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Reconstruction and calibration methods for a Mueller channeled spectropolarimeter

Guodong Zhou, Yanqiu Li, and Ke Liu

Open Access Open Access

Abstract

Channeled spectropolarimeter (CSP) measures spectrally resolved Stokes vector of light and Mueller matrix of sample from a snapshot. While reconstruction and calibration methods for Stokes CSP have been well established, their Mueller CSP counterparts are lacking. In this paper, we propose methods for Mueller spectrum reconstruction and Mueller CSP calibration. Mueller CSP is modeled as a modulation matrix, linking the Mueller spectrum to be measured and the modulated spectrum from the spectrometer. We describe an optimization problem to solve the Mueller spectrum, where both the regularizer and the residual threshold constrain the result, making our reconstruction accurate, efficient, and noise-robust. The Stokes spectrum generated by polarization state generator and the analyzing vector of polarization state analyzer are measured in situ, the convolution of which construct the calibrated modulation matrix of Mueller CSP. Total polarimetric errors and spectroscopic errors are treated as a whole and represented by the calibrated modulation matrix. Both imaging and non-imaging Mueller CSP are experimentally calibrated. Reconstruction results show high accuracy with a root-mean-square error (RMSE) of 0.0371. The proposed methods help make Mueller CSP practical and have the potential to be general reconstruction and calibration methods for imaging and non-imaging Stokes-Mueller CSP.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Stokes vector S describes the polarization state of a light beam, while Mueller matrix M describes the polarization-altering characteristics of a sample that affect the Stokes vector [1,2]. Channeled spectropolarimeter (CSP) measures spectrally resolved Stokes vector and Mueller matrix with no moving parts from a snapshot [3,4], which arouse wide interests in the fields of remote sensing [5,6], materials analysis [7–9], biomedical study [10], etc. Different from dual-rotating retarder polarimeter [11,12], CSP employs wavenumber domain amplitude modulation to encode all four components of Stokes spectrum and sixteen components of Mueller spectrum into a single modulated spectrum [13,14]. The multiple information reuse brings huge challenges to reconstruct the Stokes-Mueller spectrum from CSP. On the other hand, the imperfect manufacture and assembly of retarders and polarizers will significantly reduce the accuracy of CSP [15]. The polarization characteristics of the lens, grating, and other optical elements, as well as the noises during the measurement will interfere with the result of CSP [16]. Therefore, reconstruction and calibration are two key aspects of CSP to accurately measure Stokes-Mueller spectrum.

In the past two decades, reconstruction and calibration methods of Stokes CSP have been proposed constantly, including Fourier transform reconstruction [3], linear reconstruction [13], iterative reconstruction [17,18], compressed reconstruction [19], coherence demodulation reconstruction [20], and neural-network-based reconstruction [21–23], as well as reference calibration [24–30], self-calibration [31–33], phase rearrangement calibration [34], and linear calibration [13,35], etc. While reconstruction and calibration methods for Stokes CSP have been well established, their Mueller CSP counterparts lack. Only some of these methods have been applied in Mueller CSP, for example, Dai et al. showed simulation results of their coherence demodulation method to extract Mueller spectrum from Mueller CSP and calibrated phase deviation of retarders utilizing a vacuum and polarizer as reference samples [36]. Hagen expanded reference calibration and self-calibration for Mueller CSP in theory [37]. Dubreuil et al. studied systematic errors specific to a snapshot Mueller matrix polarimeter [38], but not a Mueller spectropolarimeter. As a result, very few quantitative experimental results of Mueller CSP have been shown.

We recently proposed linear reconstruction with double constraint and total polarimetric calibration methods for Stokes CSP [39,40]. In this paper, we expand our reconstruction method and improve our calibration method for the more complicated Mueller CSP. Inspired by [13,14], we model Mueller CSP as a modulation matrix, linking the Mueller spectrum to be measured and the modulated spectrum from the spectrometer. We describe an optimization problem to solve the Mueller spectrum, where both the regularizer and the residual threshold constrain the result, making our reconstruction accurate, efficient, and noise-robust.

We develop a non-imaging Mueller CSP and an imaging Mueller CSP for experimental calibration. The Stokes spectrum generated by polarization state generator (PSG) and the analyzing vector of polarization state analyzer (PSA) are measured in situ, the convolution of which construct the calibrated modulation matrix of Mueller CSP. Total polarimetric errors and spectroscopic errors are treated as a whole and represented by the calibrated modulation matrix, including the retardance, fast axis azimuth, and dichroism of retarders, the extinction ratio and transmission axis azimuth of polarizers, the polarization aberration of condensers, and objective lens, the polarization characteristics of the fiber and grating in spectrometer, as well as the low contrast of modulated spectrum from the spectrometer due to high modulation frequency of retarders and limited spectral resolution of the spectrometer.

2. Theoretical model

A typical Mueller CSP consists of a broad-spectrum light source, a channeled spectro-PSG (CSPSG), a channeled spectro-PSA (CSPSA), and a spectrometer, as shown in Fig. 1. The CSPSG is composed of retarders (R1, R2) and a polarizer (P1). The CSPSA is symmetrical to CSPSG. R1, R2, R3, and R4 are thick retarders usually made of the same birefringent material with a thickness ratio of 1:2:5:10. Other thickness ratio also works [41,42]. To begin with, we set the (x,y,z) coordinate system relative to the transmission axis of P2, perpendicular to the transmission axis of P2, and the direction in which the chief ray travels, respectively, as shown in Fig. 1. The transmission axis of P1 and the fast axis of R2 and R3 are parallel to the x-axis. The fast axis of R1 and R4 are oriented at 45° relative to the x-axis.

 figure: Fig. 1.

Fig. 1. A typical Mueller CSP configuration. The red dots indicate the fast axis and the transmission axis.

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The Mueller spectrum of the sample to be measured is

$${\textbf M}(\lambda ) = \left[ {\begin{array}{*{20}{c}} {{m_{00}}(\lambda )}&{{m_{01}}(\lambda )}&{{m_{02}}(\lambda )}&{{m_{03}}(\lambda )}\\ {{m_{10}}(\lambda )}&{{m_{11}}(\lambda )}&{{m_{12}}(\lambda )}&{{m_{13}}(\lambda )}\\ {{m_{20}}(\lambda )}&{{m_{21}}(\lambda )}&{{m_{22}}(\lambda )}&{{m_{23}}(\lambda )}\\ {{m_{30}}(\lambda )}&{{m_{31}}(\lambda )}&{{m_{32}}(\lambda )}&{{m_{33}}(\lambda )} \end{array}} \right],$$
which could be reshaped to a column vector
$${\textbf X} = {\left[ {\begin{array}{*{20}{c}} {{m_{00}}({\lambda_1}) \cdots {m_{00}}({\lambda_N}),}&{{m_{01}}({\lambda_1}) \cdots {m_{01}}({\lambda_N}),}& \cdots &{{m_{33}}({\lambda_N})} \end{array}} \right]^\textrm{T}} \in {{\bf {\mathbb R}}^{16N \times 1}},$$
where λ is the wavelength, T is the matrix transpose, N is the sampling points of the spectrometer, ${{\bf {\mathbb R}}^{16N \times 1}}$ indicates that X is a real matrix with 16N rows and 1 column.

The spectral irradiance measured by the spectrometer is called a modulated spectrum, which can be expressed as

$$I\textrm{(}{\lambda _n}\textrm{) = }{\textbf A}\textrm{(}{\lambda _n}\textrm{)} \cdot {\textbf M}\textrm{(}{\lambda _n}\textrm{)} \cdot {\textbf S}\textrm{(}{\lambda _n}\textrm{) = }\sum\limits_{i,j = 0}^{i,j = 3} {[{{a_i}\textrm{(}{\lambda_n}\textrm{)} \cdot {m_{ij}}\textrm{(}{\lambda_n}\textrm{)} \cdot {S_j}\textrm{(}{\lambda_n}\textrm{)}} ]\quad n = 1 \cdots N} ,$$
where A is the analyzing vector of CSPSA
$${\textbf A}\textrm{(}{\lambda _n}\textrm{)} = [\begin{array}{*{20}{c}} {{a_0}\textrm{(}{\lambda _n}\textrm{)}}&{{a_1}\textrm{(}{\lambda _n}\textrm{)}}&{{a_2}\textrm{(}{\lambda _n}\textrm{)}}&{{a_3}\textrm{(}{\lambda _n}\textrm{)}} \end{array}],$$
and S is the Stokes spectrum generated by CSPSG
$${\textbf S}\textrm{(}{\lambda _n}\textrm{)} = {[\begin{array}{*{20}{c}} {{S_0}\textrm{(}{\lambda _n}\textrm{)}}&{{S_1}\textrm{(}{\lambda _n}\textrm{)}}&{{S_2}\textrm{(}{\lambda _n}\textrm{)}}&{{S_3}\textrm{(}{\lambda _n}\textrm{)}} \end{array}]^\textrm{T}}.$$

To separate M between A and S, Eq. (3) can be also expressed as

$$I\textrm{(}{\lambda _n}\textrm{) = }\sum {[{{\textbf A}\textrm{(}{\lambda_n}\textrm{)} \otimes {\textbf S}\textrm{(}{\lambda_n}\textrm{)} \odot {\textbf M}\textrm{(}{\lambda_n}\textrm{)}} ]} ,$$
where ${\otimes}$ is convolution, ${\odot}$ is Hadamard product, ∑ is the sum of all elements in the matrix.

The modulated spectrum $I\textrm{(}\lambda \textrm{)}$ can be represented by a column vector

$${\textbf Y} = {\left[ {\begin{array}{*{20}{c}} {I({\lambda_1})}& \cdots &{I({\lambda_N})} \end{array}} \right]^\textrm{T}} \in {{\bf {\mathbb R}}^{N \times 1}}.$$

Since both the modulated spectrum and each component of Mueller spectrum have N sampling points along the wavelength $\lambda $ in our theoretical model, the reconstruction of X from Y can maintain the native spectral resolution of the spectrometer.

The Mueller CSP performs modulation as a linear operator linking X and Y, which is represented by modulation matrix Φ

$${\textbf Y}\textrm{ = }\boldsymbol{\mathrm{\varPhi}} \cdot {\textbf X},$$
where
$$\begin{array}{l} {\mathbf \Phi} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} {{{\mathbf \Phi}_{00}}}&{{{\mathbf \Phi }_{01}}}& \cdots &{{{\mathbf \Phi }_{33}}} \end{array}} \right] \in {{\bf {\mathbb R}}^{N \times 16N}},\\ {{\mathbf \Phi }_{ij}} = \textrm{diag}\left[ {\begin{array}{*{20}{c}} {{a_j}({\lambda_1}){S_i}({\lambda_1})}& \cdots &{{a_j}({\lambda_N}){S_i}({\lambda_N})} \end{array}} \right] \in {{\bf {\mathbb R}}^{N \times N}}\quad i,j = 0 \cdots 3, \end{array}$$
diag means ${{\mathbf \Phi }_{ij}}$ is a diagonal matrix with nonzero elements only in the diagonal running from the upper left to the lower right.

The reconstruction of Mueller spectrum is to solve X from Eq. (8). Since Eq. (8) is an underdetermined equation, more constraints are needed. The common practice is to expand X in a certain vector space, including a truncated Fourier basis, Legendre polynomials, and discrete cosine transform (DCT) bases [13,19]. In this paper, the vector space ${\mathbf \Psi }$ employs Legendre polynomials and a truncated discrete cosine transform basis

$${\mathbf \Psi }\textrm{ = diag}\left[ {\begin{array}{*{20}{c}} {{{\mathbf \Psi }_0}}&{{{\mathbf \Psi }_0}}& \cdots &{{{\mathbf \Psi }_0}} \end{array}} \right] \in {{\bf {\mathbb R}}^{16N \times 16(K + J)}},$$
where
$${{\mathbf \Psi }_0} = \left[ {\begin{array}{*{20}{c}} {{{\mathbf \Psi }_{\textrm{LGD}}}}&{{{\mathbf \Psi }_{\textrm{DCT}}}} \end{array}} \right] \in {{\bf {\mathbb R}}^{N \times (K + J)}}.$$
${{\mathbf \Psi }_{\textrm{LGD}}}$ is Legendre Polynomials basis
$${{\mathbf \Psi }_{\textrm{LGD}}}\textrm{ = }\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{P_1}({{x_1}} )}\\ \vdots \\ {{P_1}({{x_N}} )} \end{array}}&{\begin{array}{*{20}{c}} \cdots \\ {}\\ \cdots \end{array}}&{\begin{array}{*{20}{c}} {{P_K}({{x_1}} )}\\ \vdots \\ {{P_K}({{x_N}} )} \end{array}} \end{array}} \right] \in {{\bf {\mathbb R}}^{N \times K}},$$
where
$${P_n}(x )= {2^n}\sum\limits_{k = 0}^n {{x^k}\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\frac{{n + k - 1}}{2}}\\ n \end{array}} \right)} .$$

Figure 2 shows Legendre polynomial curves of order 0 to 4. The basis composed of Legendre polynomials is very concise and effective in describing straight lines, second and third polynomials, and low-frequency cosine components with high accuracy, which makes smaller matrix size of the entire base, leading to fewer computing resources consume and faster calculation. K should be less than 5. The larger the K, the higher the frequency represented by the curve, which theoretically contains more high-frequency details. However, as shown in Fig. 2, the deviation between the values at both ends of the Legendre polynomial curve and the center will increase with the increase of K, which is not conducive to the expansion of X. Therefore, if the K value is too large, the errors at both ends of the reconstructed Mueller spectrum will increase, which may aggravate the singularity effect [18]. In this paper, $K = 5$.

 figure: Fig. 2.

Fig. 2. Legendre polynomial curves of order 0 to 4. The five lines in the upper right corner of the coordinate system are n = 0 to 4 from top to bottom.

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${{\mathbf \Psi }_{\textrm{DCT}}}$ is truncated discrete cosine transform bases

$${{\mathbf \Psi }_{\textrm{DCT}}}(i,j) = \left\{ {\begin{array}{*{20}{l}} {\sqrt {\frac{1}{N},} }&{i = 1 \cdots N,}&{j = 1}\\ {\sqrt {\frac{2}{N}} \cos \left[ {\frac{\pi }{{2N}}({j - 1} )({2i - 1} )} \right], }&{i = 1 \cdots N,}&{j = 2 \cdots J \le N} \end{array}} \right.,$$
where J is the highest frequency of ${{\mathbf \Psi }_{\textrm{DCT}}}$ decided by the thickness and birefringence of retarders. J makes sure the reconstructed X band-limited.

X could be represented by the vector space ${\mathbf \Psi }$ and coefficients $\hat{x}$

$${\textbf X} = {\mathbf \Psi } \cdot \hat{x},$$
where $\hat{x} \in {{\mathbb R}^{16(K + J) \times 1}}$.

Finally, an optimization problem is formulated to solve the Mueller spectrum, where both the regularizer ${||{\hat{x}} ||_1}$ and the residual threshold $\varepsilon$ constrain the result, making our reconstruction accurate, efficient, and noise-robust [39].

$$\min \;{||{\hat{x}} ||_1}\quad \textrm{s}\textrm{.t}\textrm{.}\quad {||{{{\textbf Y}_{\textrm{SPEC}}} - {\mathbf \Phi }\,{\mathbf \Psi }\,\hat{x}} ||_2} \le \varepsilon ,$$
where ${{\textbf Y}_{\textrm{SPEC}}}$ is the spectral irradiance measured by the spectrometer, while ${\mathbf\Phi }\,{\mathbf \Psi }\,\hat{x}$ is the modulated spectrum calculated in theory. When X is known, ${{\textbf Y}_{\textrm{SPEC}}} - {\mathbf \Phi } \cdot {\textbf X}$ could be used to check the accuracy of calibrated modulation matrix Φ.

3. Experimental calibration and reconstruction

The modulation matrix Φ represents all the modulation effects of Mueller CSP on the Mueller spectrum to be measured and links X and Y in Eq. (8). According to Eq. (9), the modulation matrix Φ is defined by the analyzing vector A of CSPSA and the Stokes spectrum S generated by CSPSG in Eqs. (4) and (5). That is, the calibration of Mueller CSP is equivalent to accurately determining the modulation matrix Φ, which equals the accurate measurement of A and S. The major advantage of this conversion is that A and S can be measured directly using a common dual-rotating-retarder spectropolarimeter (DRRSP). By changing the complicated estimation of matrix Φ [13] to a direct measurement of A and S, the calibration of Mueller CSP is much more efficient and easier to implement.

The experimental setup of a non-imaging Mueller CSP is shown in Fig. 3. The non-imaging Mueller CSP is composed of linear polarizers P1, P2 (Newport 10LP-VIS-B), and quartz crystal retarders R1, R2, R3, and R4 (Chengdu LightPool, custom-made) with the thickness of 1, 2, 5, 10 mm, respectively. DRRSP consists of a spectro-polarization state generator (SPSG) and a spectro-polarization state analyzer (SPSA). The SPSG is composed of a fixed horizontal linear polarizer PG (Newport 10LP-VIS-B) and a rotatable achromatic quarter-wave plate QG (Newport 10RP54-1B) driven by a rotation stage (Thorlabs DDR100/M). The SPSA is symmetrical to SPSG.

 figure: Fig. 3.

Fig. 3. Configuration of a non-imaging Mueller CSP. The red dots indicate the fast axis and the transmission axis.

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The light from a warm white LED (Thorlabs MWWHLP1) is coupled into a fiber by a doublet lens then collimated by another doublet lens into free space. At the other end of the experimental setup, a doublet lens focuses the beam into a fiber that leads to a Czerny-Turner grating spectrometer (Thorlabs CCS100) which has been amplitude corrected. Due to the radiation spectrum of LED, the wavelength range in the experiment is limited to 480-700 nm with 1884 bands in the spectrometer. Since the fiber is treated as a retarder [43] and the grating in the spectrometer shows strong diattenuation, the fiber must remain static during the experiment, otherwise, it will change the spectral intensity recorded by the spectrometer and affect the experimental results. Eliminating the fiber or replacing it with a polarization-maintaining fiber can solve this problem.

Before calibrating Mueller CSP, the DRRSP must be calibrated first in order to measure A and S accurately. A DRRSP is get by removing CSPSG and CSPSA in Fig. 3. The DRRSP is calibrated with eigenvalue calibration method (ECM) [44], which is based on the measurements of four reference samples: vacuum (air), linear polarizer (Newport 10LP-VIS-B) oriented at 0° and 90°, and an achromatic quarter-wave plate (Newport 10RP54-1B) oriented at 30° [45].

The Mueller spectrum of CSPSG and CSPSA are measured by the well-calibrated DRRSP. The analyzing vector A is the first row of the Mueller spectrum of CSPSA, while the Stokes spectrum S is the first column of the Mueller spectrum of CSPSG. Results are shown in Fig. 4. g0 and a0 are direct currents. g1 and a1 are relative to the retardance of R2 and R4. Note that the horizontal axis is the wavelength λ. If the horizontal axis is changed to wavenumber ν = 1 / λ, g1 and a1 will be cosine. g2, g3, a2, and a3 contain more complex frequency components.

 figure: Fig. 4.

Fig. 4. The analyzing vector A and the Stokes spectrum S measured by DRRSP.

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Since the retarders in CSPSA are thicker than that in CSPSG, the frequency of A is higher than that of S. This raises a new problem: the maximum and minimum of a1, a2 and a3 in Fig. 4 are not ±1. The contrast of the spectrum is lower than expected, and warrants some discussion. The optical system is a low-pass filter, and so is the spectrometer. Due to the high modulation frequency generated by CSPSA and the limited spectral resolution of the spectrometer, the convolution of the point spread function of the spectrometer with the original spectrum before the slit will reduce the contrast of the recorded spectrum. The decrease in spectral contrast appears both in the analyzing vector A (Fig. 4) and the modulation spectrum recorded by the spectrometer (Fig. 5). The decrease in spectral contrast is related to the frequency of the spectrum, which can be compared to the modulation transfer function of an imaging system. The low contrast of the modulation spectrum in Fig. 5 is mainly caused by the high modulation frequency of CSPSA. Although the frequency components of the modulation spectrum are more complex than that of CSPSA, their degree of the contrast reduction is roughly the same, since the same spectrometer is used in the experiment. Similar phenomenon also appeared in [46]. Dubreuil et al. recover the spectrum in Fourier domain. We record the degree of the contrast reduction in the calibrated analyzing vector A, which could be considered as a predistortion related to the modulated spectrum.

 figure: Fig. 5.

Fig. 5. The measured spectrum and the calculated spectrum.

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The calibrated modulation matrix Φ could be get using Eq. (9) after the measurement of A and S. In order to test our reconstruction and calibration methods, the Mueller spectrum of air is measured with DRRSP and Mueller CSP after calibration respectively. Results are shown in Figs. 5 and 6. Take the Mueller spectrum measured by DRRSP calibrated with ECM as the ground truth. The measured spectrum is the spectral irradiance recorded by the spectrometer, while the calculated spectrum is the product of the calibrated Φ and the ground truth. As shown in Fig. 5, the calculated spectrum matches the measured spectrum with RMSE 0.0134, which indicates that the modulation matrix Φ is well-calibrated.

 figure: Fig. 6.

Fig. 6. The Mueller spectrum of air measured by DRRSP and Mueller CSP after calibration.

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The reconstruction error of Mueller CSP is greater at the edge of the spectral range, which is anticipated because of the deviation between the values at both ends of the Legendre polynomial curve and the center in Fig. 2. The two ends of Mueller spectrum are usually considered singularities and neglected for generic comparison [18]. The RMSE of Mueller spectrum measured by Mueller CSP compared with the ground truth is 0.0179.

Although Mueller CSP measures Mueller spectrum, we are more concerned about the polarization parameters with physical significance such as the retardance, diattenuation, and depolarization after polar decomposition [47]. For this reason, glucose solution with optical rotation is measured by DRRSP and Mueller CSP. The optical rotation can be obtained by the polar decomposition of the measured Mueller spectrum. The experimental setup is shown in Fig. 7. The concentration of the glucose solution is 100 mg/ml, and it is placed in a 20 cm polarimeter tube.

 figure: Fig. 7.

Fig. 7. Experimental setup of Mueller CSP to measure the optical rotation of glucose solution

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The results are shown in Figs. 8 and 9. Due to the scattering of glucose solution and the glass on both sides of the tube, the light intensity reaching the spectrometer is very weak. Therefore, the Mueller spectrum measured by DRRSP is affected by noise and has many burrs. Benefit from the noise-robustness of our reconstruction method, the Mueller spectrum measured by Mueller CSP is smooth and noise immunity. However, slight ringing presents in the results. The reason for this phenomenon is explained and discussed in detail in [13].

 figure: Fig. 8.

Fig. 8. The Mueller spectrum of glucose solution measured by DRRSP and Mueller CSP after calibration.

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 figure: Fig. 9.

Fig. 9. The optical rotation obtained by polar decomposition of Mueller spectrum.

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The solution concentration, the polarimeter tube length, and the illumination wavelength will affect the optical rotation. Specific rotation is defined to eliminate these effects

$$[\alpha ]= \frac{\alpha }{{c \times l}},$$
where α is the measured optical rotation (rad), c is the concentration (g/ml), l is the length of the polarimeter tube (dm).

The optical rotation is 0.1822rad at the wavelength 589.3 nm (yellow sodium D line). The specific rotation of the glucose solution is 0.911 rad, which is consistent with the theoretical value. The above experimental results of non-imaging Mueller CSP verify the correctness and effectiveness of the reconstruction and calibration methods proposed in this paper.

4. Imaging Mueller CSP

The calibration of imaging Mueller CSP is much more challenging than the calibration of non-imaging Mueller CSP, because all systematic errors in non-imaging Mueller CSP will appear in imaging Mueller CSP, which even vary with the angle of view. Besides, the polarization aberration of condenser and objective lens will interfere with the result of imaging Mueller CSP. The proposed method is competent to calibrate imaging Mueller CSP pixel by pixel with full field of view, total polarimetric errors, and spectroscopic errors, which will be verified in this section.

The imaging Mueller CSP is established by adding polarization state generator (PSG) and polarization state analyzer (PSA) into an inverted microscope (based on Thorlabs Cerna), and replacing its camera with an imaging spectrometer (Andor Kymera 328i, 300lp/mm @600 nm blazing) with an sCMOS (Andor Zyla 4.2 Plus). The PSA is composed of two thick retarders (Chengdu LightPool, custom-made), a linear polarizer (Newport 10LP-VIS-B), and a rotatable achromatic quarter-wave plate (Thorlabs AQWP10M-580) driven by a rotation stage (Thorlabs DDR100/M). The PSG is symmetrical to PSA. The instrumental coordinate system is relative to the transmission axis of P2 and the direction in which light travels, as shown in Fig. 10. In our imaging Mueller CSP, R1, R2, R3, and R4 are thick retarders made of quartz crystal with the thickness of 1, 2, 5, 10 mm to maintain the retardance ratio of 1:2:5:10. The transmission axis of P1, P2 and the fast axis of R2, R3 are parallel to the x-axis. The fast axis of R1 and R4 are oriented at 45° relative to the x-axis.

 figure: Fig. 10.

Fig. 10. Configuration of imaging Mueller CSP. P, polarizer, Q, achromatic quarter-wave plate; R, retarder; CL, condenser lens; OL, objective lens; TL, tube lens. Red dots indicate the fast axis of retarders or the transmission axis of polarizers.

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The difference between the configurations in Fig. 3 and Fig. 10 is that the CSPSG and SPSG share the same polarizer P1, while the CSPSA and SPSA share the same polarizer P2. An imaging DRRSP is get by removing R1, R2, R3, and R4 in Fig. 10. An imaging Mueller CSP is gotten by removing Q1 and Q2, or keeping their fast axis parallel to the x-axis. The fixed polarizer P2 ensures the stability of the local coordinate system of the instrument and avoids the difference in modulation spectrum caused by changing the polarizer, since the grating in the spectrometer is a diattenuation element.

Koehler illumination is usually used in microscopes to ensure the uniformity of the illumination of the sample and to match the numeral aperture (NA) of the objective lens. In order to show each line of light more clearly, the scale in Fig. 11 is exaggerated. All light passes through PSA and PSG at a certain angle. However, the retardations of thick retarders in PSG and PSA are quite sensitive to the ray direction since they are much thicker than quarter-wave plates. Such retardance errors of R1, R2, R3, and R4 brings huge error to Mueller spectrum measurement. Fortunately, the beams focused on the same pixel of the detector pass through the PSA at the same angle, so these beams are subject to the same polarization modulation of PSA, which makes it possible for the pixel by pixel calibration of the PSA. But the pixel by pixel calibration is almost impossible for PSG, since the light in a beam focused on the same pixel of detector pass through PSG at different angles and is subject to the different polarization modulation. As a result, the polarization state of light out of PSG is the superposition of various polarization states, which as shown in Fig. 12, the value of S2 is almost zero, thus loss its modulation.

 figure: Fig. 11.

Fig. 11. Koehler illumination in imaging Mueller CSP. FS, field stop; AS, aperture stop; S, sample.

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 figure: Fig. 12.

Fig. 12. Stokes spectrum out from PSG in Koehler illumination.

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The invalidation of PSG in Fig. 11 does not mean that Koehler illumination cannot be used in imaging Mueller CSP. A 4f optical system can image the first uniform illumination surface to another location to produce uniform illumination, just like the right half of Fig. 10. However, the system might be long and complex. So, we choose critical illumination in our imaging Mueller CSP, as shown in Fig. 10. The light from LED is coupled into a multimode fiber. The light is reflected multiple times in the fiber, forming a uniform light source at the exit of the fiber. By imaging the exit of the fiber to the sample, light beams falling into the same pixel of the detector pass through the PSG and PSA respectively at the same angle, which can be calibrated pixel by pixel. Note that the core diameter of the fiber is small up to hundreds of micrometers, resulting in a small illumination area. Replacing fiber with a liquid light guide can solve this problem, because its core diameter is several centimeters. What’s more, the uniform illumination affects the quality of imaging, but does not affect the accuracy of polarization measurement, because the illumination uniformity is to compare the illuminance difference of adjacent spatial positions, while the Mueller measurement at each point has nothing to do with neighbors. Nevertheless, the illumination uniformity of the sample should not be too bad, otherwise, the low signal-to-noise ratio in the low-illuminance area will reduce the measurement accuracy.

The first thing before the calibration of imaging Mueller CSP is to calibrate imaging DRRSP using ECM. The four reference samples used in section 3 are inserted between PSG and condenser, condenser and objective lens, objective lens and PSA, respectively [48]. Then, the instrument matrix of SPSG and SPSA, as well as the polarization aberration of condenser and objective lens are measured, which is represented by SSPSG, ASPSA, MCL, and MOL.

The polarization state of the light illuminating the sample is not only generated by the PSG but also affected by the condenser. So, we define a generalized PSG consisting of PSG and condenser. However, the Stokes spectrum measured by SPSA is the product of MOL and the Stokes spectrum on the sample. The Stokes spectrum S generated by the generalized PSG in Mueller imaging CSP can be obtained after dividing MOL, shown in Fig. 13.

 figure: Fig. 13.

Fig. 13. The Stokes spectrum S generated by the generalized PSG in Mueller imaging CSP.

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Similarly, taking the sample as the dividing line, we define a generalized PSA consisting of objective lens, PSA, tube lens, and the grating in the spectrometer. By rotating Q1 in SPSG to generate at least four linearly independent polarization states, the product of MCL and the analyzing vector A of generalized PSA is obtained. After dividing MCL, the pure analyzing vector A of generalized PSA is shown in Fig. 14. Since the retarders in PSA are thicker than that in PSG, the frequency of A is higher than that of S, while the contrast of the spectrum in Fig. 14 is lower than that in Fig. 13. As discussed in section 3, the degree of the contrast reduction is recorded in the calibrated analyzing vector A.

 figure: Fig. 14.

Fig. 14. The analyzing vector A of generalized PSA in Mueller imaging CSP.

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The Mueller imaging spectrum of air measured by Mueller imaging CSP after calibration is shown in Fig. 15, where the horizontal axis represents the different spatial positions of the slit of the spectrometer (also the different positions of the sample conjugation to the slit), and the vertical axis represents the wavelength. It can be seen from Fig. 15 that the measured Mueller imaging spectrum of the air is very close to the identity matrix (the Mueller matrix of air is the identity matrix), and there is no significant difference in the Mueller imaging spectrum measured at different spatial positions of the slit. This shows that even if the beams from different spatial positions of the sample passes through PSG and PSA at different angles and undergoes different polarization modulations, the Mueller imaging CSP is calibrated pixel by pixel using the method proposed in this paper. The calibrated modulation matrix contains spatial position information and its corresponding polarization information. Therefore, the difference in the field of view will not affect the accuracy of the measured spatially resolved Mueller spectrum. The RMSE of Mueller imaging spectrum measured by Mueller imaging CSP compared with the ground truth is 0.0371 after removing the singular points at both ends of the spectrum, which verifies the correctness and effectiveness of the reconstruction and calibration methods proposed in this paper.

 figure: Fig. 15.

Fig. 15. Mueller imaging spectrum of air measured by Mueller imaging CSP.

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5. Conclusion

In this paper, methods for Mueller spectrum reconstruction and Mueller CSP calibration are proposed, theoretically analyzed, and experimentally verified. The modulation matrix of Mueller CSP is constructed by the convolution of the measured Stokes spectrum generated by PSG and the analyzing vector of PSA. Total polarimetric errors and spectroscopic errors are treated as a whole and represented by the calibrated modulation matrix. Both non-imaging and imaging Mueller CSP are built and experimentally calibrated. Reconstruction results show high accuracy with RMSEs 0.0179 and 0.0371, respectively. The proposed methods help make Mueller CSP practical. Further, Stokes CSP could be regarded as a subset of Mueller CSP since they share the same PSA on hardware and follow the same modulation principle. The proposed methods have the potential to be general reconstruction and calibration methods for imaging and non-imaging Stokes-Mueller CSP.

Funding

National Natural Science Foundation of China (11627808).

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.

References

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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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Figures (15)
Fig. 1.
Fig. 1. A typical Mueller CSP configuration. The red dots indicate the fast axis and the transmission axis.

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Fig. 2.
Fig. 2. Legendre polynomial curves of order 0 to 4. The five lines in the upper right corner of the coordinate system are n = 0 to 4 from top to bottom.

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Fig. 3.
Fig. 3. Configuration of a non-imaging Mueller CSP. The red dots indicate the fast axis and the transmission axis.

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Fig. 4.
Fig. 4. The analyzing vector A and the Stokes spectrum S measured by DRRSP.

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Fig. 5.
Fig. 5. The measured spectrum and the calculated spectrum.

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Fig. 6.
Fig. 6. The Mueller spectrum of air measured by DRRSP and Mueller CSP after calibration.

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Fig. 7.
Fig. 7. Experimental setup of Mueller CSP to measure the optical rotation of glucose solution

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Fig. 8.
Fig. 8. The Mueller spectrum of glucose solution measured by DRRSP and Mueller CSP after calibration.

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Fig. 9.
Fig. 9. The optical rotation obtained by polar decomposition of Mueller spectrum.

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Fig. 10.
Fig. 10. Configuration of imaging Mueller CSP. P, polarizer, Q, achromatic quarter-wave plate; R, retarder; CL, condenser lens; OL, objective lens; TL, tube lens. Red dots indicate the fast axis of retarders or the transmission axis of polarizers.

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Fig. 11.
Fig. 11. Koehler illumination in imaging Mueller CSP. FS, field stop; AS, aperture stop; S, sample.

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Fig. 12.
Fig. 12. Stokes spectrum out from PSG in Koehler illumination.

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Fig. 13.
Fig. 13. The Stokes spectrum S generated by the generalized PSG in Mueller imaging CSP.

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Fig. 14.
Fig. 14. The analyzing vector A of generalized PSA in Mueller imaging CSP.

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Fig. 15.
Fig. 15. Mueller imaging spectrum of air measured by Mueller imaging CSP.

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Equations (17)

Equations on this page are rendered with MathJax. Learn more.

M ( λ ) = [ m 00 ( λ ) m 01 ( λ ) m 02 ( λ ) m 03 ( λ ) m 10 ( λ ) m 11 ( λ ) m 12 ( λ ) m 13 ( λ ) m 20 ( λ ) m 21 ( λ ) m 22 ( λ ) m 23 ( λ ) m 30 ( λ ) m 31 ( λ ) m 32 ( λ ) m 33 ( λ ) ] ,
X = [ m 00 ( λ 1 ) ⋯ m 00 ( λ N ) , m 01 ( λ 1 ) ⋯ m 01 ( λ N ) , ⋯ m 33 ( λ N ) ] T ∈ R 16 N × 1 ,
I ( λ n ) =  A ( λ n ) ⋅ M ( λ n ) ⋅ S ( λ n ) =  ∑ i , j = 0 i , j = 3 [ a i ( λ n ) ⋅ m i j ( λ n ) ⋅ S j ( λ n ) ] n = 1 ⋯ N ,
A ( λ n ) = [ a 0 ( λ n ) a 1 ( λ n ) a 2 ( λ n ) a 3 ( λ n ) ] ,
S ( λ n ) = [ S 0 ( λ n ) S 1 ( λ n ) S 2 ( λ n ) S 3 ( λ n ) ] T .
I ( λ n ) =  ∑ [ A ( λ n ) ⊗ S ( λ n ) ⊙ M ( λ n ) ] ,
Y = [ I ( λ 1 ) ⋯ I ( λ N ) ] T ∈ R N × 1 .
Y  =  Φ ⋅ X ,
Φ = 1 2 [ Φ 00 Φ 01 ⋯ Φ 33 ] ∈ R N × 16 N , Φ i j = diag [ a j ( λ 1 ) S i ( λ 1 ) ⋯ a j ( λ N ) S i ( λ N ) ] ∈ R N × N i , j = 0 ⋯ 3 ,
Ψ  = diag [ Ψ 0 Ψ 0 ⋯ Ψ 0 ] ∈ R 16 N × 16 ( K + J ) ,
Ψ 0 = [ Ψ LGD Ψ DCT ] ∈ R N × ( K + J ) .
Ψ LGD  =  [ P 1 ( x 1 ) ⋮ P 1 ( x N ) ⋯ ⋯ P K ( x 1 ) ⋮ P K ( x N ) ] ∈ R N × K ,
P n ( x ) = 2 n ∑ k = 0 n x k ( n k ) ( n + k − 1 2 n ) .
Ψ DCT ( i , j ) = { 1 N , i = 1 ⋯ N , j = 1 2 N cos ⁡ [ π 2 N ( j − 1 ) ( 2 i − 1 ) ] , i = 1 ⋯ N , j = 2 ⋯ J ≤ N ,
X = Ψ ⋅ x ^ ,
min | | x ^ | | 1 s .t . | | Y SPEC − Φ Ψ x ^ | | 2 ≤ ε ,
[ α ] = α c × l ,
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