Singular optics¹ primarily investigates structured light, especially vortex beams. Vortex beams exhibit a helical phase structure whose phase indeterminacy² leads to optical singularities. Specifically, for a vortex phase exp(imj) (with m being the topological charge representing quantized OAM and j being the azimuthal angle), its phase gradient along the optical axis becomes infinite³. This azimuthal phase gradient can form a high-dimensional, spatially infinite superposition state corresponding to the OAM spectrum of spatial eigenmodes^4,5, with the resulting state being referred to as a composite vortex beam. Beyond their distinctive phase structure and hollow intensity profile, vortex beams demonstrate unique energy flow characteristics manifested through the angular relationship m/kr between the Poynting vector and propagation direction (with k as the wave vector and r the radial coordinate)⁶. While numerous classical^7–9 and quantum^10–12 applications have been developed based on vortex beams’ topological charge and energy flow properties—spanning optical communication^13,14, manipulation¹⁵, and metrology^16,17, relatively few studies have explored applications centered on the singularities themselves^18–22, which motivates our present work on optical sensing, especially rotational sensing, through exploiting fundamental singularity properties.

In optical sensing, the relative motion of optical fields has a well-established physical interpretation: the Doppler effect^6,16. Whereas the linear Doppler effect is used to measure translational velocity a target object relative to a detector, the rotational Doppler effect characterizes the object's ​rotation​ by analyzing the field it emits^23–25. Studies have shown that the rotational Doppler shift magnitude is proportional to the topological charge of optical field, manifesting as time-varying phase²⁶ or frequency shifts in Fourier optics terminology. This enables OAM spectrum analysis for Doppler spectra²⁷. However, the intrinsic properties of OAM limit Doppler shift detection—different reference points selections lead to different OAM spectra, appearing as spectral broadening that causes Doppler spectrum aliasing and overlapping confusion in off-axis scenarios²⁸. Additionally, existing optical sensors cannot directly detect phase, requiring interferometric measurements (typically divided into common-path or heterodyne detection) that introduce sign degeneracy (preventing redshift/blueshift discrimination). Compared to heterodyne detection, common-path interferometry can reduce the influence of linear Doppler effects and optical path noise, with composite vortex beams being the most common choice. For detection, Doppler shifts enable time-frequency spectrum measurements in the far field, typically using Fourier transforms of photodiode (PD) signals²⁹. Single-point PD detection requires no spatial resolution, but to overcome ​alignment issues and sign degeneracy, additional spatial information—such as multi-point detection³⁰—is needed. In practice, off-axis misalignment frequently occurs at either the rotation axis or the far-field/conjugate detection axis, and although multi-point detection can mitigate this, it practically demands more stringent optical alignment of multiple detectors, thereby increasing experimental complexity. In summary, conventional PD-based methods acquire Doppler shifts through far-field detection without spatial resolution but require precise multi-point detection and alignment when addressing practical scenarios such as off-axis rotation and direction determination. In contrast, our approach measures the angular velocity of singularities through a unique surface-array gradient detection mechanism that relaxes these stringent requirements.

In this work, we focus on the characterizing role of singularities in rotational sensing, unlike traditional approaches utilizing the energy flow properties of vortex beams. As zero points in OAM density, singularities provide subwavelength-scale superoscillatory features that establish them as competitive sensing benchmarks. Recently, analogous to mass centroid and plumb lines in classic mechanics, singularities and OAM null lines have served as positioning rulers²¹ in three-dimensional static measurements. Here we extend this concept to dynamic rotating object sensing (Fig. 1a), establishing an analogy with the relationship between moment of inertia and angular momentum in center-of-mass reference frames, which we term the "optical singularity protractor (OSinP) " (Fig. 1b). By measuring the angular velocity in singularity group of the optical field, OSinP determines the rotational speed of ROs (Fig. 1c). The primary challenge in dynamically characterizing singularities is addressed through an innovative neuromorphic camera (NeCam) approach that relies on gradient-based event detection rather than absolute intensities (Fig. 1d and Fig. 1e). Furthermore, we investigate OSinP’s performance in practical rotational scenarios including off-axis rotation and non-steady-state velocities with alternating directions, demonstrating OSinP’s robustness. Our work pioneers the application of event-based sensing via structured light fields, opening new avenues for further exploration in metrology with topological light.

OAM, Rotation and Singularity. The rotation of a rigid system or an optical field is intrinsically linked to its OAM density distribution (Fig. 2a). The OAM density of an optical field, , can be derived from the energy flow , where E stands for the electric component of a monochromatic optical field with frequency w_o. For light carrying longitudinal OAM (rotation along z-axis), we primarily focus on its z-component J_z. The OAM density consists of intrinsic and extrinsic components, which are distinguished by their dependence on the reference coordinate system. While the OAM density of light varies with the choice of reference point r_o, certain invariant features persist. Specifically, regardless of the reference point selection, the null lines of OAM density always pass through both the reference point and the singularities—a key property that underlies the development of singularity rulers²¹ for static field localization. Figure 2b-2g reveals that, unlike the stable singularity lines, the OAM null lines in a singularity system exhibit more complex behavior. Off-axis singularities generate multiple OAM null lines instead of a single line passing through both the singularity and the reference point. These multiple null lines partition the OAM density into alternating positive and negative regions, which fuse or emerge depending on the reference point selection. Overall, irrespective of the reference point selection, all null lines must pass through the singularities, though not necessarily through the reference point (Fig. 2g).

We draw an analogy with the moment-of-inertia model of rigid bodies to demonstrate the unique properties of singularities in dynamic rotational speed sensing, similar to the plumb-line centroid model used in singularity rulers (Fig. 2a). In this framework, the correspondence between singularities and mass centroid is extended to singularity groups analogous to particle systems. As described by the Pappus centroid theorem³¹ or parallel axis theorem³², the angular momentum L_z of a rigid body relates to the angular velocity w_i of its mass centroids, where the choice of reference point affects the change of moment of inertia I_i+DI_i of mass centroids, consequently altering angular momentum density. The total angular momentum thus becomes . This analogy motivates us to focus on the angular velocity of singularity groups within rotating optical fields. For rotational sensing, measuring the angular velocity of singularities in these groups enables extraction of the rotating object's velocity.

For practical rotating optical fields, composite vortex beams—superpositions of vortex modes (e.g., , c_m are mode coefficient of vortex modes)—are typically employed in measurements. A canonical example is the symmetric superposition of ± m Laguerre-Gauss (LG) beams () with identical radial index p and ​opposite azimuthal index m³³, with phase pattern shown in Fig. 3d and mode spectrum in Fig. 3h, denoted as (-7,7) for m = 7. We use LG modes with p = 0 and different m, such as m = 1 (Fig. 3a, 3e) and m = 7 (Fig. 3b, 3g). For rotational sensing compatibility, we utilize (1,7) composite vortex beams but focus on singularity behavior (see Supplementary Note 1). As shown in Fig. 3c, 3f and 3i, the (1,7) composite vortex beam generates a singularity group of six off-axis singularities, each carrying unit topological charge (|m| = 1). Under this illumination condition, the rotational Doppler shift is Df = 6W/2p (For detailed calculations, refer to Supplementary Note 1). These off-axis singularities—identified by either zero-intensity points or maximum-intensity-gradient locations—exhibit a rotational speed of W. This characteristic remains invariant under off-axis misalignment. Figure 3j-3l demonstrates progressively broadened OAM spectra with increasing off-axis distance, leading to Doppler spectral overlap. Notably, singularities maintain a fixed rotational speed W, while their linear velocities scale with off-axis displacement. This characteristic holds generally for the singularity behavior in any composite vortex beam formed by two positive modes, thereby establishing a singularity-based method to characterize optical field rotation.

Rotation Velocity Detection via OSinP. Building upon the established relationship between singularity velocity and optical Doppler effects, the key remaining challenge is how to efficiently acquire singularity speeds in a fast, lightweight, and widely applicable manner. Our solution leverages gradient sensing: singularities, as abrupt transition points in optical fields, are more effectively resolved through spatial gradient detection DI than temporal integration I. Fortunately, NeCam, a novel gradient sensor that operates asynchronously, has been developed^34–37. In this architecture, each pixel in NeCam detects logarithmic intensity gradient changes through adjustable thresholds (C) and returns asynchronous data indicating the direction of change (Fig. ld). This neuromorphic mechanism provides high information throughput, wide dynamic range, and other advantages that perfectly meet singularity detection requirements. For example, the sensor's event-driven operation enables real-time tracking of singularity motion without conventional frame-based system latency, as shown in Fig. 1e. By directly capturing intensity gradients at microsecond timescales, OSinP offers a more efficient solution for dynamic singularity characterization in practical rotational sensing applications.

We first describe how to stably obtain the velocity and rotation direction of singularities in OSinP. Figure 4a displays a raw echo signal with the optical field rotation speed set at 82.86 rps. The positive and negative polarity events p represent positional changes of six off-axis singularities, exhibiting helical patterns along the time axis from 54.153 ms to 104.137 ms, where the polarity data indicates the direction of singularity linear speed vectors. To extract the speed information, we perform frame compression on event data using a 660 µs window, representing the actual temporal resolution 1515 Hz. Using Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm³⁸, we cluster events in each frame unsupervisedly to determine each singularity position and centers of opposite polarity events , then calculate singularity speed vector directions , with results shown in Fig. 4b and Fig. 4c. Circular trajectories of the six off-axis singularities are revealed in Fig. 4b, where their calculated geometric center serves as the rotation center. Unlike conventional rotation sensing, OSinP requires no strict axial alignment—only encompassing the singularities movement—to determine the rotation center from measured data. In experiments, the incident beam radius was 0.836 mm (imaging plane after f = 100 mm zoom lens), while the detector (Sony IMX636ES, 1/2.5", 6.22×3.50 mm) provided generous alignment tolerance compared to PD-based methods.

Based on the positions and speed directions of singularities, we propose two methods to calculate the rotational frequency shift. The speed-vector method utilizes (Fig. 4c), while the center-cosine method utilizes to compute the cosine angle between the singularity's motion trajectory and the geometric center (Fig. 4d). Both yield the angular velocity of singularities. Defining clockwise rotation as positive, the event data in Fig. 4 reveals the positive rotation direction (Fig. 4b-4d). For constant-speed rotation, the rotational frequency shift can be extracted through Fast Fourier Transform (FFT). Figure 4e demonstrates a measured rotational frequency shift of 13.62 Hz (relative error 1.37%) that is obtained from the FFT spectrum peak, with all singularity time series showing identical results. Notably, the FFT spectrum in Fig. 4e exhibits exceptional purity compared to the significant background interference typically observed in PD-based measurements. This superiority stems from the intrinsic alignment-free nature of OSinP, which effectively eliminates higher-order harmonics, combined with the data sparsity characteristic of the event-driven detection mechanism.

To evaluate OSinP's universality and limits, we tested Doppler shift ranging from 0 to 220 Hz, with measurement results shown in Fig. 4f. The fitted measurements exhibit a linear growth relationship with the preset Doppler shifts, fully consistent with theoretical predictions. Both methods demonstrate nearly identical results, with deviations only occurring at low speeds (first data point). This discrepancy arises from reduced event responses induced by singularities at low rotation rates, affecting clustering algorithm stability (detailed analysis in Fig. 6). Additionally, as shown in Fig. 4c and Fig. 4e, using singularity speed vectors introduces higher-frequency noise compared to trajectory-based methods, likely caused by trailing artifacts^39,40 of NeCam (see Supplementary Note 2) and subsequent clustering errors, though without impacting frequency shift calculation. All Fig. 4 results were obtained under identical experimental parameters. Adjusting background intensity and hardware thresholds (C: contrast, bias, cutoff frequency, etc.) can enhance singularity performance for different rotation velocity. Further improvements may incorporate advanced deep learning or spiking neural networks into OSinP.

Multi scene detection: off-axis and non-stationary velocity. To demonstrate OSinP’s practical utility, we examined two challenging scenarios: off-axis rotation and non-steady-state speeds with obtainable directionality. Off-axis rotation induces OAM spectral broadening, causing Doppler shift dispersion, while OSinP remains unaffected. Non-steady-state speeds showcases OSinP’s capability to track velocity variations and determine directionality. For off-axis rotation, we tested three distinct rotation axes (Fig. 5a) at identical rotational speeds, corresponding to the reference point selection in Fig. 3i. Figure 5b-5d displays singularities identification results as the rotation axis progressively shifts from the center. The misalignment between rotation and optical axis generates seven concentric circles, comprising both moving off-axis singularities and an on-axis singularity. Crucially, the rotation center is directly computable from raw data without prior knowledge of axis position. FFT analysis extracts a consistent 13.62 Hz peak from all concentric trajectories (Fig. 5e-5g), confirming rotational velocity at 13.62 rps. Compared to PD intensity measurements, singularity-derived velocity provides more direct frequency shift determination through spatial-domain analysis, circumventing OAM mode projection and its intrinsic spectral limitations.

To rigorously evaluate the practicality of OSinP in dynamic scenarios, we implemented a nonlinearly varying velocity profile simulating proportional-integral-derivative (PID) control system⁴¹ with directional reversals (Fig. 6). Such nonlinear control systems are common in real-world rotational applications⁴², with governing equations derived from damped systems. OSinP successfully characterized the velocity series (Fig. 6a) generated under a preset PID regime (P = 5.5, I = 270, D = 0.055) with preset Doppler frequency shifts of 140.5 Hz (0-1.5s), -85.5 Hz (1.5-3s), and 85.5 Hz (3-4s), which resulted in the nonlinear responses documented in Fig. 6b. For acquired singularity velocity (frequency shift) data (Fig. 6a), we applied direct FFT-based low-pass filtering and averaged all off-axis singularities to eliminate localization errors. The averaged multi-singularity velocity time-series (Fig. 6b) showed error accumulation near zero-speed transitions, inherent to the detection mechanism. As illustrated in Fig. 6c and Fig. 6d, near-zero velocities result in weak singularity responses, leading clustering and averaging inaccuracies that affect direction and velocity estimation. Integrating frame-based sensing could effectively mitigate these errors, particularly in scenarios with gradual speed variations.

In our proof-of-concept experiments, a DMD (DLP9000X, Texas Instruments) was used to generate singularity beams and simulate rotating objects^46,47. A 532 nm Gaussian laser was incident on the DMD at 12°, undergoing complex amplitude modulation via superpixel hologram encoding⁴⁸. Composite vortex beams containing off-axis singularities were obtained through + 1st-order filtering in a 4f system. To simulate rotation, we leveraged the DMD's high-speed refresh capability to load time-varying phase pattern corresponding to rotating object dynamics, allowing accurate comparison between preset and measured values—especially crucial for non-steady-state velocities. Due to the need for rapid switching across numerous holograms that generate rotational sequences, we employed Look-Up Table (LUT) mode to facilitate hologram transitions. While the DMD's refresh time is shorter than NeCam's frame compression duration, LUT storage of time-dependent phase profiles for a full rotation allowed extended simulations (e.g., 4s as shown in Fig. 6) at microsecond-level updates. The resulting optical field was described by:

where A_i (i = 1,2,…,7) denotes the amplitude of corresponding LG modes (see Supplementary Note 1), represents modulo operation, N_L is the total number of LUT entries, and n(t) is the current hologram index input as 9-bit binary data. Compared to sequential hologram transfer, this LUT-based scheme effectively circumvents data transfer rate bottlenecks between DMD and PC, enabling extended and precise simulation of rotating fields using commercially available hardware.

All data that support the findings of this study are available within the article and supplemental document, or available from the corresponding author upon reasonable request.