Polarization detection and imaging devices have attracted great and enduring attention in the past several decades for their broad applications, such as optical communications^[1,2], imaging^[3,4], remote sensing^[5], and biomedicine diagnostics^[6,7]. Since the state of polarization (SoP) is one inherent property of light, detecting SoPs can reveal the properties of light and its interaction with matter. The traditional polarization detection devices that rely on polarizers and waveplates usually suffer from having a large volume, a complex system, incomplete polarization detection functions, or a high cost^[8,9]. In contrast, the rapid development in nanophotonics facilitates new schemes toward ultra-compact, rapid, and highly stable full-Stokes polarimeters^[10]. Through properly engineering the morphology, size, and arrangement, the artificial nanostructures can effectively perceive both the linear momentum and the spin momentum of lights and exhibit polarization-dependent properties that can be detected by photodetectors, such as the polarization-dependent transmittance, reflectance, and diffractions^[11–17]. After constructing the mapping relationship S^=f(I^) between the SoP S^ of incident lights and the measured signals I^, the SoPs can be successfully extracted. In particular, the artificial intelligence (AI) algorithm employed in constructing S^=f(I^) can greatly improve the polarization resolving precision and stability, leading to high-performance polarimeters^[18–21].

However, the currently developed polarization detection and imaging devices usually rely on the well-designed structures of complex geometry. Their fabrication usually requires complex and expensive top-down methods, such as the electron-beam lithography (EBL), the laser-directing writing, and the inductively coupled plasma etching techniques^[22–26]. This greatly hinders the development and applications of polarization devices, especially for polarization imaging devices. In contrast, bottom-up methods, such as the wet-chemical synthesis^[27–29], the bio-assembled technique^[30], and the nanoshpere lithography^[31–33], usually have the advantages of simple operation process, high output, and low cost. However, the unavoidable randomness in the micro-scale self-assembled process greatly limits the uniformity of the fabricated structures and their applications. Taking microsphere lithography (MSL) as an example, a large-area, material independent, and low-cost method, it has been demonstrated in realizing a large amount of three-dimensional and two-dimensional structures, including the typical triangle array^[34], nanohole array^[35], nanodisks^[36], nanopillars^[37], nanocrescents^[38], three-dimensional nanostructures^[39], and chiral shells (CSs)^[40,41]. However, the self-assembled microsphere array usually has many micro-domains due to the interplay of various weak forces from molecular interactions to fluid dynamics in the self-assembled process. The lattice directions in different domains are random and will result in distinct structures in different domains, for example, the existence of both left- and right-handed chiral shells in one sample^[41]. This is the main drawback of MSL and limits the MSL to only being suitable for applications in labs, to date.

In this work, we demonstrate experimentally that the long-range disorder property of MSL provides a natural base for detecting the SoP of light. By two-step Ag deposition on the self-assembled microsphere monolayer, the CS samples with strong optical chirality and anisotropy are well formed. The random lattice directions lead to distinct chiral geometry and optical properties for shells in different domains. When placing one photodetector array underneath, these micro-domains with distinct optical properties will result in distinct photoelectric responses between adjacent detectors upon the injection of polarized light, where the SoP of incident light can be effectively extracted by resolving these distinct photoelectric signals. To improve the resolving precision, the convolutional neural network (CNN) is employed to establish the mapping relationship S^=f(I^) between the incident Stokes parameters S^ and the measured light intensity I^. The mean square errors (MSEs) of the polarization resolving are smaller than 1% for S1, S2, and S3 in a broad waveband ranging from 500 to 650 nm, indicating the realization of high-performance one-shot polarimeters in a broad waveband. Systematic work has been carried out to investigate the influence of the optical properties of chiral shells, the optical randomness of CSs in different domains, the reference SoP number, the exposure time and pixel number of the CCD, and the reliability and stability of the system.

High-performance CSs have been systematically developed in our previous works^[19,42–44]. By employing different materials, microsphere array with uniform lattice, or two-step deposition method, the CSs exhibit a large circular dichroism (CD) of up to 0.22, a broad working waveband from the ultraviolent to the near-infrared^[19,43], or large transmittance^[19,42], which can meet different application requirements. In contrast to the previously-reported applications that require uniform CSs in one sample^[44], the long-range disorder CSs with strong optical chirality and random optical anisotropy in different domains are highly preferred for the one-shot polarimeters. Thus, the spin-coated self-assembled method was used in this work to prepare a microsphere monolayer with several micro-domains, and the two-step material deposition method was used to improve the optical chirality and anisotropy of the CSs. The detailed fabrication process can be found in the Supplement 1, Figure S1 and Methods. As shown in Fig. 1(a), the material vapors are deposited on the sphere monolayer with a deposition angle θ referring to the substrate normal and an azimuthal angle φ1,2 referring to the vapor projection and the reference line. The relative azimuthal angle Δφ is defined as Δφ=φ2−φ1. The random lattice directions in different domains lead to the chiral shells with φ1 of different values, i.e., different geometrical morphologies and optical properties, as shown by the geometrical models of 8 typical CSs in Fig. 1(b) with Δφ=±45° and φ1=0°, 15°, 30°, and 45°. The two structures of the same colored box are a pair of enantiomers, namely, the left- and right-handed structures. The scanning electron microscope (SEM) images shown in Fig. 1(c) indicate the successful realization of long-range disorder chiral shells.

The optical chirality of the CSs is characterized by the CD spectrum defined as CD=(TRCP−TLCP)/(TRCP+TLCP), where TRCP and TLCP denote the transmittance of right circularly polarized (RCP) and left circularly polarized (LCP) lights, respectively. As shown in Fig. 1(d), the CSs exhibit large CD values in the 500 to 700 nm waveband, and the maximum value can reach about 0.22 at 612 nm. To investigate the optical anisotropy of the CSs, we measured the transmittance upon the injection of the linearly polarized light under different rotation angles. As shown in Fig. 1(e), the maximum and minimum transmittance appear at about 312° and 204°, respectively. The linear polarization extinction ratio (LPER) is larger than 1.6 in the investigated waveband, and the maximum LPER can reach about 2.37 at 602 nm [Fig. 1(f)], where LPER is defined as LPER=Imax/Imin, and Imax and Imin denote the maximum and minimum transmittance, respectively. These optical properties indicate the high performance of the CSs.

The polarization detection device is realized by simply covering the photodetector array with the fabricated long-range disorder CSs, and an optical system is also built up to collect the reference datasets [Fig. 2(a)]. To investigate the performance in a broad waveband, the nearly parallel white light was generated by two achromatic lenses and illuminated on a blazed grating at the blazing angle. The diffraction light was illuminated on the CS-covered CCD camera through one polarizer, one half-wave plate (HWP), and one quarter-wave plate (QWP), where the polarizer and waveplates were used to generate arbitrary polarized lights. Due to the long-range disorder property of the CSs, the CSs in different micro-domains have distinct optical properties and thus lead to distinct photoelectric signals between different pixels.

As shown in Fig. 2(b), the CS samples exhibit colors between the adjacent micro-domains in their micrographs, and the colors change significantly as we rotate the angle of linearly polarized lights from 0° to 45°, 90°, and 135°. These micro-domains will lead to different photo-currents for the photodetectors placed underneath. As shown in Fig. 2(c), the photo-current intensities change significantly between the adjacent pixels. When we rotate the HWP and QWP to change the SoPs of the incident lights, these pixels also show distinct responses to the incident SoPs. As shown in Fig. 2(d), the three different pixels show distinct photo-current mappings as a function of the HWP and QWP angles. According to the traditional theory^[9], at least four signals are enough to extract the SoPs of the incident lights, i.e., at least CSs in four domains or photo-currents from four pixels. Thus, these distinct photo-currents in different pixels provide an ideal basic for realizing one-shot polarimeters.

To improve the polarization resolving precision, we have made great efforts to search an optimized AI algorithm to establish the mapping relationship S^=f(I^) between the SoPs of the incident lights S^ and the measured photo-currents I^. The residual neural network (ResNet) is employed due to its high prediction accuracy in this work. In fact, the ResNet can effectively preserve the integrity of input information by incorporating shortcut connection channels and addressing the issues related to information loss and gradient disappearance, thus enhancing the network accuracy with increasing depth^[45]. As shown in Fig. 3(a), the ResNet18 model is employed as the optimized ResNet model in this work, which is composed of 18 convolutional layers featuring four residual modules with skip connections between every two convolutional layers.

In the training or predicting process, the speckle images collected by the CCD camera were used as the input layers and were convolved with a 7×7 kernel and subsampled via a max pooling layer. After four residual blocks and one average pooling layer were used to extract the average value in the pooling kernel, a fully connected layer was added to convert an image to a Stokes vector. In the forwarding propagation process, each convolution layer was activated by the nonlinear activation function, rectified linear unit (ReLU), and followed by a batch normalization (BN) layer to improve convergence. To train this ResNet18 model, 2700 datasets in total were collected at each wavelength by automatically rotating the HWP from 0° to 88.5° with a step of 1.5° and rotating the QWP from 2° to 90° with a step of 2°. The relationship between the incident polarization states and the angles of HWP and QWP is shown in the Supplement 1. Then, these datasets were divided into training datasets (90%) and validation datasets (10%). As shown in Fig. 3(b), the ResNet18 model converges quickly in the training process, where the MSE is used as the loss function. Upon sufficient parameter optimization and data training, the ResNet18 model was utilized to extract the SoP of the incident lights with the measured photo-currents. As plotted on the Poincaré sphere [Fig. 3(c)], the predicted Stokes parameters coincide well with their corresponding real values at the wavelength of 551 nm, and the MSEs are 0.459% (S1), 0.557% (S2), and 0.319% (S3), respectively. These MSE values are smaller than most of the previously reported polarimeters^[16–19,21]. Meanwhile, we also investigated its polarization detection performance in a broad waveband from 500 to 650 nm by rotating the blazed grating [Fig. 2(a)]. The calculated MSEs for S1, S2, and S3 are all smaller than 1% [Fig. 3(d)] in the investigated waveband, and the average MSEs are as small as 0.495% (S1), 0.450% (S2), and 0.310% (S3), respectively. These results indicate the successful realization of high-performance one-shot full-Stokes polarimeters in a broad waveband by utilizing the long-range disorder CSs.

The randomness degree of the CSs in different domains is one key parameter to determining the polarization detection precision. Here, we make a statistical analysis on the optical chirality and anisotropy that is perceived by the underlying pixels, where the optical chirality refers to the CD value, while the optical anisotropy refers to the ratio LPER−1=Imin/Imax and the maximum transmittance direction (MTD) for the incident linearly polarized lights. To calculate the randomness degree R of these optical properties, the CD, LPER−1, and MTD signals are divided into m uniform regions in their possible ranges, then we have R=1−∫0m|f(x)−nm|dx2n(1−1m),where y=f(x) is the probability distribution function for the detected optical signals, x is the signal value, and n is the total sample number. In this case, the ideal uniform distribution is y=f(x)=n/m, and ∫0mf(x)dx=n. The maximum randomness degree is R=1. For the MTD and the ratio LPER−1 detected in this work, we have large randomness degrees of R=0.7 and R=0.5, respectively, which are beneficial for making the eigen-polarization vectors form the Platonic polyhedra described in the Poincaré sphere^[46] and for improving the polarization detection precision. While for the detected CD signal, we get a small randomness degree of R=0.08, confirming the optical chirality uniformity of the fabricated CSs^[41]. Although this uniform optical chirality is not beneficial for improving the polarization detection precision, it is still enough to perceive the spin momentum of the incident lights, and it leads to a small MSE for the Stokes vector, which is comparable to the well-designed disorder metasurfaces^[47]. These results indicate that the fabricated CSs own a stable randomness and can effectively perceive the SoPs of the incident lights.

The polarization detection precision also highly depends on the reference datasets, the pixel number, and the exposure time of the CCD camera. As expected, the ResNet18 model can learn more realistic rules from more reference datasets and thus give more precise predictions. When we increase the reference SoP number from 648 to 4050 by decreasing the rotation step of the HWP and the QWP, the MSEs decrease significantly from about 1% to smaller than 0.5% [Fig. 4(a)]. Further, the pixel number can bring a similar influence on the polarization detection precision. Increasing the pixel number indicates employing more CSs to perceive the SoP of the incident lights, indicating that more data features are included in the mapping relationship S^=f(I^). In this case, the ResNet18 model can also learn more rules from more reference datasets and can give more precise predictions. As shown in Fig. 4(b), the MSEs of S1, S2, and S3 decrease by more than 0.4% when increasing the pixel number from 50 to 300. The exposure time of the CCD camera also shows significant influences on the polarization detection precision. As shown in Fig. 4(c), when we increase the exposure time from 100 to 300 ms, the MSEs of S1, S2, and S3 decrease by more than 0.2%. The main reason for this phenomenon is that all devices or optical elements used in the system have unavoidable noises, such as thermal noises, which will bring random errors for the detected signals and decrease the detection precision. While increasing the exposure time can increase the effective signal intensity and decrease the detection MSEs. Thus, optimizing these parameters may further decrease the MSE values and increase the detection precision.

The reliability and repeatability of the system are also very important for the established polarimeters. For the polarization detection system established in this work [Fig. 2(a)], all of the datasets were collected automatically by rotating the HWP and the QWP with electric motors, which were all controlled by the home-built Python codes. However, the rotation errors of the motors and the environmental fluctuations will also affect the reliability and repeatability of the system. Thus, we further investigated the detection MSEs after the system and the ResNet18 model were well calibrated. First, we kept the system unchanged after calibration and just rotated the QWP to obtain the previously used SoPs that have been included and new SoPs that have not been included in the reference datasets, respectively. As shown in Figs. 5(a) and 5(b), the predicted Stokes parameters coincide well with the real values, and the detection MSEs are 0.16% (S1), 0.23% (S2), and 0.27% (S3) for the previously used SoPs and 0.11% (S1), 0.20% (S2), and 0.22% (S3) for the new SoPs. Then, we reset the whole system, including the motors for controlling the HWP, the QWP, and the grating, and collected the previously used SoP datasets and new SoP datasets. As shown in Figs. 5(c) and 5(d), the predicted Stokes parameters coincide well again with the real values, and the detection MSEs are 0.15% (S1), 0.16% (S2), and 0.26% (S3) for the previously used SoPs and 0.13% (S1), 0.14% (S2), and 0.25% (S3) for the new SoPs. These results indicate that the one-shot polarimeters developed in this work have high reliability and repeatability.

In summary, we have experimentally demonstrated a high-performance one-shot polarimeter based on the long-range disorder CSs and the deep learning algorithm. The long-range disorder CSs were realized by employing the spin-coating method to fabricate the long-disorder microsphere monolayer and the two-step material deposition method to fabricate the CSs with strong optical chirality and anisotropy. By simply putting one CCD camera underneath, the distinct CSs in different micro-domains lead to distinct photo-currents for different pixels underneath, and the SoPs of the incident lights can be extracted by solving the S^=f(I^) with the measured photo-currents I^. The ResNet18 model was employed to establish the mapping relationship S^=f(I^) and improved the polarization resolving precision. After calibrating the polarization detection system and the ResNet18 model, the minimum polarization detection MSEs can reach about 0.37% (S1), 0.33% (S2), and 0.19% (S3) at 566 nm, and the average MSEs within the wavelength range of 500 to 650 nm are 0.49% (S1), 0.45% (S2), and 0.31% (S3), respectively. The detection precision can also be further improved by optimizing the macro- and micro-optical properties of the CSs, the reference SoP number, the exposure time, and the pixel number of the CCD. In addition, our research also indicates that the polarimeter developed in this work is a highly reliable and stable system. Due to the numerous advantages of the bottom-up method, our work provides a novel scheme-based long-range disorder metamaterial for realizing high-precision, broadband, and low-cost one-shot polarimeters and will inspire many new polarization detection and imaging devices and new applications for both the bottom-up methods and the related structures.