1. Introduction

Polarization, a fundamental property of light, plays a crucial role in comprehending light's characteristics and interaction with matter. Measuring states of polarization (SoP) can reveal the intrinsic properties of light, as well as the morphology and materials composition of target objects, facilitating their detection and identification [1]. Therefore, polarimetric and polarization imaging devices have developed broad applications in various fields, such as remote sensing [2,3], astronomy [4], biological diagnosis [5,6], optical transmission [7], and food industry [8], etc. Traditional polarimetry and polarization imaging devices often rely on elements such as linear polarizers, half wave plates (HWPs), or quarter wave plates (QWPs) [9]. However, these methods are usually bulky, inaccurate, time-consuming, or incapable of detecting full Stokes parameters [10]. To achieve a highly efficient, compact and high-integrity full-Stokes system, considerable efforts have been dedicated to explore new schemes, such as the micro-lens array [11], liquid-crystal-based systems [12], new semiconductor materials [13,14], plasmonic photodetector structures [15,16], artificial metamaterials [17], metasurfaces [18–22], two dimensional materials [23], and chiral materials [24–27]. Particularly, the optical metasurfaces made of subwavelength structures facilitate the development of compact and real-time full-Stokes imaging devices, raising a new round of industrial technological revolution for detecting the SoP of lights.

The majority of the previously-reported polarimetry or polarization imaging devices relied on the high-performance polarization-sensitive elements or materials and the pre-established mapping-relationship $\widehat S = f(\widehat I)$. The polarization-sensitive elements or materials convert the SoP of light to a detectable physical quantity, such as light intensity, current [14–16] or voltage [21,28]. While the mapping relationship provide the Stokes parameters $\widehat S$ based on the detected information $\widehat I$. However, the inevitable errors, such as fabrication errors of polarization-sensitive elements, and the detection errors of light intensity, greatly limit the detection precision, resulting in the detected MSEs exceeding 1%, and in some cases reaching as high as 45% [16]. This situation persisted until the introduction of artificial intelligence (AI) algorithm for the polarization detection. In 2020, Michael Juhl and Kristjan Leosson demonstrated a new type of polarimetry based on the diffraction speckle images from disordered photonic structures, where a convolutional neural network (CNN) greatly improve the detection precision to about 0.1 degrees [29]. Our previous work has also confirmed the advantages of DLA in developing polarimetry with dielectric chiral shells [30]. However, the long-range disordered lattice and weak optical chirality of the dielectric chiral shells result in a very unstable detection precision within the waveband from 400 nm to 850 nm, especially for the Stokes parameter S₃ with the maximum MSE of about 15% and the minimum MSE of around 0.37%. Moreover, there are still lots of debates regarding the necessity of utilizing DLA due to limited data availability and the well-established Mueller matrix theory.

In this work, we demonstrate a high-performance and stable polarimetry in a broad waveband through employing high-performance chiral shells with a uniform lattice over a large area. The chiral shells were fabricated using an angle-dependent deposition technique and an improved microsphere self-assembly technique [31]. Through meticulously controlling the geometry, the prepared chiral shells exhibit strong optical chirality and anisotropy, resulting in small detection MSEs and fluctuations in a broad waveband. In particular, we conducted comparative studies on the polarization detection precision using the Mueller matrix theory and the DLA. A general mapping-relationship formula $\widehat S = f(\widehat I)$ was derived using Mueller matrix theory, including a transitional formula to facilitate fitting precision. To measure the SoP of incident light, the chiral shell samples were rotated multiple times to generate the transmittance matrix $\widehat I$, and the mapping relationships were established by fitting the formula $\widehat S = f(\widehat I)$ and training the DLA with the reference data collected by illuminating the sample with a known Stokes matrix $\widehat S$. The results indicate that, both the Mueller matrix theory and the DLA can achieve high detection precision, with the MSEs being far smaller than 1% at specific wavelengths, indicating the advantages of the post-established mapping relationship. However, the DLA exhibits superior detection precision and robustness in the investigated waveband of 500 nm to 750 nm, especially for weak illumination light, and the polarization-sensitive materials with low optical anisotropy and chirality. As a result, the DLA enables high-precision, broadband polarimetry. This work not only demonstrates highly precise and stable polarimetry across a broad waveband, but also highlights the advantages of using DLA to establish the mapping-relationship.

2. Polarization detection algorithm

In general, polarization detection relies on the use of polarization-sensitive materials that can convert the Stokes parameters $\widehat S$ to a detectable physical quantity $\widehat I$. The conversion is governed by a mapping relationship $\widehat S = f(\widehat I)$, which can be established by using either the Mueller matrix theory or DLA, as shown in Fig. 1. Subsequently, the SoP of incident light can be extracted by calculating the relationship $\widehat S = f(\widehat I)$ with the detected physical quantity $\widehat I$. Therefore, both the polarization-sensitive materials and the mapping relationship are crucial factors in achieving high-performance polarization detection. These polarization-sensitive materials encompass various options, such as the combination of polarizers and wave-plates, artificial structures, two-dimensional materials [23], or chiral perovskites [14]. To achieve highly-precise full-Stokes polarimetry, materials with strong optical anisotropy and chirality are required to detect both the linear and spin components of light, respectively, which have been extensively studied in previous works [32]. In this work, we chose chiral shells as the polarization-sensitive materials, which will be comprehensively introduced in subsequent sections. In terms of establishing the mapping-relationship, both the Mueller matrix theory and DLA have been employed, and their performances will be discussed below.

 

Fig. 1. Schematic illustration of the normal polarization detection principle with the Muller matrix theory or the artificial intelligence (AI) algorithm.

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2.1 Deep learning algorithm for polarization detection

DLA is one of data-driven AI approaches and stands as the most advanced machine learning algorithm to date [33]. Leveraging its powerful data-processing capabilities, DLA is able to discern intricate correlations between the input and target spaces from a large amount of data and has been successfully used in imaging systems [34,35], target detection [36], speech recognition [37], natural language processing [38], nanophotonics [39], and computer vision [40]. In this article, we applied artificial neural networks (ANNs) to establish the mapping relationship $\widehat S = f(\widehat I)$. The measured physical quantity $\widehat I$ and the Stokes parameters $\widehat S$ were assigned as the input and output datasets, respectively. Three hidden layers were employed between them to learn progressively higher-level features according to the complexity of polarization detection. Each layer comprises numerous neurons, and each neuron connects to all neurons in the adjacent layer through specific weights. To train the ANNs, the reference data were divided into the training and validation datasets. The training process is to optimize the weight parameters using the training dataset, which is supervised by the loss function, also known as the objective function. Once the loss function converged stably, the ANNs model was well-trained and capable of accurately predicting unknown data. To enhance the training process, we incorporated various techniques, such as nonlinear activation functions, batch normalization and dropout. Detailed parameters of our model are provided in the subsequent section.

2.2 Mueller matrix theory for polarization detection

In general, the polarization materials can be described by a Mueller matrix $\widehat M$, and then the Stokes parameters $\widehat {S^{\prime}}$ of transmitted light can be calculated by

Because M₁₁, M₁₂, M₁₃ and M₁₄ in the Mueller matrix are determined by the known polarization-sensitive materials, the Stokes parameters $\widehat S$ of incident light can be easily solved by constructing four equations similar to Eq. (2) with four different polarization sensitive materials or one polarization sensitive materials of different states. Then, we have $\widehat {S_0^{\prime}} = \widehat {M^{\prime}}\cdot \widehat S$ and $\widehat S = {\widehat {M^{\prime}}^{ - 1}}\cdot \widehat {S_0^{\prime}}$, where

For the case that the detectors possess inherent polarization-sensitive properties, we also establish an equation similar to Eq. (2) to describe the relationship between the detected physical quantity $\widehat I$ and the incident Stokes parameters $\widehat S$. Then, we can employ four different detectors or one detector under four different rotation angles to realize the detection of the incident Stokes parameters.

To derive the coefficients M₁₁, M₁₂, M₁₃ and M₁₄, we can also solve Eq. (2) using the reference data measured by illuminating reference lights with known Stokes parameters. In general, four sets of reference data are enough. However, the inevitable detection errors, such as the noise of detectors, usually bring large errors to the measured reference data, leading to an inaccurate coefficients and the final detection results. To improve detection precision, abundant reference data should be employed to fit the Eq. (2), and the reference data are collected by the incidence of polarized lights generated by a HWP at a rotating angle of α and a QWP at a rotating angle of β (see Supplement 1, Figures S1-2 for details). Meanwhile, we incorporated the Eq. (4) to improve the fitting efficiency, i.e.,

3. Realization and characterization of polarization sensitive materials

Chiral shells have been selected as the polarization sensitive materials to perceive the SoP of light, owing to their strong optical chirality, anisotropy and fabrication advantages. The micro-sphere lithography technique is a material-independent, large area, high-efficient and low-cost fabrication method, and has been successfully extended to fabricate chiral shells in our previous work [41]. Notably, we have recently realized the large-area micro-sphere monolayer with a uniform lattice by an improved self-assembly technology, and developed chiral shell of one specific enantiomer in one sample in a large area [42]. These uniform chiral shells exhibit large optical chirality and transmittance, and greatly facilitate their potential in polarization detections and other applications.

The fabrication process of chiral shells involves the self-assembly of a monolayer of micro-spheres, and the deposition of silver onto the micro-sphere array (see Figs. S3-S4 for details). The deposition angle θ is set to 60 degrees and the azimuthal angle ϕ is set at 45 degrees (Fig. 1(a)), where the azimuthal angle endows the shells with chirality. The deposition thickness is 90 nm, which is defined as the thickness along the deposition direction. The diameter of polystyrene colloidal spheres (PS spheres) used here is 500 nm. The three-dimensional model is presented in Fig. 2(a), which illustrates both the top view and side view of the chiral geometry. This geometry is further confirmed by the SEM images shown in Figs. 2(b) and S4, demonstrating the successful fabrication of uniform chiral shells. The inset image showcases a sample with a large area of about 4 cm². To characterize the optical anisotropy, the linearly polarized light was illuminated on the sample, and the polarization direction was gradually rotated from 0 to 180 degrees, where the transmission intensity was measured by a spectrometer. As shown in Fig. 2(c), the maximum and minimum transmission intensity appear at around 160 degrees and 80 degrees, respectively. The optical anisotropy spectrum calculated by the formula L = T_max/T_min is shown in Fig. 2(d), showing a maximum linear anisotropy of approximately 1.7 at a wavelength of around 670 nm. The optical chirality is characterized by measuring transmittances of left- and right-circularly polarized lights (LCP and RCP) and calculating the circular dichroism (CD) by the formula CD = (T_RCP-T_LCP)/(T_RCP+ T_LCP). As shown in Fig. 2(e), the maximum CD value can reach approximately -0.21 at around 594 nm. We also investigate the CD spectra from different areas of samples, which are similar to each other and indicate the uniformity of samples (Supplement 1, Fig. S5). The significant optical anisotropy and chirality make chiral shells suitable for detecting both the linear and spin momentums of incident lights, enabling the realization of high-performance full Stokes polarimeters.

 

Fig. 2. Fabrication and characterization of metal chiral shells (MCSs). (a) Schematic diagram of one-step glancing angle deposition on the monolayer of microsphere. The deposition angle θ is set to 60°, defined as the angle between the vapor direction and the normal of substrate. The relative azimuthal angle φ is set to 45°, defined as the angle between the projection of the vapor and the crystal orientation of microsphere array. (b) Scanning Electron Micrograph (SEM) image of MCSs sample. The inset shows a photograph of large-area uniform MCSs. (c) The transmitted light intensity mapping as a function of the wavelength and the polarization rotation angle of the incident linearly-polarized light. (d) The optical anisotropy spectrum of the chiral shells extracted from (c). (e) The measured transmittance spectra of MCSs under RCP and LCP light incidence, respectively, along with the corresponding CD spectrum.

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4. Polarization detection with Mueller matrix theory and DLA

The polarization detection starts with the collection of a substantial amount of reference data for calibrating the matrix $\widehat {M^{\prime}}$ and training the DLA. The polarization detection system is shown in Fig. 3(a), where the parallel white light illuminates on the samples normally after transmitting through the linear polarizer, HWP and QWP sequentially, and the transmitted light through the samples is collected by a portable spectrometer. The reference SoP was generated by rotating the HWP and QWP at certain angles, and the transmittance $S_0^i$ was measured by rotating the sample to 0 degrees, 45 degrees, 90 degrees and 135 degrees. The rotation angle is defined as the angle between the vertical downward direction and the sample reference line in deposition process. Certainly, it should be noted that the sample can be rotated to any other angles to perceive the SoP of incident light. Figure 3(b) shows a typical transmitted intensity mapping image when the sample was rotated to 45 degrees (see Supplement 1, Fig. S6 for more data at different rotation angles), in which a periodicity of 90 degrees along the HWP axis and 180 degrees along the QWP axis is clearly exhibited. The observed periodicity indicates that the chiral shells work like a linear grating with a single optical axis.

 

Fig. 3. Polarization detection using the Mueller matrix theory and DLA. (a) Schematic diagram of the optical setup for detecting the SoP. (b) The transmitted intensity mapping image of chiral shells as a function of the HWP and QWP angles. The sample rotation angle is 45 degrees, and the wavelength is about 710 nm. (c) Schematic diagram of ANNs used in this work, which contains one input layer, three hidden layers and one output layer. (d) The true (black hollow circles) and calculated (green solid circles) values of S₁, S₂, and S₃ for 20 random polarization states using the Mueller matrix theory at 710 nm. (e) The true (black hollow circles) and predicted (red solid circles) values of S₁, S₂, and S₃ for 20 random polarization states through ANNs at 710 nm. For comparison purposes, the true values in figures (d) and (e) are the same. OFC is the optical fiber coupler. The yellow arrow on the right side of the sample indicates the clockwise rotation.

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The ANNs model used in this work consists of one input layer, three hidden layers, and one output layer, as depicted in Fig. 3(c). The input layer are the measured optical intensity, while the output layer corresponds to the incident Stokes parameters. The hidden layers, situated in the middle, form the core of the ANNs algorithm and comprise numerous neurons. The parameters for the ANNs model in this work were configured as follows: optimizer = Adamax, learning_rate = 0.1%, batch_size =32, loss function = mean squared error (MSE). In order to train the ANNs model, the total 414 datasets at each wavelength were divided into a training dataset (80%) and a verification dataset (20%). After successful completion of the training process (see the epoch curve in Supplement 1, Fig. S7), the verification results are shown in Fig. 3(e). By employing the reference data and the Eq. (4), we can also get the matrix $\widehat {M^{\prime}}$, and then calculate the verification results with $\widehat S = {\widehat {M^{\prime}}^{ - 1}}\cdot \widehat {I^{\prime}}$, which are displayed in Fig. 3(d). Notably, both the Mueller matrix theory and DLA can yield accurate predictions, indicating the successful realization of polarization detection (see Figures S8-9 for more results at 520 nm and 650 nm).

However, the ANNs exhibit higher detection precision in comparison to the Mueller matrix theory. The evaluated MSEs for S₁, S₂, and S₃ using ANNs are 0.10%, 0.41%, and 0.24% at a wavelength of 710 nm, respectively, while the calculated MSEs using the Mueller matrix theory are 0.14%, 0.46%, and 0.48%, respectively. To comprehensively reveal the prediction difference between the DLA and the Mueller matrix theory, we plot the real and predicted Stokes parameter image as a function of the HWP and QWP angles, as shown in Fig. 4. It is obvious that the DLA can give better prediction results, especially for the Stokes parameter S₃.

 

Fig. 4. The Stokes parameters S₁, S₂ and S₃ as a function of the HWP and QWP angles at 710 nm. (a)-(c) The real values of Stokes parameters. (d)-(f) The predicted Stokes parameters by ANNs. (g)-(i) The calculated Stokes parameters by the Mueller matrix theory.

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It is worth noting that the small MSEs achieved by the Mueller matrix theory are attributed to the employment of a fitting procedure with abundant reference data to derive the matrix $\widehat {M^{\prime}}$. This approach can effectively eliminate the system errors, such as those caused by detector noise. Otherwise, large MSEs will arise if the pre-designed matrix $\widehat {M^{\prime}}$ or a limited number of reference datasets are used to solve the theoretical Eq. (4). In this case, the Mueller matrix theory also requires abundant reference data to improve detection precision, which is similar to that required in the DLA. Thus, the collection of abundant reference data is imperative for high detection precision.

To further compare the polarization detection precision between the Mueller matrix theory and DLA, we have conducted investigations on combination modes at different wavelengths (Fig. 5). The combination modes refer to the number and types of chiral shells that are used in polarization detection. Since the incident light is completely polarized, it is possible to determine the Stokes parameters using only three equations (i.e., i in Eq. (4) is 1, 2, and 3). This simplification reduces the four unknown quantities in Stokes parameters to three. In this case, we selected three or four rotation angles of chiral shells to detect the SoP, denoted as combination modes A, B, C and D. These modes correspond to the chiral shells with rotation angle groups of {0° 45° 90°}, {0° 90° 135°}, {45° 90° 135°}, and {0° 45° 90° 135°}, respectively. As shown in Fig. 5, the DLA consistently outperforms the Mueller matrix theory, as indicated by smaller MSE values across all investigated wavelengths and combination modes, which is consistent with the results shown in Figs. 3 and 4. Specifically, at a wavelength of about 520 nm, the chiral shells exhibit small optical anisotropy and weak transmitted light intensity, leading to large MSEs for both the Mueller matrix theory and DLA. However, the MSEs obtained from the DLA are approximately 10 times smaller than those from the Mueller matrix theory. In addition, we can see that different combination modes yield different MSE values, and increasing the number of elements in the transmittance matrix $\widehat I$ generally decrease the MSE values in the DLA. This implies that the DLA can learn more features with additional input datasets. This attribute is advantageous for improving the detection precision with a detector array, such as CCD, where each pixel may detect signals from one polarization-sensitive material. While for the Mueller matrix theory, increasing the number of input datasets may increase the MSE values and decrease the detection precision, indicating that employing more polarization-sensitive materials will not improve the detection precision with the Mueller matrix theory. More comparisons are shown in the Supplement 1 (Figure S10).

 

Fig. 5. Influence of combination modes at different wavelengths. (a)-(c) The MSEs of S₁, S₂, and S₃ calculated by the Mueller matrix theory. (d)-(f) The MSEs of S₁, S₂, and S₃ calculated by ANNs. Each column has the identical wavelength of 520 nm, 650 nm, and 710nm, respectively. The combination modes A, B, C and D correspond to the chiral shells with rotation angle groups of {0° 45° 90°}, {0° 90° 135°}, {45° 90° 135°}, and {0° 45° 90° 135°}, respectively.

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Certainly, the precision of polarization detection is influenced by many factors, such as optical chirality, anisotropy and the transmitted light intensity. In Fig. 6, we present the MSE spectra of the Stokes parameters calculated by the Mueller matrix theory and the DLA. It is obvious that both methods can realize broadband polarimetry in the wavelength range of about 450 nm to 750 nm. However, the Mueller matrix theory shows large detection fluctuations, especially for the Stokes parameter S₃, where the maximum MSEs vary from an value of 0.48% to an unacceptably high value (cannot be depicted in Fig. 6(a)). Thus, the averaged MSEs over the waveband of 500 nm to 750 nm are 0.45% (S₁), 1% (S₂), and 39.8% (S₃) for Mueller matrix theory. These values decrease to 0.16% (S₁), 0.46% (S₂), and 0.61% (S₃) for the DLA employed in this work, indicating the superior robustness of the DLA method. Upon comparing these results with the optical properties of chiral shells presented in Fig. 2, it becomes evident that small optical chirality typically yields large MSE values for the Mueller matrix theory. In contrast, the DLA does not exhibit large MSE values under conditions of low optical chirality, indicating that it can effectively operate under such conditions. The large transmitted light intensity also is benefit for improving the polarization detection precision. As shown in Fig. 2(c), the light intensity at 400 nm is extremely weak, resulting in large MSE values for S₁ and S₂, despite the optical anisotropy at this wavelength not being the smallest. The large detection precision at around 600 nm should arise from the light source, which appears a sharp peak at this wavelength.

 

Fig. 6. The MSE spectra of S₁, S₂ and S₃ for the combination mode D. (a) The MSE spectra by the Mueller matrix theory. (b) The MSE spectra by the DLA.

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Therefore, the polarization detection precision depends on the optical properties of chiral shells and the incident light intensity for both the Mueller matrix theory and DLA. Fortunately, the DLA can still work well under conditions of low light intensity and small optical chirality and anisotropy.

5. Discussion

As discussed above, the Mueller matrix theory and the DLA approach exhibit different detection precision for Stokes parameters under the same conditions. This disparity can primarily be attributed to their different working principles. For the Mueller matrix theory, the incident and transmitted waves are assumed to ideal plane waves, and the material is also simplified to be a uniform chiral medium with a transmitted matrix $\widehat M$. These ideal assumptions are obviously different with the real cases in experiments, where the incident and transmitted white lights are approximately parallel, and the materials are also non-uniform. In particular, various factors, such as the fluctuation of light sources and photodetectors, the rotational errors of motors, as well as thermal fluctuations and vibrations from the environment, can significantly decrease the detection precision of the system. In contrast, the MLA approach is an algorithm that can learn all of the features between the input and output datasets, and then construct a mapping relationship between the incident wave $\widehat S$ and the transmitted wave $\widehat {S^{\prime}}$ with complex ML codes. With this approach, all potential influencing factors can be considered, resulting in a high detection precision. The results presented above demonstrate that the MLA approach consistently provides highly precise and stable detection, particularly within wavebands characterized by weak optical chirality, weak optical anisotropy, and weak light sources.

6. Conclusion

In conclusion, we have successfully achieved a high-precision, stable, and broadband polarimetry based on high-performance chiral shells. The large-area chiral shells with high optical anisotropy and chirality were fabricated by depositing silver on a large-area PS sphere monolayer of uniform lattice, and then rotated multiple times to perceive the SoP of the incident light. The mapping relationship $\widehat S = f(\widehat I)$ was established using both the Mueller matrix theory and DLA. We derived a general theoretical equation by the Mueller matrix theory, and fitted this theoretical equation with abundant reference data, which were also used in the training process of DLA. Our results indicate that both the Mueller matrix theory and DLA can achieve high detection precision, with the detected MSEs being less than 1% at certain wavelengths. However, the DLA exhibits more advantages under adverse conditions, such as weak light intensity, low optical chirality and anisotropy, leading to a broadband polarimetry across a wavelength range of about 500 nm to 750 nm, where the averaged MSE for S₃ is about 26 times smaller than that obtained with the Mueller matrix theory. In addition, the detection precision of DLA can be further improved by employing more polarization materials to perceive the polarization features of incident light. Our work not only demonstrates a new paradigm in high-performance polarimetry but also highlights the benefits of the established mapping relationship using the Mueller matrix theory and DLA. The developed polarimetry may inspire new ideas in realizing highly precise polarimetry and polarization imaging devices, and facilitate their applications in various fields, such as remote sensing and industrial vision.