a Schematic of the generation of 3D PVVBs with customized 3D intensity trajectories and arbitrary polarization orders in spatially separated parallel channels via a monolithic metasurface. b The two spin-dependent phase profiles imparted by the metasurface for generating two RCP and LCP 3D perfect vortices with different topological charges. c Schematic of the meta-atom of the metasurface. d The intensity (top) and phase (bottom) of the simulated complex cross-polarization transmission coefficients for the 6561 SiC nanopillars. The green dots indicate the location of 16 selected SiC nanopillars. e The SEM images of nanopillar arrays of a fabricated SiC metasurface.

To achieve a 3D PVVB, we first consider a theoretical model of a 3D perfect vortex with the topological charge of l and a customized 3D intensity trajectory composed of a series of disjoint light points. Here, the complex amplitude of a RCP or LCP 3D perfect optical vortex, modulated by the Dirac function, can be defined as:

where ?0(??) and ?0(??) are the parametric coordinates of a 3D’s closed trajectory, ??=φ, and n is the number of discrete light points. To generate the light field expressed by Eq. (2), one can use the Fresnel diffraction integral to calculate the field distribution in the initial (aperture) plane at z = 0. The complex amplitude of the required initial scalar field is approximatively derived as: ???(?,?)∝∑???[??0(??)?]?−??[?2+?0(??)2]/[2?0(??)]???? (see details in supplementary Section S1). k = 2π/λ, and ?? is a l-order Bessel function of the first kind with a parametric radial wavevector of ??0(??)=??0(??)/?0(??). The light field expressed by the function ?? can be approximatively created by using a deformed axicon phase element with a phase profile of ??0(??)?. According to ???(?,?), it is possible to generate a 3D perfect vortex by utilizing a pure-phase optical element to encode a superposed phase profile of φtotal=φdfd-axicon+φdfd-lens+φsprial. These phase profiles are given by:

Here, the 3D closed rounded polygonal curve in this work are given by ?0(??)=?0(1−?cos????) and ?0(??)=?0(1−?sin????). This curve can be regarded as a 3D stretching deformation of a 2D circle with a radius of ?0 at a distance of ?0 in space. The parameters s and p control the corner smoothness and the number of sides for the polygonal curve, respectively; the parameters u and v control the longitudinal height and the times of fluctuations in the polygonal curve, respectively. For given an initial light field of ??φtotal at z = 0, we numerically analyze its diffracted light field at a certain propagation distance by using Fresnel diffraction integral. As a simple example, the simulated transverse intensity and phase distributions of different perfect vortices with topological charges of l = 1, 20 and 30 in 3D space are shown in supplementary Fig. S1. The transverse intensity profiles of these perfect vortices are nearly identical at the same z propagation distances, demonstrating their perfect properties that the beam sizes are nominally independent of their topological charges. Therefore, thanks to the free adjustment of the light intensity and helical phase distribution in 3D spatial structure for the perfect vortex, any 3D PVVB with customized intensity trajectory and arbitrary polarization order can be constructed on a HyOPS via the linear superposition of RCP and LCP 3D perfect vortices carrying different topological charges.

To create 3D PVVB in a highly efficient and integrated manner, a monolithic dielectric metasurface is used to provide two distinct spin-dependent phase profiles φtotal,RCP and φtotal,LCP (Fig. 1b) of different topological charges ?? and ??, without additional amplitude modulation and other optical elements. This polarization-multiplexed metasurface would enable the simultaneous generation and coaxial superposition of arbitrary RCP and LCP 3D perfect vortices, free of theoretical energy loss. Based on the synchronous modulation principle of propagation and geometric phase, a series of meta-atoms with varying in-plane dimensions is required to act as subwavelength half-wave plates with high transmission efficiencies over a full 2π phase coverage at the operating wavelength and arranged at specific spatial rotation angles to construct the metasurface, completely decoupling the two different phase profiles of incident orthogonal circular polarizations. As illustrated in Fig. 1c, the meta-atom of the metasurface is composed of a rectangular SiC nanopillar with a fixed height H of 1000 nm arranged on a silica substrate with a square lattice period P of 380 nm. Here a group of 6561 SiC nanopillars with varying lengths (L) and widths (W) ranging from 80 nm to 280 nm is numerically simulated via the finite-difference time domain algorithm (see details in Methods). Figure 1d shows the intensity (top) and phase (bottom) of the simulated complex cross-polarization transmission coefficients for the above SiC nanopillars. A set of 16 SiC nanopillars (denoted by green dots) are selected to cover a full 2π phase range and exhibit high cross-polarization conversion efficiency (PCE) at the operation wavelength of 630 nm. The average cross-PCE and co-PCE of these selected SiC nanopillars are 90.82% and 0.09%, respectively, which ensures the generation efficiency of these vortices. We fabricate two SiC metasurface devices for the generation of different 3D PVVBs by using a method compatible with the semiconductor process. This process mainly includes the plasma enhanced chemical vapor deposition of SiC thin films, the electron beam lithography (EBL) for patterning metasurface arrays and the inductively coupled plasma (ICP) reactive ion etching of SiC nanopillar arrays (see details in Methods). Figure 1e shows scanning electron micrographs (SEM) of nanopillar arrays of the fabricated SiC metasurface.

Experimental demonstration of 3D PVVBs

We design and fabricate two monolithic polarization-multiplexed SiC metasurfaces to create two 3D PVVBs (PVVB1 and PVVB2) along a customized 3D curve with the parameters of s = 1/20, p = 2, u = 7.5 and v = 1. PVVB1, with polarization order ??=+5, is the superposition of RCP perfect vortex 1 with ?1=+15 and LCP perfect vortex 2 with ?2=+5; while PVVB2, with polarization order ??=−15.5, is the superposition of RCP perfect vortex 3 with ?3=−23 and LCP perfect vortex 4 with ?4=+8. Both metasurfaces share the same parameters: ?0=60?m and ?0=300 μm.

We first characterize two pairs of RCP and LCP 3D perfect vortices generated by the two SiC metasurfaces by using an experimental setup shown in supplementary Fig. S2. We capture 201 transverse intensity images of each 3D perfect vortex at 2 ?m intervals along the z direction, ranging from z = 0 ?m (i.e., the surface of the metasurface) to z = 400 ?m. The 3D light fields of the four 3D perfect vortices reconstructed from these image sequences are shown in Fig. 2a. The 3D contours of these perfect vortices appear very similar. Additionally, we quantitatively compare the sizes of the x-y plane intensity distributions (Fig. 2b) of the four 3D perfect vortices at z = 300 μm and the cross sections of these in-plane intensity images along the white dashed lines are shown in Fig. 2c. A size growth factor η of two orthogonally polarized perfect vortices with topological charges of ?? and ?? is defined as: η = (???−???)/???, where |??|<|??|, and d is the distance between the two points with the highest intensity on the white dashed line path, representing the size of optical beams along a certain direction. The distance d in the y direction is measured as, for SiC metasurface 1: 134.51 μm for perfect vortex 1 with ?1= +15, 132.63 μm for perfect vortex 2 with ?2= +5; for SiC metasurface 2: 134.95 μm for perfect vortex 3 with ?3= −23, and 133.85 μm for perfect vortex 4 with ?4= +8, respectively. The size growth factor η is calculated as: 1.42% for perfect vortex 1 and perfect vortex 2, and 0.82% for perfect vortex 3 and perfect vortex 4. To determine their topological charge numbers, we measure the 3D light fields of the interference patterns for the four generated 3D perfect vortices by using a setup shown in supplementary Fig. S3. The corresponding 3D views of these reconstructed interference patterns and their projections on the x-y plane (labelled XY view) are presented in Fig. 2d. The numbers of petals of the four interference patterns are 15 for perfect vortex 1, 5 for perfect vortex 2, 23 for perfect vortex 3, and 8 for perfect vortex 4, corresponding to the absolute values of their topological charges l of 15, 5, 23, and 8, respectively. The petal arrangement direction of perfect vortex 3 is opposite to that of the other three perfect vortices, indicating that its topological charge has the opposite sign compared to the others. These experimental results show that the sizes of the 3D perfect vortices generated by our method exhibit minimal change with increasing topological charge, suggesting that the customized intensity trajectories are largely independent of the vortices’ topological charges.

Fig. 2: Experimental generation of RCP and LCP 3D perfect vortices based on monolithic SiC metasurfaces.

a The measured and reconstructed 3D light fields of two pairs of generated 3D perfect vortices including RCP 3D perfect vortex 1 with l of +15, LCP 3D perfect vortex 2 with l of +5, RCP 3D perfect vortex 3 with l of −23, LCP 3D perfect vortex 4 with l of +8, respectively. For simplicity, perfect vortex 1 to perfect 4 in the figure are labeled as “PV1” to “PV4”. b, c The measured transverse intensity images of four 3D perfect vortices at z = 300 μm and their cross sections along the white dashed lines. d The measured 3D interference patterns corresponding to the four 3D perfect vortices in (a).

By adjusting the polarization state of the light incident on the fabricated metasurfaces, we can achieve controlled generation of various states of the 3D PVVBs. Here, six incident polarization states, including RCP, LCP and four linearly polarized states, are selected, and their corresponding coordinates (denoted by green dots) are displayed on the HyOPS in supplementary Fig. S4. To exhibit their 3D light fields, we experimentally capture a series of transverse intensity distributions of various states for the generated 3D PVVBs at 2 μm intervals along the light propagation direction. Figure 3a, b show the reconstructed 3D intensity profiles of the states I to VI after filtering through a horizontal linear polarizer (denoted by green arrows) for PVVB1 and PVVB2, respectively. As expected, the measured intensities of the states II to V exhibit 3D petal-like patterns with 2|??| lobes, i.e., 10 lobes for PVVB1 (?? = +5) and 31 lobes for PVVB2 (?? = -15.5), which arises from the azimuth distributions of their spatially anisotropic linear polarization along the customized 3D trajectories. Furthermore, the 3D light fields of the state II of PVVB1 and PVVB2 are reconstructed from the corresponding image sequences captured without any polarization analyzers, as shown in Fig. 3c, f. The XY views of reconstructed 3D intensity profiles for PVVB1 and PVVB2 are also presented in Fig. 3d, g. We observe that the slanted, continuous elliptical intensity profiles of the generated 3D PVVBs with different polarization orders exhibit highly similar contours. The x-y plane intensity profiles at z = 300 ?m and their cross sections along the y direction are shown in Fig. S5, with the measured distances d being 134.4 μm for PVVB1 and 134.4 μm for PVVB2, which are close to theoretical values of 133.0 μm.

Fig. 3: Experimental demonstration of 3D PVVBs with customized intensity trajectories.

a, b The measured and reconstructed 3D spatial intensity distributions of 6 states for the two 3D PVVBs (namely PVVB1 and PVVB2) with polarization orders ?? of +5 and −15.5 after filtering through a horizontal linear polarizer (denoted by green arrows). These output states, I to VI, are generated by illuminating metasurfaces with RCP, four linearly polarized, and LCP states of lights (denoted by white arrows). c, f The measured and reconstructed 3D spatial intensity distributions of the state II for the PVVB1 and PVVB2 without any polarization analyzer. d, g The projections of the 3D spatial intensity distributions in (c, f) on the x-y plane. e, h The measured and reconstructed 3D distributions of polarization azimuth of the state II for the PVVB1 and PVVB2. The polarization orientation parallel to the x-axis is defined as 0 or π rad.

Moreover, the 3D spatial distributions of the polarization azimuth for state II of both PVVB1 and PVVB2 are quantitatively measured using Stokes polarimetry, as shown in Fig. 3e, h. It can be seen that the linear polarization orientations rotate by rotate +10π and -31π per round trip in the clockwise direction for PVVB1 and PVVB2, respectively. The average cross-polarization conversion efficiencies of the two polarization-multiplexed metasurfaces are measured as 65.91% for RCP to LCP, and 65.37% for LCP to RCP, respectively (see details in Methods). These results demonstrate that the proposed monolithic SiC metasurfaces can generate arbitrary PVVBs with 3D intensity trajectories, independent of their topological charges or polarization orders.

Optical information encryption using 3D PVVBs

The metasurface-based 3D PVVB can offer a compact, low-loss and high-capacity platform to advance optical information security technology. A variety of distinct 3D PVVBs featuring multiple parameters could be used as different information carriers for optical encoding. As a simple example of demonstration, a series of 3D PVVBs, characterized by three geometrical parameters v, ?0, p and a polarization order ??, are chosen to encode different 8-digit binary numerals. Specially, the four parameters—v with 2 values, ?0 with 2 values, p with 4 values, and ?? with 16 values—are set to denote the first digit, the second digit, the third to fourth digits, and the fifth to eighth digits of an 8-digit binary numeral, respectively. As such, the binary numerals from 00000000 to 11111111 (corresponding to the hexadecimal numerals from 00 to FF) can be represented by 256 different kinds of 3D PVVBs (Fig. 4a). The value ranges and the number of possible values for the four parameters are also shown in Fig. 4a. A code chart related to the hexadecimal serial numbers of 3D PVVBs and corresponding beam parameters is constructed and shown in supplementary Table S1. For instance, Fig. 4b shows the intensity and polarization distribution of a 3D PVVB with v = 1, ?0 = 16 μm, p = 5 and ??= 8.0. It features a 3D rounded pentagon intensity trajectory with a smaller outer profile and its linear polarization orientations rotate 16π rad per round trip in the clockwise direction. We can know that it denotes a binary numeral “00111111” (or a hexadecimal numeral “BF”) according to the code chart.

Fig. 4: Deign principle and experimental demonstration of optical encoding and image encryption using a metasurface-based 3D PVVB array.

a A series of 3D PVVBs, characterized by four parameters v, ?0, p, and ??, represent a group of 8-digit binary numerals, from 00000000 to 11111111 (corresponding to the hexadecimal numerals from 00 to FF). v, ?0, p and ?? denote the first digit (blue), the second digit (green), the third to fourth digits (yellow), and the fifth to eighth digits (red) of an 8-digit binary numeral, respectively. b shows an example of the intensity and polarization distribution of a 3D PVVB with v = 1, ?0 = 16 μm, p = 5 and ??= 8.0, denoting a binary numeral “00111111” (or a hexadecimal numeral “BF”). c The image of the generated QR code with 24 ? 24 black and white blocks. d A 24-by-3 binary matrix transformed from the QR code. e An 8-by-9 binary matrix reshaped from (d). f An 8-by-9 array of hexadecimal serial numbers of required 3D PVVBs. g The values of four parameters for the 3D PVVB array. h The SEM image of a portion of the fabricated metasurface ciphertext. i, j The two measured and reconstructed 3D light fields (namely State 1 and State 2) of the 3D PVVB array by using RCP and x-polarized states of incident lights. k The identified values of v, ?0, p, and ?? for the generated 3D PVVB array. v, ?0, p and ?? of each PVVB are labeled by the square blue, green, yellow and red background. l The decrypted image of the QR code.

As a proof of concept, we demonstrate an information encryption scheme for optical anti-counterfeiting by using metasurface-generated 3D PVVBs. QR code is a widely used two-dimensional matrix barcode that stores data information. As an encrypted image, a 2D QR code with 24 ? 24 black and white blocks (Fig. 4c) is transformed into a 24 ? 24 binary matrix where each element, based on the block color, is either 0 or 1. Grouping every eight elements from each row into 8-digit binary numbers, the 24 ? 24 matrix is condensed into a 24 ? 3 binary matrix (Fig. 4d), and reshaped into an 8 ? 9 binary matrix (Fig. 4e) for facilitating subsequent metasurface fabrication and characterization. This matrix is then converted into a hexadecimal matrix, where each 2-digit numeral represents the serial numbers of the required 3D PVVBs (Fig. 4f). By referencing the code chart, the user obtains four parameters of the 3D PVVB array (Fig. 4g) and calculates the RCP and LCP phase profiles. A SiC metasurface, termed ciphertext (Fig. 4h), is designed and fabricated based on these phase profiles, encoding the QR code image onto this minimalist device.

To decode the message, RCP and x-polarized light are incident onto the metasurface ciphertext, and two 3D optical fields, corresponding to two states of the 3D PVVB array (State 1 and State 2), are captured after filtering through an LCP analyzer and a y-direction linear polarizer, as reconstructed in Fig. 4i, j. Projections of these states onto the x-y plane are also shown in supplementary Fig. S6. From State 1, by analyzing the height along the z-direction, the maximum outer ring size, and the annular shape on the x-y plane, the values of v, ?0, and p can be identified. For State 2, by counting the number of lobes, N, in the petal-like intensity pattern, the value of ?? can be determined as N/2. For instance, a 3D PVVB in the first row and ninth column of State 1 has a height of ~18 μm in the z-direction, a maximum outer diameter of ~42 μm, and a rounded triangular ring shape, with the corresponding values of v, ?0, and p recognized as 1, 21 μm, and 3, respectively. The intensity pattern of the 3D PVVB at the same position in State 2 has eight petals, indicating ?? = 4.0. In this way, the four parameters of the entire 3D PVVB array are determined and displayed in Fig. 4k, where v,?0, p, and ?? of each PVVB are labeled with blue, green, yellow, and red backgrounds, respectively. The hexadecimal serial number of the 3D PVVB in the first row and ninth column of the array is recognized as “57” by querying the code chart, and the decoded number of the entire 3D PVVB array is shown in supplementary Fig. S7. By sequentially converting this hexadecimal matrix into a binary matrix, reshaping it into a 24 ? 3 matrix, and splitting each 8-digit binary numeral into single-digit elements, a 24 ? 24 binary matrix is decrypted. Finally, this matrix can be transformed into the original 2D QR code (Fig. 4l).