The COronal Diagnostic EXperiment (CODEX): pre-flight polarimetric characterization

The COronal Diagnostic EXperiment (CODEX) is an externally occulted solar coronagraph designed to observe the linearly polarized K-corona to simultaneously measure the electron density, kinetic temperature, and speed. The objective of this work is to provide a detailed description of the characterization of the CODEX coronagraph from a polarimetric point of view. The final goal is to minimize the uncertainties associated with the outcome of the observation data analysis by removing the contributions of artifacts introduced by the optical elements within the instrument. This aspect is particularly critical for the CODEX coronagraph, since it is provided with a polarization image sensor, the IMX253MZR manufactured by Sony, that spatially modulates the incoming light beam. Moreover, the optical path of the instrument is composed of several elements including two fold-mirrors positioned at 90°, a system that is notoriously a source of polarization aberrations. The methodology used consists of introducing a light beam with a known polarization state into the instrument and measuring its response. Repeated measurements for different states of polarization of the incoming beam allow one to derive the matrix that connects the Stokes vector of the input light to the intensity measured by the detector. The outcome of this analysis is represented by this matrix, which will be used to derive the Stokes vector of the coronal light from the data acquired by CODEX during its mission and that takes into account the instrumental effects. This will minimize the instrumental effects on coronal polarization that occur if one simply derives the Stokes vector by combining the observed polarized images.

1 Introduction

The COronal Diagnostic EXperiment (CODEX) is a next-generation externally occulted solar coronagraph that aims to address unresolved questions related to the solar corona heating mechanism and the dynamics of the coronal features. CODEX will collect new breakthrough data to enable simultaneous measurements of the electron density, temperature, and speed in the Sun’s atmosphere for the first time, providing a snapshot of the physical conditions of the plasma composing the solar corona.

CODEX is the result of a scientific and technical collaboration led by NASA-Goddard Space Flight Center (GSFC) and supplemented with expertise from the Naval Research Laboratory (NRL), the Korea Astronomy and Space Science Institute (KASI), and the Italian National Institute for Astrophysics (INAF). The instrument was launched in November 2024 to observe the solar corona as an external payload of the International Space Station (ISS) with a nominal mission of about 6 months.

The scientific experiment carried out by the CODEX coronagraph is based on the theory formulated by Cram (1976) for the determination of the electron temperature and on its extension to the speed presented by Reginald et al. (2023). Cram’s intuition was to use measurements of the intensity of the solar corona in polarized light to derive the temperature of the electrons. The polarized radiation incoming from the white-light corona is mainly a consequence of Thomson scattering in the solar atmosphere. One of the components of the atmosphere of the Sun, the so-called K-corona (i.e., Kontinuerlich for continuum in German) is the result of the photospheric radiation scattering off the free coronal electrons (Billings, 1966). This physical process links the polarization state of the coronal light to the electron properties such as density, temperature, and speed, and makes the polarization analysis a means of investigating the state of the electrons in the plasma, therefore of fundamental importance for diagnosing the solar corona (Van de Hulst, 1950; Cram, 1976; Reginald et al., 2017). In particular, the observations performed with the CODEX coronagraph target the differential smoothing in the spectrum of the continuum corona (near the Ca II and K Fraunhofer lines) due to the high thermal velocities of the hot coronal electrons. These hot electrons cause the broadening or smearing of the strong absorption lines of the photospheric spectrum, hence measurements of the integrated signal in these areas of the spectrum will reflect the temperature of the electrons in the solar corona.

CODEX is designed to collect images of the solar corona in polarized light within the wavelength range spanning from 385 to 440 nm to measure the coronal electron density, temperature, and speed between 3 and 8 solar radii.

To measure the polarization state of the observed signal, the CODEX coronagraph implements a spatial modulation (Tyo et al., 2006; Snik et al., 2014) of the intensity of incident radiation. The CODEX camera comprises a layer of micropolarizing elements placed on the focal plane of the detector, allowing the desired polarization information to be captured in each frame by collecting four different polarized signals in a single image. The advantage of using focal-plane assemblies is represented by the possibility of having a compact solution for polarimetric measurements; however, they sacrifice spatial resolution and generate misregistration among the different polarized images. Moreover, such polarimeters are affected by imperfections that result in spatial variation in the optical and electronic response to a uniform polarized signal. Every pixel has a different sensitivity to light that needs to be characterized. Although ground-based observations of the solar corona have been previously acquired with a polarization camera (Reginald et al., 2017; Vorobiev et al., 2020; Gopalswamy et al., 2021; Liang et al., 2023; Caspi et al., 2024), CODEX (Newmark, 2026; Newmark et al., 2020; Cho et al., 2020) is the first space-based coronagraph to implement this technology (Reginald et al., 2019).

The CODEX coronagraph was designed with the objective of avoiding any alteration of the incident polarized light. However, in reality, all optical interfaces can change the polarization state of light to some degree, and the presence of lenses, mirrors, coatings, and diffractive elements in an instrument optical path is likely to introduce the so-called instrumental polarization. This is a well-known problem with many studies in the literature (Chipman, 1988; 1990; McGuire and Chipman, 1994; Chipman et al., 2018). In an instrument such as CODEX, that is used to measure the polarization of light, this contribution might lead to inaccurate measurements of the characteristics of the observed polarized light such as the Degree of Linear Polarization (DoLP), which is fundamental for the K-corona diagnostics (Reginald and Rastaetter, 2019). For this reason, the polarimetric characterization of the instrument is a fundamental requirement. In this paper, we describe the method applied to characterize the polarimetric properties of CODEX. The obtained results will be used to measure the polarization state of the coronal light observed by CODEX during its mission.

Section 2 gives a brief overview of the CODEX design to introduce the reader to the complexity of the instrument and to justify an extensive polarization analysis. The coronagraph polarization analysis is detailed in Section 3, where data collection and the mathematical method are described. Finally, our conclusions are described in Section 4.

2 The CODEX coronagraph

The CODEX optical design is the result of the trade-off made to satisfy the measurement goals and the constraints imposed by the ISS on external payloads in terms of volume and mass. The desired observational wavelength range, which is far from the maximum value of the solar spectrum, and the object of the observations, i.e., the faint solar corona, make it difficult to achieve the necessary signal-to-noise ratio (SNR; effective SNR of about 50) required to measure the temperature and the speed of coronal electrons (Reginald et al., 2021). Moreover, the measurement of these two quantities is obtained by combining K-coronal observations performed in two narrow wavelength ranges, i.e., 393.5$\pm$5 nm and 405.0$\pm$5 nm for the temperature and 398.7$\pm$5nm and 423.3$\pm$5 nm for the speed measurements (Cram, 1976; Reginald et al., 2023). This results in an even lower signal. Aiming at increasing the signal as much as possible to overcome the expected level of noise, the following solutions were implemented in the instrument design: (i) a large frontal aperture (i.e., about 170 mm): to increase the flux of photons entering the instrument, (ii) a system of stops in the optical path to limit the amount of instrumental stray light reaching the detector, and (iii) a polarization image sensor to capture only the coronal polarized signal. These features are combined with a two-fold mirror solution to create a two-stage coronagraph and satisfy the constraint on the instrument volume imposed by the ISS. Table 1 reports the main parameters of the final design.

Table 1

Table 1. Design parameters of the CODEX coronagraph.

The result is a coronagraph with a fairly complex design composed of several optical elements. Figure 1 shows the optical layout of the CODEX coronagraph. The downside of this design is represented by the fact that every element in the optical path has a potential effect on the incident polarization, in terms of diattenuation and retardance. Residual polarization, that is, the polarization introduced by elements such as mirrors and lenses, especially in the case of the CODEX experiment, is an undesired effect that must be characterized and accounted for. For instance, a second fold mirror was introduced in the instrument design with a plane of incidence orthogonal to the first fold, which compensates for the polarization state changes in one point of the field of view (Crandall and Chipman, 1995; Lam and Chipman, 2015). In addition to this, due to some difficulties faced during the instrument integration, the optics within the optical path are not in the nominal position; the first fold mirror has been integrated with a tilt angle not foreseen in the original optical design. One of the consequences is that within the second stage of the coronagraph the chief ray has a diverging angle that moves away from the instrument optical bench. Since the most important requirement for a coronagraph is to make sure that the minimum possible amount of diffracted light would reach the detector, the optical elements have been mounted to ensure that this requirement would be satisfied, even if that implies an increase of aberrations.

Figure 1

Figure 1. CODEX optical system layout. The light enters the instrument through the frontal aperture (A0), where the Sun disk is blocked by the external occulter (EO, it is not shown in this layout). After passing through the telescope boom, the coronal light and the light diffracted by EO is collected by the lens group (A1) and directed towards the two-fold mirrors system that rotates the beam of 90°. A series of lenses and stops ensure that the coronal light is focused on the detector, while the diffracted light is blocked.

A detailed description of the instrument optical design is beyond the scope of this paper and can be found in Gong et al. (2024).

The CODEX camera was developed by the Korea Astronomy and Space Science Institute (KASI). The detector selected for the CODEX instrument is a commercial polarization image sensor manufactured by Sony, the IMX253MZR. This detector consists of a CMOS array and a layer of four-directional (0, 45, 90, and 135°) wire-grid polarizers characterized by high transmission and extinction in the visible-light spectrum. The CMOS has $4096\times 3000$ pixels of 3.45$\mu$m x 3.45$\mu$m. CODEX utilizes this detector to simultaneously image each of the four polarizer angles in a single observation, therefore eliminating any evolution of the scene between polarization observations. The limit on the spatial resolution deriving from this solution is not a problem in the case of the CODEX coronagraph, since the scientific objectives do not involve small-scale features in the solar corona, but rather are of the global and mesoscale structures in the solar corona and nascent solar wind. The Sony IMX253MZR (https://www.sony-semicon.com/en/products/is/industry/polarization.html) CMOS has been designed with particular attention to preventing any cross-talk phenomena between neighboring pixels, enhancing the extinction ratio. The polarizing element on top of each pixel is a grid of parallel conducting wires, with the transmission axis perpendicular to the wires. We use the term “super-pixel” as the group of four adjacent pixels that will be used to evaluate the polarimetric characteristics of the same area resolved by the instrument.

3 Materials and methods

Before the polarimetric characterization of the whole instrument, KASI performed a calibration campaign dedicated to the standalone camera before its delivery for integration within the coronagraph. The campaign was not limited to the polarimetric aspect of the camera, but also involved characterization of the performance of the camera according to EMVA (2021), release 4.0. A full description of the results obtained will be the subject of another paper; therefore, we limit our discussion to the results strictly related to understanding the polarimetric characterization of the coronagraph.

Among the tests performed on the camera, the measurement of the detector response to the incident polarized light and of its effective quantum efficiency are of higher interest for the CODEX polarization analysis. The first gives an idea of the polarimetric capabilities of the detector, while the second one is useful to account for the differences in response to unpolarized light that exist among the pixels. In this matter, it is necessary to clarify that, as was mentioned before, the CODEX detector is a commercial focal plane assembly (FPA), composed of a CMOS device and a layer of wire grid micro-polarizers topped by a layer of on-chip lenses; which was provided as a full system. Consequently, the results of the quantum efficiency measurements include the effects of all components of the FPA on the photon-to-ADU conversion, as the lens and polarizer transmission and detector gain. The quantum efficiency test highlighted a difference in response to light among the four groups of polarized pixels, meaning that the amount of digital numbers (DN) measured for the same input photon flux is different.

The polarimetric characterization of the CODEX coronagraph is a mandatory step for achieving a correct interpretation of the instrument observations. As described in the previous sections of this paper, there are different elements within the instrument that can potentially modify the polarization state of the incoming light, leading to erroneous diagnostics of the electrons in the solar corona. In this section, we describe the setup we used to collect the data to characterize the CODEX coronagraph from a polarimetric point of view. The mathematical model is outlined in Section 3.2, while the data analysis and its results are discussed in Section 3.3.

3.1 Calibration setup and data acquisition

The full characterization of the CODEX coronagraph took place at the INAF Optical Payload System (OPSys) Facility (Fineschi et al., 2019) hosted by Altec S. p.a, in Turin, Italy. The OPSys facility was selected for these activities because it is equipped with a calibration chamber especially designed for the characterization of solar coronagraphs. The objective of the data acquisition for the polarization test was to collect images of an extended polarized light source, uniform over the instrument field of view. The setup was composed of a uniform light source, a linear polarizer, and the instrument under test. As described in Section 2, the CODEX coronagraph is working in the wavelength range spanning from 385 nm to 440 nm. This wavelength range requires specialized optical equipment as compared to typical visible light systems; i.e., light sources emit few photons, and polarizer extinction ratios are very low. To overcome this last effect, we select a linear polarizer manufactured and commercialized by the Bolder Vision Optik (BVO), INC having its effective bandwidth between 320 nm and 780 nm. Due to its large dimensions, i.e., 200 mm diameter, to ensure no changes in the polarizing film shape would take place, we placed it on a 5 mm thick UV-grade cell cast acrylic (CCA) treated with an anti-reflective coating. Before and after data acquisition, a photodiode was used to measure the light source emittance. The setup is shown on the left part of Figure 2, while on the right side there is an example of an acquired image with a magnified area to show the impact of the micropolarizer array on the acquired data. In this image, it is also possible to see the vignetting caused by the presence of the three pylons and the external occulter. The occulter is not positioned at the center of the image due to an alignment error in one of the two mirrors.

Figure 2

Figure 2. Left: Setup configuration for the polarization test of the CODEX coronagraph. The uniform light source, i.e., flat panel, is positioned in front of the instrument aperture. A linear polarizer covering the whole aperture of the instrument, was placed on a rotation fixture between CODEX and the light source, allowing the selection of the polarization angle of the light entering the instrument. Right: One of the images acquired by the CODEX coronagraph during the test.

To perform this test, we rotated the linear polarizer 180° from the initial position and acquired images every 15° (details of angles described in Section 3.3 below). The collected dataset is composed of ten images for each position of the linear polarizer plus a set of dark frames, used to estimate the background noise. We present here the results obtained with the data acquired using the instrument broadband filter, i.e., 412.5$\pm$27.5. This filter was selected as the resulting characterization is representative of the narrow band filters within the CODEX experiment. Specifically, the broadband filter fully overlaps the wavelength range of the narrowband filters, and all filters were fully characterized with high spectral resolution, thus ensuring no unknown effects. Furthermore, the optics and anti-reflective coatings in this wavelength range are well behaved in terms of their transmissions, i.e., free of the sharp spectral spikes seen in the far UV. Moreover, the light source used in this setup would not provide a sufficient level of light to perform the same test for all filters. We acquired a complementary set of data with all filters by using another setup, and a detailed comparison will be provided in Casti (2026).

3.2 Mathematical model

The most appropriate technique to derive the polarimetric properties of an optical instrument is one that exploits the Mueller calculus (Collett, 1992), which links the polarization state of the incident light to the exiting one by means of a linear system of four equations as shown in Equation 1.

${\mathbf{S}}^{\prime }=\mathbf{M}\cdot \mathbf{S}=\left[\begin{array}{cccc}\hfill m_{00}\hfill & \hfill m_{01}\hfill & \hfill m_{02}\hfill & \hfill m_{03}\hfill \\ \hfill m_{10}\hfill & \hfill m_{11}\hfill & \hfill m_{12}\hfill & \hfill m_{13}\hfill \\ \hfill m_{20}\hfill & \hfill m_{21}\hfill & \hfill m_{22}\hfill & \hfill m_{23}\hfill \\ \hfill m_{30}\hfill & \hfill m_{31}\hfill & \hfill m_{32}\hfill & \hfill m_{33}\hfill \end{array}\right]\cdot \mathbf{S}(1)$

In the above equations S = $(S_0,S_1,S_2,S_3)={(I,Q,U,V)}_{IN}$ is the so-called Stokes vector, that describes the polarization state of the incoming light, M is the Mueller matrix that describes the polarization-altering characteristics of the optical elements between the two beams, the incident and the exiting one, and S’ = $(S_0^{\prime },S_1^{\prime },S_2^{\prime },S_3^{\prime })={(I,Q,U,V)}_{\mathit{OUT}}$ describes the exiting light beam. I, Q, U, and V are measurable quantities that represent four physically independent features of the polarized radiation, i.e., intensity (I), linear (Q and U), and circular (V) polarization. CODEX will acquire images of the polarized solar K-corona, but these data will provide only information related to the intensity of the light, while the polarization state cannot be directly measured. As a consequence, we have access only to the first element of the Stokes’ vector, i.e., $I_{\mathit{OUT}}$, which is a measure of the amount of light reaching the detector. However, as it is possible to see from Equation 2, the total amount of measured light is a function of the elements of the Stokes’ vector associated to the incident light.

$I_{\mathit{OUT}}=m_{00}\cdot I_{IN}+m_{01}\cdot Q_{IN}+m_{02}\cdot U_{IN}+m_{03}\cdot V_{IN}(2)$

It is then possible to measure the polarization state of the incident light, in our case the observed solar corona, by acquiring a set of measurements (at least three since the K-corona is linearly polarized, so V = 0) and solving a new system of equations composed by these measurements. In particular, the set of measurements has to be acquired under different circumstances. In the case of CODEX, this is done by spatially dividing the observed scene into four different polarized images. Equation 3 shows the resulting system of equation.

$\left[\begin{array}{c}\hfill I^{po{l}_1}\hfill \\ \hfill I^{po{l}_2}\hfill \\ \hfill I^{po{l}_3}\hfill \\ \hfill I^{po{l}_4}\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill m_{00}^{po{l}_1}\hfill & \hfill m_{01}^{po{l}_1}\hfill & \hfill m_{02}^{po{l}_1}\hfill \\ \hfill m_{00}^{po{l}_2}\hfill & \hfill m_{01}^{po{l}_2}\hfill & \hfill m_{02}^{po{l}_2}\hfill \\ \hfill m_{00}^{po{l}_3}\hfill & \hfill m_{01}^{po{l}_3}\hfill & \hfill m_{02}^{po{l}_3}\hfill \\ \hfill m_{00}^{po{l}_4}\hfill & \hfill m_{01}^{po{l}_4}\hfill & \hfill m_{02}^{po{l}_4}\hfill \end{array}\right]\cdot \left[\begin{array}{c}\hfill I\hfill \\ \hfill Q\hfill \\ \hfill U\hfill \end{array}\right]=\mathbf{O}\cdot \mathbf{S}(3)$

$I_{\mathit{meas}}=\mathbf{O}\cdot \mathbf{S}(4)$

As shown in Equation 4, the so-called modulation matrix, i.e., O, links the measured intensity to the Stokes vector of the observed polarized light. It follows that to derive a means to evaluate the polarization state of the solar corona from the measurements acquired by CODEX, we need to invert the system shown in Equation 3. The problem reduces to one of finding the inverse of the modulation matrix, i.e., the demodulation matrix, D. Since the modulation matrix is full column rank, it is necessary to make use of mathematical methods to derive the Moore-Penrose pseudo-inverse, which can be expressed as per Equation 5.

${\mathbf{O}}^{†}=\mathbf{D}={\left({\mathbf{O}}^T\mathbf{O}\right)}^{-1}{\mathbf{O}}^T(5)$

To obtain D, it is necessary first to derive the modulation matrix from the dataset acquired in the laboratory and then to calculate its inverse (Vedel et al., 2011). To do so, it is possible to solve four different systems of equations, one for each polarized measurement performed by the detector, where the unknown parameters are the elements of the modulation matrix. The solution of each of these systems is a row of the final modulation matrix. Equation 6 shows how these systems of equations are composed.

$\left[\begin{array}{c}\hfill I^{\left(0\right)}\hfill \\ \hfill I^{\left(1\right)}\hfill \\ \hfill :\hfill \\ \hfill I^{\left(N\right)}\hfill \end{array}\right]=\left[{\left(S_{IN}\right)}^T\right]\left[\begin{array}{c}\hfill m_{00}^i\hfill \\ \hfill m_{01}^i\hfill \\ \hfill m_{02}^i\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill I_{IN}^{\left(0\right)}\hfill & \hfill Q_{IN}^{\left(0\right)}\hfill & \hfill U_{IN}^{\left(0\right)}\hfill \\ \hfill I_{IN}^{\left(1\right)}\hfill & \hfill Q_{IN}^{\left(1\right)}\hfill & \hfill U_{IN}^{\left(1\right)}\hfill \\ \hfill :\hfill & \hfill :\hfill & \hfill :\hfill & \hfill :\hfill \\ \hfill I_{IN}^{\left(N\right)}\hfill & \hfill Q_{IN}^{\left(N\right)}\hfill & \hfill U_{IN}^{\left(N\right)}\hfill \end{array}\right]\cdot \left[\begin{array}{c}\hfill m_{00}^i\hfill \\ \hfill m_{01}^i\hfill \\ \hfill m_{02}^i\hfill \end{array}\right](6)$

where i indicates the row number of the modulation matrix and, therefore, the polarization-altering characteristics of the instrument for that specific acquisition, and N refers to the number of acquisitions. Once the four systems are solved, it is possible to derive the elements of the demodulation matrix. The hypothesis behind this strategy is that the matrix elements used in the laboratory to generate the set of known polarization states are capable of generating a perfectly polarized light beam. If this is verified, the Stokes vectors of the incoming beam will be the theoretical ones.

To account for the differences in the pixels sensitivity pointed out by the QE analysis on the detector and due to the presence of the grid, the images are normalized before solving the system by dividing the measurements acquired as individual polarized images, for the total intensity in input.

3.3 Results

The first result obtained from the data analysis is the Malus curve shown in Figure 3. This plot was obtained after averaging the frames acquired with the same experimental conditions, i.e., exposure time and polarization state of the incoming polarized light. The resulting average image was then divided into the four polarized images to evaluate the digital number (DN) measured within the frames. The plot shows the DN average measured for the four groups of pixels over the angular positions of the linear polarizer during the test. The error bars represent the total detector noise. This plot provides a first evaluation of the instrument’s polarimetric performance. The maximum value reached by each curve indicates that the four groups of pixels have a different sensitivity to incident light. This difference in response was already highlighted by the previous test performed to measure the detector quantum efficiency. With respect to this latest, which was performed with unpolarized light, in this case, the difference between the groups of pixels is due not only to the electronics and to the physical presence of the grid in front of the detector, but also to the efficiency of the grid in transmitting the input polarized signal. Figure 3 shows the Malus curve in terms of intensity to highlight the different responses measured for each orientation of the grid in front of each group of pixels.

Figure 3

Figure 3. Malus curve obtained for the CODEX broadband filter, i.e., 412.5$\pm$27.5 nm, from the analysis performed on the measurements acquired during the CODEX polarimetric characterization. The colored symbols indicate the group of polarized pixels on the detector. Each point represents the average of the value distribution over each considered image excluding the occulted area and the corners, while the error bars account for the noise. The x-axis indicates the angular position of the axis of maximum transmittance of the used linear polarizer in the laboratory reference frame. The values on the y-axes are measured in digital numbers (DN).

The obtained Malus curve provides an understanding of the orientation of the grid in front of each group of pixels. The frame of reference of the CODEX coronagraph is defined as follows: the z-axis is parallel to the geometrical axis of the telescope tube with a direction that goes from A0 to A1 aperture, the y-axis is parallel to the local vertical with a down-to-up direction, and the x-axis is oriented in a right-handed sense. At the zero position of the laboratory reference frame considered for this test, the angle between the maximum transmission axis of the polarizer and the x-axis of the CODEX reference frame is approximately 45°, in the negative direction, that is, clockwise around the z-axis. As can be seen in Figure 3, at the initial position, that is, 135° in the CODEX reference frame, the ${\text{pol}}_1$ pixels measured the maximum level of intensity, while the average measured value for ${\text{pol}}_4$ is close to zero. After a positive 45-degree rotation of the linear polarizer in front of the instrument, the ${\text{pol}}_2$ measurements peaked, while ${\text{pol}}_3$ ones reached the lowest level. With this information and knowledge of the reflections inside the instrument, it is possible to derive the orientation (in the CODEX reference frame) of the transmission axis for the four polarized pixels, as shown in Table 2.

Table 2

Table 2. Correspondence between the naming convention used for the group of polarized pixels and the orientation of their maximum transmission axis within the CODEX frame of reference.

To derive a means of measuring the Stokes vector of the incident light, the mathematical approach described in Section 3.2 has been applied to the collected data. The flat panel and the linear polarizer are considered as an ideal extended source of uniform polarized light, with the angle of linear polarization given by the physical position of the polarizer maximum transmittance axis. As a result, every row of the matrix in Equation 6 is composed of the elements of the theoretical Stokes vector describing the incident polarized light for each associated measurement. The obtained result accounts not only for the orientation of the polarizing grid on top of each pixel, but also for the instrumental effect.

Equation 6 has been applied to each pixel of the acquired images to first derive the modulation matrix and then the demodulation matrix. In particular, the measurements of intensity acquired by the four pixels belonging to the same superpixel are used to derive the modulation matrix of the correspondent pixel in the reduced image. This is done by solving Equation 6 for the four pixels and by collecting the results in a single matrix, that is, the modulation matrix. The latter is then inverted to derive the elements of the demodulation matrix. It is possible to combine the results obtained for each pixel in two sets of 12 2048x1500 arrays, which we refer to as the modulation and the demodulation tensor of the instrument. Each array composing these two sets collects the value of one modulation/demodulation matrix element obtained for each pixel of the final image. The demodulation tensor obtained from the polarization analysis of the CODEX coronagraph data is shown in Figure 4. To derive the instrument demodulation tensor, the acquired data were first normalized with the measured total brightness. The difference in QE observed in the four groups is taken into account.

Figure 4

Figure 4. Demodulation tensor obtained from the polarimetric analysis of the CODEX coronagraph. Each plot shows the collection of values of a single element of the demodulation matrix, i.e., ${\text{d}}_{ij}$, with i = 0, … ,2 and j = 0, … ,3, obtained for all the pixels on the detector. The x- and y-axis represent the pixel number and show the position of the pixel (0,0), while the z-axis as well as the color bar refers to the values of the demodulation element in the pixels.

Figure 4 shows the 12 arrays that compose the demodulation tensor in a three-dimensional plot. In these images of the demodulation tensor elements, the two dark corners and the occulted area are well-visible since the mathematical method runs into problems when trying to solve the equation due to the lack of light, hence modulation. The histograms reporting the distribution of the values over each one of these arrays are reported in Figure 5. The two corners and the occulted area are excluded from these distributions. As can be seen, the images of the elements in the two last rows of the demodulation tensor show a wider distribution of values, resulting in a lower peak when compared with those in the first row. Table 3 reports the theoretical value for each one of the elements of the tensor as well as the average and standard deviation of the value distribution in the images.

Figure 5

Figure 5. Distributions of the demodulation tensor elements. Each plot shows the distribution of values in the images shown in Figure 4.

Table 3

Table 3. Average of the demodulation tensor values.

Polarization image sensors are designed to provide a fast method for determining the linearly polarized Stokes vector of the incident light. Equation 7 shows the relationship between the elements of the Stokes vector and intensity level measured with the four pixel groups.

$S=\left[\begin{array}{c}\hfill S_0\hfill \\ \hfill S_1\hfill \\ \hfill S_2\hfill \end{array}\right]=\left[\begin{array}{c}\hfill I_0+I_{90}\hfill \\ \hfill I_0-I_{90}\hfill \\ \hfill I_{45}-I_{135}\hfill \end{array}\right](7)$

where ${\text{I}}_0$, ${\text{I}}_{45}$, ${\text{I}}_{90}$, and ${\text{I}}_{135}$ are the intensities of the linear polarization components 0, 45, 90, and 135°. However, as we mentioned in the previous sections, when the detector is integrated into an instrument, the measured signal is the result of a combination of incident polarized light and instrumental polarization. The comparison between the results obtained exploiting Equation 7 and those derived by applying the demodulation tensor, can then be used as an indication of the level of instrumental polarization. As mentioned previously, the effects of polarization aberrations caused by elements within the optical path of the instrument result in a change in diattenuation and retardance. To assess the instrumental polarization, we compared the degree and the angle of linear polarization derived from the Stokes vector elements calculated (i) applying Equation 7 and (ii) by means of the obtained demodulation tensor. This verification has been done by using one image acquired during the test in the laboratory, but not used to derive the demodulation tensor. In particular, we considered a set of frames acquired with the maximum transmittance axis of the linear polarizer angle within the setup oriented at 210° in the laboratory frame of reference, i.e., −15° in the CODEX frame of reference. In the ideal world, both methods should generate an image of angle of linear polarization (AoLP) where all the sensitive pixels, i.e., not in the shadow of the occulting element or in the two upper corners, are equal to −15°. Regarding the degree of linear polarization (DoLP), we expect a result indicating that the incoming light is 100$\%$ polarized.

Figure 6 shows the differences between the pixel-by-pixel evaluation of the Angle of Linear Polarization (AoLP) obtained with the two methods. To derive the AoLP we used the known mathematical expression reported in Equation 8.

$AoLP\left[rad\right]=0.5\cdot \arctan \frac{S_2}{S_1}(8)$

On the left, Figure 6 reports the image of the difference between the two AoLP images, while on the right, it is possible to observe the probability density distribution of the values over the two AoLP images, together with their mean value and standard deviation. The solution obtained with the demodulation tensor has an average value very close to the expected value (−14.71° versus −15°). When the four images are used without any processing, the distribution of the AoLP measured over the full detector has a higher standard deviation and the mean value is further from the expected one.

Figure 6

Figure 6. Comparison between the angle of linear polarization obtained (i) by means of the demodulation tensor and (ii) using the four polarized images. On the left: image of the pixel-by-pixel difference between the two strategies of AoLP retrieval. The circular shape is due to the presence of the coronograph occulting elements, that block the light from reaching this part of the detector. On the right: probability density distribution of the values in the two considered images of AoLP.

The DoLP has been derived by means of Equation 9.

$DoLP=\frac{\sqrt{S_1^2+S_2^2}}{S_0}(9)$

The result of the comparison performed on the images showing the measured DoLP is shown in Figure 7. The left side shows the pixel-by-pixel difference between the two images, while the right side presents the distribution of values over the pixels in the images. The average and standard deviation of the two distributions are shown as well. The method used to derive the demodulation tensor assumes that the incoming light during the tests was 100$\%$ polarized. The average value of measured DoLP in the image obtained applying the demodulation tensor is equal to 99$\%$, with a standard deviation of about 1$\%$. Some of the pixels report a value greater than one, which is physically impossible, we believe this is due to the mathematical method used to solve the system of equations. A detailed analysis on the error associated with this method will be object of a future work. On the other hand, the average value of DoLP is about 96$\%$ when it is directly obtained with the four polarized images, without any pre-processing. The difference between the images obtained with the two different methods shows higher values at the edges of the image, which is what would be expected in a two-fold mirror system.

Figure 7

Figure 7. Comparison between the degree of linear polarization obtained (i) by means of the demodulation tensor and (ii) using the four polarized images. On the left: image of the pixel-by-pixel difference between the two strategies of DoLP retrieval. The circular shape is due to the presence of the coronograph occulting elements, that block the light from reaching this part of the detector. On the right: Value distribution of in the two considered DoLP images.

4 Conclusion

This paper describes the polarimetric characterization of the CODEX coronagraph. The instrument is equipped with several optical elements and with a detector sensitive to the polarization of light. Polarimetric characterization is achieved by deriving the demodulation tensor, which is defined as a set of 12 images that collect the value of the related demodulation matrix element and obtain it for each pixel on the detector. The tensor was obtained by exploiting the Mueller calculus and it accounts for the differences in sensitivity to polarized light among the pixels and the instrumental polarization. A first assessment of the instrumental polarization is given by deriving the degree and the angle of linear polarization first by applying the demodulation tensor and then by using the four polarized images acquired by the instrument without performing any further process than background removal. The comparison between these two results shows (i) a different value of AoLP on the image pixels with the second method providing a tilt of about three degrees, (ii) a difference in the averaged value of measured DoLP, which is the one derived with the four polarized images lower by about 4$\%$. These results show that the described calibration process and the derived demodulation tensor enable the retrieval of the polarization state of the solar corona with less related uncertainties than in the case in which the four separated images are combined. The capability of deriving the angle and the degree of linear polarization of the incident light is here selected as reference because these two quantities can be easily compared to the test conditions in the laboratory. These quantities result from the combination of the derived Stokes vector, which elements are used to derive the polarized brightness to measure the temperature of the electrons in the solar corona. Therefore, the capabilities of retrieving the elements of the Stokes vector with the lowest possible uncertainties reflects on a more precise temperature measurements. The quantitative results will be provided in the paper that contains all the calibration results and which is in preparation. Finally, the described pixel-by-pixel polarimetric characterization allows one to derive the orientation of each micropolarizer in a fixed frame of reference, which is in this case the mechanical one, facilitating the retrieval of the polarization state of the solar corona in the more familiar solar reference frame, which will be used in the CODEX scientific data.

The demodulation tensor will be available to the community together with the CODEX data and software to measure the polarization state of the K-corona from the images acquired by the CODEX coronagraph during its mission.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

MC: Writing – original draft. JN: Writing – original draft. Y-HK: Writing – review and editing. GC: Writing – review and editing. HH: Writing – review and editing. DS: Writing – review and editing. S-HP: Writing – review and editing. S-CB: Writing – review and editing. KC: Writing – review and editing. SC: Writing – review and editing. QG: Writing – review and editing. J-HB: Writing – review and editing. JP: Writing – review and editing. JK: Writing – review and editing. HY: Writing – review and editing. NR: Writing – review and editing. NV: Writing – review and editing. SF: Writing – review and editing. FL: Writing – review and editing. DL: Writing – review and editing. LZ: Writing – review and editing. LA: Writing – review and editing.

Funding

The authors declare that financial support was received for the research and/or publication of this article. This work was supported by NASA ROSES NNH18ZDA001N-HTIDS (Heliophysics) funds. This work was supported by NASA Goddard Space Flight Center through Cooperative Agreement 80NSSC21M0180 to the Catholic University of America, Partnership for Heliophysics and Space Environment Research (PHaSER).All KASI authors (Y-HK, DS, S-HP, S-CB, KC, SC, J-HB, JP, JK, and HY) were supported by KASI grants 2025-1-850-02 and 2025-1-850-05.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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