Single-shot polarimetry of vector beams by supervised learning

The precise generation, manipulation, and detection of the optical state of polarization (SOP) are crucial in numerous fields, including optical communication, sensing, microscopy, and quantum information and computation. While advancements in material science and nanotechnology continue to improve active polarization control, measuring light polarization remains challenging due to its vectorial nature. Complete SOP determination typically requires at least four individual measurements, each projecting the state onto a distinct vector. Traditional polarimetry methods, employing time or space replication of the polarization analyzer, often result in bulky setups or expensive compact polarimeters based on metasurfaces. Although manageable for uniformly polarized light, this limitation becomes a significant obstacle when dealing with beams possessing a spatial polarization structure. Light with non-uniform polarization across its transverse plane exhibits inseparable correlations between polarization and spatial modes, a characteristic of vector beams. These vector beams have shown considerable potential in diverse applications, including metrology, communication, optoelectronics, optomechanics, and quantum information. However, their characterization is currently reliant on bulky polarization optics, highlighting the need for faster, more accurate, and compact polarization measurement techniques to fully exploit their capabilities in widespread applications. This article presents a novel approach to achieve this, utilizing compact single-shot measurements of multiple polarizations via photonic machine learning.

Conventional polarimetry techniques rely on projective measurements, where the SOP, represented by a four-component Stokes vector, is analyzed by distinct detectors, each measuring one component. This approach requires multiple measurements to fully characterize the polarization state. Existing methods, such as those using metasurfaces, while offering compactness, often suffer from limitations in bandwidth or complexity. The need for multiple measurements becomes particularly problematic for vector beams, which exhibit non-uniform polarization across the transverse plane. Characterizing these beams typically involves bulky polarization optics, hindering their widespread application. Previous attempts at single-shot polarimetry have employed techniques such as using disordered media, but often require determining the transmission operator, adding complexity. The presented work differs by leveraging supervised learning to extract polarization information directly from a single intensity distribution, thus avoiding the need for direct polarization manipulation or complex calculations of the transmission operator.

The proposed single-shot polarimetry method maps the beam's polarizations into a complex spatial intensity distribution, creating a high-dimensional feature space. Supervised learning is then employed to extract the polarization information from this intensity data. Critically, this method avoids projecting onto a polarization basis and performing any direct operation on the polarization state, eliminating the need for polarization optics and engineered devices in polarization imaging. The experimental setup consists of two parts: a vector beam generator and a single-shot polarization analyzer. The vector beam generator utilizes a phase-only spatial light modulator (SLM) to shape the wavefront of a laser beam, creating multiple SOPs in spatially separated modes. The single-shot analyzer uses light scattering from a glass diffuser to map the SOPs into intensity data. The scattered speckle pattern is imaged on a camera, with no polarization filters. The entire polarization information is embedded within this intensity distribution. The data is then processed using an Extreme Learning Machine (ELM) algorithm. ELM is chosen for its speed and efficiency in training, enabling fast calibration with a large number of network nodes. The training involves constructing a readout layer with randomly selected camera channels, each with a linear weight. These weights form a calibration matrix, which is determined via ridge regression using a labeled dataset. The method is experimentally validated by comparing the results with those from conventional polarimetry using a rotating-waveplate polarimeter. The accuracy of the single-shot measurement is evaluated by varying the number of network channels and the size of the training dataset, revealing the double descent phenomenon in the testing error as the number of channels increases.

The study demonstrates highly accurate single-shot polarimetry of various beams encoding multiple SOPs. The method achieves accuracy exceeding 95% on each Stokes parameter when characterizing structured light encoding up to nine polarizations. Remarkably, the technique also allows the classification of beams with an unknown number of polarization modes, surpassing the capabilities of conventional techniques. The experimental results show a strong agreement between the single-shot measurements and the results obtained from conventional multiple-projection polarimetry. The analysis of the error reveals the double-descent phenomenon, where the accuracy increases even with a large number of channels, indicating high precision measurements. The fidelity of the single-qubit tomography reaches 0.99 ± 0.01, confirming the high accuracy of the single-shot method. Experiments with vector beams containing up to nine SOPs (a 27-dimensional phase space) were successfully performed, showcasing the scalability of the method. The study also demonstrates the ability to determine the number of polarizations in an unknown multiple-polarization state with over 98% accuracy, a functionality unavailable in conventional methods. The overall accuracy of the single-shot polarimetry across various experiments, even with varying numbers of polarization modes, consistently remained high (above 90%), validating the robustness and effectiveness of the proposed technique.

The findings present a significant advancement in polarization measurement, offering a compact, single-shot approach that eliminates the need for bulky polarization optics. This method combines the physical transformation of polarization into spatial intensity distributions with the power of machine learning, enabling unprecedented information extraction from a single detection. The broadband nature and lack of moving parts make it superior to existing metasurface-based polarization cameras. The ability to determine the number of polarizations in a beam adds functionality unavailable in conventional methods, opening avenues for analyzing complex polarization states in various applications. Furthermore, the efficiency of the method suggests potential extensions to other optical degrees of freedom, potentially revolutionizing optical devices in areas such as sensing, imaging, and computing.

This research successfully demonstrates single-shot polarimetry of vector beams without polarization optics, utilizing a novel combination of light scattering and supervised learning. The technique achieves high accuracy in measuring multiple polarizations, even when the number of modes is unknown. The method's compact nature, broadband operation, and lack of moving parts offer significant advantages over conventional polarimetry, promising advancements in various optical applications. Future research could explore the use of unsupervised learning methods and the extension of this technique to other optical degrees of freedom and the quantum regime.

While the method demonstrates high accuracy and scalability, the reliance on a trained machine learning model necessitates a calibration phase. The accuracy of the measurements depends on the quality of the training data and the chosen machine learning algorithm. The current experimental setup uses a specific type of diffuser, and the performance might vary with different scattering media. Furthermore, while the method is shown to be broadband, the specific spectral response might require adjustments to the calibration depending on the target wavelength range.