Discontinuous orbital angular momentum metasurface holography

Domain: physics
Published: 2025-11-27
Editorial outcome: major_revision

Abstract
Orbital angular momentum (OAM) multiplexing holography has emerged as a pivotal technology for high-capacity optical communication, encryption and display, but it requires multiple inputs for decoding and its security remains constrained due to the rotational symmetry of topological charge (TC) distribution in conventional OAM modes. Here, we introduce a general paradigm of OAM multiplexing holography that enables multi-channel holographic encoding using a single incident light. Our methodology leverages a discontinuous OAM with a spatially varying TC across the azimuth, which breaks the rotational symmetry and imposes angular selectivity for information retrieval. Notably, by rationally designing the TC distribution, the discontinuous OAM exhibits self-orthogonality at different rotation angles, laying the foundation for multiplexed holography. A modified weighted Gerchberg-Saxton algorithm is developed to calculate the holographic phase profile, which can then be encoded onto a pure geometry-phase metasurface. By further integrating different pairs of discontinuous OAMs, we successfully expand the channel capacity for holographic multiplexing, significantly advancing high-security and high-capacity optical information encryption. Our work establishes discontinuous OAM as a versatile platform for secure optical communications, high-density data storage, and dynamic holographic displays, bridging the gap between structured light manipulation and cryptographic robustness.

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 phase1–6, amplitude5,6, and polarization7,8, through intense light–matter interactions9–11. This capability enables metasurfaces to facilitate advanced multiplexing applications, particularly in metasurface holography, also referred to as meta-hologram12–14.

Recently, a variety of multiplexing techniques for metasurface holography have been explored, including polarization multiplexing15–17, wavelength multiplexing18–20, incident angle-multiplexing21–23, and spatial multiplexing24,25. Among these, orbital angular momentum (OAM) multiplexing has emerged as an especially promising technique26–29, as OAM offers theoretically unlimited orthogonal channels. This enables the multiplexing of a large number of OAM-dependent information channels within a single meta-hologram30–40. 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(θ). 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.

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(r,ϕ) = C \sum_{j=-1}^{0} a_j e^{i(l+j)(ϕ∓π/2)} \int J_{l+j}(krρ/f) ρ dρ (1)

where i^2 = −1, r and ϕ are the polar coordinates on the output image plane, ρ is the radial coordinates on the metasurface plane, f is the focal length, λ is the operating wavelength, k = 2π/λ is the wave-vector, a_j is the coefficient related to the rectangular function, l denotes the number of TC, J(·) 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 = −10, while the blue segment encodes a positive TC of l = 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 ∝ cos(10ϕ), ϕ ∈ [π/2, 3π/2] (2)

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.

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.

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.

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 = ±15, l = ±22 are employed to encode four distinct images. Similarly, for right-circularly polarized (RCP) light, the set of OAM modes l = ±15, l = ±10 encodes another four independent images. The superimposed phase of LCP light denoted by σ+ 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,

φ = arg [\sum_{n=1}^{N} e^{i(φ_n^{σ-} + ϕ_n^{r1})} + \sum_{m=1}^{M} e^{i(φ_m^{σ+} + ϕ_m^{r2})}] (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, σ- and σ+ denote RCP and LCP, respectively, φ_n^σ and φ_m^σ is the calculated phase by the WGS algorithm. The phase of discontinuous OAM ϕ_n^r1 and ϕ_n^r2 is then added to the holographic phase.

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 = ±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 = ±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.

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 onto 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.

Data availability
The data that support the figures and other findings of this study are available from the corresponding authors upon reasonable request.

Code availability
The code that support the figures and other findings of this study is available from the corresponding authors upon reasonable request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors of manuscript NCOMMS-25-25303 proposed and demonstrated a novel kind of metasurface holography based\non discontinuous orbital angular momentum (OAM). This discontinuous OAM has a spatially varying TC across the\nazimuthal angle, and thus breaks the rotational symmetry of the beam for information retrieval. Generally, this study\ndifferentiates itself from traditional OAM-based holographic approaches through the employment of discontinuous OAM\nmodes. However, after careful evaluation, I am currently hesitant to recommend this study, based on the following reasons.\n1. Their proposal provides significance such as using a single beam in information retrieval. However, this method may also\nbring along limitations, such as extra complexities in precisely controlling the rotation angle and spatial position of the input\nbeam in retrieving the information.\n2. The conventional OAM holography has significant merit of information encryption. I’m doubt of the scenarios with\nnecessity to further enhance the security of the information, especially at the cost of adding extra complexity in information\nretrieval, because the authors said ‘any deviation in the rotation angle, or illumination with conventional constant OAM, will\ntrigger severe inter-channel crosstalk,’.\n3. In the text, the authors mentioned that ‘We prove that by strategically arranging the azimuthally distributed TCs, the\ndiscontinuous OAM exhibits self-orthogonality at different rotation angles, ensuring minimal cross-talk between channels.’\nThe comparison of cross-talk with the conventional metasurface holography and state-of-the-art holography techniques with\nsmall cross-talk has not been fully discussed.\n4. The authors mentioned that ‘This enables multiple holographic images to be encoded at different rotation angles of the\nincident OAM mode, significantly enhancing the multiplexing channel numbers under single beam illumination.’ However, in\nthe experiment, the detail of how to precisely control the rotation angle in coding and decoding has not been fully discussed.\n5. Does the phase distribution necessarily require topological charges with opposite signs, or could the same functionality\nbe achieved using same-sign charges with discontinuous phase profiles (such as TC=1, 2, 3, 4)?\n6. Are the light spots at the letter edges in the experimental optical field of Figure 4d manually traced? If so, this image\nprocessing procedure should be explicitly declared in the main text.", "reviewer_2": "This manuscript proposes a new paradigm of metasurface holography based on discontinuous orbital angular momentum\n(OAM), which enables multi-channel holographic encoding under a single incident beam by breaking the rotational\nsymmetry of conventional OAM modes. The work demonstrates certain innovation in enhancing optical information security\nand capacity, particularly in the design and application of spatially varying topological charge (TC) distributions. However,\nconsidering the high academic standards and scientific rigor expected by Nature Communications, the current manuscript\nstill exhibits certain shortcomings in theoretical depth, experimental validation, and practical feasibility, falling short of the\npublication threshold. To improve the overall quality of the manuscript, the authors are advised to systematically address the\nfollowing key issues.\nFirst, traditional OAM multiplexing holography has shown great potential in high-capacity communication due to its\npropagation-invariant and stable characteristics. However, the proposed discontinuous OAM mode, while breaking the\ncentral symmetry, may adversely affect the beam’s transmission properties. For example, it remains insufficiently verified\nwhether the structural integrity of the beam can be preserved after transmission through optical fibers (such as few-mode or\nring-core fibers) or through free space. Therefore, the application prospects emphasized in the manuscript require further\nsubstantiation, and it is recommended that the authors supplement the manuscript with relevant experimental or simulation-\nbased analyses of transmission performance.\nSecond, although the authors claim that the method effectively enhances the capacity of encryption channels, the\nexperimental results presented in the manuscript demonstrate only a limited number of multiplexed channels—just a few in\ntotal. In contrast, traditional OAM holography has already achieved tens or even hundreds of multiplexed channels, with\nsome metasurface-based schemes reaching over ten channels. As such, the advantage of the current approach in terms of\ncapacity enhancement is not yet clearly demonstrated. The authors are encouraged to more explicitly clarify the core\ncompetitiveness of their method compared to traditional techniques and to further validate its potential by expanding the\nnumber of channels or integrating additional degrees of freedom.\nThird, although conventional centrally symmetric OAM multiplexing holography imposes certain alignment requirements on\nthe devices, its inherent symmetry to some extent reduces the difficulty of alignment. After breaking this symmetry, however,\nthe precision required for alignment may significantly increase, especially when rotational operations are introduced, where\naccurate centering becomes critical to decoding performance. The manuscript should therefore include a discussion on\nalignment error tolerance and assess its impact on overall system performance.\nFourth, while the manuscript emphasizes the advantage of discontinuous OAM in resisting spiral spectrum analysis, it lacks\nthorough security analysis under more complex attack scenarios, such as phase reconstruction based on machine learning\nor partial information leakage. It is recommended that the authors supplement the manuscript with corresponding\nexperimental or simulation data to verify the system’s robustness under different attack models and explore potential\nmechanisms for enhancing security.\nFinally, the manuscript mentions the possibility of further expanding channel capacity by integrating other degrees of\nfreedom, such as polarization and wavelength. However, specific technical pathways and challenges have not been\nadequately addressed, especially regarding possible crosstalk between different degrees of freedom. It is suggested that the\nauthors include preliminary experimental or simulation results on multi-degree-of-freedom multiplexing—for instance,\nvalidating the feasibility of combined polarization-OAM encoding—and analyze the coupling mechanisms between different\ndegrees of freedom (e.g., polarization-dependent phase modulation), proposing effective decoupling strategies to\ndemonstrate the future potential of this technology in high-dimensional multiplexing.\nIn summary, this manuscript presents a forward-looking technical concept, but there remains significant room for\nimprovement in terms of theoretical completeness, depth of experimental validation, and practical feasibility. If the authors\ncan systematically revise and enhance the manuscript in response to the above-mentioned concerns, the scientific value\nand engineering relevance of the work could be substantially strengthened, making it a stronger candidate for\nreconsideration for publication.", "reviewer_3": "This manuscript presents a novel approach for multi-channel holographic imaging by encoding multiple holographic images\ninto a single metasurface and decoding them using structured light generated by a spatial light modulator (SLM).\nCompared to conventional methods—where a single orbital angular momentum (OAM) pattern generated by the SLM typically decodes\nonly one hologram—this work utilizes discontinuous OAM beams with spatially separated topological charges. This design\nallows different holographic images to be selectively decoded by rotating the input beam, thereby enabling dynamic and\nsecure multiplexing. Overall, this work demonstrates an innovative and technically sound advancement in the areas of OAM\noptics, holography, cryptographic imaging, and multifunctional metasurface design. Considering the following technical\nissues, I believe the manuscript is suitable for publication in Nature Communications after minor revision.\n1. In the manuscript, does Equation (2) specifically refer to the angular range [π/2, 3π/2]? If so, would it be more appropriate\nto explicitly label this range near the equation for clarity? Additionally, I reviewed Supporting Information Equation (6), but I\ncouldn’t clearly see how this expression explains why the electric field is nearly zero in the angular range [-π/2, π/2]. Why\nthe “petal-like” intensity pattern only appears on the left side? Could the authors clarify how this behavior is captured by Eq.\n(6), or whether any assumptions or approximations are involved?\n2. In Figures 1(c) and 1(d), the bottom panels are labeled as \"electric field distributions.\" However, based on the visual\npatterns, they appear to represent intensity profiles rather than complex field amplitudes. Should these be labeled as\n“intensity” instead of “electric field” for accuracy and clarity?\n3. I recommend that the authors consider adding a colorbar next to the phase distributions in Figure 1 of the main text for\nclarity. Additionally, I noticed a potential inconsistency in the phase color mapping. In Figure 1c, the brown-colored phase\nprogresses from dark brown to white as the azimuthal angle increases, which corresponds to a positive topological charge\n(ℓ=+10). However, in the brown segment of Figure 2a, the same color progression—from dark brown to white with increasing\nazimuth—appears to represent a negative topological charge (ℓ=−10). For clarity and consistency, I suggest the authors\nensure that the color mapping of phase is consistently represented throughout the figures, especially for readers interpreting\nthe sign of the OAM from the color gradient.\n4. While Supplementary Note 3 provides insight into the orthogonality behavior via inner product analysis, it would be\nhelpful if the authors could clarify the specific method used to compute the SNR values shown in Figures 3c and 3d—such\nas how the signal and noise regions are defined, and whether any averaging or normalization is applied. In particular, the\nnear-perfect Gaussian spot distribution appears only when angular orthogonality is achieved. However, even in the absence\nof orthogonality, there is still observable power forming a petal-like intensity pattern. This raises the question of how exactly\nthe SNR and orthogonality are defined and processed in the data analysis.\n5. In Figure 2a, I noticed that the phase distributions corresponding to images B and C appear highly similar. Since the\nauthors mention that the meta-atoms are based on geometric phase, and the Methods section only specifies LCP\nillumination, the polarization state of the incident light is not fully described throughout the manuscript. Could the authors\nclarify whether different circular polarizations (e.g., LCP vs. RCP) were used to access additional holographic channels?\nIf so, it would be helpful to elaborate on the polarization dependence and specify which channels were designed for which\nincident polarization state.\n6. In Figure 5b, the experimental holographic images for the six-channel configuration appear noticeably clearer than those\nin the four-channel case shown in Figure 4d. Interestingly, in the six-channel experiment, each decoded holographic image\nis encoded using two OAM values (i.e., a pair of ±ℓ). Does this suggest that encoding each holographic image with a pair of\nOAM values inherently leads to better reconstruction quality? It would also be helpful if the authors could provide\nquantitative metrics (e.g., SNR) for the six-channel case to facilitate a direct comparison with the four-channel results."}, "author_response": "Reply report to reviewer #1\nComment: The authors of manuscript NCOMMS-25-25303 proposed and\ndemonstrated a novel kind of metasurface holography based on discontinuous orbital\nangular momentum (OAM). This discontinuous OAM has a spatially varying TC across\nthe azimuthal angle, and thus breaks the rotational symmetry of the beam for\ninformation retrieval. Generally, this study differentiates itself from traditional OAM-\nbased holographic approaches through the employment of discontinuous OAM modes.\nHowever, after careful evaluation, I am currently hesitant to recommend this study,\nbased on the following reasons.\nResponse: We would like to thank the reviewer for the affirmation of the novelty of\nour work. We are happy to address all the concerns raised by the reviewer.\nComment 1.1: Their proposal provides significance such as using a single beam in\ninformation retrieval. However, this method may also bring along limitations, such as\nextra complexities in precisely controlling the rotation angle and spatial position of the\ninput beam in retrieving the information.\nResponse 1.1: As pointed out by the reviewer, compared to traditional OAM\nholography, controlling the rotation angle and spatial position of the input beam is\nnecessary in our discontinuous OAM holography. However, slight deviations within a\ncertain range of angles and positions are acceptable in practical implementation, as\nexplained now in Supplementary Note 7.\n“Supplementary Note 7: Numerical characterization of the impact of angular and\nlateral misalignment in discontinuous OAM holography.\nOur investigation into the effects of angular and lateral misalignment in\ndiscontinuous OAM holography reveals important operational characteristics of this\ntechnique. Using a four-channel discontinuous OAM hologram as an example, we find\nthat this holographic approach demonstrates practical tolerance to angular\nmisalignment, maintaining a clear reconstruction of image “A” with near-perfect\nGaussian spots within a ±4° rotation range as shown in Fig. S7a. This indicates the\ninherent robustness of the technique in image decoding. However, at a rotation angle of\n±5°, the Gaussian spots in the reconstructed image begin to deteriorate. Fig. S7b shows\nthat as the angular deviation increases significantly, the reconstructed image completely\nshifts to the output pattern of a different channel.\nRegarding positional deviations, Fig. S8 shows that lateral misalignment up to 2%\n(~ 6 𝜇𝑚) preserves the near-perfect Gaussian spot characteristics of image “A”, while\nat 3% (~ 9 𝜇𝑚) deviation, the Gaussian profile deteriorates. This tolerance also extends\nto combined x-y plane misalignments, where the holographic reconstruction\nsuccessfully decodes image “A” with recognizable Gaussian spots at 1-2% composite\ndeviation.\nThese findings demonstrate that discontinuous OAM holography maintains\nfunctionality under practical alignment variations while providing enhanced security\nthrough its angular dependence and misalignment tolerance characteristics.\nFig. S7. Numerical results demonstrating the impact of angular misalignment in\ndiscontinuous OAM holography. (a) Holographic reconstruction based on a\ndiscontinuous OAM meta-hologram with slight angular deviation. (b) Holographic\nreconstruction based on a discontinuous OAM meta-hologram with large angular\ndeviation. When the incident angle is rotated at different angles, the decoded\nholography appears clear “A”, “B”, “C” and “D” patterns at the rotation angle of 0°,\n90°, 180°, and 270°, respectively. However, for other rotation angles, significant\ncrosstalk is observed.\nFig. S8. Numerical results of the reconstructed images demonstrating the impact\nof lateral misalignment in discontinuous OAM holography.”\nComment 1.2: The conventional OAM holography has significant merit of information\nencryption. I’m doubt of the scenarios with necessity to further enhance the security of\nthe information, especially at the cost of adding extra complexity in information\nretrieval, because the authors said ‘any deviation in the rotation angle, or illumination\nwith conventional constant OAM, will trigger severe inter-channel crosstalk.’\nResponse 1.2: The authors thank the reviewer for this point. Although conventional\nOAM holography significantly enhances security compared to polarization-encrypted\nholography, it remains vulnerable to brute-force attacks using randomly incident\ninteger-OAM vortex beams that can fully decode encrypted information. In stark\ncontrast, our discontinuous OAM holography imposes dual-decryption prerequisites:\nsimultaneous acquisition of both topological charge (TC) combination and azimuthal\nphase distribution for successful image reconstruction.\nFig. R1-1 The possibility of conventional OAM and discontinuous OAM modes\nformed by N topological charges.\nAs a proof of concept, to quantitatively analyze the information security of these\ntwo techniques, we use N = 200 (with topological charge ranging from -100 to 100) as\nan example. The commercialized SLM (model: HOLOEYE PLUTO-2.1-NIR-135) generates vortex beams with a refresh rate of 70 𝑚𝑠 . oor conventional OAM\nholography, decoding the information requires inputting 200 distinct OAM modes,\nwhich takes only 14 seconds in total. However, for discontinuous OAM holography,\nthe azimuthal partitioning is uncertain. Assuming only four possibilities of equal\npartitions (1/2/3/4), the total number of possible combinations is calculated as:\n𝑁+𝑁∗(𝑁−1)+𝑁∗(𝑁−1)∗(𝑁−2)+𝑁∗(𝑁−1)∗(𝑁−2)∗(𝑁−3) (R1)\nWith 200 TCs, this results in 1560359200 possible configurations, requiring 1264\ndays—or approximately 3.5 years—to decode by using a commercialized SLM with\nthe same refresh rate of 70 𝑚𝑠. This doesn’t even account for the effects of topological\ncharge mismatches during rotation. Therefore, our discontinuous OAM holography\noffers a security enhancement of at least 7 orders of magnitude over traditional OAM\nholographic encryption.\nThis discussion has been added in Supplementary Note 1 as:\n“Supplementary Note 1: Security enhancement of discontinuous OAM\nholography compared to conventional OAM holography.\nAlthough conventional OAM holography significantly enhances security\ncompared to polarization-encrypted holography, it remains vulnerable to brute-force\nattacks using randomly incident integer-OAM vortex beams that can fully decode\nencrypted information. In stark contrast, our discontinuous OAM holography imposes\ndual-decryption prerequisites: simultaneous acquisition of both topological charge\ncombination and azimuthal phase distribution for successful image reconstruction.\nFig. S1. The number of multiplexing possibilities of conventional OAM and\ndiscontinuous OAM modes formed by N topological charges.\nAs a proof of concept, to quantitatively compare the information security of\nconventional and discontinuous OAM holography, we use N = 200 (with topological\ncharge ranging from -100 to 100) as an example. The commercialized SLM (model:\nHOLOEYE PLUTO-2.1-NIR-135) generates vortex beams with a refresh rate of\n70 𝑚𝑠 . oor conventional OAM holography, decoding the information requires\ninputting 200 distinct OAM modes, which takes only 14 seconds in total. However, for\ndiscontinuous OAM holography, the azimuthal partitioning is uncertain. Assuming only\nfour equal partitions (1/2/3/4), the total number of possible combinations is calculated\nas:\n𝑁+𝑁∗(𝑁−1)+𝑁∗(𝑁−1)∗(𝑁−2)+𝑁∗(𝑁−1)∗(𝑁−2)∗(𝑁−3) (S1)\nWith 200 TCs, this results in 1560359200 possible configurations, requiring 1264\ndays— or approximately 3.5 years—to decode by using the commercialized SLM with\nthe same refresh rate of 70 𝑚𝑠. This doesn’t even account for the effects of topological\ncharge mismatches during rotation. Therefore, our discontinuous OAM holography\noffers a security enhancement of at least 7 orders of magnitude compared to traditional\nOAM holographic encryption.”\nComment 1.3: In the text, the authors mentioned that ‘We prove that by strategically\narranging the azimuthally distributed TCs, the discontinuous OAM exhibits self-\northogonality at different rotation angles, ensuring minimal cross-talk between\nchannels.’ The comparison of cross-talk with the conventional metasurface holography\nand state-of-the-art holography techniques with small cross-talk has not been fully\ndiscussed.\nResponse 1.3: The authors thank the reviewer for this point. The current state-of-the-\nart for minimizing crosstalk in conventional OAM holography is the complex-\namplitude OAM holography developed by Ren Haoran et al. (Nat. Nanotechnol.15,\n948–955, 2020). This method achieves optimal crosstalk suppression through precise\nmodulation of metasurface amplitude and phase profiles. However, due to fabrication\nconstraints that prevent reliable amplitude control in our laboratory, we adopted a\nphase-only metasurface design for discontinuous OAM holography. We systematically\ncompared the decoded image quality between our phase-only discontinuous OAM\nholography and conventional phase-only OAM holography, which is discussed in\nSupplementary Note 8 as:\n“Supplementary Note 8: Image quality analysis between conventional OAM\nholography and discontinuous OAM holography based on a phase-only\nmetasurface\nBoth discontinuous OAM holography and conventional OAM holography can\nreconstruct high-quality images featuring Gaussian-like spot distributions when fully\ndecoded as shown in Fig. S9. We employ the signal-to-noise ratio (SNR) as the\nquantitative metric for image quality assessment, calculated as:\n𝑆𝑁𝑅 = 10𝑙𝑔\n𝑃𝑠\n(S17)\n𝑃𝑛\nP represents the signal power of the image, and here we use the intensity of the\ns\nencoded image instead, while P denotes the noise power. To quantitatively compare\nn\ncross-talk levels between the two methods, we selected decoded single-channel image\n“A” within a four-channel holography system for quality analysis. This approach\nenables direct comparison of cross-talk magnitudes generated by both methods as\nshown in Fig. S10. As shown by the blue curve, SNR remains essentially constant as\nthe incident light rotates, maintaining relatively good image quality and low cross-talk.\nIn discontinuous OAM holography, high-quality decoding of Image “A” occurs around\n0° rotation angle, while reconstruction fails at other angles. However, under full\ndecoding conditions, both methods exhibit comparable cross-talk levels.\nConsequently, our proposed discontinuous OAM holography methodology\nachieves multi-channel switching under single-incident-beam illumination without\nincreasing cross-talk.\nFig. S9. The decoded holographic images of discontinuous OAM holography and\nconventional OAM holography with a phase-only metasurface. (a) Simulation\nresults of discontinuous OAM holography. (b) Simulation results of conventional\nholography.\nFig. S10. Relationship between SNR and incident light rotation angle for\nconventional OAM holography and discontinuous OAM holography.”\nComment 1.4: The authors mentioned that ‘This enables multiple holographic images\nto be encoded at different rotation angles of the incident OAM mode, significantly\nenhancing the multiplexing channel numbers under single beam illumination.’\nHowever, in the experiment, the detail of how to precisely control the rotation angle in\ncoding and decoding has not been fully discussed.\nResponse 1.4: The authors thank the reviewer for this point. We would like to answer\nfrom the following two aspects regarding encoding and decoding processes.\n1. During the encoding process, we achieve precise control over the angular\npartitioning of the encoded phase. Taking the four-channel discontinuous OAM\nholography proposed in this manuscript as an example, defining azimuthal sectors\nwhere discontinuous OAM phases carry specific TC. We equally divide the 2π\nazimuthal range into four sectors, encoding TC of 10, 20, -10, and -20 respectively. The\nphases for the remaining three channels are generated by rotating the obtained\ndiscontinuous OAM by 90°, 180°, and 270° as shown in Fig. R1-2. The TC and its\ncorresponding azimuthal region in discontinuous OAM can be arbitrarily designed.\nFig. R1-2 Four-channel discontinuous OAM phase design method.\n2. oor the decoding process, the angle of the rotating incident light can be\ncontrolled by either rotating the SLM or the metasurface. In our experiment, we fix the\nSLM and rotate the metasurface. Specifically, the metasurface is mounted on a manual\nrotation platform (model: CMRM1 from LBTEK) with a minimum scale of 2°, which\nfully meets the precision requirements of this study. To ensure that the center of the\nsample coincides with the rotation axis of the rotating platform, an XY translator (model:\nTXY1SQ from LBTEK) is added, as shown in Fig. R1-3c. The fully assembled unit,\nincluding the rotation platform and XY translator, is placed on a translation stage\n(model: LSS40-13C3 from LBTEK) to align the metasurface with the discontinuous\nOAM.\nThe alignment process consists of two stages: coarse adjustment and fine\nadjustment.\n(i). Coarse adjustment\nTo facilitate coarse alignment of metasurface and incident OAM beam, we\nconstructed a direct imaging path with two-lens relay to magnify both the metasurface\nand the incident OAM beam. This configuration allows for clear visualization on the\nCMOS, as illustrated Fig. R1-3a. The original holographic optical setup is shown in\nFig. R1-3b, where an additional lens is inserted in front of the CMOS to perform a\noourier transform, thereby enabling the acquisition of holographic images.\nThe coarse adjustment results are shown in Fig. R1-3d. The first row displays the\nmagnified metasurface and incident OAM beam captured by the CMOS. The red\ndashed circle marks the incident discontinuous OAM beam, while the blue dashed circle\noutlines the metasurface. The second row shows the corresponding holographic images.\nWhen the incident beam and metasurface show no overlap, the beam remains\nunmodulated by the metasurface, resulting in no holographic output. Partial overlap\nbetween the beam and metasurface generates a superposition of image “0” and “1” in\nthe imaging plane. Only when the incident beam fully overlaps with the metasurface\ndoes a relatively clear reconstructed image of “0” emerge.\n(ii). oine adjustment\nOnce coarse alignment is achieved, fine alignment is then performed by precisely\nadjusting the translation stage, ensuring that the image spot evolves into a well-defined\nGaussian profile. After fine adjustment, a clear resolved image “0” with a Gaussian spot\nis shown in Fig. R1-3e.\nIt should be noted that the angle of the rotating incident light is equivalently\nrepresented by changing the phase distribution of the SLM, which is the most\nstraightforward and simplest method that can be used in the experiment.\nFig. R1-3. Optical setup of discontinuous OAM holography. (a) Optical setup for\ncoarse adjustment. (b) Optical setup of discontinuous OAM holography. (c) The manual\nrotation platform added with an XY translator. (d) oirst row: the alignment of incident\ndiscontinuous OAM and metasurface. Second row: the corresponding holographic\nimages. The red dashed circle marks the incident discontinuous OAM beam, while the\nblue dashed circle outlines the metasurface. (e) Holographic image after fine\nadjustment.”\nThis discussion has been added in Supplementary Note 11 as:\n“Supplementary Note 11: Alignment of metasurface with the discontinuous OAM\nduring experiments.\nIn discontinuous OAM holography, precise alignment between the discontinuous\nOAM beam and the metasurface center is critical for successful decoding. oor the\ndecoding process, the angle of the rotating incident light can be controlled by either\nrotating the SLM or the metasurface. In our experiment, we fix the SLM and rotate the\nmetasurface. Specifically, the metasurface is mounted on a manual rotation platform\n(model: CMRM1 from LBTEK) with a minimum scale of 2°, which fully meets the\nprecision requirements of this study. To ensure that the center of the sample coincides\nwith the rotation axis of the rotating platform, an XY translator (model: TXY1SQ from\nLBTEK) is added, as shown in Fig. S16c. The fully assembled unit, including the\nrotation platform and XY translator, is placed on a translation stage (model: LSS40-\n13C3 from LBTEK) to align the metasurface with the discontinuous OAM.\nThe alignment process consists of two stages: coarse adjustment and fine\nadjustment.\n1. Coarse adjustment\nTo facilitate coarse alignment of the metasurface and incident OAM beam, we\nconstructed a direct imaging path with two-lens relay to magnify both the metasurface\nand the incident OAM beam. This configuration allows for clear visualization on the\nCMOS, as illustrated Fig. S16a. The original holograph optical setup is shown in Fig.\nS16b, where an additional lens is inserted in front of the CMOS to perform a oourier\ntransform, thereby enabling the acquisition of holographic images.\nThe coarse adjustment results are as shown in Fig. S16d. The first row displays\nthe magnified metasurface and incident beam captured by the CMOS. The red dashed\ncircle marks the incident discontinuous OAM beam, while the blue dashed circle\noutlines the metasurface. The second row shows the corresponding holographic images.\nWhen the incident OAM beam and metasurface show no overlap, the beam remains\nunmodulated by the metasurface, resulting in no holographic output. Partial overlap\nbetween the beam and metasurface generates a superposition of image “0” and “1” in\nthe imaging plane. Only when the incident beam fully overlaps with the metasurface\ndoes a relatively clear reconstructed image emerge.\n2. oine adjustment\nOnce coarse alignment is achieved, fine alignment is then performed by precisely\nadjusting the translation stage, ensuring that the image spot evolves into a well-defined\nGaussian profile. After fine adjustment, a clear resolved image “0” with a Gaussian spot\nis shown in Fig. S16e.\nIt should be noted that the angle of the rotating incident light is equivalently\nrepresented by changing the phase distribution of the SLM, which is the most\nstraightforward and simplest method that can be used in the experiment.\nFig. S16. Optical setup of discontinuous OAM holography. (a) Optical setup for\ncoarse adjustment. (b) Optical setup of discontinuous OAM holography. (c) The manual\nrotation platform added with an XY translator. (d) oirst row: the alignment of incident\ndiscontinuous OAM and metasurface. Second row: the corresponding holographic\nimages. The red dashed circle marks the incident discontinuous OAM beam, while the\nblue dashed circle outlines the metasurface. (e) Holographic image after fine\nadjustment.”\nComment 1.5: Does the phase distribution necessarily require topological charges\nwith opposite signs, or could the same functionality be achieved using same-sign\ncharges with discontinuous phase profiles (such as TC=1, 2, 3, 4)?\nResponse 1.5: As pointed out by the reviewer, the same functionality can be achieved\nusing same-sign charges with discontinuous phase profiles. Specifically, when different\nsectors carry distinct TCs – whether all positive, all negative, or mixed – the system\ninherently achieves orthogonality, enabling robust channel isolation, as experimentally\nvalidated in Supplementary Note 9. To reduce crosstalk, we employed OAM modes\nwith larger TC intervals for encoding the same-sign charges discontinuous OAM\nholography.\n“Supplementary Note 9: Discontinuous OAM holography with same-sign TCs\nTo expand the application scope of discontinuous OAM holography, we\ncomputationally and experimentally demonstrated that discontinuous OAM holography\ncan be achieved using same-sign TCs. Specifically, we encoded two-channel\ndiscontinuous OAM holography with two all-positive TCs 𝑙 = 1/𝑙 = 11 and four\nazimuthal regions with four all-positive TCs 𝑙 = 1/𝑙 = 8/ 𝑙 = 15/𝑙 = 22. The phase\ndistributions, simulated and experimental results of the two-channel discontinuous\nOAM modes are shown in Fig. S11. In addition, the discontinuous OAM modes with\nfour-channel holography are shown in Fig. S12a, with the corresponding simulation\nand experimental results presented in Figs. S12b and Fig. S12c, respectively. The\nresults demonstrate that discontinuous OAM holography encoded with all-positive TCs\ncan also achieve multi-channel multiplexing.\nFig. S11. Simulated and experimental results of two-channel discontinuous OAM\nholography with same-sign topological charges.\nFig. S12. Simulated and experimental results of four-channel discontinuous OAM\nholography with same-sign topological charges. (a) The phase distributions of the\nfour discontinuous OAM modes. (b) The simulation and (c) experimental results of\nfour-channel discontinuous OAM holography use same-sign charges.”\nComment 1.6: Are the light spots at the letter edges in the experimental optical field\nof Figure 4d manually traced? If so, this image processing procedure should be\nexplicitly declared in the main text.\nResponse 1.6: The authors thank the reviewer for this point. Since OAM holography\nrequires sampling the original image, the experimentally obtained pictures are all\ncomposed of Gaussian dots. To facilitate visibility, we have added frame dash lines\naround the edge of the letters to make the display clearer. This clarification has been\nadded in the revised manuscript as “To facilitate visibility, we have added frame dash\nlines around the edge of the letters to make the display clearer” in Line 201-202 Page\n8.\nIn summary, we have addressed all the comments of reviewer #1.\nReply report to reviewer #2\nComment 2: This manuscript proposes a new paradigm of metasurface holography\nbased on discontinuous orbital angular momentum (OAM), which enables multi-\nchannel holographic encoding under a single incident beam by breaking the rotational\nsymmetry of conventional OAM modes. The work demonstrates certain innovation in\nenhancing optical information security and capacity, particularly in the design and\napplication of spatially varying topological charge (TC) distributions. However,\nconsidering the high academic standards and scientific rigor expected by Nature\nCommunications, the current manuscript still exhibits certain shortcomings in\ntheoretical depth, experimental validation, and practical feasibility, falling short of the\npublication threshold. To improve the overall quality of the manuscript, the authors are\nadvised to systematically address the following key issues.\nResponse 2: We would like to thank the reviewer for the affirmation of the novelty of\nour work. We are happy to address all the concerns raised by the reviewer.\nComment 2.1: First, traditional OAM multiplexing holography has shown great\npotential in high-capacity communication due to its propagation-invariant and stable\ncharacteristics. However, the proposed discontinuous OAM mode, while breaking the\ncentral symmetry, may adversely affect the beam’s transmission properties. For\nexample, it remains insufficiently verified whether the structural integrity of the beam\ncan be preserved after transmission through optical fibers (such as few-mode or ring-\ncore fibers) or through free space. Therefore, the application prospects emphasized in\nthe manuscript require further substantiation, and it is recommended that the authors\nsupplement the manuscript with relevant experimental or simulation-based analyses of\ntransmission performance.\nResponse 2.1: The authors thank the reviewer for this point. Taking two-channel\ndiscontinuous OAM as an example. Within the oresnel diffraction regime, a vortex\nbeam carrying half-integer OAM exhibits a semi-vortex morphology. The opening\ndirection of this semi-vortex rotates with propagation distance until attaining a 90°\nphase offset relative to its initial state. The rotation direction is determined by the sign\nof the TC as explained in Supplementary Note 2. The discontinuous OAM proposed\nin this manuscript arises from the superposition of two such semi-vortices, which\ngenerate a petal-like beam profile. After the oresnel diffraction regime, i.e., in the far\nfield, the discontinuous OAM beam propagates stable in free space as shown in Fig.\nS21.\nFig. S21. The simulated propagation characteristics of the discontinuous OAM\nbeam in free space. After the oresnel diffraction regime, i.e., in the far field, the\ndiscontinuous OAM beam propagates stable in free space.\nComment 2.2: Second, although the authors claim that the method effectively enhances\nthe capacity of encryption channels, the experimental results presented in the\nmanuscript demonstrate only a limited number of multiplexed channels—just a few in\ntotal. In contrast, traditional OAM holography has already achieved tens or even\nhundreds of multiplexed channels, with some metasurface-based schemes reaching\nover ten channels. As such, the advantage of the current approach in terms of\ncapacity enhancement is not yet clearly demonstrated. The authors are encouraged to\nmore explicitly clarify the core competitiveness of their method compared to traditional\ntechniques and to further validate its potential by expanding the number of channels or\nintegrating additional degrees of freedom.\nResponse 2.2: The authors thank the reviewer for this valuable point. Conventional\nOAM metasurface holography achieving tens or hundreds of multiplexed channels\nrelies on complex-amplitude metasurfaces. Compared to phase-only metasurfaces,\ncomplex-amplitude designs exhibit lower crosstalk and higher image fidelity. However,\ndue to fabrication constraints in realizing complex-amplitude structures in our\nlaboratory, we adopted a phase-only metasurface approach utilizing Pancharatnam-\nBerry (PB) phase for multi-channel multiplexing. In the previous paper presented by\nRen Haoran et al. (Nat. Commun., 10, 2986, 2019), they demonstrated ten-channel\nphase-only OAM holography in simulations. oor comparison, we present both\nsimulation and experimental results for ten-channel discontinuous OAM holography.\nIn Fig. S17 and Fig. S18, we show that both eight-channel and ten-channel\ndiscontinuous OAM holography can be successfully achieved.\nThe key advantage over discontinuous OAM holography lies in enabling channel\nswitching through metasurface rotation under single-beam illumination, which is an\nunattainable feature in traditional systems. Moreover, it offers enhanced security as\ndecoding requires precise knowledge of both the encoded azimuthal sectors and their\ncorresponding TC.\nThis discussion has been added in Figs. S17 & S18 as,\n“\nFig. S17. Eight-channel discontinuous OAM holography. (a) Schematic of a\ndiscontinuous OAM-coded image target and its resulting multiplexing meta-hologram.\nPseudo colors are used to visualize rotation angle of discontinuous OAM modes. (b)\nThe phase distribution of ten-channel discontinuous OAM holography. (c) The\ndistribution of TC across azimuth angle. (d) Simulation holographic images\nreconstructed from eight-channel discontinuous OAM meta-hologram.\nFig. S18. Ten-channel discontinuous OAM holography. (a) The phase distribution\nof ten-channel discontinuous OAM holography. (b) The distribution of TC across\nazimuth angle. (c) OAM-encoded image multiplexing based on one discontinuous\nOAM states with ten rotation angles. (d) Simulated and (e) experimental results of\nholographic images reconstructed from ten-channel discontinuous OAM meta-\nhologram.”\nComment 2.3: Third, although conventional centrally symmetric OAM multiplexing\nholography imposes certain alignment requirements on the devices, its inherent\nsymmetry to some extent reduces the difficulty of alignment. After breaking this\nsymmetry, however, the precision required for alignment may significantly increase,\nespecially when rotational operations are introduced, where accurate centering\nbecomes critical to decoding performance. The manuscript should therefore include a\ndiscussion on alignment error tolerance and assess its impact on overall system\nperformance.\nResponse 2.3: The authors thank the reviewer for this point. In contrast to conventional\nOAM holography, our proposed discontinuous OAM holography requires angular\nalignment—a deliberate design feature that enhances information security. Random\nillumination with integer-OAM modes cannot decode the information, while the system\nmaintains robust tolerance to both angular misalignment and center-position offsets.\nThe discussion has been added in Supplementary Note 7.\n“Supplementary Note 7: Numerical characterization of the impact of angular and\nlateral misalignment in discontinuous OAM holography.\nOur investigation into the effects of angular and lateral misalignment in\ndiscontinuous OAM holography reveals important operational characteristics of this\ntechnique. Using a four-channel discontinuous OAM hologram as an example, we find\nthat this holographic approach demonstrates practical tolerance to angular\nmisalignment, maintaining a clear reconstruction of image “A” with near-perfect\nGaussian spots within a ±4° rotation range as shown in Fig. S7a. This indicates the\ninherent robustness of the technique in image decoding. However, at a rotation angle of\n±5°, the Gaussian spots in the reconstructed image begin to deteriorate. Fig. S7b shows\nthat as the angular deviation increases significantly, the reconstructed image completely\nshifts to the output pattern of a different channel.\nRegarding positional deviations, Fig. S8 shows that lateral misalignment up to 2%\n(~ 6 𝜇𝑚) preserves the near-perfect Gaussian spot characteristics of image “A”, while\nat 3% (~ 9 𝜇𝑚) deviation, the Gaussian profile deteriorates. This tolerance also extends\nto combined x-y plane misalignments, where the holographic reconstruction\nsuccessfully decodes image “A” with recognizable Gaussian spots at 1-2% composite\ndeviation.\nThese findings demonstrate that discontinuous OAM holography maintains\nfunctionality under practical alignment variations while providing enhanced security\nthrough its angular dependence and misalignment tolerance characteristics.\nFig. S7. Numerical results demonstrating the impact of angular misalignment in\ndiscontinuous OAM holography. (a) Holographic reconstruction based on a\ndiscontinuous OAM meta-hologram with slight angular deviation. (b) Holographic\nreconstruction based on a discontinuous OAM meta-hologram with large angular\ndeviation. When the incident angle is rotated at different angles, the decoded\nholography appears clear “A”, “B”, “C” and “D” patterns at the rotation angle of 0°,\n90°, 180°, and 270°, respectively. However, for other rotation angles, significant\ncrosstalk is observed.\nFig.S8. Numerical results of the reconstructed images demonstrating the impact\nof lateral misalignment in discontinuous OAM holography.\nComment 2.4: Fourth, while the manuscript emphasizes the advantage of\ndiscontinuous OAM in resisting spiral spectrum analysis, it lacks thorough security\nanalysis under more complex attack scenarios, such as phase reconstruction based on\nmachine learning or partial information leakage. It is recommended that the authors\nsupplement the manuscript with corresponding experimental or simulation data to\nverify the system’s robustness under different attack models and explore potential\nmechanisms for enhancing security.\nResponse 2.4: We sincerely appreciate the reviewer's suggestions, which have provided\nvaluable insights for our research. Indeed, there are some related researches using deep\nlearning to recover the conventional holography (such as Opt. Commun., 529(15),\n129100, 2023). We will further investigate the potential vulnerabilities of both\nconventional and discontinuous OAM holography using machine learning techniques\nin our future work.\nAt this stage, however, we can demonstrate that the proposed discontinuous OAM\noffers superior anti-attack capability compared to conventional OAM under simple\nattach scenarios. oor example, conventional OAM holography remains vulnerable to\nbrute-force attacks using randomly incident integer-OAM vortex beams that can fully\ndecode encrypted information. In stark contrast, our discontinuous OAM holography\nimposes dual-decryption prerequisites: simultaneous acquisition of both TC\ncombination and azimuthal phase distribution for successful image reconstruction.\nFig. R2-1 The number of conventional OAM and discontinuous OAM modes\nformed by N topological charges.\nAs a proof of concept, to quantitatively compare the information security of\nconventional and discontinuous OAM holography, we use N = 200 (with topological\ncharge ranging from -100 to 100) as an example. The commercialized SLM (model:\nHOLOEYE PLUTO-2.1-NIR-135) typically generates vortex beams with a refresh rate\nof 70 𝑚𝑠 . oor conventional OAM holography, decoding the information requires\ninputting 200 distinct OAM modes, which takes only 14 seconds in total. However, for\ndiscontinuous OAM holography, the azimuthal partitioning is uncertain. Assuming only\nfour equal partitions (1/2/3/4), the total number of possible combinations is calculated\nas:\n𝑁+𝑁∗(𝑁−1)+𝑁∗(𝑁−1)∗(𝑁−2)+𝑁∗(𝑁−1)∗(𝑁−2)∗(𝑁−3) (R1)\nWith 200 TCs, this results in 1560359200 possible configurations, requiring 1264\ndays—or approximately 3.5 years—to decode. Therefore, our discontinuous OAM\nholography offers a security enhancement of nearly 7 orders of magnitude over\ntraditional OAM holographic encryption.\nThis discussion has been added in Supplementary Note 1 as:\n“Supplementary Note 1: Security enhancement of discontinuous OAM\nholography compared to conventional OAM holography.\nAlthough conventional OAM holography significantly enhances security\ncompared to polarization-encrypted holography, it remains vulnerable to brute-force\nattacks using randomly incident integer-OAM vortex beams that can fully decode\nencrypted information. In stark contrast, our discontinuous OAM holography imposes\ndual-decryption prerequisites: simultaneous acquisition of both topological charge\ncombination and azimuthal phase distribution for successful image reconstruction.\nFig. S1. The number of multiplexing possibilities of conventional OAM and\ndiscontinuous OAM modes formed by N topological charges.\nAs a proof of concept, to quantitatively compare the information security of\nconventional and discontinuous OAM holography, we use N = 200 (with topological\ncharge ranging from -100 to 100) as an example. The commercialized SLM (model:\nHOLOEYE PLUTO-2.1-NIR-135) generates vortex beams with a refresh rate of\n70 𝑚𝑠 . oor conventional OAM holography, decoding the information requires\ninputting 200 distinct OAM modes, which takes only 14 seconds in total. However, for\ndiscontinuous OAM holography, the azimuthal partitioning is uncertain. Assuming only\nfour equal partitions (1/2/3/4), the total number of possible combinations is calculated\nas:\n𝑁+𝑁∗(𝑁−1)+𝑁∗(𝑁−1)∗(𝑁−2)+𝑁∗(𝑁−1)∗(𝑁−2)∗(𝑁−3) (S1)\nWith 200 TCs, this results in 1560359200 possible configurations, requiring 1264\ndays— or approximately 3.5 years—to decode by using the commercialized SLM with\nthe same refresh rate of 70 𝑚𝑠. This doesn’t even account for the effects of topological\ncharge mismatches during rotation. Therefore, our discontinuous OAM holography\noffers a security enhancement of at least 7 orders of magnitude compared to traditional\nOAM holographic encryption.”\nComment 2.5: Finally, the manuscript mentions the possibility of further expanding\nchannel capacity by integrating other degrees of freedom, such as polarization and\nwavelength. However, specific technical pathways and challenges have not been\nadequately addressed, especially regarding possible crosstalk between different\ndegrees of freedom. It is suggested that the authors include preliminary experimental\nor simulation results on multi-degree-of-freedom multiplexing—for instance,\nvalidating the feasibility of combined polarization-OAM encoding—and analyze the\ncoupling mechanisms between different degrees of freedom (e.g., polarization-\ndependent phase modulation), proposing effective decoupling strategies to\ndemonstrate the future potential of this technology in high-dimensional multiplexing.\nResponse 2.5: The authors thank the reviewer for this point. In the revised manuscript,\nwe have incorporated the combination of polarization with discontinuous OAM as an\nadditional degree of freedom. As a proof of concept, we designed a two-channel\ndiscontinuous OAM holography integrated with polarization multiplexing (LCP and\nRCP) as shown in Fig. S19. Fig. S19a demonstrates the design principle of using a pure\nPB phase metasurface for the decoupling of LCP and RCP (see detailed design principle\nin Ref. Adv. Mater. 37, 2415142, 2025). The simulated and experimental results of the\nreconstructed images are shown in Fig. S19b, which clearly show the images for the\ncorresponding channels.\nFig. S19. Polarization-OAM multiplexing holography. (a) The design principle of a\ngeometry-phase metasurface supporting the polarization-OAM hybrid encoding with a\nnon-interleaved method in each channel. o represents the oourier transform. (b) Phase\ndistribution of the incident discontinuous OAM beam and corresponding simulation\nresults of reconstructed images.”\nTo further expand the multiplexing dimension and increase the number of\nmultiplexed channels, we introduced the OAM dimension into polarization-\ndiscontinuous OAM multiplexing. oor each polarization state, two sets of two-channel\nholograms were encoded, achieving a total of eight channels. Both simulation and\nexperimental results validate the feasibility of this polarization-OAM hybrid encoding\nscheme, which are shown in the main text oig. 5 as,\n“Experiments of multiple polarization and discontinuous-OAM hybrid\nmultiplexing holography\nTo further expand multiplexing dimensions and enhance channel capacity, we\nincorporated polarization and multiple pairs of discontinuous OAMs as an additional\ndegree of freedom. Specifically, by integrating two pairs of discontinuous OAM modes\nwith two orthogonal polarization states, we have designed an eight-channel metasurface\nmultiplexed holography in Fig. 5. oor left-circularly polarized (LCP) light, two pairs of\northogonal discontinuous OAM modes 𝑙 = −1/𝑙 = −15,𝑙 = −8/𝑙 = −22 are\nemployed to encode four distinct images. Similarly, for right-circularly polarized (RCP)\nlight, the set of OAM modes 𝑙 = −7/𝑙 = −15,𝑙 = −10/𝑙 = 10 encodes another four\nindependent images. The superimposed phase of LCP light denoted by 𝜎 is inverted\n+\ndue to the opposite phase response between RCP and LCP, as shown in Fig. 5a.\nConsequently, the phase profile can be expressed as,\n𝜑 = 𝑎𝑟𝑔[∑𝑁 𝑛=1 𝑒𝑗(𝜑𝜎 𝑛 − +𝜙 𝑟 𝑛 1 ) +∑𝑀 𝑚=1 𝑒𝑗(−𝜑𝜎 𝑚 + −𝜙 𝑟 𝑚 2 )] (3)\nwhere channel n = 1, 2,..., N, with N being the channel number of discontinuous OAM\nholography for RCP, m = 1, 2,..., M, with M being the channel number of discontinuous\nOAM holography for LCP, 𝜎 and 𝜎 denote RCP and LCP, respectively, 𝜑𝑛 and\n− + 𝜎−\n𝜑𝑚 is the calculated phase by WGS algorithm. The phase of discontinuous OAM 𝜙𝑛\n𝜎+ 𝑟1\nand 𝜙𝑛 is then added to the holographic phase.\n𝑟2\nThe demonstration of a geometry-phase metasurface supporting the polarization-\ndiscontinuous OAM hybrid encoding is based on a non-interleaved method[36]. We\nintegrated two-channel discontinuous OAM encoding with polarization multiplexing.\nWhen illuminated with an LCP beam carrying TC of 𝑙 = 1/𝑙 = 15, the letter “M” is\nsuccessfully decoded. When the metasurface is rotated by 90°, the decoded image\nswitches from letter “M” to letter “E”, demonstrating dynamic channel switching\nthrough angular control as shown in Fig. 5b. Similarly, when illuminated with LCP\nbeam carrying 𝑙 = 8/𝑙 = 22 , the letter “T” is decoded, and a 90° rotation of the\nmetasurface enables switching from “T” to “A”. oor RCP beam illumination, four\ndistinct images “1”, “2”, “3”, and “4” can likewise be decoded as shown in Fig. 5c,\ndemonstrating polarization multiplexing capability.\nFig. 5. Polarization-OAM multiplexing holography. (a) The design principle of a\ngeometry-phase metasurface supporting the polarization-OAM hybrid encoding with a\nnon-interleaved method for each channel. o represents the oourier transform. Simulated\nand experimental results of reconstructed images when (b) LCP and (c) RCP light\ncarried discontinuous OAM is incident to the metasurface.”\nIn summary, we have addressed all the comments of reviewer #2.\nReply report to reviewer #3\nComment 3: This manuscript presents a novel approach for multi-channel holographic\nimaging by encoding multiple holographic images into a single metasurface and\ndecoding them using structured light generated by a spatial light modulator (SLM).\nCompared to conventional methods—where a single orbital angular momentum (OAM)\npattern generated by the SLM typically decodes only one hologram—this work utilizes\ndiscontinuous OAM beams with spatially separated topological charges. This design\nallows different holographic images to be selectively decoded by rotating the input\nbeam, thereby enabling dynamic and secure multiplexing. Overall, this work\ndemonstrates an innovative and technically sound advancement in the areas of OAM\noptics, holography, cryptographic imaging, and multifunctional metasurface design.\nConsidering the following technical issues, I believe the manuscript is suitable for\npublication in Nature Communications after minor revision.\nResponse 3: We are grateful to the reviewer’s positive feedback and recommendation\nto publish in Nature Communications. We are happy to address all the concerns raised\nby the reviewer.\nComment 3.1: In the manuscript, does Equation (2) specifically refer to the angular\nrange [π/2, 3π/2]? If so, would it be more appropriate to explicitly label this range near\nthe equation for clarity? Additionally, I reviewed Supporting Information Equation (6),\nbut I couldn’t clearly see how this expression explains why the electric field is nearly\nzero in the angular range [-π/2, π/2]. Why the “petal-like” intensity pattern only\nappears on the left side? Could the authors clarify how this behavior is captured by Eq.\n(6), or whether any assumptions or approximations are involved?\nResponse 3.1: Thank you very much for your valuable comment. oollowing your\nsuggestions, we have implemented the corresponding modifications in the manuscript,\nlabeling the range near the equation.\nThe corresponding changes were made in the main text as follows:\n“The spatial electric field distributions of the two partitioned regions overlap\ncoherently in free space as shown in the bottom-left panel of oig. 1c, which can be\ndescribed as,\n𝜋 3𝜋\n𝐸 ∝ cos (10𝜙), 𝜙 ∈ [ , ] (2)\n2 2\n” in Line 119, Page 5.\nThrough more detailed calculations, we have demonstrated that the “petal-like”\npattern only appears on the left side in Supplementary Note 2 as:\n“oor discontinuous OAM, each partition can be considered as a partially blocked spiral\nphase plate. The phase of one partition is written as,\n𝑒−𝑖𝑙𝜑, 0 < 𝜑 < 𝜋\n𝑉(𝜌,𝜑) = { (S2)\n𝑙 0, 𝜋 < 𝜑 < 2𝜋\nwhere ρ and 𝜑 are the radial distance and azimuth of the polar coordinates in the input\nplane. 𝑙 is the TC of one partition.\nThe oraunhofer diffraction pattern in the focal plane of a lens produced by this\nphase pattern is given by,\n𝐸(𝑟,𝜑) = 𝐶∫ 𝑅 𝜌𝑑𝜌∫ 2𝜋 𝑑𝜑𝑉(𝜌,𝜑)𝑒−𝑖(𝑘𝑟𝜌/𝑓)𝑐𝑜𝑠(𝜑−𝜙) (S3)\n0 0 𝑙\nwhere the variables 𝑟 and 𝜙 are polar coordinates in the output plane, 𝑘 = 2𝜋/𝜆\nand 𝑓 is the focal length of lens.\nEach partition of the discontinuous OAM can be considered as a partially blocked\nspiral phase plate, which will produce a partial vortex output pattern that is rotated by\n90° relative to the input according to the oraunhofer diffraction theory. The sign of the\n90° rotation depends on the sign of the TC. oor a phase plate with negative TC in the\nangular [0,𝜋] range as shown in Fig. S2a, the output E of the half pattern OAM can be\nwritten as,\n𝑘𝑟𝜌\n𝐸(𝑟,𝜑) = 𝐶∫ 𝑅 𝜌𝑑𝜌∫ 2𝜋 𝑑𝜑 ∗𝑒−𝑖𝑙𝜑 ∗𝑒 −𝑖( 𝑓 )cos(𝜑−𝜙) , 0 < 𝜑 < 𝜋 (S4)\n0 0\n𝐸(𝑟,𝜑) = 𝑒 −𝑖𝑙(𝜑+ 𝜋 2 ) 𝐶∫ 0 𝑅 𝐽 𝑚 (𝑘𝑟𝜌/𝑓)𝜌𝑑𝜌, 0 < 𝜑 < 𝜋 (S5)\norom Eq. S5, it can be seen that the output electric field is rotated\n𝜋\ncounterclockwise by (i.e. 90°) relative to the original phase distribution and then is\n2\n𝜋 3𝜋\ndistributed in the region of [ , ]. The electric field intensity on the opposite side is\n2 2\n𝜋\nzero as shown in Fig. S2b. By performing the coordinate transformation of 𝜙 = 𝜑+ ,\n𝑒−𝑖𝑙𝜙𝐶∫ 𝑅 𝐽 ( 𝑘𝑟𝜌 )𝜌𝑑𝜌, 𝜋 < 𝜙 < 3𝜋\n𝐸(𝑟,𝜙− 𝜋 ) = { 0 𝑚 𝑓 2 2 (S6)\n2 0, − 𝜋 < 𝜙 < 𝜋\n2 2\noor the bottom-half partition as shown in Fig. S2c, the phase distribution spans\nthe angular range of [𝜋,2𝜋] with a positive TC. The phase of this part is written as,\n0, 0 < 𝜑 < 𝜋\n𝑉(𝜌,𝜑) = { (S7)\n𝑙 𝑒𝑖𝑙𝜑, 𝜋 < 𝜑 < 2𝜋\nThe electric field distribution is given by:\n𝐸(𝑟,𝜑) = 𝑒 −𝑖𝑙(𝜑− 𝜋 2 ) 𝐶∫ 0 𝑅 𝐽 𝑚 (𝑘𝑟𝜌/𝑓)𝜌𝑑𝜌, 𝜋 < 𝜑 < 2𝜋 (S8)\n𝜋\norom Eq. S8, it can be seen that the output electric field is rotated clockwise by\n2\n(i.e. −90°) relative to the original phase distribution and then is distributed in the region\n𝜋 3𝜋\nof [ , ] as shown in Fig. S2d. By performing the coordinate transformation of 𝜙 =\n2 2\n𝜋\n𝜑− , Eq. 8 can be rewritten as:\n2\n𝜋 𝜋\n0, − < 𝜙 <\n𝜋 2 2\n𝐸(𝑟,𝜙+ ) = { (S9)\n2 𝑒−𝑖𝑙𝜙𝐶∫ 𝑅 𝐽 ( 𝑘𝑟𝜌 )𝜌𝑑𝜌, 𝜋 < 𝜙 < 3𝜋\n0 𝑚 𝑓 2 2\nTherefore, for the full dual-partition discontinuous OAM as shown in Fig. S2e,\nthe electric field distribution is the superposition of both partitions, occupying the same\n𝜋 3𝜋 𝜋 𝜋\n[ , ] range as shown in Fig. S2f. The electric field distribution in range [− , ] is\n2 2 2 2\n𝜋 3𝜋\n0, while the electric field distribution in range [ , ] is the sum of both partitions:\n2 2\n𝑅 𝑘𝑟𝜌 𝑅 𝑘𝑟𝜌\n𝐸(𝑟,𝜙) = 𝑒𝑖𝑙𝜙𝐶∫ 𝐽 ( )𝜌𝑑𝜌+𝑒−𝑖𝑙𝜙𝐶∫ 𝐽 ( )𝜌𝑑𝜌\n𝑚 𝑓 𝑚 𝑓\n0 0\n= (𝑒𝑖𝑙𝜙 +𝑒−𝑖𝑙𝜙)𝐶∫ 𝑅 𝐽 ( 𝑘𝑟𝜌 )𝜌𝑑𝜌\n0 𝑚 𝑓\n𝑅 𝑘𝑟𝜌\n= (cos(𝑙𝜙)+𝑖𝑠𝑖𝑛(𝑙𝜙)+cos(−𝑙𝜙)+𝑖𝑠𝑖𝑛(−𝑙𝜙)) 𝐶∫ 𝐽 ( )𝜌𝑑𝜌\n𝑚 𝑓\n0\n𝑅 𝑘𝑟𝜌\n= 2cos (𝑙𝜙)𝐶∫ 𝐽 ( )𝜌𝑑𝜌 (S10)\n0 𝑚 𝑓\nThe final electric field distribution of discontinuous OAM can be expressed as,\n𝜋 𝜋\n0, − < 𝜙 <\n𝜋 2 2\n𝐸(𝑟,𝜙+ ) = { (S11)\n𝑅 𝑘𝑟𝜌 3𝜋\n2 2cos(𝑙𝜙)𝐶∫ 𝐽 ( )𝜌𝑑𝜌, 𝜋 < 𝜙 <\n0 𝑚 𝑓 2\nIt can be seen that the electric field distribution of the dual-partition discontinuous\nOAM is proportional to cos (𝑙𝜙). By plotting cos (𝑙𝜙) as shown in Fig. S3, we can\n𝜋 3𝜋\nobserve ten zeros in the range of 𝜙 = [ , ], corresponding to ten dark areas with zero\n2 2\nintensity in the electric field distribution, resulting in a “petal-like” field distribution as\nshown in Fig. S2f. ”\nIn Fig. S21, we demonstrate the propagation characteristics of the discontinuous OAM\nbeam in free space. Numerical simulations demonstrate the evolution of intensity\ndistributions for both half-OAM and discontinuous OAM beams, which exhibit stable\npatterns during propagation.\nFig. S21. The simulated propagation characteristics of the discontinuous OAM\nbeam in free space. After the oresnel diffraction regime, i.e., in the far field, the\ndiscontinuous OAM beam propagates stable in free space.\nComment 3.2: In Figures 1(c) and 1(d), the bottom panels are labeled as \"electric field\ndistributions.\" However, based on the visual patterns, they appear to represent intensity\nprofiles rather than complex field amplitudes. Should these be labeled as “intensity”\ninstead of “electric field” for accuracy and clarity?\nResponse 3.2: Thank you very much for your significant comment. We've modified the\nlabel of Figs. 1c and 1d accordingly.\n“\nFig. 1. Schematic illustration of the discontinuous OAM metasurface holography.”\nComment 3.3: I recommend that the authors consider adding a colorbar next to the\nphase distributions in Figure 1 of the main text for clarity. Additionally, I noticed a\npotential inconsistency in the phase color mapping. In Figure 1c, the brown-colored\nphase progresses from dark brown to white as the azimuthal angle increases, which\ncorresponds to a positive topological charge (ℓ=+10). However, in the brown segment\nof Figure 2a, the same color progression—from dark brown to white with increasing\nazimuth—appears to represent a negative topological charge (ℓ=−10). For clarity and\nconsistency, I suggest the authors ensure that the color mapping of phase is consistently\nrepresented throughout the figures, especially for readers interpreting the sign of the\nOAM from the color gradient.\nResponse 3.3: Thank you very much for your significant comment. The sign definition\nfor TC is shown as follows: a positive TC corresponds to phase increasing in the\nclockwise direction, while a negative TC indicates phase increasing counterclockwise,\nas illustrated in Fig. R3-1.\nFig. R3-1. The sign definition for TC.\nIn oig. 1c, the brown-colored phase denoted negative topological charge (𝑙 = −10).\noollowing the reviewers' suggestions, we have added a colorbar next to the phase\ndistributions in Fig. 1c and Fig. 2a.\nFig. 1. Schematic illustration of the discontinuous OAM metasurface holography.\nFig 2. Algorithm flowchart and setup of the discontinuous OAM metasurface\nholography.\nComment 3.4: While Supplementary Note 3 provides insight into the orthogonality\nbehavior via inner product analysis, it would be helpful if the authors could clarify the\nspecific method used to compute the SNR values shown in Figures 3c and 3d—such\nas how the signal and noise regions are defined, and whether any averaging or\nnormalization is applied. In particular, the near-perfect Gaussian spot distribution\nappears only when angular orthogonality is achieved. However, even in the absence\nof orthogonality, there is still observable power forming a petal-like intensity pattern. This\nraises the question of how exactly the SNR and orthogonality are defined and processed\nin the data analysis.\nResponse 3.4: Thank you very much for your significant comment. In Supplementary\nNote 8, we provide a detailed explanation of the SNR calculation methodology, where\nthe noise is obtained by subtracting the reconstructed image from the sampled original\nimage. Since the original image has a binary distribution, we normalized the\nreconstructed image to quantify crosstalk. Although decoding remains possible even\nwith non-Gaussian spot formation when the encoded and decoded discontinuous OAM\nmodes are misaligned, the SNR under such conditions is significantly low. Maximum\nSNR is achieved only when perfect mode matching occurs during decoding.\n“Supplementary Note 8: Image quality analysis between conventional OAM\nholography and discontinuous OAM holography based on a phase-only\nmetasurface\nBoth discontinuous OAM holography and conventional OAM holography can\nreconstruct high-quality images featuring Gaussian-like spot distributions when fully\ndecoded as shown in Fig. S9. We employ the signal-to-noise ratio (SNR) as the\nquantitative metric for image quality assessment, calculated as:\n𝑆𝑁𝑅 = 10𝑙𝑔\n𝑃𝑠\n(S17)\n𝑃𝑛\nP represents the signal power of the image, and here we use the intensity of the\ns\nencoded image instead, while P denotes the noise power. To quantitatively compare\nn\ncross-talk levels between the two methods, we selected decoded single-channel image\n“A” within a four-channel holography system for quality analysis. This approach\nenables direct comparison of cross-talk magnitudes generated by both methods as\nshown in Fig. S10. As shown by the blue curve, SNR remains essentially constant as\nthe incident light rotates, maintaining relatively good image quality and low cross-talk.\nIn discontinuous OAM holography, high-quality decoding of Image “A” occurs around\n0° rotation angle, while reconstruction fails at other angles. However, under full\ndecoding conditions, both methods exhibit comparable cross-talk levels.\nConsequently, our proposed discontinuous OAM holography methodology\nachieves multi-channel switching under single-incident-beam illumination without\nincreasing cross-talk.\nFig. S9. The decoded holographic images of discontinuous OAM holography and\nconventional OAM holography with a phase-only metasurface. (a) Simulation\nresults of discontinuous OAM holography. (b) Simulation results of conventional\nholography.\nFig. S10. Relationship between SNR and incident light rotation angle for\nconventional OAM holography and discontinuous OAM holography.”\nComment 3.5: In Figure 2a, I noticed that the phase distributions corresponding to\nimages B and C appear highly similar. Since the authors mention that the meta-atoms\nare based on geometric phase, and the Methods section only specifies LCP\nillumination, the polarization state of the incident light is not fully described throughout the manuscript.\nCould the authors clarify whether different circular polarizations (e.g., LCP vs. RCP) were used to access additional holographic channels?\nIf so, it would be helpful to elaborate on the polarization dependence and specify which channels were designed for which\nincident polarization state.\nResponse 3.5: Thank you very much for your significant comment. The four-channel\ndiscontinuous OAM patterns shown in Fig. 2a originate from the same vortex beam but\nwith different rotation angles. This fundamental characteristic enables holographic\nchannel switching through metasurface rotation.\nIn our original manuscript, we solely utilized the PB phase to control LCP beam,\nresulting in a single polarization channel where all incident beams were LCP carrying\ndiscontinuous OAM. Inspired by the reviewers’ insightful comments, to enhance the\ncapacity and expand multiplexing dimensions, we have now implemented PB phase\ncontrol for both LCP and RCP beams, achieving polarization-OAM hybrid encoding.\noor PB metasurface, the phase modulation exhibits opposite responses for LCP and\nRCP. When encoding LCP light, the outgoing RCP beam is deflected to the left side.\noor RCP beam encoding, the total phase must be inverted (multiplied by -1). This\nensures that both polarization channels (LCP and RCP) are steered to the same spatial\nposition under linear polarization incidence. Therefore, it is able to use pure PB phase\nto decouple LCP and RCP. As a proof of concept, we designed a two-channel\ndiscontinuous OAM holography integrated with polarization multiplexing (see more\ndetailed design principle in Ref. Adv. Mater. 37, 2415142, 2025). In Fig. S17, we\ndemonstrated a two-channel discontinuous OAM holography combined with\npolarization multiplexing.\nFig. S19. Polarization-discontinuous OAM encoding holography. (a) The\ndemonstration of a geometry-phase metasurface supporting the polarization-OAM\nhybrid encoding with a non-interleaved method in each channel. o represents the\noourier transform. (b) Phase distribution of the incident discontinuous OAM beam and\ncorresponding simulation results of reconstructed images.”\nTo further expand the multiplexing dimension and increase the number of multiplexed\nchannels, we introduced the OAM dimension into polarization-discontinuous OAM\nmultiplexing. oor each polarization state, two sets of two-channel holograms were\nencoded, achieving a total of eight channels. Both simulation and experimental results\nvalidate the feasibility of this polarization-OAM hybrid encoding scheme, which are\nshown in the main text oig. 5 as:\n“Experiments of multiple polarization and discontinuous-OAM hybrid\nmultiplexing holography\nTo further expand multiplexing dimensions and enhance channel capacity, we\nincorporated polarization and multiple pairs of discontinuous OAMs as an additional\ndegree of freedom. Specifically, by integrating two pairs of discontinuous OAM modes\nwith two orthogonal polarization states, we have designed an eight-channel metasurface\nmultiplexed holography in Fig. 5. oor left-circularly polarized (LCP) light, two pairs of\northogonal discontinuous OAM modes 𝑙 = −1/𝑙 = −15,𝑙 = −8/𝑙 = −22 are\nemployed to encode four distinct images. Similarly, for right-circularly polarized (RCP)\nlight, the set of OAM modes 𝑙 = −7/𝑙 = −15,𝑙 = −10/𝑙 = 10 encodes another four\nindependent images. The superimposed phase of LCP polarization denoted by 𝜎 is\n+\ninverted due to the opposite phase response between RCP and LCP, as shown in Fig.\n5a. Consequently, the phase profile can be expressed as,\n𝜑 = 𝑎𝑟𝑔[∑𝑁 𝑛=1 𝑒𝑗(𝜑𝜎 𝑛 − +𝜙 𝑟 𝑛 1 ) +∑𝑀 𝑚=1 𝑒𝑗(−𝜑𝜎 𝑚 + −𝜙 𝑟 𝑚 2 )] (3)\nwhere channel n = 1, 2,..., N, with N being the channel number of discontinuous OAM\nholography of RCP, m = 1, 2,..., M, with M being the channel number of discontinuous\nOAM holography of LCP. 𝜎 and 𝜎 denote RCP and LCP, respectively, 𝜑𝑛 and\n− + 𝜎−\n𝜑𝑚 is the calculated phase by WGS algorithm. The phase of discontinuous OAM 𝜙𝑛\n𝜎+ 𝑟1\nand 𝜙𝑛 is then added to the holographic phase.\n𝑟2\nThe demonstration of a geometry-phase metasurface supporting the polarization-\ndiscontinuous OAM hybrid encoding is based on a non-interleaved method[36]. We\nintegrated two-channel discontinuous OAM encoding with polarization multiplexing.\nWhen illuminated with an LCP beam carrying TC of 𝑙 = 1/𝑙 = 15, the letter “M” is\nsuccessfully decoded. When the metasurface is rotated by 90°, the decoded image\nswitches from letter “M” to letter “E”, demonstrating dynamic channel switching\nthrough angular control as shown in Fig. 5b. Similarly, when illuminated with LCP\nbeam carrying 𝑙 = 8/𝑙 = 22 , the letter “T” is decoded, and a 90° rotation of the\nmetasurface enables switching from “T” to “A”. oor RCP beam illumination, four\ndistinct images “1”, “2”, “3”, and “4” can likewise be decoded as shown in Fig. 5c,\ndemonstrating polarization multiplexing capability.\nFig. 5. Polarization-OAM multiplexing holography. (a) The design principle of a\ngeometry-phase metasurface supporting the polarization-OAM hybrid encoding with a\nnon-interleaved method for each channel. o represents the oourier transform. Simulated\nand experimental results of reconstructed images when (b) LCP and (c) RCP light\ncarried discontinuous OAM is incident to the metasurface.”\nComment 3.6: In Figure 5b, the experimental holographic images for the six-channel\nconfiguration appear noticeably clearer than those in the four-channel case shown in\nFigure 4d. Interestingly, in the six-channel experiment, each decoded holographic\nimage is encoded using two OAM values (i.e., a pair of ±ℓ). Does this suggest that\nencoding each holographic image with a pair of OAM values inherently leads to better\nreconstruction quality? It would also be helpful if the authors could provide\nquantitative metrics (e.g., SNR) for the six-channel case to facilitate a direct\ncomparison with the four-channel results.\nResponse 3.6: Thank you very much for your significant comment. In Supplementary\nNote 9, to enhance image clarity and enable quantitative characterization of inter-\nchannel crosstalk magnitude and intensity, we spatially separated the encoded images\nof each channel into distinct regions. This deliberate spatial isolation results in\nsignificantly improved visual discernibility, with detailed crosstalk patterns for other\nchannels provided in oig. S14.\nAs rightly noted by the reviewers, the four-channel OAM holography achieves\nhigher SNR compared to the six-channel configuration. We calculated the SNR for each\nchannel in both four-channel and six-channel discontinuous OAM holography. As\nshown in oig. S20, the four-channel system exhibits higher SNR values than the six-\nchannel system, indicating superior image quality in the four-channel configuration.\nHowever, the reviewer observed that the experimental results appeared to show better\nquality in the six-channel system. This apparent contradiction stems from our design\nmethodology, the six channels were spatially distributed across different physical\nlocations, thereby dispersing crosstalk to other spatial regions as shown in oig. S13. In\ncontrast, the encoded images in two-channel and four-channel discontinuous OAM\nholography overlap spatially.\n“Supplementary Note 11: The design of multiple-discontinuous-OAM\nmultiplexing holography.\nTo further expand multiplexing dimensions and enhance channel capacity, we\nincorporated multiple pairs of discontinuous OAMs as an additional degree of freedom.\nSpecifically, we designed a six-channel multiplexed metasurface holography by\nintegrating three pairs of discontinuous OAM. The design workflow of the holographic\nphase information is detailed in Fig. S13. In each channel, the input image is multiplied\nby a 2D Dirac comb function spatially positioned at distinct locations across the optical\nfield. oollowing a modified WGS algorithm for phase extraction, the acquired phase is\nencoded by three orthogonal discontinuous OAM modes: 𝑙 = 10/𝑙 = −10,𝑙 =\n15/𝑙 = −15,𝑙 = 20/𝑙 = −20 . The superposition of all channels enables the\nrealization of multiple-discontinuous-OAM multiplexing holography. The\ndiscontinuous OAM azimuthal distributions of these modes are shown in the left panel\nof Fig. S14a. This spectral complexity inherently enhances security by preventing\nunauthorized access through brute-force OAM mode matching. Experimental results in\nFig. S14b demonstrate the successful decoding of all six channels at different locations,\nwith the reconstructed images exhibiting near-perfect Gaussian spot profiles.\nFig. S13. The algorithm flowchart of multiple-discontinuous-OAM multiplexing\nholography.\nFig. S14. Multiplexing OAM holography by integrating various pairs of\ndiscontinuous OAM modes. (a) Schematic diagram. When the incident discontinuous\nOAM mode is 𝑙 = 10/𝑙 = −10, the output images are “0” and “1”; when the incident\ndiscontinuous OAM mode is 𝑙 = 15/𝑙 = −15 , the output images are “2” and “3”;\nwhen the incident discontinuous OAM mode is 𝑙 = 20/𝑙 = −20, the output images\nare “4” and “5”. The TC distributions of these three modes are shown in the left panel.\n(b) Experiment results of the six channels under proper illumination of TC distribution\nand rotation angle of incident beam.\nThe comprehensive experimental results of multiple-discontinuous-OAM\nmultiplexing holography are shown in Fig. S15. To better visualize the spatial intensity\nprofiles of non-decoded channels under discontinuous OAM illumination, we\nstrategically positioned six channels at distinct areas. This design enables clear\ndifferentiation: only the fully decoded image exhibits a Gaussian spot (indicating\nperfect orthogonality), while non-decoded channels display semi-annular or hybrid\nintensity patterns, resulting from partial overlap between incident discontinuous OAM\nand encoded phase profiles.\nFig. S15. Experimental results of multiple-discontinuous-OAM holography. The\nred boxes indicate the fully decoded images under proper discontinuous OAM\nillumination.”\nFig. S20. SNR Comparison between four-channel and six-channel discontinuous\nOAM Holography. The blue dots represent the SNR of each channel image in the\nfour-channel discontinuous OAM holography. The red dots represent the SNR of\neach channel image in the six-channel discontinuous OAM holography.\nIn summary, we have addressed all the comments of reviewer #3."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Thanks to authors for their effort with improved revised manuscript. The authors have addressed most of questions raised in\nprevious round reviewing. Therefore, I recommend considering potential publication of this study after addressing two\nremaining questions still unclear.\n1. The discontinuous OAM modes are used as a new dimension to enhance the capacity of holographic multiplexing.\nHowever, the capacity limit of the proposed method remains unclear, when the inter-channel crosstalk and deviations are\nsmall and acceptable. Please provide discussions and quantitatively estimate this limit.\n2. The vortex beam with discontinuous OAM in this study is normal, which means that the beams with varied TCs also have\ndifferent beam profile, such as beam size and central area with null field. Is the proposed method still effective in case of\nfractional vortex beams and perfect vortex beams such as (Yao, J. et al. A spatial-frequency patching metasurface enabling\nsuper-capacity perfect vector vortex beams. eLight 4, 21 (2024))?", "reviewer_2": "The revised manuscript demonstrates the authors’ considerable efforts in addressing the questions and concerns raised by\nmyself and the other reviewers. Overall, the manuscript has been significantly improved. Nevertheless, one critical issue\nremains that precludes me from recommending acceptance: the quality and signal-to-noise ratio of the experimental results\nare substantially inadequate. Without the reference dashed boxes, it would be difficult to identify which letter the decrypted\nexperimental image represents. The specific concerns are outlined below:\n1. In Figure 5, stray speckles are still present in non-letter regions. What criteria were used to distinguish these artifacts from\nactual characters, and how were they excluded? Explicit justification should be provided. Moreover, certain areas within the\nletter and number regions suffer from severely diminished brightness or a complete lack of signal. What may have caused\nthese irregularities?\n2. The problems in Figure S18 are more pronounced. Some numerals exhibit extensive missing data points, and the font\nstyle in the experimental results diverge noticeably from that in the simulations. Additionally, the experimental images\ndisplay inconsistent rotational misalignment across different numerals. What is the origin of these rotational errors, and how\nwere the angles determined during the experiments?", "reviewer_3": "After a thorough review of the revised manuscript and supporting materials, I confirm that the authors have satisfactorily\naddressed the technical issues raised by Reviewers 1, 2, and myself (Reviewer 3). The relevant technical clarifications are\nappropriately documented in the Supplementary Information. This work demonstrates an innovative and technically sound\nadvancement in OAM optics, holography, cryptographic imaging, and multifunctional metasurface design. I believe the\nmanuscript is suitable for publication in Nature Communications."}, "author_response": "Reply report to reviewer #1\nComment: Thanks to authors for their effort with improved revised manuscript. The\nauthors have addressed most of questions raised in previous round reviewing.\nTherefore, I recommend considering potential publication of this study after addressing\ntwo remaining questions still unclear.\nResponse: We would like to thank the reviewer for recognizing our effort in\nimproving the manuscript. We are happy to address the remaining concerns raised by the reviewer.\nComment 1.1: The discontinuous OAM modes are used as a new dimension to enhance\nthe capacity of holographic multiplexing. However, the capacity limit of the proposed\nmethod remains unclear, when the inter-channel crosstalk and deviations are small and\nacceptable. Please provide discussions and quantitatively estimate this limit.\nResponse 1.1: The authors thank the reviewer for this point. To present the capacity\nlimit of our proposed method, we have calculated the SNR for different channels. As\nthe number of topological charges encoded by discontinuous OAM increases, the\nnumber of azimuthal partitions rises, making the crosstalk more significant. When the\nnumber of channels in discontinuous OAM holography reaches forty, the SNR drops to\napproximately 0. This indicates that the noise (i.e., crosstalk) becomes almost equal to\nthe signal (the decoded hologram), demonstrating that the quality of the decoded\nholographic signal is severely degraded and multi-channel encryption can no longer be\neffectively achieved.\nThe channel capacity of discontinuous OAM holography is analyzed in\nSupplementary Note 12.\n“Supplementary Note 12: The capacity limit of discontinuous OAM holography.\nTheoretically, the number of channels in discontinuous OAM holography is\nunlimited because the 2π azimuthal angle can be infinitely divided. However, the\ncrosstalk between different channels will increase. As shown in Fig. S16, we calculated\nthe SNR of discontinuous OAM holography with different numbers of channels. When\nthe number of channels in discontinuous OAM holography increases to forty-two, the\nSNR drops to approximately 0. This indicates that the noise (i.e., crosstalk) becomes\nnearly equivalent to the signal (the decoded hologram), demonstrating that the quality\nof the decoded holographic signal is severely degraded and multi-channel encryption\ncan no longer be effectively achieved.\nFig. S16. The SNR of discontinuous OAM holography with different numbers of\nchannels. The error bars represent the range between the maximum and minimum SNR\nvalues across all channels in the multi-channel discontinuous OAM holography.”\nComment 1.2: The vortex beam with discontinuous OAM in this study is normal, which\nmeans that the beams with varied TCs also have different beam profile, such as beam\nsize and central area with null field. Is the proposed method still effective in case of\nfractional vortex beams and perfect vortex beams such as (Yao, J. et al. A spatial-\nfrequency patching metasurface enabling super-capacity perfect vector vortex beams.\neLight 4, 21 (2024))?\nResponse 1.2: The authors thank the reviewer for this point. Our approach of\ndiscontinuous OAM holography is universal—it is applicable not only to conventional\nOAM beams, but also to fractional vortex beams and perfect vortex beams. The only\nrequirement is that the OAM modes encoded at different angular positions must be\nmutually orthogonal.\nThis discussion has been added in Supplementary Note 13 as:\n“Supplementary Note 13: Discontinuous fractional and perfect OAM holography.\nThe concept of discontinuous OAM holography can be extended to other types of\nbeams. This method is universal—it is applicable not only to conventional OAM beams,\nbut also to fractional vortex beams and perfect vortex beams. The only requirement is\nthat the OAM modes encoded at different angular positions must be mutually\northogonal. Here, we demonstrate the implementation of two-channel discontinuous\nOAM holography using fractional vortex beams and perfect vortex beams as an\nexample.\nThe discontinuous OAM is encoded with fractional TC of +10.5 in the upper half\nand –10.5 in the lower half, as shown in Fig. S17a. These two regions correspond to\ntwo independent holographic channels. The decoded reconstructed results are shown in\nFig. S17b. When illuminated by the discontinuous fractional OAM beams with this\nspecific configuration, the image “0” is successfully decoded, after rotating the beam\nby 180 degrees, the decoded image switches from image “0” to image “1”. Also, the\ndiscontinuous OAM encoded with perfect vortex beams is shown in Fig. S17c. The\ndecoded reconstruction results are shown in Fig. S17d. Both channels of discontinuous\nfractional OAM holography and discontinuous perfect OAM holography demonstrate\nexcellent performance.\nFig. S17. Discontinuous OAM holography using fractional vortex beams and\nperfect vortex beams. (a) Phase distribution of discontinuous fractional vortex beams.\n(b) Experimental results of two-channel discontinuous fractional OAM holography. (c)\nPhase distribution of discontinuous perfect vortex beams. (d) Experimental results of\ntwo-channel discontinuous perfect OAM holography.”\nIn summary, we have addressed all the comments of reviewer #1.\nReply report to reviewer #2\nComment 2: The revised manuscript demonstrates the authors’ considerable efforts in\naddressing the questions and concerns raised by myself and the other reviewers.\nOverall, the manuscript has been significantly improved. Nevertheless, one critical\nissue remains that precludes me from recommending acceptance: the quality and\nsignal-to-noise ratio of the experimental results are substantially inadequate. Without\nthe reference dashed boxes, it would be difficult to identify which letter the decrypted\nexperimental image represents.\nResponse 2: We would like to thank the reviewer for recognizing our effort in\nimproving the manuscript. We are happy to address the concerns raised by the reviewer.\nIn our new results, we have removed all reference dashed boxes for better clarification.\nComment 2.1: In Figure 5, stray speckles are still present in non-letter regions. What\ncriteria were used to distinguish these artifacts from actual characters, and how were\nthey excluded? Explicit justification should be provided. Moreover, certain areas\nwithin the letter and number regions suffer from severely diminished brightness or a\ncomplete lack of signal. What may have caused these irregularities?\nResponse 2.1: We sincerely appreciate the reviewer's constructive feedback and have\nconducted a comprehensive analysis of the factors contributing to the suboptimal\nresults in our previous experiment. The stray speckles observed in non-letter regions\nresult from imperfect alignment between the sample and the incident light, as well as\nspherical aberrations introduced by the previous lens. In addition, certain areas suffer\nfrom severely diminished brightness or a complete lack of signal, resulting from\nmultiple holographic channels designed to occupy the same spatial location, causing\ncrosstalk and overlapping decoded images. When the beam is not parallel or properly\naligned, the crosstalk at certain points will exhibit intensity comparable to that of the\ndecoded Gaussian spots.\nTo improve the experimental results, we repeated the experiment by replacing the lens\nwith a high Numerical Aperture (NA) objective (NA=0.6) and adjusted the pitch angle\nbetween the lens and the metasurface to achieve precise alignment. The final results\ndemonstrate excellent agreement with simulations, as shown in the updated Fig. 5\nbelow.\nFig. 5. Polarization-OAM hybrid encoding. (a) The demonstration of a geometry-\nphase metasurface supporting the polarization-OAM hybrid encoding with a non-\ninterleaved method in each channel. F represents the Fourier transform. Simulated and\nexperimental results of reconstructed images when (b) LCP and (c) RCP light beam\ncarried discontinuous OAM is incident to the metasurface.\nComment 2.2: The problems in Figure S18 are more pronounced. Some numerals\nexhibit extensive missing data points, and the font style in the experimental results\ndiverge noticeably from that in the simulations. Additionally, the experimental images\ndisplay inconsistent rotational misalignment across different numerals. What is the\norigin of these rotational errors, and how were the angles determined during the\nexperiments?\nResponse 2.2: The authors thank the reviewer for this valuable point. Similar to the\nResponse 2.1, the missing data spots in the sample result from imperfect alignment\nbetween the sample and the incident light, which induces crosstalk that obscures the\nGaussian signal spots. Additionally, the inconsistent rotational misalignment observed\nacross different numerals in the experimental results occurred because, in the previous\nexperiment, the sample was not properly leveled and was tilted at an angle. This caused\nsome angular discrepancies when manually adding the reference dashed boxes, which\nwere not aligned correctly with the original experimental numerals, leading to\nnoticeable variations in angles. We are sorry for this oversight and have now carefully\nreconduct the experiment. The new experimental results show clear visualization of the\noutcomes without the need for any postprocessing of the image or adding of reference\nboxes. The experimental results, shown in the revised Supplementary Fig. S19 below,\nare in excellent agreement with the simulated results.\nWe sincerely appreciate the reviewer’s insightful comments, which have allowed us to\nsignificantly improve the experimental results and the manuscript.\nFig. S19. Ten-channel discontinuous OAM holography. (a) The phase distribution\nof ten-channel discontinuous OAM holography. (b) The distribution of TC across\nazimuth angle. (c) OAM-encoded image multiplexing based on one discontinuous\nOAM states with ten rotate angles. (d) Simulated holographic images reconstructed\nfrom ten-channel discontinuous OAM meta-hologram. (e) Experimental holographic\nimages reconstructed from ten-channel discontinuous OAM meta-hologram.\nIn summary, we have addressed all the comments of reviewer #2.\nReply report to reviewer #3\nComment 3: After a thorough review of the revised manuscript and supporting\nmaterials, I confirm that the authors have satisfactorily addressed the technical issues\nraised by Reviewers 1, 2, and myself (Reviewer 3). The relevant technical clarifications\nare appropriately documented in the Supplementary Information. This work\ndemonstrates an innovative and technically sound advancement in OAM optics,\nholography, cryptographic imaging, and multifunctional metasurface design. I believe\nthe manuscript is suitable for publication in Nature Communications.\nResponse 3: We are grateful to the reviewer’s positive feedback and recommendation\nto publish in Nature Communications."}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors have almost addressed all of my concerns. There is only one point that is unclear. In the reply, the authors said\nthe SNR approaches zero when the topological charge reaches 40. However, in the Supplementary notes 12, the limit of TC\nis given as 42. Moreover, the authors should also include some discussions on the acceptable level of SNR so that the\nauthors can get a clear concept of this limit. Finally, some classic references would be necessary to help the audience\nunderstand PVB and FVB mentioned in the Supplementary Notes. After fully addressing these minor issues, I would be\ndelighted to recommended this article for publication in Nature Communications.", "reviewer_2": "Based on the revised manuscript, the author has addressed the relevant questions and concerns I raised, and it can now be\nrecommended for publication.", "reviewer_3": "In this revision, the authors have carefully addressed the reviewers’ comments. For example, the SNR analysis for non-\ninteger OAM requested by Reviewer 1, and the improved measurement methodology raised by Reviewer 2, have both been\nclearly incorporated. These updates allow the experimental data to better reflect the target designs, with or without reference\ndashed boxes. In my opinion, the manuscript could be accepted after addressing a few minor points:\n1. In Response 1.1 and Supplementary Note 12, the statement “SNR drops to approximately 0” seems unusual. Typically,\nwhen SNR approaches 1, the signal becomes indistinguishable from noise, which should already indicate the capacity limit.\nThe choice of “SNR ~ 0” as the threshold feels inconsistent and should be clarified.\n2. In the revised experimental holograms of Figure 5b and 5c, the target designs are not as visually clear as in Figure S19 if\nthe dashed boxes are removed. I assume this is because Figure 5 involves multiple polarization channels and fewer\nsampling points per channel, while Figure S19 provides denser sampling for each holographic image. Overall, this does not\nundermine the authors’ contribution to the field but could be clarified in the discussion."}, "author_response": "Reply report to reviewer #1\nComment 1.1: The authors have almost addressed all of my concerns. There is only\none point that is unclear. In the reply, the authors said the SNR approaches zero when\nthe topological charge reaches 40. However, in the Supplementary notes 12, the limit\nof TC is given as 42.\nResponse 1.1: We would like to thank the reviewer for recognizing our effort in\nimproving the manuscript. There is a typo in the original response letter. When the\nnumber of channels in discontinuous OAM holography increases to forty-two, the SNR\nof all holographic channels drops to the range of 0-2 which indicates that the signal and\nnoise have become inseparable.\nThis modification is detailed in Supplementary Note 12 as:\n“Supplementary Note 12: The capacity limit of discontinuous OAM holography.\nTheoretically, the number of channels in discontinuous OAM holography is\nunlimited because the 2π azimuthal angle can be infinitely divided. However, the\ncrosstalk between different channels will increase. As shown in Fig. S16, we calculated\nthe SNR of discontinuous OAM holography with different numbers of channels. As the\nnumber of channels grows, the SNR of the decoded image gradually increases, while\nthe crosstalk also intensifies. When the number of channels in discontinuous OAM\nholography increases to 42, the SNR of all holographic channels drops to the range of\n0-2 which indicates that the signal and noise have become inseparable. This indicates\nthat the noise (i.e., crosstalk) becomes nearly equivalent to the signal (the decoded\nhologram), demonstrating that the quality of the decoded holographic signal is severely\ndegraded and multi-channel encryption can no longer be effectively achieved.\nFig. S16. The SNR of discontinuous OAM holography with different numbers of\nchannels. The error bars represent the range between the maximum and minimum SNR\nvalues across all channels in the multi-channel discontinuous OAM holography.”\nComment 1.2: Moreover, the authors should also include some discussions on the\nacceptable level of SNR so that the authors can get a clear concept of this limit.\nResponse 1.2: The authors thank the reviewer for this point. When the SNR is 1 (i.e.,\n0 dB), it indicates that the average power of the signal is equal to the average power of\nthe noise. Therefore, we have modified the threshold for evaluating image quality,\ndetermining that when the SNR reaches 1, the signal and noise become inseparable.\nThis discussion is detailed in Supplementary Note 8 as:\n“Supplementary Note 8: Image quality analysis between conventional OAM\nholography and discontinuous OAM holography based on a phase-only\nmetasurface\nBoth discontinuous OAM holography and conventional OAM holography can\nreconstruct high-quality images featuring Gaussian-like spot distributions when fully\ndecoded as shown in Fig. S9. We employ the signal-to-noise ratio (SNR) as the\nquantitative metric for image quality assessment, calculated as:\n𝑆𝑁𝑅 = 10𝑙𝑔\n𝑃𝑠\n(S17)\n𝑃𝑛\nP represents the signal power of the image, and here we use the intensity of the\ns\nencoded image instead, while P denotes the noise power. When the SNR is 1 (i.e., 0\nn\ndB), it indicates that the average power of the signal is equal to the average power of\nthe noise. To quantitatively compare cross-talk levels between the two methods, we\nselected decoded single-channel image “A” within a four-channel holography system\nfor quality analysis. This approach enables direct comparison of cross-talk magnitudes\ngenerated by both methods as shown in Fig. S10. As shown by the blue curve, SNR\nremains essentially constant as the incident light rotates, maintaining relatively good\nimage quality and low cross-talk. In discontinuous OAM holography, high-quality\ndecoding of Image “A” occurs around 0° rotation angle, while reconstruction fails at\nother angles. However, under full decoding conditions, both methods exhibit\ncomparable cross-talk levels.\nConsequently, our proposed discontinuous OAM holography methodology\nachieves multi-channel switching under single-incident-beam illumination without\nincreasing cross-talk.”\nComment 1.3: Finally, some classic references would be necessary to help the\naudience understand PVB and FVB mentioned in the Supplementary Notes. After fully\naddressing these minor issues, I would be delighted to recommended this article for\npublication in Nature Communications.?\nResponse 1.3: The authors thank the reviewer for this point. We cited some classic\nreferences related to PVB and FVB in the supplementary material.\n“3. Zhu J, Wang L, Zhao S. Security-enhanced and high-resolution fractional orbital\nangular momentum multiplexing holography. ACS Photonics 11, 4626-4634\n(2024).\n4. Zhu G, Bai Z, Chen J, et al. Ultra-dense perfect optical orbital angular momentum\nmultiplexed holography. Opt. Express 29, 28452-28460 (2021).”\nIn summary, we have addressed all the comments of reviewer #1.\nReply report to reviewer #2\nComment 2: Based on the revised manuscript, the author has addressed the relevant\nquestions and concerns I raised, and it can now be recommended for publication.\nResponse 2: We are grateful to the reviewer’s positive feedback and recommendation\nto publish in Nature Communications.\nIn summary, we have addressed all the comments of reviewer #2.\nReply report to reviewer #3\nComment 3: In this revision, the authors have carefully addressed the reviewers’\ncomments. For example, the SNR analysis for non-integer OAM requested by Reviewer\n1, and the improved measurement methodology raised by Reviewer 2, have both been\nclearly incorporated. These updates allow the experimental data to better reflect the\ntarget designs, with or without reference dashed boxes. In my opinion, the manuscript\ncould be accepted after addressing a few minor points:\nResponse 3: We are grateful to the reviewer’s positive feedback and recommendation\nto publish in Nature Communications.\nComment 3.1: In Response 1.1 and Supplementary Note 12, the statement “SNR drops\nto approximately 0” seems unusual. Typically, when SNR approaches 1, the signal\nbecomes indistinguishable from noise, which should already indicate the capacity limit.\nThe choice of “SNR ~ 0” as the threshold feels inconsistent and should be clarified.\nResponse 3.1: The authors thank the reviewer for this point. When the SNR is 1 (i.e.,\n0 dB), it indicates that the average power of the signal is equal to the average power of\nthe noise. Therefore, we have modified the threshold for evaluating image quality,\ndetermining that when the SNR reaches 1, the signal and noise become inseparable.\nThis discussion is detailed in Supplementary Note 8 as:\n“Supplementary Note 8: Image quality analysis between conventional OAM\nholography and discontinuous OAM holography based on a phase-only\nmetasurface\nBoth discontinuous OAM holography and conventional OAM holography can\nreconstruct high-quality images featuring Gaussian-like spot distributions when fully\ndecoded as shown in Fig. S9. We employ the signal-to-noise ratio (SNR) as the\nquantitative metric for image quality assessment, calculated as:\n𝑆𝑁𝑅 = 10𝑙𝑔\n𝑃𝑠\n(S17)\n𝑃𝑛\nP represents the signal power of the image, and here we use the intensity of the\ns\nencoded image instead, while P denotes the noise power. When the SNR is 1 (i.e., 0\nn\ndB), it indicates that the average power of the signal is equal to the average power of\nthe noise. To quantitatively compare cross-talk levels between the two methods, we\nselected decoded single-channel image “A” within a four-channel holography system\nfor quality analysis. This approach enables direct comparison of cross-talk magnitudes\ngenerated by both methods as shown in Fig. S10. As shown by the blue curve, SNR\nremains essentially constant as the incident light rotates, maintaining relatively good\nimage quality and low cross-talk. In discontinuous OAM holography, high-quality\ndecoding of Image “A” occurs around 0° rotation angle, while reconstruction fails at\nother angles. However, under full decoding conditions, both methods exhibit\ncomparable cross-talk levels.\nConsequently, our proposed discontinuous OAM holography methodology\nachieves multi-channel switching under single-incident-beam illumination without\nincreasing cross-talk.”\nComment 3.2: In the revised experimental holograms of Figure 5b and 5c, the target\ndesigns are not as visually clear as in Figure S19 if the dashed boxes are removed. I\nassume this is because Figure 5 involves multiple polarization channels and fewer\nsampling points per channel, while Figure S19 provides denser sampling for each\nholographic image. Overall, this does not undermine the authors’ contribution to the\nfield but could be clarified in the discussion.\nResponse 3.2: The authors thank the reviewer for this point. In fact, the images of\ndifferent channels in Fig. 5 are designed at the same position, whereas in Fig. S19,\nthey are placed at distinct positions. Therefore, Fig. S19 exhibits superior image quality with\nenhanced clarity compared to Fig. 5. A detailed explanation of this aspect is provided\nin Supplementary Note 10. “It should be noted that the numerals and letters in the\nholographic images across different channels are designed to occupy the same spatial\nposition in all figures of the main text and Supplementary materials, except for Figs.\nS13, S14, S16 and S19, where they are intentionally placed at different positions to\ndemonstrate the versatility of our approach for improving the channel numbers.”\nIn summary, we have addressed all the comments of reviewer #3."}]

Decision letter