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Abstract
Channeled spectropolarimetry is a snapshot technique for measuring the spectra of Stokes parameters of light by demodulating the measured spectrum. As an indispensable part of the channeled spectropolarimeter, the spectrometer module is far from being perfect to reflect the real modulation spectrum, which further reduces the polarimetric reconstruction accuracy of the channeled spectropolarimeter. Since the modulation spectrum is composed of many continuous narrow-band spectra with high frequency, it is a challenging work to reconstruct it effectively by existing methods. To alleviate this issue, a convolutional neural network (CNN)-based spectral reconstruction solver is proposed for channeled spectropolarimeter. The key idea of the proposed method is to first preprocess the measured spectra using existing traditional methods, so that the preprocessed spectra contain more spectral features of the real spectra, and then these spectral features are employed to train a CNN to learn a map from the preprocessed spectra to the real spectra, so as to further improve the reconstruction quality of the preprocessed spectra. A series of simulation experiments and real experiments were carried out to verify the effect of the proposed method. In simulation experiments, we investigated the spectral reconstruction accuracy and robustness of the proposed method on three synthetic datasets and evaluate the effect of the proposed method on the demodulation results obtained by the Fourier reconstruction method. In real experiments, system matrices are constructed by using measured spectra and reconstructed spectra respectively, and the spectra of Stokes parameters of incident light are estimated by the linear operator method. Several other advanced demodulation methods are also used to demodulate the measured spectrum in both simulation and real experiments. The results show that compared with other methods, the accuracy of the demodulation results can be much more improved by employing the CNN-based solver to reconstruct the measured spectrum.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Spectropolarimetry could quantify the polarization state and spectral content of light [1]. It has been widely applied in many fields such as remote sensing [2–4], material characterization [5–7], biomedical diagnosis [8–10], and environmental monitoring [11–13]. In the past few decades, a variety of methods for polarization and spectral measurements have been developed [14–17]. Among these methods, channeled spectropolarimetry, which was almost simultaneously proposed by Oka et al. [18] and Iannarilli et al. in 1999 [19], is a more remarkable and attractive approach. It could simultaneously measure the spectra of the four Stokes parameters in the wavenumber domain. This approach is notable for its snapshot nature, and the polarimetric modulation elements commonly used to implement channeled spectropolarimetry comprise an independent module called polarimetric spectral intensity modulation (PSIM) module, which is composed of mechanically fixed thick retarders and linear polarizers and can be easily incorporated into various spectrometers and imaging spectrometers.
To make the channeled spectropolarimetry better development and application, many scholars have conducted numerous researches [20–22]. Mu et al. [20], Yang et al. [21] and Ju et al. [22] have calibrated and compensated alignment errors by reference beam or additional components in laboratory environment. Specifically, Mu et al. theoretically analyzed the alignment and delay errors of the channel spectropolarimeter and used two linearly polarized reference beams oriented at 22.5° and 45° to calculate the orientation errors of two retarders; Yang et al. proposed a calibration method with an auxiliary high-order retarder and a reference beam to determine the alignment errors, and developed a correction algorithm without any precise mechanical adjustments to compensate for them; Ju et al. proposed a method to reduce the effects of the alignment errors of high-order retarders using the amplitude items in the results of reference beam calibration technique. To reduce the effects of temperature variation on retardations in a channeled spectropolarimeter, researchers have proposed several methods. For instance, Snik et al. [23] and Craven-Jones et al. [24] proposed two similar methods using two uniaxial crystals or biaxial retarders, respectively, to produce thermally stable retarders; Taniguchi and Oka [25] proposed a self-calibration method that can reduce the effects of temperature change; Chrysler et al. shew how to adaptively estimate and correct the phase ambiguity using a polynomial curve-fitting algorithm, extending the temperature range to virtually all practical scenarios. Moreover, Locke et al. [26] used a 3:1 ratio of the retarder thickness to check the system alignments. Xing et al. [27] proposed a self-correction method to calibrate and compensate the alignment errors and retardations of the retarders simultaneously by measuring the target light in orbit.
However, the existing research works on channeled spectropolarimetry mainly focused on the PSIM module. As another important part of the channeled spectropolarimeter, the spectrometer module also has an important influence on the overall performance of it, but it is rarely paid attention to. In fact, the linear error factors [28–37] in the spectrometer, mainly including spectral broadening [29–31], the nonlinearity response of CCD [32,33], stray light [34,35], and noise [36,37], will affect the measured spectrum to varying degrees, causing it to deviate from the real spectrum and further affect the reconstruction of Stokes parameters. At present, there are three kinds of classical methods for obtaining the spectra of Stokes parameters from measured spectra, namely the Fourier reconstruction method [20], optimization problem-based method [38], and linear operator method [39]. The Fourier reconstruction method obtains the spectra of Stokes parameters by taking the Fourier transform of the wavenumber spectrum, applying a set of rectangular window functions to separate the autocorrelation function in the optical path difference (OPD) domain, and taking the inverse Fourier transform. The optimization problem-based method transforms the demodulation problem into an optimization problem by creating the mathematical model of channel spectropolarimeter, and solves the objective function to obtain the spectra of Stokes parameters. From the detailed description of these two methods, it can be concluded that the measurement error of the spectrum will directly affect the accuracy of the solution of the polarization state. Different from the previous two kinds of methods, the linear operator method models channel spectropolarimeter as a linear operator, represented in practice by a matrix, which is estimated in the calibration, and pseudo inverted subject to a constraint on object space for reconstructions. Since the system matrix already contains the characterization of the spectrometer, Sabatke et al. [39] believed there is unnecessary to correct the measured spectrum. However, the system matrix contains the information of PSIM module and spectrometer module simultaneously, which makes it necessary to input a large number of high-precision data in the process of calibration to ensure the accuracy of it. In view of this defect, if the measured spectrum can be calibrated to make its error within the allowable range, the system matrix can be simplified to include only the information of PSIM module. Accordingly, the amount of data to be input in the calibration process can be reduced, or the accuracy of the spectra of Stokes parameters can be improved under the same amount of data. In summary, no matter which demodulation method is employed, the effective reconstruction of the measured spectrum of channeled spectropolarimeter is a matter with great significance.
Nonetheless, the modulation spectrum is often composed of many continuous narrow -band spectra. According to the previous experience of spectral reconstruction [30,31], compared with broad-band spectra such as light-emitting diode (LED), it is a challenging work to effectively reconstruct narrow-band spectra due to the fact that undercorrection and oscillations occur normally. Recent advances in deep learning (DL) have been successfully applied to the fields of image classification [40] and natural language processing [41,42], which open up new opportunities for designing effective and robust algorithms for DL-based reconstruction. For example, in [43], a deep neural network (DNN) is trained to predict non-uniformly sampled nuclear magnetic resonance (NMR) spectra. Later on, a peak picking and spectral deconvolution approach based on DNN [44] is introduced, which can semi automatically analyze two-dimensional NMR spectra. In [45] and [46], a DL-based adaptive filtering is proposed for channeled spectropolarimeter to improve the quality of demodulation results. Lv et al. trained neural networks (NNs) [47] to calibrate the channeled imaging spectropolarimeter. The established NNs can effectively learn the forward conversion procedure through minimizing a loss function, subsequently enabling a stable output containing spectral, polarization, and spatial information. In addition, the application of combining the neural network (NN) and existing algorithms greatly inspires us. Kanarachos et al. [48] propose a detection algorithm combining wavelets, NN and Hilbert transform for time series data. To handle the problem of classifying imbalanced data, Jeatrakul et al. [49] propose a method combining synthetic minority over-sampling technique (SMOTE) and Complementary Neural Network (CMTNN). In [50], a method combining a NN with the genetic algorithm and particle swarm optimization for permeability estimation of the reservoir is proposed. Zhang et al. [51] propose a method based on regularized non-negative least squares and NN for spectrum reconstruction of colloidal quantum dot spectrometer.
Motivated by the above research findings, a novel reconstruction method by combining existing spectral reconstruction methods and the convolutional neural network (CNN) [52–54] is proposed for the channeled spectropolarimeter. Specifically, a set of existing spectral preprocessing methods are used to extract spectral features from the measured spectra, and then the spectral features are employed to train a CNN to learn a map from the preprocessed spectra to the real spectra. We refer to this reconstruction method as the CNN-based spectrum reconstruction solver. The key difference and advantage between the proposed method and general reconstruction methods is that it establishes a map from preprocessed spectra to real spectra instead of merely deconvoluting, making the preliminary correction closer to the real spectrum. Compared with the direct employment of CNN, the proposed reconstruction method first uses the traditional reconstruction methods to make the CNN input data closer to the real spectrum, so as to enhance the capability of the trained CNN to make better predictions. We demonstrated the feasibility and performance of the proposed method on simulation data and real experimental data collected by a channeled spectropolarimeter.
The remainder of this work is organized as follows. Section 2 provides an overview of the principle of channeled spectropolarimetry and the measurement error factors of the spectrometer. In Section 3, we discuss the proposed CNN-based spectral reconstruction solver in detail. We then illustrate the effectiveness and performance of the proposed methods via several simulations and experiments in Section 4. Section 5 concludes this paper.
2. Background
2.1 Principle of channeled spectropolarimetry
Channeled spectropolarimetry is a technique that converts a spectrometer into a spectropolarimeter by incorporating PSIM module into various spectrometers and imaging spectrometers. Based on this theory, a channeled imaging spectropolarimeter for airborne remote sensing was developed, and its optical system is shown in Fig. 1.
The whole system consists of two parts: the front telescope and the imaging spectrometer. The front telescope includes the PSIM module to realize the polarization function of the imaging target, and the spectrometer adopts Offner structure, which consists of two spherical mirrors and a convex grating to satisfy Rowland circle to correct coma aberration. Incident light from the ground enters the PSIM module after the expansion of the fore optical system, and then focuses on a slit through the imaging system. Thereafter, the light is reflected to a convex diffraction grating by a spherical mirror, diffracted into different wavelengths. The emergent light from the convex diffraction grating incidents to another spherical mirror, and is focused on the detector.
The schematic of the PSIM module is shown in Fig. 2. It consists of two high-order retarders R1 and R2 with thicknesses d1 and d2, and a linear polarizer P to encode the polarization state onto spectral carriers. The fast axes of R1 and R2 are 45° apart, while the transmission axis of P is aligned with the fast axis of R1. A rectangular coordinate system is established with the directions parallel and perpendicular to the transmission axis of the linear polarizer P as the x-axis and the y-axis, respectively. From the Mueller matrix of each optical device, the Mueller matrix of the PSIM module under ideal conditions can be obtained:
The CCD can only respond to the total radiation intensity ${S_0}$ in the Stokes vector, and the intensity of the illumination onto the CCD can be expressed as
2.2 Effects of the error factors of the spectrometer on the measured spectrum
The demodulation process in Section 2.1 is only applicable to ideal conditions. The non-ideality of the PSIM module and spectrometer module will introduce errors to the demodulation results. The non-ideality of the PSIM module mainly depends on the alignment errors of the high-order retarders and linear polarizer and the calibration errors of phase factors, which have been deeply studied. This paper mainly explores the non-ideality of the spectrometer. The spectrum measured by the spectrometer is not the real spectrum of incident light, but the result of the interaction between the real spectrum and the error factors of the spectrometer, which mainly include spectral broadening, stray light, nonlinearity response of CCD and noise. The relationship between the measured spectrum and real spectrum can be expressed as
where $M(\lambda )$ is the measured spectrum, $R(\lambda )$ is the real spectrum, $LSF(\lambda )$ is the spectral line spread function (LSF) [55], and ${\otimes}$ indicates the convolution operation. $LSF(\lambda )$ shows the relative response of the instrument caused by excitation with monochromatic laser radiation, and includes the effects of spectral broadening and stray light. $Nl(.)$ is the nonlinearity response function, and reflects the nonlinear relationship between the photon signal incident on the CCD and the output digital signal. $N(\lambda )$ represents measurement noise, which can be classified into 3 types of Gaussian white noise: readout noise, dark noise and photoelectron noise.To more intuitively reflect the influence of the error factors on the measured spectrum and demodulation results, the process of modulation and demodulation is simulated by the Matlab software. It is assumed that the working band of the channeled spectropolarimeter is 400 ∼ 800 nm, the corresponding wavenumber range is 1.25 ∼ 2.50 cm−1, and the number of sampling points is 2000. The two high-order retarders in PSIM module are quartz crystals with a birefringence index of 0.0092, and their physical thicknesses are 0.58 cm and 1.16 cm respectively. The spectral resolution of the spectrometer module is 1 nm, the stray light level is 5×10−4, the nonlinearity response of the CCD is 0.99, and the signal-to-noise ratio (SNR) of the measured spectrum is 50. The Stokes parameters of incident light are set as constants in the whole wavenumber range, i. e. S0 = 1, S1 = 0.5, S2 = 0.7 and S3 = 0.1, for the convenience of calculation and comparison. The real spectrum and the measured spectrum are shown in Fig. 3. As can be seen from the figure, due to the influence of error factors, the measured spectrum is deformed compared with the real spectrum. Moreover, due to the higher frequency, the distortion of the spectrum corresponding to the short wave is larger than that corresponding to the long wave.
Fig. 3. Real spectrum and measured spectrum of a channeled spectropolarimeter obtained by simulation.
Then, the real spectrum and the measured spectrum are spline interpolated to meet the uniform sampling in the wavenumber domain, and the inverse Fourier transform is performed to obtain the autocorrelation functions in OPD domain, as shown in Fig. 4. The measurement errors are reflected in the autocorrelation function as high-frequency signals, as well as the peak value of each channel is much lower than the real value.
Finally, the demodulation results obtained by Eq. (6) and the real values are shown in Fig. 5. As can be seen from the figure, except that S0 deviates slightly from the real value, the demodulation results of S1, S2 and S3 are greatly different from the real values, and the errors increase with the increase of wavenumber, which corresponds to the fact that the measured spectrum at the short wave in Fig. 3 is more different from the real spectrum.
3. Spectrum reconstruction method
As analyzed in Section 2, the error factors of the spectrometer module of the channeled spectropolarimeter cause the measured spectrum to deviate from the real spectrum, resulting in deviations between the demodulated spectra of Stokes parameters and the real spectra. Therefore, reconstructing the measured spectrum to eliminate measurement errors as much as possible is of great significance to improve the polarization measurement accuracy of the channeled spectropolarimeter.
3.1 Traditional reconstruction methods
A series of traditional reconstruction methods are tried to reconstruct the real spectrum. According to the mathematical model of the measured spectrum, to eliminate the influence of each error factor, the spectral reconstruction needs to carry out nonlinear correction, bandwidth correction, stray light correction and noise reduction in sequence.
For nonlinear correction, a simple and effective method is adopted: maintaining the stability of the light source, gradually and evenly increasing the integration time, and measuring the response of multiple pixels. The relative response is obtained by dividing the response value by the integration time. Taking the response value as x and the normalized relative response as y, the nonlinear correction function is obtained by seventh-order polynomial fitting. The nonlinear corrected spectrum can be obtained by Eq. (8).
In Eq. (7), LSF is employed to represent the comprehensive effects of spectral broadening and stray light, and the simultaneous correction of them can be completed by the deconvolution method where LSF acts as the convolution kernel. In practice, it is often possible to shorten a tedious tuneable laser-based characterization by interpolation of the LSFs determined at a limited number of laser wavelengths. It is known that deconvolution is an ill-posed inverse problem, some strategies with regularization are often used to relieve it in spectral deconvolution, and the noise can be suppressed effectively. Therefore, a cost function with regularization term can be established, as shown in Eq. (9).
The method employs the Euler–Lagrange with Neumann boundary conditions to perform the minimization of the cost function with respect to R, and the latent spectrum is solved by employing a successive approximations iteration
3.2 CNN-based solver
Thanks to the use of traditional reconstruction methods, the demodulation quality of the spectra of Stokes parameters have been significantly improved. Nevertheless, these methods still often result in undercorrection and oscillation in the reconstructed spectrum because the modulation spectrum is often composed of many continuous narrow-band spectra with high frequency.
To overcome these limitations, we introduce a novel reconstruction method based on the CNN framework to improve the spectrum reconstruction performance of the channeled spectropolarimeter. The key idea of the proposed method is to first preprocess the measured spectra using these traditional methods, so that the preprocessed spectra contain more spectral features of the real spectra, and then these spectral features are employed to train a CNN to learn a map from the preprocessed spectra to the real spectra, so as to further improve the reconstruction quality of the preprocessed spectra. Figure 6 shows the schematic of the spectral reconstruction process using the proposed CNN-based spectrum reconstruction solver. This system consists of three parts: CNN training process, CNN-based solver predicting process and CNN architecture.
Fig. 6. Overview of spectral reconstruction process using the proposed CNN-based spectrum reconstruction solver. (a) Schematic diagram of the training stage (b) spectral reconstruction process by CNN-based solver (c) CNN architecture.
In the CNN training process, firstly, the real spectral dataset is constructed by using the simulated and real experimental data combined with Eq. (3). Then the measured spectral dataset can be obtained by degrading the spectra of the real spectral dataset according to Eq. (7). Finally, the traditional methods are used to preprocess the measured spectral dataset as the CNN input dataset. The generation details of training data will be described in Section 4. The CNN output data are compared with the real spectra, and the parameters of the CNN are adjusted to minimize a loss function L, so as to complete the training of the CNN. The mean squared error between the CNN output data (reconstructed spectra) S and the real spectra R as the loss function:
In the CNN-based solver predicting process, the measured spectra are preprocessed by traditional reconstruction methods such as nonlinear correction and deconvolution, and then the preprocessed data are input into the trained CNN to obtain the reconstructed spectra. We refer to the reconstruction method which carries out nonlinear correction, deconvolution and CNN reconstruction of the measured spectrum successively as the CNN-based spectrum reconstruction solver. The key advantage of the proposed reconstruction method is that it establishes a map from the preprocessed spectrum to real spectrum instead of deconvoluting the nonlinear corrected measured spectrum merely, which leads the result of preliminary correction closer to the real spectrum. Compared with the direct employment of the CNN, the proposed reconstruction method first uses the traditional reconstruction methods to make the input data of the CNN closer to the real spectrum, so as to enhance the capability of the trained CNN to make better predictions.
The CNN architecture includes eight learnable layers, five of which are convolution layers, three are fully-connected layers. The convolution layers are used for feature extraction in the non-linear mapping between the CNN input data and real spectra. The fully-connected layers are used for spectrum reconstruction. Each of the convolution layers has a set of one-dimensional learnable kernels with specific window sizes. The number of kernels and the window sizes have been indicated in Fig. 6(c). After every convolutional layer, a rectified linear unit (ReLU) is used as an activation function, followed by a subsampling. Non-overlapping max-pooling is employed to down-sample the activation function. The convolutional layer, ReLU, and subsampling are stacked five times. The output of the last subsampling is flattened and then fed into the subsequent three fully-connected layers. The first two layers are followed by the ReLU and dropout, which is introduced to reduce the overfitting. The output of the last fully-connected layer is fed into a linear activation function. The number of units in each of the fully-connected layers is noted in Fig. 6(c).
4. Experiments and discussion
To demonstrate the performance of the proposed method, a series of experiments on simulated and real measured spectra were carried out. The experiments on simulated spectra are used to evaluate the effect of the proposed method on the demodulation results obtained by Fourier reconstruction method and the superiority of the proposed method over other algorithms, while the experiments on real spectra are employed to prove that the proposed method can improve the accuracy of the demodulation results obtained by the linear operator method.
4.1 Data preparation and training process
We generated three synthetic spectral datasets. All of them contain 1800 spectra and are simulated by the modulation model (Eq. (3)), but the generation approaches of the spectra of parameter S0 are different. The spectra of parameter S0 of the first synthetic dataset were set as constants in the whole wavenumber range, i. e. S0 = 1. The spectra of parameter S0 of the second synthetic dataset were generated by using the glossy Munsell color spectral dataset [57] and the spectrum of a white LED light source, while the spectra of parameter S0 of the third synthetic dataset were collected by a commercial spectrometer (Ocean Optics QE65 Pro) from a white LED light source filtered by different color plastics and their combinations and the collected spectra needed to be denoised and smoothed. The spectra of parameters S1 ∼ S3 of these three datasets were calculated by S0 combined with three randomly generated parameters: degree of polarization (DOP), agree of polarization (AOP) and ellipticity angle. Then, the real spectral datasets were constructed by using the data of S0 ∼ S3 combined with Eq. (3), and the measured spectra could be obtained by degrading the spectra of the real spectra according to Eq. (7). Next, the traditional methods were used to preprocess the measured spectra as the CNN input datasets. Finally, each spectrum was normalized such that its values vary from 0 to 1.
In each spectral dataset, the number of training, validation, and test spectra were randomly assigned using a ratio of 4: 1: 1 for the real spectral datasets and CNN input datasets, respectively. The validation spectra were used for estimating the number of epochs and tuning the hyper-parameters. The Adam optimizer implemented in Tensorflow 2.0 with the batch size of 32 was used to train the CNN. All computations were conducted on an NVIDIA GeForce GTX 1080Ti graphics processing unit (GPU).
4.2 Simulated experiments
To demonstrate the ability of the CNN-based solver to reconstruct spectra, we conducted the spectral demodulation on the three synthetic datasets described in Section 4.1 by using the traditional method (HMSBD & nonlinear correction), adaptive filtering method, NNs and CNN-based solver. The traditional method, adaptive filtering method and CNN-based solver are used to reconstruct the measured spectra and then the Fourier reconstruction method is employed to obtain the spectra of Stokes parameters. Different from them, NNs directly establishes the map between the measured spectrum and demodulation result. Three parameters are used to evaluate the quality of the reconstruction, the average root mean squared error (RMSE) of the reconstructed spectra, the standard variance of RMSE of the reconstructed spectra, and the average error of DOP. As shown in Table. 1, for all three synthetic test sets, the CNN-based solver outperforms the traditional method, the adaptive filtering method [45] and the NNs [47] in terms of these three parameters, which shows excellent spectral reconstruction effect and good robustness, and the accuracy of the demodulation results of the spectra of Stokes parameters is greatly improved.
Table 1. Average RMSE, standard variance of RMSE and average error of DOP over the synthetic test sets based on different reconstruction methods.
To further demonstrate the superiority of the proposed method compared with the other three methods, the simulated measured spectrum in Section 2.2 is reconstructed by these four methods. Figure 7 shows the reconstruction errors of the traditional method, the adaptive filtering method and the CNN-based solver. As can be seen from Fig. 7, both the traditional method and the adaptive filtering method have good reconstruction effects in the long-wave spectral band. However, for the short-wave spectral band, it is challenging for both the two methods to achieve good reconstruction effects, since the frequency of the measured spectrum in the short-wave band is higher. In contrast, CNN-based solver exhibits extremely superior reconstruction effects on the whole spectral band, only with little residual error near the 400 nm wavelength. Figure 8 shows the demodulation results of the spectra of Stokes parameters from the reconstructed spectra. Compared with the result in Fig. 5, the demodulation results of the reconstructed spectra by the four methods are improved. It can be observed that with the increase of wavenumber, the demodulation performances of the three methods other than NNs are also declining. This is because the larger the wavenumber, the higher the frequency of the corresponding modulation spectrum, and the lower the reconstruction performances of the methods, like shown in Fig. 7. Nevertheless, compared with the other three methods, the demodulation results by the CNN-based solver are most consistent with the real values.
Fig. 7. Reconstruction errors of all the three methods for processing the simulated measured spectrum in Section 2.2. (a) traditional method; (b) adaptive filtering method; (c) CNN-based solver.
As a reference, the demodulation results of the spectra in the second synthetic test set and third synthetic test set are shown in Fig. 9 and Fig. 10. Similarly, the demodulation results by the CNN-based solver are most consistent with the real values.
4.3 Real measured data experiments
The verification experiment was carried out in the laboratory to verify the effect of the proposed method on airborne remote sensing polarization detection. Since the channeled imaging spectropolarimeter is being installed and adjusted, we have built another channeled spectropolarimeter system for the experiment. The test system is shown in Fig. 11, which consists of a polarized light generator, a self-built channeled spectropolarimeter and a computer. Since the circular polarization component in the reflection of the sunlight by ground objects is very small, the Stokes parameter S3 was not considered in the experiment. The polarized light generator comprises a polarizer and a polarizing system. Due to the large volume of the polarizing system, it is not shown in Fig. 11. During the experiment, it was in the position of the polarizer in Fig. 11. The polarizer was used to generate fully polarized light. The polarizing system consists of four glass stacks, which can produce partially polarized light with a known degree of linear polarization (DoLP) and an angle of linear polarization (AoLP) by changing the relative angle of the glass stacks and the rotation angle of the motor, respectively. The glass stacks, made of K9, have an adjustable range of relative angle from 0 to 65 degrees with an uncertainty of ${\pm} 0.01$ degree. The absolute accuracy of the DoLP of the polarizing system is better than ${\pm} 0.002$ by modelling the response of the glass stacks, which takes into account the accuracy of the relative angle and refractive index of the glass stacks. The accuracy of the AoLP of the polarizing system is better than ${\pm} 0.01$ degree, which depends on the accuracy of the rotation angle of the motor. The channeled spectropolarimeter includes a PSIM module and a spectrometer for measuring the modulation spectrum. The spectrometer is a commercial spectrometer (Ocean Optics QE65 Pro), with a spectral resolution of 1 nm and a detection range of 200 ∼ 1100 nm.
The linear operator method was used to calibrate the channeled spectropolarimeter. Since the workload of calibrating the whole band is too large, and our purpose is only to verify the effectiveness of the proposed method, we only calibrated the 420 ∼ 500 nm band of the system (the corresponding wavenumber range is 2.00 ∼ 2.38 cm−1), which is the spectral band of a blue LED. Calibration states were prepared by feeding light of narrow spectral content bandwidth from a monochromator through the polarizer and polarizing system. Their polarization and spectral content were controlled via the polarizer and polarizing system and the monochromator’s center wavelength setting, respectively. Measurements were made for 66 different wavelengths at each polarization state, and 396 incident light beams with different polarization states were generated. With 514 sampling points in the spectrometer in this experiment, the calibration comprises more than 200000 individual measurements. The measured spectra were reconstructed by the adaptive filtering method and CNN-based solver, and then the reconstructed spectra and the measured spectra were used to construct the output data matrix G respectively. The standard spectra of S0 of these incident light beams were measured by a spectrometer which has been absolute radiometric calibrated, and the standard spectra of S1 and S2 were calculated. All of them were used to construct the input data matrix Q, and the system matrix H is obtained by
where a superscripted + represents the Moore-Penrose pseudoinverse [58]. The spectra of Stokes parameters of incident light $\hat{s}$ can be obtained by where g is the measured spectrum. To verify the improvement of the calibration effect of the linear operator method under the condition of a limited number of inputs, system matrices were constructed by using the information of 198 (the polarization states of incident light beams were reduced by half) and 396 incident light beams respectively, and the spectra of the Stokes parameters of a blue LED were predicted. For example, when the DoLP and AoLP of the incident light are set to 0.32 and 14°, Fig. 12 and Fig. 13 show the estimated results of the spectra of the polarization state of an incident light beam by the system matrix constructed from the information of the 198 and 396 incident light beams, respectively. For comparison, NNs was also employed to estimate the demodulation results, and the results are shown in Fig. 13. To compare the demodulation effects of various algorithms more intuitively, the ratios of S1 to S0 and S2 to S0 are given in Fig. 12 and Fig. 13. These values are not sensitive to wavenumber changes, so they look like constants, but they are not.Fig. 12. Estimated results of the spectra of polarization state of an incident light by the system matrix constructed from 198 incident lights information.
Fig. 13. Estimated results of the spectra of polarization state of an incident light by the system matrix constructed from 396 incident lights information and NNs.
As can be seen from Fig. 12, the estimated results of the spectra of the polarization state of the incident light by the system matrixes constructed by the reconstructed spectra is closer to the reference value, compared with the system matrix constructed by the measured spectra. When the number of incident light beams increases to 396, it can be seen from Fig. 13 that better results can be obtained by the system matrix constructed by the measured spectra, and the accuracy is comparable to the system matrix constructed by the 198 reconstructed spectra, but significantly weaker than the system matrix constructed by the 396 reconstructed spectra. The demodulation results obtained by NNs are close to those obtained by the linear operator method which employ a system matrix constructed by 396 incident light beams. There is no doubt that compared with other methods, the results obtained by the CNN-based solver combined with the linear operator method are closer to the reference values. The demodulation results of the other generated incident light are similar. It means that reconstructing the measured spectrum by the CNN-based solver can effectively reduce the amount of data to be input in the calibration process, or improve the accuracy of the Stokes parameter spectrum with the same amount of data.
From the above simulations and experiments, it can be found that the measurement error of spectrometer module has a great impact on the polarimetric reconstruction accuracy of channeled spectropolarimeter. The traditional reconstruction methods cannot effectively reconstruct the measured spectrum of channeled spectropolarimeter, which leads to the deviation between the demodulation results and the real value. Different from traditional methods, CNN-based solver establishes a map from the preprocessed spectra to the real spectra, which can further improve the reconstruction quality of the preprocessed spectra, so as to effectively improve the polarimetric reconstruction accuracy of channeled spectropolarimeter. As can be seen from the comparison of demodulation results from several methods, the effect of the CNN-based solver is no less than other demodulation methods proposed recently, which is of great significance to the improvement of the product quality of channeled spectropolarimeter. Of course, the CNN-based solver is essentially a spectral reconstruction method, which considers almost all the error factors in the spectrometer, and can also be effectively used in other types of spectrometers.
5. Conclusion
Channeled spectropolarimetry is a snapshot technique for measuring the spectra of Stokes parameters of light by demodulating the measured spectrum. As an indispensable part of the channeled spectropolarimeter, the spectrometer module has an important influence on the overall performance of the spectropolarimeter, but it is rarely paid attention to. The error factors of the spectrometer deviate the measured spectrum from the real spectrum, which further reduce the polarimetric reconstruction accuracy of the spectropolarimeter. In this paper, a CNN-based spectral reconstruction solver is proposed for channeled spectropolarimeters. Unlike most of the existing reconstruction methods, the key idea of the proposed method is to first preprocess measured spectra using existing traditional methods, and then train a CNN to learn a map from the preprocessed spectra to the real spectra, so as to further improve the reconstruction quality of the preprocessed spectra. Compared with the direct employment of the CNN, the proposed reconstruction method first uses the traditional reconstruction methods to make the CNN input data closer to the real spectrum, so as to enhance the capability of the trained CNN to make better predictions. To verify the effect of the proposed method, a series of experiments on simulated spectra and real spectra were carried out. The results show that compared with other methods, the accuracy of the demodulation results can be much more improved by employing the CNN-based solver to reconstruct the measured spectrum.
Funding
HFIPS Director’s Fund (YZJJ2022QN12); the Institute of Energy, Hefei Comprehensive National Science Center (21KZS205).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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