© The Authors 2026

1. Introduction

A routine data reduction step for an optical telescope is correcting for its point spread function (PSF). The PSF describes the instrument’s response to a single point source of light. For an ideal optical telescope, the PSF is the Airy function with the first minima at the diffraction limit. For space-based solar telescopes, optical aberrations and spacecraft jitter are the main sources of non-ideal imaging performance, of which one component is commonly called stray light and defined in further detail later in this section.

The Polarimetric and Helioseismic Imager on Solar Orbiter (SO/PHI, Solanki et al. 2020; Müller et al. 2020) is a such an optical telescope, specifically a space-based solar magnetograph, which, due to the orbit of Solar Orbiter, is the first to observe out of the Sun-Earth line, and over a wide range of distances to the Sun. It samples the Fe I 6173.3 Å absorption line and infers the photospheric vector magnetic field and line-of-sight (LoS) velocity. It has two telescopes: the High Resolution Telescope of SO/PHI (SO/PHI-HRT, Gandorfer et al. 2018), which performs routine observations of small portions of the solar disc at close approaches to the Sun; and the Full Disk Telescope (SO/PHI-FDT), which observes the entire disc, mostly outside the perihelion windows.

This paper concentrates solely on SO/PHI-HRT. In Sinjan et al. (2023), SO/PHI-HRT magnetic field data were compared with that from the very similar Helioseismic and Magnetic Imager on the Solar Dynamics Observatory (SDO/HMI, Schou et al. 2012; Scherrer et al. 2012; Pesnell et al. 2012). The magnetic field data was similar, especially the LoS magnetograms; however, significant differences were observed in the strong field regions. Sinjan et al. (2023) postulated that stray light could be a contributing factor that might explain these differences.

In the case of SO/PHI-HRT, the core of the PSF is estimated by phase diversity wavefront analysis, which uses combinations of focused and defocused images to estimate the wavefront error (Kahil et al. 2022, 2023; Bailén et al. 2024). The width of the PSF core is a measure of the spatial resolution. The wings of the PSF, corresponding to higher order spatial frequencies, lie outside the sensitivity range of phase diversity analysis (Gonsalves 1983, 1985) and are therefore not corrected when a PSF is taken into account via phase diversity analysis alone. This large-spatial-scale contribution to the wings of the PSF is commonly referred to as stray light.

Past works to characterise the stray light of space-based solar magnetographs include Mathew et al. (2007) for the Michelson Doppler Imager on the Solar and Heliospheric Observatory (SOHO/MDI, Scherrer et al. 1995; Domingo et al. 1995), Mathew et al. (2009), Wedemeyer-Böhm (2008) for the Solar Optical Telescope on the Hinode spacecraft (Hinode/SOT, Tsuneta et al. 2008; Kosugi et al. 2007), and Yeo et al. (2014), Couvidat et al. (2016) for SDO/HMI. They employ a variety of analysis techniques, most common of which are solar limb profile analysis and planetary transits across the solar disc. A different approach is to instead model contributions from stray light (but also the effects of insufficient spatial resolution) by adding a second non-magnetic component to the model atmosphere when performing radiative transfer inversions (e.g. Lites et al. 2007; Orozco Suárez et al. 2007; Asensio Ramos & Manso Sainz 2011).

In this paper we present a stray light analysis of SO/PHI-HRT using in-flight data and demonstrate that the observed differences with SDO/HMI reported by Sinjan et al. (2023) are in large part explained when residual stray light in SO/PHI-HRT is accounted for. The improvement in data fidelity is such that now SO/PHI-HRT data products from 2023 onwards include a stray light correction as standard. In Section 2 we first introduce the main origins of stray light in optical instruments like SO/PHI-HRT, as although the instrument has specific stray light suppression components, limitations still exist. We then estimate the stray light of SO/PHI-HRT and generate an extended PSF in Section 3 that can be applied to correct for residual stray light. In Section 4 we analyse the impact of the stray light correction on the SO/PHI-HRT data products and in Section 5 we outline an updated comparison of the inferred magnetic field data products with those from SDO/HMI. In Section 6 we summarise these results and compare them with the first comparison made with SDO/HMI in Sinjan et al. (2023).

2. Understanding stray light in SO/PHI-HRT

2.1. Origin and suppression of false light contributions in space-based solar telescopes

From a strict instrumental viewpoint, however, the term stray light (or false light) is used to describe any unwanted light that reaches the detector or image plane of a telescope. It is considered false as it can be light that does not come from the target object; for example, as the result of unwanted reflections or scattering from dust particles or imperfections on optical surfaces. It is called ‘in-field’ stray light when the false light originates from within the field of view (FoV) and ‘out-of-field’ when it originates from outside the FoV.

Both the origin and the suppression of false light in solar telescopes is rather different compared to night time observatories. The whole field from which light can be brought into the instrument is called the ‘illuminating field’ and can cover as much as a full hemisphere. In order to suppress stray light contamination, it is beneficial to prevent it from reaching the telescope optics and mechanical structure. The most efficient way of achieving this is by an aperture baffle, which limits the acceptance angle with respect to the illuminating field. When the optical system or its mechanical structure is illuminated by light from bright objects, many different mechanisms lead to contamination of the image with false light.

In the observation of the solar photosphere, the only disturbing light source is the Sun itself. This means that the illuminating field and the science FoV are at least partially coinciding; in a full disc telescope, they are by definition identical. The size of the solar disc is very small compared to the half sphere, and scattering does not contribute significantly to false light in the detector. Under such an observing geometry, even a fully dusty mirror does not produce relevant stray light. While a large number of photons are scattered out of the beam path, only very little light is scattered into the beam path towards the detector. The dominant physical mechanisms, which lead to a redistribution of photons within the very limited illuminating FoV in solar observations, is diffraction and optical aberrations, in combination with the fact that the Sun is a continuous bright object.

Even for non-aberrated systems limited only by diffraction, a significant portion of false light will be redistributed. Sunspots are one solar feature where this is very apparent: a sunspot represents a dark feature in front of a bright background, and the full Sun will contribute to filling this dark spot on the detector by sending photons, which have ‘almost’ the right direction, therefore falsifying the measured signals. This is because only 88 percent of the light will be contained within the central peak of the Airy pattern, whereas the remaining 12 percent will be distributed over the diffraction rings.

Pores and dark intergranular lanes are also greatly impacted, which reduces the overall intensity contrast, an important metric to provide information on the thermal variations and solar convection in the photosphere (e.g. Hirzberger et al. 2010; Yeo et al. 2013; Kahil et al. 2019, 2023). Furthermore the intensity contrast is also particularly important for irradiance models (Yeo & Krivova 2019). The main contributor to contrast loss in solar imaging is low- to mid-order optical aberrations. They lead not only to a broadening of the central peak, but to an enhancement of the secondary maxima, and to an increase of light of the minima, which no longer have zero energy. All this occurs at the expense of the energy content within the central peak, and therefore its amplitude. It is not only dark features but also bright features that are impacted, including their polarisation signal (e.g. Riethmüller et al. 2014).

Low-order aberrations have spatial scales that are of a similar order to the aperture of the telescope, which determines the width of the central diffraction peak of the non-aberrated image. Danilovic et al. (2008) have investigated the quantitative effect of low-order aberrations in the case of the Spectropolarimeter (SP) of the Hinode/SOT observations.

To account for this redistribution of light, the PSF of the telescope is usually either modelled as the non-aberrated Airy pattern, or parameterised through a wavefront fit based on a sum of (a limited number) of Zernike or Karhunen-Loeve polynomials and their relative amplitudes (see Noll 1976). The first Zernike terms are typically used to represent the wave front of an aberrated system in phase diversity reconstructions and are able to reproduce only spatially slowly changing distortions of the wavefront. Deformations of the wavefront that change more rapidly in space cannot be correctly represented in this way. Those spatial scales, which are macroscopically large (as compared to ‘scatterers’ such as dust, scratches, and digs), but are too small to be represented by Zernikes, are called intermediate or mid-spatial-frequency (MSF) errors. These errors can have different origins; however, the main source and their spatial size is dependent on the method used to polish the main mirrors. They are the dominating contributors to residual false light in solar images, which have been already been ‘corrected’ by deconvolution with the ‘PSF’.

Usually the false light contribution by MSF errors is modelled with a Gaussian or Lorentz distribution (or with a convolved version of the PSF with these distributions), which is typically 10–1000 times (depending on the size of the aperture) wider than the core of the diffraction-limited PSF, since the spatial frequencies correspond to physical errors, which can range from millimetres to several centimetres in size.

2.2. False light and its suppression in SO/PHI-HRT

The design of SO/PHI-HRT represents a necessary compromise between performance under demanding orbital conditions, and strong resource restrictions in volume and mass. The quite large science FoV of 0.28° ×0.28° necessitates a wide-angle optical design. The Ritchey-Chrétien telescope does not have a real intermediate focus, and thus inhibits the use of a prime field stop, which would prevent downstream optical and structural parts to be illuminated by unwanted photons, arising from the relatively large illuminating field of 2° (the full solar disc at perihelion), which is 35 times larger in area than the science FoV. The main mirrors of SO/PHI-HRT were polished using ion-beam figuring (IBF), and its corresponding measured MSF errors can be seen in Fig. A.1. To suppress the false light, a number of unsharp baffles and vanes are used, together with a field stop in the filtergraph and a Lyot-type pupil stop between the filtergraph and the camera. This system is very effective in suppressing false light from outside the FoV. For more information on the baffling architecture of SO/PHI-HRT, we here refer to Gandorfer et al. (2018). A stray light analysis taking into account all illuminated mechanical components and their optical surface properties has demonstrated that the false light contributions by scattering at mechanical parts can be safely neglected (Bischoff et al. 2014). A worst case prediction of the combined in- and out-of-field stray light (pre-launch) is presented in Fig. A.2, and is compared to the in-flight stray light measurements later in Section 3.

3. In-flight stray light analysis

3.1. Data selection

To perform the in-flight stray light analysis we used several datasets, the details of which are summarised in Table 1. A particularly useful dataset was acquired with SO/PHI-HRT on 3 January 2023, when Mercury transited across the solar disc from Solar Orbiter’s point of view. At this time Solar Orbiter was at a distance of 0.949 au, and it was off-pointed such that the full transit, which was close to the southern limb, was observed. For the specific observation we show in Fig. 1, the Mercury disc was at μ = 0.56, where μ = cos(θ) and θ is the heliocentric angle. Furthermore, to test for any stray light dependence on the solar distance, two datasets with the solar limb in the FoV during October 2023 were analysed: one near perihelion, and another at an intermediate distance. These datasets were processed using the standard data reduction pipeline, including dark and flat field corrections, but without partial reconstruction by a PSF (Sinjan et al. 2022; Kahil et al. 2023).

Fig. 1. Top row: SO/PHI-HRT intensity image from 3 January 2023 displaying the solar limb (left) and Mercury disc (right), with a radial cut indicated by the dashed red lines. Bottom row: Normalised intensities along the radial cut at the solar limb (left) and Mercury disc (right), from the observation (top row), an idealised profile, and the idealised profile convolved with the best fit PSF.

3.2. Derivation of the extended PSF

We modelled the extended PSF of SO/PHI-HRT as the linear combination of the PSF obtained from phase diversity (PD) analysis, PSF_PD, together with a Gaussian function modelling the residual stray light contribution. We then took a radial one-dimensional cut through limb observations and the Mercury transit data, and compared an observed solar profile l(x) with an idealised profile l_ideal(x), where the Mercury limb is represented by a Heaviside function. We modelled the relation between the observed and idealised profiles as follows:

$$\begin{array}{c}\hfill l(x)=l_{\mathrm{ideal}}(x)\ast (w_1{\mathrm{PSF}}_{\mathrm{PD}}(x)+w_2{g}_2(x)),\end{array}$$(1)

where x is the distance along the one dimensional cut, w₁, w₂ are weights, and g₂ is a normalised Gaussian function. The * operator denotes a convolution. In Fourier space this becomes

$$\begin{array}{c}\hfill L(k)=L_{\mathrm{ideal}}(k)·(w_1{\mathrm{MTF}}_{\mathrm{PD}}(k)+w_{2}exp(-2{\pi }^2{\sigma }_2^2{k}^2)),\end{array}$$(2)

where L, L_ideal and the exponential term are the Fourier transforms of the quantities denoted by the corresponding lower case symbols in Eq. (1) and k is the spatial frequency. The MTF is the modulation transfer function, which is the absolute value of the complex valued OTF (optical transfer function). The OTF is the 2D Fourier transform of the PSF.

The sum of the weights w₁ + w₂ = 1. The unknown or free parameters are therefore σ₂ and one of the weights. The PSF_PD was obtained as described in Bailén et al. (2024), an upgraded approach to that used for the SO/PHI-HRT data used in the Sinjan et al. (2023) comparison. A key upgrade to phase diversity method is the use of a Limited-memory Broyden-Fletcher-Goldfarb-Shannon (L-BFGS) algorithm to minimise the merit function, instead of the singular value decomposition method previously used. Disentangling the impact of this change from the inclusion of the stray light term is described in Section 4.

The idealised radial profile was generated, using a fifth-order polynomial expression to describe the centre-to-limb variation Pierce & Slaughter (1977). For the best fit parameter search, the idealised profile was convolved with varying extended PSFs and compared to the observed profile. The weights, w_i, were explored between 0 and 1, while σ₂ was explored between 0 and 1000, until a good match with the observed profiles was found, with particular attention paid to the off limb and Mercury disc pixels. An example of these radial cuts, with the best fit parameters, is shown in Fig. 1 for a solar limb and a Mercury transit observation.

The following parameters of the Gaussian stray light term fit best for all considered datasets, independent of the distance of the spacecraft to the Sun:

$$\begin{array}{cc}\hfill w_1& =1/1.11=0.9009\dots \hfill \\ \hfill w_2& =0.11/1.11=0.0991\dots \hfill \\ \hfill {\sigma }_2& =300″.\hfill \end{array}$$(3)

3.3. Interpreting the stray light analysis results

To visually indicate the impact of the best fit stray light term, we refer to Fig. 2. This figure indicates a radial cut through an Airy function for a 14 cm aperture of a perfect ideal telescope observing the Sun at 1 au. When the stray light term is added, one can see the wings of the PSF extending at a higher value, which at the lowest spatial frequencies in the current analysis yields about 2 × 10⁻⁸, below the estimated worst case prediction (see Appendix A for the pre-flight stray light testing results). The system is still dominated by diffraction up to a radius of approximately 50 arcseconds (100 pixels), before stray light dominates over diffraction. The angular resolution of one SO/PHI-HRT pixel is 0.5 arcseconds. The energy content of the wide halo with respect to the central peak is about 10%; thus, the peak amplitude is reduced accordingly. Were one to give the classical Strehl estimate, one would end up with a value of 0.9. Were one to observe a star field with this telescope, the stray light halo would be orders of magnitude below the sensitivity threshold. Only the fact that the Sun is an extended bright object leads to a significant accumulated contribution in dark parts of the image.

Fig. 2. One-dimensional cut through the centre of an Airy function for a 14 cm aperture at 1 au, and the same Airy function convolved with the best fit Gaussian stray light term. Both are normalised to have the same total energy. The Strehl ratio is 0.901.

The fact that the best fit parameters, Equations (3), are the same for all datasets in Table 1 demonstrates that the residual stray light is independent of the distance to the Sun. Were there to be a significant contribution from out-of-field light, these parameters would differ for datasets at different distances, since along the orbit the ratio of the science field to the illuminating field changes by a factor of almost 13.

We can therefore isolate the origin of the two terms in our extended PSF model and attribute them to entrance window and Ritchey-Chrétien telescope. The PSF_PD term primarily originates from the heat rejecting entrance window (HREW) and its detrimental effect on the wave-front of the system has been discussed by Kahil et al. (2023). The huge thermal gradient at perihelion and the associated thermo-optical effect produces a thermal lens of up to three peak-to-valley waves defocus (which is compensated for by the refocus mechanism in the telescope). Deviations from a pure defocus shape give rise to residual low order aberrations (mainly trefoil and spherical aberration), which have been discussed by Kahil et al. (2023) and Bailén et al. (2024). They are the limiting contributors to image sharpness and granulation contrast, as they determine the shape and energy content of the PD core. The additional Gaussian stray light distribution of 300″ width is attributed to the surface ripple of the two main mirrors of the decentred Ritchey-Chrétien telescope (M1 and M2, see Gandorfer et al. 2018, for details), which have been polished by IBF.

4. Application and impact of the extended PSF

When SO/PHI-HRT data were produced, first the MTF_PD was retrieved using a series of one focused and four defocused images taken on the same day with the method described in Bailén et al. (2024). Using the L-BFGS algorithm resulted in a much improved retrieval of the low- to medium-order Zernike terms, which greatly improves the data quality (particularly at and after close approaches to the Sun). This MTF was then extended as described in Sect. 3.1. The phase information from the PSF_PD (complex component of the OTF) was then added back to the extended MTF to form the extended OTF. The extended OTF (or PSF in real space) was then used to partially reconstruct the SO/PHI-HRT polarised Stokes data, termed an ‘aberration correction’ in Kahil et al. (2023). In this correction we reconstructed the image with a deconvolution method optimised for the phase diversity technique used in (Löfdahl & Scharmer 1994; Bailén et al. 2024) and then re-convolved with the theoretical Airy function to limit the noise increase. This partially reconstructed Stokes data is the Level 2 (L2) Stokes data that is publicly available. The L2 Stokes data then underwent through the inversion of the radiative transfer equation in a two-pass process (MILOS, see Orozco Suárez & Del Toro Iniesta 2007) to produce the L2 continuum intensity, magnetic field vector, and LoS velocity. After the first inversion, the outputs were smoothed with a 15 pixel wide median filter, and used as the initial guess for the second and final inversion. For the purposes of this paper, we refer to these L2 data products as ‘V2’ (we therefore also refer to the data processed with previous versions of the pipeline as ‘V1’) and the SO/PHI-HRT data from 2024 that are publicly released to the Solar Orbiter Archive are processed to the ‘V2’ level. Past SO/PHI-HRT datasets from 2023 have been reprocessed and 2025 spring data at the ‘V2’ have now been released¹.

Now we present the impact of applying the extended PSF to the L2 data products for a specific set of observations on 29 March 2023 from 11:40–15:40 UTC. At this time Solar Orbiter was in an inferior conjunction with the Earth, with an angular separation of 2.3° and at a solar distance of 0.39 au. The high-resolution instruments on Solar Orbiter during this time observed an active region (NOAA number 13262), with a sunspot at a μ value of 0.96. During this 4 hour observation window, SO/PHI-HRT observed on a 1 minute cadence, resulting in 240 individual datasets. Until 13:00 UTC, the datasets were compressed with the nominal 6-bit compression, and the remaining datasets were more strongly compressed with a 4-bit compression scheme. For the remainder of this study, we only considered the datasets with the standard 6-bit compression.

Further details regarding this set of observations and the impact on the LoS velocity ‘V2’ data are presented in Calchetti et al. (2026). The polarimetric signal-to-noise ratio (S/N) of the L2 Stokes data does not significantly change due to the extended PSF. Any significant change in the S/N from ‘V1’ to ‘V2’ is due to the change in the improved PD analysis, not the additional stray light correction. For these observations on 29 March 2023, the improved PD analysis does not result in significant changes, and therefore the polarimetric S/N in Q,U and V all differ by < 3 × 10⁻⁵I_c, well below the polarimetric sensitivity of order 10⁻³I_c. The S/N of Stokes I is driven by shot noise, which is not impacted by the stray light correction. A proxy parameter is the continuum intensity contrast, which is discussed later in this section.

This also means that the observed changes in these ‘V2’ SO/PHI-HRT inferred quantities that we discuss here are exclusively a result of the stray light correction alone. At or after perihelion passages, the PSF_PD significantly changes with the improved PD method, and thus so does the S/N and changes in the inferred quantities.

First we compare the continuum intensity. In the left column of Fig. 3 the ‘V1’ and ‘V2’ intensity images are shown for the first dataset of this set of observations on 29 March 2023. In a small patch of quiet Sun, the root mean square intensity contrast of the ‘V1’ image is 6.45%, while it is enhanced to 7.01% in ‘V2’. The quantitative differences we discuss here are representative of the entire 80 minute period of observations at the 6-bit compression level. The difference between the ‘V1’ and ‘V2’ intensity images is displayed in the bottom panel of the left column in Fig. 3, and shows that ‘V2’ has a lower intensity in the pores and sunspot, while also being brighter in the centres of granules and darker in the intergranular lanes. Quantitatively, the intensity in the dark cool umbra and nearby pores is reduced in ‘V2’ by 15.9%. The mean intensity in the umbra, which we define as the area in the ‘V2’ dataset where I/I_c < 0.55, is 0.43 and 0.37 in the ‘V1’ and ‘V2’ datasets, respectively.

Fig. 3. Continuum intensity (left) and LoS magnetic field (right) on 29 March 2023 at 11:40 UTC. A 620 × 620 pixel portion of the FoV surrounding the sunspot and nearby plage is selected. The ‘V1’ quantities are displayed in the top row, ‘V2’ in the middle row, and their difference in the bottom row. A zoomed inset of a small group of pores is shown for all panels.

The changes to the Stokes profiles after stray light correction result in significantly different inferred magnetic fields after inverting the radiative transfer equations. The removal of the nearby quiet Sun intensity from the profiles in the darker features results in magnetic fields of greater strength and more vertical inclination being inferred, as the Zeeman splitting becomes more apparent. The following analysis describes only the first dataset of these observations. The mean quantities of several magnetic field parameters in the sunspot umbra and the entire FoV for the two versions of the first dataset are summarised in Table 2. The mean magnetic field strength in the umbra is increased from 1657 G to 1994 G, while the field in the (negative polarity) sunspot umbra also becomes more vertical, with a mean inclination changing from 143.4° to 151.2°. In the penumbra the mean inclination only changes by 1.4°. Figures such as Fig. 3 for the magnetic field strength, inclination, and azimuth are displayed in Appendix B. By fitting a Gaussian function to the weak-field pixels, the upper estimate of the LoS magnetic field uncertainty, σ_BLOS, is almost unchanged, in fact very slightly decreasing from 9.23 G to 8.79 G. Hence, the fraction of active pixels (i.e. pixels with |B_LOS|> 3σ_BLOS) in the entire FoV does not significantly change, from 11.6% to 12.1%. Across the entire FoV the unsigned total LoS flux, Φ_USFLUX (selecting only active pixels), increases from 7.95 × 10²¹ Mx to 9.05 × 10²¹ Mx, which, given the small change in the number of active pixels, reflects the impact of the strength increase in inferred LoS magnetic fields.

The increase in magnetic field strength, and more vertical inclination, results in a significant change in the LoS fields, which is visible in the right column of Fig. 3. In the umbra the mean value increases in strength from −1308 G to −1727 G. A two-dimensional histogram of the LoS magnetic field for the entire 80 pairs of datasets is displayed in the left panel of Fig. 4. From the median lines, it is clear that on the whole stronger fields are found everywhere, but this difference dramatically increases where |B_LOS|> 1200 G. On average in this regime the ‘V2’ |B_LOS| is 253 G greater with a standard deviation of 153 G, which corresponds to a mean percentage increase of 19.7% with respect to the ‘V1’ values. The slope and offset from a first-order Deming regression are 1.19 and 0.90 G. In the right panel of Fig. 4, the difference in the ‘V1’ and ‘V2’ |B_LOS| is displayed in a histogram for the entire 80 datasets. One notices a clear positive skew, with larger absolute values found across most pixels.

Fig. 4. Left: Two-dimensional histogram comparing the ‘V1’ and ‘V2’ BLOS observations from 11:40–13:00 UTC on 29 March 2023. The log density of the pixels is given by the colour scale. The first-order Deming regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in bins of width 80 G is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars. Right: Histogram with bins of width 40 G of the difference between the absolute ‘V2’ and ‘V1’ BLOS for the same observations in the left panel. The MAD and RMSD are indicated in the overlaid text.

5. Updated comparison with SDO/HMI

We now present an updated comparison of the inferred magnetic field from SO/PHI-HRT (with the stray light correction) with that from SDO/HMI, a magnetograph similar to SO/PHI-HRT that orbits the Earth and provides near continuous full disc solar observations. In Sinjan et al. (2023) a first comparison was made, using the ‘V1’ data from March 2022 and reported very similar LoS magnetic fields, while differences were noted between the vector magnetic field components. The observational data used here were recorded during an inferior conjunction in March 2023. We used the 2023 data here to update our comparison instead of the 2022 data for several reasons. Firstly, due to a change in the PSF deconvolution method, the new method cannot be applied to the 2022 data as the required calibration files are not available. Secondly, the 2023 data, given the new PSF reconstruction method and other improvements to the processing mentioned in Sect. 4 are much more representative of the majority of the available SO/PHI-HRT data, and therefore a comparison with this data is more relevant for the wider community. Finally, the 2023 data covers a longer time frame, providing superior statistics. The 2023 data have remarkably similar characteristics to the 2022 inferior conjunction data used in Sinjan et al. (2023). The 2022 data included approximately an hour’s worth of co-observations with a short interruption, while here we have 80 minutes of co-observations without interruption. In both cases an active region with a negative polarity sunspot and plage was present in the co-spatial FoV. The angle between the two spacecraft at the time of observations in March 2023 was 2.3°, to within a degree of the 3° angular separation during the March 2022 inferior conjunction data.

There are some subtle differences. In March 2023 Solar Orbiter was closer to the Sun, 0.39 au, as opposed to near 0.5 au in March 2022. Furthermore in March 2022 the image stabilisation system (ISS) of SO/PHI-HRT was not operating, while it was during March 2023. An operating ISS helps to increase the S/N and to reduce the cross-talk. While the closer distance of approach does increase image degradation due to the aforementioned aberrations from the instrument entrance window, due to our enhanced PD analysis, the noise level of the Stokes products in March 2023 is lower than that of March 2022 (1.3 × 10⁻³V/I_c vs 1.8 × 10⁻³V/I_c). In March 2022, the data were taken just after 00:00 UTC, while the March 2023 data were recorded around 12:00 UTC. Due to its 24 hour orbital period, this meant that for both datasets SDO had a large radial velocity relative to the Sun, which greatly impacts the inferred fields (see Couvidat et al. 2016), of a similar magnitude but in the opposite direction (approximately −2 km s⁻¹ instead of +3.3 km s⁻¹ in March 2022). The impact of this on the SDO/HMI data and the resulting comparison is described in the following sections.

Similarly to SO/PHI-HRT, SDO/HMI infers the photospheric vector magnetic field and LoS components from the 6173 Å absorption line, but at different cadences (90 s and 720 s for the vector and 45 s and 720 s for the LoS component, respectively). As noted in Sinjan et al. (2023), SDO/HMI produces LoS magnetic fields via two different methods. We make a clear distinction between the LoS magnetic field computed from the inferred vector magnetic field by means of radiative transfer inversion (using the complete Stokes vector) or the LoS magnetic field computed using only the intensity and circular polarisation (I, V) via the MDI-like algorithm (Couvidat et al. 2016). The inversion scheme used by SDO/HMI is the very fast inversion of the Stokes vector code (VFISV, Borrero et al. 2011), which assumes a Milne-Eddington atmosphere. Meanwhile SO/PHI-HRT only produces the LoS magnetic fields from the full magnetic field vector, also via a Milne-Eddington inversion. To make the distinction between the two methods clear, from now on when referring to the LoS component of the inferred vector magnetic field from an inversion we use ME-B_LOS, where ME represents a Milne-Eddington inversion. We use B_LOS to denote the SDO/HMI product produced using the MDI-like algorithm.

The two instruments also produce these maps at different spatial resolutions: the spatial resolution of SO/PHI-HRT at 0.39 au is 280 km, while for SDO/HMI it is 725 km. A brief comparison of the instruments’ properties can be found in Table 1 of Sinjan et al. (2023). For more information regarding the description of the SDO/HMI data and its treatment, we refer the reader to Section 2.2 of Sinjan et al. (2023) and the references therein. Table 3 briefly summarises the observation details of the SO/PHI-HRT and SDO/HMI data employed for this updated comparison.

SDO/HMI also produces hmi.dcon and hmi.dconS data series upon request, which are complete restored images using the full known PSF that includes a stray light correction (see Norton et al. 2025, for more information.). The ‘_dcon’ series are the LoS products, where the deconvolution is applied to the filtergrams, while the ‘_dconS’ have the deconvolution applied to the Stokes images. Given that the deconvolution is done using the full PSF, these data series are not directly equivalent, as the SO/PHI-HRT ‘V2’ data is only a partial image reconstruction. In Appendix C a short comparison between ‘V2’ SO/PHI-HRT data and the ‘_dcon’ and ‘_dconS’ series is presented.

5.1. Method

The remapping and alignment procedure is almost identical to that presented in Sect. 3 of Calchetti et al. (2026). We briefly summarise the steps here. First the World Coordinate System (WCS) of SO/PHI-HRT was updated by performing a cross-correlation of the continuum intensity maps of SO/PHI-HRT and SDO/HMI, with both in the SO/PHI-HRT detector frame. Secondly geometrical distortions were compensated for by comparing sub regions in the FoV when both SO/PHI-HRT and SDO/HMI LoS magnetic field maps are in the SDO/HMI detector frame. Thirdly all the SO/PHI-HRT data products, except for the azimuth, were spatially degraded with the theoretical ideal PSF of SDO/HMI (Airy function of a 14 cm aperture at 1 au). We did not apply the spatial PSF to the azimuth as it would otherwise greatly distort the values near the ambiguous [0°,180°] boundary and at the penumbra and quiet Sun boundary. Finally, with the WCS correction applied, the SO/PHI-HRT data products were remapped onto the SDO/HMI detector frame, including a distortion correction, using ‘scipy.ndimage.map_coordinates’² with one single third-order spine interpolation. This allowed for a pixel-to-pixel comparison. When every pair of datasets from SO/PHI-HRT and SDO/HMI was compared, care was taken to ensure the difference in light travel time, and the differences between UTC and TAI were both taken into account. The middle point of each dataset was taken as the reference time: ‘DATE-AVG’ for SO/PHI-HRT and ‘T_REC’ for SDO/HMI. As the SDO/HMI 45s and SO/PHI-HRT 60s data products are at a slightly different cadence, with the SO/PHI-HRT data as the limiting factor, the closest SDO/HMI 45s dataset to each SO/PHI-HRT 60s dataset was found. This ensured a minimum time overlap of at least 30s between the two datasets. The same procedure was applied when comparing the SO/PHI-HRT 60s and SDO/HMI 90s data products, where now the SDO/HMI 90s cadence is the limiting factor. Here the minimum time overlap between the any two dataset pairs was 45s.

To compare SO/PHI-HRT data with the 720s data products from SDO/HMI, we took the mean of 12 consecutive SO/PHI-HRT Stokes datasets, each separated by 60s, centred at the middle of the time over which the 720s data product from SDO/HMI was integrated. Solar rotation was also taken into account. The resulting long-term averaged SO/PHI-HRT Stokes profiles were then inverted with MILOS following the nominal procedure, only updating the weights for the Stokes parameters to reflect the enhanced S/N. This procedure resulted in a Stokes V noise level of the order of 5 × 10⁻⁴V/I_c and LoS magnetic field maps with uncertainties of the order of 4 − 5 G, both of which are similar levels to those reported for the 720s data products from SDO/HMI (Couvidat et al. 2016).

5.2. Comparison of SO/PHI-HRT and HMI LoS magnetograms

In the top left panel of Fig. 5, six pairs of 720s B_LOS SDO/HMI (hmi.m720s) data and 720s ME-B_LOS SO/PHI-HRT data are compared and visually represented with a two-dimensional histogram. The same is indicated in the top right panel but for 80 pairs of 45s B_LOS SDO/HMI (hmi.m45s) data and 60s ME-B_LOS SO/PHI-HRT data.

Fig. 5. Top row: Two-dimensional histograms comparing pairs of SO/PHI-HRT ME-BLOS and SDO/HMI BLOS values with 200 bins along each axis. Left panel: SO/PHI-HRT 720s vs SDO/HMI 720s datasets. Right panel: SO/PHI-HRT 60s vs SDO/HMI 45s datasets. Middle row: Two-dimensional histograms comparing the vector magnetic field components from SO/PHI-HRT and SDO/HMI. The first column compares inversion results from 6 pairs of SO/PHI-HRT 720s and SDO/HMI 720s datasets, while the second column does the same for 55 pairs of SO/PHI-HRT 60s and SDO/HMI 90s datasets. The log density of the pixels is given by the colour scale. The first-order Deming regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

The slope and offset from a first-order Deming regression are indicated. The Deming regression takes into account the error from both input distributions. The two top panels show very similar distributions, with near identical fit parameters and each with a Pearson correlation coefficient of 0.99. The offset is < 1 G, indicating parity between the data products of the zero-level. The slope values from the Deming regression, 1.09 and 1.07, do not fully capture the stronger fields retrieved by SO/PHI-HRT where B_LOS < −1300 G. The mean differences in this strong field regime is 252 G and 291 G with a 1σ scatter of 67 G and 89 G. This corresponded to a mean absolute percentage difference of 16% and 19% for the top left and top right panels, respectively, using the values from SDO/HMI as the reference. As reported in Sinjan et al. (2023), the mean differences of the March 2022 data were 149 G and 151 G with a 1σ scatter of 197 G (for the 720-second comparison, no scatter was reported for the SDO/HMI 45s and SO/PHI-HRT 60 second comparison). While differences found in this work are larger, which reflects the significantly stronger fields inferred by SO/PHI-HRT due to the stray light correction, the scatter is greatly reduced. This implies that the large scatter found in Sinjan et al. (2023) was in large part due to stray light contamination.

From Couvidat et al. (2016) a SDO radial velocity near −2 km s⁻¹ results in a residual increase in B_LOS (hmi.m* series) by up to 100 G in umbrae, with up to tens of Gauss in penumbrae and no significant residual in the quiet Sun. Therefore the difference between the instruments may be larger when comparing SO/PHI-HRT data against SDO/HMI at other times of the day. Unfortunately, no SO/PHI-HRT data are available at other times during the conjunction.

5.3. Comparison of SO/PHI-HRT and HMI vector magnetic fields

Now we compare the three components of the vector magnetic field, retrieved via Milne-Eddington inversions by both instruments. First we compare the magnetic field strength, |B|. In the bottom row of Fig. 5, one notices a good agreement between SO/PHI-HRT and SDO/HMI, particularly in the left panel, where both instrument’s data products are at a 720s cadence. The concentration of pixels near [x, y]=[100 G,100 G] is near symmetrical around the y = x line, highlighting the similarity of the noise levels. The offsets derived from the Deming regression fit are 36.8 G and 61.5 G, respectively. This hints that perhaps the inversion schemes of the two instruments are differently tuned for these very weak field pixels, as the noise levels of the input Stokes data are equivalent. This also points to a wider point of discussion: while both SDO/HMI and SO/PHI-HRT use Milne-Eddington inversions, they may be implemented in different ways or have different treatment of certain pixels. A comparison between VFISV (Borrero et al. 2011) and MILOS (Orozco Suárez & Del Toro Iniesta 2007), similar to those already done in Borrero et al. (2014), is out of the scope of this paper and is reserved for future work.

The slope values are 0.97 and 0.95, but are heavily dominated by the concentration of pixels in the weak field regime (|B|< 600 G). In both bottom panels, the lines denoting the medians in bins of x and y closely follow the y = x line (with a slight deviation around 1000 G) until a value of approximately 1600 G, where SO/PHI-HRT begins to infer slightly weaker magnetic field strengths. Where |B|> 1600 G (in both SO/PHI-HRT and SDO/HMI), SDO/HMI infers stronger field strengths of 60 ± 72 G and 56 ± 78 G for the bottom left and bottom right panels, respectively, where the error here denotes the standard deviation (1σ), not the standard error of the mean. The root mean squared differences (RMSDs) in this regime are 94 G and 96 G, while the mean absolute percentage differences are 4% for both cadence comparisons.

When comparing the magnetic field inclination, first we discuss the offset at the point [90° ,90°], which is the origin of interest for the inclination in our spherical geometry co-ordinate system. Using the fits found from the Deming regression, the offset at this origin is < 0.5° when comparing the 720s data products as well as the 60s versus 90s data products. When including all pixels, as we do here in the top row of Fig. 6, we naturally include many pixels where the polarisation signal is small, which results in a broad distribution of inferred inclinations by both instruments and as a consequence produces this butterfly-like spatial distribution of pixels. As for the weak field cluster of points in the magnetic field strength comparison, the concentration of pixels near [90° ,90°] dominates the regression fit here. As described in Sinjan et al. (2023), a small part of the scatter around the first-order Deming regression fit is due to the angular separation between the two spacecraft, and could also result in a small offset of < 1°.

Fig. 6. Two-dimensional histograms comparing tow of the vector magnetic field components from SO/PHI-HRT and SDO/HMI. The first column compares inversion results from 6 pairs of SO/PHI-HRT 720s and SDO/HMI 720s datasets, while the second column does the same for 55 pairs of SO/PHI-HRT 60s and SDO/HMI 90s datasets. Top row: Magnetic field inclination (relative to the LoS). Bottom row: Magnetic field azimuth. For the azimuth, pixels where |ϕHMI − ϕHRT|> 90° or |B|HRT < 600 G are omitted and not included in the fit. See caption of Fig. 5 for details regarding the over-plotted lines.

When displaying these two-dimensional histograms, we have removed pixels where SDO/HMI infers γ > 175 or γ < 5. The unprocessed SO/PHI-HRT inclination maps also contain a similar fraction of pixels with very vertical fields; however, the post-processing (Section 5.1), specifically the application of the SDO/HMI PSF on the SO/PHI-HRT data products to reduce the spatial resolution to the SDO/HMI pixel size, mixes these vertical pixels with neighbouring inclined fields. This results in an artificial upper and lower limit in inclination in the processed SO/PHI-HRT data of approximately 15° and 165°. To avoid this, the SDO/HMI PSF should be applied to the Stokes profiles from SO/PHI-HRT and then inverted; however, as the wider scientific community will study and combine the standard inversion products we choose to compare them here in this manner instead. It is the same effect that creates an artificial lower limit in the magnetic field strength values in SO/PHI-HRT (visible in the bottom row of Fig. 5).

Of greater interest is the comparison of the pixels where there is a significant signal. Using |B|> 600 G (in both SO/PHI-HRT and SDO/HMI) as a threshold to select only strong field pixels, the resulting two-dimensional histograms are depicted in Fig. 7. There is a lack of points in the [60°–90°] range, as the only sunspot in the overlapping FoV has a negative polarity, resulting in pixels with inclinations of > 90°. The pixels in the range [20°–60°] arise from the nearby positive polarity plage. In these strong field pixels, SDO/HMI infers a magnetic field that is slightly more vertical, by 2 ± 5° in both panels. The RMSD is 5°. Where |B|> 600 G, SO/PHI-HRT infers 11% and 10% more inclined fields for the 720s and SO/PHI-HRT 60s versus SDO/HMI 90s comparisons.

Fig. 7. Two-dimensional histograms comparing pairs of magnetic field inclination from SO/PHI-HRT and SDO/HMI γ with 180 bins along each axis, for all pixels where |B| > 600 G (inferred by both SO/PHI-HRT and SDO/HMI). Left panel: SO/PHI-HRT 720s vs SDO/HMI 720s datasets. Right panel: SO/PHI-HRT 60s vs SDO/HMI 90s datasets. The log density of the pixels is given by the colour scale. The first-order Deming regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

The magnetic field azimuth is compared in the two lower panels of Fig. 6. Here we only depict pixels in the sunspot, where |B|_HRT > 600 G. Due to slightly different locations of the [0° ,180° ] ambiguous boundary and the data processing described earlier, there are two clusters of pixels with very large differences in azimuth, which would severely influence the comparison. These are therefore excluded by applying an additional pixel selection criterion where the absolute difference in azimuth is < 90°. This treatment is identical to that applied in Sinjan et al. (2023), Moreno Vacas et al. (2024). Visually it is clearly evident that the azimuths from the two datasets are in very close agreement. As was mentioned earlier, the residual angular separation of the two spacecraft could result in an offset in the azimuth of 0 − 2°. Solar north has been aligned between the two datasets to ensure the same definition is used. The over-plotted lines indicating the median value in 100 bins in both x and y follow the y = x line closely and only deviate near the [0° ,180° ] ambiguous boundary.

5.4. Comparison of SO/PHI-HRT and HMI LoS components of the full vector magnetic field

In Fig. 8 we compare ME-B_LOS from both instruments. There is a clear trend that SDO/HMI infers stronger ME-B_LOS, indicated by the slope of the Deming regression fit: 0.95 and 0.97 for the left and right panels, respectively. The median in bins of x and y also deviates from the y = x line and presents a similar picture. The RMSDs for all pixels is 13 G for both. This comparison of ME-B_LOS is simply another representation of the combined differences in |B| and γ that we obtained in the previous subsection. The slightly weaker |B| and more inclined fields of SO/PHI-HRT result in a slightly less strong ME-B_LOS being inferred. Furthermore, the absolute mean difference where the SDO/HMI ME-B_LOS < −1300 is 127 ± 2 G and 107 ± 1 G for the left and right panels, respectively. These differences correspond to mean SO/PHI-HRT fields that are 7% and 6% weaker, respectively. The scatter (1σ) on these difference distributions is 86 G and 91 G, respectively. Sinjan et al. (2023) report mean differences of 486 G and 491 G and 1σ scatter of 239 G and 247 G. The mean difference values found using the SO/PHI-HRT ‘V2’ data are therefore almost a quarter of that in Sinjan et al. (2023), while the scatter values are reduced by two thirds, indicating a much closer agreement as a result of the stray light correction, which disproportionally affects these stronger fields.

Fig. 8. Two-dimensional histograms comparing pairs of SO/PHI-HRT ME-BLOS and SDO/HMI ME-BLOS with 200 bins along each axis. Left panel: SO/PHI-HRT 720s vs SDO/HMI 720s datasets. Right panel: SO/PHI-HRT 60s vs SDO/HMI 90s datasets. The log density of the pixels is given by the colour scale. The first-order Deming regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

From Couvidat et al. (2016), the residual of the SDO/HMI ME-B_LOS due to a SDO radial velocity near −2 km s⁻¹ in the umbra and penumbra is approximately 30 − 40 G, accounting for approximately 30% of the noted difference found above in the strong field regime. The slope and offset of the Deming regression, including their associated standard errors and Pearson correlation coefficient, are summarised in Table 4 for all compared quantities in this paper.

6. Summary

We modelled the (high-order) stray light contribution to the PSF of the SO/PHI-HRT telescope. Using observations of a Mercury transit and limb observations, we found that a simple and broad Gaussian term modelled the impact of stray light rather well, with no dependence on the solar distance. Its independence of the distance implies that there is no significant contribution from out-of-field light. Additionally our findings are in line with pre-flight testing and modelling.

We employed the PSF inferred through an improved version of the phase diversity technique presented in Bailén et al. (2024) and the fitted stray light term to reconstruct the polarisation maps from the March 2023 conjunction data and found significant differences in the resulting continuum intensity and vector magnetic field data products when compared with the originally released datasets. The root mean square quiet Sun intensity contrast increases from 6.45% to 7.01%. Intensity levels in sunspot umbrae and pores drop by 0.07I_c. This in turn results in much stronger and more vertical magnetic fields being inferred (within a spectral line, stray light from the quiet Sun or the penumbra, by affecting π and σ components in different ways, is expected to have such an effect). In the sunspot umbra, a 20% stronger magnetic field was retrieved and the mean inclination increases by 5%. This combination means that the LoS fields inferred in the umbra are 31% stronger. The noise level of the magnetograms is not significantly impacted. Of greater importance for this quantity is the quality of the PD analysis and the accuracy of the retrieved PSF core. In SO/PHI-HRT observations at closer distances to the Sun, there is a greater difference between the ‘V2’ and ‘V1’ data products, as there the improved PD retrieval for the core of the PSF plays a significant role, in addition to the stray light correction.

Using very similar observational conditions, the initial comparison with SDO/HMI in Sinjan et al. (2023) was updated now using the ‘V2’ SO/PHI-HRT data. On the whole a much closer agreement was found in the vector magnetic field comparison, particularly when the SO/PHI-HRT data was post-processed to the same 720s cadence as that from SDO/HMI. With both at a 720s cadence, the noise and weak field regions are much closer aligned. Nevertheless we remind the reader that due to the highly variable observational campaigns of Solar Orbiter and the extremely limited telemetry assigned to the instrument, SO/PHI-HRT may not always observe at a one minute cadence, and therefore curating a 720s cadence dataset post facto will not always be possible.

Now SO/PHI-HRT infers LoS magnetic fields (by means of an inversion) stronger than the SDO/HMI fields inferred via the MDI-like algorithm, but slightly weaker than the LoS component retrieved from the complete magnetic field vector inferred via an inversion. For the 720s comparison for B_LOS < −1300 G SO/PHI-HRT inferred LoS fields on average 13% stronger than the SDO/HMI MDI-like algorithm derived B_LOS, while when compared to the SDO/HMI vector it derived ME-B_LOS and inferred LoS fields on average 7% weaker. Furthermore the magnetic field strengths are now more closely aligned than in Sinjan et al. (2023). While the slope values of the magnetic field inclination in the strong field pixels are similar to that found in Sinjan et al. (2023), in the umbra the agreement is now closer.

It is now evident that the reported large scatter of the strong field pixels between the ‘V1’ SO/PHI-HRT and SDO/HMI data in Sinjan et al. (2023) were most likely the result of uncorrected stray light in SO/PHI-HRT, which was discussed as a possible cause. Another potential reason discussed in Sinjan et al. (2023) for the observed differences between SDO/HMI and SO/PHI-HRT are the different sampling positions across the Fe I 6173.3 Å spectral line between the two instruments (see Table 1 of Sinjan et al. 2023). However from tests using synthetic stokes profiles and the MILOS inversion code, the different positions play no role in the inference of the magnetic field strength until values greater than approximately 3.5 kG. The different inversion schemes that SO/PHI and SDO/HMI use may play a larger role. Studying this is reserved for future work.

We would like to remind the reader that we expect these comparisons (and therefore the resulting regression fits) to vary due to multiple factors; for example, as a result of different viewing conditions, the SDO radial velocity, and the Solar Orbiter distance to the Sun. This comparison, similarly to that in Sinjan et al. (2023), Moreno Vacas et al. (2024), primarily serves to highlight that the two instruments infer very similar magnetic fields, with an understanding that some relatively small residual differences do exist.

Acknowledgments

Solar Orbiter is a space mission of international collaboration between ESA and NASA, operated by ESA. We are grateful to the ESA SOC and MOC teams for their support. The German contribution to SO/PHI is funded by the BMWi through DLR and by MPG central funds. The Spanish contribution is funded by AEI/MCIN/10.13039/501100011033/ (RTI2018-096886-C5, PID2021-125325OB-C5, PCI2022-135009-2) and ERDF “A way of making Europe”; “Center of Excellence Severo Ochoa” awards to IAA-CSIC (SEV-2017-0709, CEX2021-001131-S); and a Ramón y Cajal fellowship awarded to DOS. The French contribution is funded by CNES. The SDO/HMI data are courtesy of NASA/SDO and the HMI Science Team. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101097844 – project WINSUN).

References

Appendix A: Pre-flight stray light testing and prediction

Fig. A.1 shows the measured interferogram of the Ritchey-Chrétien telescope in the lab. The path of the ion beam (the polishing tool) can be seen as a grid-like structure in the wave-front. In addition, a central peak can be seen, which is the result of a blind spot for metrology of the mirrors during the polishing procedure. For details on the mirrors of the HRT telescope we here refer to Bischoff et al. (2014). Simulations during the instrument development, taking into account MSF roughness of the mirrors, have given a worst case stray light background which is 7-6 orders of magnitude below the peak intensity, and is presented in Fig. A.2 (see also Bischoff et al. 2014).

Fig. A.1. Measured interferogram of the HRT telescope showing the ripple pattern of the ion beam figuring of M1. (Note that the measurement has been done in double-pass autocollimation; all wavefront values must therefore be divided by a factor of 2.)

Fig. A.2. Combined in- and out-of-field stray light background from the worst case straylight prediction.

Appendix B: Impact of extended PSF

In Fig. B.1 the ‘V1’,‘V2’ magnetic field strength, inclination, and azimuth, as well as the difference between the two versions, are shown for the first dataset. There is a significant increase in the magnetic field strength, particularly in the sunspot umbra, with differences on the order of 500 G. Similarly the inclination shows more vertical fields. This is to be expected, as the stray light correction removes contamination from the neighbouring quiet Sun pixels. In the quiet Sun, noise biases the inversion towards more horizontally inclined fields, and now with this signal (and therefore also its noise) removed in the sunspot, the magnetic field vectors are inferred to be more vertical Borrero & Kobel (2011).

Fig. B.1. Magnetic field strength (left), inclination (middle) and azimuth (right) on 29th March 2023 at 11:40 UTC. A 620 x 620 pixel portion of the FoV surrounding the sunspot and nearby plage is selected. The ‘V1’ quantities are displayed in the top row, ‘V2’ in the middle row, and their difference in the bottom row.

When comparing the dominant colours in the quiet Sun azimuth panels (top right and middle right in Fig. B.1) a redistribution of the azimuth in the quiet Sun is evident. In the quiet Sun, the azimuth is purely dominated by noise, so this redistribution is purely a proxy measure of the noise redistribution in Stokes Q and U as a result of the stray light correction. In any case, the azimuth in the quiet Sun is not reliable and should not be meaningfully interpreted. More importantly are changes where the field is strong, such as the sunspot. Here the azimuth shows very little change in the penumbra, while in the umbra there are two noticeable changes in the azimuth. Firstly there is a vertical strip of increased azimuth, which spatially corresponds to the location of the 0, 180° discontinuity line. This pattern is the same for all datasets and is the result of a slight correction to the CROTA keyword (0.2°), which defines the angle of rotation between the detector and solar north, hence this is purely a geometrical effect. The smaller azimuth in the umbra spatially corresponds exactly to the region where the field becomes more vertical. As the inversion infers a differently orientated (and stronger) magnetic field vector, changes in both the inclination and azimuth are to be expected.

Appendix C: Comparison with stray light corrected SDO/HMI data

Norton et al. (2025) describe the stray light correction for the SDO/HMI data, which was modelled in similar approach to the one described in this paper for SO/PHI-HRT. One daily stray light corrected SDO/HMI dataset is immediately available for download, but any dataset can be made available on request. As described therein, there exist two stray light corrected data series with the following appended names:‘_dcon’ and ‘_dconS’. The former is for the 45s data, where the deconvolution is applied to the filtergrams, and then the derived quantities computed (without use of an inversion scheme). The latter is for the 720s data, where the deconvolution is applied to the Stokes images, and then inverted with the VFISV Milne-Eddington code. The SDO/HMI stray light correction results in a significant increase in the continuum intensity contrast and the magnetic field strength, the latter particularly so in the plage regions by a factor between 1.4 − 2.5.

However there is one implementation difference to the stray light corrected SO/PHI-HRT data we present in this work. The SDO/HMI ‘_dcon’ and ‘_dconS’ series are created through a deconvolution of the complete PSF, fully reconstructing the images. In the SO/PHI-HRT data we perform a partial reconstruction to minimise the increase in the noise levels. Therefore to put these on an equal footing and compare them in this section, we additionally convolve the stray light corrected SDO/HMI data with a theoretical PSF containing an Airy function for a 14 cm aperture at 1 au.

In Fig. C.1 the LoS magnetograms are compared: in the left panel the 720s ‘_dconS’ SDO/HMI data is compared with the 720s SO/PHI-HRT data, which both have undergone inversions. The right panel compares the 45s ‘_dcon’ SDO/HMI LoS magnetic field, which is computed via a MDI-like algorithm, with 60s SO/PHI-HRT inverted data. Both have Pearson correlation coefficients of 0.99. As expected, the stray light corrected SDO/HMI magnetograms infer enhanced LoS fields compared to their normal data product counterparts.

Fig. C.1. Two-dimensional histograms comparing pairs of ‘V2’ SO/PHI-HRT and stray light corrected SDO/HMI magnetograms with 200 bins in each axis. Left panel: SO/PHI-HRT ME-BLOS 720s vs SDO/HMI ME-BLOS dconS 720s datasets. Right panel: SO/PHI-HRT ME-BLOS 60s vs SDO/HMI BLOS ‘_dcon’ 45s datasets. The ‘_dcon’ and ‘_dconS’ data from SDO/HMI is convolved with an Airy PSF (see Appendix C text for details). The log density of the pixels is given by the colour scale. The 1st order Deming Regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

In the left panel of Fig. C.1, the fields are much enhanced in SDO/HMI throughout the entire range, resulting in a lower slope of 0.81 in the comparison with the SO/PHI-HRT 720s data, as opposed to the 0.95 found in the left panel of Fig. 8. Here the SDO/HMI data is on average 243 ± 104G stronger than SO/PHI-HRT where |B_LOS|> 1000 G for both SDO/HMI and SO/PHI-HRT, compared to 101 ± 86G with the original 720s data. Here the errors denote one standard deviation of the absolute differences, not the standard error on the mean.

From the right panel of Fig. C.1, we notice that the stray light corrected SDO/HMI 45s magnetograms are now much more closely aligned with the SO/PHI-HRT data. For stronger fields, where |B_LOS|> 1000 G for both SDO/HMI and SO/PHI-HRT, the SO/PHI-HRT 60s data still has stronger fields, as found previously with the original 45s data, but by a reduced amount: stronger on average by 80 ± 84G, compared to 119 ± 72G with the original 45s data. However, the slope from the Deming Regression fit here is just below unity at 0.99, as the SDO/HMI ‘_dcon’ data is very slightly stronger in the weak field regime, which dominates the regression due to the much larger number of pixels.

In an example sunspot Norton et al. (2025) show that the stray light corrected line of sight magnetic fields, both the ‘_dcon’ and ‘_dconS’ series, are increased by a factor of 1.2 compared to their original counterparts, which is approximately in line with the relative changes in slope we find here, compared to the slopes with the original counterparts in the main section. This however neglects the impact of the additional PSF convolution we applied to the SDO/HMI data. Without the PSF applied, the slopes will decrease further.

In Fig. C.2 the magnetic field strength and inclination are compared. In general here the magnetic field strength from SDO/HMI is greater than SO/PHI-HRT for all values above approximately 250G. The slope also indicates this with a value of 0.92 as opposed to the 0.97 value found in Fig. 5. Like with the original 720s data, SO/PHI-HRT infers weaker magnetic fields and now the mean difference, where |B|> 1600G for both SDO/HMI and SO/PHI-HRT, is 124 ± 83G, a two-fold increase compared to the 60 ± 72G with the original SDO/HMI data. There is also a larger scatter of the distribution, indicated by the larger median absolute deviation (MAD; purple and grey error bars) compared to that in Fig. 5.

Fig. C.2. Two-dimensional histograms comparing pairs of ‘V2’ SO/PHI-HRT and stray light corrected SDO/HMI magnetic field strength and inclination. Left panel: SO/PHI-HRT |B| 720s vs SDO/HMI |B| ‘_dconS’ 720s datasets. Right panel: SO/PHI-HRT γ 720s vs SDO/HMI γ ‘_dconS’ 720s datasets where |B|> 600G. The ‘_dconS’ data from SDO/HMI is convolved with an Airy PSF (see text for details). The log density of the pixels is given by the colour scale. The 1st order Deming Regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

Finally, as described in the Section 5.3, we notice the consequence of the application of the SDO/HMI PSF on the lower limits of the data. Due to the application of the theoretical PSF on the SDO/HMI data to put them on equal footing, now both the SDO/HMI and SO/PHI-HRT data have artificial lower limits of approximately 40G. This is why the offset between the two in the left panel of Fig. C.2is now only 1.5G, as opposed to the 37G found in the bottom left panel of Fig. 5, when the PSF was only applied to the SO/PHI-HRT data.

From the comparison of the magnetic field inclination in the right panel of Fig. C.2 we show only the pixels where |B|> 600G in both SDO/HMI and SO/PHI-HRT. Here the agreement is strong, slightly more so than found in the main text, with a slope of 0.95, and a reduced scatter around the line denoted by the median in 100 equally spaced bins.

All Tables

All Figures

	Fig. 1. Top row: SO/PHI-HRT intensity image from 3 January 2023 displaying the solar limb (left) and Mercury disc (right), with a radial cut indicated by the dashed red lines. Bottom row: Normalised intensities along the radial cut at the solar limb (left) and Mercury disc (right), from the observation (top row), an idealised profile, and the idealised profile convolved with the best fit PSF.
In the text

	Fig. 2. One-dimensional cut through the centre of an Airy function for a 14 cm aperture at 1 au, and the same Airy function convolved with the best fit Gaussian stray light term. Both are normalised to have the same total energy. The Strehl ratio is 0.901.
In the text

	Fig. 3. Continuum intensity (left) and LoS magnetic field (right) on 29 March 2023 at 11:40 UTC. A 620 × 620 pixel portion of the FoV surrounding the sunspot and nearby plage is selected. The ‘V1’ quantities are displayed in the top row, ‘V2’ in the middle row, and their difference in the bottom row. A zoomed inset of a small group of pores is shown for all panels.
In the text

Fig. 4. Left: Two-dimensional histogram comparing the ‘V1’ and ‘V2’ BLOS observations from 11:40–13:00 UTC on 29 March 2023. The log density of the pixels is given by the colour scale. The first-order Deming regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in bins of width 80 G is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars. Right: Histogram with bins of width 40 G of the difference between the absolute ‘V2’ and ‘V1’ BLOS for the same observations in the left panel. The MAD and RMSD are indicated in the overlaid text.

In the text

Fig. 5. Top row: Two-dimensional histograms comparing pairs of SO/PHI-HRT ME-BLOS and SDO/HMI BLOS values with 200 bins along each axis. Left panel: SO/PHI-HRT 720s vs SDO/HMI 720s datasets. Right panel: SO/PHI-HRT 60s vs SDO/HMI 45s datasets. Middle row: Two-dimensional histograms comparing the vector magnetic field components from SO/PHI-HRT and SDO/HMI. The first column compares inversion results from 6 pairs of SO/PHI-HRT 720s and SDO/HMI 720s datasets, while the second column does the same for 55 pairs of SO/PHI-HRT 60s and SDO/HMI 90s datasets. The log density of the pixels is given by the colour scale. The first-order Deming regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

In the text

Fig. 6. Two-dimensional histograms comparing tow of the vector magnetic field components from SO/PHI-HRT and SDO/HMI. The first column compares inversion results from 6 pairs of SO/PHI-HRT 720s and SDO/HMI 720s datasets, while the second column does the same for 55 pairs of SO/PHI-HRT 60s and SDO/HMI 90s datasets. Top row: Magnetic field inclination (relative to the LoS). Bottom row: Magnetic field azimuth. For the azimuth, pixels where |ϕHMI − ϕHRT|> 90° or |B|HRT < 600 G are omitted and not included in the fit. See caption of Fig. 5 for details regarding the over-plotted lines.

In the text

Fig. 7. Two-dimensional histograms comparing pairs of magnetic field inclination from SO/PHI-HRT and SDO/HMI γ with 180 bins along each axis, for all pixels where |B| > 600 G (inferred by both SO/PHI-HRT and SDO/HMI). Left panel: SO/PHI-HRT 720s vs SDO/HMI 720s datasets. Right panel: SO/PHI-HRT 60s vs SDO/HMI 90s datasets. The log density of the pixels is given by the colour scale. The first-order Deming regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

In the text

Fig. 8. Two-dimensional histograms comparing pairs of SO/PHI-HRT ME-BLOS and SDO/HMI ME-BLOS with 200 bins along each axis. Left panel: SO/PHI-HRT 720s vs SDO/HMI 720s datasets. Right panel: SO/PHI-HRT 60s vs SDO/HMI 90s datasets. The log density of the pixels is given by the colour scale. The first-order Deming regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

In the text

	Fig. A.1. Measured interferogram of the HRT telescope showing the ripple pattern of the ion beam figuring of M1. (Note that the measurement has been done in double-pass autocollimation; all wavefront values must therefore be divided by a factor of 2.)
In the text

	Fig. A.2. Combined in- and out-of-field stray light background from the worst case straylight prediction.
In the text

	Fig. B.1. Magnetic field strength (left), inclination (middle) and azimuth (right) on 29th March 2023 at 11:40 UTC. A 620 x 620 pixel portion of the FoV surrounding the sunspot and nearby plage is selected. The ‘V1’ quantities are displayed in the top row, ‘V2’ in the middle row, and their difference in the bottom row.
In the text

Fig. C.1. Two-dimensional histograms comparing pairs of ‘V2’ SO/PHI-HRT and stray light corrected SDO/HMI magnetograms with 200 bins in each axis. Left panel: SO/PHI-HRT ME-BLOS 720s vs SDO/HMI ME-BLOS dconS 720s datasets. Right panel: SO/PHI-HRT ME-BLOS 60s vs SDO/HMI BLOS ‘_dcon’ 45s datasets. The ‘_dcon’ and ‘_dconS’ data from SDO/HMI is convolved with an Airy PSF (see Appendix C text for details). The log density of the pixels is given by the colour scale. The 1st order Deming Regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

In the text

Fig. C.2. Two-dimensional histograms comparing pairs of ‘V2’ SO/PHI-HRT and stray light corrected SDO/HMI magnetic field strength and inclination. Left panel: SO/PHI-HRT |B| 720s vs SDO/HMI |B| ‘_dconS’ 720s datasets. Right panel: SO/PHI-HRT γ 720s vs SDO/HMI γ ‘_dconS’ 720s datasets where |B|> 600G. The ‘_dconS’ data from SDO/HMI is convolved with an Airy PSF (see text for details). The log density of the pixels is given by the colour scale. The 1st order Deming Regression fit and y = x line are given by the dashed black and solid grey lines, respectively. The median value in 100 equally spaced bins is indicated in both x (grey) and y (purple) as well as the MAD denoted by the respective error bars.

In the text