© The Authors 2026

1 Introduction

Dust in circumstellar disks around young stars and dust in shells around red giants have been recognized in many systems as infrared excess in the spectral energy distribution. This dust plays an important role in the formation of stars and planetary systems (Birnstiel 2024) and the mass loss of evolved stars and the dust enrichment of the interstellar medium (Ferrarotti & Gail 2006).

In the past decade, a lot of new information on circumstellar dust has been obtained with imaging observations of scattered stellar light in the near-IR and visual wavelength range. High-contrast instruments at large ground-based telescopes achieve an angular resolution of up to 20 mas, and this provides important information about the geometric distribution and the scattering properties of the circumstellar dust. In particular, the new information from scattered light observations is complementary to the extensive data already available from IR observations of the thermal emission of the dust around protoplanetary disks (e.g., Woitke et al. 2019), debris disks (e.g., Hughes et al. 2018; Chen et al. 2014), and red giants (e.g., Höfner & Olofsson 2018).

The determination of the geometric structure and the scattering properties of circumstellar dust near stars is of much interest for the investigation of the formation of planetary systems from dusty circumstellar disks around young stars. The measurement of circumstellar polarization is also important for the study of dust formation in the winds around cool, mass loosing stars. However, detection of the scattered intensity from circumstellar dust is difficult because the signal is faint and often hidden in the strong glare of the central star. This requires a careful separation of the stellar signal from the circumstellar signal, which can be achieved with high contrast techniques using space telescopes, such as the Hubble Space Telescope (HST; e.g., Kalas et al. 2013; Schneider et al. 2014; Zhou et al. 2023) or JWST (e.g., Gáspár et al. 2023; Lawson et al. 2023), or with ground-based telescopes using adaptive optics (AO) systems (Milli et al. 2017; Ren et al. 2023).

High-resolution imaging polarimetry is a very powerful method to separate the circumstellar polarization signal introduced by dust scattering from the direct light of a bright central source, which usually has no or only a small net linear polarization (e.g., Kuhn et al. 2001; Quanz et al. 2011). Great progress has been achieved with new instruments for imaging polarimetry in combination with AO systems at large ground-based telescopes, including Subaru CIAO, HiCIAO, and SCeXAO instruments (Murakawa et al. 2004; Hodapp et al. 2008; Lucas et al. 2024); Gemini GPI (Perrin et al. 2015); and VLT NACO and SPHERE (Lenzen et al. 2003; Beuzit et al. 2019). Polarimetric imaging of many objects has been obtained, including protoplanetary disks (e.g., Avenhaus et al. 2018; Benisty et al. 2023, and references therein), debris disks (e.g., Esposito et al. 2020; Crotts et al. 2024), circumstellar shells around red giants (e.g., Ohnaka et al. 2016; Khouri et al. 2020; Montargès et al. 2023), and post-AGB stars (e.g., Andrych et al. 2023).

High-resolution imaging polarimetry is a quite new and not so simple observing technique because of point spread function (PSF) smearing and polarization cancellation effects and noise in the weak differential signal (Schmid et al. 2006; Schmid 2022). In addition the polarimetric calibration of the complex high contrast imaging systems has to be considered (e.g., Tinbergen 2007; Schmid et al. 2018; de Boer et al. 2020; van Holstein et al. 2020), including the temporal PSF variations for ground-based observations using AO systems (Tschudi & Schmid 2021; Ma et al. 2023). Depite this, many impressive results on dust scattering geometry have been obtained, but for most objects, no detailed analysis of the polarized intensity has been attempted, mostly because this was not the focus of the study. For some special cases, the polarized flux is derived more accurately to constrain the properties of the scattering dust, for example, for extended circumstellar disks (Stolker et al. 2016; Monnier et al. 2019; Milli et al. 2019; Arriaga et al. 2020; Hunziker et al. 2021), axisymmetric systems (Pinilla et al. 2018; Tschudi & Schmid 2021; Ma et al. 2023), and edge-on debris disks (e.g., Graham et al. 2007; Engler et al. 2017). It is highly desirable that such detailed studies on the circumstellar polarization are applied to a larger sample and are further improved for some key objects.

Accurate polarimetric measurements for circumstellar scattering regions require a good understanding of the observational aspects. Convolution effects degrade the measurable polarization signal, as described for some cases in Schmid et al. (2006), Avenhaus et al. (2017), Heikamp & Keller (2019), Tschudi & Schmid (2021), and Ma et al. (2023, 2024b). Additional issues are polarization offsets introduced by instrumental effects, interstellar polarization, and an intrinsically polarized central star. These offsets can be eliminated with a polarimetric normalization, which sets the polarization for a certain region to zero (Quanz et al. 2011; Avenhaus et al. 2014). We call this procedure hereafter polarimetric zero-point correction (zp-correction).

A very useful and frequently used way to describe the circumstellar signal is the azimuthal polarization Q_ϕ (Schmid et al. 2006; Quanz et al. 2013). A convolution or an offset by a zp-correction introduces for Q_ϕ quite complex changes, and some of these effects have not been described in detail or have not been investigated at all. For example, it is clear that polarimetric offsets, or a zp-correction, change the polarization signal, but this work describes for the first time how this change affects the azimuthal distribution of the circumstellar Q_ϕ signal. Because of the lack of a systematic study, it is difficult to extrapolate findings from the papers cited above to a more general picture.

For compact scattering regions near the star, the mentioned effects are much stronger, and they must even be considered for a qualitative interpretation of the polarization signal. Therefore, the scattering regions at small separations to the star of r ≲ 5 λ/D, or r ≲ 200 mas for large telescopes with AO systems, are often disregarded despite the fact that these regions are scientifically very interesting. They correspond for protoplanetary disks at ≈ 100 pc to the innermost <20 AU, which is where planets form (e.g., Birnstiel 2024), and for nearby red giant stars, they correspond to a separation of a few stellar radii where dust particles condensate (e.g., Höfner & Olofsson 2018). In non-coronagraphic and non-saturated data, this region is sometimes masked in the data reduction process because it looks “noisy” or shows an unexplained polarization pattern. However, these features might just be caused by a polarization offset or a convolution effect.

In coronagraphic observations, the central region is hidden behind the focal plane mask to achieve observations more sensitive to the polarization signal further out. Also for this type of data, it is important to understand the impact of the convolution and of polarization offsets.

This work presents model simulations and a parameter framework for a systematic description of the PSF convolution effects and an investigation of the polarimetric offsets introduced by the instrument, the interstellar polarization, the central star, or by the zp-correction applied to the data. The results should provide a better understanding of these effects for the interpretation of observational results and be useful for the definition of observing and measuring strategies for polarimetric imaging of circumstellar scattering regions.

This paper is organized as follows. In the next section, we describe the model simulation and introduce the used polarimetric parameters. Section 3 describes the degradation of the scattering polarization by the PSF convolution for axisymmetric circumstellar scattering regions and for inclined disk models. The polarimetric calibration and zp-correction effects are investigated in Sect. 4 and discussed in particular for coronagraphic observation and for data with only partially resolved circum-stellar scattering regions. In Sect. 5 the results are summarized and discussed in the context of observational data. Detailed information, in particular numerical results for the presented simulations, are given in the appendix.

2 Model calculations

The block diagram in Fig. 1 gives an overview of the steps involved in imaging polarimetry using the Stokes Q = I₀ − I₉₀ and U = I₄₅ − I₁₃₅ polarization parameters as example. The full process is described by the boxes connected with red arrows, from the intrinsic polarization model defined in system coordinates Q′(x, y), U′(x, y) to the model in sky coordinates Q′(α, δ), U′(α, δ), including the contribution of interstellar polarization Q″(α, δ), U″(α, δ), to the observed signal Q(α, δ), U(α, δ) and the result after a possible polarimetric zp-correction Q^z(α, δ), U^z(α, δ). This is simplified in the simulations according to the blue path in Fig. 1 by calculating the observed polarization signal in system coordinates Q(x, y), U(x, y) considering only convolution and polarimetric offsets and a zp-correction Q^z(x, y), U^z(x, y) for the calibration of the data. This approach still considers many of the key aspects of polarimetric imaging but disregards second-order effects and particular problems of individual instruments or datasets. The parameters for the x, y coordinates are used for the description of the model simulations the x, y coordinates, while some general polarimetric principle are discussed for on sky parameters using (α, δ) coordinates. However, it is important to be aware of simplifying assumptions outlined in Fig. 1 for the interpretation of the model results.

Fig. 1 Block diagram with the simplified description of the simulated imaging polarimetry given in blue. The red arrows show the full imaging process from the intrinsic model to the on-sky model including interstellar polarization to the observed and possibly zp-corrected polarization signal.

2.1 Intrinsic scattering models

The intrinsic models for the circumstellar dust scattering region and a point-like central star at x₀, y₀ are described by 2D maps for each component of the Stokes vector $$I^{\prime }(x,y)=\left(I^{\prime }(x,y),Q^{\prime }(x,y),U^{\prime }(x,y)\right).$$(1)

The components describe the intensity, I′(x, y), and the linear polarization, Q′(x, y), U′(x, y), in x,y coordinates aligned with the object. The circular polarization signal V′ is expected to be much weaker for circumstellar scattering and is neglected. In principle, multiple scattering by dust can produce a circular polarization signal, in particular for (magnetically) aligned dust grains. Measurements for circumstellar circular polarization exist since many decades (e.g., Kwon et al. 2014; Bastien et al. 1989; Angel & Martin 1973), but the signals are weak, or originate from regions far away from the star, where interstellar magnetic fields may play a role. Considering this in our models is beyond the scope of this paper.

The models use $I^{\prime }(x,y)=I_s^{\prime }+I_d^{\prime }(x,y)$ consisting of a star, $I_s^{\prime }(x,y)=I_s^{\prime }(0,0)=I_s^{\prime },$, with or without an intrinsic polarization, $Q_s^{\prime }$, $U_s^{\prime }$, and an extended dust scattering region, $I_{\text{d}}^{\prime }(x,y)$, according to the vector components $$I^{\prime }(x,y)=I_s^{\prime }+I_d^{\prime }(x,y),$$(2) $$Q^{\prime }(x,y)=Q_s^{\prime }+Q_d^{\prime }(x,y),$$(3) $$U^{\prime }(x,y)=U_s^{\prime }+U_d^{\prime }(x,y).$$(4)

In most models the star is not polarized and $Q_s^{\prime }$ and $U_s^{\prime }$ are set to zero or $Q^{\prime }(x,y)=Q_d^{\prime }(x,y)$ and $U^{\prime }(x,y)=U_d^{\prime }(x,y)$.

Scattering geometries. We considered the axisymmetric models Ring0 and Disk0 representing circumstellar disks seen pole-on or spherical dust shells as illustrated by the intensity maps $I_d^{\prime }(x,y)$ in the upper row of Fig. 2. The intrinsic polarization flux, $P_d^{\prime }$, is proportional to the intensity, $P_d^{\prime }(x,y)=0.25\cdot I_d^{\prime }(x,y)$.

The Ring0 models have a mean radius, r₀, and a Gaussian radial profile with full width at half maximum (FWHM) of ∆r = 0.2 r₀, and r₀ is varied from 3.15 to 806.4 mas. The Disk0 models are axisymmetric scattering regions extending from an inner radius, r_in, to an outer radius, r_out, with a surface brightness described by the power law $$I_d^{\prime }(r)=A_I{\left(r/r_{\text{ref}}\right)}^{\alpha },$$(5) with reference radius r_ref. Three cases, Disk0α0, Disk0α−1, and Disk0α−2, are considered with α = 0 for a constant surface brightness and α = −1 and −2 for a brightness decreasing with distance. For all cases the outer radius is r_out = 100.8 mas while the radius of the inner cavity is varied between r_in = 3.15 mas and 50.4 mas to investigate the differences between fully resolved and partially resolved scattering regions. Figure 2 shows $I_d^{\prime }(x,y)$ maps for Disk0α−1 with r_in = 12.6 mas and 50.4 mas.

Simulations for non-axisymmetric scattering geometries are obtained by adopting the Ring0 and Disk0 geometries for inclined, flat disks with i = 60^◦. The dust density in the disk plane ρ_d(r_d) of the inclined models RingI60 and DiskI60α0, DiskI60α−1, DiskI60α−2 are described by the same radial parameters as for the axisymmetric models. The scattering angle in the inclined disk model varies as a function of azimuthal angle ϕ on the disk, and therefore the scattered intensity and polarization also depend on ϕ. This is simulated as in Schmid (2021) for flat, optically thin disks with a dust scattering phase function described by a Henyey-Greenstein function for the intensity with asymmetry parameter g = 0.6 and a Rayleigh scattering like dependence for the fractional polarization with p_max = 0.25. With these settings the resulting model maps I′(x, y) depend only on r₀ for the RingI60 model and on α and the radius of the central cavity r_in for the DiskI60 models. Figure 2 shows the maps for RingI60 and DiskI60α−1 with the same r₀ and r_in parameters as for the pole-on models. The major and minor axes of the inclined disks are aligned with the x and y coordinates and the backside of the disk is in the +y direction. Because of the strong forward scattering (g = 0.6) the intensity signal is enhanced on the front-side, and this is a frequently observed property for inclined disks (e.g., Ginski et al. 2023).

Fig. 2 Maps for the intrinsic disk intensity $I_d^{\prime }(x,y)$ for the four types of circumstellar scattering models we used.

2.2 Sky coordinates and interstellar polarization

Observations are obtained in sky coordinates (α, δ) and the intrinsic maps in system coordinates Q′(x, y), U′(x, y) can be transformed into the sky coordinates Q′(α, δ), U′(α, δ) by a rotation of the geometry and the polarization vector (e.g., Schmid 2021). The signal reaching Earth Q″(α, δ), U″(α, δ) is often affected by interstellar polarization introduced by dichroic absorption of magnetically aligned interstellar grains (e.g., Draine 2003). This introduces a fractional polarization offset, which can be described for the usual low-polarization case (Q′, U′ ≪ I′ and q, u ≪ 1) by $$Q^{″}(\alpha ,\delta )\approx Q^{\prime }(\alpha ,\delta )+q_{\text{is}}{I}^{\prime }(\alpha ,\delta ),$$(6) $$U^{″}(\alpha ,\delta )\approx U^{\prime }(\alpha ,\delta )+u_{\text{is}}{I}^{\prime }(\alpha ,\delta ),$$(7) where q_is, u_is are the components of the fractional interstellar polarization $p_{\text{is}}={\left(q_{\text{is}}^2+u_{\text{is}}^2\right)}^{-1/2}$ with position angle θ_is = 0.5 · atan2(u_is, q_is)¹ defined in sky coordinates. We used flux ratios to quantify the resulting intensity or polarization so that the interstellar transmission losses can be neglected. Extreme dichroic extinction by the interstellar medium can convert linear polarization partly to circular polarization (e.g., Kwon et al. 2014), but this is not considered in this work.

2.3 Signal degradation by ground-based AO observations

The turbulence in the Earth atmosphere leads to a strong seeing convolution of the incoming signal (I″, Q″, U″). With an AO system the seeing can be strongly reduced but there remain smearing and polarimetric cancellation effects that can strongly change the spatial distribution of the observed signal I(α, δ), Q(α, δ), U(α, δ) when compared to the incoming signal (e.g., Perrin et al. 2003; Fétick et al. 2019). The effects are particularly strong for not well resolved structures near the star. The AO correction depends on the atmospheric turbulence and instrument properties, and therefore the observational PSF changes with time and shows various types of non-axisymmetric structures (e.g., Cantalloube et al. 2019). In addition, AO instruments are complex and they usually introduce instrumental polarization offsets which are difficult to calibrate accurately (Tinbergen 2007). These observational effects are investigated in this work.

PSF convolution. Axisymmetric PSFs are adopted so that the convolved scattering signal does not depend on the orientation of the observed system. Therefore we can apply the convolution directly to the models described in disk coordinates (x, y). Real PSFs are variable and deviate from axisymmetry but assuming a stable, axisymmetric PSF is a reasonable approximation for PSFs with high Strehl ratio or for the averaged PSF obtained after a number of polarimetric cycles taken in field rotation mode.

We used PSF_AO(x, y) representing an AO system with a narrow core with FWHM or diameter of D_PSF = 25.2 mas and an extended halo. This profile is obtained by averaging azimuthally the PSF described in Schmid et al. (2018) for the standard star HD 161096 taken under excellent condition with the AO instrument VLT/SPHERE/ZIMPOL in the N_I filter (λ_c = 817 nm) with a Strehl ratio of about 0.4. The radial profile of PSF_AO is plotted in Fig. 3 together with a Gaussian profile PSF_G with the same diameter D_PSF, which is used to investigate the impact of the extended halo in PSF_AO on the convolved signal. Convolution with a Gaussion PSF with D_PSF ≈ λ/D could also represent roughly diffraction limited imaging polarimetry with a space telescope with diameter D or ground based seeing limited imaging polarimetry with D_PSF of the order ≈1000 mas.

Polarimetric calibration. Many types of instrumental effects are introduced in observations taken with high-contrast imaging polarimetry and there exist established procedures to calibrate the data, for example for SPHERE/IRDIS (de Boer et al. 2020; van Holstein et al. 2020) and SPHERE/ZIMPOL polarimetry (Schmid et al. 2018; Hunziker et al. 2020; Tschudi et al. 2024). Individual instruments show also particular effects but these aspects are beyond the scope of this paper.

A very general and important observational effect are the offsets introduced by the interstellar p_is or by the instrument polarization p_inst of the kind described in Eqs. (6) and (7). Because the intensity of the star ΣI_s is much higher than the polarization from the circumstellar dust ΣQ_d, ΣU_d, by about a factor 100 or even more, already a small fractional polarization offset for the central star of about p ≈ 0.001 can strongly disturb the circumstellar signal, while p ≈ 0.01 can completely mask that signal. In addition, there could also be a contribution from an intrinsic polarization of the central star. The effects of polarimetric offsets do not depend on the orientation of the selected coordinate system, and therefore they can also be treated in the (x, y) coordinate system.

Unfortunately, it can be quite difficult to disentangle the different contributions to the overall polarimetric offset, and therefore an ad hoc zp-correction for the polarization is often applied to the data based on the assumption that the polarization of the target in a selected integration region is zero, or at least very small (Quanz et al. 2011; Avenhaus et al. 2014). A polarimetric offset sets the integrated Stokes parameters in a certain region to zero ΣQ^z = ∫ Q^z(α, δ)dαdδ = 0 and similar for ΣU^z = 0, but this procedure can introduce a bias. For coronagraphic observations, the central star cannot be included for the determination of the zp-correction, and this adds another complication, which we also describe later in the text.

Fig. 3 Radial profiles for the extended PSFAO (red) and the Gaussian PSFG (black). In the main panel, the total flux is normalized to 106 counts. In the inset, PSFG is reduced by a factor of 0.4 for a comparison of the PSF cores. The pixel size is 3.6 mas × 3.6 mas.

2.4 Analysis of diagnostic polarization parameters

The impact of the convolution and the polarimetric zp-correction, follows from the comparisons between the simulated observational maps I(x, y), Q(x, y), U(x, y) or the corresponding zp-corrected polarization Q^z(x, y), U^z(x, y) with the initial maps I′(x, y), Q′(x, y), U′(x, y).

Circumstellar scattering produces predominantly a linear polarization in azimuthal direction Q_ϕ with respect to the central star located at x₀, y₀. Therefore Q_ϕ is a very useful polarization parameter which is defined by $$Q_{\varphi }(x,y)=-Q(x,y)\cos \left(2{\varphi }_{xy}\right)-U(x,y)\sin \left(2{\varphi }_{x,y}\right),$$(8) $$U_{\varphi }(x,y)=+Q(x,y)\sin \left(2{\varphi }_{xy}\right)-U(x,y)\cos \left(2{\varphi }_{x,y}\right),$$(9) with ϕ_xy = atan2((x − x₀), (y − y₀)) according to the description of Schmid et al. (2006) for the radial Stokes parameters Q_r, U_r and using Q_ϕ = −Q_r and U_ϕ = −U_r. The U_ϕ(x, y) parameter gives the linear polarization component rotated by 45^◦ with respect to azimuthal component Q_ϕ(x, y). The different polarized intensities for a given point (x, y) in the polarization maps are related to the polarized flux for the linear polarization according to $$P(x,y)={\left(Q^2(x,y)+U^2(x,y)\right)}^{1/2}={\left(Q_{\varphi }^2(x,y)+U_{\varphi }^2(x,y)\right)}^{1/2}.$$(10)

Figure 4 illustrates the relation between Q_ϕ and the Stokes parameters Q and U for a pole-on disk ring. In this model the U_ϕ component is zero and P(x, y) = Q_ϕ(x, y) because of the axisymmetric geometry. Non-axisymmetric geometries can produce an intrinsic $U_{\varphi }^{\prime }(x,y)$ signal, for example by multiple scattering (Canovas et al. 2015), but for the simple (single scattering) models adopted in this work the intrinsic $U_{\varphi }^{\prime }(x,y)$-signal is also zero for inclined disks. However, the convolution and polarimetric offsets will introduce also for these models a U_ϕ signal, which is equivalent to a non-azimuthal polarization component (Fig. 9).

The observational Q(x, y), U(x, y) or Q_ϕ(x, y), U_ϕ(x, y) maps have in high contrast imaging polarimetry often a lower signal per pixel than the photon noise σ(x, y) or other pixel to pixel noise sources. Therefore, the polarized flux P(x, y) is biased, because it is always positive, and the measured values follow a Rice probability distribution. For a low polarization P(x, y) ≲ σ(x_, y) the noise will introduce on average a signal of about P(x, y) ≈ +σ(x, y) (Simmons & Stewart 1985), and this can add up to a very significant spurious signal for the polarized flux ΣP in an integration region Σ containing many tens or more pixels. This noise problem is avoided by using Q_ϕ(x, y) for measuring the strength of the spatially resolved linear polarization (Schmid et al. 2006). This is a reasonable approximation, because circumstellar scattering produces predominantly a polarization in azimuthal direction with Q_ϕ(x, y) ≫ 0 and U_ϕ(x, y) ≈ 0, and therefore one can consider Q_ϕ as rough proxy for P according to $$P(x,y)={\left(Q_{\varphi }^2(x,y)+U_{\varphi }^2(x,y)\right)}^{1/2}\approx Q_{\varphi }(x,y).$$(11)

Enhanced random noise in the data does not change the mean Q_ϕ signal for a pixel region, but P is for observational data often very significantly affected by the bias problem described above. The simulations in this work do not consider statistical noise in the data. However, there are other systematic differences between Q_ϕ(x, y) and P(x, y) because of the U_ϕ(x, y) signal introduced by the PSF convolution and polarization offsets, which are described by the simulation results.

The convolution and zp-correction effects are quantified using polarization parameters integrated in well-defined apertures $$\text{Σ}X={{\displaystyle \int }}^{\text{}}X(x,y)\text{d}x\text{d}y,$$(12) where X is a place holder for the PSF convolved radiation parameters, such as X = {I, Q, U, P, Q_ϕ, U_ϕ}, and similarly for X′ or X^z for the intrinsic models or zp-correcected models, respectively. The term Σ defines a circular aperture centered on the star for the determination of the system integrated parameters. Later, we also consider other axisymmetric circular and annular apertures for the parameters of radial subregions (see Sect. 4.3.1). Axisymmetric apertures are very useful for quantifying the convolution by an axisymmetric PSF and polarization offsets for an intensity signal dominated by the axisymmetric stellar PSF. An overview on the used polarization parameters is given in Appendix A (Tables A.1 and A.2).

For a given integration region, Σ, an “aperture” polarization, (Σ) = (ΣQ)² + (ΣU)²)^1/2. with the corresponding position angle θ(Σ_i) can be defined in the same way as for aperture polarimetry. These parameters are useful for specifying the polarization of the star or the fractional polarization p = (Σ)/ΣI and the fractional Stokes parameters q, u in order to quantify polarization offsets and zp-correction factors (see Appendix A).

The polarization parameters are usually normalized to the integrated intrinsic polarization $\text{Σ}{Q}_{\varphi }^{\prime }$, such as $\text{Σ}Q\text{/Σ}{Q}_{\varphi }^{\prime }$, $\text{Σ}{Q}^{\text{z}}\text{/Σ}{Q}_{\varphi }^{\prime }$, because $\text{Σ}{Q}_{\varphi }^{\prime }$ provides a good reference for the characterization of the impact of the convolution or of offsets on polarization parameters.

For the characterization of the azimuthal distribution of the polarization Q(ϕ) and U(ϕ), the four Stokes Q quadrants, Q_xxx = {Q₀₀₀, Q₀₉₀, Q₁₈₀, Q₂₇₀}, and the four Stokes U quadrants, U_xxx = {U₀₄₅, U₁₃₅, U₂₂₅, U₃₁₅}, are used (Schmid 2021), while X_xxx stands for all eight parameters. They represent the Stokes Q and U polarization in the four wedges of 90^◦ centered on the position angle ϕ = xxx, which form for well-resolved circum-stellar scattering regions the positive-negative Q and U quadrant patterns. The signals in these quadrants are changed by the convolution and polarization offsets, and they are useful to quantify the corresponding changes for the azimuthal distribution of the polarization signal Q_ϕ(ϕ).

Fig. 4 Maps for the Ring0 models with r0 = 12.6 mas (left), 25.2 mas (middle), and 50.4 mas (right) for the intrinsic circumstellar intensities, $I_d^{\prime }(x,y)$, and the PSFG convolved intensity, Id(x, y), and polarization Qd(x, y), Ud(x, y), and Qϕ(x, y). Here, $\text{Σ}{I}_d^{\prime }$ is the same for all models, and ±1 for the color scale represents the same fluxes per pixel for all panels.

3 Polarization degradation by the PSF convolution

This section describes the basic PSF smearing and cancellation effects for imaging polarimetry of circumstellar scattering regions. The convolution is always applied to the intrinsic intensity I′ and Stokes Q′, U′ maps, or from maps with a polarization offset like the Q″, U″ maps, and from the resulting Q and U one can then derive according to Eq. (8)–(10) the convolved maps for the polarization parameters Q_ϕ, U_ϕ and P (e.g., Tschudi & Schmid 2021). Alternatively, one could also apply the convolution to the $I_0^{\prime }$, $I_{90}^{\prime }$, $I_{45}^{\prime }$, and $I_{135}^{\prime }$ polarization components and derive from this the convolved intensity and polarization maps, because the convolution operation is distributive (PSF ∗ A + PSF ∗ B = PSF ∗ (A + B)). However, a direct convolution of the intrinsic azimuthal polarization ($\text{PSF\hspace{0.17em}}*Q_{\varphi }^{\prime }$) or of the polarized flux (PSF ∗ P′) gives wrong results.

3.1 Convolution for axisymmetric scattering models

3.1.1 Narrow pole-on rings

The convolution depends strongly on the spatial resolution, specifically on the angular size of the scattering signal compared to the PSF widths. This can be described by an axisymmetric scattering geometry like for a dust disk seen pole-on or a spherical dust shell.

Simulations for the convolution of the Ring0 models are shown in Fig. 4 for three ring sizes with r₀ = 12.6, 25.2, and 50.4 mas and using the Gaussian PSF_G. Maps for the intrinsic dust scattering intensities $I_d^{\prime }(x,y)$, and convolved intensities I_d(x, y) and the polarization Q_d(x, y), U_d(x, y), Q_ϕ,d(x, y) are given. The central star is unpolarized in this model and there is Q_d = Q, U_d = U and Q_ϕ,d = Q_ϕ. All three models have the same intrinsic flux $\text{Σ}{I}_d^{\prime }$ and $\text{Σ}{Q}_{\varphi }^{\prime }$, and therefore the peak fluxes scale as max $\left(I_d^{\prime }(x,y)\right)\propto 1/r_{02}^{}$ and are higher for a smaller r₀. The star is unpolarized and therefore does not contribute to the polarization maps. The star I_s is not included in the intensity maps because for realistic cases it would dominate strongly the scattering intensity I_d. For the intrinsic model there is $Q_{\varphi }^{\prime }(x,y)=P^{\prime }(x,y)$ and $U_{\varphi }^{\prime }(x,y)=0$. For axisymmetric models, this property is not changed by the convolution, and therefore the maps for P(x, y) and U_ϕ(x, y) are not shown in Fig. 4.

Radial profiles for the intensity and azimuthal polarization I_d(r) and Q_ϕ(r) are plotted in Fig. 5 for the Ring0 model convolved with the Gaussian PSF_G and the extended PSF_AO. The profiles are normalized to the intrinsic peak flux $I_{\text{d},\max }^{\prime }=\max \left(I_d^{\prime }(r)\right)$ and $Q_{\varphi ,\max }^{\prime }=\max \left(Q_{\varphi }^{\prime }(r)\right)$ respectively, to illustrate the signal degradation. The intensity is smeared by the convolution and this reduces the peak surface brightness of the rings but the total intensity is preserved $\text{Σ}{I}_d=\text{Σ}{I}_d^{\prime }$. For the smallest model with r₀ = 12.5 mas the ring structure is no more visible (Fig. 4) and there is a strong maximum at the center.

The convolved polarization signal shows also a smearing and in addition a mutual polarimetric cancellation between the positive and negative quadrants in the Q and U maps (Schmid et al. 2006). This reduces strongly the polarization signal for Q_d(x, y), U_d(x, y), and Q_ϕ(x, y) close to the star and produces also for compact rings a central zero (Figs. 4 and 5).

The integrated polarization ΣQ_ϕ is reduced for the PSF_G convolved models by factors $\text{Σ}{Q}_{\varphi }/\text{Σ}{Q}_{\varphi }^{\prime }$ = 0.91, 0.66, and 0.28 for ring radii of r₀ = 50.4 mas, 25.2 mas and 12.6 mas, respectively (Table 1). We measure the apparent ring size using the radius for the maximum surface brightness r(max(Q_ϕ)) and the size of the central cancellation hole by the radius r_h at half maximum Q_ϕ(r_h) = 0.5 · (max(Q_ϕ)). For r₀ ≲ 0_.5 D_PSF the peak radius for the convolved ring is significantly larger than r₀. For larger rings r₀ ≳ D_PSF these two radii agree well (r(max(Q_ϕ)) ≈ r₀).

The $\text{Σ}{Q}_{\varphi }/\text{Σ}{Q}_{\varphi }^{\prime }$ values in Table 1 are significantly lower for the PSF_AO convolved models because of the much stronger smearing effects introduced by the PSF halo. For r₀ < 4 D_PSF the reduction is about a factor of 0.5 lower with respect to a convolution with PSF_G, or very roughly at the level of the Strehl ratio for PSF_AO simulated for the AO system.

For very small rings with r₀ = 0.25 D_PSF and 0.125 D_PSF only a very small amount of the Q_ϕ polarization remains and the radial Q_ϕ profile has for these two cases roughly the same shape as the convolved Ring0 with r₀ = 0.5 D_PSF with a peak at r(max(Q_ϕ)) ≈ (2/3) D_PSF and a hole radius r_h ≈ (1/3) D_PSF. This represents the simulated inner working angle for the detection of a resolved polarization signal for a circumstellar scattering region. For observational data these limits might be less good because of PSF variation and alignment errors.

Importantly, the spatial resolution is practically not reduced by the convolution with PSF_AO when compared to PSF_G and the radii r(max(Q_ϕ)) and r_h(Q_ϕ) for PSF_AO are very similar for the two cases because both PSF have the same peak width D_PSF. Therefore, the shape of the radial profiles in Fig. 5 are very similar for the two cases and the morphology of the strong polarization structures in the PSF_G model maps of Fig. 4, would look practically the same for a convolution with PSF_AO. The halo of the PSF_AO introduces faint, extended polarization artifacts for axisymmetric geometries as described in the Appendix (Appendix B), but they are very weak with a surface brightness of ≲ 1% when compared to the peak signal of the ring.

Fig. 5 Normalized radial profiles for the intrinsic $I_d^{\prime }(r)/I_{\text{d},\text{max}}^{\prime }$ (top panel) and the PSFG (thick lines) and PSFAO convolved (thin lines) intensity $I_d(r)/I_{\text{d},\max }^{\prime }$ (middle panel) for the Ring0 models with r0 = 12.6 mas (black), 25.2 mas (red), and 50.4 mas (blue). The bottom panel shows the convolved polarization $Q_{\varphi }(r)/Q_{\varphi ,\max }^{\prime }$, where the dotted black lines are the solid black lines multiplied by a factor of ten. The radii for the profile peaks r(maxQϕ(r)) and the central holes rh(Qϕ) given in Table 1 are indicated with vertical marks.

3.1.2 Radially extended scattering regions

Axisymmetric, radially extended disks or shells models are just superpositions of concentric ring signals, and the polarimetric cancellation effects will be strong for the innermost regions and much reduced further out. Therefore, the convolved polarization signal strongly underestimates the scattering near the central star and can mimic the presence of a central cavity even if no such cavity is present. The effect is illustrated in Fig. 6 with PSF_AO convolved Q_ϕ(x, y) maps for the models Disk0α0, Disk0α−1, and Disk0α−2, with radial brightness profiles $Q_{\varphi }^{\prime }(r)=A_{\varphi },A_{\varphi }{\left(r/r_{\text{ref}}\right)}^{-1}$, and A_ϕ(r/r_ref)⁻², respectively. In all three cases the intrinsic disk extends from r_in = 3.15 mas or 0.125 D_PSF to r_out = 100.8 mas. The radius of the convolution hole (r_h(Q_ϕ)) is slightly smaller than 0.4 D_PSF for the α = −2 case and slightly larger than 0.4 D_PSF for the α = −1 disk with a flatter brightness distribution (Table C.1).

Constant surface brightness. The convolution effects for Disk0α0 are shown in Fig. 7 with radial profiles for the intrinsic parameters $Q_{\varphi }(r)/I_0^{\prime }$, $I_d^{\prime }(r)=I_0^{\prime }$ and the convolved intensity I_d(r) and polarization Q_ϕ(r) for different inner disk radii r_in and for PSF_G and PSF_AO convolution. The profiles I_d(r) show for increasing cavity size r_in an increasing central dip depth and width. The surface brightness I_d(r) is strongly reduced after convolution with PSF_AO while the central cavity is less pronounced. The convolution does not change ΣI_d but for PSF_AO a lot of the signal is redistributed to radii r ≫ 100 mas.

The convolved Q_ϕ(r) profiles show for all cases a central zero Q_ϕ(0) = 0, even for the disk without central cavity. Only models with r_in ≈ D_PSF or larger show an obvious difference when compared to the model without cavity. The Q_ϕ(r) profiles do not reach the intrinsic $Q_{\varphi }^{\prime }(r)=A_{\varphi }=0.25I_0^{\prime }$ level even for the convolution with a narrow PSF_G, because of the polarimetric cancellation. For the convolution with PSF_AO there is an additional reduction of the Q_ϕ(r) level by about 40% but despite this, the radial shape of the profiles is very similar for the two cases as follows also from hole radii r_h(Q_ϕ) given in Table C.1.

Centrally bright scattering regions. Many circumstellar disks and shells show a steep surface brightness profile increasing strongly toward smaller radii. Therefore, the polarimetric cancellation suppresses efficiently the intrinsically very bright but barely resolved central signal (e.g., Avenhaus et al. 2018; Garufi et al. 2020; Khouri et al. 2020; Andrych et al. 2023).

Simulations of radial profiles are shown in Fig. 8 for PSF_AO convolved models DiskI60α−2 with intrinsic surface brightness profile for the polarization $Q_{\varphi }^{\prime }(r)\propto r^{-2}$ with r_ref = 12.6 mas and A_ϕ = 20.7 similar to Eq. (5) and for different cavity sizes r_in. The convolved profiles Q_ϕ(r) show clearly how the central hole size decreases for smaller r_in, and how the measurable peak polarization signal max(Q_ϕ(r)) increases. Despite the strongly increasing intrinsic $Q_{\varphi }^{\prime }(r)=A_{\varphi }{\left(r/r_{\text{ref}}\right)}^{-2}$ signal at small radii the convolved Q_ϕ(r) curves converge to a limiting model case with r_i ≈ 0.2 D_PSF (≈5 mas), because even large amounts of intrinsic signal in the center are fully cancelled by the convolution (Table C.1). The Q_ϕ(r) profiles are still quite sensitive for constraining the intrinsic $Q_{\varphi }^{\prime }$ signal around r ≈ D_PSF based on the location of the flux maximum r(max(Q_ϕ)) and the amount of Q_ϕ(r) signal near this location. A careful analysis of Q_ϕ(r) can therefore constrain an inner cavity for a dust scattering region and this could be potentially useful for estimates on the dust sublimation radius for disks around young stars or the dust condensation radius for circumstellar shells around mass losing stars.

Details of the profiles Q_ϕ(r) depend also on the power law index α for the surface brightness. The peak radii r(max(Q_ϕ(r)) and central hole radii r_h for a given r_in are a bit larger for Disk0α−1 than Disk0α−2 (Table C.1), because in this model the contributions of smeared signal from larger separations are more important than for Disk0α−2 and this reduces the apparent sharpness of the central cancellation hole.

Fig. 6 Central cancellation holes in the Qϕ(x, y) maps for the models Disk0α0, Disk0α−1, and Disk0α−2 after convolution with the extended PSFAO. The size of the inner cavity rin = 0.125DPSF (3.15 mas) is the same for all three models and indicated by the black dot.

Fig. 7 Normalized profiles $I(r)/I_0^{\prime }$ and $I(r)/I_0^{\prime }$ for the Disk0α0 models with r0 = DPSF (red), 0.5 DPSF (blue), and without a cavity (black) for PSFG (thick) and PSFAO (thin) convolution and for the intrinsic model (dotted).

Fig. 8 Radial profiles Qϕ(r) for the Disk0α−2 model convolved with PSFAO on a log-scale (upper panel) and a linear scale (lower panel) for different inner cavities r0 as indicated by the colors. The thin black line is the intrinsic surface brightness $Q_{\varphi }^{\prime }(r)$.

Fig. 9 Maps for the intensity Id and polarization Qϕ, Q, U, Qϕ, Uϕ, and P (from top to bottom) for RingI60 models with r0 = 12.6 mas, 25.2 mas, and 50.4 mas after convolution with the Gaussian PSFG (first three columns). The last column gives the same for the intrinsic model with r0 = 50.4 mas. Stokes quadrant parameters are indicated in some Q and U maps.

3.2 Convolution for inclined disk ring models

The scattering geometry for a rotationally symmetric system with an inclined symmetry axis is not axisymmetric and then new features appear in the intensity and polarization maps. This is illustrated in Fig. 9 by the PSF_G convolved maps for the scattered intensity I_d and the polarization parameters Q, U, Q_ϕ, U_ϕ, and P for RingI60 models with an inclination of i = 60^◦ and r₀ = 12.5 mas, 25.2 mas and 50.4 mas and including the intrinsic maps for the last case. The central star is assumed to be unpolarized, or Q = Q_d, U = U_d and P = P_d, and the stellar intensity is not included in the I_d(x, y) intensity maps. All RingI60 models have the same intrinsic polarization $Q_{\varphi }^{\prime }(r)$.

The ring front side is much brighter because of the adopted forward scattering parameter g = 0.6 for the dust and this is clearly visible for the I′(x, y) and I(x, y) maps for r₀ = 50.4 mas. This changes for the less resolved systems into a front-side intensity arc for r₀ = 25.2 mas or r₀ = D_PSF and a slightly elongated spot offset toward the front side for r₀ = 12.6 mas (0.5 D_PSF).

The inclined models show a left-right symmetry for the intensity I_d(x, y) = I_d(−x, y), for Stokes Q, the azimuthal polarization Q_ϕ, and the polarized flux P. The Stokes U-parameter has a left-right antisymmetry U_d(x, y) = −U_d(−x, y), as well as the U_ϕ-signal introduced by convolution effects.

The intrinsic Stokes parameters Q′(x, y) and U′(x, y) show positive and negative regions which can be characterized by the quadrant polarization parameters $\Sigma Q_{\varphi }^{\prime }$, $Q_{000}^{\prime }$, $Q_{090}^{\prime }$, and $Q_{180}^{\prime }$ for Stokes Q′ and $Q_{000}^{\prime }$, $U_{045}^{\prime }$, $U_{135}^{\prime }$, and $U_{225}^{\prime }$ for Stokes U′ as indicated in Fig. 9. They are useful for the characterization of the azimuthal distribution of the polarization based on the natural Stokes patterns produced by circumstellar scattering (Schmid 2021). Quadrant parameters have been calculated for simple models of debris disks (Schmid 2021) and of transition disks (Ma & Schmid 2022).

The intrinsic RingI60 models have strong positive Q₀₉₀ and Q₂₇₀ quadrants centered on the major axis of the projected disk, because of the high fractional polarization produced by scatterings with θ ≈ 90^◦. The front side quadrant Q₁₈₀ shows for well resolved systems a clear negative component, which is, however, less dominant than in intensity, because the fractional polarization of forward scattering is lower than for 90^◦ scattering.

The negative Q₀₀₀ and Q₁₈₀ components disappear for not well resolved disks because the PSF smearing of the two strong positive components Q₀₉₀ and Q₂₇₀ cancel the signal of the negative Stokes Q quadrants. For the convolved RingI60 model with r₀ = 12.6 mas there remain only two positive Q_d-spots but their relative position still indicates the orientation of the projected major axis. The brighter disk front side produces in the Stokes U_d map a left side dominated by the positive U₁₃₅ signal and a right side by the negative U₂₂₅ component. This feature is still visible for barely resolved disks and this indicates the location of the disk front-side.

The intrinsic disk polarization of our models is azimuthal everywhere, and therefore the map $U_{315}^{\prime }$ is equal P′(x, y), while $Q_{\varphi }^{\prime }(x,y)$, according to Eq. (10). The convolved maps Q_ϕ(x, y) for r₀ = 50.4 mas represents well the intrinsic map appart from the smearing, but for less well resolved disks Q_ϕ(x, y) starts to display negative values above and below the center and for r₀ = 12.6 mas a strong, central quadrant pattern for Q_ϕ(x, y) is visible as expected for an unresolved, central source with a positive Stokes ΣQ signal and ΣU = 0.

The convolution of the non-axisymmetric scattering polarization produces a U_ϕ signal and we call this effect the convolution cross talks for the azimuthal polarization² Q_ϕ → U_ϕ. This effect increases with decreasing spatial resolution, and therefore the U_ϕ(x, y) signal becomes stronger for less well resolved disks. For an unresolved disk the quadrant patterns for the azimuthal polarization Q_ϕ(x, y) = −Q(x, y) cos(2ϕ_xy) and U_ϕ(x, y) = Q(x, y) sin(2ϕ_xy) are equally strong but rotated by 45^◦. For the polarized flux P the convolved signal for large rings is roughly equal to the azimuthal polarization P(x, y) ≈ Q_ϕ(x, y), and for barely resolved rings it evolves toward P(x, y) ≈ Q(x, y) (Fig. 9).

3.2.1 Convolution and integrated polarization parameters

The convolution with a normalized PSF does not change the integrated Stokes polarization or the integrated intensity. Thus, there is for the RingI60 models ΣQ = ΣQ′ = 0.421 $P^{\prime }(x,y)$ and ΣU = ΣU′ = 0, independent of the disk size or the spatial resolution. Contrary to this, the integrated polarization parameters ΣQ_ϕ, ΣU_ϕ, and ΣP and the sums of absolute values Σ|Q| and Σ|U|, Σ|Q_ϕ| and Σ|U_ϕ| depend on the resolution and the PSF convolution. The dependencies are plotted for the RingI60 models as a function of ring radius r₀ convolved with PSF_G and PSF_AO in Fig. 10, and Table D.1 lists numerical values.

The red curves in Fig. 10 show ΣQ_ϕ, which is large for well resolved disks and approaches zero for small, unresolved disks r₀ → 0 when the Q_ϕ map shows a perfect positive-negative quadrant pattern with no net ΣQ_ϕ polarization. For the intrinsic disk there is $\text{Σ}{Q}_{\varphi }^{\prime }=100\%$ and $\text{Σ}{P}^{\prime }=\text{Σ}{Q}_{\varphi }^{\prime }=\text{Σ}\left|Q_{\varphi }^{\prime }\right|$ and these relations are still approximately valid for well resolved disks convolved with PSF_G because the convolution effects are small (Fig. 10a). The convolution with PSF_AO introduces even for very extended disks strong smearing, because of the extended PSF halo (Fig. 10b). Therefore, ΣP is for well resolved disks larger than ΣQ_ϕ by about 5% for the model with r₀ ≈ 800 mas and about 18% for r₀ = 201.6 mas (Table D.1).

For an unresolved scattering region r₀ ≪ D_PSF there remains only an unresolved polarized source with ΣP. Because of the alignment of the RingI60 models with the (x, y) disk coordinates, there is ΣQ = Σ|Q| = ΣP and ΣU = Σ|U| = 0. The azimuthal polarization is then zero ΣQ_ϕ ≈ 0 and the integrated absolute values for the azimuthal polarization are Σ|Q_ϕ| = Σ|U_ϕ| = (2/π) |ΣP| for the quadrant pattern of an unresolved source (Schmid 2021).

In the models a net signal in azimuthal polarization ΣQ_ϕ > 0 or an integrated absolute signal Σ|Q_ϕ| larger than Σ|U_ϕ| indicates the presence of at least a marginally resolved circumstellar polarization signal (Fig. 10). In observational data, ΣQ_ϕ signals can also be produced by noise effects and one needs to derive for a given dataset the limits for a significant detection of a resolved circumstellar ΣQ_ϕ-signal. However, one can expect for random noise sources introduced for example by the atmospheric turbulence, photon and read-out noise, that they produce also random positive and negative Q_ϕ and U_ϕ signals, and different systematic effects like polarization offsets (see Sect. 4) or alignment errors produce no or only small positive or negative ΣQ_ϕ and ΣU_ϕ signals. Therefore, the two criteria ΣQ_ϕ > 0 and Σ|Q_ϕ| > Σ|U_ϕ| are reliable indicators for the presence of resolved circumstellar polarization. Contrary to this, the ΣP signal is systematically enhanced by random noise and also polarization offsets change typically ΣP substantially.

Fig. 10 Integrated polarization parameters for the RingI60 model as a function of the ring radius r0 after convolution with PSFG (a) and PSFAO (b). All parameters have been normalized to the intrinsic value $\text{Σ}{Q}_{\varphi }^{\prime }$. In panel b the ΣQϕ curve from panel a is repeated as a dotted line.

Fig. 11 Large-scale halo signals for the Stokes parameters Q, U and the azimuthal polarization Qϕ(x, y), Uϕ(x, y) for the RingI60 model with r0 = 50.4 mas convolved with PSFAO. The inset on the lower-right in each panel shows the disk ring in the center with a color scale reduced by 500 times.

3.2.2 Halo signals produced by an extended PSF_AO

The extended halo in the PSF_AO of an adaptive optics system smears substantially the net Stokes polarization ΣQ, ΣU of a system over a large area producing an extended P(x, y) halo of linear polarization for r ≫ r₀. This halo effect is illustrated in Fig. 11 for the PSF_AO convolved RingI60 model with r₀ = 50.4 mas, where the smeared ΣQ signal produces a polarized speckle ring ghost around r ≈ 0.4″ and a halo. The Stokes U signal in the halo is much weaker because there is no net U-signal for this model, and only a much weaker artifact pattern of the kind described in Appendix B is visible. The Q-halo produces extended Q_ϕ(x, y), U_ϕ(x, y) quadrant patterns including the relatively strong speckle ring around r ≈ 0.4″, which shows the small scale imprint of the polarization from the bright ring.

The surface brightness of the halo is low and it can be difficult to recognize it in real data because of observational noise. Nonetheless, for the RingI60 model shown in Fig. 11 the Q-halo integrated in an annulus from r = 0.2″to 1.5″ is almost 30% of the system integrated ΣQ-signal (Table D.1). Therefore, it is important to use large integration apertures for the determination of ΣQ and ΣU for data convolved by an extended PSF.

The Q_ϕ signal in the halo integrated from r = 0.2″ to 1.5″ contributes less than 0.2% to the total ΣQ_ϕ (Table D.1), because this signal has a positive-negative Q_ϕ, U_ϕ quadrant patterns with zero net signal. Therefore, the measurement of ΣQ_ϕ for a compact circumstellar scattering region should be restricted to a circular aperture which excludes the outer halo regions containing no net Q_ϕ signal but possibly substantial observational noise.

Fig. 12 Azimuthally averaged profiles for Qϕ(r), |Qϕ(r)|, and |Uϕ(r)| for the RingI60 models with r0 = 25.2 mas and 201.6 mas convolved with PSFAO. The green curve for the ratio |Uϕ(r)|/|Qϕ(r)| provides a rough measure for the convolution cross talk, and the dashed line indicates the system integrated value Σ|Uϕ|/Σ|Qϕ| from Table D.1.

3.2.3 Convolution cross talk Q_ϕ → U_ϕ or intrinsic U_ϕ signal

The models considered in this work use a simpified description of the dust scattering which does no produce an intrinsic $\text{Σ}\left|U_{\varphi }^{\prime }\right|=0$ polarization. However, it was pointed out in Canovas et al. (2015) that multiple-scattering by dust in optically thick protoplanetary disks with a non-axisymmetric scattering geometry can produce intrinsic $U_{\varphi }^{\prime }(x,y)$ signals representing non-azimuthal polarization components. Their simulations give $U_{\varphi }^{\prime }$ signals of up to about ±5% of the azimuthal polarization $U_{\varphi }^{\prime }(x,y)$ and this U′-polarization can be useful to constrain the dust scattering properties and the disk geometry. This effect was also recognized in the disk models presented by Ma & Schmid (2022, Fig. 6).

As previously discussed, the PSF convolution can also produce very substantial U_ϕ(x, y) signals for models with zero intrinsic $U_{\varphi }^{\prime }(x,y)$. This fact was already mentioned by Canovas et al. (2015), but they did not quantify this effect and their modeling used only a Gaussian PSF for the convolution.

The RingI60 simulations can be used to quantify the Q_ϕ → U_ϕ convolution signal for different disk sizes r₀. As a simple metric for this effect, we used the ratio Σ|U_ϕ|/Σ|Q_ϕ| given in Table D.1 (see also the Σ|U_ϕ| and Σ|Q_ϕ| curves in Fig. 10). For an unresolved disk, the convolution gives as an extreme limit a ratio of Σ|U_ϕ|/Σ|Q_ϕ| = 1. The ratio is 0.22 and still high for the PSF_G convolved disk with r₀ = 50.4 mas plotted in Fig. 9. A larger disk of about r₀ = 201.6 mas is requird to reach a low ratio of 0.03 so that an intrinsic U_ϕ-signal could be detectable.

The situation is worse for disk models convolved with PSF_AO producing substantially more Q_ϕ → U_ϕ cross talk, and the ratio Σ|U_ϕ|/Σ|Q_ϕ| is 0.19 for r₀ = 201.6 mas and still larger than 0.1 for r₀ = 806.4 mas The strong smearing of the Stokes Q-signal produces in the halo a ratio Σ|U_ϕ|/Σ|Q_ϕ| ≈ 1. For a more detailed analysis the radial dependence of the ratio |U_ϕ(r)|/|Q_ϕ(r)| should be considered, and such profiles are shown in Fig. 12 for PSF_AO convolved RingI60 models with r₀ = 25.2 mas and 201.6 mas. For the compact disk the |Q_ϕ(r)| signal is only at the separation of the ring r ≈ 12−35 mas substantially larger than |U_ϕ(r)|, but nowhere more than a factor of 2. For the unresolved polarization near the center and for the smeared halo signal the ratio is |U_ϕ(r)|/|Q_ϕ(r)| ≈ 1. For the larger disk, the Q_ϕ(r) signal dominates the U_ϕ(r) cross talk signal by about a factor of 20 at the location of the inclined ring r ≈ 100−200 mas.

This indicates that an intrinsic $U_{\varphi }^{\prime }(x,y)$ polarization at the level of 0.05 $U_{\varphi }^{\prime }$ is only detectable for large disks r₀ ≳ 200 mas in high quality data for currently available AO systems, otherwise the convolution cross talks dominate. Helpful is, that the expected geometric structure of the observable U_ϕ(x, y) signal produced by multiple scattering (see Canovas et al. 2015; Ma & Schmid 2022) is different from the Q_ϕ → U_ϕ convolution artifacts, which can even be constrained strongly from the observed Q_ϕ(x, y) signal. In any case, the detection of the presence of intrinsic $Q_{\varphi }^{\prime }$ polarization requires high quality data of an extended scattering region and a very careful assessment of the convolution cross talk effects.

U_ϕ signal and Q_ϕ uncertainty. In many studies the U_ϕ signal is used as a proxy for the observational uncertainty for the measured Q_ϕ signal. This is a reasonable approach for cases with small Q_ϕ → U_ϕ cross talk, like axisymmetric or close to axisymmetric scattering geometries, and very extended systems like RingI60 models with r ≳ 200 mas. The spurious signals introduces by speckle noise, read-out and photon noise, and small-scale instrumental artifacts in the U_ϕ map are then larger than the convolution cross talk and therefore also representative for observational uncertainties in the Q_ϕ map.

However, for non-axisymmetric compact scattering regions the U_ϕ signal consist mainly of the well-defined systematic convolution cross talk signal with high ratios |U_ϕ(r)|/|Q_ϕ(r)| ≳ 0.5 like the RingI60 r₀ = 25.2 mas model in Fig. 12. Despite this the azimuthal polarization signal Q_ϕ can still be highly significant, because for high quality data the observational noise in Q_ϕ is much smaller than the systematic Q_ϕ → U_ϕ cross talk. In such cases another metric than the U_ϕ signal must be used for the assessment of the observational uncertainties, like the dispersion of the measured results for different datasets or a detailed analysis of speckle and pixel to pixel noise in the data.

Nonetheless, a low |U_ϕ| signal can be used to identify high quality data within a series of measurements taken under variable observing conditions. Because the systematic cross talk Q_ϕ → U_ϕ is anti-correlated with the quality of the observational PSF one can select the polarization cycles with low |U_ϕ| and this may provide a higher quality Q_ϕ map for a target.

3.2.4 Convolution and quadrant polarization parameters

The convolution can change for not well resolved RingI60 models strongly the azimuthal distribution of the polarization signal Q_ϕ(ϕ) and U_ϕ(ϕ). This can be quantified with the quadrant polarization parameters which provide a simple formalism for the description of polarimetric convolution effects. We use the symbol X_xxx for all quadrant parameters, and Q_xxx or U_xxx for the four Stokes Q or Stokes U quadrants, respectively. The quadrant values are related to the integrated Stokes parameters according to $$\text{Σ}Q=\text{Σ}{Q}_{000}+\text{Σ}{Q}_{090}+\text{Σ}{Q}_{180}+\text{Σ}{Q}_{270},\text{and}$$(13) $$\text{Σ}U=\text{Σ}{U}_{045}+\text{Σ}{U}_{135}+\text{Σ}{U}_{225}+\text{Σ}{U}_{315}.$$(14)

For the mirror-symmetric RingI60 models, there is also ΣQ₀₀₀ = ΣQ₂₇₀, ΣU₀₄₅ = −ΣU₃₁₅, and ΣU₁₃₅ = −ΣU₂₂₅ (Schmid 2021).

The convolution changes the flux in the different Stokes quadrants because of smearing and mutual cancellation as can be seen in Fig. 9. This degradation is also illustrated for the normalized quandrant parameters $\text{Σ}{Q}_{\varphi }^{\prime }$ for the PSF_G and PSF_AO convolved RingI60 models as a function of r₀ in the upper panels of Fig. 13 (see also Table D.1). For smaller and less resolved disks all Q-quadrants approach the same value ΣQ_xxx = ΣQ_d/4 as expected for an unresolved system with an integrated polarization ΣQ_d. For the Stokes U quadrants the effects are equivalent on both sides of the y-axis, and strong smearing turns the sign of the weaker back-side quadrants values U₀₄₅ and U₁₃₅ to the sign of the strong front side quadrants. This produces for compact disks positive signals for U₀₄₅ and U₁₃₅ “on the left side” of the star, and negative signals for U₂₂₅ and U₃₁₅ “on the right” side in Fig. 9 for the RingI60 model with r₀ = 12.6 mas.

Differential quadrant parameters. Despite the smearing and polarimetric cancellation the information about the relative intrinsic strengths of the quadrant parameters can still be recovered to some degree as long as the scattering region is partially resolved. This is possible, because the mutual compensation of the polarization signal changes opposite sign quadrants roughly by similar amounts (Fig. 13 upper panel), and in step with the degradation of the total azimuthal polarization ΣQ_ϕ shown in Fig. 10 for the same models.

Therefore, good values to constrain the intrinsic $\text{Σ}{Q}_{xxx}^{\prime }$ fluxes are the relative differential values $$\frac{\text{Δ}{Q}_{xxx}}{\text{Σ}{Q}_{\varphi }}=\frac{\text{Σ}{Q}_{xxx}-(\text{Σ}Q/4)}{\text{Σ}{Q}_{\varphi }},$$(15) which quantify how much the individual Stokes Q quadrants contribute to the total Stokes Q signal (Eq. (13)) or how much more or less than the average contribution Q/4. According to Fig. 13 (lower left and Table D.2), the ∆Q_xxx/ΣQ_ϕ values are quite independent of the resolution for disks with r₀ ≥ 50 mas, and one can still recognize for a RingI60 model disk with r₀ ≈ 25 mas that the front side deviates more from the average than the back side. The relative differential quadrant values (Eq. (15)) are also practically the same for a convolution with PSF_G and PSF_AO and in very good agreement with the intrinsic values $$\begin{array}{l}\frac{\text{Δ}{Q}_{xxx}}{\text{Σ}{Q}_{\varphi }}\approx \frac{\text{Δ}{Q^{\prime }}_{xxx}}{\text{Σ}{Q^{\prime }}_{\varphi }}.\end{array}$$(16)

The equivalent quantities for the Stokes U quadrants are just ratios (ΣU_xxx/ΣQ_ϕ) because the average value (ΣU/4) is zero. Figure 13 (lower right) shows, that the relative Stokes U quadrant ratios deviate for small disks with r₀ ≲ 25 mas substantially from a constant. This is caused by the morphology of the Stokes U map for the RingI60 model, which has one dominant positive U₁₃₅ and one dominant negative U₂₂₅ quadrant. Smearing and cancellation affect the weak quadrants ΣU₀₄₅ and ΣU₃₁₅ stronger than the reference value ΣQ_ϕ, while the effects between the strong components U₁₃₅ and U₂₂₅ are smaller than for ΣQ_ϕ, because the separation between the strong components is relatively large.

A useful alternative are the relative differential values between the U quadrants on the positive or negative x-axis side: $$\frac{\text{Δ}{U}_{+}}{\text{Σ}{Q}_{\varphi }}=\frac{\text{Σ}{U}_{135}-\text{Σ}{U}_{045}}{\text{Σ}{Q}_{\varphi }}\text{and}\frac{\text{Δ}{U}_{-}}{\text{Σ}{Q}_{\varphi }}=\frac{\text{Σ}{U}_{225}-\text{Σ}{U}_{315}}{\text{Σ}{Q}_{\varphi }}.$$(17)

There is ∆U₊ = −∆U₋ because of the symmetry of the RingI60 models. The ratio ∆U₊/ΣQ_ϕ is almost independent of the disk size r₀ and the used PSF, according to the red line and points in Fig. 13 (lower right) and the values in Table D.2, which are very similar to the parameters ∆Q_xxx/ΣQ_ϕ. This demonstrates that differential quadrant parameters are useful to push the characterization of Q_ϕ(ϕ) and U_ϕ(ϕ) for circumstellar scattering regions toward smaller separations.

Fig. 13 Quadrant polarization parameters ΣXxxx normalized to the intrinsic azimuthal polarization $U_{\varphi }^{\prime }(x,y)$ (upper panels) and differential quadrant parameters ∆Xxxx normalized to the convolved ΣQϕ (lower panels) for Ringi60 models as a function of radius r0. Full lines give the values for the convolved PSFG models, and crosses are for convolved PSFAO models. The filled diamonds are the intrinsic values. For clarity not all ∆Uxxx values are given.

3.3 Convolution of inclined extended disks

Extended disks with unresolved or partially unresolved central regions are frequently observed (e.g., Garufi et al. 2022) and it is of interest to investigate regions close to the star r < D_PSF because they correspond to the zone of terrestrial planet formation ≲ 3 AU for systems in nearby star forming regions. In particular, the polarization of the unresolved part of the disk at r ≲ 0_.5 D_PSF (_<12.6 mas) can be compared with the polarization of the resolved region r ≳ D_PSF to constrain the presence or absence of significant changes in the scattering properties between the two regions.

Polarization maps for the inclined and extended DiskI60α-2 models are plotted in Fig. 14 with radii for the inner cavities of r_in = 0.125 D_PSF, 0.5 D_PSF, and D_PSF, or 3.15 mas, 12.6 mas, and 25.2 mas, respectively. The inner disk rim is very bright for these models, and therefore the convolved disk maps look similar to the images in Fig. 9 for small disk rings. The model with r_in = D_PSF in Fig. 14 shows a resolved Q_ϕ disk image with small cross talk residuals in U_ϕ, while the disk with a very small cavitiy of r_in = 0.125 D_PSF has strong quadrant patterns for Q_ϕ and U_ϕ as expected from the net Q-signal of a bright, unresolved central disk region.

The integrated polarization parameters for the DiskI60α−2 models depend strongly on the inner disk radius r_in according to Fig. 15 or the numerical values given in Table D.3. The parameters are all normalized to the intrinsic azimuthal polarization $\text{Σ}{Q}_{xxx}^{\prime }$ for the disk with r_in = 0.5 D_PSF. The intrinsic polarization parameters $Q_{\varphi ,\text{ref}}^{\prime }=\text{Σ}{Q}_{\varphi }^{\prime }$ and ΣQ = ΣQ′ = 0.421 $\text{Σ}{Q}_{\varphi }^{\prime }=\text{Σ}{P}^{\prime }$ are much larger for disks with small inner radii, for example by a factor 1.7 for r_in = 0.125 D_PSF when compared to r_in = 0.5 D_PSF. The convolution does not change the integrated Stokes polarization ΣQ = ΣQ′ but only redistributes spatially the signal Q′(x, y) → Q(x, y), and therefore the ΣQ curves are identical in the two panels of Fig. 15 for the models convolved with PSF_G and PSF_AO.

Contrary to this, the PSF convolution reduces or even cancels the strong $Q_{\varphi }^{\prime }(r)\propto r^{-2}$-signal of the central region and the effect is more important for PSF_AO than for PSF_G. Therefore, ΣQ_ϕ reaches for a given convolution PSF a limiting value for models with a cavity smaller than r_in < 0.5 D_PSF, despite the fact that the intrinsic $\text{Σ}{Q}_{\varphi }^{\prime }$ and ΣQ′ increase steeply for r_in → 0 for the DiskI60α−2 model.

The central quadrant patterns in the convolved Q_ϕ and U_ϕ maps have a strength proportional to the net Q signal from the unresolved inner disk region. The central Stokes signal ΣQ_c ≈ ΣQ(r ≲ 0.5 D_PSF) and ΣU_c can be used to constrain the amount of polarization P_c and the averaged polarization position angle (θ_c = 0.5 · arctan2(ΣU_c, ΣQ_c)) for the unresolved part. Differences between θ_c and the averaged polarization position angle of the resolved signal θ_d for ΣQ(r≳ 0.5 D_PSF) can point to a structural change between the unresolved and the resolved part of the scattering region, or to a contribution from interstellar or instrumental polarization as discussed in the next section.

The intrinsic profile $\text{Σ}{Q}_{\varphi }^{\prime }=100\%$ for the DiskI60α−2 models is steep, and therefore the ratio of convolved parameters ΣQ/ΣQ_ϕ changes strongly with the radius of the inner cavity r_in (Fig. 10). According to Table D.3 the dependence is smaller for the DiskI60α0 and DiskI60α−1 models. Nonetheless, ΣQ/ΣQ_ϕ could be a good parameter to derive from observational data the radius r_in of an unresolved central cavity or other disk properties at r < D_PSF, in particular when using constraints about disk inclination and surface brightness profile from the resolved part of the disk at r ≳ D_PSF.

Fig. 14 Intrinsic maps $\text{Σ}{Q}_{\varphi }^{\prime }$ and PSFG convolved maps for Q, U, Qϕ, Uϕ, and P for DiskI60α−2 models with different cavity sizes, rin. The radius rout = 100.8 mas and the intrinsic surface brightness profile $\text{Σ}{Q}_{\varphi }^{\prime }$ are the same for the three models.

Fig. 15 Integrated polarization parameters ΣQϕ, ΣP, ΣQd for the inclined DiskI60α−2 model as a function of the radius of the inner cavity rin and for PSFG and PSFAO convolution. All values have been normalized to the intrinsic value $\text{Σ}{Q}_{\varphi }^{\prime }$ (rin = 0.5 DPSF) (or rin = 12.6 mas).

3.4 The contribution of a polarized central star

The simulations presented up to now assume that the central star is unpolarized ($Q_{\varphi }^{\prime }(r)$ and $Q_s^{\prime }=0$ in Eqs. (3) and (4)), and therefore it does not affect the polarization signal of the circumstellar scattering region. However, often the stars with resolved circumstellar dust scattering regions have also unresolved components, as inferred for example from the thermal emission of hot dust. This dust can produce for the central, unresolved point-like source an intrinsic polarization, if scattering occurs in a non-symmetric structure. The central star can also be polarized by uncorrected contributions from interstellar or instrumental polarization. As the star is typically much brighter than the resolved circumstellar scattering already a small fractional polarization of the order p_s = ΣQ_s/ΣI_s ≈ 0.001 can have a strong impact on the observable polarization, and the following Sect. 4 addresses the question on how to correct for this.

In this subsection we explore the impact of a polarized central source on the imaging polarimetry of a circumstellar scattering region. For this, we have to distinguish between the stellar Q_s, U_s and circumstellar Q_d, U_d polarization components. We consider a central point source with a fractional polarization $U_s^{\prime }=0$ and position angle θ_s = 0.5 · atan2(u_s, q_s) defined in disk coordinates (x, y). The convolved intensity distribution of the central source is I_s(x, y) = ΣI_s · PSF(x, y) and the corresponding Stokes parameter maps are Q_s(x, y) = q_s I_s(x, y) and U_s(x, y) = u_s I_s(x, y). The integrated azimuthal polarization signals of the star are zero ΣQ_ϕ,s = ΣU_ϕ,s = 0, but the corresponding convolved signal maps show the quadrant patterns $$Q_{\varphi ,s}(x,y)=-p_s{I}_s(x,y)\cos \left(2\left({\varphi }_{xy}-{\theta }_s\right)\right)$$(18) $$U_{\varphi ,s}(x,y)=+p_s{I}_s(x,y)\sin \left(2\left({\varphi }_{xy}-{\theta }_s\right)\right).$$(19)

The strengths of these Q_ϕ,s and U_ϕ,s quadrant patterns are independent of θ_s, but their orientation is defined by θ_s. The impact of the polarized intensity of the star ΣP_s = p_s ΣI_s on the disk polarization map in convolved data depends of course on the relative strength between ΣP_s and ΣQ_ϕ,d.

Figure 16 shows as example a disk plus star system with an intrinsic ratio of of ${\theta }_s^{\prime }={67.5}^{\circ }$ for a system with disk intensity $\text{Σ}{P}_s^{\prime }/\text{Σ}{Q}_{\varphi ,d}^{\prime }=0.2$, disk polarization $\text{Σ}{I}_d^{\prime }=0.05\text{Σ}{I}_s^{\prime }$, and a stellar polarization of $\text{Σ}{P}_s^{\prime }=0.001\text{Σ}{I}_s^{\prime }$. The scattering region is simulated with the inclined disk model DiskI60α−2 with r_in = 0.5 D_PSF. The star polarization has an orientation θ_s = 67.5^◦ (q_s = −0.71p_s, u_s = +0.71p_s) and the whole system is convolved with PSF_AO. This reduces the azimuthal polarization Q_ϕ,d of the disk and it results a ratio of ΣP_s/ΣQ_ϕ,d = 0.635 for the convolved model.

The polarization of the star has a strong impact on the polarization maps, despite the fact that ΣP_s is weaker than the circumstellar polarization ΣQ_ϕ,d. This is apparent in the maps in Fig. 16, in particular when compared to the disk maps of Fig. 14 without stellar polarization. The star produces in the center of the Stokes Q(x, y) map a strong negative signal, and in U(x, y) the antisymmetric appearance for the inner disk region is distorted. For the azimuthal polarization strong quadrant patterns for Q_ϕ(x, y) and U_ϕ(x, y) are visible around the position of the central star.

The azimuthally averaged radial profiles for p_sI_s(r) and Q_ϕ,d(r) in Fig. 17 show that the stellar polarization dominates strongly for r < 20 mas, where the stellar intensity I_s(r) is much higher than the disk intensity I_d(r), while the disk polarization is the main polarization component in the range r ≈ 30–120 mas.

There are three important properties for the interpretation of the polarization signal: (i) the integrated Stokes parameters do not depend on the PSF convolution, and therefore the disk and star signal are added together $$\text{Σ}Q=\text{Σ}{P}_s\cos \left(2{\theta }_s\right)+\text{Σ}{Q}_d,$$(20) $$\begin{array}{l}\text{Σ}U=\text{Σ}{P}_s\sin \left(2{\theta }_s\right)+\text{Σ}{U}_d;\\ \end{array}$$(21)

(ii) ΣQ_ϕ and ΣU_ϕ do not depend on the polarization signal of an unresolved central object because the introduced quadrant patterns add no net signal, i.e., ΣQ_ϕ,s = 0 and ΣU_ϕ,s = 0; (iii) the orientation of the central Q_ϕ and U_ϕ quadrant patterns are defined by θ_s of the central source.

If the polarization of the central source is aligned with the resolved disk θ_s = θ_d, then we expect a left right symmetry Q_ϕ(x, y) = Q_ϕ(−x, y) for the overall Stokes Q_ϕ map and an antisymmetry U_ϕ(x, y) = −U_ϕ(−x, y) for Stokes U_ϕ like for the partially resolved disks shown in Fig. 14. Deviations from these properties are indicative for a more complicated polarization structure with a stellar polarization component not aligned with the disk polarization.

Interstellar and instrumental polarization. The impact of polarization offsets from interstellar p_isI(x, y) or instrumental p_instI(x, y) polarization are proportional to the total intensity, while an offset from an intrinsically polarized star p_sI_s(x, y) is proportional to the PSF of the star. However, for many typical cases one can approximate p_s I_s(x, y) ≈ p_s I(x, y) because the difference between stellar and total intensity, which is equivalent to the scattered intensity from the dust I_d(x, y), is very small for circumstellar scattering regions.

This is is supported by the azimuthal profiles in Fig. 17 for the disk plus star model described above (Fig. 16). In this example the disk polarization in the range r ≈ 30–120 mas is much larger than the difference between the polarization offsets of p = 0.001 for the total intensity and the stellar intensity Q_ϕ(r) ≫ (p(I(r) − I_s(r)). Therefore it makes for this example practically no difference for the disk polarization signal Q_ϕ(x, y) whether one uses p I(x, y) or p I_s(x, y) for the polarization offset correction.

This approximation is also valid for faint circumstellar scattering regions, if the fractional scattering polarization is at the same level ΣQ_ϕ ≈ 0.1 ΣI_d as in the example shown above, because the difference for the two cases of polarization offsets scales still with ΣI_d according to p (I(r) − I_s(r)) = p I_d(r). The approximation I ≈ I_s is less good for a bright scattering region I_d, in particular if the polarization offsets is large p ≳ 0.01 and the scattering region weakly polarized ΣQ_ϕ/ΣI_d ≲ 0.1, and combination of these cases.

We treat in this study an intrinsic stellar polarization like an interstellar polarization offset. Interstellar and instrumental polarization offsets can be corrected using p_is I(x, y) or p_inst I(x, y) with the advantage, that the system intensity I(x, y) can be derived from unsaturated observations of the target. In the best case I(x, y) is obtained simultaneously with the polarization signal Q(x, y) and U(x, y) so that atmospheric variations can be taken accurately into account (e.g., Tschudi & Schmid 2021). Using the system intensity I(x, y) also as approximation for the correction of an intrinsic stellar polarization p_s I(x, y) ≈ p_s I_s(x, y) overestimates slightly the offset, but avoids the difficult procedure of splitting I(x, y) into a stellar PSF component I_s(x, y) and a disk component I_d(x, y). Using I_s(x, y) instead of I(x, y) would provide in many cases only a marginal improvement for the offset correction when considering other PSF calibration issues in high contrast imaging polarimetry.

Fig. 16 Polarization maps Q, U, Qϕ, Uϕ for the model DiskI60α−2 (rin = 0.5 DPSF or 12.6 mas) and a polarized central star with $p_s=\text{Σ}{Q}_s/\text{Σ}{I}_s\approx 0.001$ and $\text{Σ}{P}_s^{\prime }=0.2\cdot \text{Σ}{Q}_{\varphi }^{\prime }$ convolved with PSFAO.

Fig. 17 Azimuthally averaged intensity profiles for the disk, Id(r); the star, Is(r); and the total, I(r), for the disk polarization, Qϕ(r), and the intrinsic stellar polarization, psIs(r), or interstellar polarization, pisI(r), introduced by a fractional offset of p = 0.1%. The same disk model, polarization offset psIs(r), and PSFAO convolution as in Fig. 16 is used, while the interstellar offset for pisI(r) is applied to the total intensity.

4 Polarimetric calibration and zp-correction

4.1 The impact of a fractional polarization offset

Already a small fractional polarization offset of the order p ≈ 0.1% from an intrinsic polarization of the central star, from interstellar polarization, or from instrumental polarization can have a strong impact on the observed polarization maps of circumstellar scattering regions (Fig. 16). Unfortunately, the different polarization effects are often not well known, and after a calibration there can remain non-negligible polarization residuals, p_res, with an arbitrary position angle, θ_res, or a residual fractional Stokes values, q_res and u_res: $$Q_{obs}(\alpha ,\delta )\approx Q(\alpha ,\delta )+q_{\text{res}}I(\alpha ,\delta ),$$(22) $$U_{obs}(\alpha ,\delta )\approx U(\alpha ,\delta )+u_{\text{res}}I(\alpha ,\delta ).$$(23)

This can also be expressed by the integrated parameters $$\text{Σ}{Q}_{\text{obs}}=\text{Σ}Q+q_{\text{res}}\text{Σ}I,$$(24) and equivalent for the Stokes U polarization component. The residual offset p_res ΣI can strongly disturb or even dominate the weak polarization signal Q_d, U_d of the circumstellar scattering region. The standard method to improve the situation is a polarimetric zp-correction, which cancels the integrated Stokes signals, $$\begin{array}{l}\text{Σ}{Q}^{\text{z}}=\text{Σ}{Q}_{\text{obs}}-\langle q_{\text{obs}}\rangle \text{Σ}I=0,\\ \end{array}$$(25) $$\text{Σ}{U}^{\text{z}}=\text{Σ}{U}_{\text{obs}}-\langle u_{\text{obs}}\rangle \text{Σ}I=0,$$(26) in a certain apperture, Σ, using the fractional Stokes parameters ⟨q_obs⟩ = ΣQ_obs/ΣI and ⟨u_obs⟩ = ΣU_obs/ΣI (e.g., Quanz et al. 2011; Avenhaus et al. 2014). This is a powerful method to find the circumstellar polarization component in data dominated by residual interstellar, instrumental, or stellar polarization offsets.

The zp-correction is also very useful for data, where a circumstellar polarization is detected, but where the signal might be affected by an unknown offset q_res. Because ⟨q_obs⟩ = ΣQ/ΣI + q_res according to Eq. (24) and using Eq. (25) gives $$\text{Σ}{Q}^{\text{Z}}=\text{Σ}Q-\langle q\rangle \text{Σ}I=0,$$(27) with ⟨q⟩ = ΣQ/ΣI and equivalent for ΣU^z. This means, that the zp-corrected signals of a Stokes map with a fractional polarization offset, e.g. because of calibration uncertainties, is equal to the zp-corrected signal of the map without offsets like for perfectly calibrated data. Therefore, the zp-correction provides data with a well-defined offset correction, which can be re-calibrated later Q^z(α, δ) → Q(α, δ) in a second step once the intrinsic offset value ⟨q⟩ is known from more accurate polarimetry or constraints from scattering models. For example, for a given object the same zp-corrected Stokes signals ΣQ^z, ΣU^z should result for data affected by different instrumental polarization offsets p_res. The Stokes maps Q^z(α, δ), U^z(α, δ) still depend on the observational PSF.

The zp-correction has practically no impact on the integrated azimuthal polarization ΣQ_ϕ, and there is $$\text{Σ}{Q}_{\varphi }^{\text{z}}\approx \text{Σ}{Q}_{\varphi }$$(28) because a fractional Stokes polarization offset introduces for the Q_ϕ(α, δ) and U_ϕ(α, δ) maps positive-negative quadrant patterns with a zero net signal. Therefore, ΣQ_ϕ is a very robust quantity, which is hardly affected by polarization offsets introduced by calibration uncertainties. However, it is important to note that the zp-correction changes the signal distribution in the Stokes maps Q^z(α, δ), and U^z(α, δ) and this can produce a strong bias for the measured azimuthal distribution of the $Q_{\varphi }^{\text{Z}}(\varphi )$ polarization which must be considered for the interpretation of zp-corrected data.

4.2 Cases without zp-correction bias

Polarimetric calibration uncertainties can be corrected with a zp-correction without introducing a bias, if the central source is intrinsically unpolarized ($\text{Σ}{Q}_s^{\prime }\approx 0$, $\text{Σ}{U}_s^{\prime }\approx 0$ or ΣQ_d = ΣQ, ΣU_d = ΣU) and can be used as zero polarization reference. The zp-correction value ⟨q₁⟩ = Σ₁Q/Σ₁I derived for a small stellar aperture Σ₁ then accounts for the polarization offsets, q_res, introduced by the interstellar and instrumental polarization according to $$Q^{z1}(\alpha ,\delta )=Q(\alpha ,\delta )+q_{\text{res}}I(\alpha ,\delta )-\langle q_1\rangle I(\alpha ,\delta )\approx Q(\alpha ,\delta ),$$(29) and equivalent for Stokes U^z1 component. This approximation assumes that the scattering region does not contribute to the polarization signal in Σ₁.

The central star cannot be used as calibration source for coronagraphic observations. However, if the object has an axisymmetric scattering geometry with zero or close to zero net polarization Σ₂Q ≈ 0 and Σ₂U ≈ 0, measured in an annular aperture Σ₂, then this signal can be used as zero polarization reference. Good examples for such axisymmetric systems are circumstellar disks seen pole-on, like for TW Hya, HD 169142 or RX J1604 (Rapson et al. 2015; van Boekel et al. 2017; Poteet et al. 2018; Tschudi & Schmid 2021; Ma et al. 2023).

4.3 Bias introduced by the polarimetric zp-correction

The zp-correction is usually applied to the Q(α, δ) and the U(α, δ) frames in the sky coordinate system. We investigate the zp-correction effects in the (x, y) coordinate system of inclined disks, because the impact of polarization offsets does not depend on the orientation of the selected coordinate system. This simplifies the comparison of Q^z(x, y), U^z(x, y) with the signal of the convolved models Q(x, y), U(x, y) representing perfectly calibrated data. The corrected signal depends on the selected correction region Σ_zp and different cases are considered including the zp-correction of coronagraphic data, or systems with partly unresolved disks.

4.3.1 ZP-correction for an inclined disk ring model

We considered the inclined model RingI60 with r₀ = 100.8 mas in (x, y) coordinates convolved with the extended PSF_AO, applied a zp-correction, and compared the corrected Stokes signal Q^z(x, y) with the corresponding disk signal Q(x, y) without offsets. An overview on the used polarization parameters for zp-corrected models is given in Table A.2.

The zp-correction offset derived from a large aperture Σ_zp = Σ with a radius of r = 1.5″ yields for the corrected signal integrated in the same aperture ΣQ^z = ΣQ − ⟨q⟩ ΣI = 0 according to Eq. (27). This procedure compensates the positive disk polarization ΣQ by a negative offset −⟨q⟩ ΣI. The spatial distribution of the Stokes Q signal differs between the zp-corrected and the initial signals by $$\text{Δ}Q(x,y)=Q^z(x,y)-Q(x,y)=-\langle q\rangle I(x,y).$$(30)

For Stokes U there is U^z(x, y) = U(x, y) for the RingI60 models because ΣU = 0, and therefore ⟨u⟩ = 0. Thus, a Stokes Q zp-correction has no impact on the Stokes U signal and vice versa. Therefore the zp-correction effects described for Stokes Q can be generalized to an offset with Q and U components.

The zp-correction effects are illustrated in Fig. 18 with RingI60 maps of the polarization parameters X = {Q, U, Q_ϕ, U_ϕ, P} for the convolved model X(x, y), the Σ_zp = Σ corrected model X^z(x, y), and the Σ₂ corrected case X^z2(x, y) to be discussed later (Sect. 4.3.3). For Stokes Q a strong negative signal is introduced in the Q^z(x, y) map at the position of the star, and because of the extended PSF_AO, there are also negative contributions further out. Therefore, the corrected polarization Q^z is also lower than Q at the location of the disk ring, while there is no difference between U^z(x, y) and U(x, y). The zp-corrected azimuthal polarizations $Q_{\varphi }^{\text{Z}}(x,y)$ and $U_{\varphi }^{\text{Z}}(x,y)$ show the additional quadrant patterns for a convolved central point source with a negative Stokes Q polarization. For the polarized intensity P the offset −⟨q⟩ ΣI adds predominantly a positive component to the P^z(x, y) map.

The impact of the zp-correction in the Stokes polarization maps depends on the separation from the center, and therefore in addition to the total system integration region Σ_int = Σ, we can also consider three radial integration regions: Σ₁, Σ₂, and Σ₃. These regions are indicated in Fig. 18 in the Q_ϕ(x, y) panel. They represent the star, the disk, and the halo regions, and they cover together the total system integration region Σ₁ + Σ₂ + Σ₃ = Σ.

In the following a detailed description for the system-corrected map Q^z(x, y) in Fig. 18 is given. The negative signal Σ₁Q^z in the stellar aperture Σ₁ with r < 0.027″ introduced by the zp-correction can be estimated using Σ₁Q^z = Σ₁Q − ⟨q⟩ Σ₁I, which is Eq. (27) applied to the integration region Σ₁. The disk contribution is very small Σ₁Q ≈ 0 and the PSF_AO convolved intensity Σ₁I is about 40% of the total intensity ΣI (see Table E.1 for accurate values), and we obtained $${\text{Σ}}_1{Q}^Z\approx -\langle q\rangle {\text{Σ}}_{1}I\approx -0.4\text{Σ}Q.$$(31)

In the halo region Σ₃ (0.2″ < r < 1.5″) there is roughly everywhere the same fractional polarization Q(x, y)/I(x, y) ≈ Σ₃Q/Σ₃I ≈ ⟨q⟩ for the convolved model, because far from the center the PSF_AO smearing of the intrinsic Stokes signal Q′(x, y) of a compact disk is very similar to the smearing of the total intensity I′(x, y) dominated by the central star. Therefore, the halo signal is practically cancelled by the zp-correction and Eq. (27) for Σ₃ can be approximated by $${\text{Σ}}_3{Q}^{\text{Z}}\approx 0.$$(32)

Of importance for the analysis of the disk polarization is the annular aperture Σ₂ with 0.027″ < r < 0.2″ covering the disk, and there is (using Eq. (27) for Σ₂) $${\text{Σ}}_2{Q}^{\text{z}}={\text{Σ}}_{2}Q-\langle q\rangle {\text{Σ}}_{2}I.$$(33)

Because ΣQ^z = 0 for the entire system r < 1.5″ and Σ₃Q^z ≈ 0 for the halo, the polarization for the zp-corrected disk region Σ₂Q^z is equal to the negative signal of the central star, or $${\text{Σ}}_2{Q}^{\text{Z}}\approx -{\text{Σ}}_1{Q}^{\text{Z}}\approx \langle q\rangle {\text{Σ}}_{1}I,$$(34) where the second relation is equal to Eq. (31). The Stokes Q polarization is after zp-correction Σ₂Q^z ≈ +0.44 ΣQ, which is significantly lower than for the non-corrected disk Σ₂Q ≈ +0.71 ΣQ (Table E.1). Thus, the zp-correction introduces for the PSF_AO convolved RingI60 model a strong bias for Stokes Q, changing also the angular distribution of the azimuthal polarization $Q_{\varphi }^{\text{Z}}(\varphi )$.

Fig. 18 Polarization maps for the RingI60 model with r0 = 100.8 mas convolved with PSFAO. The columns shows, from left to right, the convolved maps X(x, y), the maps Xz(x, y) for a system zp-correction, and Xz2(x, y) for disk zp-correction. The circles describe the integration regions Σ1, Σ2, and Σ3 as indicated in the Qϕ(x, y) panel and quadrant regions Σ2Xxxx are identified in the Q(x, y) and U(x, y) panels.

Fig. 19 Relative quadrant polarization parameters Σ2Xxxx (♢) and their azimuthal values Σ2Xxxx|ϕ (horizontal bars) for the disk integration region Σ2 for RingI60 with r0 = 100.8 mas: (a) instrinsic model, (b) convolved model, (c) Σ corrected model, and (d) Σ2 corrected model. The green and grey lines in (b) indicate the change of Σ2Xxxx and Σ2Xxxx|ϕ, respectively, with respect to panel a, and in the panels c and d with respect to panel b. All vallues are normalized to $\text{Σ}{Q}_{\varphi }^{\prime }$.

4.3.2 ZP-correction and quadrant polarization parameters

The intrinsic angular distribution of the azimuthal polarization $Q_{\varphi }^{\prime }(\varphi )$ is changed in polarimetric observations, first by the instrumental convolution as described by the quadrant parameters ΣX_xxx in Sect. 3.2.4, and, if applied, also by the polarimetric zp-correction. A Stokes Q zp-correction adds a signal −⟨q⟩ I(x, y), and because we adopted only axisymmetric PSFs and approximate I(x, y) ≈ I_s(x, y), this introduces for all Stokes quadrants Σ_intQ_xxx practically the same offset. These simple dependencies of the quadrant parameters is very useful to describe the convolution and zp-correction effects of the observed azimuthal polarization signal Q_ϕ(ϕ), which contains important information about the scattering geometry and the dust scattering properties.

The impact of the convolution and zp-correction are illustrated in Fig. 19 (upper panels) for the RingI60 (r₀ = 100.8 mas) model with the quadrant polarization parameters Σ₂X_xxx (diamonds) measured inside the integration region Σ₂ extending from r = 27 mas to 200 mas as defined in Fig. 18 (panels Q(x, y) and U(x, y)). The quadrant values X_xxx represent different azimuthal parts ϕ_xxx of the polarizaton signal Q_ϕ(ϕ), but one needs also to consider whether the sign of X_xxx corresponds to a positive or negative contributions to the azimuthal polarization Q_ϕ. Therefore, we defined azimuthal quadrant values Σ_iX_xxx|_ϕ, that account for the sign of the Stokes polarization with respect to Q_ϕ(ϕ_xxx) as follows:

• For the Stokes Q quadrants, Σ_iQ₀₀₀|_ϕ = −Σ_iQ₀₀₀, Σ_iQ₀₉₀|_ϕ = +Σ_iQ₀₉₀, Σ_iQ₁₈₀|_ϕ = −Σ_iQ₁₈₀, and Σ_iQ₂₇₀|_ϕ = +Σ_iQ₂₇₀.

• For the Stokes U quadrants, Σ_iU₀₄₅|_ϕ = −Σ_iU₀₄₅, Σ_iQ₁₃₅|_ϕ = +Σ_iU₁₃₅, Σ_iU₂₂₅|_ϕ = −Σ_iU₂₂₅, and Σ_iU₃₁₅|_ϕ = +Σ_iU₃₁₅.

The Σ_iX_xxx|_ϕ values are plotted in Fig. 19 as horizontal bars. For the intrinsic disk model they are just equal to the absolute value of the quadrant values $\text{Σ}{X^{\prime }}_{xxx}|\varphi =\text{Σ}\left|{X^{\prime }}_{xxx}\right|$, but this is not always the case for convolved and corrected models. From the change of the Σ_iX_xxx|_ϕ values for the different cases one can estimate the change of the azimuthal distribution of Q_ϕ(ϕ) for angles ϕ_xxx representative for a particular quadrant.

The convolution changes the relative strengths of the quadrants as already described in Sect. 3.2.4 for the total system integration region Σ. In this section, we concentrate on the quadrant signals in the disk integration region Σ₂ of the model RingI60 (r₀ = 100.8 mas) and in Fig. 19 (panel b) the differences between the intrinsic and the convolved quadrant values Σ₂X_xxx (diamonds) are illustrated by the vertical green lines and for the azimuthal quadrant values Σ₂X_xxx|_ϕ (bars) by grey lines. The mutual cancellation and the smearing of signal into the halo region Σ₃ reduces the signal in Σ₂ for the positive quadrants and enhances it for the negative quadrants and this corresponds to a substantial reduction of the absolute quadrant values |Σ₂X_xxx| and of the corresponding azimuthal values Σ₂X_xxx|_ϕ in step with reduction for ΣQ_ϕ (see also Table E.1).

The zp-correction compensates the disk polarization Σ₂Q by adding a negative signal −Σ₂Q with a distribution like I(x, y) and this reduces the polarization in Σ₂ by ∆₂Q = Σ₂Q^z − Σ₂Q = −⟨q⟩ Σ₂I according to Eq. (33). For the RingI60 model with r₀ = 100.8 mas the effect is equal to ${\text{Δ}}_{2}Q/\text{Σ}{Q}_{\varphi }^{\prime }=-11.2\text{\%}$ (Table E.1). All the Stokes Q quadrant values are changed by roughly the same amount, $${\text{Σ}}_2{Q}_{xxx}^{\text{Z}}\approx {\text{Σ}}_2{Q}_{xxx}+\left({\text{Δ}}_{2}Q\right)/4,$$(35) with $\left({\text{Δ}}_{2}Q/4\right)/\text{Σ}{Q}_{\varphi }^{\prime }\approx -2.8\text{\%}$. The Stokes U quadrants are not changed, because Σ₂U = 0 for the used model. Also the integrated azimuthal polarization Σ₂Q_ϕ is practically not changed by the zp-correction offset.

Because of the correspondence between Q_ϕ(ϕ) for different ϕ-wedges and the azimuthal quadrant parameters, the negative Σ₂Q/4 contribution is equivalent to a positive contribution for Σ₂Q₀₀₀|_ϕ and Σ₂Q₁₈₀|_ϕ, and a negative contribution for Σ₂Q₀₉₀|_ϕ and Σ₂Q₂₇₀|_ϕ as illustrated by the grey vertical lines in panel (c) of Fig. 19. This zp-correction enhances the relative signal of the disk front side by more than 50% or from ${\text{Σ}}_2{Q}_{180}{|}_{\varphi }/\text{Σ}{Q}_{\varphi }^{\prime }=5.6\%$ to ${\text{Σ}}_2{Q}_{180}^{{}^Z}{|}_{\varphi }/\text{Σ}{Q}_{\varphi }^{\prime }=8.6\%$ between the “non-corrected” and the zp-corrected disk maps (Table E.1). If one compares the ratio Σ₂Q₁₈₀|_ϕ/Σ₂Q₀₉₀|_ϕ between the signal on the front side with respect to the signal in the left or right quadrants, which are reduced by the zp-correction then the initial ratio is boosted from Σ₂Q₁₈₀|_ϕ/Σ₂Q₀₉₀|_ϕ = 0.32 to ${\text{Σ}}_2{Q}_{180}^{{}^Z}{|}_{\varphi }/{\text{Σ}}_2^Z{Q}_{090}{|}_{\varphi }=0.60$, or almost a factor of two. This example illustrates that the azimuthal distribution of the polarization signal Q_ϕ(ϕ) can be very significantly changed by a polarimetric zp-correction.

4.3.3 ZP-correction for coronagraphic observations

Annular appertures must be used for the zp-correction of coronagraphic observations or data with detector saturation at the position of the bright central star. Thus, one needs to consider other zp-correction regions than Σ_zp = Σ for the whole system. The resulting zp-corrected maps, integrated Stokes parameters, or quadrant parameters are identified with a superscript like Q^z2, Q^z3 or Q^z2+3 for zp-corrections applied to the annuli Σ_zp = Σ₂, Σ₃ or Σ₂₊₃ representing the disk region, the halo region, or the disk plus halo region, respectively. Table E.1 gives an overview on how the polarization parameters for different integration regions Σ_i depend on the used correction region Σ_zp for the model RingI60 with r₀ = 100.8 mas. Important are the resulting polarization values for the annular aperture Σ₂ which includes the disk ring.

The zp-correction applied to a given annulus Σ_zp = Σ_i sets the corrected Stokes value in this region Σ_iQ^zi to zero according to Σ_iQ^zi = Σ_iQ − ⟨q_zi⟩ Σ_iI = 0, which is a generalization of Eq. (27). Selecting different zp-correction regions introduces different offsets ⟨q_zi⟩ factors, and for the RingI60 models, the ⟨q_zi⟩ values behave as follows: $$\langle q_2\rangle >\langle q_{2+3}\rangle >\langle q_3\rangle \text{and}\langle q_3\rangle \approx \langle q\rangle .$$(36)

ZP-correction for the disk. The largest offset ⟨q₂⟩ results for a zp-correction based on the disk region with a strong positive Σ₂Q polarization. This sets Σ₂Q^z2 polarization to zero, or the positive quadrants ($\left({\text{Σ}}_2{Q}_{090}^{Z2}+{\text{Σ}}_2{Q}_{270}^{Z2}\right)$) equal to the negative quadrants $-\left({\text{Σ}}_2{Q}_{000}^{Z2}+{\text{Σ}}_2{Q}_{180}^{Z2}\right)$ and produces apparently a very strong ${\text{Σ}}_2{Q}_{180}^{Z2}{|}_{\varphi }$ signal for the disk front side as shown in panel d of Fig. 19 (Table E.1). This is only a zp-correction effect and should not be interpreted as real azimuthal distribution of the Q_ϕ(ϕ) signal, as would be expected for a disk with highly forward scattering dust.

Moreover, the derived correction offset ⟨q₂⟩Σ₂I = Σ₂Q depends on the quality of the PSF. It is difficult to account for this, particularly for extended disks without sharp structures as in the RingI60 models. Thus, whenever possible one should avoid a zp-correction based on the disk region. This can be problematic for coronagraphic observation of faint disks with large outer radii r₀, where the halo region r > r₀ has not enough signal for a well-defined offset correction.

A zp-correction based on the whole region outside the coronagraphic mask Σ_zp = Σ₂₊₃ reduces the correction offset (Table E.1). In the halo the smeared stellar intensity dilutes more efficiently the smeared disk Stokes signal, and there is ⟨q₂₊₃⟩ < ⟨q₂⟩. Thus, the zp-correction offset is reduced, but it depends still significantly on the PSF profile.

ZP-correction for the halo. For the halo there is ⟨q₃⟩ ≈ ⟨q⟩ because the smearing of a ΣQ signal of a compact disk model like RingI60 with r₀ = 100.8 mas is very similar to the smearing of the intensity, which is dominated by the star. Therefore the halo zp-corrected disk polarization is practically identical to the system zp-corrected signal (Table E.1), or $${\text{Σ}}_2{Q}^{\text{Z}3}\approx {\text{Σ}}_2{Q}^{\text{Z}}.$$(37)

The offset defined by the system zp-correction −⟨q⟩ΣI does not depend on the PSF profile, and this is also the case for the halo corrected models. This also applies, at least approximately, for coronagraphic observations and they should be corrected based on the halo region Σ_zp = Σ₃ outside the disk because of the well-defined and relatively small bias offset.

For coronagraphic observations or data with detector saturation at the position of the star there is the fundamental issue that the PSF profile and the total intensity of the system ΣI cannot be derived from the same data. Flux and PSF calibrations are therefore required for quantitative measurements to allow for a correction of the convolution effects and to relate the measured polarization signal to the intensity of the central star, which is typically a good flux reference.

Coronagraphic observations of a large ring. A RingI60 model with r₀ = 403.2 mas convolved with PSF_AO provides a good example for the effects of a zp-correction for coronagraphic observations of an extended disk taken with a currently available instrument. Figure 20 shows the Stokes Q(x, y) map, and the zp-corrected maps Q^z3(x, y), Q^z2+3(x, y), and Q^z2(x, y) while corresponding numerical values are given in Table E.2. The Σ₁ region with the central star is defined by r < 100 mas and assumed to be covered by a coronagraphic mask. The region Σ₂ with the circumstellar disk is described by the annulus 0.1″ < r < 0.5″ and Σ₃ for the halo by 0.5″ < r < 1.5″. A very narrow color scale was selected to illustrate the weak extended signal.

The intrinsic disk has a positive net signal ΣQ and this produces in the convolved map Q(x, y) an extended halo with a faint, positive Q signal (Fig. 20a). A zp-correction applied to the coronagraphic data Σ_zp = Σ₂₊₃ cancels the net positive polarization in the Σ₂₊₃ region and this turns the positive background in Q(x, y) into a negative background in the corrected map for Q^z2+3 (Fig. 20c). Moreover, the subtraction −⟨q₂₊₃⟩ I(x, y) produces a quite significant ring of negative Q polarization at the position of the PSF speckle ring typical for AO systems. This correction artifact can disturb significantly the analysis despite the fact that the disk signal is very well resolved. The bias effects is smaller, if the halo region is used for the zp-correction Σ_zp = Σ₃ and larger, if only the disk region is used Σ_zp = Σ₂ (see Figs. 20b,d or Table E.2).

Determining the zp-offset ⟨q₃⟩ from the halo is therefore also for coronagraphic observations of extended disks, like the example in Fig. 20, a good approach for a well-defined ⟨q₃⟩ ≈ ⟨q⟩ zero point correction. For this, one should select a region which represents well the average fractional polarization of the system, like the region outside the AO speckle ring. This feature should be avoided, because it overrepresents the stellar contribution.

Speckle ring as stellar polarization signal. The AO speckle ring in high contrast coronagraphic data can be strong and could be used to measure or estimate the polarization of the star ΣQ_s/ΣI_s and ΣU_s/ΣI_s. The requirement is, that stellar light in the speckle ring can be separated well from the circumstellar scattering signal.

This could be the case for a small scattering region located clearly between coronagraphic mask and the stellar speckle ring in particular if the disk polarization has a small net Stokes Q and U signal so that the smeared halo of the disk is almost unpolarized. The polarization of the stellar speckle ring can probably also be measured quite well for a circumstellar regions similar to the coronagraphic model in Fig. 20 because the speckle halo at ∆y ± 450 mas above and below the star is well separated from the disk signal.

The derived fractional polarization from the stellar speckles can then be used as approximation for ΣQ_s/ΣI_s and ΣU_s/ΣI_s and be used for a polarimetric zp-correction. If the star is unpolarized, then one would get well calibrated coronagraphic polarization data Q(x, y) and U(x, y) like for the case described in Sect. 4.2. Applying specific procedures for a given dataset must probably be considered to achieve the best results, but this is beyond the scope of this paper.

Fig. 20 Effects of the zp-correction for Stokes Q for the PSFAO convolved RingI60 model with r0 = 403.2 mas for observations with a coronagraphic mask, indicated by the central hatched area. Panel a shows the convolved signal Q(x, y), b shows the map after a correction based on the halo Qz3, c shows the map after correction based on the disk plus halo Qz2+3, and d shows the map after correction based on the disk region Qz2. The peak Q signal is in all panels between 22 and 23 units with respect to the indicated color scale.

4.4 Partially resolved circumstellar scattering regions

Many circumstellar disks and shells are too small to be fully separated from the star. The PSF_AO convolved DiskI60α−2 model with a small inner cavity r_in = 3.15 mas represents such a situation. The circumstellar scattering in this model produces a substantial amount of unresolved Stokes Q_d signal in the center (Fig. 21, left panels). The zp-correction for the system (middle panels) sets the integrated ΣQ^z signal to zero and therefore produces a strong −Q signal in the center and correspondingly strong central quadrant patterns for the Q_ϕ and U_ϕ map with opposite sign when compared to the convolved model. In both cases the strong central Q signal disturbs substantially the relatively faint signal of the spatially resolved part of the disk, and it is difficult to separate the two polarization components.

4.4.1 ZP-correction for the center

The polarization offset introduced by the central source can be removed with a star peak or center zp-correction where the offsets, ⟨q₀⟩ = Σ₀Q/Σ₀I and ⟨u₀⟩ = Σ₀Q/Σ₀I, are derived from a few pixels centered on the intensity peak. This sets the Stokes Q polarization signal at the center to zero, $$Q^{z0}(0,0)=Q(0,0)-\langle q_0\rangle I(0,0)=0,$$(38) and equivalent for the Stokes U component U^z0(0, 0). In Fig. 21 it removes the strong central Stokes Q-component and quadrant patterns in the Q_ϕ and U_ϕ maps. The center correction leaves an extended polarization signal from circumstellar scattering, called hereafter Q^ex, U^ex, according to $$Q^{z0}(\alpha ,\delta )=Q(\alpha ,\delta )-\langle q_0\rangle I(\alpha ,\delta )=Q^{ex}(\alpha ,\delta )$$(39) $$U^{z0}(\alpha ,\delta )=U(\alpha ,\delta )-\langle u_0\rangle I(\alpha ,\delta )=U^{ex}(\alpha ,\delta )$$(40)

The polarization in the PSF peak ⟨q₀⟩ I(0, 0) and ⟨u₀⟩ I(0, 0) can include contributions from the following components: (i) offsets introduced by (residual) interstellar and instrumental polarization, (q_is + q_inst) I, (u_is + u_inst) I, and (ii) the intrinsic central polarization, Q_c = q_sI_s + q_d,cI_s and U_c = u_sI_s + u_d,cI_s, from the star and the unresolved part of the scattering region.

The offset from the center correction is proportional to I(α, δ) and therefore it corrects accurately the signal introduced by the interstellar and instrumental polarization. In addition also the contribution from the intrinsic stellar polarization Q_c, U_c are corrected, but only approximately because the intensity distribution of the convolved central signal is ∝PSF(α, δ) and differs slightly from I(α, δ) used for the correction offset. This introduces a small overcorrection at the level of (q_s + q_d,c) I^ext(α, δ)/I_s(α, δ) and similar for Stokes U, where I^ext corresponds to the resolved part of the scattering intensity. There is typically (q_s + q_d,c)I^ext(r) ≪ Q_ϕ(r) based on simular arguments as discussed in Sect. 3.4, and therefore we can assume that the center correction accounts for many cases well for all contributions to the polarization of the central source.

The center-corrected maps X^ex(x, y) for the DiskI60α−2 model with small cavity r_in = 0.125 D_PSF in Fig. 21 look very similar to the non-corrected DiskI60α−2 model with a larger inner cavity with r₀ = 0.5 D_PSF plotted in Fig. 14. Thus, the effect of the center zp-correction is very similar to the removal of the polarization signal from the non-resolved inner disk region r < 0.5 D_PSF (Table E.3).

The zp-correction for the center splits the integrated polarization into a central unresolved component and an extended resolved component: $$\text{Σ}Q=\text{Σ}{Q}_0+\text{Σ}{Q}^{\text{ex}}\text{and\hspace{0.17em}Σ}U=\text{Σ}{U}_0+\text{Σ}{U}^{\text{ex}}.$$(41)

For simulations with a given intrinsic model but using different PSFs for the convolution the total Stokes signals, ΣQ and ΣU are conserved, but the splitting between the ΣQ₀ and ΣQ^ex or the ΣU₀ and ΣU^ex components depends on the PSF profile. For a more extended PSF a larger fraction of the polarization signal will contribute to the unresolved central source and a smaller fraction to the extended (resolved) component.

The impact of the center correction for the ΣQ splitting is illustrated in the left panel of Fig. 22 for DiskI60α−2 models with different r_in. For large central cavities r_in > D_PSF (>25 mas) there is practically no difference between ΣQ^ex and ΣQ because of the lack of a central component. For small cavitity r_in < D_PSF the unresolved central signal ΣQ₀ ≈ ⟨q₀⟩ ΣI plotted by the green curve increases strongly for r_in → 0, while the extended polarization ΣQ^ex approaches a limiting value.

Equation (41) for Stokes Q can also be written as ΣQ^ex/ΣI = ΣQ/ΣI − ΣQ₀/ΣI = ⟨q⟩ − ⟨q₀⟩ ΣI₀/ΣI and the same applies for Stokes U. In observational data, one can often assume that the central object dominates the total intensity of the system, and use the rough approximation ΣI₀/ΣI = 1 − ϵ ≈ 1. For such cases one can then derive the approximative Stokes flux Q^ex for the resolved scattering region from the total intensity and the difference between the fractional polarization measured for the whole system and the center according to $$\text{Σ}{Q}^{ex}\approx \left(\langle q_{\text{obs}}\rangle -\langle q_{0,\text{obs}}\rangle \right)\text{Σ}I,$$(42) and equivalent for the ΣU^ex. The two fractional polarization values ⟨q_obs⟩ and ⟨q_0,obs⟩ depend equally or practically equally on the various polarization offsets listed above. Therefore the differential value is barely affected by polarimetric offset, even if their nature is unclear. This could provides for many cases useful Stokes ΣQ^ex and ΣU^ex parameters for the spatially resolved circumstellar scattering polarization.

The resulting values for the Stokes parameters ΣQ^ex and ΣU^ex describe like ΣQ_ϕ properties of the resolved part of the circumstellar scattering and all three parameters depend on the spatial resolution of the observations. Simulations of the convolution can be used to obtain from the ratios of convolved or observed polarization values ΣQ^ex/ΣQ_ϕ and ΣU^ex/ΣQ_ϕ stronger constraints on the intrinsic polarization Q′(x, y), U′(x, y), and $Q^{\prime }(x,y),U^{\prime }(x,y)$.

Fig. 21 Zero-point correction effects for DiskI60α−2 with a small rin = 0.125DPSF (3.15 mas) and strong unresolved scattering polarization. The panel columns give, from left to right, the polarization maps for the PSFAO convolved disk X(x, y), system-corrected maps Xz(x, y), and center-corrected maps for the extended polarization region Xex(x, y).

Fig. 22 Dependence of the polarization ΣX/Qϕ,ref and quadrant parameters ΣXxxx/Qϕ,ref for DiskI60α−2 as a function of rin for the PSFAO convolved, system-corrected Xz and the center-corrected Xex model parameters. All disks have the same outer boundary of rout = 100.8 mas. For Stokes U quadrants, there is $\text{Σ}{U}_{xxx}=\text{Σ}{U}_{xxx}^{\text{z}}=\text{Σ}{U}_{xxx}^{\text{ex}}$.

Fig. 23 Radial profiles for the polarization signal in the Stokes quadrants for the DiskI60α−2 model with rin = 0.125 DPSF (3.15 mas). The panels give Xxxx(r) for the convolved, $X_{xxx}^n(r)$ for the system-corrected, and $X_{xxx}^{\text{ex}}(r)$ for the center-corrected models. For all three cases the Stokes Uxxx(r) profiles are practically the same.

4.4.2 Quadrant parameters for the extended polarization

The maps for the extended polarization Q^ex and U^ex are useful for the characterization of the azimuthal polarization Q_ϕ(ϕ). This is illustrated for the DiskI60α−2 models in the right panels of Fig. 22, which show ΣX_xxx/Q_ϕ,ref for the convolved models, and also for the system-corrected $\text{Σ}{X}_{xxx}^{\text{Z}}/Q_{\varphi ,\text{ref}}$, and center-corrected $\text{Σ}{X}_{xxx}^{\text{ex}}/Q_{\varphi ,\text{ref}}$ models plotted by black dotted, blue dashed and solid red lines respectively (see also Table E.3).