Introduction

Metasurfaces, composed of sub-wavelength artificial structure arrays, have made a profound impact on the fields of optics and photonics due to their exceptional performance, such as compact size and high resolution, ushering optical components into the era of planar optics. They possess unprecedented ability to manipulate various degrees of freedom, such as phase1,2,3,4,5,6, amplitude5,6, and polarization7,8, through intense light–matter interactions9,10,11. This capability enables metasurfaces to facilitate advanced multiplexing applications, particularly in metasurface holography, also referred to as meta-hologram12,13,14. Recently, a variety of multiplexing techniques for metasurface holography have been explored, including polarization multiplexing15,16,17, wavelength multiplexing18,19,20, incident angle-multiplexing21,22,23, and spatial multiplexing24,25. Among these, orbital angular momentum (OAM) multiplexing has emerged as an especially promising technique26,27,28,29, as OAM offers theoretically unlimited orthogonal channels. This enables the multiplexing of a large number of OAM-dependent information channels within a single meta-hologram30,31,32,33,34,35,36,37,38,39,40. However, conventional OAM-multiplexing holography relies on azimuthally symmetric OAM encoding. By randomly inputting regular OAM states, the encoded information can be easily retrieved, thereby significantly compromising the security of the information. In addition, each channel is encoded with a single constant topological charge (TC) of OAM, which inevitably requires different input vortex beams with distinct TCs of OAM to decode separate holographic images, thereby complicating the generation of multiple holographic channels.

Here, we propose a general method for OAM multiplexing holography using a single input beam that is azimuthally encoded with discontinuous TCs of OAM. Our methodology relies on the fact that the discontinuous arrangement of TC breaks the rotational symmetry of the OAM mode, laying the foundation to encode multiple holographic channels by simply rotating the single input beam. We prove that by strategically arranging the azimuthally distributed TCs, the discontinuous OAM exhibits self-orthogonality at different rotation angles, ensuring minimal cross-talk between channels. Additionally, various discontinuous OAMs can be multiplexed on a single device, further enhancing the holographic channels. To achieve this, a modified weighted Gerchberg–Saxton (WGS) algorithm is introduced for the generation of holographic phase information, which is then encoded in a pure geometry-phase metasurface. Notably, the encrypted holographic images using the discontinuous OAM cannot be decrypted by illuminating with conventional continuous OAM, significantly enhancing the security of the information. Our work paves the way for high-capacity and high-security information transfer and optical networks, addressing the escalating demands of optical and quantum communications, as well as big-data photonics.

Results

Design principle of discontinuous OAM holography

The design principle of discontinuous OAM metasurface holography is illustrated in Fig. 1. In conventional OAM holography, the illuminated OAM mode exhibits rotational symmetry due to the constant TC, resulting in an output holographic image that remains invariant regardless of the incident beam’s rotation angle as shown in Fig. 1a. In contrast, the proposed discontinuous OAM holography leverages a rotational symmetry broken OAM mode, featuring azimuthally encoded TCs, which exhibit azimuth-dependent channel selectivity as shown in Fig. 1b. The encoded holographic image can only be reconstructed when the azimuth of the incident OAM mode aligns precisely with the encoded phase. This enables multiple holographic images to be encoded at different rotation angles of the incident OAM mode, significantly enhancing the multiplexing channel numbers under single beam illumination. Our discontinuous OAM holography offers a security enhancement of at least 7 orders of magnitude compared to traditional OAM holographic encryption (see more details in Supplementary Note 1). Specifically, we encode distinct TCs across varying azimuth, with each TC value assigned to a specific angular region defined as \({l}_{\theta }\). Switching between multiple independent channels—from image “A” to “D”—can be achieved by rotating the input OAM mode. Compared to constant OAM, discontinuous OAM generates multimodal spectral interference due to its azimuthal phase partitioning. This inherent complexity fundamentally resists detection and decryption through spiral spectrum analysis. Consequently, any attempt to reconstruct the encrypted hologram via exhaustive constant OAM modes fails to achieve meaningful reconstruction.

Fig. 1: Schematic illustration of the discontinuous OAM metasurface holography.
Fig. 1: Schematic illustration of the discontinuous OAM metasurface holography.
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a, b Comparison between conventional constant OAM holography and our proposed discontinuous OAM holography. a For conventional OAM holography, the TC remains constant with azimuth. When the incident light is rotated, the output image remains unchanged. b For discontinuous OAM holography, the TC is azimuthally encoded to vary with the angular position, enabling dynamic image switching by rotating the incident beam. c The discontinuous OAM phase (top row) and the resultant electric field distribution (bottom row) of multiplexing Mode 1. d The discontinuous OAM phase (top row) and the resultant electric field distribution (bottom row) of multiplexing Mode 2. The first column of (c, d) shows the original discontinuous OAM. The second column of (c, d) shows the new OAM phase that is rotated anticlockwise by 180° and 90°, respectively. The red dashed boxes in (d) show the overlap regions with unchanged TC distribution after rotation. e The calculated orthogonality of the two discontinuous OAM modes. Mode 1 shows complete orthogonality. Mode 2 shows partial orthogonality due to the partial overlap of the TCs.

To investigate the optical properties of discontinuous OAM, we design a dual-partition OAM by evenly dividing the azimuth into two regions, each encoded with distinct TCs. Generally, the electric field distribution of the output beam by illuminating the discontinuous OAM can be expressed as41,

$$E\left(r,\phi \right)=C{\sum }_{j=-\infty }^{\infty }{a}_{j}{e}^{i(l+j)(\phi \mp \frac{\pi }{2})}{\int }_{0}^{R}{J}_{l+j}({kr}\rho /f)\rho d\rho$$
(1)

where i2 = −1, r and ϕ are the polar coordinates on the output image plane, \(\rho\) is the radial coordinates on the metasurface plane, \(f\) is the focal length, \(\lambda\) is the operating wavelength, \(k=2\pi /\lambda\) is the wavevector, \({a}_{j}\) is the coefficient related to the rectangular function, \(l\) denotes the number of TC, \(J(\cdot )\) is the Bessel function, C is a constant. In the example illustrated in Fig. 1c, the azimuthal plane is divided into two distinct regions (top-left panel of Fig. 1c: the brown segment encodes a negative TC of \({l}_{0-\pi }=-10\), while the blue segment encodes a positive TC of \({l}_{\pi -2\pi }=10\). The electric field distribution of the output light undergoes a characteristic 90° rotation relative to the encoded phase pattern (bottom-left panel of Fig. 1c), with the rotation direction governed by the sign of TC (see more details of the rotation rules in Supplementary Note 2 and Fig. S2). Specifically, a negative TC induces a −90° rotation of the spatial electric field distribution, whereas a positive TC generates a +90° rotation. Therefore, for the dual-partition discontinuous OAM with opposite sign of TCs, the spatial electric field distributions of the two partitioned regions overlap coherently in free space as shown in the bottom-left panel of Fig. 1c, which can be described as,

$$E\propto \cos (10\phi ),\, \phi \in \left[\frac{\pi }{2},\frac{3\pi }{2}\right]$$
(2)

It can be seen that a second-order vortex beam exhibiting a unique “petal-like” intensity pattern is produced, resulting from the interference between the two opposite TCs (see details in Supplementary Note 2 and Fig. S3). However, the phase discontinuity between the two regions and the mismatch in TC at the partitioned boundaries introduce a half-period gap in the petal pattern, breaking the rotational symmetry and resulting in an asymmetric intensity profile.

We further investigate the self-orthogonality of the discontinuous OAM mode at different rotation angles. As illustrated in Fig. 1c, d, two sets of multiplexing configurations with different relative rotation angles—Multiplexing Mode 1 and Multiplexing Mode 2 are analyzed. In Mode 1, the two discontinuous OAMs (first two columns of Fig. 1c) exhibit a relative rotation angle of 180°, showing no overlap in their azimuthal TC distribution, thereby ensuring complete orthogonality. This orthogonality is also manifested in real space as the beam propagates (bottom panel of Fig. 1c). Each discontinuous OAM contributes symmetrically to half of the ring pattern, and their superposition yields a uniform annular intensity profile. In contrast, Mode 2 introduces a 90° azimuthal shift between the two OAMs, showing partial overlap of the TC distribution (red dashed box in the top panel of Fig. 1d), which leads to crosstalk in the spatial electric field distributions. This overlap breaks orthogonality, and the superposition in real space generates an interference pattern with crosstalk. Figure 1e quantifies the orthogonality of the two modes (see calculation details in Supplementary Notes 3 and 4), highlighting the superior orthogonality of Mode 1 compared to Mode 2. As a result, Mode 1 is suitable for multiplexing due to its complete orthogonality. These findings underscore the critical role of spatially non-overlapping discontinuous OAM designs in achieving high-fidelity multiplexing holography.

Design processes of discontinuous OAM holography

The algorithm process for the discontinuous OAM multiplexing holography is shown in Fig. 2a. The holographic phase information for different channels is first generated from sampled images using a WGS algorithm (see details in Supplementary Note 5). The phase is subsequently superimposed with the discontinuous OAM at different rotation angles. The final phase profile is obtained by superimposing all holographic channels, leading to multi-channel OAM holography on a single phase profile. The metasurface is composed of subwavelength TiO2 nanopillar arrays with a fixed height of 600 nm on a SiO2 substrate. These nanopillars are designed as half-waveplates to convert the handedness of circular polarized (CP) light and impose a geometry phase when the nanopillars are rotated at various angles. Therefore, the final phase profile of the discontinuous OAM holography can be encoded pixel by pixel in the geometry-phase metasurface. The width and length of the nanopillars are chosen as L1 = 100 nm and L2 = 290 nm, showing optimized CP conversion efficiency in the visible range (see more details in Supplementary Note 6). The experimental optical setup (see more details in the “Methods” section) and the scanning electron microscopic image of the fabricated metasurface are illustrated in Fig. 2b, c, respectively.

Fig. 2: Algorithm flowchart and experimental optical setup of the discontinuous OAM metasurface holography.
Fig. 2: Algorithm flowchart and experimental optical setup of the discontinuous OAM metasurface holography.
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a The algorithm flowchart for the design of four-channel discontinuous OAM metasurface holography. In each channel, the sampled image is multiplied by a 2D Dirac comb function, followed by a modified WGS algorithm to acquire the holographic phase information. The obtained phase is added with the corresponding discontinuous OAM helical phase. The final phase profile is obtained by superimposing all holographic channels, leading to the design of multi-channel OAM holography in a single phase profile. b Experimental setup of the holographic measurement. L1–L4 lens, P polarizer, QWP quarter-wave plate, SLM spatial light modulator, BS beam splitter, CCD charge-coupled device. c SEM image of the fabricated metasurface under tilted view.

Experiments of single-discontinuous-OAM multiplexing holography

Figure 3a, b show the decoded holograms of two discontinuous OAM multiplexing modes, with simulation results in the second row and experimental results in the third row. In Mode 1, when two discontinuous OAM beams with rotation angles of 0° and 180° are illuminated, a minimal crosstalk between the two images is observed due to their complete orthogonality as discussed in Fig. 1c, and near-perfect Gaussian spot distribution are obtained as shown in the first two columns of Fig. 3a. It should be noted that slight crosstalk between the two channels appears in the experimental results due to fabrication errors. When illuminated by a plane wave, as shown in the third column of Fig. 3a, the two images of “0” and “1” do not overlap, each showing opposite half-ring vortices (see the zoomed-in figures in the bottom-right panel of Fig. 3a): Image “0” shows a right-half vortex, while Image “1” shows a left-half vortex, with their opening directions matching those in Fig. 1c. The overlapping regions of Image “0” and “1” produce a seamless annular “petal-like” intensity profile formed by their coherent superposition. In contrast, Mode 2, with a relative rotation angle of 90° between the two discontinuous incident OAMs, suffers from non-orthogonality as discussed in Fig. 1d, leading to significant crosstalk that degrades image quality as shown in the first two columns of Fig. 3b. Although the decoded images still yield a Gaussian-like spot distribution, the non-overlapping regions for Image “0” retain the right-half vortex, while Image “1” displays a top-half vortex. As a result, under plane wave illumination, the overlapping regions generate an incomplete three-quarter annular pattern.

Fig. 3: Comparison of the orthogonality and the SNR of decoding holograms between the two Modes.
Fig. 3: Comparison of the orthogonality and the SNR of decoding holograms between the two Modes.
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a The input beam (first row), and the corresponding simulated (second row) and experimental results (third row) of dual-channel holography for multiplexing Mode 1. b The input beam (first row), and the corresponding simulated (second row) and experimental results (third row) of dual-channel holography for multiplexing Mode 2. The enlarged view in the bottom panel of (a, b) shows the detailed intensity distribution of each “petal-like” pattern. c SNR of dual-channel holograms for Mode 1 and Mode 2. (d) Normalized SNR of decoded holograms under various rotation angles for Mode 1.

To quantify the imaging fidelity, the signal-to-noise ratio (SNR) is calculated for both modes as shown in Fig. 3c. It is evident that Mode 1 achieves a higher SNR than Mode 2. Figure 3d further demonstrates the azimuth sensitivity of discontinuous OAM. Rotating the incident beam reduces the SNR of the original channel while progressively activating the complementary channel (see more details in Supplementary Note 4). Specifically, at a 0° rotation, Image “0” is fully decoded, while at a 180° rotation, Image “1” is fully decoded, highlighting the rotation-angle selectivity of the incident beam inherent in the discontinuous OAM encoding. This behavior underscores the multidimensional security and dynamic channel-switching capabilities enabled by discontinuous OAM holography.

To validate the scalability of our method, a four-channel discontinuous OAM holography is designed as shown in Fig. 4, where distinct information channels are selectively activated by rotating the incident discontinuous OAM beam with 90° intervals. Figure 4a illustrates the rotational encoding scheme, and Fig. 4b quantifies the SNR of the decoded images across the four channels as a function of rotation angle. At 0°, 90°, 180°, and 270° rotations, the SNR of the decoded images “A”, “B”, “C”, and “D” are normalized to 1, respectively, with the maximal SNR occurring at these angles, while intermediate angles exhibit severe crosstalk. Both simulated (Fig. 4c) and experimental results (Fig. 4d) confirm the azimuth selectivity. The reconstructed patterns at intermediate angles, characterized by hybrid intensity profiles, are detailed in Supplementary Note 7.

Fig. 4: Simulation and experimental results of the four-channel discontinuous OAM holography.
Fig. 4: Simulation and experimental results of the four-channel discontinuous OAM holography.
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a Schematic illustration of the four-channel discontinuous OAM holography. Rotating the incident discontinuous OAM beam enables dynamic switching of the four output images. b Normalized SNR of decoded holographic images under different rotation angles of the incident beam. c Simulated and d experimental results of four-channel holography.

Experiments of multiple polarization and discontinuous-OAM hybrid multiplexing holography

To further expand multiplexing dimensions and enhance channel capacity, we incorporated polarization and multiple pairs of discontinuous OAMs as an additional degree of freedom. Specifically, by integrating two pairs of discontinuous OAM modes with two orthogonal polarization states, we have designed an eight-channel metasurface multiplexed holography in Fig. 5. For left-circularly polarized (LCP) light, two pairs of orthogonal discontinuous OAM modes \(l=-1/l=-15,l=-8/l=-22\) are employed to encode four distinct images. Similarly, for right-circularly polarized (RCP) light, the set of OAM modes \(l=-7/l=-15,l=-10/l=10\) encodes another four independent images. The superimposed phase of LCP light denoted by \({\sigma }_{+}\) is inverted due to the opposite phase response between RCP and LCP, as shown in Fig. 5a. Consequently, the phase profile can be expressed as,

$$\varphi={\arg }\left[{\sum }_{n=1}^{N}{e}^{\;j\left({\varphi }_{{\sigma }_{-}}^{n}+{\phi }_{r1}^{n}\right)}+{\sum }_{m=1}^{M}{e}^{\;j\left({-\varphi }_{{\sigma }_{+}}^{m}-{\phi }_{r2}^{m}\right)}\right]$$
(3)

where channel n = 1, 2,…, N, with N being the channel number of discontinuous OAM holography for RCP, m = 1, 2,…, M, with M being the channel number of discontinuous OAM holography for LCP, \({\sigma }_{-}\) and \({\sigma }_{+}\) denote RCP and LCP, respectively, \({\varphi }_{{\sigma }_{-}}^{n}\) and \({\varphi }_{{\sigma }_{+}}^{m}\) is the calculated phase by the WGS algorithm. The phase of discontinuous OAM \({\phi }_{r1}^{n}\) and \({\phi }_{r2}^{n}\) is then added to the holographic phase.

Fig. 5: Polarization–OAM hybrid encoding.
Fig. 5: Polarization–OAM hybrid encoding.
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a The demonstration of a geometry-phase metasurface supporting the polarization-OAM hybrid encoding with a non-interleaved method in each channel. F represents the Fourier transform. Simulated and experimental results of reconstructed images when b LCP and c RCP light beams carrying discontinuous OAM are incident on the metasurface.

The demonstration of a geometry-phase metasurface supporting the polarization-discontinuous OAM hybrid encoding is based on a non-interleaved method35. We integrated two-channel discontinuous OAM encoding with polarization multiplexing. When illuminated with an LCP beam carrying TC of \(l=1/l=15\), the letter “M” is successfully decoded. When the metasurface is rotated by 180°, the decoded image switches from letter “M” to letter “E”, demonstrating dynamic channel switching through angular control as shown in Fig. 5b. Similarly, when illuminated with LCP beam carrying \(l=8/l=22\), the letter “T” is decoded, and a 180° rotation of the metasurface enables switching from “T” to “A”. For RCP beam illumination, four distinct images “1”, “2”, “3”, and “4” can likewise be decoded as shown in Fig. 5c, demonstrating polarization multiplexing capability.

The reconstructed images exhibit near-perfect Gaussian spot profiles. The polarization selectivity and azimuthal selectivity of discontinuous OAM ensure that each channel is activated only when the polarization and the alignment of the incident beam match the encoded phase profile, thereby enabling dynamic, high-capacity, and high-security holography with cryptographic robustness. Notably, the numerals and letters in all the holographic images shown in Figs. 3–5 are designed in the same spatial position, yet exhibit almost no crosstalk between different channels.

Discussion

We have successfully demonstrated the discontinuous OAM holography, which fundamentally advances optical information encoding by integrating azimuthal selectivity with OAM multiplexing. Unlike conventional OAM systems, where rotational symmetry allows for unrestricted decoding, our proposed discontinuous OAM introduces an enhanced security mechanism by breaking the rotational symmetry of the TC distributions. The holograms can only be decoded when both the TC and azimuth of the incident discontinuous OAM mode precisely match the encoded phase. The scalability of discontinuous OAM is exemplified through four- and ten-channel holography, where rotating the incident beam sequentially activates distinct holographic images. This work also explores the relationship between spatial electric field distributions overlap and orthogonality in different discontinuous OAM modes, which provides a general design methodology for multi-channel irregular OAM holography. By further integrating additional degrees of freedom, such as polarization and wavelength, the channel capacity can be exponentially expanded, paving the way for ultra-high-dimensional multiplexing. Our proposed discontinuous OAM holography redefines the paradigm of secure optical information systems. Its angular selectivity, scalability, and compatibility with multi-dimensional encoding make it a cornerstone technology for next-generation secure communications, quantum cryptography, and information storage. By combining structured light manipulation with cryptographic principles, this work may open new frontiers in photonic security and high-capacity data transmission.

Methods

Sample fabrication

The metasurface samples are prepared using electron beam lithography (EBL). First, a 600 nm-thick titanium dioxide (TiO2) film is deposited on a silicon dioxide (SiO2) substrate by sputtering evaporation. A polymethyl methacrylate (PMMA) layer is evenly spin-coated onto the TiO2 film. After EBL, the PMMA resist is developed, and a Cr film is deposited on the PMMA using electron beam evaporation (EBE). The undeveloped PMMA is removed, and the TiO2 is etched using reactive ion etching. The process end with the removal of the Cr mask by chemical etching.

Optical setup

The experimental optical setup for the discontinuous OAM holography is shown in Fig. 2b. The laser from a supercontinuum laser source (SC-PRO, Wuhan Yangtze Soton Laser Co., Ltd., 430–2400 nm) filtered by an acousto-optic tunable filter (AOTF, Wuhan Yangtze Soton Laser Co., Ltd., 400–2000 nm) is initially expand the beam through the objective lens (L1, NA = 0.65) and then directed through a beam splitter, projected perpendicularly onto a spatial light modulator (SLM, HOLOEYE PLUTO-2.1-NIR−135, German). This is to generate a vertically emitted discontinuous OAM beam. Then, lens L2 (f = 150 mm), lens L3 (f = 50 mm), and lens L4 (f = 75 mm) are used to focus the discontinuous OAM on the metasurface. A set of polarizer and quarter-wave plate is used to convert the polarization of the incident beam from linear polarization to LCP light. A scientific complementary metal-oxide-semiconductor camera (Dhyana 400BSI V2, Tucsen, China) is used to extract the holographic images.