2.1

Liquid Crystals Variable Retarders

As explained in Section 1 and at the beginning of Section 2, along the optical path of the visible channel, just before the VL detector, is set a polarimeter assembly with the presence of a couple of electro-optically modulating Liquid Crystals Variable Retarders (LCVRs).^8,9

Figure 3:

Schematic representation of a Liquid Crystal Variable Retarders operation.

The LCVRs of the Metis polarimeter consist of optically anisotropic liquid crystal molecules embedded between two glasses with a conductive film. These molecules have been accommodated in the cells with an ordered orientation and the effective birefringence of the cell can be changed by applying an electric field that rotates the molecules. Consequently, a change in orientation of the molecules introduce an optical retardance in the orthogonal polarization components of the incoming light. The advantages introduced by this technology are several. In this respect, retarders based on liquid crystals permits to avoid the use of mechanical parts, reducing noise, failure probability and mass. Moreover, the power consumption of the liquid crystal-based retarders is reduced and their response is very quick, providing a fast modulation of the polarization state of light. In the other hand, LCVR devices are sensitive to temperature: retardance depends on temperature cells. For this reason, Metis PMP has been provided with a temperature controller, in order to guarantee the temperature uniformity on liquid crystal cells. Furthermore, the LCVR cells that compose the Metis PMP have been assembled with the fast axes aligned and the pre-tilt angle of the liquid crystal molecules in opposite direction, in order to obtain a wider acceptance angle (that results to be equal to ± 4°).^10,11

Obviously, to evaluate the effective polarization angle corrispondent to a certain applied tension, we have to consider the entire polarimeter assembly (which includes, in addition to the PMP, also a LP and a QW). In particular, in order to analyze the data acquired for the polarimetric VL channel calibration, a new reference frame has been introduced; the Polarimeter Instrument Level System (PILS) reference frame.¹¹ Therefore, the on-ground data analysis is based on the PILS reference frame (Fig. 4) which is define as follows:

Figure 4:

Schematic view of the set up used for data acquisition and Metis polarimeter optical elements relative orientations in PILS reference frame [compared with the Unit Reference Frame (URF)]. The arrows in the optical elements represent, respectively: the fast axes for both the QW and the LP, and the PMP polarization axis.

Basically, the difference between the “standard” reference frame [i.e. as reported in literature] used to retrieve the Stokes vector reported in literature and our PILS reference frame is a 90° rotation around the y-axis. This means that, for retrieving the theoretical Stokes vector in our case is necessary to apply a rotation matrix to them. Fig. 5 shows the two reference frames to be compared. In particular, as we will see in Subsec. 2.2, during on ground calibration we considered polarization angles equals to 0°, 45°, 90° and 135°.

Figure 5:

Stokes vectors for linearly polarized radiation in the PILS reference frame (bottom row) compared to the “standard” one (upper row).

Looking at the PILS reference frame, we can define the Stokes parameters (S₀, S₁, S₂) = (I, Q, U) as:

2.2

On-ground calibration

Being this paper specifically dedicated to in-flight calibration, not many details will be provided about on-ground one. Anyway, some information can be useful to the fully understanding of the in-flight calibration and, if necessary, more information about on-ground calibration can be found in Refs. 6, 11 and 12.

Part of the Metis on-ground calibration aims at characterizing the polarimetric response in visible light. This goal is achieved by retrieving the Metis polarimeter demodulation tensor X^† (obtained by computing and inverting the modulation matrix X associated with each pixel of the acquired polarimetric images). This tensor will allow us to polarimetrically characterize the incoming light from the intensity. A full polarimetric characterization requires to find the Stokes vector S = (S₀, S₁, S₂, S₃) = (I, Q, U, V). Being the S₃ (≡ V) parameter associated with circular polarization, we are interested just in the first three Stokes parameters (for the study of the linear polarization present in the solar corona due to the Thomson scattering - see Sec. 1). If we modulate the incoming light by varying the voltage applied to the LCVRs tacking four images m = [m₁, m₂, m₃, m₄] at different retardance values and knowing the demodulation tensor, we can obtain the Stokes vector S of the incoming radiation by solving the linear system:

The modulation matrix X, as we are about to see, come from the Mueller matrix M_POL associated with Metis polarimeter. In particular, it is obtained as the product of the Mueller matrices associated with the polarizing elements inside the polarimeter as follow:

where θ is the rotation angle of the linear polarization vector and δ = 2θ the retardance. It is possible to see that the form obtained for Eq. 2 is equivalent to M_POL = M_LP M_rot(2θ = δ) where M_rot is the Mueller matrix of a rotating element. More details and a step by step description about how to obtain M_POL are shown in Ref. 11.

The last column of the retrieved Mueller matrix is composed of elements equal to zero; this proves that this configuration does not suffer from the possible presence of circularly polarized radiation. Now it is possible to retrieve the Metis polarimeter modulation matrix. In fact, each row of this matrix corresponds to the first row of the Mueller matrix, assuming a different retardance angle and, consequently, a different applied voltage. Then, we have:

If we consider the optical retardance angles δ_i = 0°, 90°, 180°, 270° (see Tab. 2), we can obtain a theoretical modulation matrix X that assumes the following values:

Table 2:

Positions of the Pre-Polarizer during data acquisition and the correspondent angles in the considered reference frame (PILS). In the last column, the associated theoretical Stokes vector. In this respect, it is necessary to analyze the Pre-Polarizer fast axis position with respect to the polarimeter analyzer, the PILS z-axis.11

Now we can retrieve the theoretical demodulation tensor X^† as Moore-Penrose inverse of the theoretical modulation matrix X just found. What we obtain is:

Then, the theoretical Stokes parameters can be easily obtained just solving Eq. 1:

However, with the intention of deriving the real demodulation tensor associated with the Metis polarimeter, during the on-ground calibration activities we acquired data related to different well-known polarized incoming light states and we modulated them by applying different voltages to the LCVR cells (the considered pre-pol. position are reported in Tab.2). For each pre-pol. position, the left part of Eq. 1 assumes the following expression:

where xdeg = [-45°, 0°, 45°, 90°] is the angular position of the linear pre-polarizer acceptance axis referred to the PILS reference frame, and:

where i the voltage applied to the Metis polarimeter corresponding to a polarimeter rotation angle 0°, 45°, 90° and 135° in this order.

To solve the linear system shown in Eq. 7, we need at least 12 equations (because we have 12 unknown quantities: x_ij with 1 < i, j < 4). In particular, this analysis is performed considering four different voltages ; because we get for i = 1, 2, 3, 4), in order to find the quadruplet related to the most efficient demodulation matrix.

The final form of the linear system is shown in Eq. 9 where refers to measure performed when the rotation angle correspondent to the applied voltage is ≈ 0°; refers to measure performed when the rotation angle correspondent to the applied voltage is ≈ 45°; refers to measure performed when the rotation angle correspondent to the applied voltage is ≈ 90° and refers to measure performed when the rotation angle correspondent to the applied voltage is ≈ 135°.

Considering the elements of the theoretical Stokes vectors associated to the linear pre-polarizer position (Tab. 2) with respect to the PILS reference frame, is shown in Eq. 10. In order to compute the modulation matrix, Eq. 10 has been solved for each detector pixel.

The resulting modulation matrix is the following:

The associated demodulation matrix X^†, can then be obtained by inverting X, obtained computing Eq. 11 pseudo-inverse. The resulting X^† is shown in Eq. 12:

Finally, Metis was thought as an externally occulted coronagraph. The highly vignetted aperture of the externally occulter coronagraph at lower heliocentric height compensates in part the high dynamic range, making possible the observation of the corona at higher heights. Anyway, to be able to see the real corona profile, a vignetting function (VF) for the visible channel, obtained from the average of four polarimetric flat field images at different polarization angle, were acquired in order to be applied to the in-flight VL images m_j [DN/pix] (where j = 0, 1, 2, 3 is the polarization angle the image was taken with). The on-ground measured vignetting function is shown in Fig. 6. In detail, the dimensionless vignetting function VF is given from on ground calibration by:

Figure 6:

Metis vignetting function obtained during the on-ground calibration (on the left) and its diagonal cut, from (0, 0) to (2047, 2047), on the right.

where φ_i are the polarimetric flat field images [DN/pix], M_ij the demodulation tensor elements and I₀ [ph/s/sr/cm²] is the input radiance (obtained from the diode current used during calibration). The other parameters are: average gain 〈G〉 = 0.119 [DN/e^–], average efficiency 〈η〉 = 0.118 ± 0.002 [e^–/ph] (inclusive of linear analyzer 1/2 transmissivity and the average quantum efficiency), VL detector plate scale P_L = 2.5 × 10^–9 [sr/pix], entrance aperture area A_T = 12.6 [cm²] and exposure time T_exp [s]. In particular, the input radiance I₀ is obtained as:

where are considered: diode current C_d [A], responsivity (@610 nm)R [A/(J/s)], diode sensitive area A_d [cm²], filter transmittivity T_filter and the diode field of view FoV_d; obtained as:

considering the filter-diode distance D_f [cm] and the filter radius r_f [cm].

The final VF was obtained as an average of 4 VFs, each one, with a fixed voltage and a different pre-polarizer orientation (the same procedure was performed keeping fixed the prepolarizer and changing the voltage of the LCVR obtaining a comparable result). This VF was then obtained with the theoretical demodulation tensor and the one obtained during on-ground calibration as well. Comparing them, the difference was at the order of ~ 1‰. In order to keep “indipendent” the VF and the demodulation tensor, we choose to use the VF obtained using the theoretical X^†.

The UV vignetting function is the same of the VL channel, since visible and UV light share the same optical path in the telescope, which is the only vignetting optical element. However, some considerations need to be taken into account. In the next section we will explain more in detail how the VL vignetting function could be adapted to the UV channel.

3.1

Vignetting function recentering

Before discussing about different Metis in-flight calibration, a step took place before the actual calibration of the instrument. In particular, during the Metis first light, the images were dominated by stray light. That was due to a misalignment of the IO. This required an adjustment of the position of the IO in order to obtain a correct occultation of the solar disk (Fig. 8 - left). For this reason, also the VF acquired during the on-ground calibration required a “shift” of the same quantity in the same direction in order to be applied to flight images. To evaluate this shift, we have adopted a solution based on the use of the isophotes. In fact, it is possible to fit the points of equal brightness, near to the IO, using an ellipse or a circumference. Repeating this process for different brightness thresholds it is possible to obtain the IO center from the average of these fits output. By carrying out this process for the VF pre and post launch, can be possible to evaluate the IO shift. With the same procedure it is also possible to check on the in-flight position of the coronagraph field stop (FS) comparing its position before and after the launch (Fig. 7).

Figure 7:

IO and FS center evaluation respect to the image center using isophotes with two different thresholds (indeed, the real final center -on the right-value is obtained by the average of 40 thresholds). In this partiular example we considerd on-ground VL vignetting function but the same procedure was performed also for in-flight VL and UV images in order to evaluate the respective pre- / post-launch shifts.

Figure 8:

Left: IO center shift between on-ground VF (white occulter) and a post adjustment in-flight VF (black occulter). It is possible to see that this shift seems to be almost along a spider. Right: VL vignetting function shifted of d = 26 pixels to obtain the pre- / post-launch IO center recentering. This shift was not applied to the VF areas near to the FS (black belt).

The VF can be shifted so that the “on-ground IO” coincides with the flying position. However, the internal occulter shift did not change the field stop and spiders position. A net shift of the entire vignetting would therefore have an over-/under-correction effect in the proximity to the external field of view. To avoid that, it was decided to stop for the amount of shifted vignetting function near the FS. This “partial shift” involves the creation of an “empty” area in which the data is missing; a “black belt” all around the frame inside the FS (Fig. 8 - right).

The values in this “black belt” are extrapolated trying to connect the internal and external values of this “empty” area. In first approximation, a linear function was considered for the data fitting (future adjustments can be performed by increasing the order of the fit). For each fit was considered data belonging to angle ranges of 1 degree (future adjustments can be made by decreasing the angle range step). An example of the linear fit extrapolation and the final VF after 360 fit (1 fit for each degree) are shown in Fig. 9.

Figure 9:

Left: An example of linear fit (for the angle range 11°-12°). The blue dots are the vignetting function radial values in the considered angle range and the green ones are the points considered for the fit extrapolation (the red line). Right: VF after in-flight IO recentering.

The VL vignetting function can be used, under some considerations, for the UV images as well. In fact, considering the vignetting function obtained on ground for the VL channel, the UV one can be obtained by centering the on-ground VL IO with the center of the in-flight UV IO (both centers can be obtained using the isophothes). As shown before (Fig. 7), the on-ground VL IO center coordinates are: (x, y) = (12 ± 1, –62 ± 1) with respect to the image center (1023, 1023). However, in order to use the visible VF as the UV VF, a necessary pre-step is to rebin the visible VF from 2048 × 2048 to 1024 × 1024. After the rebinning, the center of the on-ground visible IO is: (x, y) = (7 ± 1, –31 ± 1) respect to the image center (512, 512). With the same method used to evaluate the VL IO center, it is possible to obtain the in-flight UV IO center. Summarizing the results, we have that the IO center for on-ground VL image (1024 × 1024) at (x, y) = (7 ± 1, –31 ± 1) and the IO center for in-flight UV image (1024 × 1024) at (x, y) = (–5 ± 1, –3 ± 1) both relative to the center of the image. Then, shifting the on-ground VL vignetting function of 12 pixels (from +7 to -5) to the left (W → E) and 28 pixels (from -31 to -3) upward (S → N) we obtain the in-flight UV vignetting function (Fig. 10).

Figure 10:

Left: VF for UV channel. It was obtained through the shift of VL vignetting function to recenter the UV and VL IO centers. The black pixels on side of the frame, are due to this shift (SW → NE direction). Right: VF for UV channel obtained after the VL vignetting function shift.

3.1.1

Polarized Brigthness

After the in-flight IO adjustment, it was possibile to obtain images not dominates by stray-light and test the new, recentered, VF on them. In Fig. 11 it is shown an example of polarized brightness (pB) obtained as . The Q and U Stokes parameters come from the 4 images of the nominal quadruplet (which were subtracted for the dark, divided by the flat field and corrected by the recentered vignetting function).

Figure 11:

Left: An example of Metis polarized brightness (pB). Right: Comparison of pB values along the West streamer between Metis, LASCO and K-Cor instruments during an almost Sun-S/C-Earth alignment.

It is possible to compare the pB value along a radial distance as measured by Metis with the pB obtained, the same day, by other instruments (Fig. 11). In particular, we compared the Metis, LASCO C2, LASCO-C3 and K-Cor pB of May 15^th, 2020 because, at that time, Solar Orbiter was almost aligned with the Sun-Earth axis. As it is possible to see, the Metis pB values are consistent with the other instruments pB. This result is also a symptom of a good radiometric calibration.

3.2

Polarized flat field

During IT-7 campaign (carried out on June 8^th; distance to the Sun: 0.52A.U.) a complete roll was performed by SolO S/C. During the roll maneuver, the Metis polarimeter acquired several sets of images of the K-corona polarized brightness for a total of 8 roll positions (Fig. 12). The goal is to check the polarized flat-field: the pB image of the same coronal structure was measured in different locations of the detector focal plane. The comparison of these images will give indications of the differences in the polarimetric response from different locations across the polarimeter and the detector. To perform this study, we considered the Degree of Linear Polarization (DoLP) obtained dividing the pB for the first Stokes parameter I.

Figure 12:

Degree of linear polarization (pB/I) of the K-corona for the 8 different S/C roll positions.

Quantitatively, we can plot the value of the pB/I variation for each roll in a fixed region on the frame (for example along a streamer). Considering this region and following it the S/C rotation during the 8 different rolls (Fig. 13) maintaining the fixed distance from the Sun center (found by astrometry), we expect to find almost the same values of pB/I.

Figure 13:

Considered region (black rectangles) during the 8 rolls. The white circle inside the internal occulter show the Sun size and position.

Fig. 14 shows the result. The pB/I values are consistent with the expected values (i.e. the pB/I value at the same heliocentric height is almost constant for different roll). The average value is: pB/I = 0.25 ± 0.01 with an averaged percentage error ~ 4% (all the values are reported in 3). In particular, the dispersion of the pB/I mean values using the theoretical demodulation tensor is higher (pB/I = 0.21 ± 0.02 with an averaged percentage error ~ 10%) than using the demodulation tensor obtained during the on-ground calibration. These results confirm the value of using the demodulation tensor obtained from the ground-based calibration. With the theoretical demodulation tensor there is an accuracy on the pB/I values of about 10% (whereas a ~ 3% is due to the dispersion of the pB/I in the considered area) instead of a 4% accuracy obtained with the demodulation tensor from on-ground calibration (with a ~ 2% due to the dispersion). Then, in conclusion, by using the demodulation tensor from ground calibration the accuracy in the polarization measurement improves by a factor of ~ 4.

Figure 14:

pB/I average over the pixels inside the selected regions (for each roll) using the demodulation tensor obtained during the ground calibration (on the left) and with a theoretical one (on the right). The bars represent the dispersion inside the considered area.

The same kind of study can be performed using the quantity Q/I and U/I instead of pB/I. What we expect to obtain is a Malus curve behaviour with a shift between them of ~ 45°. Plotting these quantities, we obtain the expected trend for both Q/I and U/I (Fig. 15).

Table 3:

pB/I average values with standard deviation (pB and I obtained through the on-ground demodulation tensor [left] and theoretical one [right]). Dispersion column contains the standard deviation obtained considering the pB/I values in the region at fixed heliocentric distance (black squares in Fig. 13) for each roll.

Figure 15:

Q/I and U/I averages over the pixels inside the selected regions (for each roll) using the demodulation tensor obtained during the on-ground calibration.

3.3

In-flight calibration of the LCVR

3.3.1

Method 1

Considering the geometric properties of the linearly polarized light in the K-corona (Fig. 16) it is possible to perform an in-flight calibration of the LCVR retardances. It is necessary to consider 4 corona images at different linear polarizations. In particular, considering the Stokes vector in the reference system of the image, for a point P with coordinates (r, α), we can rewrite the recorded signal for each sensor element such as:¹³

Figure 16:

Direction of K-corona linear polarization (tangent to the solar limb) for different distance from the Sun center (SC).

where m_i are the single images of a quadruplet, I is the first Stokes parameter and δ_i are the retardance values that can be estimated through the 3-parameters regression:

considering y_i = (2m_i – I)/pB, P₀ is the bias of fitting curve, P₁ is the altitude modulation and P₂ = δ_i.

Obtained the Sun center (SC) position by astrometry, it is possible to perform the 4 regressions (i.e. one for each y_i obtained from the respective image of the quadruplet m_i), for a fixed distance from the center of the Sun. What we obtain is shown in Fig. 17. It is possible to see that in some regions the error bars are much higher than elsewhere. This is due to the fact that the error is proportional to ~ 1/pB. For this reason, the less noisy areas are those with the most polarized signal (along the streamers). The reduced χ² is always < 1 (oscillating between values of 0.3 and 0.6). Looking at the P₂ parameters, we can obtain the polarization angles ϑ. In particular, we performed this study for different heliocentric heights. We considered 10 distances from 580 to 670 pixels from the Sun center in order to avoid noisy regions too near or too far away from the IO edges. The expected values from ground calibration for the nominal quadruplet polarization angles are: (49.1°, 84.3°, 133.2°, 181.1°) with an error or 1.4° each. From the average of the values at different heliocentric heights, we obtain: (45.0° ± 0.1°, 81.4° ± 0.1°, 128.7° ± 0.1°, 175.4° ± 0.1°) that are consistent with the expected ones.

Figure 17:

Plot of the 4 regressions to perform the in-flight calibration of the LCVR retardances. The considered data come from a quadruplet acquired during the “roll n.0” of the IT-7 campaign (see Fig. 12) for a fixed heliocentric height (620 pixels from the Sun center in this particular case).

3.3.2

Method 2

It is possible to obtain an in-flight calibration of the LCVR using data from the IT-7 rolls as well. In fact, considering always the same region in the solar corona, the S/C rotation gives the same result as “rotate the polarizer”. We therefore expect that the recorded intensities follow the Malus’s law (Fig. 18). By making a fit with this function we can have an estimation of the delays and compare them with those expected from the ground calibration and what we obtained in the previous method (Tab. 4). What we obtain is: (46° ± 4°, 82° ± 4°, 130° ± 5°, 176° ± 5°) that are again consistent with the expected one.

Figure 18:

Malus curve obtained from y_i = (2m_i − I)/pB average values over the pixels in a fixed region in the solar corona during each roll.

Table 4:

Comparison between the LCVR polarization angles from on-ground and in-flight calibration.

On-ground	In-flight [method 1]	In-flight [method 2]
49°± 1°	45.0° ± 0.1°	46° ± 4°
84°± 1°	81.4° ± 0.1°	82° ± 4°
133°± 1°	128.7° ± 0.1°	130° ± 5°
181°± 1°	175.4° ± 0.1°	176° ± 5°

3.4

Different quadruplets

During the first remote sensing checkout window (RSCW1 - June 18^th, 2020 - distance to the Sun: 0.52 A.U.) campaign, polarimetric sequences with different quadruplets of PMP voltages were acquired. The goal is to check the polarimeter’s response for different voltage configurations. All these quadruplets consist of 4 polarization angles separated by ~ 45° from each other. All the considered quadruplets, with an effective angle error of ~ 1°, are summarized in Tab.

Table 5:

Comparison between the LCVR polarization angles from on-ground and in-flight calibration.

Using the demodulation tensor associated with each quadruplet (obtained during on ground calibration), it is possible to get the corresponding pB (in particular, 5 pB were obtained; two acquisitions with “quadruplet 4” were performed). Comparing them, we expect to not observe huge variations. To check that, we considered 4 different pB regions (along streamers, coronal holes, etc…) and we compared the average of the pixels inside them (Fig. 19). We obtain that the differences between the pB are ≤ 2%.

Figure 19:

Comparison between the average of the four pB regions for the different quadruplets. The differences between the different pB are less than 2%.