Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 20, Number 1, 2024
EOSAM 2023



Article Number  5  
Number of page(s)  16  
DOI  https://doi.org/10.1051/jeos/2024003  
Published online  12 March 2024 
Research Article
Mueller matrix imaging polarimeter with polarization camera selfcalibration applied to structured light components
^{1}
Instituto de Bioingeniería, Universidad Miguel Hernández de Elche, 03202 Elche, Spain
^{2}
Departamento de Física Aplicada, Universidad Miguel Hernández de Elche, 03202 Elche, Spain
^{3}
Departamento de Ciencias Físicas, Universidad de La Frontera, Temuco, Chile
^{4}
Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
^{5}
Departamento de Ciencia de Materiales, Óptica y Tecnología Electrónica, Universidad Miguel Hernández de Elche, 03202 Elche, Spain
^{*} Corresponding author: mar.sanchez@umh.es
Received:
8
August
2023
Accepted:
25
January
2024
This work presents a complete Mueller matrix imaging polarimeter that uses three liquidcrystal retarders and a pixelated polarization camera. The polarimeter is characterized and optimized with a standard correction procedure here adapted to be performed fully insitu, without any additional element, based on considering the polarization camera as the reference. The accuracy limit caused by the extinction ratio in the camera micropolarizers is analyzed. Finally, the imaging polarimeter is tested experimentally by analyzing wellknown samples for structured light applications such as patterned retarders, a patterned polarizer, and a liquidcrystal depolarizer. The work is presented in a tutorial style useful to reproduce the procedure by nonexperts in polarimetry.
Key words: Polarimetry / LiquidCrystals Retarders / Structured Light / Polarization Camera
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Among the different properties of light, polarization has been demonstrated to provide extremely useful information [1]. This is becoming especially relevant in bioimaging, where several techniques harnessing biological and clinical research are based on the vectorial properties of light [2, 3]. In the last decades, important technological advances related to the generation and measurement of polarized light have been produced, that provide tools and devices like liquidcrystal variable retarders, spatial light modulators (SLM) or polarization cameras that have become common in optics laboratories, leading to the development of precise imaging polarimeters [4, 5].
Another field developed in parallel is the generation and detection of structured light. This denomination was introduced to describe light beams with a spatial and temporal control of their amplitude, phase and state of polarization [6]. Light beams with nonuniform polarization, known as vector beams, are prompting important advances in areas such as microscopy, materials processing, metrology, or optical communications [7, 8]. Structured light is typically produced with patterned polarization elements [9, 10]. SLM devices show the advantage of being reconfigurable, although requiring bulk optical systems [11, 12]. Alternatively, flat patterned polarization components can be fabricated with metamaterials [13] or with liquidcrystal (LC) geometricphase elements [14, 15]. Patterned polarizers with arbitrary spatial distributions of the transmission axis are of great interest especially for developing micropolarizers in polarization cameras [16].
Research in patterned polarization elements and research in polarimetric imaging are closely related [17]. While patterned polarization elements are providing new tools for advanced polarization imaging [18, 19], polarimetric imaging relying on the Mueller matrix has proven very worthy to evaluate the quality of the fabricated components [20, 21]. Although Mueller matrix imaging polarimetry has been employed for almost three decades [22], it is a technique in constant evolution, where multiple variants have been introduced over the years [23, 24]. In this context, this work presents the realization of a complete Mueller matrix (MM) imaging polarimeter whose performance is tested on polarization elements commonly applied to generate structured light. This polarimeter is based on our previous system [21], but here we provide the following improvements: (1) the use of a multiwavelength LED light source to avoid interference and speckle noise, (2) the realization of a setup on a rotatable breadboard to easily change the polarimeter from a transmission to a backreflection configuration, (3) the use of spectrally calibrated liquidcrystal retarders (LCR), both in the polarization state generator (PSG) and in the polarization state analyzer (PSA), and (4) the application of a wellstablished sequential polarimeter calibration and optimization method [25], here adapted to consider the polarization camera as the reference for polarization measurements (selfcalibration). This way, the method can be applied without any additional elements simply by properly tuning the LCR devices in the system. These improvements allow us to present a multifunctional complete MM imaging polarimeter that can be applied to a variety of samples.
The paper is organized as follows: after this introduction, Section 2 describes the polarimeter components. Then, Section 3 presents the calibration steps required to correctly operate the system, including the correct tuning of the LCR devices and the calibration and compensation procedure of the PSG and the PSA systems. Finally, in Section 4 we discuss the results obtained to evaluate different elements used in structured light like spatially patterned retarders and polarizers, and a liquidcrystal depolarizer.
2 Description of the imaging polarimeter
The developed imaging polarimeter system is shown in Figure 1. As mentioned, it represents an upgraded version of our previous system [21] with larger potential for sample imaging and quality control applications. The light source is a multiwavelength RGB LED (Thorlabs LED4D067) with controllable intensity level. A light guide (Thorlabs LLG054H) directs the light to the polarimeter entrance. A 50 mm photographic objective lens is used to image the lightguide output onto the sample. A diffuser (Edmund Holo 30 deg) is added to improve the intensity uniformity on the sample plane. The PSG is composed by a vertically oriented linear polarizer, and two liquidcrystal variable retarders (LCR1 and LCR2) from ARCOptix. LCR1 is oriented at 45° while LCR2 is oriented vertically, such that upon adjusting their retardances, an arbitrary fullypolarized state of polarization (SOP) can be generated.
Fig. 1 Scheme of the Mueller matrix imaging polarimeter. D: diffuser. SF: Spectral filter. P: linear polarizer. LCR: liquidcrystal retarders. (a) Transmission geometry. (b) Reflection geometry. (c) Scheme of the micropolarizers pattern on a macropixel of the polarization camera. (d) Spectra of the three LED sources with central wavelengths of 470 nm, 565 nm and 660 nm, and after being filtered by the corresponding spectral filter (SF). Pictures of the system in (e) the transmission configuration and (f) the reflection configuration. 
The polarized light illuminates the sample and after lightmatter interaction the output is analyzed with a PSA. A circular iris diaphragm of variable diameter is placed just in front of the sample plane and kept in the calibration procedure. The PSA is an imaging detector that comprises a polarization camera and another variable retarder (LCR3), also from ARCOptix. The sample and the PSA are positioned on a rotating breadboard (Thorlabs RBB300A/M), so the detection system can be rotated an angle θ, changing from a transmission configuration (Fig. 1e) to a reflection configuration (Fig. 1f). The transmission configuration is used in the calibration process, and it is useful for analyzing highly transparent samples. The reflection configuration is suitable for samples with high scattering and for reflective devices.
The polarization camera (Thorlabs CS505MUP Kiralux) is a monochrome sensor with 12 bit resolution (4096 intensity digital levels) and with 2448 × 2048 square pixels of 3.45 μm side. The sensor includes an integrated micropolarizer array attached to the pixel detectors. Thus, images for horizontal (H), vertical (V), diagonal (D) and antidiagonal (A) linear analyzers can be captured in a single shot (Fig. 1c). An objective macro zoom lens (Computar MLH 10X) is attached to the polarization camera; thus, the sample can be imaged with relatively large magnification at distances around 20 cm. If larger magnification is required, an extension tube is added.
The three RGB LEDs of the light source have central wavelengths at 660 nm (red light), 565 nm (green light) and 470 nm (blue light) respectively. Figure 1d shows the spectra of the three LEDs, measured with a spectrometer (StellarNet, STNBLKCSR) in the spectral range from 400 to 700 nm, with 2 nm resolution). A spectral filter (SF) filters each of these LED spectra, resulting in filtered spectra with full width at half maximum (FWHM) of about 10 nm, as the measured narrow RGB bands shown in Figure 1d. The SF is useful to reduce depolarization values that might have been artificially added to the final measurements due to the LCRs wavelength retardance dependence.
3 Calibration procedure
This section presents the calibration procedures required to accurately operate the MM imaging polarimeter. First, we review the method for calibrating the LCR devices. Then, the following subsection describes the calibration and optimization of the polarimeter.
3.1 Calibration of the LCR devices
The first step to properly operate the polarimeter is to calibrate the retardance of the LCR devices that will be used to provide the required SOPs, both in the PSG and in the PSA. For that purpose, we follow a simple yet accurate enough procedure that consists in placing the LCR device between crossed/parallel polarizers, oriented at 45° with respect to the LC director axis [26, 27]. The normalized transmission for crossed polarizers is given by$${T}_{\perp}\left(\lambda ,V\right)={\mathrm{cos}}^{2}\left[\frac{1}{2}\phi \left(\lambda ,V\right)\right],$$(1)where the LC retardance φ(λ, V) = (2π/λ) · Δn · t depends on the thickness of the LC layer (t), Δn(V) is the voltagedependent birefringence, λ is the wavelength and V is the applied voltage. This relation shows the expected oscillatory behavior of T_{⊥}(λ, V) both as a function of λ and V.
We measured the LCRs retardance versus the applied voltage for the three wavelength bands. As an example, Figure 2 shows the results obtained for LCR1. Figure 2a illustrates the normalized transmission curves T_{⊥}(V) for the three wavelength bands centered at λ_{R} = 660 nm, λ_{G} = 565 nm and λ_{B} = 470 nm. The curves feature the typical oscillatory behavior, with more oscillations for the shortest blue wavelength. Figure 2b shows the retrieved retardance function φ(V) for the three wavelengths. For the longer wavelength λ_{R} = 660 nm the retardance variation is slightly greater than 2π, while it almost reaches 3π and 3.5π for the green and blue bands respectively.
Fig. 2 Example of calibration of an LCR device in the MM imaging polarimeter. (a) Transmission T_{⊥}(V) between crossed polarizers oriented at 45° with respect to the LC director. (b) LCR retardance φ(V). Both graphs show the results for the three bands, with central wavelengths 470 nm (B), 565 nm (G) and 660 nm (R). 
3.2 Calibration of the PSG
We operate the polarimeter to generate and detect the wellknown polarimetric basis consisting of six standard SOP: namely horizontal (H), vertical (V), diagonal (D) and antidiagonal (A) linear states, and right (R) and left (L) circular states. These six SOP define an octahedron in the Poincaré sphere and provide the polarimeter ideal conditional number (CN = 1.732) [25], thus ensuring an optimal performance in terms of noise amplification minimization from intensity measurements to polarimetric measurements. The LCR calibration provides the required voltages; typically, the quarterwave retardance, the halfwave retardance or the fullwave retardance. Considering that the PSG polarizer (P) is oriented vertically, LCR1 is at 45° and LCR2 is vertical, Table 1 provides the retardances required to achieve the input SOPs.
However, variable LCRs feature effects that might introduce inaccuracy in the polarimeter, being the most relevant the nonuniform retardance on the clear aperture [26], the retardance temperature dependance, and multiple reflection interference effects that cause intensity variations coupled to the retardance modulation [28]. These effects must be compensated to achieve accurate polarimetric values. Among the different methods [29–31], here we propose and apply a modification of a wellstablished sequential calibration and optimization method [32, 33]. In the typical procedure, a calibrated polarimeter serves as the reference to measure the SOP of the PSG states, and the results are compared to those measured with the developed polarimeter. Here the technique is adapted to consider the polarization camera as the reference. This way, a selfreference procedure is applied that does not require any additional external element. Of course, the accuracy of this selfcalibration depends on the quality of the micropolarizers in the polarization camera. To this goal Appendix includes an analysis of the limits of this procedure considering the extinction ratio of the micropolarizers in the camera. We measured extinction ratios over 120:1. Although other sources of error have been identified [34] (such as spatial variations of the extinction ratio or misalignments in the orientation of the micropolarizers), here we assume this simplified model with a limited but uniform extinction ratio. According to the approach described in the Appendix, the selfcalibration procedure leads to an error in the measured MM elements below 1.7%.
Figure 3 shows the intensity captured for the circular iris diaphragm under red light illumination. The polarization camera provides a fourquadrant image where quadrants i, ii, iii and iv correspond to the detection of the V, D, A and H linear states. The input intensity is adjusted to ensure the nonsaturation of the camera and it is maintained in the sample characterization. Each picture in Figure 3 corresponds to one of the six standard input SOP generated with the PSG (H, V, D, A, R, L). In each case, the voltage values derived from the LCR1 and LCR2 calibrations are taken as the starting point, but a fine adjustment must be then performed to provide images like those in Figure 3. The successful generation of the linear states H, V, A and D is verified when the following conditions are simultaneously fulfilled: the image quadrant of the orthogonal detector becomes the darkest, the image quadrant corresponding to the given input state is the brightest, and the other two quadrants feature the same intensity. On the contrary, the generation of the circular R and L states is verified when the four quadrants appear with equal weight. The information provided by the LCR calibration in Figure 2 allows differentiating R and L states. On the right, the intensity profiles of the four quadrants for inputs D and L are shown.
Fig. 3 (a) Calibration of the PSG system: experimental fourquadrant images captured by the polarization camera under the red channel illumination for input SOPs as linear horizontal (H), vertical (V), antidiagonal (A), diagonal (D), circular left (R) and circular right (L). (b) The intensity profiles for inputs D and L. (c) Calculations of average values are performed in the central circle with diameter of 600 pixels. 
The polarization camera is an incomplete polarimeter, since only the linear polarization components are measured. The first three Stokes parameters S_{0}, S_{1}, S_{2} can be retrieved from the intensity images I_{i}(x, y), I_{ii}(x, y), I_{iii}(x, y), and I_{iv}(x, y) captured in quadrants i, ii, iii, and iv as$${S}_{0}\left(x,y\right)=\frac{1}{2}\left[{I}_{i}\left(x,y\right)+{I}_{\mathrm{ii}}\left(x,y\right)+{I}_{\mathrm{iii}}\left(x,y\right)+{I}_{\mathrm{iv}}\left(x,y\right)\right],$$(2a) $${S}_{1}\left(x,y\right)={I}_{\mathrm{iv}}\left(x,y\right){I}_{i}\left(x,y\right),$$(2b) $${S}_{2}\left(x,y\right)={I}_{\mathrm{ii}}\left(x,y\right){I}_{\mathrm{iii}}\left(x,y\right).$$(2c)
To retrieve S_{3}, a quarterwave plate must be placed in front of the polarization camera, to measure the circular polarization components. However, since LCR2 is oriented vertical, a quarterwave retardance can be added to this retarder to achieve an equivalent situation and make quadrants ii and iii behave as equivalent R and L circular analyzers. Therefore, the PSG calibration can be completed with$${S}_{3}\left(x,y\right)=I{\mathrm{\prime}}_{\mathrm{ii}}\left(x,y\right)I{\mathrm{\prime}}_{\mathrm{iii}}\left(x,y\right),$$(2d)where now I′ indicates that the additional quarterwave retardance is added to LCR2. Note that LCR1 and LCR2 devices remain within the PSG system in all measurements, and they simply change voltage, so there are not differences due to reflection or absorption losses.
Table 2 presents the average value of the intensity measured for each of the 6 × 4 images shown in Figure 3. The error is given by the standard deviation. These values are calculated in a circle centered on the image and with diameter 600 pixels, as indicated in Figure 3c. These error values are also affected by noise in the detector. The camera was calibrated following a standard procedure [35] and it was operated in the regime limited by shot noise. The impact of this noise can be reduced by averaging several captures of each image, at the cost of increasing the time required for acquisition. Since we are using relatively high levels of intensity and samples with high transmission, we take single captures.
Retardances required on LCR1 and LCR2 to provide SOP in the PSG.
The measurement of the Stokes parameters for each PSG input state defines the S matrix, whose columns are given by images S_{ig} where index i = 0, 1, 2, 3 indicates the Stokes parameter, and index g denotes the state generated by the PSG (in our case. g = H, V, D, A, R, L). Thus, S is a 4 × 6 matrix defined as:$$\mathbf{S}=\left(\begin{array}{cccccc}{S}_{0H}& {S}_{0V}& {S}_{0D}& {S}_{0A}& {S}_{0R}& {S}_{0L}\\ {S}_{1H}& {S}_{1V}& {S}_{1D}& {S}_{1A}& {S}_{1R}& {S}_{1L}\\ {S}_{2H}& {S}_{2V}& {S}_{2D}& {S}_{2A}& {S}_{2R}& {S}_{2L}\\ {S}_{3H}& {S}_{3V}& {S}_{3D}& {S}_{3A}& {S}_{3R}& {S}_{3L}\end{array}\right)=\left(\begin{array}{cccccc}1& 1& 1& 1& 1& 1\\ 1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 1\end{array}\right),$$(3)where the second part of equation (3) denotes the ideal values. Since S is not a square matrix, its pseudoinverse matrix is a 6 × 4 matrix given by:$${\mathbf{S}}^{\mathbf{1}}={\mathbf{S}}^{T}{\left[\mathbf{S}\xb7{\mathbf{S}}^{T}\right]}^{\mathbf{1}}=\left(\begin{array}{cccc}\frac{1}{6}& \frac{1}{2}& 0& 0\\ \frac{1}{6}& \frac{1}{2}& 0& 0\\ \frac{1}{6}& 0& \frac{1}{2}& 0\\ \frac{1}{6}& 0& \frac{1}{2}& 0\\ \frac{1}{6}& 0& 0& \frac{1}{2}\\ \frac{1}{6}& 0& 0& \frac{1}{2}\end{array}\right),$$(4)where superindex T indicates the transposed matrix and where the matrix on the right corresponds again to the ideal values. This expression of the pseudoinverse matrix is valid when it is considered to multiply a matrix on the right.
Figure 4 shows matrices S(x, y) and S^{ −1 }(x, y) retrieved from the data in Figure 3. We use a color map to represent values from +1 to −1. To avoid presenting calculations in pixels where the input light is blocked by the iris diaphragm, we calculated the average I_{mean} (x, y) of the 24 images in Figure 3a and searched for its maximum value M = max[I_{mean} (x, y)]. Then, polarimetric matrices and parameters are calculated at pixels (x, y) where I_{mean} (x, y) > 0.2 M, while pixels not fulfilling this threshold are represented in black. The value 0.2 was tested to provide a visualization of the polarimetric parameters within the iris aperture, while the outside region, which otherwise would appear with random values, appears pitch black. The average values of each element in matrices S(x, y) and S^{ −1 }(x, y) were also calculated, together with its corresponding standard deviation, again within the circle defined in Figure 3c, leading to the results indicated in Table 3. These numerical results agree well with the expected theoretical values in equations (3) and (4).
Fig. 4 Experimental images for the PSG polarimeter calibration under the red channel illumination. (a) S(x, y) matrix and (b) its pseudoinverse matrix S^{ −1 }(x, y). 
Average values of the matrix 〈S(x, y)〉_{exp}.
Average values of the matrix 〈S^{−1}(x, y)〉_{exp}.
3.3 Calibration of the PSA
Once the PSG has been calibrated, the polarimeter calibration is completed by adding the variable retarder LCR3 to the PSA and measuring again for each input SOP generated with the PSG. LCR3 is set with its axes oriented at 45°. LCR3 retardance is switched between zero or fullwave retardance, where the polarization camera measures H, V, D and A states, and a quarterwave retardance, where circular R and L states can be measured at the micropixels with vertical and horizontal polarizers. This way, the PSA operates within an octahedron and it thus corresponds to the optimal solution named “I” in [36].
Setting the PSA to detect different SOPs yields the intensity matrix I defined by the images I_{ag} (x, y), where a denotes the analyzer in the PSA and g denotes the input state at the PSG. We also select the six standard SOP for the PSA. Hence, I is a 6 × 6 matrix where each element I_{ag} is the intensity measured for each input state g = H, V, D, A, R, L when being detected through analyzer a = H, V, D, A, R, L, i.e.:$$\mathbf{I}=\left(\begin{array}{ccccc}{I}_{\mathrm{HH}}& {I}_{\mathrm{HV}}& {I}_{\mathrm{HD}}& {I}_{\mathrm{HA}}& {I}_{\mathrm{HL}}\\ {I}_{\mathrm{VH}}& {I}_{\mathrm{VV}}& {I}_{\mathrm{VD}}& {I}_{\mathrm{VA}}& {I}_{\mathrm{VL}}\\ {I}_{\mathrm{DH}}& {I}_{\mathrm{DV}}& {I}_{\mathrm{DD}}& {I}_{\mathrm{VA}}& {I}_{\mathrm{DL}}\\ {I}_{\mathrm{AH}}& {I}_{\mathrm{AV}}& {I}_{\mathrm{AD}}& {I}_{\mathrm{AA}}& {I}_{\mathrm{AL}}\\ {I}_{\mathrm{RH}}& {I}_{\mathrm{RV}}& {I}_{\mathrm{RD}}& {I}_{\mathrm{RA}}& {I}_{\mathrm{RL}}\\ {I}_{\mathrm{LH}}& {I}_{\mathrm{LV}}& {I}_{\mathrm{LD}}& {I}_{\mathrm{LA}}& {I}_{\mathrm{LL}}\end{array}\right)=\left(\begin{array}{cccccc}1& 0& 1/2& 1/2& 1/2& 1/2\\ 0& 1& 1/2& 1/2& 1/2& 1/2\\ 1/2& 1/2& 1& 0& 1/2& 1/2\\ 1/2& 1/2& 0& 1& 1/2& 1/2\\ 1/2& 1/2& 1/2& 1/2& 1& 0\\ 1/2& 1/2& 1/2& 1/2& 0& 1\end{array}\right),$$(5)where again the matrix values on the right stand for the ideal case.
Let us emphasize that matrix I is formed by images so each element I_{ag} (x, y) depends on the pixel location. Also, note that the polarization camera only requires two shots to capture the six images per input SOP. Thus the 36 images required in matrix I are acquired after only 12 shots corresponding to 6 PSG configurations by 2 PSA configurations. Figure 5a shows the experimental result obtained for I(x, y) under the red channel illumination. For each input SOP, the images were normalized to the image where the analyzer matches the input polarization. The results bear good agreement with the ideal values in equation (6) despite slight variations along the diaphragm aperture. These variations are due to lighting nonuniformities (as shown in the intensity profiles in Fig. 3b), but also to the insertion of the LCR3 device in the PSA. Again, we calculated the average values and standard deviation of the intensity images in Figure 5a characterizing the intensity matrix. The result given in Table 4, although approximates reasonably well the expected values indicated in equation (5), shows significant differences that suggest the necessity of performing the compensation procedure to achieve accurate results.
Fig. 5 Experimental images for the polarimeter PSA calibration under the red channel illumination. (a) Intensity matrix I(x, y). (b) Detector matrix A(x, y). (c) Pseudoinverse detector matrix A^{ −1 }(x, y). 
Average values of the matrix 〈I(x, y)〉_{exp}.
Matrices I in equation (5) and S in equation (3) are related as I = A · S, where for simplicity we omit the pixel dependence. Here A is the detector matrix that characterizes the PSA. For the selected SOP, A is a 6 × 4 matrix that can be calculated as:$$\mathbf{A}=\mathbf{I}\xb7\mathrm{}{\mathbf{S}}^{\mathbf{1}}=\left(\begin{array}{cccc}1/2& 1/2& 0& 0\\ 1/2& 1/2& 0& 0\\ 1/2& 0& 1/2& 0\\ 1/2& 0& 1/2& 0\\ 1/2& 0& 0& 1/2\\ 1/2& 0& 0& 1/2\end{array}\right),$$(6)where the pseudoinverse matrix S^{ −1 } in equation (4) was applied. The corresponding pseudoinverse matrix A^{ −1 } is a 4 × 6 matrix given by$${\mathbf{A}}^{1}={\left[{\mathbf{A}}^{T}\xb7\mathbf{A}\right]}^{1}\xb7{\mathbf{A}}^{T}=\left(\begin{array}{cccccc}1/3& 1/3& 1/3& 1/3& 1/3& 1/3\\ 1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 1\end{array}\right),$$(7)where the matrices on the right in equations (6) and (7) correspond to the ideal values. This expression in equation (7) is valid when it is considered to multiply a matrix on the left.
Figures 5b and 5c show the experimental images of A(x, y) and A^{−1}(x, y), which again bear a good agreement with the expected values in equations (6) and (7). However, note some elements of A^{−1} in Figure 5c appreciably differ from the ideal zero value in equation (7), indicating the correction that the PSA experimental matrix requires. Let us remark that these matrices are calculated at every pixel, thus providing a PSA calibration at pixel level. Again, for each element on these matrices, we calculated the average value and its standard deviation on the circle defined in Figure 3c. The results are given in Table 5. These values differ appreciably from the ideal values in equations (6) and (7) since they account for the nonideal behaviour of the components involved in the PSG and the PSA.
Average values of the matrix 〈A(x, y)〉_{exp}.
Average values of the matrix 〈A^{−1}(x, y)〉_{exp}.
4 Mueller matrix imaging
Once the PSG and PSA matrices, S(x, y) and A(x, y), have been determined, the sample can be introduced in the polarimeter. For each input SOP generated by the PSG, the images given by the PSA are captured. As a result, a new set of 36 images are obtained that define a new intensity matrix I_{S}(x, y), like in equation (5), now related to the sample Mueller matrix M_{S}(x, y) as$${\mathbf{I}}_{\mathrm{S}}\left(x,y\right)=\mathbf{A}\left(x,y\right)\xb7{\mathbf{M}}_{\mathrm{S}}\left(x,y\right)\xb7\mathbf{S}\left(x,y\right).$$(8)
M_{S}(x, y) is then obtained with the aid of the pseudoinverse matrices A^{−1} and S^{−1} as:$${\mathbf{M}}_{\mathrm{S}}\left(x,y\right)={\mathbf{A}}^{1}\xb7{\mathbf{I}}_{\mathrm{S}}\xb7{\mathbf{S}}^{1}=\left\{{\left[{\mathbf{A}}^{T}\xb7\mathbf{A}\right]}^{1}\xb7{\mathbf{A}}^{T}\right\}\xb7{\mathbf{I}}_{\mathrm{S}}\xb7\left\{{\mathbf{S}}^{T}\xb7{\left[\mathbf{S}\xb7{\mathbf{S}}^{T}\right]}^{1}\right\}.$$(9)
We illustrate the procedure by considering different polarization devices that modify the input SOP by three distinct physical mechanisms: a pure retarder, a pure diattenuator, and a pure depolarizer. These samples of welldefined properties thus serve as validation tests to probe the accuracy of the imaging polarimeter. For each sample, the experimental MM was retrieved following equation (9) and the Lu–Chipman decomposition M_{S} = M_{Δ} · M_{R} · M_{D} [37] was applied to calculate relevant polarimetric parameters using standard formulas [21]. We applied the very wellknown Lu–Chipman decomposition because these samples, each of distinct physical mechanism, imply relatively simple Mueller matrices. Note that more complex Mueller matrices, simultaneously dealing with retardance, depolarization and diattenuation responses could require other decompositions, like for instance the inverse decomposition, the arrow form decomposition or the symmetric decomposition, among others [38].
As a first example, Figure 6 shows the results for two patterned retarders: a birefringent NBS 1963A resolution test (Thorlabs R2L2S1B), and a qplate device (Thorlabs WPV10633), both under the red channel illumination. Figures 6a and 6c show the retrieved MM normalized to m_{00}. They show the expected result for a retarder, where m_{0j} = m_{j0} ≈ 0, for j = 1, 2, 3, indicating null polarizance and null diattenuation. The retarder information is contained in the 3 × 3 bottom right submatrix. The retardance vector [21, 37] is calculated and some representative parameters of the retarder are shown in each case.
Fig. 6 (a) Experimental MM of the birefringent resolution test under red light channel illumination. (b) Retrieved horizontal linear retardance (R_{L}) and total retardance (R), and azimuth (α) and ellipticity (ε) of fast eigenstates. (c) and (d) show equivalent results for the qplate element under red channel illumination. 
Figures 6a and 6b show the results for the birefringent resolution test. This is a linear retarder designed to have a uniform retardance but a different orientation in the patterns compared to the background. The total retardance R(x, y) appears uniform all over the image, with average value 〈R〉 = 131° ± 1° (the value is obtained by averaging over the significant pixels and the error is given by its standard deviation, where again, the significant pixels are those comprised within a circle as defined in Fig. 3c). However, the horizontal component of the retardance vector R_{H}(x, y) clearly shows a difference between the patterns and the background. The azimuth α(x, y) of the fast eigenstate confirms the two regions, with the background oriented approximately horizontal, 〈α_{b}〉 = −8.5° ± 0.5°, and the resolution test pattern oriented at 〈α_{t}〉 = −55.8° ± 1.0°. The ellipticity image ε(x, y) of the fast eigenstate remains uniform over the entire image, with a very small value, 〈ε〉 = 2.6° ± 1.2° as expected for a linear retarder.
Figures 6c and 6d show equivalent results for the qplate component, a patterned retarder widely used to generate orbital angular momentum (OAM) beams and vector beams [39]. This is a linear retarder with fixed retardance and whose axes orientation changes azimuthally. This qplate was designed as a halfwave retarder for the 633 nm wavelength and with value q = 1, so the retarder axes make a complete azimuthal rotation. In this case, since the retarder is very close to the halfwave condition, the most relevant information in the MM is in the m_{11}, m_{12}, m_{21,} and m_{22} elements, which show an azimuthal variation in the complete range from −1 to +1, and in the m_{33} element, which appears uniform with a value around −1. In this case Figure 6d illustrates the linear R_{L} and circular R_{C} components of the retardance vector, which confirm the linear retarder condition of this component. The total retardance is uniform with an average value 〈R〉 = 178° ± 1.1° in agreement with the halfwave retardance design of this qplate for red light. The azimuth of the fast eigenstate orientation α(x, y) shows a continuous azimuthal variation, as corresponding to a qplate with q = 1 and the eigenstate ellipticity ε(x, y) shows a null constant value with an average value 〈ε〉 = 1° ± 3°, as expected for a linear retarder.
The second type of structured polarization sample is a pure diattenuator: a radial polarizer from the company Codixx (ColorPol VIS500 BC3), consisting of 12 segments where the transmission axis is aligned radially, thus being shifted 30° between adjacent segments. Figure 7a shows the corresponding MM, again using the red channel. In this case the last row and column become null, m_{3j} = m_{j3} ≈ 0 with j = 0, 1, 2. The other MM elements clearly show the segmentation of the polarizer. Figure 7b shows as parameters of interest the linear and circular components of the diattenuation, D_{L}(x, y) and D_{C}(x, y). The average total diattenuation is 〈D〉 = 0.85 ± 0.13, where again the value is averaged over the significant pixels and the error is given by its standard deviation. We also calculate the polarizer transmission angle α(x, y), which confirms the expected rotation.
Fig. 7 (a) Experimental MM of a radial polarizer under red channel illumination. (b) Linear diattenuation (D_{L}), circular diattenuation (D_{C}), and azimuth (α) of the transmission axis. (c) Experimental MM for a depolarizing FLC modulator under the green channel illumination. (d) Linear depolarizance (Δ_{L}), circular depolarizance (Δ_{C}), and total effective retardance (R). 
Finally, as a pure depolarizing sample we consider a ferroelectric liquid–crystal (FLC) modulator. This type of modulator behaves as a linear retarder whose director axis switches between two stable orientations within the modulator plane [40]. When operated at a high frequency compared to the detector integration time, FLC modulators have proven to be useful to generate an effective depolarization [41]. In this work, we use a FLC modulator (CRL Opto LCS2G) showing a switching angle of 45° and a retardance of 180° for green light [40]. We operate the device at a switching frequency of 500 Hz, much higher than the frame rate of the polarization camera. Therefore, each measurement with the camera corresponds to the incoherent superposition of the two states emerging from the FLC [42]. The measured MM can be regarded as the average of M_{A} and M_{B} the two matrices corresponding to the two stable states of the modulator.
The MM of a retarder with retardance φ and orientation α, corresponding to the first FLC stable position is given by [1]:$${\mathbf{M}}_{\mathrm{A}}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {c}_{2\alpha}^{2}+{c}_{\phi}{s}_{2\alpha}^{2}& \left(1{c}_{\phi}\right){s}_{2\alpha}{c}_{2\alpha}& {s}_{\phi}{c}_{2\alpha}\\ 0& \left(1{c}_{\phi}\right){s}_{2\alpha}{c}_{2\alpha}& {s}_{2\alpha}^{2}+{c}_{\phi}{c}_{2\alpha}^{2}& {s}_{\phi}{c}_{2\alpha}\\ 0& {s}_{\phi}{s}_{2\alpha}& {s}_{\phi}{c}_{2\alpha}& {c}_{\phi}\end{array}\right),$$(10)where c_{x} = cos(x) and s_{x} = sin(x). The second FLC stable position is given by M_{B} = (−45°) · M_{A} · R(+45°) where R stands for the Mueller rotation matrix. Then, the average MM given by $\langle {\mathbf{M}}_{\mathrm{FLC}}\rangle =\frac{1}{2}\left({\mathbf{M}}_{A}+{\mathbf{M}}_{B}\right)$ results in$$\langle {\mathbf{M}}_{\mathrm{FLC}}\rangle =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& {c}_{2\phi}^{2}& 0& \frac{1}{2}{s}_{\phi}\left({c}_{2\alpha}+{s}_{2\alpha}\right)\\ 0& 0& {c}_{2\phi}^{2}& \frac{1}{2}{s}_{\phi}\left({c}_{2\alpha}+{s}_{2\alpha}\right)\\ 0& \frac{1}{2}{s}_{\phi}\left({c}_{2\alpha}+{s}_{2\alpha}\right)& \frac{1}{2}{s}_{\phi}\left({c}_{2\alpha}+{s}_{2\alpha}\right)& {c}_{\phi}\end{array}\right].$$(11)
In the case of a halfwave retardance, φ = 180°, this average matrix 〈M_{FLC}〉 becomes:$$\langle {\mathbf{M}}_{\mathrm{FLC}}\rangle =\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right).$$(12)
This result can be regarded as a perfect depolarizer for input linear polarizations, whereas circular polarizations retain the full degree of polarization but change handedness.
Figure 7c shows the experimental MM results obtained with the polarimeter. In this case we use the green channel where the FLC layer behaves closely to the halfwave retardance condition [40]. Now the experimental MM elements are uniform images since the FLC modulator is a singlepixel element. The retrieved MM matches quite well the expected ideal result in equation (12). All the elements are close to zero, except m_{00} ≈ 1 and m_{33} ≈ −1. Elements m_{32} and m_{23} slightly differ from zero, due to the nonperfect halfwave retardance of the FLC layer. Figure 7d shows images of the depolarization and retardance parameters are calculated. The linear depolarizance Δ_{L}(x, y) shows high values, with an average value 〈Δ_{L}〉 = 0.88 ± 0.03, while the circular depolarization Δ_{C}(x, y) has very low values, with 〈Δ_{C}〉 = 0.04 ± 0.03. This confirms the ability of the FLC modulator to depolarize linearly polarized input light. Finally, the effective retardance R(x, y) shows an average value 〈R〉 = 158° ± 3°, not far but nonperfect halfwave retardance. Again, the error given in these average values is calculated as the standard deviation of the values in all pixels in the aperture.
5 Conclusion
In summary, this work presents an imaging complete MM polarimeter comprised of a polarization camera and three LCRs. The use of polarization cameras in imaging systems is quite recent and it is arising high interest in a wide range of applications. The presented polarimeter is an improved version of our previous system [21] with significant advances. First, a multiwavelength LED light source replaced the laser source, thus avoiding interference and speckle derived from the coherent source. A rotatable breadboard is added to allow changing from the transmission configuration to a configuration useful for reflective or backscattering samples. The system incorporates two LCRs in the PSG and another one in the PSA, to allow fully automated measurements without any moving element. All LCR devices were calibrated for the three spectral bands of the LED source.
The polarimeter was calibrated and compensated by modifying the time sequential method [25, 32]. The procedure has been illustrated step by step. The novelty here is that the polarization camera included in the polarimeter is itself the calibration reference. This way no additional external elements are required and the polarimeter can be fully optimized in situ. Furthermore, the procedure is applied at pixel level, therefore compensating the residual nonuniformities in the illumination and in the PSA system, because of the pixelwise calculation of the normalized Mueller matrix.
This approach of course relies on the quality of the polarization camera. To this aim, the Appendix incorporates a calculation of the error in the measurement of the MM elements caused by the limited extinction ratio (ER) of the micropolarizers, which shows an error lower than 1.7% for the minimum value of ER = 120 measured in our camera. The calibration and compensation procedure has been illustrated step by step by displaying the S(x, y) and A(x, y) images that define the PSG and the PSA. They are compared with the theoretical expected values, showing a very good agreement in all cases.
Finally, three different samples used in structured light applications with wellknown polarization properties were evaluated to verify the accuracy of the polarimeter: (1) two patterned pure retarders with constant retardance but different orientation of the optical axis (a birefringent resolution test and a qplate), (2) a radial linear polarizer, and (3) a linear depolarizer based on a fastswitching FLC modulator with 45° switching angle. The retrieved experimental MM agree very well with the expected results and, after performing the Lu–Chipman decomposition, the polarization parameters of interest (retardance, diattenuation and depolarization) were derived in each case.
Funding
This work was financed by Ministerio de Ciencia e Innovación, Spain (refs.: PID2021126509OBC22, C21 and PDC2022133332C22) and Generalitat Valenciana (ref. CIAICO/2021/276). EN acknowledges Generalitat Valenciana (ref. GRISOLIAP/2020/004). AV acknowledges Fondecyt N°1191811.
Conflicts of Interest
This work has no financial or nonfinancial competing interests.
Data availability statement
Data will be made available on request.
Author contribution statement
All coauthors contributed to the paper. EN and AV contributed with the realization of the experiments. EN and IM contributed with the computer calculations. MMSL, AL, JC and IM contributed to the conceptualization, design of the experiments, and analysis of the results. MMSL contributed to writing the manuscript, and all authors contributed to the revision of the manuscript.
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Appendix
A1 Evaluation of the micropolarizers limited extinction ratio
The polarization camera has wiregrid micropolarizers attached to the pixels of the CMOS detector. In this appendix we evaluate the accuracy of the polarimeter selfcalibration procedure in terms of their extinction ratio. Since the polarization camera is used as the reference, the accuracy of the polarimeter calibration is dictated by the quality of its polarizers. Although other sources of errors have been identified (such as the spatial variation of the extinction ratio, or spatial variations in the orientation angle of the micropolarizers transmission axes) [34], here we assume a simplified model where we consider the micropolarizers perfectly oriented (horizontal, vertical, diagonal and antidiagonal), but having a limited and spatially uniform extinction ratio ER = T_{}/T_{⊥} where T_{} and T_{⊥} are the transmission for parallel and crossed polarizers respectively. Information about this main source of error in the polarization camera is provided by the supplier, indicating in our case values ER > 120:1 in the three colour bands. To take this into account, the MM of these four linear polarizers with limited ER is calculated analytically. The MM of a polarizer aligned horizontal and with limited extinction ratio is given by [1]:$${\mathbf{P}}_{0}=\frac{{T}_{\parallel}}{2}\left(\begin{array}{cccc}1+e& 1e& 0& 0\\ 1e& 1+e& 0& 0\\ 0& 0& 2\sqrt{e}& 0\\ 0& 0& 0& 2\sqrt{e}\end{array}\right),$$(A1a)where e ≡ 1/ER = T_{⊥}/T_{∥}. The limit e → 0 leads to the ideal polarizer with normalized MM:$${\mathbf{P}}_{0}\left(e=0\right)=\frac{{T}_{\parallel}}{2}\left(\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$(A1b)
We measured values ER > 120:1 for our camera polarizers in the three spectral bands, in agreement with the manufacturer data. The MM of the micropolarizers oriented vertical and at ±45° are simply obtained by applying the corresponding rotation transformation [1].
Assuming ideal input states (ideal values in the S matrix in Eq. (3)), and that the circular components R and L are measured using an ideal QWP in front of the camera, the intensity measured for each input state when being analyzed with these limited linear micropolarizers can be calculated analytically, leading to an intensity matrix as in equation (5) which now takes the following form:$${\mathbf{I}}_{\mathrm{air}}\left(e\right)={T}_{\parallel}\left(\begin{array}{cccccc}1& e& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)\\ e& 1& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)\\ \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& 1& e& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)\\ \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& e& 1& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)\\ \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& 1& e\\ \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& \frac{1}{2}\left(1+e\right)& e& 1\end{array}\right),$$(A2)where the ideal matrix in equation (5) is recovered for the limit e → 0 and T_{∥} → 1. Hence, equation (A2) is the intensity matrix of the polarimeter calibration, which contains the errors induced by the limited ER of the micropolarizers. The analytical expression of the calibrated PSA matrix that compensates the limited ER is thus obtained as A(e) = I_{air}(e) · S^{−1}) using equation (A2) and the ideal S^{−1} matrix in equation (4). Its pseudoinverse is given by:$${\mathbf{A}}^{1}\left(e\right)=\frac{1}{{T}_{\parallel}}\left(\begin{array}{cccccc}\frac{1}{3\left(1+e\right)}& \frac{1}{3\left(1+e\right)}& \frac{1}{3\left(1+e\right)}& \frac{1}{3\left(1+e\right)}& \frac{1}{3\left(1+e\right)}& \frac{1}{3\left(1+e\right)}\\ \frac{1}{1e}& \frac{1}{1e}& 0& 0& 0& 0\\ 0& 0& \frac{1}{1e}& \frac{1}{1e}& 0& 0\\ 0& 0& 0& 0& \frac{1}{1e}& \frac{1}{1e}\end{array}\right).$$(A3)
This matrix recovers the ideal case in the right part of equation (7) in the limit when e → 0 and T_{∥} → 1.
To evaluate analytically the impact of the limited extinction ratio on the polarimeter measurement, we first consider as sample the air and apply equation (9), M = A^{−1} I S^{−1}, to obtain the MM of air. Here we use the nonideal intensity matrix I_{air} in equation (A2), but we consider the ideal values of the matrices S^{−1} and A^{−1} given in the right part of equation (4) and equation (7) respectively. This way we assume that the polarizers extinction ratio error is present in the experiment, but it is ignored in the polarimetric calculation. The analytical calculation leads to a MM matrix affected by the error e given by:$${\mathbf{M}}_{\mathrm{air}}\left(e\right)={\mathbf{A}}^{1}\xb7{\mathbf{I}}_{\mathrm{air}}\left(e\right)\xb7{\mathbf{S}}^{1}={T}_{\parallel}\left(\begin{array}{cccc}1+e& 0& 0& 0\\ 0& 1e& 0& 0\\ 0& 0& 1e& 0\\ 0& 0& 0& 1e\end{array}\right),$$(A4)which recovers the expected identity matrix of air when e → 0 and T_{∥} → 1. The MM in equation (A4) can be rewritten as$${\mathbf{M}}_{\mathrm{air}}\left(e\right)={T}_{\parallel}\left(1+e\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1\delta & 0& 0\\ 0& 0& 1\delta & 0\\ 0& 0& 0& 1\delta \end{array}\right),$$(A5)which shows that the MM normalized to m_{00} has diagonal elements m_{11} = m_{22} = m_{33} = (1 − e)/(1 + e) ≡ 1 − δ. This result reflects that the limited extinction ratio of the camera micropolarizers induces an error δ in the diagonal elements of the normalized MM for air given by:$$\delta =1\frac{1e}{1+e}=\frac{2e}{1+e}=\frac{2}{1+\mathrm{ER}}\simeq \frac{2}{\mathrm{ER}},$$(A6)where we used that e = 1/ER and ER ≫ 1 in the final approximation. Figure A1 shows the evolution of this error parameter as a function of the extinction ratio. Since ER > 120:1 in our polarization camera, the deviation expected from this problem is δ < 1.7%.
Fig. A1 Evolution of the error parameter δ as a function of the polarizers extinction ratio ER. The arrow indicates the limit value ER = 120 measured for our polarization camera. 
Obviously, the identity MM of air is also recovered when the PSA calibration compensates for the limited ER i.e., when using the PSA pseudoinverse matrix A^{−1}(e) in equation (A3). Then equation (9) results directly on the identity matrix since A^{−1}(e) · I_{air}(e) · S^{−1} = A^{−1}(e) · A(e).
We can now analytically show that, for any other sample, the procedure described in Section 4 compensates the limited ER of the micropolarizers if the PSA pseudoinverse matrix A^{−1}(e) in equation (A3) is the one employed to retrieve the Mueller matrix. To illustrate this, we consider as sample an ideal linear polarizer with horizontal transmission axis, whose MM is given by equation (A1b). The 6 × 6 elements of the intensity matrix I_{p}(e) of this polarizer sample are analytically obtained by calculating the intensity I_{ag} expected for each input state, g = H, V, D, A, R, L (assumed ideal), when being detected through analyzers a = H, V, D, A, R, L affected by a limited extinction ratio (e ≠ 0). In this case, the obtained intensity matrix is:$${I}_{p}\left(e\right)={T}_{\parallel}\left(\begin{array}{cccccc}1& e& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ e& 1& \frac{1}{2}e& \frac{1}{2}e& \frac{1}{2}e& \frac{1}{2}e\\ \frac{1}{2}(1+e)& 0& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)\\ \frac{1}{2}(1+e)& 0& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)\\ \frac{1}{2}(1+e)& 0& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)\\ \frac{1}{2}(1+e)& 0& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)& \frac{1}{4}(1+e)\end{array}\right)$$(A7)
The MM of this polarizer sample is then obtained from equation (9). If the calculation is made ignoring the error in the micropolarizers, i.e., using the ideal A^{−1} matrix defined equation (7), the analytical expression of such noncompensated MM is:$${\mathbf{M}}_{p}\left(e\right)={\mathbf{A}}^{1}\xb7{\mathbf{I}}_{p}\left(e\right)\xb7{\mathbf{S}}^{1}=\frac{{T}_{\left\right}}{2}\left(1+e\right)\left(\begin{array}{cccc}1& 1& 0& 0\\ 1\delta & 1\delta & 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$(A8)
Thus, the normalized MM contains errors in the m_{10} and m_{11} elements. These errors are completely compensated by using the pseudoinverse matrix A^{−1}(e) of equation (A3). A simple calculation shows that the matrix product A^{−1}(e) · I_{p}(e) · S^{−1} results in the expected MM in equation (A1b).
Finally, the same kind of calculation can be generalized to an arbitrary sample described by a generic MM:$${\mathbf{M}}_{s}=\left(\begin{array}{cccc}{m}_{00}& {m}_{01}& {m}_{02}& {m}_{03}\\ {m}_{10}& {m}_{11}& {m}_{12}& {m}_{13}\\ {m}_{20}& {m}_{21}& {m}_{22}& {m}_{23}\\ {m}_{30}& {m}_{31}& {m}_{32}& {m}_{33}\end{array}\right).$$(A9)
We again evaluate analytically the intensity matrix I_{S}(e) when the sample is introduced in the polarimeter affected by the error e. We calculate the intensity I_{ag} expected for each input state g = H, V, D, A, R, L (assumed ideal), when being detected through analyzers a = H, V, D, A, R, L affected by a limited extinction ratio (e ≠ 0). Again, the impact of the limited extinction ratio on the polarimeter measurement can be analytically evaluated by considering equation (9), M_{S}(e) = A^{−1} · I_{S}(e) · S^{−1}, where the ideal A^{−1} is applied (thus ignoring the error in the micropolarizers). It is a long but straightforward calculation to obtain that the retrieved MM reads:$${M}_{s}\left(e\right)={T}_{\parallel}\left(1+e\right)\left(\begin{array}{cccc}{m}_{00}& {m}_{01}& {m}_{02}& {m}_{03}\\ {m}_{10}(1\delta )& {m}_{11}(1\delta )& {m}_{12}(1\delta )& {m}_{13}(1\delta )\\ {m}_{20}(1\delta )& {m}_{21}(1\delta )& {m}_{22}(1\delta )& {m}_{23}(1\delta )\\ {m}_{30}(1\delta )& {m}_{31}(1\delta )& {m}_{32}(1\delta )& {m}_{33}(1\delta )\end{array}\right),$$(A10)which recovers the ideal matrix in equation (A9) in the limit e → 0 (δ → 0) and T_{∥} → 1. This shows that all elements in the normalized MM except those in the first row are affected by the same factor 1 − δ.
However, it can also be demonstrated that the sample matrix in equation (A10) is recovered when using the PSA matrix in equation (A3) to calculate A^{−1}(e) · I_{S}(e) · S^{−1} = M_{S}, i.e., the ER error of the micropolarizers is completely compensated.
Finally, let us note that we are applying this compensation method to a simplified model where only a uniform limited extinction ratio of the micropolarizers is considered. The effect of spatial variations of the extinction ratio or errors in the orientation of the micropolarizers can be considered as indicated in [34].
All Tables
All Figures
Fig. 1 Scheme of the Mueller matrix imaging polarimeter. D: diffuser. SF: Spectral filter. P: linear polarizer. LCR: liquidcrystal retarders. (a) Transmission geometry. (b) Reflection geometry. (c) Scheme of the micropolarizers pattern on a macropixel of the polarization camera. (d) Spectra of the three LED sources with central wavelengths of 470 nm, 565 nm and 660 nm, and after being filtered by the corresponding spectral filter (SF). Pictures of the system in (e) the transmission configuration and (f) the reflection configuration. 

In the text 
Fig. 2 Example of calibration of an LCR device in the MM imaging polarimeter. (a) Transmission T_{⊥}(V) between crossed polarizers oriented at 45° with respect to the LC director. (b) LCR retardance φ(V). Both graphs show the results for the three bands, with central wavelengths 470 nm (B), 565 nm (G) and 660 nm (R). 

In the text 
Fig. 3 (a) Calibration of the PSG system: experimental fourquadrant images captured by the polarization camera under the red channel illumination for input SOPs as linear horizontal (H), vertical (V), antidiagonal (A), diagonal (D), circular left (R) and circular right (L). (b) The intensity profiles for inputs D and L. (c) Calculations of average values are performed in the central circle with diameter of 600 pixels. 

In the text 
Fig. 4 Experimental images for the PSG polarimeter calibration under the red channel illumination. (a) S(x, y) matrix and (b) its pseudoinverse matrix S^{ −1 }(x, y). 

In the text 
Fig. 5 Experimental images for the polarimeter PSA calibration under the red channel illumination. (a) Intensity matrix I(x, y). (b) Detector matrix A(x, y). (c) Pseudoinverse detector matrix A^{ −1 }(x, y). 

In the text 
Fig. 6 (a) Experimental MM of the birefringent resolution test under red light channel illumination. (b) Retrieved horizontal linear retardance (R_{L}) and total retardance (R), and azimuth (α) and ellipticity (ε) of fast eigenstates. (c) and (d) show equivalent results for the qplate element under red channel illumination. 

In the text 
Fig. 7 (a) Experimental MM of a radial polarizer under red channel illumination. (b) Linear diattenuation (D_{L}), circular diattenuation (D_{C}), and azimuth (α) of the transmission axis. (c) Experimental MM for a depolarizing FLC modulator under the green channel illumination. (d) Linear depolarizance (Δ_{L}), circular depolarizance (Δ_{C}), and total effective retardance (R). 

In the text 
Fig. A1 Evolution of the error parameter δ as a function of the polarizers extinction ratio ER. The arrow indicates the limit value ER = 120 measured for our polarization camera. 

In the text 
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