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Abstract
In this paper, we present a Mueller matrix imaging system consisting of a spatially modulated polarization light source (SMPL) and a dual division-of-focal-plane (DoFP) polarimeters as the PSA and 2D detector. The system does not contain moving parts such as a rotating stage, which leads to more robust and reliable operations for applications in hostile settings. By taking Muller matrix images at variable distances between the SMPL and the target, we examine in details errors due to different spatial distributions in angle and intensity of different polarized lights. A calibration method is proposed to reduce such errors introduced by SMPL. The performances of the new imaging technique and the calibration method are tested in Mueller matrix imaging of different samples.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
While light scatter from samples, its polarization state changes according to samples’ anisotropic microstructure. Therefore, polarization imaging can obtain the structural information and composition of scattering samples. Mueller matrix (MM) provides a comprehensive characterization of sample’s polarization properties. MM imaging has been used in many fields, including target detection [1], material identification [2,3] and medical diagnosis [4–7].
MM measurement is performed by illuminating samples with different polarized light and measuring the polarization state of the outgoing light. Polarization state generator (PSG) and polarization state analyzer (PSA) are needed for generating and identifying at least four specific polarization states of lights respectively. Many different designs have been reported for polarization modulations in PSA and PSG, including fixing linear polarizers and rotating quarter-wave plates (FPRQ) [8], rotating linear polarizers and rotating quarter-wave plates (RPRQ) [9] or fixing linear polarizers and changing the phase delay of liquid crystal phase variable retarders (LCVR) [1,10] and so on. DoFP polarimeters use pixeled micro-polarizer array (MPA) in front of the ordinary CCD sensor to take the linear polarization states in a single shot [11]. A dual DoFP module can serve as a Stokes PSA and detector for fast Mueller matrix imaging [12].
Most MM measurement systems adopt a single light source and perform polarization modulation by rotatable or other electric controlled polarization optics. By adopting multiple light sources with fixed PSG, the structure of polarization modulation components can be simplified. For applying MM examination to endoscopes, previous studies reported multichannel optical fibers respectively equipped with fixed polarizers to compose a flexible polarimetric probe [13]. The polarimetric probe is designed for $3 \times 3$ MM point measurement and consider no distribution errors caused by multiple light sources. Though multiple light sources contribute to some applications by simplifying the structure of PSG, it brings differences in angle and intensity distributions of different polarized lights. For performing the imaging of MM instead of point measurement and maintaining a high MM imaging accuracy, these systematic errors should be calibrated.
In this paper, we establish a MM imaging system consisting of a SMPL, and a dual DoFP polarimeters as PSA [12]. The system accomplishes MM imaging without rotatable modules, which would benefit some specific applications, such as underwater detection in the future. As multiple light sources structure, SMPL introduces angle and intensity distributions. We propose a system calibration scheme to calibrate the inhomogeneous distributions of light intensities from SMPL characteristics. By taking MM images at variable distances between the SMPL and the target, we examine in details the errors caused by the angle and intensity distributions of lights from different regions of SMPL, meanwhile, test the calibration method performance. To demonstrate the application potential, our system is applied to measure samples made of different materials in free space to prove its ability in material classification.
2. Materials and methods
2.1 Experiment setup
The system consists of a SMPL and a dual DoFP polarimeters module as PSA. The SMPL is composed of a spatially switching four-quadrant LED light source, and a PSG with one polarizer and four quarter wave plates of different orientations.
(1). The structure of SMPL:
The LED light source and PSG are shown in Fig. 1 as two parts of SMPL.
Fig. 1. (a) Photograph of the four quadrant LED light source. (b) Schematic of the structural combination of the SMPL.
As shown in Fig. 1(a), the LED light source is spatially divided into four regions, quadrant one(Q1), quadrant two(Q2), quadrant three(Q3), and quadrant four(Q4). The divergent lights from the four quadrants illuminate the same sample and always keep an overlapped area at different distance. The light source is 100 mm in diameter. Each of the four regions is made up of 36 LEDs ($\mathrm{\lambda } = 633\; \textrm{nm}$ with 3 mm diameter) working simultaneously, but the four quadrants work independently.
Figure 1(b) gives the structural combination of the light source and the PSG. The PSG generates four independent incident polarization states. It consists of a linear polarizer (WP140HE, #14-723, Edmund Optics Inc., USA) and quarter wave plates (#86-179, Edmund Optics Inc., USA) at specific angles corresponding to the four quadrants of the SMPL. Condition number (CN) can be used to characterize the optimization degree of PSG and PSA [14]. A smaller CN indicates a smaller error transfer for more accurate MM measurements.
In SMPL, if we direct both the linear polarizers and the quarter wave plates for the four quadrants to specific angles, the instrument matrix of PSG can reach a minimum CN of 1.73 [15]. In our experiment, considering the operation accuracy and convenience, the linear polarizer is oriented at ${0^{\circ}}$ for all four quadrants. The fast axis orientations of the four quarter wave plates are set to ${\pm} 15.1^\circ $ and ${\pm} 51.7^\circ $ respectively. This configuration ensures the theoretical CN of PSG instrument matrix reaches 3.4, which is the minimum CN of the PSG based on such fixed polarizer and rotating quarter wave plate (FPRQ) [16]. For convenience of alignment, a 100 mm diameter non-polarization glass plate is used to fix together the linear polarizer and retarders. For experimental convenience, we use a square PSG with the same diameter rather than a circular PSG in schematic Fig. 1(b). The square PSG also satisfies previous experimental setup.
SMPL changes the polarized light by lighting different quadrant. This process is triggered by electrical signal, which means the switch of polarization states is timesaving.
(2) The PSA and 2D detector:
As shown in Fig. 2(a), we build a PSA and 2D detector based on Ref. [12]. The PSA includes two 16-bit DoFP polarimeters (PHX050S-PC, Lucid Vision Labs Inc., Canada, DoFP-CCD1 and DoFP-CCD2), with resolution of 2048 × 2448 pixels and a frame rate of 21 fps. Two DoFP polarimeters are fixed to the transmission end and the reflection end of the 50:50 non-polarized beam splitter (NBS) prism (CCM1-BS013/M, Thorlabs Inc., USA), respectively, and a fixed-angle phase quarter-wave plate (R1) is installed between the transmission end of the prism and DoFP-CCD1. The dual DoFP polarimeters module acquires a Stokes vector through one acquisition. Therefore, like SMPL, it saves time and works without any moving parts.
Fig. 2. (a) Schematic of PSA. (b) Configuration schematic: the front view of SMPL-DoFPs system. (c) Configuration schematic: the vertical view of SMPL-DoFPs system. The inhomogeneous distributions of four quadrants vary with measurement distance.
(3) The experimental set-up:
The system configuration is shown in Fig. 2(b) and Fig. 2(c). The SMPL is placed closely alongside the PSA. Both the coordinate axes and central axes of PSG and PSA are parallel. Only in this configuration, the angles of incident light and outgoing light keep approximately parallel as measurement distance varies. The whole MM imaging system is defined as SMPL-DoFPs system according to the structure of its PSG and PSA.
When SMPL-DoFPs system works, the SMPL switches its lighting area in sequence, then PSA detects the complete Stokes vector of corresponding outgoing light. The measurement is finished when four quadrants are traversed.
The MM measured by SMPL-DoFPs system can be calculated as follow:
where ${\textbf{S}_{\textrm{out}}}$ is combined by four column Stokes vectors of the output light, ${\textbf{S}_{\textrm{in}}}$ is combined by four column Stokes vectors of the input light. The four column Stokes vectors correspond to four quadrants of SMPL respectively.Without any moving parts, the system MM acquisition time can be expressed as ${\textrm{t}_\textrm{s}} = {\textrm{t}_{\textrm{constant}}} + 4 \times {\textrm{t}_{\textrm{exposure}}}$, where ${\textrm{t}_{\textrm{exposure}}}$ is the exposure time of PSA and ${\textrm{t}_{\textrm{constant}}}$ is the constant response time of system and hardware. For the SMPL-DoFPs system we constructed, ${\textrm{t}_{\textrm{constant}}} = 2\; \textrm{s}$, which is also the theoretical minimal acquisition time of our system.
(4) A non-spatially modulated polarization (Non-SMP) system for measurement accuracy evaluation:
Since different quadrants of SMPL are separated in space, the MM imaging of transparent samples cannot be performed. For instance, when we measure a linear polarizer or a retarder film, the imaging regions of four quadrants detected by PSA are separated and concentrated at different pixels. The calculation of MM at each pixel cannot be performed without detecting four outgoing lights. Since most standard samples used for calibration or validation are transparent optics, the measurement accuracy of SMPL-DoFPs system cannot be validated by measuring standard samples.
Therefore, we construct a calibrated Non-SMP system as control group. As shown in Fig. 3(a), the Non-SMP system is transformed by adding a fixed slide capable of both horizontal and vertical movement to PSG of SMPL-DoFPs system. The Non-SMP system achieves MM imaging by translating PSG, but not through switching SMPL lighting area. As shown in Fig. 3(b), each of the four quadrants can take one MM by translating PSG four times. For each MM measured by Non-SMP system, its light source adopts only one quadrant of SMPL. Therefore, it avoids the angle and intensity differences caused by switching SMPL lighting area in SMPL-DoFPs system and can serve as the ideal control group.
Fig. 3. (a) The Non-SMP system achieved by adding a fixed slide to the PSG of SMPL-DoFPs system. (b) The demonstration of achieving MM imaging through Q1 by translating PSG.
For each quadrant, the calculation of its MM can be expressed as:
where ${\textrm{S}_{\textrm{out}}}$ is combined by four column Stokes vectors of the output light, ${\textrm{S}_{\textrm{in}}}$ is combined by four column Stokes vectors of the input light, and i represents the MM is measured by quadrant i. We add the ${\textrm{S}_{\textrm{out}}}$ of four quadrants together, then there is the MM represents the whole four quadrants:This MM eliminates the SMPL interference and equates to the MM measured by all four quadrants simultaneously. We adopt this MM as the true value to evaluate the measurement accuracy of SMPL-DoFPs system.
2.2 Calibration algorithm
Since the SMPL consists four quadrants for polarized light illumination, the intensities and angles of incident lights of different polarization states vary at different points of the sample. Therefore, the instrument matrix of PSG is different for different pixels of the image. The inhomogeneous distribution of PSG instrument matrix has to be calibrated for accurate MM imaging.
We divide the calibration method into three steps: (a) Calibration of PSA: calibrate and calculate the instrument matrix of PSA. (b) Average intensity calibration of PSG: calibrate and calculate the average instrument matrix of PSG. (c) Intensity distribution calibration of PSG: calibrate and calculate the distribution of PSG instrument matrix.
- (a) Calibration of PSA
By combining two DoFP polarimeters with a fixed retarder, the polarization states could be measured through one acquisition [12]. Here we adopt the same calibration method of PSA as Ref. [12] and briefly describe the process.
According to [12], to make the PSA optimized, R1 can be a quarter-wave plate simply fixed between NBS and any of the DoFP polarimeters at any fast axis orientation. Without loss of generality, when R1 is fixed between NBS and DoFP1 at ${0^{\circ}}$ fast axis orientation, the 8 $\times$ 4 instrument matrix of the PSA can be calculated by:
$${\textbf{A}_{\textrm{PSA}}} = \left[ \begin{array}{l} {\textbf{A}_{\textrm{DoFP}}}{\textbf{M}_{\textrm{R1}}}\textrm{(}{0^ \circ }\textrm{,}\frac{\pi }{2}\textrm{)}{\textbf{M}_{\textrm{trans}\_\textrm{NBS}}}\\ {\textbf{A}_{\textrm{DoFP}}}{\textbf{M}_{\textrm{reflect}\_\textrm{NBS}}} \end{array} \right],$$where ${\textbf{M}_{\textrm{trans}\_\textrm{NBS}}}$ and ${\textbf{M}_{\textrm{reflect}\_\textrm{NBS}}}$ are the MMs of NBS’s transmission end and reflection end, respectively. ${\textbf{A}_{\textrm{DoFP}}}$ is the instrument matrix of DoFP polarimeter, when the micro polarizers in front of every four adjacent pixels in DoFP polarimeter have the polarization orientation of ${0^{\circ}}$, ${45^{\circ}}$, ${90^{\circ}}$ and ${135^{\circ}}$, ${\textbf{A}_{\textrm{DoFP}}}$ can be expressed as: - (b) Average intensity calibration of PSG
Since different quadrants of SMPL are separated in space, its instrument matrix distribution cannot be directly calibrated by measuring standard samples. Therefore, we utilize the calibrated PSA as a point measurement tool and measure the air to calculate the PSG average instrument matrix instead of its instrument matrix distribution.
Place PSG in the center of the field of view (FOV) of PSA and make the coordinates of the two coincide. Light up the four quadrants of the PSG sequentially under dark conditions to reduce interference by background light. Then the Stokes vector of each pixel detected by PSA can be expressed as:
$$\textbf{S}^{\prime}_{\textrm{in}} = \textrm{pinv}({\textbf{A}_{\textrm{PSA}}}) \cdot \left[ {\begin{array}{c} {{\textbf{I}_{\textrm{CCD}1}}}\\ {{\textbf{I}_{\textrm{CCD}2}}} \end{array}} \right],$$${\textbf{I}_{\textrm{CCD}1}}$ and ${\textbf{I}_{\textrm{CCD}2}}$ correspond to the intensity values detected by two DoFP polarimeters at the same pixel, respectively. For each of the two DoFP polarimeters in PSA, the intensity detected by four channels is expressed as:$${\textbf{I}_{\textrm{CCD}}} = \left[ {\begin{array}{cccc} {{\textbf{I}_{0^{\circ}}},}&{{\textbf{I}_{45^\circ }},}&{{\textbf{I}_{90^\circ }},}&{{\textbf{I}_{135^\circ }}} \end{array}} \right].$$The four quadrants of PSG detected by PSA is shown as Fig. 4(a). Then integrate the Stokes vectors of all pixels, we obtain the average Stokes vector of a single quadrant:
$${\textbf{S}_{\textrm{in}}} = \textrm{sum}({{{[{\textbf{S}^{\prime}_{\textrm{in}}} ]}_{\textrm{width} \times \textrm{height}}}} ).$$${\textbf{S}_{\textbf{in}}}$ is the average Stokes value of a single quadrant, and $\textbf{S}_{\textrm{in}}^{\prime}$ is the Stokes value of each pixel.Calculate the average Stokes value of four quadrants and combine them into the following matrix by column:
$$\textbf{A}^{\prime}_{\textrm{PSG}} = \left[ {\begin{array}{cccc} {{\textbf{S}_{\textrm{in}1}},}&{{\textbf{S}_{\textrm{in}2}},}&{{\textbf{S}_{\textrm{in}3}},}&{{\textbf{S}_{\textrm{in}4}}} \end{array}} \right] = \left[ {\begin{array}{cccc} {{\textrm{I}_1}}&{{\textrm{I}_2}}&{{\textrm{I}_3}}&{{\textrm{I}_4}}\\ {{\textrm{Q}_1}}&{{\textrm{Q}_2}}&{{\textrm{Q}_3}}&{{\textrm{Q}_4}}\\ {{\textrm{U}_1}}&{{\textrm{U}_2}}&{{\textrm{U}_3}}&{{\textrm{U}_4}}\\ {{\textrm{V}_1}}&{{\textrm{V}_2}}&{{\textrm{V}_3}}&{{\textrm{V}_4}} \end{array}} \right].$$The $\textbf{A}_{\textrm{PSG}}^{\prime}$ is the average instrument matrix of PSG and it is a constant value. When intensity distribution is nearly homogeneous or the requirement of measurement accuracy is not strict, this matrix can be used to represent the instrument matrix of the PSG for the whole imaging plane in MM imaging.
- (c) Calibration of light intensity distribution
SMPL causes different angle and intensity distribution for different pixels of the image, which also vary with measurement distance as shown in Fig. 4(b) and Fig. 4(c). The maximum intensity points of four quadrants gradually get closer and concentrate in the center of FOV as measurement distance increase. For higher measurement accuracy, an average instrument matrix of PSG is no longer appropriate. Although it is difficult to take into account the effects due to variations in illumination angles, intensity distribution can be considered in the calibration process to obtain a PSG instrument matrix distribution in the whole imaging plane.
Fig. 4. (a) Four images of four quadrants detected by PSA. (b) and (c) The intensity distribution on an A4 white paper from four quadrants respectively at (b) 5 meters. (c) 0.5 meters.
The first row of $\textbf{A}_{\textrm{PSG}}^{\prime}$ corresponds to the intensity values of four quadrants. Extract the first row of the matrix as the intensity scaling coefficient ${\textrm{P}_1}$ for the four quadrants:
Different from the single light source system, the PSG used in different quadrants of SMPL are separated and independent, so we must consider the optical transmittance difference of four PSGs. Therefore, we remove the PSG off the SMPL and make a point calibration of the light source alone. After a same point calibration method mentioned above, we obtain the instrument matrix of the light source $\textbf{A}_{\textrm{LED}}^{\prime}$. For a non-polarized light source, only the first row of $\textbf{A}_{\textrm{LED}}^{\prime}$ is not zero. Then extract the first row of $\textbf{A}_{\textrm{LED}}^{\prime}$ as the intensity scaling factor:
Reset the whole system and sample to origin status as shown in Fig. 2(c). Remove the PSG, light up the light source in sequence, then the PSA can acquire the light intensity distribution from four quadrants of PSG on the sample. The intensity distribution matrix of the four quadrants measured by PSA is ${[{{\textrm{k}_1}} ]_{\textrm{width} \times \textrm{height}}}$, ${[{{\textrm{k}_2}} ]_{\textrm{width} \times \textrm{height}}}$, ${[{{\textrm{k}_3}} ]_{\textrm{width} \times \textrm{height}}}$, ${[{{\textrm{k}_4}} ]_{\textrm{width} \times \textrm{height}}}$, then by matching the intensity distribution of the four quadrants pixel by pixel, the overall intensity distribution matrix of the four quadrants of the SMPL can be reformed:
Compared with transmission MM imaging system, the actual coordinate of PSG or PSA in backscattering MM imaging system is flipped. To balance the transformation of SMPL-DoFPs system coordinate chirality, the instrument matrix distribution of the PSG corresponding to the sample at this measurement distance should be expressed as:
The expansion of ${\textbf{M}_{\textrm{flipped}}}$ is:
Return PSG to its original position. Turn on the four quadrants of SMPL in sequence, and measure the Stokes of the corresponding outgoing light at PSA:
The calibration and measurement method mentioned above is suitable for MM imaging in darkrooms without background interferences by stray light. For applications where background light cannot be ignored, we need to perform a sample imaging before turning on SMPL. The light intensity from sample detected by PSA when turning off SMPL can be expressed as: $[{\textbf{I}_{\textrm{CCD}1}^\textrm{T}}^{\prime}\; \; {\textbf{I}_{\textrm{CCD}2}^\textrm{T}}^{\prime} ]$. Then Eq. (17) is rewritten as:
Then the MM calculated by Eq. (19) and Eq. (18) can eliminate the interference of background light. We need to note that though background light can be eliminated from MM measurement, its appearance might reduce the signal-to-noise ratio of the system. The system still needs to avoid imaging in an environment with strong background light interference.
To evaluate measurement accuracy when adopting different imaging system or calibration method, the average root mean squared error (RMSE) of the measured MMs of the sample is calculated according to Eq. (20).
3. Experimental results
3.1 Validation experiment
In this section, we validate the performance of different calibration methods by experiments. MM images are taken at different measurement distance and then calculated without calibration, calibrated by average calibration and intensity distribution calibration respectively. The MM measured by Non-SMP system is considered as the true value.
Figure 5 gives the average MM element values of an A4 white paper, which should have uniform polarization property, measured by different calibration methods. The range of the measurement distance varies from 0.5 m to 5 m in 0.5 m steps.
Fig. 5. Average MM element values of an A4 white paper measured by different calibration method respectively at different measurement distance. MM1(black) is the MM measured by Non-SMP system with the maximum experimental measurement distance 5 meter as the true value. MM2 (blue) is the MM measured by SMPL-DoFPs system without calibration. MM3 (green) is the MM measured by SMPL-DoFPs system after average intensity calibration. MM4 (red) is the MM measured by SMPL-DoFPs system after intensity distribution calibration. The range of the measurement distance varies from 0.5 m to 5 m in 0.5 m steps.
The Non-SMP system avoids the angle and intensity distribution interferences caused by SMPL different quadrants. Therefore, the MM measured by Non-SMP system serves as the true value to evaluate the measurement accuracy of SMPL-DoFPs system.
As shown in Fig. 5, the errors caused by SMPL mainly appear in the first row of the MM. This phenomenon is closely related to SMPL and sample polarization characteristics. For any MM measurement, the theoretical model has a simplified expression:
where S is the Stokes vector of the output light, M is the MM of sample, G is the Stokes vector of the input light and I is the light intensity of light source. For our SMPL-DoFPs system, S and G can be expressed as a $4 \times 4$ matrix combined by corresponding four column Stokes vectors. I can be expressed as $4 \times 4$ diagonal matrix corresponds to the intensities of four quadrants. Then the MM of sample is calculated by:Since the sample we select possess a high degree depolarization characteristic, the polarization degree of output light is low. Hence the first row of S is significantly larger than the other rows. When light intensities of four quadrants fluctuate, the diagonal matrix I changes its diagonal elements and becomes the source of error. The error appears in each row of M can be expressed as:
Then let us focus on the comparison between MMs measured by SMPL-DoFPs system after different calibration methods. If the measurement distance is large enough, the angle and intensity distributions of the four quadrants tend to be the same. This means the systematic error caused by SMPL will decrease as the measurement distance increase.
As shown in Fig. 5, the MM curve measured after intensity distribution calibration of PSG maintain the lowest error and quickly approaches to the true value. For MM measured only after intensity average calibration, both angle and intensities distribution affect its measurement accuracy. Hence it suffers from extra error sources than MM measured after intensity distribution calibration. A larger measurement distance is necessary to reduce errors in MM for intensity average calibration.
For MM measured without any calibration, it possesses an invariable PSG instrument matrix like average intensity calibration but with a different value. This explains the similar MM curve shape as MM curve measured after average intensity calibration. However, besides the angle and intensities distribution factors, the incorrected average intensity ratio leads to another systematic error. The former will decline as the increase of measurement distance, the latter will not. Therefore, its curve would never coincide with the true value due to its incorrected substitution of four quadrants intensity ratio.
To further examine the systematic error caused by SMPL and test calibration method performance, the average RMSE is used to characterize the deviation degree of the 16 MM elements measured by SMPL-DoFPs system:
Figure 6 reflects the systematic errors caused by SMPL vary with measurement distance. A smaller RMSE signifies less systematic errors and the MM is closer to the true value. For MM after intensity distribution calibration of PSG, only the angle differences of four quadrants affect its measurement accuracy. Hence the RMSE after intensity distribution calibration is always the lowest and is close to zero at large measurement distance. For MM after average intensity calibration, its measurement accuracy is determined by both angle and intensity distributions of four quadrants. Thereby it possesses a higher RMSE and a longer distance to approach to the true value than distribution calibration method. For MM without calibration, besides the variable uncalibrated angle and intensity distribution effects, it possesses a constant systematic error. It is predictable that its RMSE curve would approach to a reasonable large constant even at the longest measurement distance.
Fig. 6. The average RMSE of 16 MM elements measured by SMPL-DoFPs system without calibration (blue), after average intensity calibration (green) and after intensity distribution calibration (red). The range of the measurement distance varies in 0.5 m steps.
These results demonstrate when using SMPL for MM imaging, the intensity calibration of PSG must be implemented. An intensity distribution calibration of PSG is the most effectiveness method to improve measurement accuracy. And if measurement distance is large enough compared to the size of SMPL, it is also acceptable to calibrate the average of PSG instrument matrix only. Above conclusions is based on the premise that intensity distribution of each quadrants is similar. Clearly, if one quadrant of SMPL possesses an apparent different distribution compared with the others, only intensity distribution calibration of PSG is advisable.
Above experiments used a piece of white uniform paper as sample to study the systematic errors caused by SMPL and the performance of calibration method. Actually, the conclusions above and the calibration method are applicable to any sample. Here we exhibit the MM imaging results and the average RMSE of metal and wooden board measured by SMPL-DoFPs system after different calibration methods. Similarly, the MM measured by Non-SMP system is considered as the true value.
As shown in Fig. 7, the systematic errors caused by SMPL emerge when measuring any sample. Due to metal’s strong polarization-maintaining characteristic, the systematic errors not only concentrate in the first row of its MM. By comparing the MM image and RMSE of metal and wooden after different calibration methods, it is clearly that the systematic error caused by SMPL is variable for different samples. Nevertheless, the intensity distribution calibration of PSG achieves the minimum RMSE for both metal and wooden board. It is an effective method to improve the measurement accuracy of SMPL-DoFPs system.
Fig. 7. The MM imaging of (A) metal and (B) wooden board measured at 1 meter by (a) Non-SMP system as the true value. (b) SMPL-DoFPs system after intensity distribution calibration. (c) SMPL-DoFPs system after average intensity calibration. (d) SMPL-DoFPs system without calibration. The rmse_A and the rmse_B represent the RMSE of metal and wooden board respectively.
3.2 Using SMPL-DoFPs system for material characterization
SMPL-DoFPs system can be used for free space polarization imaging. To explore potential applications of SMPL-DoFPs system, we design an experiment with samples of different structures and demonstrate the performance and capabilities of MM imaging using SMPL.
As shown in Fig. 8(a), we design a composite sample consists of two pieces of paper with micro structural differences (paper 1 and paper 2), two compound metals plates with micro structural differences (metal 1 and metal 2), a plastic sheet and a wooden board. When we use SMPL-DoFPs system for polarization imaging of different materials, polarization features extracted from MM can be used to differentiate materials of different microstructural features [7], or samples of different surface morphology. As shown in Fig. 8(b) and Fig. 8(c), samples made of different materials are easily discriminated when adopting polarization parameters with physical meanings derived from the MM such as MM Trace(${\textrm{m}_{11}} + {\textrm{m}_{22}} + {\textrm{m}_{33}} + {\textrm{m}_{44}}$) and MMD Delta($\Delta $) [17].
Fig. 8. (a) Photograph of six selected samples: (A) paper 1. (B) paper 2. (C) metal 1. (D) metal 2. (E) plastic sheet. (F) wooden board. (b) and (c) The polarization parameters derived from MM elements measured by SMPL-DoFPs system.
Both MM Trace and MMD Delta can characterize the depolarization power of a MM. A MM with a stronger depolarization power tends to have a lower MM Trace and a higher MMD Delta. From the MM Trace and MMD Delta value of this composite sample, we demonstrate paper possess the strongest depolarization power, while metal have an opposite performance. Even paper 1 and 2, and metal 1 and 2 are discriminated due to the micro differences of their depolarization capabilities. The results demonstrate the strong relevance between MM and sample structure and that polarization measurement contributes to make classification in accordance with our purpose, which indicates potential applications of the system.
4. Conclusion
In this paper, we establish a free space MM imaging system based on SMPL. The system can eliminate rotating parts through combining a SMPL and a dual DoFP polarimeters-based PSA. The application of SMPL simplifies the system structure, increases the measurement speed and expands application scope. We report the calibration methods for eliminating the systematic error introduced by SMPL. And we study the effect of adopting SMPL and performance of different calibration methods through comparing MMs measured at different distance.
Preliminary results demonstrate when using SMPL for MM imaging, the intensity calibration of PSG must be implemented. When measurement distance is large enough compared to the size of SMPL, an average intensity calibration of PSG instrument matrix is acceptable for measurement convenience. By contrast, a complete intensity distribution calibration of PSG is always the most effectiveness method for eliminating the systematic error caused by SMPL. The application demonstration shows that the SMPL-DoFPs system can perform MM imaging of samples in free space, which is conductive to achieve target detection and material classification.
Funding
National Key Research and Development Program of China (2018YFC1406600); Guangdong Development Project of Science and Technology (2020B1111040001); National Natural Science Foundation of China (41527901, 61527826, 61975088).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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