respectively. Considering that the S0 patterns result from a superposition of LCP and RCP PVBs, they manifest as doughnut-shaped total intensities, with radially, π/4, −π/4, and azimuthally polarized states overlapping within their patterns, correlating to the theoretical phase difference Δδ/2.

To enhance the understanding of the polarization characteristics of the vector fields, Fig. 2(f) visually presents the positions of the four-mode PPBs on the HyOPS by calculating the spherical coordinates (χ,ψ) using the Stokes parameters: $$\begin{gathered} χ=arccos(S3/S0), \\ ψ=arctan(S2/S1). \end{gathered}$$(11)

This graphical representation aligns perfectly with the theoretical framework, indicating that precise manipulation of the phase difference between two orthogonal CP channels allows for the generation of arbitrary linear polarization states located on the equator of the HyOPS. In essence, all these findings validate that the proposed geometric metasurface platform effectively produces PPBs with diverse polarization orders and topological Pancharatnam charges by controlling the assigned TCs to two orthogonal polarization eigenstates. Moreover, adjusting the phase difference Δδ between the two polarization eigenstates facilitates the realization of various linear polarization states of the PPBs, highlighting the versatility and efficiency of the proposed metasurface design.

To showcase the robustness of our innovative methodology, we have developed a spin-multiplexed metasurface capable of generating multiple PPBs with diverse polarizations along arbitrary 3D spatial paths when illuminated by LP light. As a prime illustration, we have engineered a spin-multiplexed metasurface, denoted as MF5, specifically aimed at producing eightfold PPBs along a cylindrical helical path. The MF5 sample spans an area of π×3002μm2. The phase distribution for the MF5 sample can be derived from the specified Eqs. (6), (9), and (10), featuring particular parameters: the operational wavelength λo = 4.4 µm, the spin-multiplexed order s=8, the weight factor assigned to each PPB as C1=1/10, C2=1/8, C3=1/6, C4=1/5, C5=1/4, C6=1/3, C7=1/2, and C8=1, respectively, and the TCs of the resulting RCP and LCP PVBs being lL1=lL2=lL3=lL4=lL5=lL6=lL7=lL8=1 and lR1=lR2=lR3=lR4=lR5=lR6=lR7=lR8=−1. To achieve the desired eightfold PPBs with diverse polarizations, the amplitude coefficient ratios of the two orthogonal polarization eigenstates are set as M1:N1=2:1, M2:N2=1:2, M3:N3=1:2, M4–7:N4–7=1:1, and M8:N8=2:1, respectively. The initial phase differences between the LCP and RCP PVB components are carefully configured as Δδ1=π/4, Δδ2=π, Δδ3=7π/4, Δδ4=0, Δδ5=π/2, Δδ6=π, Δδ7=3π/2, and Δδ8=0, respectively. Importantly, non-unity amplitude ratios are utilized to create PPBs with elliptically polarized states, aligning with points between the north/south poles and the equator on the HyOPS. For enabling the eightfold PPBs along a cylindrical helical path, t-dependent functions are characterized by x(t)=60cos(t)(μm), y(t)=60sin(t)(μm), and f(t)=50*(t+3π/4)(μm), with t taking values at intervals of 0, π/4, 2π/4, 3π/4, 4π/4, 5π/4, 6π/4, and 7π/4. To maintain uniformity in the annular intensity radius of the generated eightfold PPBs (herein R=20μm), the decisive parameters d are computed as d1=25.92μm, d2=34.56μm, d3=43.20μm, d4=51.84μm, d5=60.48μm, d6=69.12μm, d7=77.75μm, and d8=86.39μm, respectively, as per the formula d=f(t)*λ/R.

Figure 3(a) depicts the cross-sectional intensity profiles of Ex-components in the prescribed focal planes for the produced eightfold PPBs following the designated cylindrical helical path. The observed patterns indicate that the generated eightfold PPBs exhibit nearly identical radii, in accordance with the theoretical design portrayed in Fig. 3(b). Furthermore, the Ex-component patterns comprise two lobes with varying orientations separated by dark lines for the eightfold PPBs along the cylindrical helical path, indicating the diverse polarization states carried by these beams, spanning from 0 to 7/8π. In order to visually represent the 3D spatial locations of the eightfold PPBs more effectively, Fig. 3(c) provides a clear illustration of the precise spatial positions of these beams in 3D space. This depiction efficiently partitions the x–y plane into eight sections in the right-side view.

To comprehensively evaluate the unique polarizations exhibited by the eightfold PPBs, Fig. 3(e) presents their component intensities of RCP, LCP, x, y, π/4, and 3π/4 polarizations. When LCP or RCP interacts with the specified metasurface, the intensity profiles of the eightfold beams observed through a horizontal linear polarizer reveal doughnut-shaped patterns with central dark cores, confirming them as two PVBs with orthogonal polarization states. While under LP illuminations, the component intensities of x, y, π/4, and 3π/4 polarizations of the eightfold beams exhibit two lobes with varying orientations separated by the dark lines, signifying them as eightfold PPBs with diverse polarizations along the designated cylindrical helical path. This aligns with the theoretical conclusion that the polarization states of the PPBs are determined by the initial phase difference between the two orthogonal polarization eigenstates. To delve deeper into the polarization behavior of the generated PPBs, Stokes parameters (S0, S1, S2, and S3) and polarization orientations (Ω) were calculated and visualized in rows 7–11 in Fig. 3(e). The patterns observed in S1 and S2 of the eightfold PPBs showcased four lobes with varying orientations dictated by the specific polarization states. It is worth noting that, for the PPBs associated with longitudinal coordinates z4, z5, z6, and z7, the Stokes parameter S3 hovers around 0, hinting an equally weighted superposition of two CP PVBs. Consequently, these four synthesized beams displayed LP characteristics, aligning with points on the equator of the HyOPS. Conversely, an examination of PPBs linked to longitudinal coordinates z1, z2, z3, and z8 revealed a departure from zero in the Stokes parameter S3, signaling an unevenly weighted superposition of two CP PVBs, which resulted in the formation of elliptically polarized PPBs aligned with points on the upper/lower hemisphere of the HyOPS. An intriguing observation was the opposing trend in the S3 value distributions between z1 and z8 compared to z2 and z3, positioning the PPBs at z1 and z8 in the lower hemisphere and those at z2 and z3 in the upper hemisphere. By projecting the resulting eightfold PPBs onto the HyOPS, full coverage of the HyOPS was achieved through a single-layer metasurface design, as illustrated in Fig. 3(d). Despite the consistent 4π rotation per round trip in polarization orientations across all eightfold PPBs, reflective of a polarization order of 1, distinct spatial polarization distributions emerged for the eightfold PPBs, as shown in row 11 of Fig. 3(e).

Essentially, the successful creation of eightfold PPBs with varying polarization states along arbitrary 3D spatial trajectories via a single-layer non-interweaving metasurface highlights the effectiveness and efficiency of the design approach. This breakthrough establishes a theoretical groundwork for encoding and demultiplexing multiple PPBs in forthcoming applications. While maintaining the alignment in topological order and topological Pancharatnam charge of the eightfold PPBs in this section, the adaptable nature of the suggested metasurface platform allows for the bespoke generation of multiple PPBs possessing varied OAM and polarizations.

Our innovative metasurface concept not only enables the simultaneous generation of multiple PPBs with tailored polarization states along arbitrary 3D spatial propagation paths but also allows for precise control over the topological order and topological Pancharatnam charge of these diverse PPBs. To showcase this capability, a spin-multiplexed metasurface variant, designated as MF6, was developed utilizing pure geometric phase modulation. This MF6 can concurrently produce 32-fold PPBs with individualized topological orders and dual polarization states, forming a 4×4 matrix across two misaligned focal planes. The MF6 sample occupies an area of π×2402μm2. The phase profiles necessary for the MF6 configuration can be derived from Eqs. (6)–(8), with specific parameters as follows: the operational wavelength λo = 4.4 µm, the spin-multiplexed order s=32, and the amplitude ratios of two orthogonal polarization eigenstates for the 32-fold PPBs uniformly defined as M1–32:N1–32=1:1. To ensure the non-intersecting arrangement of the 32-fold PPBs in a structured 4×4 matrix layout across the two focal planes, precise parameters were meticulously crafted. To be more specific, the spin-multiplexed order (S), the weight factor (C) assigned to each PPB, the TCs of the resulting RCP and LCP PVBs (lL and lR), and the values for the horizontal displacement (Δx) and vertical displacement (Δy) of each PPB can be seen in Table 1. The focal lengths are designated as follows: f1–16=200μm and f17–32=300μm. For simplicity, we set an initial phase difference (Δδ) 0 to the first 16-fold PPBs on the focal plane at z=200μm, while assigning a phase difference π to the last 16-fold PPBs on the focal plane at z=300μm. This distinction ensures the unique polarization characteristics of each set of PPBs. To maintain a consistent annular intensity radius (R=30μm herein) for all 32-fold PPBs, the parameter d was meticulously fine-tuned. Specifically, a value of d=30μm was assigned to the first set of 16-fold PPBs on the focal plane at z=200μm, while d=44μm was allocated to the remaining 16-fold PPBs on the focal plane at z=300μm. This precise calibration of the parameter d effectively harmonizes the annular intensity radii of all 32-fold PPBs, ensuring uniformity in performance while accommodating the distinct focal lengths of the optical system.

Figure 4(a1) illustrates the operational principle of the proposed MF6 design, aiming at crafting 32-fold PPBs. The input LP illuminations are modulated by the proposed MF6 configuration, and then the output waves propagate to diverse directions and are converged at discrete off-axis positions on two target focal planes. Within each target focal plane, a set of 16-fold output PPBs is housed, characterized by distinct polarization orders and topological Pancharatnam charges structured in a 4×4 matrix format. The polarization orders can be discerned by examining the lobe numbers present in the component intensity of the resultant PPBs, correlating to half of the total lobe numbers, as elucidated in Fig. 4(a2). Theoretically, the polarization orientations of each PPB offer the possibility of infinite and arbitrary adjustments, mirroring half of the initial phase difference (Δδ), as illustrated in Fig. 4(a3).

Figures 4(b) and 4(c) display the simulated cross-sectional intensity profiles of RCP beams across two target focal planes, generated by the MF6 setup when subjected to LCP incident light. The observation unveils uniform radii and thicknesses in the annular intensities of all 32-fold RCP beams, regardless of their varied TCs, signifying the generation of 32-fold RCP PVBs. Upon scrutinizing the Ex and Ey components of the 32-fold beams produced by the MF6 configuration under LP irradiation in Figs. 4(b) and 4(c), all the generated 32-fold beams exhibit petal-shaped field distributions with nearly identical contours yet varying lobe numbers. These variations indicate the emergence of PPBs characterized by distinct topological orders and topological Pancharatnam charges. This empirical finding aligns seamlessly with the projected theoretical configurations. An examination of the electric field vector distribution and polarization orientation will validate specific polarization states for all 32-fold PPBs. Notably, in this scenario, two selected PPBs serve as prototypes, revealing distinct polarization states. Specifically, within the focal plane at z=200μm, 16-fold PPBs exhibit radial polarization, whereas those positioned in the focal plane at z=300μm exhibit azimuthal polarization, in accordance with the specified theoretical parameters. It is worth emphasizing that the polarization states of the 32-fold PPBs in these two focal planes can be individually tailored without interplay. Consequently, theoretically, these PPBs can be configured to manifest up to 32 unique polarization states.

We have demonstrated the precise design of an all-dielectric metasurface to produce 32-fold unique PPBs with independent polarization states and specific polarization orders. This innovative approach, which integrates both OAM and polarization attributes of light into PPBs through parallel pathways, holds significant promise for applications in optical information encryption. The diverse PPBs, exhibiting varying polarization orders and states, can effectively encode various double-digit hexadecimal numbers. To further elaborate, consider a scenario where the absolute value of the polarization order |p| and the polarization orientation Ω of the intensity pattern represent the first and second digits of a double-digit hexadecimal number, respectively. Consequently, each PPB essentially symbolizes a byte of data. By combining n1 distinct values of |p| with n2 different values of Ω for the PPBs, we can encode a total of n1*n2 double-digit hexadecimal numbers. This innovative methodology highlights the flexibility and effectiveness of utilizing metasurfaces for optical data encryption.

For simplicity, we have set the polarization orientations Ω to only two values, 0 and π, representing the first hexadecimal digits 0 and F, respectively. Meanwhile, the polarization orders |p| spans from 1 to 8.5 with increments of 0.5, corresponding to the second hexadecimal digit from 0 to F. By combining these settings—2 values for Ω and 16 values for |p|—for the PPBs, we can represent 32 double-digit hexadecimal numbers, as illustrated in Fig. 4(d). This encoding scheme can effectively map the 26 English letters and 6 function keys on a standard keyboard. The encoding methods and the code chart are explicitly elucidated in Fig. 4(e). Theoretically, the code chart has the capacity to support the encryption and transmission of any textual information, showcasing the potential versatility and applicability of this innovative technique in information security and communication.

As a compelling proof-of-concept demonstration, we utilized the meticulously crafted code chart and character encoding system to implement optical encoding and decoding for information encryption. Let us delve into the scenario where User 1 intends to securely transmit a dataset of sensitive information to User 2, depicted in Fig. 4(g), by translating the plaintext into combinations of two hexadecimal numbers, as detailed in Fig. 4(f). Subsequently, leveraging the assigned hexadecimal numbers, 32-fold PPBs with different values of |p| and Ω are meticulously designed and encrypted on the metasurface, forming what we term the ciphertext [as illustrated in Fig. 4(a)]. Following the encryption process, User 1 sends the metasurface sample alongside a bespoke key to User 2. Upon illuminating the metasurface with horizontally LP light at a wavelength of 4.4 µm, two arrays of PPBs are captured through a linear polarizer [as shown in the middle panel of Figs. 4(b) and 4(c)]. The decryption methodology revolves around deducing that the first hexadecimal digit corresponds to the calculated polarization orientation Ω of the PPBs patterns, while the second digit is inferred from the lobe numbers of petal-shaped patterns. Consequently, through image identification, the 32 double-digit hexadecimal numbers can be effortlessly decrypted. Relying on the code chart, the 115 double-digit hexadecimal digits outlined in Fig. 4(f) can be adeptly deciphered into the desired information, subsequently transmitted as delineated in Figs. 4(g) and 4(h). This seamless encryption and decryption process underscores the efficacy and robustness of the proposed optical information security framework.

Notably, in the encryption process, a double-digit hexadecimal number is represented by combining two states (OAM and polarizations) of PPBs’ annular intensity images. This method effectively conceals the encrypted information, bolstering security measures. Accessing the original data is contingent upon possessing all essential hardware and software components, including the metasurface device serving as the encrypted information carrier, specialized keys for extracting beam intensities, a code chart for deciphering numbers, and a character encoding system for linking numbers to plaintext comprised of diverse characters. Even if an individual were to illicitly obtain the metasurface device and keys, and capture the two states of PPBs’ annular intensity patterns, decoding the accurate hexadecimal numbers would remain unattainable without access to the code chart. The pivotal role played by the code chart and character encoding system is underscored by their adaptable nature—they can be customized and are not rigidly predetermined. Consequently, achieving access to the encrypted information without a precise one-to-one mapping relationship presents an almost insurmountable challenge.

In summary, we have introduced and validated an innovative technique for generating and controlling multiple PPBs with diverse polarizations and OAM precisely beating along customized 3D spatial paths using a specialized single-layer dielectric geometric metasurface for mid-infrared light. Through precise adjustments of the TCs assigned to two perpendicular PVBs, we can craft PPBs capable of carrying arbitrary polarization orders and topological Pancharatnam charges, enabling meticulous control over the OAM properties. By finely tailoring the phase difference and amplitude ratios between LCP and RCP PVB components, the resulting PPBs can embody all possible polarization states across the entire HyOPS. By tailoring and strategizing the trajectories of these diverse PPBs, we can achieve precise manipulation of PPBs in the realm of 3D space. It is important to emphasize that the tripartite modulation mechanism described above operates independently. By harnessing the encoded OAM and polarization states within the multiple PPBs, we have successfully demonstrated a proof-of-concept implementation of optical information encryption. We envision that this research will spark the development of compact flat nanophotonic components for efficient generation and manipulation of structured beams, driving advancements in practical applications such as optical encryption, particle control, and quantum optics.