Speckle-driven single-shot orbital angular momentum recognition with ultra-low sampling density

Domain: physics
Published: 2025-12-12
Editorial outcome: minor_revision

Abstract
Orbital 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 light passes through scattering media, such as multimode fibers (MMF), and often rely on high-resolution sensors with tens of thousands of pixels to record detailed intensity profiles. Here, by harnessing scattering media as intrinsic encoders rather than detrimental factors, we introduce a speckle-driven OAM recognition technique termed spatially multiplexed points detection (SMPD). This method extracts intensity information from a few spatially distributed points in a speckle plane, where object feature is naturally multiplexed, thereby transforming scattering from a detrimental effect into an efficient encoding mechanism while drastically reducing sampling requirements. Remarkably, it achieves over 99% retrieval accuracy for OAMs recognition using just 16 sampling points, corresponding to a sampling density of 0.024% compared with conventional imaging-based approaches. Furthermore, spatiotemporally interleaved vortex beams decoding, high-capacity OAM-multiplexed communication, MNIST, and Fashion-MNIST classification are implemented to verify the versatility of SMPD. This work establishes a scalable strategy for efficient optical information processing and fiber-based sensing in complex environments.

Introduction
The precise manipulation of light’s orbital angular momentum (OAM) 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 limits1–3.

To decode OAM modes, approaches such as interferometry19–21, vortex diffraction grating22–24, and machine learning25,26 have proven effective in free-space environments. While demonstrating success, they generally depend on high-resolution imaging to capture well-defined full-field light structures, necessitating thousands of pixels or more and computationally intensive processing. Such reliance on dense spatial sampling imposes stringent trade-offs; it throttles system bandwidth, complicates scalability, and obstructs real-time operation27,28. 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 content29–31. Unlike direct imaging systems, which localize spatial information to discrete 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 media32, diffuser-based super-resolution imaging via sub-Nyquist sampling33, and OAM recognition5,9,34–36 from arbitrary speckle subregions.

Yet, leveraging this redundancy for ultra-low-sampling detection remains underexplored. Although indeed utilizing a portion of the speckle field, current implementations still often rely on capturing a high-resolution intensity profile pattern (comprising thousands to millions of data points). 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 in diversity OAM recognition scenarios. Beyond OAM-mode retrieval, SMPD generalizes to classify mode-dependent features and distinguish complex optical fields, including handwritten digit recognition, grayscale images of clothing classification, 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 pathway toward 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 various complex scenarios of OAM recognition and its applicability to pattern classification. Below, we present the key findings that highlight the method’s performance, robustness, and scalability.

High-accuracy OAMs recognition with ultra-low sampling density
Figure 1 illustrates the concept of SMPD, where two orthogonally polarized vortex beams transmitted through a multimode fiber (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 was 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 (7.42 μm2) in the CCD camera. For instance, an SPD area of 52 px indicates 5 by 5 neighboring CCD elements are combined to form one SPD, and a total of 9 points mean that 9 such SPDs are imposed on the detecting plane 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 1000 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. 2a (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 methods9 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 between SPD count and SPD size: reducing either the number of sampling points (which is critical for capturing multiplexed information and speckle correlations) or the sensing area (which helps resist perturbations and avoids sampling low-intensity or uninformative zones, as further discussed in Supplementary Note 1.4) led to degraded performance. For instance, with only 6 SPDs (1 × 1 px), accuracy dropped below 50%, highlighting the necessity of using sufficient sampling points and an adequately large sensing area to ensure robust performance, mitigate noise, and reduce sensitivity to pixel displacement. To better visualize this trend, Fig. 2b plots recognition accuracy as a function of the total SPD area across different measurement sets.

In addition to recognizing orthogonally polarized vortex beams, we also conducted simulations involving pairs of linearly polarized vortex beams with varying polarization angle differences (see details in Supplementary Note 6). The proposed method successfully distinguished overlapping vortex beams with polarization angle differences as small as 5°, achieving a recognition accuracy of approximately 90%. As the angular separation decreased to 3°, the recognition accuracy declined to around 80%, reflecting the increasing challenge of resolving nearly co-polarized beams. These results further demonstrate the sensitivity and robustness of our method in scenarios involving subtle polarization differences. It indicates that multiplexing OAM inputs across different polarization states in polarization space, combined with the proposed SMPD method, can largely reduce the amount of stored data while significantly improving the channel capacity of OAM transmission. This approach enables the potential for ultra-large capacity OAM transmission.

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.

While our experimental results indicate that more sparsely distributed sampling regions yield slightly higher reconstruction accuracy than densely clustered ones, despite having the same number of sampling points and total area, this advantage is not absolute. Specifically, under conditions of strong scattering, where the information content can be fully scrambled and uniformly distributed across the speckle field, the choice of sampling region configuration (dense or sparse) has minimal effect on recognition accuracy (see details in Supplementary Note 1.4).

However, in practical scenarios, such as with speckles generated by a small-core MMF, a strong scrambled condition is difficult to achieve. In these cases, the speckle patterns exhibit larger grain sizes, leading to high redundancy in proximal sampling. Sparse sampling, by contrast, collects information from more uncorrelated regions of the speckle field and thus significantly improves reconstruction accuracy.

Crucially, sparse sampling shows robust performance across different scattering regimes. It remains effective whether the speckle grains are large (indicative of insufficient scattering) or small (indicative of strong scattering), offering broad applicability and better practical performance under varying physical conditions.

Spatiotemporally interleaved OAMs decoding via SMPD
Figure 3 illustrates the conceptual framework of the experiment (see Supplementary Note 3 for details). Each vortex beam, carrying a randomly selected OAM value alongside its corresponding temporal phase masks, is transmitted through the MMF in chronological order. At the distal end of the MMF, a CCD camera captures a single speckle image representing the superposition of the eight individual speckle patterns. This process is synchronized using a function generator, which triggers the digital micromirror device (DMD) eight times and the camera once during each acquisition cycle.

To enable single-pixel detection, eight masks with varying point densities (ranging from 20 to 49 points) are applied to the speckle patterns, effectively serving as virtual SPDs arrays, with each SPD covering an area of 10 × 10 pixels. Notably, the densest mask (7 × 7 SPD array, 49 sampling points) corresponds to just 0.07% of the 256 × 256 speckle image. Using this configuration, 20,000 speckle-label pairs were collected, each containing eight OAM values randomly selected from the 2.1~2.8 range. These data were split into training, validation, and testing sets in an 8:1:1 ratio, and processed through a modified ROAM-ANN (see Supplementary Note 4).

Figure 4 presents recognition performance under various sampling conditions, reporting both collective and individual OAM detection accuracy, as well as test loss. Collective recognition accuracy refers to the success rate of correctly identifying all OAM values within a single multi-OAM input. In this context, an input consists of eight distinct OAM modes, and a prediction is considered correct only if all eight are accurately classified. Using 49 sampling points (i.e., a 7 × 7 SPD array), the collective recognition accuracy reaches 98.6%, indicating highly reliable multi-mode detection. Individual recognition accuracy, on the other hand, measures the accuracy of identifying each OAM value independently, regardless of the correctness of other values in the same input. Under the same 49-point sampling configuration, the individual recognition accuracy achieves 99.82%, with only 28 misclassifications out of 16,000 individual OAM values in the test set. Even under more constrained conditions, using only 20 sampling points, the method maintains a robust individual recognition accuracy of approximately 85%, demonstrating strong performance even with limited spatial information.

High-capacity optical communication via SMPD
To demonstrate real-world applicability, we implemented an OAM-multiplexed optical communication system (Fig. 5). Two spiral phase images (100 × 100 pixels, 8-bit color depth) were encoded using OAM bases ranging from 2.1~2.8 and multiplexed via the optimized Lee hologram37 (see Supplementary Note 5 for details). A 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. 5c). 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 and clothing classification using the MNIST dataset (Binary images)38 and Fashion-MNIST dataset (8-bit grayscale images)39.

Conventional speckle-based digit classification relies on high-resolution speckle imaging and computationally intensive spatial feature extraction. In contrast, as illustrated in Fig. 6a, SMPD employed simulated scattering transmission of MNIST digits or Fashion-MNIST (28 × 28 pixels), generating speckle patterns (256 × 256 pixels) that were subsampled via programmable SPD arrays. Three configurations were tested for MNIST and Fashion-MNIST: 7 × 7, 6 × 6, and 5 × 5 SPD arrays, where each detector integrated intensity over a 10 × 10 px region. This resulted in sparse sequences of 49, 36, or 25 intensity values per image. For both datasets, the 60,000 images were partitioned into training, validation, and test sets in an 8:1:1 ratio, and processed through a modified ROAM-ANN (see Supplementary Note 4).

Remarkably, SMPD achieved 92%, 89.45%, and 83.28% recognition accuracy on the MNIST test dataset (6000 samples) for 7 × 7, 6 × 6, and 5 × 5 SPD arrays, respectively (Fig. 6b). The confusion matrix (Fig. 6c) reveals minimal misclassification trends, with errors predominantly occurring between morphologically similar digits (e.g., 3 ↔ 5, 4 ↔ 9).

For the more visually complex Fashion-MNIST dataset, SMPD achieved test accuracies of approximately 84%, 82%, and 81% for each SPD configuration. As shown in the confusion matrix (Fig. 6c), misclassifications were concentrated among upper-body garments (such as shirts, T-shirts, pullovers, and coats), whose silhouettes exhibit high visual similarity; representative examples are highlighted in Fig. 6a.

These results confirm that spatially sparse, non-locally encoded speckle measurements encapsulate sufficient information for high-accuracy classification across diverse, non-orthogonal image sets, without resorting to pixel-dense sensors. By reducing sensor-pixel requirements to a low subsampling rate (e.g., 5 × 5 arrays vs. 28 × 28 pixels, 3.19% sampling density), SMPD offers a computationally efficient, hardware-lean pathway to deploy optical classifiers in scenarios where pixel-dense cameras are impractical or where speed, power, and footprint are at a premium.

Methods
Speckle generation of orthogonally polarized vortex beams
A polarized femtosecond laser passes through a half-wave plate to adjust its polarization to 45°. A polarizing beam splitter (PBS) divides the 45° polarized beam into two orthogonally polarized paths. Each path is independently modulated by a spatial light modulator (SLM) to generate vortex beams with different OAMs. Then, the vortex beams are recombined by the PBS and coupled into an MMF using a 10× objective. At the distal end of the fiber, speckle patterns are collected by a 20× objective and recorded by a CCD camera (7.4µm pixel size). (More details can be found in Supplementary Materials Note 2).

Speckle generation of Spatiotemporally interleaved OAMs
Eight vortex beams carrying OAM values (ℓ=2.1–2.8) are sequentially modulated by a DMD illuminated with a continuous-wave laser. Each vortex beam is generated by Lee hologram37 and temporally modulated by a distinct chronological mask, and subsequently coupled into an MMF through a 10× objective. The output speckle field is collected by a 20× objective lens and recorded by a CCD camera. The entire process is synchronized using a function generator, ensuring temporal alignment between DMD modulation and CCD exposure. During each speckle acquisition period, the CCD remains continuously exposed to integrate the intensity of the speckle field generated by the eight sequentially transmitted vortex beams.

Multiplexed OAM transmission and speckle generation
A continuous-wave laser is expanded and directed onto the DMD, which sequentially displays the CGHs corresponding to the encoded gray values. The modulated beam is relayed through the 4f system and coupled into an MMF using a collimator. The output field is collected by a 20× objective lens and recorded by a CCD camera. In the data acquisition process, the DMD sequentially loads CGHs corresponding to grayscale values ranging from 0 to 255, each repeated ten times to improve signal stability. Two spiral phase images of 100 × 100 pixels are encoded into 20,000 CGHs, forming the testing dataset. To further mitigate environmental noise, all CGHs associated with grayscale values 0–255 are re-captured in ten repetitions after image transmission. In total, 5120 intensity-label pairs constitute the training dataset, while 20,000 CGHs serve as the testing set for evaluating multiplexed-OAM transmission (More details can be found in Supplementary Materials Note 5).

Speckle transformation of MNIST and Fashion-MNIST datasets
To numerically simulate the optical speckle transformation of standard image datasets, we modeled a transmission process through a random scattering medium using a complex-valued transmission matrix. Each 28 × 28 pixels image from the MNIST38 or Fashion-MNIST39 datasets was first vectorized into a column vector and normalized to the range [0, 1]. A random complex-valued transmission matrix T (with row of 282 rows and column of 2562) was then generated to represent the scattering process. The resulting out field (256 × 256) was obtained by multiplying T with the input image vector, treating the normalized values as the input amplitudes. The speckle intensity pattern was calculated as the squared magnitude of this complex field. Finally, the speckle intensity was partitioned into a grid of localized regions (5 × 5, 6 × 6, or 7 × 7), and the summed intensity within each region was used as a representative feature vector.

Neural network architecture and training
A dual-path neural network (ROAM-ANN) is designed to extract both local and global correlations within the intensity sequences. One path employs multiple 1D convolutional layers to capture spatially associated features, while the other uses fully connected layers to extract distributed sequence characteristics. The features from both paths are concatenated and fed into output modules with softmax activation. The network was implemented in TensorFlow 2.6 and trained with the Adam optimizer using binary cross-entropy loss. All experiments were performed on an NVIDIA P620 (More details can be found in Supplementary Note 4).

Data availability
All relevant data are available from the corresponding author upon request.

Code availability
All relevant code is available from the corresponding author upon request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors present a method for detecting OAM states of light using speckle patterns produced when superpositions of OAM modes, marked with polarization states, are transmitted through a multimode fiber. The method exploits sampling of judiciously selected points in the output pattern. While the methods and techniques, as well as supporting results, seem promising, I don't see any significant advancements either than incrementing to their previous iteration of the work as reported in Chinese Optics Letters, titled “Recognizing the orbital angular momentum (OAM) of vortex beams from speckle patterns”, where even the example images in their Fig. 5 (above SLM1) also appears in the previous article – same image, different colourmap. Similarly, work around the same theme was published [ J. Opt. Soc. Am. A 39, 759-765 (2022)] by Venugopal Raskatla in 2022, however, they applied the same method to LG and HG modes. Hence, this does not feel like original work, but rather an incremental effort, and cannot support it for publication in Nature Communications.\nMoreover, crediting pixel density and sparsity for this performance can be misleading. By explicitly defining their sampling regions, but now in different areas, the authors are ultimately doing the same thing as using a small neighbourhood of pixels; strong scattering will ensure information redundancy across the speckle pattern. As long as the sampling satisfies the Nyquist criterion and the scattering is sufficiently strong, any selection of regions (in proximity or scattered) will contain the same information content.\nNonetheless, the fact that this method successfully classifies handwritten digits, whose optical encodings are inherently non-orthogonal, makes it especially compelling. Unlike OAM and polarization modes, these digit-shaped fields overlap in the transverse plane. Yet, the technique still achieves high accuracy, suggesting real promise for structured-light approaches to non-orthogonal image data. This has also been reported before.", "reviewer_2": "The captioned manuscript presents a novel method for orbital angular momentum (OAM) single-shot recognition and OAM multiplex decoding at an ultra-low sampling rate based on spatially multiplexed speckle correlations. The proposed technique demonstrates comparable recognition accuracy to traditional methods while significantly reducing data sampling requirements. This challenges the conventional notion that dense pixel sampling is essential for accurate recognition. The method's versatility across various applications further enhances its promising impact to the field.\nThe manuscript is well structured, and the results are solid and convincing to support the claims. That said, there are a few points that require further clarifications or justifications to avoid potential confusions before acceptance by the journal.\n1. In Table 1, the denser spatial allocation of SPDs on the left shows inferior recognition performance compared to the sparser spatial distributions. The results indicate that a more uniformly distributed SPD configuration leads to higher recognition accuracy. Is this due to increased correlation between neighboring speckles, reduced spatial diversity, or other factors? The author should explain the underlying reasons for this observation.\n2. Figure 2 indicates that increasing both the number and area of SPDs improves recognition accuracy. In practical applications, how should this trade-off be optimized or balanced?\n3. Does the size of the speckle grains (e.g., as determined by the scattering medium or system optics) influence the final recognition accuracy? Please quantify or discuss this impact.\n4. As environmental vibrations are inevitable in real-world settings, it is important to assess the impact of speckle pattern translations on recognition accuracy. How sensitive is the system to such perturbations, and what is its robustness threshold?\n5. The recognition accuracy for OAM2 is consistently lower than that for OAM1 by ~10%. What causes this difference? Is it related to the intrinsic properties of the OAM states, their spatial frequency content, or potential overlaps in speckle features?\n6. Under the current experimental conditions, what is the finest OAM resolution (i.e., minimal discernible OAM spacing) that can be reliably achieved? What factors fundamentally limit this resolution?\n7. When using 9 or 16 SPDs (with 1 px area), the recognition accuracy falls below 90%, yet the OAM multiplexed transmission accuracy exceeds 93%. The authors should explain this discrepancy.\n8. Given that multimode fibers (MMFs) are sensitive to environmental factors such as temperature and airflow, which reportedly limit data stability to ~30 minutes, the authors are encouraged to discuss potential mitigation strategies.\n9. Other minor suggestions:\n1) In “High-Accuracy OAM Recognition with Ultra-Low Sampling Density,” Paragraph 2, the authors state that “SPDs are imposed on the CCD.” This phrase is ambiguous and is suggested to be further improved.\n2) The first sentence of Paragraph 2 in the Discussion section appears disconnected from the content that follows. Paragraph 4 contains redundant expressions. Please revise to improve the conciseness and clarity in these two paragraphs.\n3) Figure 6 shows a clear trend between single-point speckle intensity and OAM order. Does this trend generalize to other types of input, such as handwritten digits or axicons shown in the manuscript?"}, "author_response": "We sincerely appreciate the reviewers’ insightful feedback and the opportunity to address concerns regarding our manuscript. The critiques have significantly strengthened our work, prompting new experiments and deeper analysis that unequivocally demonstrate the conceptual and technical advances of our scattering-enhanced OAM detection method. Below, we provide a point-by-point response showing how the revised manuscript now addresses all concerns and establishes a paradigm shift in OAM sensing by transforming scattering media into high-efficiency encoders, orders-of-magnitude improvements in data efficiency (0.024% sampling density), and generalizability to non-orthogonal, spatiotemporal, and complex image classification tasks.\nReviewer #1 (Remarks to the Author)\nComment 1:\nThe authors present a method for detecting OAM states of light using speckle patterns produced when superpositions of OAM modes, marked with polarization states, are transmitted through a multimode fiber. The method exploits sampling of judiciously selected points in the output pattern. While the methods and techniques, as well as supporting results, seem promising, I don't see any significant advancements either than incrementing to their previous iteration of the work as reported in Chinese Optics Letters, titled “Recognizing the orbital angular momentum (OAM) of vortex beams from speckle patterns”, where even the example images in their Fig. 5 (above SLM1) also appears in the previous article – same image, different colourmap. Similarly, work around the same theme was published [ J. Opt. Soc. Am. A 39, 759-765 (2022)] by Venugopal Raskatla in 2022, however, they applied the same method to LG and HG modes. Hence, this does not feel like original work, but rather an incremental effort, and cannot support it for publication in Nature Communications.\nResponse:\nWe acknowledge the reviewer’s concerns regarding the novelty of the work and its relationship to prior literature. In this point-by-point response, we have carefully clarified our advances (Sections 1.1 and 1.2) with additional experiment designs and analyses (Sections 1.3 and 1.4). We have also provided detailed justifications addressing the concern about the selection of speckle regions and their information content (Sections 2.1 and 2.2). We trust this revision could appropriately address the reviewer’s all comments. The corresponding clarification and additional experiment have been added to Pages 8-10 in the manuscript (titled ‘Spatiotemporally Interleaved OAMs Decoding via SMPD’), Pages 7-9, 17-21, and 30-31 in the supplementary Notes 1.3, 4, and 6. These contents are marked with red color for your attention.\n1.1 Novelty and originality\nWe thank the reviewer for underscoring the need to clarify our advances. The core novelty lies in actively exploiting scattering physics to enable ultra-efficient OAM detection—a fundamental departure from conventional free-space approaches. While prior works (including our own) used scattering as a constraint to overcome, we repurpose it as an encoding resource, achieving:\na) A new detection paradigm: Traditional methods (Fig. R1a) require full-field imaging, reference beams, or specialized apertures. In contrast, our method (Fig. R1b) leverages scattering to compress OAM information into sparse speckle features, enabling high-accuracy detection with 16 sampling points (96.8-99.2% accuracy; Table R1). Crucially, this is impossible in free space—scattering is essential for information concentration.\nb) Unprecedented data efficiency: Compared to Raskatla et al. [J. Opt. Soc. Am. A 39, 759, 2022)] (512×512 pixels), we reduce sampling volume by 10⁴× (to 4×4 pixels) while resolving fractional OAM states (0.1 interval vs. integer orders), non-orthogonal polarizations (5° separation; Section 1.3), and spatiotemporal superpositions (8 beams; Section 1.4).\nc) Cross-domain generalizability: Beyond OAM, we achieve 83.98% classification accuracy on Fashion-MNIST (Fig. R8), demonstrating applicability to complex, non-orthogonal data, which is unsolved by traditional mode-based approaches.\nConsequently, this approach offers a substantial reduction in data volume (enabling lightweight data acquisition), lower computational requirement (facilitating real-time and embedded implementations), and potential for ultra-fast optical communication and sensing applications.\nFig. R1 Comparison of OAM decoding methods in (a) free space and (b) through scattering media. Unlike traditional imaging-based approaches, the proposed SMPD method does not require full-field detection or an external reference beam. Its effectiveness relies on the scattering effect, which enables high information encoding. Using the same sampling mask (16 sampling points, 102 px sampling area), SMPD achieves a recognition accuracy of 99%—a level difficult to attain in free-space conditions.\nIt should be highlighted that while scattering media are typically regarded as obstacles that disrupt information transmission in conventional scenarios, they function as efficient encoders in our proposed speckle-driven strategy. By actively introducing scattering, we transform the detection task into interpreting spatially sparse features within speckle patterns, enabling ultra-efficient sampling that yields significant reductions in both data volume and computational complexity. Fig. R1 compares the recognition accuracy for decoding fractional OAM via the SMPD method with and without scattering media. We demonstrate that direct free-space transmission using SMPD yields less than 60% recognition accuracy regardless of sampling area (Table R1). In contrast, with scattering media, recovery from just 16 positions achieves over 96% accuracy for fractional-order OAM in both polarization directions. This confirms that actively introducing scattering media combined with SMPD provides distinct advantages in the field of OAM detection.\nTable R1 Recognition results using SMPD with/without scattering.\nIntensity pattern (example: OAM1=1.5, OAM2=1.5)\nSampling mask\nRecorded intensity profile by SMPD\nAccuracy for OAM1 (%) 56.2 58 99.2\nAccuracy for OAM2 (%) 55.9 56.5 96.8\nLast but not the least, the proposed method can also be used for achieving high-accuracy classification for handwritten digits (MNIST1), grayscale images of clothing (Fashion-MNIST2, see details in Response 3), and axicon angles, demonstrating its generalization capability to diverse target recognition tasks.\n1.2 Addressing overlap with prior works\nRegarding the reviewer’s concern about similarity with our previous work— “Recognizing the orbital angular momentum (OAM) of vortex beams from speckle patterns” [Science China Physics, Mechanics & Astronomy 65 (4), 244211], the explanation is as follows:\nFirst of all, essential contributions of these two studies are fundamentally distinct. The previous work addressed limitations imposed by modal dispersion in multimode fibers (MMFs) for vortex beam transmission and communication, which inherently compromises vortex beam utility in MMFs. Consequently, it designed a network to reconstruct the original OAM information, enabling vortex beam application within MMFs. Here, we propose a novel OAM detection framework representing significant advances over traditional free-space strategies; it introduces a scattering medium in detection, enabling OAM recognition with a substantially reduced dataset (0.024% of the original data volume). New experiments (spatiotemporal/non-orthogonal decoding) and cross-domain validation (image classification) substantiate this advance. This method greatly reduces both data requirements and computational complexity for OAM detection.\nDuring this study, we utilized the same experimental platform as the prior work to assess the method's applicability and parameter limits to ensure consistent comparisons, which, however, triggers the reviewer’s concern of this work’s innovation. We sincerely appreciate this insight and have made the following modifications to the manuscript in response:\n1. Result 1: We investigated OAM detection performance for inputs under different non-orthogonal polarization conditions. The proposed method can still achieve high-precision OAM detection and decoding (~90%) in their corresponding polarization directions, even with a minimal 5-degree angular separation. The reconstruction results for these non-orthogonal inputs demonstrate that effective OAM detection on each polarization component is attainable even for inputs exhibiting significant polarization overlap. This finding indicates that multiplexing OAM inputs across different polarization states in polarization space, combined with the proposed method, can largely reduce the amount of stored data while significantly improving the channel capacity of OAM transmission. This approach enables potentials for ultra-large capacity OAM transmission. See more details in 1.3 Additional Experiments Part 1.\n2. Result 2: We demonstrate that the proposed method can be utilized to achieve high-accuracy OAM detection under spatiotemporally overlapping conditions. The digital micromirror device (DMD) and detector are synchronized via a function generator. During a single detector integration period, the DMD successively generates eight combined beams, each containing a fractional OAM phase alongside their corresponding temporal phase masks. After propagating through the MMF, the detector collects the resultant speckle patterns with superimposed intensities within this single exposure time, yielding a spatiotemporally superimposed speckle pattern. Compared to sequential transmission and intensity detection, this spatiotemporally superimposed speckle introduces greater complexity in the temporal dimension, presenting increased decoding challenges. Nevertheless, the proposed method maintains high-accuracy OAM detection for each temporal sequence. This confirms the method's applicability across diverse OAM transmission scenarios, suggesting that OAM transmission exploiting spatiotemporal superposition could enable single-channel, large-capacity optical encryption transmission. See more details in 1.4 Additional Experiments Part 2.\n3. The proposed method is a versatile that is not only capable for OAM measurement but also can be generalized for image classification. To further showcase its effectiveness in recognizing more complex objects, we have extended its application beyond handwritten digits and axicon angle recognition. The revised manuscript also includes results on the more challenging gray-scale Fashion-MNIST classification task, achieving a recognition accuracy of 82.15%.\n4. We have also simulated the applicability of the method under different speckle grain sizes (SGS). The results show that nearly 99% of the dual-polarization fractional OAM detection can be achieved under different speckle particle sizes, proving excellent detection capability of the method under different scattering scenarios and parameters. The simulations varied the SGS from 2 to 16 pixels by adjusting the transmission matrix in the simulation program. Table R2 summarizes the recognition accuracy for two OAM modes (OAM1 and OAM2), with a fixed number of sampling points (16) and sampling area (10² pixels).\n5. We have updated the example images in Fig. 5 (now moved to the supplementary material as Fig. S4 due to space constraints introduced by additional experiments) to better highlight the differences from prior work.\nTable R2 Recognition accuracy for OAM1 and OAM2 at different SGS (Sampling points = 16, sampling area = 102 pixels)\nSGS 2 4 6 8 16\nOAM1 accuracy (%) 98.7 99.4 98.9 99.2 99.3\nOAM2 accuracy (%) 98.9 99.2 99.6 98.9 98.7\nCollectively, these enhancements enable our method to achieve high-accuracy recognition of orthogonal polarization fractional vortex beams, spatiotemporally superimposed fractional vortex beams, and 8-bit information encoding via fractional OAM multiplexing. These results rigorously validate the method’s applicability across diverse, complex OAM detection scenarios. We contend this work represents a significant technical advance beyond prior works—both in enabling new capabilities (e.g., spatiotemporal decoding at 98.6% accuracy) and expanding the operational scope of scattering-enhanced photonics.\nRegarding the concerned overlap with the other work mentioned by the reviewer, while Raskatla et al. [J. Opt. Soc. Am. A 39, 759-765 (2022)] demonstrated partial speckle utilization for mode classification, our approach achieves fundamental advances on the following two aspects:\n1. Revolutionary data efficiency: The prior method requires high-resolution speckle images (512×512 pixels), but our technique achieves comparable accuracy using 10⁴× less data—only 4×4 sampled points. This ultra-sparse sampling enables real-time processing in resource-constrained systems impossible with conventional approaches.\n2. Unprecedented detection complexity: Unlike the prior work limited to orthogonal integer-order modes, we resolve fractional OAM states at 0.1 intervals (extendable to 0.01 in simulations) despite inherent non-orthogonality challenges. This represents 250× higher OAM resolution per channel, classification of 2,000 fractional states vs. 8 integer states, and robust decoding of spatially overlapping speckle patterns with >99% accuracy.\nIn summary, our method delivers the following advantages over prior works:\na) Ultra-lightweight acquisition (0.024% data volume);\nb) Real-time embedded operation via reduced computation;\nc) Scalability for ultra-fast optical communications;\nd) Wavelength/modality independence demonstrated across diverse structured light systems.\n1.3 Additional Experiments Part 1: Non-orthogonal Polarization Decoding\nThis section is added to address novelty concerns and demonstrate method robustness.\nTo rigorously validate our method’s capability in non-ideal scenarios, we conducted new simulations decoding non-orthogonal vortex beam superpositions with polarization-dependent transmission matrices. Crucially, distinct matrices were pre-generated for X/Y polarization states to model realistic MMF behavior.\nIn the experiment, OAM1 was fixed at a polarization angle of 45°, while OAM2 varied in polarization angle from 0° to 90°, introducing an angular separation ranging from 0° to 45° between OAM1 and OAM2. For all cases, we employed a sparse sampling strategy using 16 spatially SPDs with a pixel density of 10×10 per SPD.\nOur results (Fig. R2) show that the proposed SMPD method can achieve unprecedented discrimination of polarization angle difference as small as 5° for overlapping vortex beams. Performance remains robust at 80% for 3° separation but declines below 50% for <2° where polarization crosstalk dominates. This demonstrates the first sparse-sampling technique capable of resolving orthogonal-polarized vortex beams as well as non-orthogonal structured-light fields with ≤ 5° polarization variations. This establishes new potential for high-density polarization multiplexing in scattering media.\nFig. R2 Recognition accuracy and test loss for OAM1 and OAM2. The polarization angle of OAM1 was fixed at 45°, while the angle of OAM2 varied from 0° to 90°.\nTable R3 Recognition of OAM1 and OAM2. The polarization angle of OAM1 was fixed at 45°.\nAngle of OAM2 (°) 0 15 30 35 40 42\nAccuracy for OAM1 (%) 98.9 99 97.1 96.9 89.7 81.3\nAccuracy for OAM2 (%) 98.8 98.6 96.5 95.7 89.9 79.7\nAngle of OAM2 (°) 43 44 45 46 47 48\nAccuracy for OAM1 (%) 51.1 24.3 22.8 26.4 56.8 80.5\nAccuracy for OAM2 (%) 49.5 24.5 22.7 24.4 56.6 81.7\nAngle of OAM2 (°) 50 55 60 75 90 50\nAccuracy for OAM1 (%) 90.7 95.3 96.3 98.7 98.8 90.7\nAccuracy for OAM2 (%) 91.1 96.9 97.6 98.9 99.1 91.1\n1.4 Additional Experiments Part 2: High-Capacity Spatiotemporal OAM Decoding\nTo further address the reviewers’ concerns regarding the originality of the proposed method, an additional decoding experiment using SMPD is conducted to resolve spatiotemporal superimposed vortex beams. In this experiment, SMPD is employed to simultaneously decode eight vortex beams whose OAM values are randomly distributed between 2.1 and 2.8. These beams are sequentially transmitted through an MMF, and their intensities are spatiotemporally superimposed, resulting in a composite speckle pattern that encodes all eight beams. We demonstrate the versatility of SMPD in spatiotemporal resolving by exploiting the redundancy of spatial speckle field.\nFigure R3 illustrates the conceptual framework of the experiment. Each vortex beam, carrying a randomly selected OAM value alongside their corresponding temporal phase masks, is transmitted through the MMF in chronological order. At the distal end of the MMF, a CCD camera captures a single speckle image representing the superposition of the eight individual speckle patterns. This process is synchronized using a function generator, which triggers the DMD eight times and the camera once during each acquisition cycle.\nFig. R3 Schematic of the encoding and decoding process for spatiotemporally interleaved vortex beams transmission. Each vortex beam with a different OAM is injected into the MMF sequentially. Eight vortex beams with different OAM values are sequentially transmitted through the MMF. The CCD camera records a single speckle image, representing the spatiotemporally superposition of all beams, synchronized with the DMD. A sampling mask is applied, and the sampled intensities are input into ROAM-ANN to decode the OAM information.\nTo enable single-pixel detection, a sampling mask is applied to the full-resolution speckle image, effectively serving as a virtual array of single-pixel detectors (SPDs). Among the various sampling configurations tested, the densest mask corresponds to a 7×7 SPD array, totaling 49 sampling points. Notably, this represents an effective sampling rate of just 0.07% compared to the full 256×256-pixel speckle image.\nThe sampled intensity sequences are normalized and fed into a modified ROAM-ANN. In this architecture, the original output module is replaced with convolutional layers, culminating in a final convolutional layer with a sigmoid activation function. The output of this layer produces a probability distribution, where the position of the highest probability corresponds to the predicted OAM value.\nThe sampling mask design is illustrated in Fig. R4, where eight masks with varying point densities (ranging from 20 to 49 points) are employed. The only difference among the masks lies in the number of active sampling points.\nFig. R4 Sampling masks used for decoding the spatiotemporally interleaved vortex beams. (a~h) show sampling profiles of mask with different numbers of sampling points.\nFigure R5 illustrates the recognition performance of the proposed method in terms of collective and individual OAM detection accuracy, along with the corresponding test loss under various sampling configurations. Collective recognition accuracy refers to the success rate of correctly identifying all OAM values within a single multi-OAM input. In this context, an input consists of eight distinct OAM modes, and a prediction is considered correct only if all eight are accurately classified. Using 49 sampling points (i.e., a 7×7 SPD array), the collective recognition accuracy reaches 98.6%, indicating highly reliable multi-mode detection. Individual recognition accuracy, on the other hand, measures the accuracy of identifying each OAM value independently, regardless of the correctness of other values in the same input. Under the same 49-point sampling configuration, the individual recognition accuracy achieves 99.82%, with only 28 misclassifications out of 16,000 individual OAM values in the test set. Even under more constrained conditions—using only 20 sampling points—the method maintains a robust individual recognition accuracy of approximately 85%, demonstrating strong performance even with limited spatial information.\nFig. R5 Recognition accuracy and test loss vs. number of sampling points. Collective accuracy: the proportion of test samples in which all OAM modes are correctly identified within a single multi-mode input; Individual accuracy: the accuracy of identifying each OAM mode independently, regardless of the correctness of other modes in the same sample.\nThe confusion matrix in Fig. R6 shows that most misclassifications occur between neighboring OAM categories, and that accuracy steadily improves with more sampling points.\nFig. R6 Confusion matrices for individual vortex beam classification. (a~h) correspond to sampling points of 20, 24, 28, 32, 36, 40, 45, and 49, respectively.\nThe experimental setup is depicted in Fig. R7. A continuous-wave laser is collimated using reflectors R1 and R2, then directed to the DMD. The DMD sequentially loads eight OAM phase patterns that generated by LEE hologram3, each modulated by a corresponding chronological mask. The modulated beam passes through a spatial filter (4f system and aperture A) and is coupled into the MMF using objective lens O1. At the output, objective lens O2 collects the resultant speckle pattern, which is recorded by a CCD camera. The entire process is synchronized via a function generator.\nFig. R7 Experimental setup for collecting speckle patterns of spatiotemporally interleaved vortex beams. Abbreviations: R1 & R2, reflector; P, polarizer; DMD, digital micromirror device; L1&L2, lens; A, aperture; MMF, multimode fiber; O1&O2, objective; CCD, charge couple device.\nDetailed recognition accuracy that demonstrated in Fig. R5 is listed in Table R4. As seen, when the number of sampling points exceeds 36, the collective accuracy surpasses 90% and the individual accuracy exceeds 99%. Even with only 20 sampling points, the system maintains an individual classification performance of approximately 85%.\nTable R4 Recognition results for different sampling densities\nNumber of points 20 24 28 32 36 40 45 49\nCollective accuracy (%) 33.92 58.3 76 86.4 93.1 94.84 97.37 98.6\nIndividual accuracy (%) 84.98 92.643 96.45 98.09 99.1 99.32 99.67 99.82\nThis experiment establishes the originality and rigor of the proposed method by demonstrating its capacity to reliably decode spatiotemporally superimposed vortex beams through a single-shot, ultra-low-resolution measurement. The results clearly validate the feasibility of our approach for high-dimensional optical information encoding and decoding, offering a compact, cost-effective, and scalable alternative to traditional imaging-based methods.\nComment 2:\nMoreover, crediting pixel density and sparsity for this performance can be misleading. By explicitly defining their sampling regions, but now in different areas, the authors are ultimately doing the same thing as using a small neighbourhood of pixels; strong scattering will ensure information redundancy across the speckle pattern. As long as the sampling satisfies the Nyquist criterion and the scattering is sufficiently strong, any selection of regions (in proximity or scattered) will contain the same information content.\nResponse:\nWe sincerely thank the reviewer for the insightful comment regarding the interpretation of pixel density, sparsity, and the role of sampling in our approach. We fully agree that, under the condition of sufficiently strong scattering, as long as the sampling satisfies the Nyquist criterion, any selection of regions—whether proximal or scattered—should contain comparable information content. This is a valid and important observation. That said, practical systems face constraints associated with a) sensor density within the sampling regions, which impacts local noise robustness, and b) spatial sparsity of the sampling regions, which helps mitigate redundancy by capturing uncorrelated speckle grains.\nThe reviewer’s opinion holds well in the idealized case of strong scattering, where speckle grains are small and densely distributed. But in scenarios involving insufficient scattering, such as with multi-mode fibers (MMFs) of small cores, the output exhibits larger grain sizes and proximal sampling becomes highly redundant. In such cases, scattered sampling significantly improves reconstruction accuracy, as it captures more independent speckle information. More importantly, scattered sampling is robust across both regimes: it performs well regardless the size of the speckle grains, thus offering broader applicability and better performance across various physical conditions.\nWe have incorporated both the reviewer’s perspective and our extended discussion—covering the influence of sampling density, spatial sparsity, and speckle characteristics—into the revised manuscript to provide a more balanced and comprehensive analysis. Detailed mathematical analysis and experimental validation are also presented in the following sections for your information: 2.1 Pixel Density within Sampling Regions, 2.2 Sparsity of Sampling Regions across the Speckle Field, and 2.3 Experimental Validation.\n2.1 Pixel Density within Sampling Regions\nThere are two practical considerations that justify the use of higher pixel density in a given sampling region: noise reduction and sampling effectiveness in sparse speckle fields. In theory, a single pixel in each sampling region is effective. However, in practice, individual pixels are susceptible to noise from various sources, such as mechanical vibrations (causing pixel displacement) or sensor noise (e.g., dark current). Sampling over a denser pixel region helps to average out such noise, thereby increasing the reliability and stability of the recorded intensity.\nFig. S2 Different categories presents similar intensity distribution in the output field: (a) the speckle pattern of two encoded pixels (gray value of 102 and 189, respectively); (b) the difference between the two speckle patterns.\nAs shown in Fig. S2a in the supplementary, the speckle field distribution is inherently random and sparse, with large areas exhibiting zero or near-zero intensity due to destructive interference. This sparsity means that if a sampling region is too small, there is a significant chance it may fall in a low-intensity or uninformative zone. A higher pixel density (corresponding to a larger active area) increases the likelihood of capturing useful information embedded in the speckle pattern.\nThus, while a single pixel may theoretically suffice, a higher-density (larger area) sampling region provides practical robustness and ensures greater information reliability, especially under real-world conditions.\n2.2 Sparsity of Sampling Regions across the Speckle Field\nWhile it is true that, under strong scattering, any selection of regions (in proximity or scattered) will contain the same information content, the spatial correlation structure of the speckle pattern reveals practical advantages of sparse sampling. As shown in Table 1 of the manuscript, more sparsely distributed sampling regions yield higher recognition accuracy compared to densely clustered ones, despite the same number of sampling points and total area.\nTable 1 Recognition with different spatial deployments of SPDs (fixed number of sampling points and total area)\nSpatial allocations of 9 SPDs\nRecognition accuracy of OAM1 (%) 89.40 95.15 93.85\nRecognition accuracy of OAM2 (%) 78.05 80.55 79.70\nThis phenomenon can be explained through the spatial autocorrelation function of the speckle field. The output field from a MMF is a coherent superposition of guided modes:\nE(r)=∑ a φ (r)e−iβ m L, (1)\nm m\nm\nwhere φ ( r ) is the modal field distribution, β is the propagation constant of mode m m m, and L is the fiber length. The modal interference at the output plane generates a complex speckle pattern. The statistical correlation in the speckle field is characterized by:\nJ(r,r )=〈E*(r)E(r )〉,\n1 2 1 2 (2)\nC(r,r )=〈I(r)I(r )〉,\n1 2 1 2\nwhere J(r,r ) is the mutual coherence function, C(r,r )is the intensity correlation 1 2 1 2\nfunction, and I(r)=|E(r)|2.\nFor a fully developed speckle field, the Gaussian moment theorem allows us to express the intensity correlation as:\n2\n〈I(r)I(r+∆r)〉 =〈I〉2 + 〈E*(r)E(r+∆r)〉 , (3)\nand we define the normalized intensity correlation function as:\n2\n〈I(r)I(r+∆r)〉 〈E*(r)E(r+∆r)〉\nC (∆r)= −1= . (4)\nI 〈I〉2 〈I〉\nThis expression indicates that C (0)=1, and as the displacement ∆r increases, the field correlation decays, leading to a reduction in intensity correlation. Hence, neighboring points in a speckle pattern exhibit higher correlation, and the decay of C (∆r)reflects the speckle grain size (SGS).\nI\nFor statistically stationary fields, we define the spatial autocorrelation function,\nG(∆r)=〈E*(r)E(r+∆r)〉 . The intensity correlation is then related to the modulus 2\nG(∆r)\nsquare of this function C (∆r)= . According to the Van Cittert–Zernike I 〈I〉\ntheorem, the spatial correlation function is the Fourier transform of the source distribution limited by the fiber’s numerical aperture (NA). As a result, G(∆r)peaks at ∆r =0and decays rapidly with increasing ∆r . The width of G(∆r)is approximately the SGS, estimated by the diffraction limit:\nλ\nδr ≈ , (5)\n2⋅NA\nwhere λ is the wavelength, and NA is the numerical aperture of the fiber.\nPhysically, the spatial frequencies in the output field are limited by the fiber’s aperture and NA, which effectively acts as a low-pass spatial filter. This limits the high-frequency content and leads to relatively large speckle grains, hence stronger correlation between neighboring points.\nMathematically, the normalized correlation function C (∆r) is approximately 1 for I\nsmall displacements|∆r|<δr , but approaches 0 (correlation decays rapidly) when |∆r|>δr . This is also supported by modal interference: nearby output points are dominated by similar sets of modes with smooth spatial variation, resulting in similar interference patterns. This means that points located within one speckle grain are highly correlated, and their intensity values provide redundant information. Conversely, sampling points spaced farther apart are more likely to capture independent modal interference patterns, thereby increasing the overall information diversity available for classification.\nTherefore, while the total information content may be preserved under proximity regions, the practical effectiveness of sampling is significantly enhanced when sampling regions are placed beyond the correlation width of the speckle field. Given the approximately 65 µm aperture of the optical fiber used, the resulting SGS is relatively large, making correlation a significant factor in adjacent sampling. The closer the sampling points, the more correlated—and hence redundant—the information becomes. To prevent overlap within this correlation range, the most straightforward strategy is to enlarge the sampling regions and distribute them sparsely. This is why in our study we define sampling regions in separate areas rather than in close neighborhoods.\n2.3 Experimental Validation\nIn simulation, two vortex beams (polarization angles of 30° and 70°, respectively) carrying same/different OAM transmitted through an MMF, generating speckle patterns in the output field. The grain size of the speckle field was set to 6 pixels.\nTo evaluate the impact of different sampling strategies, we applied masks with varying sampling areas (pixel densities) and spatial sparsity to the speckle field. Each mask contained 16 spatially positioned detectors (SPDs). The configurations included both wide and narrow spatial distributions, as well as varying pixel densities (from 6×6 down to 1×1). The profiles and corresponding recognition accuracies are summarized in the following table (Table R5).\nAs shown in Table R5, the performance of a wider deployment of SPDs array is better than the narrow one when the sampling size is 2×2, which is smaller than SGS of the speckle field. When the sampling area of the mask is larger than SGS, the narrow and the wide deployments lead to similar but better performance than the strategy with a smaller sampling area. Notably, when the sampling area is reduced to 1×1 pixels, recognition accuracy drops significantly. As mentioned above, this is attributed to the higher probability of sampling in intensity-null or low-information regions, due to the limited spatial footprint of individual detectors.\nTable R5 Recognition accuracy with different spatial deployments for SPDs (different sampling areas and sampling sparsity)\nSpatial allocations of 9 SPDs\nPixel density 6×6 2×2 6×6 2×2 1×1\nSparsity wide wide narrow narrow wide\nRecognition accuracy of OAM1 (%) 96.9 93.7 96.4 64.5 38.8\nRecognition accuracy of OAM2 (%) 97.1 92.3 96.6 74.5 42.1\nWe hope these further clarify the motivation and technical rationale behind our sampling approach. Once again, we thank the reviewer for the valuable opportunity to elaborate on these points. These contents have been added to Pages 9-13 in the Supplementary Note 1.4 (titled ‘Spatial correlation in speckle field’) marked with red color for your information.\nComment 3:\nNonetheless, the fact that this method successfully classifies handwritten digits, whose optical encodings are inherently non-orthogonal, makes it especially compelling. Unlike OAM and polarization modes, these digit-shaped fields overlap in the transverse plane. Yet, the technique still achieves high accuracy, suggesting real promise for structured-light approaches to non-orthogonal image data. This has also been reported before.\nResponse:\nWe fully appreciate the reviewer’s insightful comment regarding the classification of optically encoded non-orthogonal data, such as handwritten digits. Indeed, unlike orthogonal mode sets like orbital angular momentum (OAM) and polarization, digit-shaped fields significantly overlap in the transverse plane, making them inherently more challenging to distinguish. The fact that our method achieves high classification accuracy under such conditions demonstrates its robustness and highlights its potential for structured-light-based recognition in complex, real-world scenarios.\nAs mentioned in the additional simulations (1.3 Additional Experiments Part 1), the proposed SMPD method can successfully distinguish between overlapping vortex beams with a polarization angle difference as small as 5°, achieving recognition accuracy of approximately 90%. These results confirm that the proposed method is not only capable of recognizing digit-shaped fields but is also effective in resolving non-orthogonal structured-light fields with fine polarization variations. This reinforces the general applicability and robustness of our technique in complex optical encoding scenarios.\nTo further demonstrate the versatility of our method in non-orthogonal domains, we also conducted a simulation using the Fashion-MNIST dataset to classify grayscale images of clothing. Three configurations were tested for MNIST and Fashion-MNIST: 7 × 7, 6 × 6, and 5 × 5 arrays, where each detector integrates intensity over a 10 × 10 px region. This results in sparse sequences of 49, 36, or 25 intensity values per image. The dataset of 60,000 samples was split into training, validation, and test sets in an 8:1:1 ratio.\nRemarkably, SMPD achieved 92%, 89.45%, and 83.28% recognition accuracy on the MNIST test dataset (6,000 samples) for 7 × 7, 6 × 6, and 5 × 5 SPD arrays, respectively (Fig. R8b). The confusion matrices (Fig. R8c) reveal minimal misclassification trends, with errors predominantly occurring between morphologically similar digits (e.g., 3↔5, 4↔9).\nOn the Fashion-MNIST test set, which contains 6,000 intensities sequences, SMPD achieved test accuracies of approximately 84%, 82%, and 81% for each SPD configuration. As shown in the confusion matrix (Fig. R8c), misclassifications primarily occurred among upper-body clothing items—such as shirts, T-shirts, pullovers, and coats—due to their high visual similarity, examples of which are illustrated in Fig. R8a.\nThese results confirm that spatially sparse, non-locally encoded speckle measurements encapsulate sufficient information for high-accuracy classification across diverse, non-orthogonal image sets, without resorting to pixel-dense sensors. By reducing sensor-pixel requirements to an extremely low subsampling rate (e.g., 5 × 5 arrays vs. 28× 28 pixels, 3.19% sampling density), SMPD offers a computationally efficient, hardware-lean pathway to deploy optical classifiers in scenarios where pixel-dense cameras are impractical or where speed, power and footprint are at a premium.\nFig. R8 Grayscale image and handwritten digits recognition using SMPD with different sampling masks. (a) Simulated scattering transmission and sparse detection: Images from the MNIST (handwritten digits) and Fashion-MNIST (clothing) datasets (28 × 28 pixels) are numerically propagated through a scattering medium (random transmission matrix), generating speckle patterns (256 × 256 pixels). Programmable sampling masks (7 × 7, 6 × 6, or 5 × 5 SPD arrays) extract mean intensity values from 10 × 10 px regions, emulating sparse single-pixel detection. (b) Performance metrics: Recognition accuracy and for each SPD configuration on the MNIST and Fashion-MNIST test datasets (6,000 samples each). For MNIST, SMPD achieved recognition accuracies of 92.00%, 89.45%, and 83.28% with 7 × 7, 6 × 6, and 5 × 5 arrays, respectively. For Fashion-MNIST, the corresponding accuracies are 83.98%, 81.93%, and 81.17%. (c) Confusion matrices: Classification results for MNIST and Fashion-MNIST datasets using the 7 × 7 SPD array (49 detectors).\nWhile similar attempts have been previously reported, those methods typically require large-scale speckle patterns or dense spatial sampling to achieve comparable results. In contrast, our SMPD approach accomplishes OAM recognition, data transmission, and image classification tasks using only a very limited number of sampling points. This not only simplifies the system design but also significantly enhances its practicality and efficiency for real-world applications.\nThis simulation has been included in Pages 12-14 the manuscript (Generalizable Pattern Recognition via Ultra-Sparse Speckle Sampling), marked in red color.\nReferences:\nReviewer #2 (remarks to the authors):\nThe captioned manuscript presents a novel method for orbital angular momentum (OAM) single-shot recognition and OAM multiplex decoding at an ultra-low sampling rate based on spatially multiplexed speckle correlations. The proposed technique demonstrates comparable recognition accuracy to traditional methods while significantly reducing data sampling requirements. This challenges the conventional notion that dense pixel sampling is essential for accurate recognition. The method's versatility across various applications further enhances its promising impact to the field.\nThe manuscript is well structured, and the results are solid and convincing to support the claims. That said, there are a few points that require further clarifications or justifications to avoid potential confusions before acceptance by the journal.\nResponse:\nWe sincerely thank the reviewer for the positive and encouraging feedback on the manuscript, and we are grateful for the recognition of the potential impact of this work to the field. In response to the reviewer’s suggestion for further clarifications to avoid possible confusion, we have carefully revised the manuscript to address all points that may require additional explanation or justification. These clarifications include:\na) A detailed discussion on the correlation between neighboring speckles;\nb) Clarification on the spatial deployment optimization strategy;\nc) Analysis of the influence of speckle grain size (SGS) on recognition accuracy;\nd) Evaluation of the robustness of the proposed method under vibration and rotational perturbations;\ne) Additional minor clarifications regarding experimental improvements and differences in experimental results.\nWe have incorporated these clarifications into the revised manuscript and believe they can further strengthen the clarity and completeness of the presentation. Once again, we thank the reviewer for the thoughtful feedback and constructive suggestions, which have helped us improve the quality of the work.\n1. In Table 1, the denser spatial allocation of SPDs on the left shows inferior recognition performance compared to the sparser spatial distributions. The results indicate that a more uniformly distributed SPD configuration leads to higher recognition accuracy. Is this due to increased correlation between neighboring speckles, reduced spatial diversity, or other factors? The author should explain the underlying reasons for this observation.\nResponse:\nThis phenomenon can be explained through the spatial autocorrelation function of the speckle field. The output field from a multimode fiber (MMF) is a coherent superposition of guided modes:\nE(r)=∑ a φ (r)e−iβ m L, (6)\nm m\nm\nwhere φ ( r ) is the modal field distribution, β is the propagation constant of mode m m m, and L is the fiber length. The modal interference at the output plane generates a complex speckle pattern. The statistical correlation in the speckle field is characterized by:\nJ(r,r )=〈E*(r)E(r )〉,\n1 2 1 2 (7)\nC(r,r )=〈I(r)I(r )〉,\n1 2 1 2\nwhere J(r,r ) is the mutual coherence function, C(r,r )is the intensity correlation 1 2 1 2\nfunction, and I(r)=|E(r)|2.\nFor a fully developed speckle field, the Gaussian moment theorem allows us to express the intensity correlation as:\n2\n〈I(r)I(r+∆r)〉 =〈I〉2 + 〈E*(r)E(r+∆r)〉 , (8)\nand we define the normalized intensity correlation function as::\n2\n〈I(r)I(r+∆r)〉 〈E*(r)E(r+∆r)〉\nC (∆r)= −1= (9)\nI 〈I〉2 〈I〉\nThis expression indicates that C (0)=1, and as the displacement ∆r increases, the field correlation decays, leading to a reduction in intensity correlation. Hence, neighboring points in a speckle pattern exhibit higher correlation, and the decay of C (∆r)reflects the SGS.\nI\nFor statistically stationary fields, we define the spatial autocorrelation function,\nG(∆r)=〈E*(r)E(r+∆r)〉 . The intensity correlation is then related to the modulus 2\nG(∆r)\nsquare of this function C (∆r)= . According to the Van Cittert–Zernike I 〈I〉\ntheorem, the spatial correlation function is the Fourier transform of the source distribution limited by the fiber’s numerical aperture (NA). As a result, G(∆r)peaks at ∆r =0and decays rapidly with increasing ∆r . The width of G(∆r)is approximately the SGS, estimated by the diffraction limit:\nλ\nδr ≈ , (10)\n2⋅NA\nwhere λ is the wavelength, and NA is the numerical aperture of the fiber.\nPhysically, the spatial frequencies in the output field are limited by the fiber’s aperture and NA, which effectively acts as a low-pass spatial filter. This limits the high-frequency content and leads to relatively large speckle grains, hence stronger correlation between neighboring points.\nMathematically, the normalized correlation function C (∆r) is approximately 1 for I\nsmall displacements|∆r|<δr , but approaches 0 (correlation decays rapidly) when |∆r|>δr . This is also supported by modal interference: nearby output points are dominated by similar sets of modes with smooth spatial variation, resulting in similar interference patterns.\nThis means that points located within one speckle grain are highly correlated, and their intensity values provide redundant information. Conversely, sampling points spaced farther apart are more likely to capture independent modal interference patterns, thereby increasing the overall information diversity available for classification.\nTherefore, the practical effectiveness of sampling is significantly enhanced when sampling regions are placed beyond the correlation width of the speckle field.\nThis discussion has included in the Pages 9-11, Supplementary Note 1.4 marked with red color.\n2. Figure 2 indicates that increasing both the number and area of SPDs improves recognition accuracy. In practical applications, how should this trade-off be optimized or balanced?\nResponse:\nTo better visualize the influence of the number and area of SPDs on recognition accuracy, we conducted a series of simulations. In these simulations, two vortex beams (with polarization angles of 30° and 70°, respectively) carrying either the same or different orbital angular momentum (OAM) values were transmitted through a MMF, generating speckle patterns at the output. The grain size of the resulting speckle field was set to 6 pixels. To evaluate the impact of different sampling strategies, we applied masks with varying sampling areas (i.e., pixel densities) and spatial sparsity to the speckle field. Each mask contained 16 SPDs, with configurations varying in both spatial sparsity (wide vs. narrow distribution) and pixel density (from 6×6 down to 1×1). The profiles and corresponding recognition accuracies are summarized in Table R5.\nTable R5 Recognition accuracy with different spatial deployments of SPDs (different sampling areas and sampling sparsity)\nSpatial allocations of 16 SPDs\nPixel density 6×6 2×2 6×6 2×2 1×1\nSparsity wide wide narrow narrow wide\nRecognition accuracy of OAM1 (%) 96.9 93.7 96.4 64.5 38.8\nRecognition accuracy of OAM2 (%) 97.1 92.3 96.6 74.5 42.1\nAs shown in Table R5, when the sampling area is smaller than the SGS (e.g., 2×2 pixels), a wider spatial deployment of SPDs yields noticeably better recognition performance than a narrow deployment does. However, when the sampling area exceeds the SGS (e.g., 6×6 pixels), both wide and narrow deployments achieve comparably high recognition accuracies, outperforming configurations with smaller sampling areas. Notably, when the sampling area is reduced to 1×1 pixels, recognition accuracy drops significantly. As mentioned above, this is attributed to the higher probability of sampling in intensity-null or low-information regions, due to the limited spatial footprint of individual detectors.\nIn practical applications, this trade-off can be solved by considering two perspectives: SGS of the speckle field and effective sampling points. As mentioned earlier, the sampling area of each detector should ideally be larger than half the SGS. This ensures that each SPD captures meaningful intensity information, rather than noise or null regions. Furthermore, the minimum spatial distance between SPDs should be larger than the SGS to reduce spatial correlation and capture diverse speckle features. On ther other hand, while a larger sampling area helps improve robustness and noise resistance, the total number of SPDs plays a more critical role in capturing a broader range of usable information. Therefore, increasing the number of well-distributed SPDs generally yields greater improvements in recognition accuracy than merely increasing their size.\nIn conclusion, for optimal system design, the detector size should be matched to the SGS (preferably slightly larger), and the SPDs should be distributed with sufficient spacing to avoid redundancy. Under constraints on the number of detectors, maximizing spatial coverage (i.e., using a wide distribution) can significantly improve recognition performance.\nThese results have been incorporated throughout the supplementary to enhance the clarity, completeness, and overall presentation of the work. Specifically, the detailed discussion can be found on Pages 12–13 in Supplementary Note 1.4, where the changes are clearly marked in red for the reviewer’s convenience.\n3. Does the size of the speckle grains (e.g., as determined by the scattering medium or system optics) influence the final recognition accuracy? Please quantify or discuss this impact.\nResponse:\nAs mentioned in Responses 1 and 2, SGS influences the correlation distance within the speckle field. Consequently, the spacing between each single-pixel detector (SPD) should be larger than the SGS to avoid signal redundancy and ensure effective sampling. To validate this concept, we conducted several simulations where speckle patterns were generated, and the corresponding intensity sequences were recorded. The simulations varied the SGS from 2 to 16 pixels by adjusting the transmission matrix in the simulation program. Table R6 summarizes the recognition accuracy for two OAM modes (OAM1 and OAM2), with a fixed number of sampling points (16) and sampling area (10² pixels).\nTable R6 Recognition accuracy for OAM1 and OAM2 at different SGS (sampling points = 16, sampling area = 102 pixels)\nSGS 2 4 6 8 16\nOAM1 accuracy (%) 98.7 99.4 98.9 99.2 99.3\nOAM2 accuracy (%) 98.9 99.2 99.6 98.9 98.7\nAs shown in Table R6, the recognition accuracies are consistently high (around 99%) across different SGS values. These minor fluctuations fall within the range of expected experimental variation. Therefore, we conclude that, for a given scattering system, as long as the sampling region is larger than approximately half the SGS, the impact on recognition accuracy is minimal.\nThese results have been added on Pages 13 in Supplementary Note 1.4, where the changes are clearly marked in red for the reviewer’s convenience.\n4. As environmental vibrations are inevitable in real-world settings, it is important to assess the impact of speckle pattern translations on recognition accuracy. How sensitive is the system to such perturbations, and what is its robustness threshold?\nResponse:\nTo assess the system’s robustness against environmental perturbations, we conducted a series of simulations focusing on two main types of disturbances: linear translations—referred to as \"speckle moving width\"—of 2 and 4 pixels, and rotational shifts randomly ranging from 0° to 3°. These simulations were performed under a variety of configurations, including different speckle grain sizes (SGS), sampling areas, and numbers of sampling points.\nTables R7–R11 summarize the recognition accuracy for two OAM states under different speckle moving widths (2 and 4 pixels), SGS (6 and 16), and sampling areas (102~302).\nWhen the speckle moving width was limited to 2 pixels, recognition accuracy experienced only a minor decline. For instance, with SGS set to 6 and a sampling area of 102 pixels, the accuracy for OAM1 and OAM2 was 75.1% and 70.0%, respectively. As the sampling area increased to 302 pixels, the accuracy improved significantly, reaching 88.5% and 88.6%. This trend was even more pronounced when using a larger SGS of 16, where the accuracy under the same conditions rose to 93.7% for OAM1 and 92.7% for OAM2. These findings highlight that both a larger sampling area and a larger SGS can effectively mitigate the impact of minor translations.\nHowever, when the speckle moving width increased to 4 pixels, the system’s performance exhibited more noticeable degradation, particularly with smaller sampling areas and lower SGS. For example, under SGS = 6 and a 102 pixel sampling area, recognition accuracy dropped to 56.3% for OAM1 and 47.3% for OAM2. Nevertheless, increasing the sampling area to 302 pixels raised the accuracy to over 70% for both states. The use of a larger SGS (16) significantly improved robustness in these cases as well. Even at a 4-pixel displacement, accuracy for OAM1 and OAM2 reached 79.1% and 83.4%, respectively, with a 302 pixel sampling area. Larger SGS (16) provides greater robustness to speckle translation, as larger speckle grains are less sensitive to small positional shifts.\nFurther analysis demonstrated that increasing the number of sampling points also enhances resilience to translation. In scenarios with a 4-pixel speckle moving width and SGS = 16, increasing the sampling points from 25 to 36 yielded noticeable improvements. For instance, with a 202 pixel sampling area, recognition accuracy improved from 89.0% to 92.5% for OAM1, and from 88.3% to 92.7% for OAM2. This suggests that denser sampling can effectively compensate for inaccurate measurement caused by larger displacements and reinforce the system’s robustness.\nTable R7 Recognition results for OAM1 and OAM2 (sampling points=16, speckle moving width= 2, SGS = 6)\nSampling area (px) 102 202 302\nOAM1 accuracy (%) 75.1 84.8 88.5\nOAM2 accuracy (%) 70 85.8 88.6\nTable R8 Recognition results for OAM1 and OAM2 (sampling points=16, speckle moving width= 2, SGS = 16)\nSampling area (px) 102 202 302\nOAM1 accuracy (%) 83.7 90.5 93.7\nOAM2 accuracy (%) 83 89.6 92.7\nTable R9 Recognition results for OAM1 and OAM2 (sampling points=16, speckle moving width= 4, SGS = 6)\nSampling area (px) 102 202 302\nOAM1 accuracy (%) 56.3 68.7 73.5\nOAM2 accuracy (%) 47.3 65 72.5\nTable R10 Recognition results for OAM1 and OAM2 (sampling points=16, speckle moving width= 4, SGS = 16)\nSampling area (px) 102 202 302\nOAM1 accuracy (%) 64.3 75.2 79.1\nOAM2 accuracy (%) 66 81.4 83.4\nTable R11 Recognition result for OAM1 and OAM2 (speckle moving width= 4, SGS = 16)\nSampling points 25 36\nSampling area (px) 102 202 102 202\nOAM1 accuracy (%) 78.3 89 86.6 92.5\nOAM2 accuracy (%) 75 88.3 87.1 92.7\nTable R12 Recognition results for OAM1 and OAM2 (sampling points=16, randomly rotated from 0~3°)\nSGS 6 16\nSampling area (px) 102 202 302 102 202 302\nOAM1 accuracy (%) 84.3 87.8 91.9 88.5 90.3 93.3\nOAM2 accuracy (%) 83.9 88.3 91.8 91.1 94.8 95.4\nRotational disturbances proved to have a less detrimental effect overall. As shown in table R12, when speckle patterns were randomly rotated within a range of 0° to 3°, the system maintained high recognition accuracy across all configurations. Even with SGS = 6 and a small sampling area of 102 pixels, accuracy remained above 84% for both OAM states. With SGS increased to 16 and larger sampling areas, accuracy consistently exceeded 93%, indicating that the system is highly resilient to minor angular variations.\nIn summary, the system can tolerate speckle displacements of up to 2 pixels with minimal loss in accuracy, especially when using a larger SGS and larger sampling areas. Although a 4-pixel displacement introduces more substantial performance degradation, this can be effectively mitigated by increasing the sampling area and the number of sampling points. Rotational perturbations up to 3° have a relatively minor impact on recognition accuracy, particularly when the system is configured with a high SGS and an adequate sampling region.\nThese results have been incorporated into the revised manuscript on Pages 33–35 in Supplementary Note 8, with the corresponding changes clearly marked in red for the reviewer’s convenience.\n5. The recognition accuracy for OAM2 is consistently lower than that for OAM1 by ~10%. What causes this difference? Is it related to the intrinsic properties of the OAM states, their spatial frequency content, or potential overlaps in speckle features?\nResponse:\nThaks a lot for noticing the details. The observed difference in recognition accuracy between OAM1 and OAM2 can be attributed to a combination of experimental imperfections and slight misalignments in the optical setup. These differences are not primarily due to the intrinsic properties of the OAM states themselves, but rather to practical factors affecting the generation and coupling of the vortex beams.\nFirstly, imperfections in the experimental components, such as the half-wave plate and polarizing beam splitter, contribute to discrepancies in the intensity and quality of the two vortex beams. Manufacturing tolerances and non-ideal optical coatings can result in uneven polarization modulation and splitting efficiency, which in turn causes variations in the output intensity of the two beams. Since the proposed recognition method relies heavily on the intensity distribution of the speckle patterns, such differences can lead to unbalanced datasets and reduced recognition performance for the lower-intensity beam—typically OAM2 in this case.\nSecondly, the optical alignment of the two vortex beams introduces another source of asymmetry. The system requires independent modulation of each beam using separate spatial light modulators (SLMs), followed by coupling into the same MMF. This process involves multiple reflective and refractive adjustments, making it difficult to achieve identical coupling efficiency for both beams. Even slight deviations in beam alignment or focus can lead to differences in the resulting speckle patterns, further contributing to the accuracy gap between OAM1 and OAM2.\nIn summary, the lower recognition accuracy for OAM2 is primarily due to experimental factors. That includes non-ideal optical components and imperfect alignment during beam modulation and coupling, but not fundamental differences in the spatial frequency content or intrinsic properties of the OAM states. Addressing these issues through improved component quality and more precise alignment procedures may help reduce the recognition gap between different OAM channels.\n6. Under the current experimental conditions, what is the finest OAM resolution (i.e., minimal discernible OAM spacing) that can be reliably achieved? What factors fundamentally limit this resolution?\nResponse:\nUnder the current experimental conditions, the finest reliably achievable OAM resolution is approximately 0.1. This means that the system can distinguish between two OAM states whose topological charges differ by as small as 0.1. In contrast, simulations suggest that the system has the potential to achieve an even finer resolution of down to 0.01 under ideal, noise-free conditions. In simulation tests where the topological charges of two vortex beams varied continuously from 1.00 to 1.99, the recognition accuracy reached 99.4% for OAM1 and 99.9% for OAM2, demonstrating the method’s theoretical capability to resolve extremely fine differences in OAM.\nHowever, achieving this level of resolution experimentally is significantly more challenging due to the presence of noise and imperfections in the optical system. Under current conditions, a resolution of 0.1 represents a reliable and achievable benchmark for OAM discrimination.\n7. When using 9 or 16 SPDs (with 1 px area), the recognition accuracy falls below 90%, yet the OAM multiplexed transmission accuracy exceeds 93%. The authors should explain this discrepancy.\nResponse:\nThis discrepancy arises primarily from two key differences between the experimental setups: light source stability and speckle grain size (SGS).\na) Light source stability\nExperiment 1 employed a sub-femtosecond pulsed laser (Origami-10XP, 1028 nm central wavelength, 400 fs pulse duration, 1 MHz repetition rate). Experiment 2 used a continuous-wave (CW) laser (EXLSR-532-300-CDRH, Spectra Physics, USA). The former one, the pulsed laser, is inherently more sensitive to environmental fluctuations such as temperature, vibration, and power supply instability. These factors lead to variations in the average power and output intensity of the speckle patterns. Since the recognition method relies on ultra-sparse intensity sampling, even minor fluctuations can introduce inconsistencies in the dataset, degrading recognition accuracy. In contrast, the CW laser used in Experiment 2 provides a more stable output, leading to more consistent speckle intensity profiles and improved recognition performance. As a result, Experiment 1 requires more SPDs or a larger sampling area to compensate for the instability of the light source.\nb) Speckle grain size (SGS)\nAs discussed in Responses 1–3, SGS significantly influences the recognition accuracy. In this study, Experiment 1 used a MMF with a 62.5 µm core diameter, resulting in larger speckle grains. Experiment 2 used a MMF with a 200 µm core diameter, producing smaller speckle grains. When the detector area is much smaller than the SGS, sampling becomes more susceptible to noise and less representative of the overall speckle pattern. In Experiment 1, the larger SGS reduces the effectiveness of sparse sampling with 1 px SPDs, leading to lower recognition accuracy. Furthermore, larger SGSs are generally more sensitive to positional shifts, reducing robustness in practical setups.\n8. Given that multimode fibers (MMFs) are sensitive to environmental factors such as temperature and airflow, which reportedly limit data stability to ~30 minutes, the authors are encouraged to discuss potential mitigation strategies.\nResponse:\nWe thank the reviewer for pointing out the practical limitations imposed by the environmental sensitivity of MMFs. Indeed, unmitigated fluctuations in temperature, airflow, and mechanical stress can alter the fiber’s transmission matrix and hence the speckle patterns, compromising data stability over extended periods. To address these challenges, we propose the following mitigation strategies:\na) Periodic reference calibration: By intermittently injecting a known reference beam (e.g., a Gaussian mode) and recording its speckle fingerprint, we can track slow drifts in the transmission matrix and apply a real-time correction to the classifier inputs.\nb) Sliding-Window training: Implementing an adaptive classifier that continuously updates its training set using only the most recent data can compensate for gradual changes, effectively “retraining on the fly” to maintain accuracy beyond 30 minutes.\nc) Data augmentation with perturbations: During classifier training, we can synthetically augment the speckle dataset by applying small translations, scaling, and rotations that mimic environmental shifts. This “stress-testing” of the network promotes inherent resilience to similar perturbations in practice.\nd) Ensemble classification: Combining the outputs of multiple independently calibrated classifiers (each trained on speckle patterns acquired under slightly different conditions) can average out residual errors resulting from transient fiber fluctuations.\nBy integrating the strategies outlined above, hopefully, it may significantly extend the stable operating window of our MMF-based system and ensure consistent recognition accuracy in practical, real-world environments.\n9. Other minor suggestions:\n1) In “High-Accuracy OAM Recognition with Ultra-Low Sampling Density,” Paragraph 2, the authors state that “SPDs are imposed on the CCD.” This phrase is ambiguous and is suggested to be further improved.\nResponse:\nThanks for the kind reminder, we have carefully checked it and revised this sentence as follows:\n“… a total of 9 points means that 9 such SPDs are imposed on the detecting plane …”\nThe modification can be found in Page 5 of the manuscript, marked with red color for your attention.\n2) The first sentence of Paragraph 2 in the Discussion section appears disconnected from the content that follows. Paragraph 4 contains redundant expressions. Please revise to improve the conciseness and clarity in these two paragraphs.\nResponse:\nThank you for the insightful comment. In response, we have carefully reorganized the Discussion section to improve the logical flow and coherence. Specifically, we revised the first sentence of Paragraph 2 to better align with the content that follows, and we eliminated redundancies in Paragraph 4 to enhance conciseness and clarity.\nAdditionally, we have incorporated some of the reviewer’s valuable perspectives into the revised text to strengthen the overall discussion.\nThese revisions are reflected on Pages 14–16 in the Discussion section, with all changes clearly marked in red for your reference.\n3) Figure 6 shows a clear trend between single-point speckle intensity and OAM order. Does this trend generalize to other types of input, such as handwritten digits or axicons shown in the manuscript?\nResponse:\nThank you for the thoughtful question. The trend shown in Figure 6 illustrates the evolution of speckle intensity by treating OAM1 and OAM2 as independent and dependent variables, respectively. Because this experiment involves two well-defined, orthogonal variables (i.e., different OAM modes), it is possible to generate a 2D intensity map that captures the full relationship between them.\nIn contrast, inputs such as handwritten digits and axicons represent single-variable classes. As a result, it is not feasible to construct a similar 2D intensity profile, since the input space is not defined by two controllable parameters. Instead, the speckle intensity evolution can only be visualized as 1D curves—one per category—based on the temporal or spatial sampling sequence.\nTo further investigate this, we analyzed the intensity profiles of the first four sampling points for different MNIST digit classes using SMPD. As shown in Fig. R9, only the digit ‘1’ exhibits a noticeable difference in intensity profile, likely due to its unique structure. For most other digit categories, there is no significant variation in single-point intensity, making it difficult to extract clear trends as seen in the OAM case.\nFig. R9 Speckle intensity profiles for handwritten MNIST digits (0–9) at the first four sampling points. (Each category includes 5,000 samples.)"}, {"round": 2, "reviewer_comments": {"reviewer_2": "In this revised manuscript, the authors have addressed my concerns raised in the last review, and thus I recommend that it be published without further revision."}, "author_response": ""}]

Decision letter