1.1

Introduction to polarimetric measurements

This Subsection is meant to provide to the reader an introduction to polarimetry by getting in all the essential elements necessary for the rest of the manuscript. Anyhow, there are numerous sources of reference for a detailed introduction to polarimetry; more information about polarimetry and its application can be found, for example, in Ref. 2.

In general, a complete polarimetric study of the light incident to the detector sensor requires the evaluation of the Stokes vector S. The Stokes vector elements are defined as follow:

where I₀, I₉₀, I₄₅, I₁₃₅, are the intensities of the linear polarisation components at 0, 90, 45, 135 degrees and I_LHC, I_RHC are the left hand and right hand circular polarisation intensities respectively. Then, results to be . Commonly, the Stokes elements S₀, S₁, S₂, S₃ are also reported in literature with the notation I, Q, U, V respectively. In particular, the linearly polarised light requires the measurement of the I, Q, U quantities to be fully characterised. The V parameters is associated to the circular polarisation and it is not relevant to the work being done in this manuscript. There are several equations that relate all these quantities. In particular:

where p is the Degree of Linear Polarisation (DoLP) and θ is the Angle of Linear Polarisation (AoLP). Then, this two quantities can be expressed, relatively to the detector axis as:

Another quantity often used in polarimetric studies is the polarimetric brightness defined as:

There are other possible representations that will not be discussed in this paper. Anyhow, all these intrinsic properties of the radiation should not be confused with the properties of the polarising elements (see Subsec. 3.4).

Finally, it is good to introduce the concept of demodulation tensor X^†. As we will see in the next Sections, part of the PolarCam characterisation aims to achieve the camera X^†. This tensor (obtained from a modulation matrix X associated with each pixel of the acquired polarimetric images) allow us to polarimetrically characterise the incoming light founding the associated Stokes vector S* as shown in Eq. 7 (where m is the vector of the acquired images at different polarisation).

According to the equations showed before, it is necessary to acquire images at 4 different polarizations and, as mentioned, the PolarCam, with a single shot, can give us images at θ = (0°, 45°, 90°, 135°). Then, in our case, we have m = (m₀, m₄₅, m₉₀, m₁₃₅).

From the Mueller matrix² it is possible to obtain a theoretical modulation matrix X. In particular, considering the angles θ_i = (0°, 45°, 90°, 135°), it results to be equal to:

Now we can retrieve the theoretical demodulation matrix X^† as Moore-Penrose inverse of the theoretical modulation matrix X. We obtain:

Then, the theoretical Stokes parameters can be easily obtained just solving Eq. 7:

Performing this study pixel by pixel we obtain the full theoretical demodulation tensor X^† instead of a single matrix. The results in the next Section were obtained by using both the theoretical demodulation tensor and the one we got during the PolarCam calibration campaign (Sec. 3.4).

2.1

Raw image demosaicing

To obtain the images at different polarisation (0°, 45°, 90°, 135°) from the original raw, a demosaic process is required. In Fig. 2 an example of a possible way to perform this process is shown. In particular, once a certain polarization angle has been chosen (e.g. 0°), the method used during our analysis was to consider the 3 remaining pixels, for each super-pixel, as the average between the considered pixel and the values that the pixels with the chosen polarisation have in each adjacent super-pixel (i.e. “Output 2” in Fig. 2). The same procedure can be applied to get the images with the other polarisation.

Figure 2.

Demosaic example to obtain a polarised image from the original raw. In this particular example it is shown how to obtain the image with a polarisation angle of 0 degrees (I₀). Output 1: each super-pixel value was obtained from the pixel value with the micropolarizer at 0 degrees. Output 2: each super-pixel value was obtained considering also the pixel values with the micropolarizer at 0 degrees in the adjacent super-pixels.

3.1

Detector resolution

To evaluate the detector resolution we used the PolarCam looking at a resolution target USAF-1951³ illuminated by a white led source with a light diffuser, powered at 3.2 V (Fig. 3). A collimator was placed between the detector and the resolution target. The collimator aperture is ≈ 50 mm with a focal length of ≈ 300 mm.

Figure 3.

An image of a resolution target as seen by the PolarCam (i.e. the AntarctiCor detector) with gain = 1 and T_exp = 71.86 ms. To avoid values conditioned by the polarisation, this image is obtained as the sum of the intensities of the linear polarisation components at 0 and 90 degrees (i.e. the first element of the Stokes vector).

Thanks to a Modulation Transfer Function, it is possible to evaluate the resolution through the Rayleigh criterion (Figure 4). In particular, when the modulation is equal to 0.2 we obtain that the frequency is almost 22 [linepairs/mm] (i.e. ≈ 22 μm) that corresponds to the group 4 element 4 of the resolution target (Table 2).

Figure 4.

PolarCam Modulation Transfer Function (MTF).

Table 2.

Line pairs per millimetre for each group element of the resolution target.3

3.2

Detector linearity

Changing the exposure time it was possible to check the PolarCam linear response (Fig. 5). During these acquisitions we set an analog gain of AG = (44, 44) and a digital gain DG = 1 (it is possible to set a digital gain from a minimum of DG = 1 until a maximum DG = 16, engineering units). In particular, the camera linear response was evaluated by averaging over the entire frame and by separating the contributions given by the pixels with different orientation of the micropolarizers. It is possible to check also the average dark for different exposure time and different digital gain. What we obtain is shown in Fig. 6. As expected, the average dark values increase for higher DG. Moreover, it is almost constant in the considered exposure time range.

Figure 5.

PolarCam linear response averaging over the whole frame and considering the four different micropolarizer array orientation (in the grey box).

Figure 6.

PolarCam dark measurement for different digital gain and fixed analog gain.

3.3

Angle and Degree of Linear Polarisation

As introduced in Sec. 1, with a single shot, we can evaluate the Stokes I, Q and U parameters using the theoretical demodulation tensor (as explained in Subsec. 2.1) and then calculate the DoLP and AoLP from them (Eqs. 4 and 5).

Considering a flat-field panel as light source (i.e., unpolarized light source), what we expect to obtain looking at the DoLP is an almost zero degree of polarisation. The measured DoLP could be interpreted as instrumental polarisation, or the polarisation introduced by the PolarCam itself and as a consequence it will not be possible to measure a polarisation lower than this one. The results are shown in Fig. 7 (right side). It is possible to see that the polarisation introduced by the camera on the left side of the frame is at the order of ~ 4% on the Stokes Q and U parameters and as shown in the right panel, the DoLP increase up to the ~ 10%. The average of the degree of linear polarisation on the whole frame is: (6 ± 2)%.

Figure 7.

Test with flat-field (unpolarised light). Left side: Stokes parameters obtained with the theoretical demodulation tensor (Stokes vector = [1, Q/I, U/I]). Right side: Degree of Linear Polarisation obtained with a theoretical demodulation tensor.

In addition to this unpolarized flat-field measurements, we performed also the same kind of measurement by introducing a linear polarizer (pre-polarizer) between the detector and the flat-field panel, obtaining a polarimetric flat-field. By rotating the pre-polarizer, at the detector plane, the light is fully linearly polarised in the direction defined by the orientation of the acceptance axis of the pre-polarizer. Therefore, what we expect to observe is a DoLP of ~ 100% and the AoLP having the same angle of the acceptance axis of the pre-polarizer. As an example, in Figs. 8 and 9 are shown the DoLP and the AoLP measured by the PolarCam when the pre-polarizer is oriented at the four main angles of 0°, 45 °, 90 °and 135 °. The results are summarised in Tab. 3.

Table 3.

unpolarised flat-field (UFF) and polarimetric flat-field (PFF) DoLP and AoLP evaluated by the application of the theoretical demodulation tensor.

Figure 8.

Measured Angle of Linear Polarisation (AoLP) for different polarised incoming light using the theoretical demodulation tensor. In particular, in this figure we show the 4 angles of 0°, 45°, 90°, 135°.

Figure 9.

Measured Degree of Linear Polarisation (DoLP) for different polarised incoming light using the theoretical demodulation tensor.

3.4

Micro-polarizers orientation and demodulation tensor

As show in the previous Section, the degrees and angles of linear polarisation results to be not much consistent with the expected theoretical ones. To improve these results we need to consider a demodulation tensor different from the theoretical one. This new demodulation tensor must take into account different aspects not considered in the theoretical X^†, like, for example, the effective orientation of each micro-polarizer and other characteristics of the polarising elements upon each pixel. As first step, is useful to point out that, exactly like for the study of polarised light (Subsec. 1.1), to characterise the behaviour of a linearly polarising element, it is necessary to have three quantities as well. These quantities are:

There are different conventions commonly used to present these three quantities. Considering I, Q, and U as the Stokes parameters polarimetrically describing the incident light beam, in this work we adopt the convention that the output beam intensity passing through a polarising element is given by:^{4, 5}

where S_k is the measured signal, A_k, B_k, C_k are transmissivity terms and ∊_k the polarising efficiency that describe each polarizer (i.e., each pixel of the PolarCam). The subscript k denotes the polarizer orientation. Considering Eqs. 2 and 3, the above equation can be rewritten as:

where I is the total intensity, p is the intrinsic fractional polarisation of the source and ψ is the intrinsic polarisation angle. The transmissivity terms can now be written in terms of a generic throughput t_k:

where ϕ_k are the position angles of the polarizers. It is possible to simplify the equation by set p = 1. In this way, equation results to be:

By fitting to the response curves it is possible to determine the efficiency ∊_k and orientation ϕ_k of every (micro) polarizer. This fit was performed for each PolarCam pixel, since each pixel correspond to a different micropolarizer. The transmissivity t_k was evaluated before the data fitting. These values have been obtained searching the pixel with the highest transmissivity with the same input, considered as , and normalising the others t_k in function of it. An example of what we obtain from a fit is shown in Fig. 10. This study was performed setting the camera analog gain AG = (44, 44), the digital gain DG = 1 and and exposure time of t_exp = 71.36 ms. In particular, instead of S_k, we considered a normalised irradiance S defined as:

where k = [0°, 45°, 90°, 135°] are the micropolarizers orientations.

Figure 10.

Example of curve fit for a particular super-pixel [in this case, the super-pixel associated to the pixel (0, 0)]. It is possible to see that the obtained values for the ϕ_к parameters are pretty consistent with the expected ones.

At that point is possible to estimate the Stokes parameters of the incident light. To do that, as illustrated in Subsec. 1.1, we have to solve the following equation:

where m is the vector of the images with the four different linear polarisation, X is the modulation matrix and S_input is the Stokes vector associated to the incoming light. In particular, considering the ∊_i and ϕ_і obtained from the data fit, we get the matrix in Eq. 17.

By (pseudo)inverting the X matrix it is possible to obtain the demodulation matrix X^†:

Then, from Eq. 16 and Eq. 18 we can obtain S_input as:

Performing this process for the entire frame (pixel by pixel) we obtain a demodulation tensor X^† where each of the 12 elements is a matrix (Fig. 11). A summary of this process is shown in Fig. 12.

Figure 11.

Demodulation tensor element x_ij obtained during PolarCam calibration.

Figure 12.

Schematic step-by-step procedure for the experimental evaluation of the demodulation tensor X^†.

Using this specific demodulation tensor (specific for our camera), we obtain, using the same procedure already applied for the theoretical demodulation tensor (Subsec. 3.3), the Stokes parameters, the DoLP and the AoLP. The results for the unpolarized light case (flat-field source) are shown in Fig. 13. In Fig 14 and Fig 16 are depicted the results for the polarised light. As expected, the use of a calibrated demodulation tensor improves significantly the accuracy in the measurements of the polarisation state of the light detected by the PolarCam. In particular, it is possible to see how the new demodulation tensor remove (almost totally) the residual instrumental polarisation. Moreover, the frames acquired with the polarimetric flat-field, show an almost perfect agreement between the flat-field polarizer orientation and the measured one (the differences between the expected angles of linear polarisation and the obtained ones are shown in Fig. 15).

Figure 13.

Test with flat-field (unpolarised light). Left side: Stokes parameters obtained with the calibrated demodulation tensor (Stokes vector = [1, Q/I, U/I]). Right side: Degree of Linear Polarization obtained with the calibrated demodulation tensor.

Figure 14.

Measured Angle of Linear Polarization (AoLP) for different polarized incoming light using the calibrated demodulation tensor. In particular, in this picture, the results for the angles of 0°, 45°, 90°, 135°are shown.

Figure 15.

Histograms of the differences between the measured Angle of Linear Polarization (ψ^exp) and the orientation of the acceptance axis of the pre-polarizer (ψ^theo).

Figure 16.

Measured Degree of Linear Polarization (DoLP) for different polarised light using the calibrated demodulation tensor.

A summary and comparison of these results is reported in Tab. 4. Finally, in Fig. 17, we show the maps for the different micropolarizers throughput t_k for the 4 orientations (the same study was performed for the efficiency ϵ_k as well). Moreover, looking at Fig. 18, it is possible to see that the throughput seems to be systematically different for pixels with different orientations. In Fig. 19, a map of the effective micro-polarizer orientation is shown.

Table 4.

Unpolarised flat-field (UFF) and polarimetric flat-field (PFF) DoLP and AoLP retrieved by applying the calibrated demodulation tensor.

Figure 17.

Throughput t_k (for each pixel orientation) normalised to the maximum throughput of all pixels.

Figure 18.

Histograms of throughput t_k (for each pixel orientation) normalised to the maximum throughput of all pixels.

4.1

ESCAPE Project

The Antarctica plateau of Dome C (coord: 75°06’S; 123°20’E) offers the unique opportunity for ground-based observations of the solar corona thanks to the high altitude of the site (3, 233m a.s.l.), the almost total absence of human pollution and the uninterrupted daily hours for observations during the antarctic summer.⁶ In order to take advantage of this opportunity, the Italian Piano Nazionale Ricerche Antartico (PNRA)⁷ has selected our proposal for the “Extreme Solar CoronagraphyAntarctic Program Experiment” (ESCAPE) for the installation of an Antarctic coronagraph (AntarctiCor) at the Italian-French Concordia base at Dome C. More information about ESCAPE Project and its scientific objectives can be found in Ref 8.

Figure 19.

Maps of the micro-polarizer orientations, ϕ_k.

4.2

Antarctic Coronagraph

The Antarctica solar coronagraph (AntarctiCor) for the ESCAPE program is a classical internally-occulted Lyot coronagraph (Fig. 20) based on the optical design of the ASPIICS coronagraph of the PROBA-3 ESA mission.⁹ The chosen detector for this telescope is the PolarCam. The main features of the instrument are summarized in Tab. 5.

Figure 20.

Up: AntarctiCor in the INAF Optical Payload Systems facility (OPSys), Turin-Italy, for tests and calibrations. In the right panel, the main sub-assembly is shown. In particular it contains the following items: 1. Objective lens assembly; 2. Inner main barrel assembly; 3. Internal Occulter assembly; 4. Lenses assembly; 5. Filter assembly; 6. Light trap assembly; 7. Microscope assembly. Down: AntarctiCor optical ray trace.

Before the Antarctic campaigns, instrument calibrations were carried out. However, being this paper not specifically dedicated to the AntarctiCor, only the aspects of the calibration concerning the camera will be treated here.

4.2.1

Point Spread Function

To evaluate the Point Spread Function (PSF) we used a pinhole (50 μm) on the Illumination System Visible Light (ISVL)¹¹ in the Optical Payload System - OPSys - Facility¹² in ALTEC, Turin, Italy. In particular, 10 images are summed to increase the SNR. The image region with the pinhole (Fig. 21) is selected as Region Of Interest (ROI) to evaluate the PSF. For these measurements, the following settings of the PolarCam were used: Exposure time equal to 71.82 ms and digital gain equal to 16. Then, for the two dimensions, rows (red) and columns (blue), the, pixel values are plotted and a best-fit (“horizontal” and “vertical” fit respectively) has been performed with a Gaussian function. The results are shown in Fig. 22.

Figure 21.

Sum of 10 pinhole images with region of interest (ROI). A light leak from the edge of the pinhole mask is also visible around the region [1750, 1250] as a thin line.

Figure 22.

Gaussian best-fit of the ROI for rows and columns. On the right side, a 3D representation of the considered fit-geometry. The error bars are negligible and not visible in the plots.

From the variance is possible to obtain the Half Width at Half Maximum (HWHM) for both the fit, obtaining: and respectively. Averaging between them, we obtain: . Then, considering the Full Width at Half Maximum (FWHM) we can considered that a “point-like signal” is spread over pixels (≡ 20.28 μm).

Table 5.

Main AntarctiCor features.8

Telescope design	Classic internally-occulted Lyot coronagraph10
Aperture	50 mm
Eff. Focal Length	700 mm
f/ratio	14
Spectral Ranges	(591 ± 5) nm
Camera type	Interline transfer CCD PolarCam (model: U4)1
Camera format	1950 × 1950 pixels
Pixel size	7.4 μm × 7.4 μm
Plate scale	4.3 arcsec/pixel
Field of View	±0.6° = ±2.24 R⊙
Polarization analysis	spatial modulation by linear micropolarizer array on CCD sensor

Looking at Tab. 5 we can see that the wavelength λ = 591 nm and the f-ratio F/# = 14. Then we can evaluate the telescope diffraction limit:

Then, we can conclude that the telescope is diffraction limited. Anyhow, it is good to remember that, being the super pixel composed by 4 different orientation (whereas each pixel has dimension 7.4 × 7.4 μm), the pixel dimension for a frame post-demosaicing (i.e. with a single polarizer orientation; see Fig. 2) results to be twice the original frame: 14.8 × 14.8 μm.

4.3

Antarctic campaign

During the Antarctic campaign, in addition to coronal images (still under analysis), systematic images of the sky were acquired. The goal is to study it from a polarimetric point of view and try to evaluate its brightness.

In Fig. 23 an example of the Intensity (Stokes I parameter) of the data acquired from the sky and from the Sun with an opal (i.e., diffuser) in front of the telescope are shown.

Figure 23.

Left: Example of first Stokes parameter (i.e., I) obtained from an image acquired pointing to the sky. Right: Example of first Stokes parameter (i.e., I) obtained from an image acquired pointing to the Sun with a diffuser in front of the coronagraph.

The sky brightness (B [B_⊙]) was measured for the full data acquisition campaign at almost regular intervals during the day. To obtain the sky brightness in unit of solar disk brightness (B_sky [B_odot]) a “sun-centered + diffuser” (I_opal) and a sky (I_sky) image are needed. Indeed, from the ratio of these quantities we can obtain:

whereas are the exposure times. Then, multiplying by the diffuser transmissivity T_diff (~ 28%):

Then, considering geometric factor K = 1.82 × 10⁻⁵ (light scattered over the solid angle by the diffuser) it is possible to evaluate the B_sky [B_⊙] as:

Moreover, the obtained B_sky [B_⊙] frame was divided in four different pad and the final B was obtained by averaging them (Fig. 24).

Figure 24.

Example of the measured sky brightness (Dome C, Concordia Base - Antarctica). As shown, the entire frame is divided into 4 different pads. The final sky brightness is obtained as the average of the 4 pads.

A summary of the obtained sky brightness for different days and different hours is shown in Fig. 25. In particular, averaging over the different values and considering also that the I_sky results to be at the limit of the camera sensitivity (~ 10 DN), we get that: . This means that Dome C sky can be considered as “coronagraphic” sky.

Figure 25.

Sky brightness [B_⊙] measurements obtained during the XXXV Italian Antarctic Campaign (2019-2020), Concordia Base - Dome C, ~ 3230m a.s.l., Antarctica. On x-axis, the acquisition UTC time from January 1, 2020.