Speckle-Driven Single-Shot Orbital Angular Momentum Recognition with Ultra-Low Sampling Density

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However, conventional approaches designed for free-space propagation struggle when vortex beams propagate within or through complex media, such as multimode fibers (MMF), and often rely on high-resolution imaging sensors with tens of thousands of pixels to record dense intensity profiles. Here, we introduce a speckle-driven OAM recognition technique that exploits the intrinsic correlation between speckle patterns and OAM states, circumventing the limitations of scattering media while drastically reducing sampling requirements. Our method, termed spatially multiplexed points detection (SMPD), extracts intensity information from spatially distributed points in a multiplexed speckle plane. Remarkably, it achieves > 99% retrieval accuracy for OAMs recognition using just 16 sampling points, corresponding to a sampling density of 0.024%—4096 times lower than conventional imaging-based approaches. Furthermore, high-capacity OAM-multiplexed communication decoding with an error rate of < 0.2% and handwritten digit recognition with an accuracy of 89% are implemented to verify the versatility of SMPD. This work transcends the trade-off between sampling density and accuracy, establishing a scalable platform for resource-efficient photonic applications like quantum communication and endoscopic sensing. Physical sciences/Optics and photonics/Optical techniques/Imaging and sensing Physical sciences/Optics and photonics/Applied optics/Fibre optics and optical communications Physical sciences/Mathematics and computing/Information technology Single pixel detection Multimode fiber Speckle Orbital angular momentum Wavefront shaping Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Introduction The precise manipulation of light’s orbital angular momentum (OAM) has emerged as a cornerstone of modern optics 1 – 4 , unlocking an infinite-dimensional basis for applications ranging from high-capacity optical communication 5 – 12 and quantum information protocols 13 – 15 to multidimensional sensing 16 – 18 . Central to this paradigm is the vortex beams, whose helical phase structure – characterized by an azimuthal phase dependence of exp( iℓθ ) – enables robust encoding of information into discrete OAM modes. These modes form the backbone of spatial multiplexing strategies that transcend classical transmission limits 1 – 3 . Yet, when propagating through complex media such as multimode fibers (MMFs), which are commonly used for light information transmission, these beams undergo severe distortions due to modal coupling and dispersion, transforming their deterministic phase profiles into disordered speckle fields 9 , 19 – 22 . This wavefront scrambling erases the intrinsic OAM signatures, creating a critical bottleneck for reliable OAM retrieval at the fiber output, a limitation that stifles the deployment of OAM-based systems in real-world scenarios. To address this challenge, computational strategies such as transmission matrix (TM) inversion 5 , deep learning 9 , 19 , 23 , and Fourier covariance analysis 10 have been developed to decode OAM states from speckle-corrupted outputs. While these methods demonstrate success, they generally depend on high-resolution imaging to capture full-field speckle intensity maps, necessitating thousands of or more pixels and computationally intensive processing. Such reliance on dense spatial sampling imposes stringent trade-offs; it throttles system bandwidth, complicates scalability, and obstructs real-time operation 24 , 25 . Consequently, applications demanding compact form factors, high-speed detection, or operation in photon-limited scenarios — cornerstones of next-generation photonic technologies — remain largely out of reach. Recent advances reveal that the ostensibly disordered nature of speckle fields marks a profound redundancy in their information content 26 – 28 . Unlike direct imaging systems, which localize spatial information to discreate regions, multiple scattering in complex media redistributes input-field contributions across the entire speckle pattern (see Supplementary note 1 for more details). This non-local encoding enables information retrieval based on partial-field sampling for tasks such as imaging through scattering media 29 , diffuser-based super-resolution imaging via sub-Nyquist sampling 30 , and OAM recognition from arbitrary speckle subregions 9 . Yet, leveraging this redundancy for ultralow-sampling detection, particularly in OAM-based pattern classification, remains underexplored. A pivot challenge persists: What constitutes the minimal spatial sampling required to maintain accurate, scalable mode discrimination? To address this, we propose spatially multiplexed points detection (SMPD), a framework for single-shot pattern classification with unprecedented sampling efficiency. Instead of recording full-field speckles, SMPD spatially samples intensity fluctuations at a sparse array of spatially multiplexed single-pixel detectors (SPDs), each capturing globally encoded information from the scattered field. A custom-designed neural network deciphers these sub-Nyquist intensity sequences, achieving robust OAM recognition with a sampling density of 0.024%—4096 times lower than conventional imaging-based methods—while maintaining over 99% accuracy. Beyond OAM-mode retrieval, SMPD generalizes to classify mode-dependent features and distinguish complex optical fields, including handwritten digit recognition and axicon angle identification, demonstrating its versatility across diverse scenarios. By refining the spatial sampling-information capability trade-off, SMPD transcends the limitation of imaging-centric detection paradigms. This approach enables high-dimensional optical sensing and communication systems that are both compact and bandwidth-efficient, critical for real-time operation in photon-starved environments. Furthermore, its ultralow sampling requirements may unlock applications for non-line-of-sight (NLOS) sensing, low-power LIDAR, and quantum-enhanced sensing, where conventional pixel-dense sensors, constrained by their limited bandwidth and spectral ranges, prove impractical. SMPD thus establishes a new roadmap for scalable photonic technologies, bridging the gap between theoretical information redundancy and practical, resource-efficient detection. Results The proposed SMPD framework was rigorously evaluated through a series of experiments to validate its efficiency in OAM recognition and its applicability to optical communication systems. Below, we present the key findings that highlight the method’s performance, robustness, and scalability. High-Accuracy OAM Recognition with Ultra-Low Sampling Density Figure 1 illustrates the concept of SMPD, where two orthogonally polarized vortex beams transmitted through an MMF generate a speckle pattern at the distal end of the fiber. Instead of conventional full-field imaging, SMPD employs spatially distributed SPDs to sample sparse intensity points. These measurements are processed by a customized neural network termed as Recognizing OAM Artificial Neural Network (ROAM-ANN), which secures two paths to extract the features of the inputs in different dimensions and eventually outputs the encoded OAMs from the pointwise intensity sequences. To quantify the impact of SPD parameters on recognition accuracy, we systematically varied the number and size of SPDs (Fig. 2 ). A charge-coupled device (CCD) camera with various masks is used to create different SPDs of varying numbers and sizes. Here, SPD is defined in units of ‘px’, and 1 px corresponds to the area of a single element (7.4 2 µm 2 ) in the CCD camera. For instance, an SPD area of 5 2 px indicates 5 by 5 neighboring CCD elements are combined to form one SPD, and a total of 9 points means that 9 such SPDs are imposed on the CCD via the mask. For each SPD, the light intensities detected by CCD’s light-sensitive elements are accumulated, and the recorded intensity sequence of the SPD array is used as the input of the network. The recognition results for the test sets, each containing 1,000 intensity sequences randomly selected from a dataset of 10,000 combinations of two orthogonally polarized vortex beams (both OAMs varying from 1 to 10.9 with an interval of 0.1) and excluded from the training process, are shown in Fig. 2 a (more detailed results can be found in Supplementary Note 2 ). As seen, 16 SPDs (each spanning 30 × 30 px) achieved > 99% accuracy for both OAM modes. Critically, the total sampling density of this configuration is approximately 0.024% (16 vs. 256 × 256), corresponding to 4096 times lower than that of conventional imaging-based methods 9 that usually require 256 × 256 or more pixel arrays. Even with SPDs reduced to 2 × 2 px (retaining 16 detectors), accuracy remained above 93%. That said, a clear trade-off emerged: reducing SPD count or size degraded performance. For instance, with only 6 SPDs (1 × 1 px), accuracy dropped below 50%, highlighting the necessity of more sampling points to capture speckle correlations, as well as a larger sensing area to mitigate the noise and pixel displacement. To better visualize this trend, Fig. 2 b plots recognition accuracy as a function of the total SPD area across different measurement sets. Spatial Deployment Flexibility A key advantage of SMPD lies in its insensitivity to the spatial deployment of SPDs. As shown in Table 1 , three distinct spatial deployments of 9 SPDs (with a fixed total area of 10× 10 px) yielded minimal accuracy fluctuations: 89.40-95.15% for OAM1 and 78.05–80.55% for OAM2. This flexibility arises from the non-local encoding of information inherent to speckle patterns, where each SPD integrates contributions from the entire input field. Unlike traditional imaging systems requiring fixed source-detector alignment, SMPD’s adaptability enables flexible deployment in spatially restricted or geometrically constrained environments, such as endoscopic probe or miniaturized sensors. High-Capacity Optical Communication via SMPD To demonstrate real-world applicability, we implemented an OAM-multiplexed optical communication system (Fig. 3 ). Two spiral phase images (100 × 100 pixels, 8-bit color depth) were encoded using OAM bases ranging from 2.1 ~ 2.8 (see Supplementary Note 3 for details). A digital micro-mirror device (DMD) was employed to effectively generate multiplexed vortex beams carrying the encoded data, which were then coupled into the MMF. As the beams propagated through the MMF, speckle patterns formed at the distal end. At the receiver, SMPD decoded speckle patterns using three configurations: 16, 9, or 4 SPDs. With 16 SPDs (10 × 10 px each), decoding error rates remained < 0.2% (Fig. 3 c). Reducing SPD size to 1 × 1 px slightly increased errors (due to localized noise and pixel displacement) but maintained robust performance. Even with only 4 SPDs (10 × 10 px each), the system achieved 74% accuracy, underscoring SMPD’s resilience under extreme under-sampling. This experiment validates SMPD’s potential for high-speed, low-bandwidth optical communication, particularly in resource-limited settings. Generalizable Pattern Recognition via Ultra-Sparse Speckle Sampling To demonstrate SMPD’s versatility beyond OAM retrieval, we extended its application to handwritten digit recognition using the MNIST dataset 31 . Conventional speckle-based digit classification relies on full-field speckle imaging and computationally intensive spatial feature extraction. In contrast, as illustrated in Fig. 4 a, our method employs simulated scattering transmission of MNIST digits (28 × 28 pixels), generating speckle patterns (256 × 256 pixels) that are subsampled via programmable SPD arrays. Three configurations were tested: 6 × 6, 5 × 5, and 4 × 4 SPD arrays, where each detector integrates intensity over a 10 × 10 px region. The resulting intensity sequences (comprising 36, 25, or 16 values, respectively) were fed into a neural network trained on 48,000 intensity-label pairs and validated on 2,000 pairs. Remarkably, SMPD achieved 89.42%, 83.21%, and 71.78% recognition accuracy on the test dataset (10,000 samples) for 6 × 6, 5 × 5, and 4 × 4 SPD arrays, respectively (Fig. 4 b). The confusion matrices reveal minimal misclassification trends, with errors predominantly occurring between morphologically similar digits ( e.g. , 3↔5, 4↔9). This performance underscores SMPD’s capacity to generalize across diverse optical patterns despite extreme subsampling (e.g., 16 SPDs for 4 × 4 arrays vs. 28× 28 pixels, 2.04% sampling density). The results validate that spatially sparse, non-locally encoded speckle correlations retain sufficient information for high-accuracy classification tasks—even in photon-efficient regimes—without resorting to pixel-dense sensors. Discussion The success of SMPD in achieving high-accuracy OAM recognition with ultra-low sampling density stems from the intrinsic properties of speckle patterns formed in complex media. Unlike free-space propagation, model coupling and superposition processes in multimode fibers (MMFs) redistribute input wavefront information across the entire speckle field, creating spatially redundant correlations. That is, the information projected onto a single speckle grain on the output plane is the superposition of the entire incident light field 29 , 30 , 32 , 33 . Such non-local encoding enables partial-field sampling to retain sufficient information for OAM retrieval—a principle validated by transmission matrix theory and prior work on scattering media 26 , 27 . By exploiting this redundancy, SMPD circumvents the need for high-resolution imaging, which has long constrained the scalability of OAM-based technologies 9 , 19 , 23 . A pivotal insight from our experiments is that spatial sampling density, rather than absolute detection area, governs recognition accuracy. While conventional methods 8 , 9 , 19 , 23 require dense pixel arrays (e.g., over 256 × 256 pixels) to resolve OAM-dependent features, SMPD achieves comparable accuracy with just 16 SPDs (0.024% sampling density). This efficiency arises because each SPD integrates contributions from the entire input field, effectively compressing global correlations into sparse measurements. The neural network (ROAM-ANN) further amplifies this advantage by learning discriminative features from multiplexed intensity sequences. Consequently, with the proposed method, spatially resolved cameras are no longer necessary for OAM recognition. Instead, speckle intensity data from a minimal amount of SPDs is sufficient, opening venues for high-speed yet low-cost single-shot recognition solutions to a wide range of field applications. To demonstrate its versatility, we present an additional experiment in Supplementary Notes 4 , including axicon angle recognition, further highlighting the broad applicability of SMPD. The flexibility of SPD spatial deployment (Table 1 ) underscores SMPD’s adaptability. Unlike ballistic imaging or ghost imaging, which demand rigid detector-source alignment, SMPD’s performance remains robust across arbitrary SPD arrangements. This property is particularly advantageous for miniaturized systems, such as endoscopic probes or wearable sensors, where physical constraints preclude traditional imaging setups. Furthermore, the method’s compatibility with single-pixel detectors extends its applicability to spectral regimes (e.g., infrared, ultraviolet) where high-resolution cameras are costly or unavailable. While SMPD excels in pattern recognition tasks, its current framework faces limitations in high-entropy scenarios, such as reconstructing detailed images. These challenges stem from the inherent trade-off between sampling sparsity and information capacity. Meanwhile, SMPD currently prioritizes pattern recognition over full-field reconstruction, limiting its utility in high-entropy scenarios (e.g., detailed image recovery). The results in Fig. 2 b show that the number of SPDs (a measure of sampling density) has a more significant impact on recognition performance than the effective detection area; (which influences noise resistance). This indicates that while SMPD demonstrates the effectiveness of ultra-low sampling density for relative regular tasks, high sampling density remains the most straightforward solution for more complex challenges, such as image reconstruction and intricate object detection. However, integrating SMPD with emerging technologies—such as metasurface-based SPD arrays or compressed sensing algorithms—could mitigate these limitations, enabling real-time, high-fidelity imaging through scattering media. Beyond OAM recognition, our supplementary experiments (handwritten digit classification, axicon angle detection) demonstrate SMPD’s versatility in decoding diverse optical information. This generalizability positions SMPD as a universal platform for speckle-based sensing, with transformative potential for quantum communication 14 , low-power LIDAR 17 , and NLOS sensing 7 , 34 . Future work could explore hybrid frameworks combining SMPD with quantum-enhanced detectors 35 , adaptive optics 36 , or wavefront shaping 37 , further bridging the gap between theoretical information redundancy and practical, resource-efficient photonic systems. Conclusion In summary, we have demonstrated a speckle-driven framework, spatially multiplexed points detection (SMPD), that achieves efficient OAM recognition with unprecedented sampling efficiency. By leveraging the non-local correlations inherent to speckle patterns, SMPD extracts OAM information from sparse intensity measurements—using as few as 16 SPDs and a sampling density of 0.024%. This represents a 4096-fold improvement over conventional imaging-based methods, while maintaining > 99% accuracy. The method’s insensitivity to SPD spatial arrangements, combined with its compatibility with low-cost fast detectors, enables deployment in resource-constrained environments, from endoscopic imaging to high-speed optical communication. Experiments on OAM-multiplexed data transmission further validate SMPD’s practicality, achieving < 0.2% decoding error rates with 16 SPDs and robust performance under extreme under-sampling. The synergy between sparse sampling and machine learning not only addresses the ill-posed challenges of MMF transmission but also establishes a blueprint for scalable photonic technologies. Furthermore, by utilizing single detectors without spatial resolution, SMPD extends the range of its applications compared to conventional methods relying on high-resolution sensors. Looking ahead, SMPD’s modular design invites integration with advanced computational algorithms and compact detector arrays, promising breakthroughs in high-dimensional optical sensing, real-time quantum state tomography, and miniaturized medical diagnostics. By redefining the spatial sampling-information capacity trade-off, this work advances the frontier of optical technologies, enabling efficient light-matter interaction control in complex, scattering environments. Methods The experimental setup, illustrated in Fig. 5 , comprises three key stages: vortex beam generation, multimode fiber (MMF) transmission, and speckle detection with different masks. A vertically polarized femtosecond laser output (Origami-10XP, 1028 nm central wavelength, 400 fs pulse duration, 1 MHz repetition rate) passes through a half-wave plate (HWP) to adjust its polarization direction to 45°. A polarizing beam splitter (PBS) then splits the beam is divided into two orthogonally polarized paths. Each path is independently modulated by a spatial light modulator (SLM, X13138-03 and X13138-09, HAMAMATSU) to impose spiral phase profiles, generating vortex beams with tunable OAM. The modulated beams are recombined by the PBS and coupled into a 20-meter MMF (core diameter = 62.5 µm, NA = 0.275, M31L20, Thorlabs) through an objective (O1, 10×, NA = 0.25). At the MMF output, speckle patterns are collimated by a 20× objective (O2, NA = 0.4) and recorded by a CCD camera (Pike F421B, 7.4 µm pixel size, AVT). To simulate single-pixel detectors (SPDs), numerical masks with predefined apertures are applied to the CCD, enabling programmable sampling of sparse speckle regions. Each SPD accumulates intensity values within its designated aperture (e.g., 10×10 pixels), emulating a multiplexed SPD array. On the SLM, the OAM of spiral phases varies from 1 to 10.9 with an interval of 0.1, generating 100 phase variations. As the two OAMs alternate, 10,000 different intensity-value sequences of the speckle field are recorded. Upon imposing different masks on the speckle patterns, a simulated several SPDs strategy is realized. Finally, these intensity sequences of corresponding detectors are used as the input of the designed neural network. Figure 6 illustrates the evolution profiles of light intensity at one sampled SPD region in the speckle field. As shown in Fig. 6 a, the speckle field is imposed with a mask containing 6 points, resulting in the sampled light intensity distribution that is recorded in experiment. Figure 6 b displays the corresponding intensity values at the rightmost position of the speckle light field in (a) when two orthogonally polarized vortex beams of different OAM1 and OAM2 transmit through the MMF, which exhibits fluctuations with changes in OAMs (OAM1 = 1-10.9, and OAM2 = 1-10.9). In Fig. 6 c, the evolution of intensity values at individual SPDs reveal periodic fluctuations correlated with OAM changes. For example, intensity profiles at one SPD region as OAM modes vary ( ℓ = 1–10.9). While single-point measurements alone could not resolve modes (due to the ill-posed nature of MMF transmission), aggregating data from multiple SPDs enabled ROAM-ANN to disentangle complex correlations. This synergy between sparse sampling and machine learning bypasses the need for high-resolution imaging, addressing a critical bottleneck in scattering media applications. Declarations Disclosures The authors declare no conflicts of interest. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 62375092, 62005086, 12150410318, 81930048), Key Project of Natural Science Foundation of Fujian Province (No. 2023J02020), Hong Kong Research Grant Council (15217721, 15125724), Guangdong Science and Technology Commission (2019BT02X105), Shenzhen Science and Technology Innovation Commission (JCYJ20220818100202005), and Hong Kong Polytechnic University (P0038180, P0039517, P0043485, P0045762, P0049101). Code and Data Availability The code and data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. References Shen, Y. et al. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light: Science & Applications 8 , 90 (2019). Yang, Y., Ren, Y.-X., Chen, M., Arita, Y. & Rosales-Guzmán, C. Optical trapping with structured light: a review. 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University","correspondingAuthor":false,"prefix":"","firstName":"Qi","middleName":"","lastName":"Zhao","suffix":""},{"id":449934325,"identity":"fe3bbb67-2136-4d07-81b8-dbf5535264ac","order_by":5,"name":"Vinu R.V","email":"","orcid":"https://orcid.org/0000-0001-8361-5200","institution":"Huaqiao University","correspondingAuthor":false,"prefix":"","firstName":"Vinu","middleName":"","lastName":"R.V","suffix":""},{"id":449934326,"identity":"86494a6b-c06a-4eb5-a4df-af41925fff44","order_by":6,"name":"Huanhao Li","email":"","orcid":"","institution":"The Hong Kong Polytechnic University","correspondingAuthor":false,"prefix":"","firstName":"Huanhao","middleName":"","lastName":"Li","suffix":""},{"id":449934327,"identity":"6c6b7e11-d1a4-4f8d-81c4-e67709ec3613","order_by":7,"name":"Zhipeng Yu","email":"","orcid":"https://orcid.org/0000-0003-4668-0649","institution":"Hong Kong Polytechnic University","correspondingAuthor":false,"prefix":"","firstName":"Zhipeng","middleName":"","lastName":"Yu","suffix":""},{"id":449934328,"identity":"264ad1b7-01fc-4221-a6fc-40e20d9ef671","order_by":8,"name":"Jixiong Pu","email":"","orcid":"","institution":"Huaqiao University","correspondingAuthor":false,"prefix":"","firstName":"Jixiong","middleName":"","lastName":"Pu","suffix":""},{"id":449934329,"identity":"b3c211e2-55d5-4a80-bae1-a712fa5b2f24","order_by":9,"name":"Ziyang Chen","email":"","orcid":"","institution":"College of Information Science and Engineering, Fujian Key Laboratory of Light Propagation and Transformation","correspondingAuthor":false,"prefix":"","firstName":"Ziyang","middleName":"","lastName":"Chen","suffix":""},{"id":449934330,"identity":"4de73c5f-9795-4271-a209-eee5e1f73f4e","order_by":10,"name":"Xiaocong Yuan","email":"","orcid":"https://orcid.org/0000-0003-2605-9003","institution":"Shenzhen University","correspondingAuthor":false,"prefix":"","firstName":"Xiaocong","middleName":"","lastName":"Yuan","suffix":""}],"badges":[],"createdAt":"2025-04-02 16:30:52","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6363105/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6363105/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41467-025-66074-3","type":"published","date":"2025-12-12T05:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":81966304,"identity":"a0189051-e3a0-4fef-9444-5b786205d0e2","added_by":"auto","created_at":"2025-05-05 11:35:39","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":257598,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSMPD Framework for Single-Shot OAM Recognition via Ultra-Sparse Speckle Sampling.\u003c/strong\u003e\u003cbr\u003e\nTwo orthogonally polarized vortex beams propagate through a multimode fiber (MMF), generating a scrambled speckle pattern at the distal end. The intensities values are sampled by several strategically distributed single-pixel detectors in the speckle field and input into a designed neural network responsible for recognizing the OAMs of the input vortex beams.\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-6363105/v1/37117fccd9efcdaf6bf52421.png"},{"id":81966302,"identity":"6c519bdb-e656-49dc-9e78-b910297f4b15","added_by":"auto","created_at":"2025-05-05 11:35:39","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":286788,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eRecognition accuracies of SPD configurations on test set. \u003c/strong\u003e(a) Impact of SPD size and count: Recognition accuracy for OAM1 and OAM2 modes (\u003cem\u003eℓ\u003c/em\u003e = 1–10.9) versus individual SPD area (1×1 px to 30×30 px) and number of SPDs (6–16). (b) Sampling efficiency analysis: Accuracy versus total effective detection area.\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-6363105/v1/35723f6299820650543c2884.png"},{"id":81967782,"identity":"b63ff15c-b549-4aee-9e70-0cfe55201b24","added_by":"auto","created_at":"2025-05-05 11:43:39","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":618312,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eHigh-capacity OAM-multiplexed optical communication via SMPD.\u003c/strong\u003e (a) Encoding process: Two spiral phase images (100 × 100 pixels, 8-bit depth) are encoded into OAM bases (\u003cem\u003eℓ\u003c/em\u003e = 2.1–2.8) and multiplexed via orthogonally polarization. (b) Transmission and speckle generation\u003cem\u003e:\u003c/em\u003e The multiplexed beams propagate through a 20-meter MMF, generating mode-scrambled speckle patterns at the output. (c) Decoding performance\u003cem\u003e:\u003c/em\u003e Reconstructed images and error rates for three SPD configurations, 16 SPDs (10 × 10 px each) achieve \u0026lt;0.2% decoding error, while 4 SPDs retain 74% accuracy.\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-6363105/v1/df07c511f1b1909f91245428.png"},{"id":81966307,"identity":"f7a94717-0ce9-497e-93a8-f4a8042c7a1a","added_by":"auto","created_at":"2025-05-05 11:35:39","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":482602,"visible":true,"origin":"","legend":"\u003cp\u003eHandwritten digit recognition using SMPD with different sampling masks. (a) Simulated scattering transmission and sparse detection: Handwritten digits from the MNIST dataset (28 × 28 pixels) are numerically propagated through a scattering medium, generating speckle patterns (256 × 256 pixels). Programmable sampling masks (6 × 6, 5 × 5, or 4 × 4 SPD arrays) extract mean intensity values from 10 × 10 px regions, emulating sparse single-pixel detection. (b) Performance metrics: Recognition accuracy and confusion matrices for each SPD configuration. SMPD achieves 89.42%, 83.21%, and 71.78% accuracy with 36, 25, and 16 SPDs, respectively.\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-6363105/v1/d3f8a97ce86f755559976bb6.png"},{"id":81966305,"identity":"8088a2f5-1320-4b69-a06c-b296dfec68cf","added_by":"auto","created_at":"2025-05-05 11:35:39","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":599988,"visible":true,"origin":"","legend":"\u003cp\u003eExperimental setup. Two orthogonally polarized vortex beams are coupled into an MMF, generating speckle patterns at the distal end of the MMF, which are captured by a spatially masked CCD camera that functions as an array of single-pixel detectors. Abbreviations: BS, beam splitter; CCD, charge-coupled device; HWP, half-wave plate; MMF: multimode fiber; O1 and O2, microscope objective; PBS, polarizing beam splitter; SLM1 and SLM2, spatial light modulator.\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-6363105/v1/564b1b365c8ed7b9904717dc.png"},{"id":81968156,"identity":"113c7b78-05cd-4f42-a9c2-452ca302565a","added_by":"auto","created_at":"2025-05-05 11:51:39","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":283331,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eIntensity-Encoded OAM Signatures via Non-Local Speckle Correlations\u003c/strong\u003e. (a) Experimental speckle sampling: A speckle pattern generated by two orthogonally polarized vortex beams (OAM1 and OAM2) transmitted through a 20-meter MMF. A 6-SPD mask samples intensity values from spatially distributed regions (red circles), with one SPD highlighted for analysis. (b) Intensity fluctuations at a single SPD: Measured intensity variations at the selected SPD region (10 × 10 px) as OAM1 and OAM2 modes independently span \u003cem\u003eℓ\u003c/em\u003e = 1–10.9. (c) Evolution profiles: Periodic intensity oscillations correlated with incremental OAM changes (Δ\u003cem\u003eℓ\u003c/em\u003e = 0.1), demonstrating how non-local speckle correlations encode OAM information into sparse SPD measurements. Aggregating these distributed fluctuations enables ROAM-ANN to resolve OAM modes with \u0026gt;99% accuracy, bypassing the need for pixel-dense detection.\u003c/p\u003e","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-6363105/v1/57be4a17d55c27b38cdaf6df.png"},{"id":98118558,"identity":"12c086cf-a706-4330-bb25-6fd4d378256a","added_by":"auto","created_at":"2025-12-13 08:06:54","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3163608,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6363105/v1/28045da0-af75-42bf-ae6a-982a484c77a9.pdf"},{"id":81967783,"identity":"7473408f-5414-4fd8-ba1d-8202ce467e68","added_by":"auto","created_at":"2025-05-05 11:43:39","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":1645234,"visible":true,"origin":"","legend":"Supplementary Material for: \u0026#x201C;Speckle-Driven Single-Shot Orbital Angular Momentum Recognition with Ultra-Low Sampling Density\u0026#x201D;","description":"","filename":"SMPDSupplementary0328.docx","url":"https://assets-eu.researchsquare.com/files/rs-6363105/v1/509deacfb9dc1c2277e29ad8.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Speckle-Driven Single-Shot Orbital Angular Momentum Recognition with Ultra-Low Sampling Density","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe precise manipulation of light\u0026rsquo;s orbital angular momentum (OAM) has emerged as a cornerstone of modern optics\u003csup\u003e\u003cspan additionalcitationids=\"CR2 CR3\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e, unlocking an infinite-dimensional basis for applications ranging from high-capacity optical communication\u003csup\u003e\u003cspan additionalcitationids=\"CR6 CR7 CR8 CR9 CR10 CR11\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e and quantum information protocols\u003csup\u003e\u003cspan additionalcitationids=\"CR14\" citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/sup\u003e to multidimensional sensing\u003csup\u003e\u003cspan additionalcitationids=\"CR17\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e. Central to this paradigm is the vortex beams, whose helical phase structure \u0026ndash; characterized by an azimuthal phase dependence of exp(\u003cem\u003eiℓθ\u003c/em\u003e) \u0026ndash; enables robust encoding of information into discrete OAM modes. These modes form the backbone of spatial multiplexing strategies that transcend classical transmission limits\u003csup\u003e\u003cspan additionalcitationids=\"CR2\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u003c/sup\u003e. Yet, when propagating through complex media such as multimode fibers (MMFs), which are commonly used for light information transmission, these beams undergo severe distortions due to modal coupling and dispersion, transforming their deterministic phase profiles into disordered speckle fields\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e,\u003cspan additionalcitationids=\"CR20 CR21\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e. This wavefront scrambling erases the intrinsic OAM signatures, creating a critical bottleneck for reliable OAM retrieval at the fiber output, a limitation that stifles the deployment of OAM-based systems in real-world scenarios.\u003c/p\u003e \u003cp\u003eTo address this challenge, computational strategies such as transmission matrix (TM) inversion\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e, deep learning\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e,\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e,\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e, and Fourier covariance analysis\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e have been developed to decode OAM states from speckle-corrupted outputs. While these methods demonstrate success, they generally depend on high-resolution imaging to capture full-field speckle intensity maps, necessitating thousands of or more pixels and computationally intensive processing. Such reliance on dense spatial sampling imposes stringent trade-offs; it throttles system bandwidth, complicates scalability, and obstructs real-time operation\u003csup\u003e\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e,\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e. Consequently, applications demanding compact form factors, high-speed detection, or operation in photon-limited scenarios \u0026mdash; cornerstones of next-generation photonic technologies \u0026mdash; remain largely out of reach.\u003c/p\u003e \u003cp\u003eRecent advances reveal that the ostensibly disordered nature of speckle fields marks a profound redundancy in their information content\u003csup\u003e\u003cspan additionalcitationids=\"CR27\" citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/sup\u003e. Unlike direct imaging systems, which localize spatial information to discreate regions, multiple scattering in complex media redistributes input-field contributions across the entire speckle pattern (see \u003cem\u003eSupplementary note 1\u003c/em\u003e for more details). This non-local encoding enables information retrieval based on partial-field sampling for tasks such as imaging through scattering media\u003csup\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e, diffuser-based super-resolution imaging via sub-Nyquist sampling\u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e, and OAM recognition from arbitrary speckle subregions\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e. Yet, leveraging this redundancy for ultralow-sampling detection, particularly in OAM-based pattern classification, remains underexplored. A pivot challenge persists: What constitutes the minimal spatial sampling required to maintain accurate, scalable mode discrimination?\u003c/p\u003e \u003cp\u003eTo address this, we propose spatially multiplexed points detection (SMPD), a framework for single-shot pattern classification with unprecedented sampling efficiency. Instead of recording full-field speckles, SMPD spatially samples intensity fluctuations at a sparse array of spatially multiplexed single-pixel detectors (SPDs), each capturing globally encoded information from the scattered field. A custom-designed neural network deciphers these sub-Nyquist intensity sequences, achieving robust OAM recognition with a sampling density of 0.024%\u0026mdash;4096 times lower than conventional imaging-based methods\u0026mdash;while maintaining over 99% accuracy. Beyond OAM-mode retrieval, SMPD generalizes to classify mode-dependent features and distinguish complex optical fields, including handwritten digit recognition and axicon angle identification, demonstrating its versatility across diverse scenarios.\u003c/p\u003e \u003cp\u003eBy refining the spatial sampling-information capability trade-off, SMPD transcends the limitation of imaging-centric detection paradigms. This approach enables high-dimensional optical sensing and communication systems that are both compact and bandwidth-efficient, critical for real-time operation in photon-starved environments. Furthermore, its ultralow sampling requirements may unlock applications for non-line-of-sight (NLOS) sensing, low-power LIDAR, and quantum-enhanced sensing, where conventional pixel-dense sensors, constrained by their limited bandwidth and spectral ranges, prove impractical. SMPD thus establishes a new roadmap for scalable photonic technologies, bridging the gap between theoretical information redundancy and practical, resource-efficient detection.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003eThe proposed SMPD framework was rigorously evaluated through a series of experiments to validate its efficiency in OAM recognition and its applicability to optical communication systems. Below, we present the key findings that highlight the method\u0026rsquo;s performance, robustness, and scalability.\u003c/p\u003e\n\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n \u003ch2\u003eHigh-Accuracy OAM Recognition with Ultra-Low Sampling Density\u003c/h2\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e illustrates the concept of SMPD, where two orthogonally polarized vortex beams transmitted through an MMF generate a speckle pattern at the distal end of the fiber. Instead of conventional full-field imaging, SMPD employs spatially distributed SPDs to sample sparse intensity points. These measurements are processed by a customized neural network termed as Recognizing OAM Artificial Neural Network (ROAM-ANN), which secures two paths to extract the features of the inputs in different dimensions and eventually outputs the encoded OAMs from the pointwise intensity sequences.\u003c/p\u003e\n \u003cp\u003eTo quantify the impact of SPD parameters on recognition accuracy, we systematically varied the number and size of SPDs (Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e). A charge-coupled device (CCD) camera with various masks is used to create different SPDs of varying numbers and sizes. Here, SPD is defined in units of \u0026lsquo;px\u0026rsquo;, and 1 px corresponds to the area of a single element (7.4\u003csup\u003e2\u003c/sup\u003e \u0026micro;m\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e) in the CCD camera. For instance, an SPD area of 5\u003csup\u003e2\u003c/sup\u003e px indicates 5 by 5 neighboring CCD elements are combined to form one SPD, and a total of 9 points means that 9 such SPDs are imposed on the CCD via the mask. For each SPD, the light intensities detected by CCD\u0026rsquo;s light-sensitive elements are accumulated, and the recorded intensity sequence of the SPD array is used as the input of the network.\u003c/p\u003e\n \u003cp\u003eThe recognition results for the test sets, each containing 1,000 intensity sequences randomly selected from a dataset of 10,000 combinations of two orthogonally polarized vortex beams (both OAMs varying from 1 to 10.9 with an interval of 0.1) and excluded from the training process, are shown in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ea (more detailed results can be found in \u003cem\u003eSupplementary Note 2\u003c/em\u003e). As seen, 16 SPDs (each spanning 30 \u0026times; 30 px) achieved\u0026thinsp;\u0026gt;\u0026thinsp;99% accuracy for both OAM modes. Critically, the total sampling density of this configuration is approximately 0.024% (16 vs. 256 \u0026times; 256), corresponding to 4096 times lower than that of conventional imaging-based methods\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e that usually require 256 \u0026times; 256 or more pixel arrays. Even with SPDs reduced to 2 \u0026times; 2 px (retaining 16 detectors), accuracy remained above 93%. That said, a clear trade-off emerged: reducing SPD count or size degraded performance. For instance, with only 6 SPDs (1 \u0026times; 1 px), accuracy dropped below 50%, highlighting the necessity of more sampling points to capture speckle correlations, as well as a larger sensing area to mitigate the noise and pixel displacement. To better visualize this trend, Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003eb plots recognition accuracy as a function of the total SPD area across different measurement sets.\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003eSpatial Deployment Flexibility\u003c/h3\u003e\n\u003cp\u003eA key advantage of SMPD lies in its insensitivity to the spatial deployment of SPDs. As shown in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, three distinct spatial deployments of 9 SPDs (with a fixed total area of 10\u0026times; 10 px) yielded minimal accuracy fluctuations: 89.40-95.15% for OAM1 and 78.05\u0026ndash;80.55% for OAM2. This flexibility arises from the non-local encoding of information inherent to speckle patterns, where each SPD integrates contributions from the entire input field. Unlike traditional imaging systems requiring fixed source-detector alignment, SMPD\u0026rsquo;s adaptability enables flexible deployment in spatially restricted or geometrically constrained environments, such as endoscopic probe or miniaturized sensors.\u003c/p\u003e\n\u003cp\u003e\u003cimg 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\"\u003e\u003c/p\u003e\n\u003cp\u003eHigh-Capacity Optical Communication via SMPD\u003c/p\u003e\n\u003cp\u003eTo demonstrate real-world applicability, we implemented an OAM-multiplexed optical communication system (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e). Two spiral phase images (100 \u0026times; 100 pixels, 8-bit color depth) were encoded using OAM bases ranging from 2.1\u0026thinsp;~\u0026thinsp;2.8 (see \u003cem\u003eSupplementary Note 3\u003c/em\u003e for details). A digital micro-mirror device (DMD) was employed to effectively generate multiplexed vortex beams carrying the encoded data, which were then coupled into the MMF. As the beams propagated through the MMF, speckle patterns formed at the distal end. At the receiver, SMPD decoded speckle patterns using three configurations: 16, 9, or 4 SPDs. With 16 SPDs (10 \u0026times; 10 px each), decoding error rates remained\u0026thinsp;\u0026lt;\u0026thinsp;0.2% (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ec). Reducing SPD size to 1 \u0026times; 1 px slightly increased errors (due to localized noise and pixel displacement) but maintained robust performance. Even with only 4 SPDs (10 \u0026times; 10 px each), the system achieved 74% accuracy, underscoring SMPD\u0026rsquo;s resilience under extreme under-sampling. This experiment validates SMPD\u0026rsquo;s potential for high-speed, low-bandwidth optical communication, particularly in resource-limited settings.\u003c/p\u003e\n\u003cp\u003eGeneralizable Pattern Recognition via Ultra-Sparse Speckle Sampling\u003c/p\u003e\n\u003cp\u003eTo demonstrate SMPD\u0026rsquo;s versatility beyond OAM retrieval, we extended its application to handwritten digit recognition using the MNIST dataset\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/sup\u003e. Conventional speckle-based digit classification relies on full-field speckle imaging and computationally intensive spatial feature extraction. In contrast, as illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003ea, our method employs simulated scattering transmission of MNIST digits (28 \u0026times; 28 pixels), generating speckle patterns (256 \u0026times; 256 pixels) that are subsampled via programmable SPD arrays. Three configurations were tested: 6 \u0026times; 6, 5 \u0026times; 5, and 4 \u0026times; 4 SPD arrays, where each detector integrates intensity over a 10 \u0026times; 10 px region. The resulting intensity sequences (comprising 36, 25, or 16 values, respectively) were fed into a neural network trained on 48,000 intensity-label pairs and validated on 2,000 pairs.\u003c/p\u003e\n\u003cp\u003eRemarkably, SMPD achieved 89.42%, 83.21%, and 71.78% recognition accuracy on the test dataset (10,000 samples) for 6 \u0026times; 6, 5 \u0026times; 5, and 4 \u0026times; 4 SPD arrays, respectively (Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003eb). The confusion matrices reveal minimal misclassification trends, with errors predominantly occurring between morphologically similar digits (\u003cem\u003ee.g.\u003c/em\u003e, 3\u0026harr;5, 4\u0026harr;9). This performance underscores SMPD\u0026rsquo;s capacity to generalize across diverse optical patterns despite extreme subsampling (e.g., 16 SPDs for 4 \u0026times; 4 arrays vs. 28\u0026times; 28 pixels, 2.04% sampling density). The results validate that spatially sparse, non-locally encoded speckle correlations retain sufficient information for high-accuracy classification tasks\u0026mdash;even in photon-efficient regimes\u0026mdash;without resorting to pixel-dense sensors.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe success of SMPD in achieving high-accuracy OAM recognition with ultra-low sampling density stems from the intrinsic properties of speckle patterns formed in complex media. Unlike free-space propagation, model coupling and superposition processes in multimode fibers (MMFs) redistribute input wavefront information across the entire speckle field, creating spatially redundant correlations. That is, the information projected onto a single speckle grain on the output plane is the superposition of the entire incident light field\u003csup\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e,\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e,\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e,\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e\u003c/sup\u003e. Such non-local encoding enables partial-field sampling to retain sufficient information for OAM retrieval\u0026mdash;a principle validated by transmission matrix theory and prior work on scattering media\u003csup\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e,\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/sup\u003e. By exploiting this redundancy, SMPD circumvents the need for high-resolution imaging, which has long constrained the scalability of OAM-based technologies\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e,\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e,\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eA pivotal insight from our experiments is that spatial sampling density, rather than absolute detection area, governs recognition accuracy. While conventional methods\u003csup\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e,\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e,\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e,\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e require dense pixel arrays (e.g., over 256 \u0026times; 256 pixels) to resolve OAM-dependent features, SMPD achieves comparable accuracy with just 16 SPDs (0.024% sampling density). This efficiency arises because each SPD integrates contributions from the entire input field, effectively compressing global correlations into sparse measurements. The neural network (ROAM-ANN) further amplifies this advantage by learning discriminative features from multiplexed intensity sequences. Consequently, with the proposed method, spatially resolved cameras are no longer necessary for OAM recognition. Instead, speckle intensity data from a minimal amount of SPDs is sufficient, opening venues for high-speed yet low-cost single-shot recognition solutions to a wide range of field applications. To demonstrate its versatility, we present an additional experiment in \u003cem\u003eSupplementary Notes 4\u003c/em\u003e, including axicon angle recognition, further highlighting the broad applicability of SMPD.\u003c/p\u003e \u003cp\u003eThe flexibility of SPD spatial deployment (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) underscores SMPD\u0026rsquo;s adaptability. Unlike ballistic imaging or ghost imaging, which demand rigid detector-source alignment, SMPD\u0026rsquo;s performance remains robust across arbitrary SPD arrangements. This property is particularly advantageous for miniaturized systems, such as endoscopic probes or wearable sensors, where physical constraints preclude traditional imaging setups. Furthermore, the method\u0026rsquo;s compatibility with single-pixel detectors extends its applicability to spectral regimes (e.g., infrared, ultraviolet) where high-resolution cameras are costly or unavailable.\u003c/p\u003e \u003cp\u003eWhile SMPD excels in pattern recognition tasks, its current framework faces limitations in high-entropy scenarios, such as reconstructing detailed images. These challenges stem from the inherent trade-off between sampling sparsity and information capacity. Meanwhile, SMPD currently prioritizes pattern recognition over full-field reconstruction, limiting its utility in high-entropy scenarios (e.g., detailed image recovery). The results in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb show that the number of SPDs (a measure of sampling density) has a more significant impact on recognition performance than the effective detection area; (which influences noise resistance). This indicates that while SMPD demonstrates the effectiveness of ultra-low sampling density for relative regular tasks, high sampling density remains the most straightforward solution for more complex challenges, such as image reconstruction and intricate object detection. However, integrating SMPD with emerging technologies\u0026mdash;such as metasurface-based SPD arrays or compressed sensing algorithms\u0026mdash;could mitigate these limitations, enabling real-time, high-fidelity imaging through scattering media.\u003c/p\u003e \u003cp\u003eBeyond OAM recognition, our supplementary experiments (handwritten digit classification, axicon angle detection) demonstrate SMPD\u0026rsquo;s versatility in decoding diverse optical information. This generalizability positions SMPD as a universal platform for speckle-based sensing, with transformative potential for quantum communication\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e, low-power LIDAR\u003csup\u003e\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/sup\u003e, and NLOS sensing \u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e,\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e\u003c/sup\u003e. Future work could explore hybrid frameworks combining SMPD with quantum-enhanced detectors\u003csup\u003e\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e, adaptive optics\u003csup\u003e\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e\u003c/sup\u003e, or wavefront shaping\u003csup\u003e\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/sup\u003e, further bridging the gap between theoretical information redundancy and practical, resource-efficient photonic systems.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eIn summary, we have demonstrated a speckle-driven framework, spatially multiplexed points detection (SMPD), that achieves efficient OAM recognition with unprecedented sampling efficiency. By leveraging the non-local correlations inherent to speckle patterns, SMPD extracts OAM information from sparse intensity measurements\u0026mdash;using as few as 16 SPDs and a sampling density of 0.024%. This represents a 4096-fold improvement over conventional imaging-based methods, while maintaining\u0026thinsp;\u0026gt;\u0026thinsp;99% accuracy. The method\u0026rsquo;s insensitivity to SPD spatial arrangements, combined with its compatibility with low-cost fast detectors, enables deployment in resource-constrained environments, from endoscopic imaging to high-speed optical communication. Experiments on OAM-multiplexed data transmission further validate SMPD\u0026rsquo;s practicality, achieving\u0026thinsp;\u0026lt;\u0026thinsp;0.2% decoding error rates with 16 SPDs and robust performance under extreme under-sampling. The synergy between sparse sampling and machine learning not only addresses the ill-posed challenges of MMF transmission but also establishes a blueprint for scalable photonic technologies.\u003c/p\u003e \u003cp\u003eFurthermore, by utilizing single detectors without spatial resolution, SMPD extends the range of its applications compared to conventional methods relying on high-resolution sensors. Looking ahead, SMPD\u0026rsquo;s modular design invites integration with advanced computational algorithms and compact detector arrays, promising breakthroughs in high-dimensional optical sensing, real-time quantum state tomography, and miniaturized medical diagnostics. By redefining the spatial sampling-information capacity trade-off, this work advances the frontier of optical technologies, enabling efficient light-matter interaction control in complex, scattering environments.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003eThe experimental setup, illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, comprises three key stages: vortex beam generation, multimode fiber (MMF) transmission, and speckle detection with different masks. A vertically polarized femtosecond laser output (Origami-10XP, 1028 nm central wavelength, 400 fs pulse duration, 1 MHz repetition rate) passes through a half-wave plate (HWP) to adjust its polarization direction to 45\u0026deg;. A polarizing beam splitter (PBS) then splits the beam is divided into two orthogonally polarized paths. Each path is independently modulated by a spatial light modulator (SLM, X13138-03 and X13138-09, HAMAMATSU) to impose spiral phase profiles, generating vortex beams with tunable OAM. The modulated beams are recombined by the PBS and coupled into a 20-meter MMF (core diameter\u0026thinsp;=\u0026thinsp;62.5 \u0026micro;m, NA\u0026thinsp;=\u0026thinsp;0.275, M31L20, Thorlabs) through an objective (O1, 10\u0026times;, NA\u0026thinsp;=\u0026thinsp;0.25).\u003c/p\u003e \u003cp\u003eAt the MMF output, speckle patterns are collimated by a 20\u0026times; objective (O2, NA\u0026thinsp;=\u0026thinsp;0.4) and recorded by a CCD camera (Pike F421B, 7.4 \u0026micro;m pixel size, AVT). To simulate single-pixel detectors (SPDs), numerical masks with predefined apertures are applied to the CCD, enabling programmable sampling of sparse speckle regions. Each SPD accumulates intensity values within its designated aperture (e.g., 10\u0026times;10 pixels), emulating a multiplexed SPD array.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eOn the SLM, the OAM of spiral phases varies from 1 to 10.9 with an interval of 0.1, generating 100 phase variations. As the two OAMs alternate, 10,000 different intensity-value sequences of the speckle field are recorded. Upon imposing different masks on the speckle patterns, a simulated several SPDs strategy is realized. Finally, these intensity sequences of corresponding detectors are used as the input of the designed neural network.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates the evolution profiles of light intensity at one sampled SPD region in the speckle field. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ea, the speckle field is imposed with a mask containing 6 points, resulting in the sampled light intensity distribution that is recorded in experiment. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eb displays the corresponding intensity values at the rightmost position of the speckle light field in (a) when two orthogonally polarized vortex beams of different OAM1 and OAM2 transmit through the MMF, which exhibits fluctuations with changes in OAMs (OAM1\u0026thinsp;=\u0026thinsp;1-10.9, and OAM2\u0026thinsp;=\u0026thinsp;1-10.9). In Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ec, the evolution of intensity values at individual SPDs reveal periodic fluctuations correlated with OAM changes. For example, intensity profiles at one SPD region as OAM modes vary (\u003cem\u003eℓ\u003c/em\u003e = 1\u0026ndash;10.9). While single-point measurements alone could not resolve modes (due to the ill-posed nature of MMF transmission), aggregating data from multiple SPDs enabled ROAM-ANN to disentangle complex correlations. This synergy between sparse sampling and machine learning bypasses the need for high-resolution imaging, addressing a critical bottleneck in scattering media applications.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003e\u003cstrong\u003eDisclosures\u003c/strong\u003e\u003c/h2\u003e\n\u003cp\u003eThe authors declare no conflicts of interest.\u003c/p\u003e\n\u003ch2\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/h2\u003e\n\u003cp\u003eThis work was supported by the National Natural Science Foundation of China (Grant No. 62375092,\u0026nbsp;62005086, 12150410318, 81930048), Key Project of Natural Science Foundation of Fujian Province (No. 2023J02020), Hong Kong Research Grant Council (15217721, 15125724), Guangdong Science and Technology Commission (2019BT02X105), Shenzhen Science and Technology Innovation Commission (JCYJ20220818100202005), and Hong Kong Polytechnic University (P0038180, P0039517, P0043485, P0045762, P0049101).\u003c/p\u003e\n\u003ch2\u003e\u003cstrong\u003eCode and Data Availability\u003c/strong\u003e\u003c/h2\u003e\n\u003cp\u003eThe code and data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.\u003c/p\u003e\n"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eShen, Y.\u003cem\u003e et al.\u003c/em\u003e Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. \u003cem\u003eLight: Science \u0026amp; Applications\u003c/em\u003e \u003cstrong\u003e8\u003c/strong\u003e, 90 (2019).\u003c/li\u003e\n\u003cli\u003eYang, Y., Ren, Y.-X., Chen, M., Arita, Y. \u0026amp; Rosales-Guzm\u0026aacute;n, C. 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Opportunities and Challenges in Quantum-Enhanced Optical Target Detection. \u003cem\u003eACS Photonics\u003c/em\u003e (2025).\u003c/li\u003e\n\u003cli\u003eRao, C.\u003cem\u003e et al.\u003c/em\u003e Astronomical adaptive optics: a review. \u003cem\u003ePhotoniX\u003c/em\u003e \u003cstrong\u003e5\u003c/strong\u003e, 16 (2024).\u003c/li\u003e\n\u003cli\u003eYu, Z.\u003cem\u003e et al.\u003c/em\u003e Wavefront shaping: a versatile tool to conquer multiple scattering in multidisciplinary fields. \u003cem\u003eThe Innovation\u003c/em\u003e \u003cstrong\u003e3\u003c/strong\u003e, 100292 (2022).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Single pixel detection, Multimode fiber, Speckle, Orbital angular momentum, Wavefront shaping","lastPublishedDoi":"10.21203/rs.3.rs-6363105/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6363105/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eOrbital angular momentum (OAM) recognition of vortex beams is critical for applications ranging from optical communications to quantum technologies. However, conventional approaches designed for free-space propagation struggle when vortex beams propagate within or through complex media, such as multimode fibers (MMF), and often rely on high-resolution imaging sensors with tens of thousands of pixels to record dense intensity profiles. Here, we introduce a speckle-driven OAM recognition technique that exploits the intrinsic correlation between speckle patterns and OAM states, circumventing the limitations of scattering media while drastically reducing sampling requirements. Our method, termed spatially multiplexed points detection (SMPD), extracts intensity information from spatially distributed points in a multiplexed speckle plane. Remarkably, it achieves\u0026thinsp;\u0026gt;\u0026thinsp;99% retrieval accuracy for OAMs recognition using just 16 sampling points, corresponding to a sampling density of 0.024%\u0026mdash;4096 times lower than conventional imaging-based approaches. Furthermore, high-capacity OAM-multiplexed communication decoding with an error rate of \u0026lt;\u0026thinsp;0.2% and handwritten digit recognition with an accuracy of 89% are implemented to verify the versatility of SMPD. This work transcends the trade-off between sampling density and accuracy, establishing a scalable platform for resource-efficient photonic applications like quantum communication and endoscopic sensing.\u003c/p\u003e","manuscriptTitle":"Speckle-Driven Single-Shot Orbital Angular Momentum Recognition with Ultra-Low Sampling Density","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-05-05 11:35:34","doi":"10.21203/rs.3.rs-6363105/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-communications","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"NCOMMS","sideBox":"Learn more about [Nature Communications](http://www.nature.com/ncomms/)","snPcode":"","submissionUrl":"https://mts-ncomms.nature.com/","title":"Nature Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Communications","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"651e0e31-2683-474e-970c-bd1897ced739","owner":[],"postedDate":"May 5th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":47878322,"name":"Physical sciences/Optics and photonics/Optical techniques/Imaging and sensing"},{"id":47878323,"name":"Physical sciences/Optics and photonics/Applied optics/Fibre optics and optical communications"},{"id":47878324,"name":"Physical sciences/Mathematics and computing/Information technology"}],"tags":[],"updatedAt":"2025-12-13T08:06:48+00:00","versionOfRecord":{"articleIdentity":"rs-6363105","link":"https://doi.org/10.1038/s41467-025-66074-3","journal":{"identity":"nature-communications","isVorOnly":false,"title":"Nature Communications"},"publishedOn":"2025-12-12 05:00:00","publishedOnDateReadable":"December 12th, 2025"},"versionCreatedAt":"2025-05-05 11:35:34","video":"","vorDoi":"10.1038/s41467-025-66074-3","vorDoiUrl":"https://doi.org/10.1038/s41467-025-66074-3","workflowStages":[]},"version":"v1","identity":"rs-6363105","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6363105","identity":"rs-6363105","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}