Single-shot polarimetry of vector beams by supervised learning

Measuring and controlling the optical state of polarization (SOP) is central to applications in communications, sensing, microscopy, and quantum information. Conventional polarimetry relies on at least four projective measurements to retrieve a single SOP and typically uses replicated analyzers in time or space, leading to bulky or costly systems. While this is manageable for uniformly polarized light, it becomes a major challenge for beams with spatially varying polarization (vector beams), which exhibit non-separable polarization–spatial correlations. Existing characterization methods for vector beams largely depend on polarization optics and multiple projections, limiting speed, scalability, and compactness. The authors aim to demonstrate a compact, fast, single-shot method for measuring multiple polarizations embedded in vector beams without polarization optics. Their approach maps polarization content into a high-dimensional spatial intensity feature space via light scattering and recovers the Stokes parameters through supervised learning, addressing scalability to many polarization modes and even enabling inference of the number of polarization components.

Prior work established that full SOP determination requires multiple projections and often bulky optical setups or specialized metasurface-based polarimeters. Metasurfaces and integrated devices can perform compact polarimetry but often operate over narrow bandwidths and still rely on engineered polarization-selective structures. Various studies have explored polarization generation and control, optimal polarimeter designs, and projective tomography, as well as advanced single-shot or generalized measurements for specific configurations. Disordered and complex media have been used to implement optical instruments and computations (e.g., lenses, spectrometers, polarimeters) via transmission matrix methods, but typically require characterizing the medium. Photonic machine learning frameworks, including kernel methods, reservoir computing, and extreme learning machines (ELMs), have shown scalability and speed in optical implementations. Building on these, the present work uses scattering as a polarization-sensitive transformation and supervised learning to avoid explicit polarization projections and detailed transmission matrix recovery, aiming for single-shot, scalable polarimetry of vector beams.

Concept: The method maps the incident beam’s SOP(s) to a high-dimensional vector of intensity features obtained after propagation through a polarization-dependent optical system (here, a glass diffuser). The input SOP s = (S1, S2, S3, S0) is transformed into an n-dimensional feature vector x = (x1, x2, …, xn) corresponding to camera-sampled intensities. A linear readout recovers the Stokes parameters via s = B x, where B (the calibration matrix) is learned from data. For vector beams with D spatially separated polarization modes, the overall state comprises D SOPs and the calibration matrix scales to size 4D × n, enabling single-shot retrieval of all component SOPs from a single intensity image. Physical setup: A 532 nm CW laser is expanded and linearly polarized. A reflective phase-only SLM, sandwiched by waveplates (HWP before; QWP+HWP after on motorized stages), partitions the beam into D spatial modes by grouping L×L SLM pixels; each mode carries a programmable phase ϕi and a target SOP controlled by waveplate angles (α, β). The polarization-modulated beam is focused onto a ground-glass diffuser (1500 grit). The transmitted speckle is imaged onto a CMOS/CCD camera. No polarization analyzers are used in the detection path. From the camera, 4M output channels (binned pixels, size comparable to speckle grains) are randomly selected as the readout features. Reference measurements for validation use a commercial rotating-waveplate polarimeter for single SOP and a custom rotating-waveplate analyzer (QWP + linear polarizer + camera) for vector beams with projections along H, V, D, and R components. Supervised learning and calibration: Training uses a dataset of randomly selected target Stokes vectors S (size N_train × 4D) and corresponding intensity measurements X (size N_train × 4M). Ridge regression (extreme learning machine readout) determines the calibration weights β by solving argmin_β ||Xβ − S||^2 + c^2||β||^2, yielding β = (X^T X + c I)^{-1} X^T S. For inference on an unknown input, the measured intensity vector x is mapped via s = β^T x (single SOP case S_i = Σ_k β_{ik} x_k) or via the learned 4D × 4M matrix for vector beams. The approach exploits redundancy (M can be large) to achieve high accuracy and scalability without knowledge of the medium’s transmission operator. Training datasets cover the PB sphere uniformly by varying waveplate angles. For unknown D, the dataset includes samples with varying D and an additional output channel to regress D, enabling simultaneous SOP estimation and dimensionality identification. Scalability and coupling: For D=1, some polarization–spatial coupling in the scatterer is required so distinct SOPs yield distinguishable intensity patterns. For D>1, the input partitioning provides sufficient interaction, and comparable performance is observed across diffusers with varying depolarization; experiments reported small coupling with transmitted light having a degree of polarization near one. The number of readout channels M is user-selectable; hundreds suffice for effective measurements, while tens of thousands further enhance accuracy and reveal double-descent behavior in error vs. model size. Metrics: Accuracy is quantified via mean absolute error E(S_i) between measured and generated Stokes components, distance d on the PB sphere (sum of squared errors), overlap/accuracy matrices α relative to projective measurements, and fidelity F between single-shot and multi-projection reconstructions in the single-SOP case. Regularization parameter c and channel count M are tuned empirically; typical M values range from hundreds to >10,000 (up to ~52,000 channels used in some D=4 measurements; training feasibility for D=9 limited to M≈13,000).

- Single SOP (D=1): The single-shot analyzer achieves high accuracy over random SOPs on the PB sphere. Mean distance between measured and generated SOP: d = 0.014 ± 0.002. Stokes error decreases with M, reaching E(S) ≈ 0.0075 for large M. Experimental error vs. M exhibits a clear double-descent peak at the interpolation threshold (maximum error near M ≈ N_train), after which accuracy improves without overfitting. Comparison with a rotating-waveplate polarimeter yields fidelity F(ρ_s, ρ_m) = 0.99 ± 0.01 and strong agreement in the accuracy matrix including the degree of polarization. - Vector beams with D=4: Single-shot reconstructions of four spatially partitioned SOPs closely match conventional polarization tomography. Accuracy matrix gives Tr(α)/(4D) ≈ 0.95, indicating ≳95% agreement. Error decreases with the number of channels M (up to ~52,000 total channels), with localized peaks marking interpolation thresholds. - Vector beams with D=9: The method reconstructs nine SOPs in a single shot (overall state lies in a 27D phase space). The single-shot results closely match multiple projections, with an overlap of 0.91 from a 1296-entry accuracy matrix. Distance d decreases rapidly with M and saturates at a plateau, reflecting state complexity; an interpolation threshold is not observed within accessible M, suggesting more channels would be needed. - Unknown dimensionality D: When trained on mixed D∈{1,4,9}, the system identifies the number of polarization partitions with >98% accuracy at M=10,000 and simultaneously estimates Stokes parameters with precision comparable to the known-D case (E(S_i) ≈ 0.057 for D=9). This capability—inferring D—is not directly feasible with standard multiple projective measurements. - Overall: Using light scattering plus supervised learning enables single-shot measurements of vector beams encoding up to nine polarizations with accuracy beyond 95% on each Stokes parameter, while revealing and exploiting double-descent behavior to enhance precision.

The study addresses the core challenge of scalable, compact polarimetry for vector beams by replacing multiple polarization projections with a single intensity measurement followed by learned linear decoding. Mapping SOPs into a high-dimensional feature space through scattering provides redundancy that, together with supervised learning, yields accurate Stokes parameter estimation for many concurrent polarizations. The results demonstrate that photonic machine learning can deliver precise polarization imaging without polarization optics, offering broadband operation and a compact, static setup. Observing double descent highlights that larger feature spaces can improve generalization in this physical ELM, guiding system sizing for optimal accuracy. Importantly, the method extracts otherwise inaccessible beam properties, notably the number of polarization partitions D, expanding the scope of polarimetric analysis beyond traditional tomography. These advances are relevant to sensing, imaging, optical communications, and on-chip or edge devices, and suggest extensions to other degrees of freedom and spectral regimes.

This work demonstrates a compact, single-shot polarimeter for vector beams that requires no polarization optics, combining polarization-to-intensity mapping via scattering with supervised learning to decode multiple simultaneous SOPs. The method scales to at least nine polarizations with >95% per-Stokes accuracy, identifies the unknown number of polarization modes with ~98% accuracy, and achieves high fidelity relative to conventional tomography. The discovery of double-descent behavior informs the design of high-dimensional readouts that enhance measurement precision. Future directions include broadband operation via wavelength-dependent calibration, extension across the electromagnetic spectrum and to subwavelength or topological fields, application to other optical degrees of freedom, deployment in edge and integrated photonic platforms, incorporation of unsupervised learning (e.g., variational autoencoders) for generative modeling of Stokes distributions, and exploration of quantum regimes where partitioned vector beams encode multiple qubits.

- For high-dimensional states (e.g., D=9), the accessible number of output channels limited the observation of the interpolation threshold; more channels are likely required to fully characterize double-descent behavior at larger D. - Training feasibility constrained M in some experiments (e.g., capped at ~13,000 for D=9), which can bound ultimate accuracy improvements. - The approach requires a supervised calibration phase with a representative dataset; performance depends on coverage of the PB sphere and stability of the optical system. - Reported accuracy comparisons include uncertainties from both the SOP generator and the conventional analyzer used for reference, which may slightly bias error estimates. - While polarization–spatial coupling in the scatterer is unnecessary for D>1, some coupling is required for D=1 to ensure distinguishable intensity patterns, potentially constraining scatterer choice for single-SOP scenarios.