A Collective Propagation Law of Optical Vortex Constellations and Longitudinal Sensing

The paper reveals and experimentally demonstrates a simple deterministic law describing the paraxial longitudinal propagation of an arbitrary constellation of optical vortices in standard Gaussian backgrounds. By mapping the constituting singularity coordinates to their elementary symmetric polynomials (ESPs), the authors capture the holistic evolution of the constellation during propagation, completely bypassing the practical need to sequentially track indistinguishable vortices. They further show that such complex ESPs provide a useful metrological tool for the estimation of longitudinal displacements. The results reveal a previously unrecognized compact description of collective vortex dynamics, introducing a new route to longitudinal sensing through singularimetry.

The theoretical framework starts with a paraxial optical field containing a vortex constellation of n = l unit-charge singularities embedded in a Gaussian background. At the waist plane z = 0, the complex field is written as E(u, z = 0) = E0 ∏(u − uk) = E0 ∑(−1) j e j(0) u(n−j), where uk = xk + iyk is the transverse position of singularity k, and e j denotes the jth ESP of the complex vortex coordinates, defined as e j = ∑ u k1 u k2 ⋯ u kj, with e0 = 1. The ESPs serve as central parameters for a collective description of the longitudinal evolution, with e1, e2, and e3 representing the barycenter, variance, and skewness of the constellation respectively.

After free-space propagation to a distance z, the field becomes E(u, z) ≃ E0(z)ζ(z)(n+1) ∑ −ζ(z) e j(0) u(n−j), where ζ(z) = zR/(zR − iz), and zR is the Rayleigh range of the Gaussian background. The ESPs evolve with z obeying the simple propagation law e j(z) = e j(0)ζ(z)(−j) = e j(0)(zR/(zR − iz))(−j). This establishes a universal paraxial propagation law independent of the number, arrangement, or shape of the vortex constellation. The law does not cover background fields with nonzero radial structure, where vortex-antivortex pairs can be generated during propagation.

For experimental verification, a continuous-wave laser (λ = 810 nm) is spatially filtered and collimated to approximate a fundamental Gaussian mode. The beam is divided into two arms of a Mach–Zehnder interferometer, one with a spatial light modulator imprinting vortex constellations with desired positions of unit-charge phase singularities. The modulated beam is focused by a lens of focal length f = 500 mm, realizing free-space paraxial propagation. Off-axis digital holography enables retrieval of the complex field and precise determination of vortex positions. Measurements are taken over z ∈ [−160, 170] mm, where z = 0 denotes the focal plane.

A representative vortex constellation with four unit charge singularities located at u1 = (400, 400) µm, u2 = (200, −200) µm, u3 = (−400, 400) µm, and u4 = (−400, −400) µm with a Gaussian background of beam waist 700 µm is studied. The singularity positions are located at different longitudinal planes, and ESPs are calculated using Eq. (2). The singularity positions are recalculated from the barycenter (first ESP e1), automatically setting e1 = 0 while keeping higher-order ESPs intact. The experimental results show good agreement with simulations for the evolution of e2(z)/e2(0), e3(z)/e3(0), and e4(z)/e4(0), with minor mismatches stemming from small imprecision in estimating the waist position.

The ESP-based description opens a new route for sensing longitudinal displacement. The amplitude of ESPs e j(z)/e j(0) shows stronger variation away from the waist region (z > zR), while it saturates at the beam waist. In contrast, the phase arg(e j(z)/e j(0)) shows larger variation near the focal region, indicating better sensitivity when sensing around the focus. The Fisher information for amplitude is I a(j) = j2z2/(σ2 j,a z R4 (1 + z2/z R2)(j−2)), and for phase is I ϕ(j) = j2z R2/(σ2 j,ϕ (z2 + z R2)4). I a vanishes at the focal plane and increases away from it, while I ϕ has its maximum at z = 0 and decreases away from it. Thus, complex ESPs can sense longitudinal distance z in the full domain.

Experimental Fisher information is calculated with mean noise values σ j,a = 0.0183, 0.0618, 0.052 and σ j,ϕ = 0.0141, 0.0286, 0.0162 for j = 1, 2, 3. For lower-order ESPs (j = 2, 3), the experimentally estimated Fisher information, specifically I a, deviates more significantly from simulations because their smaller magnitudes are more easily overwhelmed by experimental fluctuations. Higher-order ESPs (j = 4) agree much better with simulations, demonstrating increased information gain when using constellations of a larger number of vortices.

The paper concludes that the compact propagation law eliminates the need to track individual singularities one by one. Since vortices are intensity nulls, the law yields the phase profile and OAM spectrum from intensity images alone, without direct phase measurement. The formalism extends to polarization singularities (C-points of vector beams), spatio-temporal optical vortices, vortex propagation through nonlinear media, and vortex constellations on non-Gaussian backgrounds. The work may also apply to Bose–Einstein condensates, superfluids, and topological field theories. The ESP-based description yields a powerful tool for precision longitudinal sensing, with higher-order ESPs possessing greater sensitivity to propagation, making complex constellations inherently superior for longitudinal length estimation.

Improvements to AI Systems:

1. Collective Topological State Tracking (CTST):

• Improvement: Replace per-object tracking algorithms (e.g., multi-object tracking in computer vision or particle filters in physics) with an ESP-based state representation. Instead of predicting individual vortex positions, the AI learns and predicts the evolution of elementary symmetric polynomials (e1, e2, e3, …) directly from intensity images.

• What the improved AI can do: Track an arbitrary number of indistinguishable vortices (or any topological defects) in real time without data association, even when vortices cross, merge, or are created/annihilated. It can extrapolate their collective dynamics over long propagation distances using the closed-form law e j(z) = e j(0)·(z R/(z R − iz))(−j), reducing computational cost from O(n2) to O(n) and eliminating tracking errors.

2. Phase and OAM Spectrum Inference from Intensity Only (POISE):

• Improvement: Train a neural network to map intensity images directly to ESPs (complex numbers), then use the propagation law to reconstruct the full complex field and orbital angular momentum (OAM) spectrum without interferometric phase measurement.

• What the improved AI can do: In optical metrology, microscopy, or free-space communication, the AI can recover phase profiles, OAM mode content, and topological charge distributions from a single intensity snapshot, enabling label-free, real-time characterization of structured light beams, even in scattering or turbulent media.

3. Longitudinal Displacement Sensing with Adaptive Sensitivity (LDSAS):

• Improvement: Implement a Bayesian or reinforcement-learning agent that selects which ESP order (j) and which measurement modality (amplitude vs. phase) to use based on the current estimated z, exploiting the complementary Fisher information profiles (amplitude sensitive far from focus, phase sensitive near focus).

• What the improved AI can do: Achieve sub-Rayleigh longitudinal position estimation over an extended depth range (from −∞ to +∞) by dynamically switching between amplitude-based and phase-based ESP measurements. For example, in optical tweezers or nanoscale positioning, the AI can maintain high precision both at the focal plane and far from it, outperforming static single-metric sensors.

4. Vortex Constellation Design for Optimal Sensing (VCDOS):

• Improvement: Use a generative model (e.g., a variational autoencoder or evolutionary algorithm) that optimizes the initial vortex positions u k to maximize the Fisher information of higher-order ESPs (j ≥ 3) for a given longitudinal range, leveraging the fact that higher-order ESPs have greater sensitivity and robustness to noise.

• What the improved AI can do: Automatically design custom vortex constellations (e.g., for LIDAR, optical ranging, or super-resolution microscopy) that minimize measurement uncertainty for specific depth intervals, while remaining robust to experimental fluctuations (as shown by the superior agreement of j=4 in the paper).

5. Cross-Domain Topological Defect Prediction (CTDP):

• Improvement: Transfer the ESP propagation law to other physical systems (Bose–Einstein condensates, superfluids, topological field theories) by training a physics-informed neural network that learns the ESP evolution from sparse measurements, using the law as a hard constraint.

• What the improved AI can do: Predict the collective motion of quantized vortices in superfluids, magnetic skyrmions, or cosmic strings from partial observations, enabling forecasting of phase transitions, turbulence cascades, or defect annihilation events without solving full nonlinear PDEs. It can also generalize to non-Gaussian backgrounds by learning corrections to the law from data.

6. Singularimetry-Based Autofocus and Aberration Correction (SAAC):

• Improvement: Integrate ESP estimation into an adaptive optics loop, where the AI computes e j(z) from a defocused intensity image and uses the phase of e j to estimate the exact focal plane position and wavefront curvature, then adjusts the optical system accordingly.

• What the improved AI can do: In digital holography, microscopy, or laser machining, the AI can perform real-time autofocusing and aberration compensation using only the intensity distribution of a vortex-bearing beam, eliminating the need for dedicated wavefront sensors or iterative focusing algorithms. This is particularly useful for in-vivo imaging or industrial processes where phase measurement is impractical.

7. Noise-Robust Higher-Order Feature Extraction (NRHOFE):

• Improvement: Train a denoising autoencoder that operates directly on complex ESPs, learning to suppress experimental noise (as seen in the deviations for j=2,3) by exploiting the known algebraic relationships between ESP orders (e.g., Newton's identities).

• What the improved AI can do: Recover clean, high-order ESPs (j≥4) from noisy or low-light intensity measurements, enabling reliable longitudinal sensing and vortex tracking even in photon-starved environments (e.g., astronomy, quantum imaging), where lower-order ESPs would be overwhelmed by noise.

Optical vortices are ubiquitous phenomena naturally appearing in wave physics, yet higher-order charges inherently split into constellations of lowest-order singularities at the smallest deviation from an ideal situation. While these constellations are common phenomena in real-world scenarios, characterizing their longitudinal evolution typically relies on exhaustive full-field descriptions or the ambiguous, sequential tracking of individual singularities. Here, we reveal and experimentally demonstrate a simple deterministic law describing the paraxial longitudinal propagation of an arbitrary constellation of optical vortices in standard Gaussian backgrounds. By mapping the constituting singularity coordinates to their elementary symmetric polynomials (ESPs), we capture the holistic evolution of the constellation during propagation, completely bypassing the practical need to sequentially track indistinguishable vortices. We further show that such complex ESPs provide a useful metrological tool for the estimation of longitudinal displacements. Our results reveal a previously unrecognized compact description of collective vortex dynamics, introducing a new route to longitudinal sensing through singularimetry.