Tomographic Imaging of the Sagittarius Spiral Arm's Magnetic Field Structure

The Galactic global magnetic field is thought to play a vital role in shaping Galactic structures such as spiral arms and giant molecular clouds. However, our knowledge of magnetic field structures in the Galactic plane at different distances is limited, as measurements used to map the magnetic field are the integrated effect along the line of sight. In this study, we present the first ever tomographic imaging of magnetic field structures in a Galactic spiral arm. Using optical stellar polarimetry over a 17′×10′ field of view, we probe the Sagittarius spiral arm. Combining these data with stellar distances from the Gaia mission, we can isolate the contributions of five individual clouds along the line of sight by analyzing the polarimetry data as a function of distance. The observed clouds include a foreground cloud (d < 200 pc) and four clouds in the Sagittarius arm at 1.23, 1.47, 1.63, and 2.23 kpc. The column densities of these clouds range from 0.5 to 2.8 × 1021 cm−2. The magnetic fields associated with each cloud show smooth spatial distributions within their observed regions on scales smaller than 10 pc and display distinct orientations. The position angles projected on the plane of the sky, measured from the Galactic north to the east, for the clouds in increasing order of distance are 135°, 46°, 58°, 150°, and 40°, with uncertainties of a few degrees. Notably, these position angles deviate significantly from the direction parallel to the Galactic plane.


INTRODUCTION
Magnetic fields significantly contribute to the hydrostatic balance in the interstellar medium (Boulares & Cox 1990;Ferrière 2001;Cox 2005;Han 2017).Magnetic pressure and magnetic tension caused by magnetic fields are both non-uniform forces acting perpendicular to the magnetic field lines.Therefore, magnetic fields are believed to introduce anisotropy in the gas motion and consequently have a significant impact on structure formation and evolution in the interstellar medium (ISM), ranging from galaxy formation to the formation of filamentary molecular clouds within a single star-forming region (Heiles & Crutcher 2005;Boulanger et al. 2018).Indeed, magnetic field lines are expected to be influenced by the motion of the interstellar medium, leading to their dragging or bending (e.g., Doi et al. 2021a;Tahani 2022;Tahani et al. 2023).As a result, the interstellar magnetic field structure is expected to be inscribed with a history of deformation of the ISM (Gómez et al. 2018).In other words, by revealing the structure of the interstellar magnetic field, we can elucidate the formation history of the ISM structure (e.g., Tahani et al. 2022a,b;Tahani 2022).Mapping the distribution of magnetic fields from the spatial scale of individual molecular clouds to Galactic scales (10 pc -1 kpc scale) may therefore provide critical information for understanding the role of, for example, Galactic spiral arms in the formation of giant molecular clouds and the subsequent star formation inside them (e.g., Han 2017;Zucker et al. 2018;Stephens et al. 2022).
The structure of magnetic fields can be studied by observing polarized radiation arriving from astronomical objects.Asymmetric dust particles irradiated by incoming radiation fields align their rotation axes parallel to the ambient magnetic field direction (radiative alignment torques; Lazarian & Hoang 2007).This process causes polarized light from both extincted background stars and thermal dust emission from the grains themselves (Stein 1966;Hildebrand 1988).Thus, the planeof-sky (POS) component of the magnetic field (B POS ), associated with dust particles that are primarily in cold neutral ISM (≲ 100 K;McKee 1995), can be observed with both stellar optical/near-infrared polarimetry and sub-mm polarimetry (Lazarian 2007).However, one of the limitations of these observational techniques, particularly polarimetry of optically thin dust emission, is that it can only obtain the average value of the superimposed magnetic field components along the line of sight (LOS).Especially for regions close to the Galactic plane, multiple clouds can be along the LOS and that can complicate the inferred B POS from optically thin dust.
In recent years, Gaia data have provided accurate distances to stars (Gaia Collaboration et al. 2016, 2022;Bailer-Jones et al. 2021) and interstellar extinction values for these stars (Andrae et al. 2022;Babusiaux et al. 2022).By combining these pieces of information with stellar polarimetry data, it becomes possible to reveal the three-dimensional (3D) distribution of the interstellar medium (ISM) and its associated magnetic field up to distances of a few kpc (e.g., Panopoulou et al. 2019;Doi et al. 2021b;Pelgrims et al. 2023).
The Galactic magnetic field is expected to be nearly parallel to the Galactic disk, i.e., B Z ≃ 0, and correlated with the spiral arms (Beck 2013;Beck & Wielebinski 2013;Beck 2015;Haverkorn 2015;Han 2017).Polarimetry of dust emission shows a magnetic field distribution that is generally parallel to the Galactic plane (Novak et al. 2003;Li et al. 2006;Bierman et al. 2011;Bennett et al. 2013;Planck Collaboration et al. 2016).On the other hand, the magnetic field of the neutral ISM traced by stellar polarimetry is not always parallel to the Galactic plane (Heiles 2000;Clemens et al. 2020;Choudhury et al. 2022), and a variation of the position angle along the LOS has been observed (Pavel 2014;Zenko et al. 2020).We need more detailed observational information to reveal the magnetic field structure along the LOS (e.g., Jaffe 2019).
The Sagittarius arm is one of the four major spiral arms of the Galaxy and is observed in −14 • ≲ l ≲ +50 • of the Galactic plane (Vallée 2022).This structure is the closest major spiral arm in the inner Galactic plane and harbors massive star-forming regions such as M8, M16, M17, and M20 (Kuhn et al. 2021).The l ≳ +20 • region is heavily obscured by the Aquila Rift in the foreground (approximately 200 to 500 pc along the LOS), but there are no noticeable foreground clouds in l ≲ +20 • .In addition, the arm is almost entirely in the POS in the smaller Galactic longitude range (l ≲ +20 • ), allowing us to estimate the large-scale magnetic field structure that follows the Galactic arm structure with good approximation from the observed position angle of B POS .Furthermore, target stars are more abundant in the inner Galactic plane than in the outer Galactic plane, making it a good target for obtaining the 3D magnetic field from stellar polarimetry.
To reveal the magnetic field structure along the LOS in the Sagittarius arm by a stellar polarimetric survey, this paper, as a first step, will demonstrate that we can identify multiple ISM clouds and their associated local magnetic field structure along the LOS, including the amplitude of the direction dispersion of the turbulent magnetic fields.
This paper is organized as follows.In Section 2, we describe the selection of the observation area within the Sagittarius arm, the observations, and the data reduction procedure.Section 3 provides a detailed analysis of the distance dependence of the observed magnetic field position angles along the LOS.It discusses the identification of clouds through statistical analysis of the polarimetry data, as well as the magnetic field characteristics specific to each cloud.Section 4 discusses the relationship between the observed distance dependence and the magnetic field traced by sub-mm polarimetry observed by the Planck satellite, which is integrated along the LOS, as well as the amplitude of the turbulent magnetic field in each cloud.In Section 5, we summarize the results.

Target Selection
We selected the Sagittarius arm, the nearest major spiral arm in the inner Galactic plane with abundant observable stars in optical polarimetry, as our first target to create a tomographic image of the magnetic field in a spiral arm.We observed a target field between +10 • < l < +20 • to avoid the Aquila Rift and to have a good sky position from the Higashi-Hiroshima Observatory (see Section 2.2).
To define the target region, in addition to the above constraints, we also imposed the following conditions, referring to the Gaia Data Release 2 (DR2, Gaia Collaboration et al. 2018) catalog, which was the latest Gaia release when the observation was planned: 1.A sufficient number of stars (≳ 100 stars) with Gaia distances are distributed across all distances up to ∼ 3 kpc; 2. The interstellar extinction increases gradually with distance along the LOS, rather than experiencing concentrated increases at specific distances.
These conditions impose a continuous sampling of the magnetic field across the Sagittarius arm along the LOS.Consequently, we selected a 17 ′ × 10 ′ field centered at l = +14 • .15,b = −1 • .47.

Observations
We obtained linear polarimetry in the Cousins R band (R C -band: λ = 0.65 µm) using the Hiroshima Optical and Near-InfraRed camera (HONIR; Akitaya et al. 2014) on the 1.5-m Kanata Telescope, Higashi-Hiroshima Observatory, on August 5, 2021.The optics of the HONIR instrument consists of a rotating halfwave plate, a focal mask of five equally spaced slits with a 50% opening ratio, and a Wollaston prism that splits the incident light into two orthogonally polarized images next to each other on the detector focal plane (see Section 5 of Akitaya et al. 2014).As a result, five pairs of images with orthogonal polarizations are exposed across the entire surface of the detector.To cover the 7 ′ .0× 9 ′ .6 detector's field-of-view (FOV) with multiple exposures, we made 3 × 3 spatial dithers with a 31 ′′ .2-step in the East-West direction and a 20 ′′ .0-step in the North-South direction.
To measure the polarization parameters q ≡ Q/I and u ≡ U/I of each star, we acquired photometry with four position angles of the half-wave plate at 0 • , 45 • , 22 • .5, and 67 • .5 (Kawabata et al. 1999).As a result, we obtained a total of 36 exposures, with each exposure lasting 75 seconds.

Calibration
We calibrated the instrumental polarization by observing the unpolarized standard star G 191-B2B on July 27, 2021.The measured instrumental polarization, an offset vector to the origin in the q-u parameter space, is q inst = 0.01 ± 0.02% and u inst = −0.04 ± 0.02%, which is negligible for our measurements.The stability of the instrumental polarization, measured over a period of 10 months including the observational period, is consistently better than 0.1%, and is thus considered negligible for our measurements.The variation of the instrumental polarization across the detector is better than 0.1% and can also be considered negligible (Akitaya et al. 2014).

Doi et al.
We calibrated the polarization position angle by observing the strongly polarized standard stars BD+64 106, BD+59 389, and HD 204827 (Schmidt et al. 1992) on July 27 and August 30, 2021.The achieved calibration accuracy is better than 0 • .4 and the stability during the observational period is estimated to be better than 0 • .3.
We calibrated the polarization efficiency of the instrument by observing an artificially polarized star through a wire-grid polarizer inserted before the half-wave plate.The measured efficiency is 99.1 ± 0.01 %, by which we scaled the observed polarization fractions.
We converted the measured normalized Stokes parameters, q and u, defined in equatorial coordinates, into the Galactic coordinates, q Gal and u Gal .This transformation allows us to align the polarization measurements with the Galactic coordinate system for further analysis and interpretation.The details of the coordinate conversion process are described in Appendix A.

Gaia Identification and Selection
We referred to the Gaia Data Release 3 (DR3, Gaia Collaboration et al. 2022) catalog and cross-match the observed stars with detections of polarization within a search radius of 1 ′′ .We referred to a Gaia-based catalog by Bailer-Jones et al. (2021) for the distances of each star.Among their distance estimations, we adopted 'geometric' distances, including distance estimates for all our observed stars.
We limited our search by applying the condition that the renormalized unit weight error ('ruwe') ⩽ 1.4 and 'parallax over error' ⩾ 3 in the Gaia DR3 catalog, and distance uncertainty (a 68% confidence interval) ⩽ 20% of the stellar distance.In addition, we selected data with an estimated error δP ⩽ 0.3% for the fractional polarization, which was typically achieved by the stars with R C ⩽ 15.5 mag.Following this procedure, we identified 184 stars within the observed field.In investigating the interstellar extinction in the observed region, we referred to 259 stars meeting the criteria of distance uncertainty ⩽ 20% and A G values available in the DR3 catalog.There were 130 stars found in both datasets.We analyzed all available data for both polarization and extinction, regardless of their availability in the other dataset.We summarize the identified stars in Table 6 in Appendix B.

Spatial and Distance Distribution of Polarimetry Data
Figure 1 shows the spatial distribution of the observed polarization pseudo-vectors (white segments), indicating position angles (PA) and polarization fraction (P).The derivation of these values from the observed q and u values is detailed in Appendices A and B. Of the 184 stars applied in the following analyses, 105 stars with a polarization position angle uncertainty δPA ⩽ 10 • are plotted in the figure.The distribution of B POS traced by stellar polarimetry appears to be a perfect mix of various PAs in space.The histogram of PA shown in Figure 2 shows a bimodal distribution centered around 30 • and 140 • .PA = 90 • , which is the direction parallel to the Galactic plane, corresponds to a minimum of the distribution.Thus the observed B POS is not parallel to but predominantly tilted from the Galactic plane.The PAs and their distribution do not show particular variations or trends with sky coordinates (Figure 1).
Figures 1 and 2 also show Planck's observed magnetic field PA for the same region (orange segments).In the following, we refer to the polarimetry data observed by the Planck satellite at 353 GHz (data release 3 ("PR3"); Planck Collaboration et al. 2020a), as provided by IRSA (Planck Team 2020), with a resolution set to 10 ′ .
Given that the Stokes parameters in the Planck data products are provided in the HEALPix convention rather than the IAU convention, we estimate the polarization position angle (PA) of the data using the following equation: We estimate the PA of the magnetic field by rotating the polarization PA observed by Planck by 90 • .We will use the terms "Planck's observed magnetic field" or simply the "Planck magnetic field" for simplicity.Similar to our stellar polarimetry data, the Planck magnetic field also shows PA deviating from 90 • (140 • .0,71 • .1,121 • .0,and 66 • .0 from north to south in Figure 1).However, the angle offset from 90 • is generally larger for the stellar polarimetry magnetic field orientations.
The distance dependence of the optical polarimetry data is shown in Figure 3. Specifically, we show how PA and P vary as a function of Gaia stellar distances estimated by Bailer-Jones et al. (2021).Note that PA is mostly non-parallel to the Galactic plane, as shown in Figure 2.    A G ) values taken from the Gaia DR3 catalog.We find an apparent increase of about two mag in A G at distances beyond ∼ 1.2 kpc.It further becomes A G ⩾ 2.5 mag beyond ∼ 2 kpc.We can attribute this increase in interstellar extinction at distances of about 1.2 -2 kpc to the dust in the Sagittarius arm.The foreground component of A G < 1 mag (0.53 mag or 1.5 × 10 21 H-atom cm −2 for d < 1.23 kpc, see Table 4) can be attributed to the cloud(s) in the outskirt of the Aquila Rift at d < 200 pc (Section 1), and it is likely related to the Local Bubble shell (Lallement et al. 2019;Pelgrims et al. 2020, also see Figure 5).

Identification of Four Dust Clouds along the LOS
using Breakpoint Analysis Doi et al. (2021b) showed that a breakpoint analysis, a statistical technique that detects the points at which data values make stepwise changes, can effectively recover the distance dependence of stellar polarimetry data.Based on this breakpoint analysis, Doi et al. (2021b) characterized the distribution of dust clouds as a function of distance along the LOS and the 3D structure of the magnetic field associated with those clouds.The details of the breakpoint analysis are described in Appendix C. We apply the breakpoint analysis to our observed q Gal and u Gal , assuming a step change at each breakpoint and constant values between them, as was done by Doi et al. (2021b).We identify four breakpoints, as shown in Table 1 (Polarimetry) and the dashed lines in Figure 3 (top two panels), together with 68% confidence intervals (C.I.) of the estimation.
We also perform the breakpoint analysis for A G values similar to that for the polarimetry data.The results are shown in Table 1 (A G ) and Figure 3 (the bottom panel).We can find reasonable agreement between the two independent evaluations.In particular, the three breakpoint distances on the near side show good consistency.On the other hand, the A G analysis finds an extra breakpoint at larger distances and these are roughly on either side of the polarimetry breakpoint.We note that there are fewer stars with Gaia-estimated A G values than those with polarimetry data in this distance range (d > 1.6 kpc; the number of stars in each distance range is listed in Table 2).Also, we find significant step change in PA at 2.2 kpc (Figure 3).As the focus of this paper is on the magnetic field structure inferred from the polarization data, we utilize the breakpoints detected in the polarization data analysis, which are expected to directly trace changes in the magnetic field structure, in the subsequent analyses.
In the breakpoint analysis, as in Doi et al. (2021b), we assume that the values of q Gal and u Gal are both constant between the neighboring breakpoints.To validate this assumption, we perform a linear fitting on each parameter between breakpoints to test if the slope is statistically consistent with a value of 0. We perform the Student t-test for q Gal and u Gal , and A G as well.Statistical p-values, which are for the null hypothesis that the slope of the distribution is equal to 0, are shown in Table 2.The null hypothesis that the slope of the distribution is equal to 0, i.e., a constant value, cannot be rejected as the p-values are all greater than 5% for the tested cases.
Among the test results, the two distance ranges of d ⩾ 1.63 kpc, where the breakpoint estimate of A G dif- fers from that of the polarimetry data, the A G values show smaller p-values.However, the p-values are larger than 15% and are still consistent with the assumption of constant A G values at each distance range defined by the polarimetry.Therefore, these analysis results can be considered as supporting evidence for the validity of the breakpoint analysis of the polarimetry data.
The constancy of the q Gal and u Gal values within each distance range implies that there is a discrete contribution of polarizing dust sheets at the breakpoints, while there is no significant contribution between breakpoints.In Figure 4, we visually confirm this discrete polarization along the LOS by presenting a cumulative sum plot of the q Gal -u Gal vectors with increasing distance.In this plot, the sum vector defines a straight line while the position angles of the polarization remains constant if the position angles of the vectors are aligned.This is because the vector sum averages out the random component of each vector.On the other hand, if they are not aligned, a change in the polarization PA turns the direction of the path of the cumulative sum plot.As shown in Figure 4, the cumulative sum can be described by the combination of five sections, including four line segments and a clump between 1.23 kpc and 1.47 kpc.The four line segments indicate that the q Gal -u Gal vectors are well-aligned in each distance range.The phase angle of the q Gal -u Gal vector on the q Gal -u Gal plane corresponds to twice the PA, and therefore, it should be noted that vectors pointing in opposite directions (e.g., the green and light green vectors in the figure) differ by 90 • in PA.The clump between 1.23 kpc and 1.47 kpc shows that the length of q Gal -u Gal vectors is 0 on average, which indicates that the vectors are aligned in one orientation in this distance range (due to the complete depolarization by its foreground cloud in this case).In summary, Figure 4 shows that the polarization vectors as a whole are well aligned in a specific direction for each of the five distance ranges, with discrete contributions of thin polarizing dust sheets at the breakpoints.Different colors depict the distance range between the breakpoints, which correspond well to each line segment and clump.
The scattered distribution of dust clouds and their discrete contribution to the polarization (Figure 4) is comparable to the finding for the Perseus and (foreground) Taurus molecular clouds (Doi et al. 2021b) and the thinlayer model developed for high Galactic latitude clouds (Pelgrims et al. 2023).This suggests that the thin-layer model is also applicable to LOSs at low Galactic latitudes.Therefore, in the following, we will assume that the discrete dust sheets/clouds at the four breakpoints, in addition to a foreground component before the first breakpoint, generate polarization in each distance range.That is, 'foreground', '1.23 kpc cloud', '1.47 kpc cloud', '1.63 kpc cloud', and '2.23 kpc cloud'.
Within the distance range where we identify dust clouds along the LOS (d = 1.2 -2.2 kpc from the Sun), the vertical offset from the Galactic plane is |Z| = 32 -57 pc.This vertical distance is comparable to or less than the scale height of the Galactic thin disk component (50 -70 pc, Nakanishi & Sofue 2006;Kalberla et al. 2007;Yao et al. 2017) and well below that of the disk component of Galactic magnetic field models (100 -400 pc, Sun et al. 2008;Jansson & Farrar 2012;Jaffe et al. 2013;Han et al. 2018).Therefore, we are likely observing the Galactic disk component of the magnetic field.
We estimate the 3D distribution of dust clouds in the Galactic disc by applying the breakpoint analysis to A G values, as described in Appendix D. The color scale in Figure 5  The high dust surface density structure observed around the 1.23 kpc, 1.47 kpc, and 1.63 kpc clouds in Figure 5 corresponds to the Sagittarius arm.The surface density around the 2.23 kpc cloud appears to be relatively low.However, this does not necessarily imply the absence of dust clouds or the absence of the Sagittarius and Scutum arm structures, as dust clouds located on the far side within the Sagittarius arm may go undetected, being hidden behind the dust clouds on the near side of the Sagittarius arm or other foreground clouds.
In summary, our observations identify multiple clouds in the Sagittarius arm and detect polarization at each distance range.

Magnetic Field Structure of Each Dust Cloud
We show the distribution of polarization pseudovectors and their PA for each distance range in Figures 6 and 7.As in Figures 1 and 2, we plot only the data q q q q q q q q q q q q q q q q q 0 1 23 P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) q q q q q q q q q q q q q q q q 0 1 23 2 pc @ 1.47 kpc q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 1 23 P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) 2 pc @ 2.23 kpc q q q q q 0 1 23 2 pc @ 1.23 kpc q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 1 23  points with good PA determination (δPA ⩽ 10 • ), which corresponds to 105 objects.The overall distribution of PA in Figures 1 and 2 appeared spatially uncorrelated with a large scatter.However, if we plot the data independently for each distance bin, as shown in Figures 6 and 7, the polarization pseudo-vectors instead show a well-ordered pattern.
We present the mean orientation and angular dispersion of the PA for each distance range in Table 3, cal-  culated using the circular mean and circular standard deviation.The circular mean and circular standard deviation (hereafter σPA ) account for the 180 • ambiguity of the polarization pseudo-vectors.This approach allows for an unbiased estimation of the standard deviation of PA, even if the deviation exceeds 50 • , and is capable of capturing a wider, though not infinite, range of deviations in the PA measurements compared to the usual arithmetic standard deviation, which saturates at π/ √ 12 (rad) = 51 • .96(Doi et al. 2020) 1 .We utilize all 184 objects selected according to the criteria described in Section 2.4, including those with large δPA, for estimating the circular mean and σPA .This is in contrast to Figures 1, 2, 6 and 7, which display data from only 105 objects.To estimate the uncertainty of each parameter, we perform 10,000 Monte Carlo simulations.In each simulation, we add Gaussian random errors independently to the relative Stokes parameters q and u based on their respective uncertainties.From the generated samples, we calculate P and PA and obtain the required quantities for the analysis.We show the median value of the 10,000 estimates as the maximum likelihood value and the 15.9% and 84.1% quantiles as the negative and positive errors in Table 3 and the following estimations in this paper.
The angular dispersion (σ PA ) of the observed polarization pseudo-vectors are found in the 'Observed' column in Table 3. Except for the 1.23 -1.47 kpc distance range, where the polarization pseudo-vectors are almost zero length due to the geometrical depolarization, the angular dispersion for each distance range is significantly smaller than that of the total data, confirming that the polarization pseudo-vectors of each distance bin are better aligned.
To accurately evaluate the magnetic field structure associated with each cloud, it is important to consider that the observed polarization is a result of integrating all contributions along the optical path to the stars.The relative Stokes parameters q Gal and u Gal can be approximated as an addition of the contributions from each element along the LOS, particularly in the case of low polarization levels (say, ≪ 10%; e.g., Patat et al. 2010;Panopoulou et al. 2019;Pelgrims et al. 2023).By subtracting the foreground contribution from the observed polarization in each distance range, we can obtain a more reliable approximation of the intrinsic magnetic field structure associated with each cloud.This allows us to isolate the specific magnetic field characteristics within each cloud, independent of the foreground effects.
The observed q Gal and u Gal data for the n th distance range on the LOS are the sum of the contributions from all distance ranges from the 1 st to the n th distance Doi et al. ranges.Similarly, the observed q Gal and u Gal data for the (n − 1) th distance range are the sum of the contributions from the 1 st to the (n − 1) th distance ranges.Therefore, to obtain the q Gal and u Gal values of the n th distance range, we can subtract the (n − 1) th data from the n th data, i.e., we can differentiate the observed q Gal and u Gal values of each distance range.
Figure 8 shows the q Gal -u Gal data distribution for all distance groups.We also plot the 1-σ contours of the q Gal -u Gal data scatter for each distance range.We can see that the data are discriminated by distance.We estimate the average intrinsic polarization of each interstellar cloud by subtracting the average observed data of the immediately preceding cloud from the average observed data of a particular cloud.The average intrinsic q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q −0.02 −0.01 0.00 0.01 −0.01 0.00 0.01 q Gal u Gal q q q q q d < 1. Distribution of q Gal -u Gal by distance range.Colored contours are the 1-σ contours of the q Gal -u Gal data scatter for each distance range.The black line segments connect the average q Gal -u Gal values of individual distance ranges and indicate the intrinsic polarization of each cloud.polarization vector is represented by each black line segment in Figure 8.For each data point, similarly, we can obtain a better approximation of the q Gal and u Gal values of individual clouds by subtracting the average values of q Gal and u Gal of the immediately preceding cloud, which represents the integration of the contributions of foreground clouds.The subtraction of the foreground contributions is thus equivalent to shifting the coordinate origin of the q Gal -u Gal plane to the average of the q Gal and u Gal values of the immediately preceding cloud.We will discuss the connection between this shift of origin and an anti-correlation between P and σPA in Section 4.1.
Figures 9 and 10 depict the distribution of polarization pseudo-vectors specific to each distance range, obtained by subtracting the mean foreground polarization.Comparing them to the raw observed values plotted in Fig-q q q q q q q q q q q q q q q q q 0 123 P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) q q q q q q q q q q q q q q q q 0 123 P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) 2 pc @ 1.47 kpc q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 123 P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) 2 pc @ 2.23 kpc q q q q q q q q q q 0 123 P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) P (%) 2 pc @ 1.23 kpc q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 123  ures 6 and 7, we observe that the polarization pseudovectors in each distance range exhibit better alignment.This alignment enhancement can be attributed to the subtraction of the mean foreground polarization, which effectively shifts the origin of the q Gal -u Gal plane to the average foreground value, as discussed earlier.Consequently, this adjustment often elongates the q Gal -u Gal vectors (resulting in increased P ) and aligns them more coherently.The improved alignment of these polarization pseudo-vectors indicates a well-ordered magnetic field associated with each dust cloud.The spatial scale of the observed region is approximately 5-10 pc, as indicated by the scales shown in Figure 9.This suggests that the spatial structure of the magnetic field associated with each cloud appears smooth at scales smaller than 5-10 pc, with a scale length of the magnetic field structure larger than 10 pc.
However, it is important to note that in Figure 9, we only subtract the mean foreground polarization, which means that the depicted vectors are corrected for the mean foreground contributions and not their variances.The contribution of the foreground component to the variance of PA and P can only be estimated statistically, and individual polarization pseudo-vectors cannot be corrected individually for this contribution.
Additionally, the observed variance of PA, or (σ PA )2 , does not arise from a linear sum of contributions from each element along the line of sight (LOS), as will be discussed in Section 4.Moreover, the observed values of σPA are positively biased due to observation errors.Instead, we compute the variance of q Gal -u Gal vectors ((σ q,u ) 2 ) specific to individual clouds by removing the foreground cloud's contribution as follows: where the variance of q Gal -u Gal for the nth cloud is denoted as σ n q,u 2 , and the variance of q Gal -u Gal for the immediately preceding cloud is denoted as Subsequently, this derived variance of q Gal -u Gal vectors is employed to determine the variance in PA specific to individual clouds.For a more precise evaluation of the variance of PA, we will provide further discussion in Section 4.2.
Table 3 presents the circular mean and circular standard deviation of the polarization position angles for both the raw observed values (listed in the 'Observed' columns) and the differential values.The differential values of the circular means are considered intrinsic to the magnetic field associated with each cloud, and we label these estimations as 'Intrinsic' values in the table.
On the other hand, the differential values of σPA in Table 3 do not represent the angular dispersions specific to individual clouds, as explained previously.Therefore, in Table 3, we label the differential values of σPA as 'Differential' instead of 'Intrinsic'.
In the subsequent discussions, our primary focus will be on the intrinsic properties of the magnetic field associated with each cloud, unless stated otherwise.

Polarization Fraction and Polarization Efficiency of Each Dust Cloud
Table 4 shows the polarization fraction (P ) for each cloud.To obtain these intrinsic P values, we subtract the average observed q Gal and u Gal values of the immediately preceding cloud from the average observed q Gal and u Gal values of the specific cloud, and subsequently convert them into the polarization fraction (P ).This estimation can be visualized as the length of the black line segment in Figure 8.The average P values of the raw observed data used for evaluating the intrinsic P values are listed in Appendix E.
To estimate the column density of each cloud, we utilize the Gaia DR3-cataloged interstellar extinction (A G ; Andrae et al. 2022).We calculate the average A G within the ranges corresponding to each cloud and subtract the average A G value of the immediately preceding cloud from the average A G value of the specific cloud.
We estimate the column density (N H ) of each cloud based on these A G values, assuming The estimated intrinsic A G of each cloud ranges from 0.17 -0.98 mag, corresponding to relatively low column densities of N H ≲ 2.76 × 10 21 (H-atom cm −2 ).This is because we have selected an observational field of view with relatively low interstellar extinction and with high accuracy measurements from Gaia's optical trigonometry.In other words, the observed magnetic field is not associated with star-forming regions within dense molecular clouds, but rather with the diffuse gas that likely surrounds the molecular gas in isolated clouds.In fact, no corresponding CO molecular cloud is found in our field of view in catalogs (Rice et al. 2016;Miville-Deschênes et al. 2017), indicating that the gas is primarily atomic.HI surveys (e.g., Kalberla et al. 2005;Kalberla & Haud 2015) do not resolve the clouds due to low spatial and spectral resolutions, so the velocity dispersion of each dust cloud is unknown.and 2.23 kpc), weighted by their column densities, is estimated to be 1767.0+1.3 −1.3 pc.This average distance is almost identical to the distances of clouds No. 505 and No. 506.We find more overlapping clouds in the LOS than in the literature, suggesting that we have detected tenuous dust clouds thanks to the distinct change in the magnetic fields' position angles as a function of distance.
We estimate the polarization efficiency (e.g., Whittet 2022) by dividing P by A G , as tabulated in Table 4.The estimated polarization efficiency specific to individual clouds is 0.4% mag −1 for the foreground cloud and 1.0-1.4% mag −1 for the clouds in the Sagittarius arm.In a similar analysis, Doi et al. (2021b) estimated a polarization efficiency of 1.5% mag −1 for the Taurus and Perseus molecular clouds.Taking into account the difference between the two observations (0.7625 µm for Taurus and Perseus; Goodman et al. 1990, and 0.65 µm for this work) and assuming a wavelength dependence of the fractional polarization as P ∝ λ −1.8 (Mathis 1990), it corresponds to approximately 2.0% mag −1 .Therefore, the observed efficiencies in our study are relatively smaller than those estimated for the Taurus and Perseus molecular clouds using the same method.
The pitch angle (the angle relative to the direction of the Galactic rotation) of the Sagittarius arm around the observed region is estimated to be ψ ≃ 17 • (Reid et al. 2019).Assuming the magnetic field follows the spiral arm structure, the magnetic field in the observation field is inclined to the POS by i = 35 • (see Figure 5).In this case, the expected polarization fraction is approximately 0.7 times the maximum value, or 1.4% mag −1 , if the maximum value is ∼ 2.0% mag −1 as observed in Taurus and Perseus clouds, based on the relation P ∝ cos 2 i.That is, the smaller polarization efficiency found in the Sagittarius arm compared to Taurus and Perseus molecular clouds may be partly due to the tilted magnetic field orientation to the POS in the Sagittarius arm, while it is nearly parallel to the POS in Taurus and Perseus (e.g., Jansson & Farrar 2012).The difference in polarization efficiency of clouds in the Sagittarius arm may indicate that the magnetic field structure in the arm has a substantial variation in the in-plane direction of the Galaxy in addition to the direction perpendicular to the Galactic plane.This variation in polarization efficiency may arise from a combination of factors, including differences in the alignment of dust particles and the intricate geometry of the magnetic field.2020b) reported an anticorrelation between the dispersion of polarization angles and the polarization fraction (also see Fissel et al. 2016).They attributed this anti-correlation to variations in the magnetic field structure along the LOS. Figure 11 illustrates the mean observed polarization fraction (P ) and position angle dispersion (σ PA ) estimated for each distance range in the optical polarimetry data.These values, presented as the 'Observed' values of σPA and P in Tables 3 and 4, represent measurements of multiple magnetic field components superimposed along the LOS at their respective distances.In other words, Figure 11 showcases the relationship between P and σPA associated with different numbers of magnetic field layers along the LOS.
The data presented in Figure 11 show an anticorrelation, albeit with a slightly shallower slope compared to the correlation reported by Planck Collaboration et al. (2020b) as σPA × P = const.In the following analysis, we will investigate whether this shallower anticorrelation can be attributed to the same correlation reported in Planck Collaboration et al. (2020b).

Doi et al.
) σ q ,u = 0 .8 % σ q ,u = 0 .8 % σ q ,u = 0 .8 % σ q ,u = 0 .8 % σ q ,u = 0 .8 % σ q ,u = 0 .8 % 0 .6 % 0 .6 % 0 .6 % 0 .6 % 0 .6 % 0 .6 % 0 .4 % 0 .4 % 0 .4 % 0 .4 % 0 .4 % 0 .4 % 0 . 2 % 0 . 2 % 0 . 2 % 0 . 2 % 0 . 2 % 0 . 2 % A geometrical depolarization caused by multiple magnetic field layers along the LOS is equivalent to shifting the coordinate origin of the q-u plane, as discussed in Section 3.3.When the origin of the coordinate system deviates further from the distribution of q and u data, the polarization fraction P increases proportionally.At the same time, the polarization position angle dispersion σPA decreases approximately inversely, particularly when P is sufficiently large.This dependence of σPA on P is the same as that of the estimation error of PA derived from the observed q Gal and u Gal when the standard deviations of q Gal and u Gal (σ q , σ u ) are interpreted as uncertainties in q Gal and u Gal , respectively, rather than standard deviations.For isotropic uncertainty distributions where σ q ≈ σ u ≡ σ q,u , the marginal probability distribution G of PA can be expressed as (Naghizadeh-Khouei & Clarke 1993; Quinn 2012): Here, P 0 and PA 0 represent the average values of P and PA, respectively, and "erf" denotes the Gaussian error function.
We can estimate the angular dispersion σPA from the probability distribution of PA based on the function G (hereafter σG(PA) ), for each value of σ q,u , or more precisely, for each value of P 0 /σ q,u (see Equation 2).Since we cannot solve the function G analytically, we numerically estimate the dependence of σG(PA) on P , shown in Figure 11.The dashed lines in Figure 11 show the σG(PA) dependence on P for several example σ q,u values.We observe a general agreement between the angular dispersion σPA obtained from observations and the theoretical σG(PA) values within the range of σ q,u values ranging from 0.2% to 0.8%.
In the q-u plane, the angular dispersion σG(PA) corresponds to the spread of q-u data, measured in radians from the origin of the q-u plane.This angle can be approximated by the tangent of σ q,u with respect to P .This is why σG(PA) in Equation ( 2) is a function of P 0 /σ q,u .We illustrate the comparison between P 0 /σ q,u and σG(PA) in Figure 12.When normalizing the mean observed polarization fraction (P ) estimated for each distance range in the optical polarimetry data shown in Figure 11 by the values of σ q,u for the same distance range, this normalization removes the dependence of all the observed σPA values and the data points should fall on the same theoretical curve of σG(PA) represented by the solid line in Figure 12.
The theoretical curve of σG(PA) follows the relation σPA × P/σ q,u = 0.5 radian when P is sufficiently large and σPA ≪ 10 • .This is because the phase angle standard deviation of the q-u vectors in radians is approximately equal to the ratio between σ q,u and P if P is sufficiently large compared to σ q,u .Thus, σPA is approximately 0.5 × σ q,u /P radians.On the other hand, when P is small and σPA ≫ 10 • , the slope of the theoretical curve becomes larger than -1 and closer to 0.

Influence of Non-Isotropic σq and σu Distributions
Inset is the distribution of the observed optical polarimetry data, whose symbols are the same as in Figure 11.The shaded area illustrates how the dependence of σPA on P/σq,u deviates from the theoretical curve of σG(PA) when σq,u is not perfectly isotropic and the aspect ratio of its distribution is 1.54.Please refer to the main text for details.
Equation (2) or the solid line in Figure 12 assumes isotropic uncertainty distributions (σ q ≈ σ u ≡ σ q,u ).However, the observed distributions of σ q,u are not perfectly isotropic (Figure 8).When the distribution is non-isotropic, the dependence of σPA on P/σ q,u deviates from the theoretical curve of σG(PA) .Due to the small sample size in our observations (minimum of 24 objects, Table 1), it is not possible to distinguish whether this bias stems from a physical background or from a bias in the observation sampling itself.In the following, we demonstrate that the deviation from an isotropic Gaussian distribution observed in the data has a minimal impact on the estimation of σPA and does not affect the discussions in this paper.
We note, however, that it is important for future observations to increase the sample size and determine the precise shape of the σ q,u distributions, as these distributions contain information about the position angle distribution of turbulent magnetic fields and the spatial variation of dust properties; the σ q,u value measured in the direction perpendicular to each cloud's mean q-u vector (hereafter σ q,u⊥ ) approximates σPA and reflects the PA dispersion of the magnetic field on the POS, and the σ q,u value measured in the direction parallel to each cloud's mean q-u vector (hereafter σ q,u∥ ) is considered to arise from the angular dispersion in the LOS direction of the magnetic field, as well as the fluctuations in polarization efficiency for each region within the polarizing cloud (such as variations in column density and dust alignment efficiency, also see Pelgrims et al. 2023).
If we approximate the distribution of σ q and σ u as an ellipse and estimate the aspect ratio of the major and minor axes, the aspect ratio of the observed σ q,u distribution ranges from a minimum of 1.22 (for the d > 2.23 kpc distance range) to a maximum of 1.54 (for the 1.63-2.23 kpc distance range).In Figure 12, we illustrate how the dependence of σPA on P/σ q,u deviates from the theoretical curve of σG(PA) when σ q,u is not perfectly isotropic, represented by the shaded area.We indicate the deviation corresponding to the maximum aspect ratio of the observed data (1.54).
When the major axis of the distribution aligns with σ q,u∥ , σPA is maximized, corresponding to the upper boundary of the shaded area.This is because the proportion of data closer to the q-u coordinate origin increases.Conversely, when the major axis aligns with σ q,u⊥ , σPA is minimized, corresponding to the lower boundary of the shaded area.This is because the variability of data with distance from the q-u coordinate origin decreases, reducing the proportion of data closer to this origin.In cases where the major axis of the distribution is oblique to both σ q,u∥ and σ q,u⊥ , an intermediate dependence is observed.When σPA ≪ 10 • , the deviation from the theoretical curve due to the non-isotropic σ q,u distribution can be ignored.
In the inset of Figure 12, we normalize the observed P with σ q,u⊥ , because σ q,u⊥ closely approximates σPA .The estimated values of σ q,u⊥ are provided in Table 8 in Appendix E. As shown in the figure, the data align well with the theoretical curve of σG(PA) within the expected range of deviation, which arises from the anisotropy of the observed σ q and σ u values.
The asymmetric distribution around the mean positions of σ q and σ u data generally offsets the value of σPA from the predicted σG(PA) based on Equation (2).In cases where the data follows a non-Gaussian distribution, a non-zero kurtosis does not have an effect, but if non-zero skewness is present, it influences the estimation of σPA .
In Appendix F, we further check the deviation from the theoretical σG(PA) curve caused by the non-isotropic σ q and σ u distribution including the oblique and skewed σ q,u .We present a comparison of estimated values of σPA with and without consideration of the anisotropic Doi et al.
distribution of σ q and σ u .Even when considering the anisotropic distribution, it is emphasized that the difference from the case without consideration falls within the range of estimated uncertainties.
According to the discussion above, we conclude that the observed optical polarimetry data are consistent with the theoretical curve of σG(PA) taking into account the influence of the non-isotropic σ q,u distribution.In the following, we proceed with the discussion using the intrinsic σPA for each cloud, considering the influence of the non-isotropic distribution of σ q,u .

Anti-Correlation Induced by Superposition of Multiple Magnetic Field Layers
We find that for the optical polarimetry data, σG(PA) is significantly larger than 10 • , and the slope of the theoretical curve in Figure 12 is shallower than -1.On the other hand, in the study by Planck Collaboration et al. (2020b), they referred to the data with σPA ≲ 10 • when discussing the anti-correlation, where the theoretical curve exhibits a linear anti-correlation with a slope of -1, which is consistent with their findings.In their study, Planck Collaboration et al. (2020b) demonstrated a general agreement of their observed values with a single anti-correlation: σ PA × P = 31 (deg • %).This suggests that the value of σ q,u from Planck does not vary significantly across different observed sources and is estimated to be σ q,u = 1.08%.However, it is worth noting that there is an order of magnitude variation in the observed σ PA × P values from Planck, which is comparable to the variation we observe in optical polarimetry, where σ q,u ranges from 0.2% to 0.8% (Figure 11, also see Table 7).
According to the above discussion, we can interpret the anti-correlation of optical polarimetry data shown in Figure 11 and the anti-correlation of Planck data discussed by Planck Collaboration et al. (2020b) as a distribution that follows the same function, σG(PA) .In other words, the anti-correlation observed by Planck can be created by the variation of cloud superposition along the LOS that causes the variation of geometrical depolarization due to the superposition of multiple magnetic field components along the LOS.Our observations thus suggest that this multi-component geometrical depolarization is likely the primary cause of the anti-correlation observed along the LOS in the Sagittarius arm, which confirms the discussion by Planck Collaboration et al. (2020b).
In the Planck Collaboration et al. (2020b) model, the intensity ratio between the turbulent magnetic field (B turb , or different components of the magnetic field between layers) and the uniform component (B unif ) is 0.9, and the fluctuation of the turbulent magnetic field within the Planck beam is negligible.As a result, the PA differs significantly between layers in the LOS in their model, but a well-aligned magnetic field is required within a single layer.We note that in our observation, the magnetic field of individual clouds (Figure 9) is well aligned with position angles that vary significantly from one cloud to another and are notably different from those observed by Planck.This alignment remains consistent even at scales less than 10 ′ , which approximately corresponds to the native resolution of Planck's polarization data.This observation is also in line with the discussion by Planck Collaboration et al. (2020b).
The smooth magnetic field structure of each cloud, even at spatial scales below those resolved by Planck observations, along with the significant variation in position angles from one cloud to another, suggests that for diffuse clouds with N H ≲ 3 × 10 21 cm −2 in this study, the discrepancies between Planck and stellar polarization are likely attributed to differences in probed distances rather than differences in beam sizes.Planck captures the superposition of all ISM along the LOS, whereas stellar polarization only probes the ISM located in front of each individual star.

Intrinsic σPA Values of Each Cloud
As described in Section 4.1, σ q,u is a function of σPA and P derived from Equation (2) and can be approximated as σ q,u ≃ (σ PA /0.5 rad) × P .Thus, we see that σ q,u is a function of three physical quantities: magnetic turbulence amplitude (= σPA ), dust alignment efficiency (∝ P/A G ), and the extinction or gas column density (∝ A G ∝ log N H ) as follows: If we estimate σ q,u , P , and A G from observations, we can evaluate these physical quantities, including fluctuations in the position angle of the turbulent magnetic field in the POS.
The intrinsic values of P and A G for individual clouds can be found in Table 4.We can estimate the intrinsic σ q,u values specific to each cloud by subtracting the contributions from foreground polarization and observational uncertainties from the observed values (Equation ( 1)).This estimation assumes that the observed σ q,u is the squared sum of the intrinsic σ q,u and the contributions from foreground and observational uncertainties.We measure σ q,u⊥ , which represents the σ q,u values in the direction perpendicular to each cloud's mean q-u vector.The estimated intrinsic σ q,u⊥ values are presented in Table 5.The σ q,u⊥ values of the raw observed 47.8 +9.0 −6.9 Notes.
The standard deviation of q and u is measured in the direction perpendicular to the mean q-u vector (σ q,u⊥ ) of each cloud, and the amplitude of the turbulent magnetic field (σPA) is estimated from σ q,u⊥ and the polarization fraction (P ) a , by referencing the theoretical function σG(PA) (Equation 2), considering the influence of the non-isotropic distribution of σq,u.
a Intrinsic P value of each cloud is taken from Table 4.
data and their observational uncertainties used for evaluating the intrinsic σ q,u⊥ values are listed in Appendix E. Similar to σ q,u , the observed σPA is determined by the summation of contributions from multiple clouds along the LOS.However, it should be noted that the addition of these contributions is not a simple linear sum of squares, as evident from the deviation of σG(PA) from the relation σPA × P = const.in Figures 11 and 12. Therefore, we estimate the intrinsic σPA values specific to individual clouds by referencing the σ q,u and P values and the theoretical function σG(PA) .
In the evaluation of σG(PA) , we also take into account the non-isotropic distribution of σ q,u , as described in Section 4.1.2.For each cloud, we determine the aspect ratio of the σ q,u distribution's major and minor axes, the rotation angle between the major axis and the average direction of the q-u vectors, and the skewness of the σ q,u distribution both in radial and tangential directions.The obtained results are listed in Table 9 in Appendix F. We numerically calculate the deviation from the theoretical curve given by Equation (2), taking into account the non-isotropic distribution of measured σ q and σ u , and use the obtained theoretical curve to calculate σPA .The values of σPA obtained considering the anisotropy of σ q and σ u vary within the range of estimation errors compared to the case where this consideration is omitted.The comparison of σPA estimation values with and without considering anisotropy is shown in Table 9.Finally, we present the obtained values of σPA , taking into account the anisotropy in the distribution of σ q and σ u , in Table 5.

Ratio
The estimated intrinsic σPA values shown in Table 5 represent the amplitude of the turbulent magnetic field on the POS and can be used as indicators of the turbulent-to-uniform magnetic field intensity ratio B turb /B unif when expressed in radians (e.g., Zweibel 1996;Falceta-Gonçalves et al. 2008;Skalidis et al. 2021).
When B unif is not on the POS (i.e., for angles i from the POS with i > 0 • ), the estimated value of B turb /B unif from σPA can be overestimated depending on the value of i.This is because the uniform magnetic field component projected onto the POS (B unif, POS ) has a dependence of B unif, POS = B unif • cos(i) with respect to i, while the random component B turb , if its distribution is isotropic, does not have a dependence on i.As a result, the observed angular dispersion σPA roughly increases proportionally to [cos(i)] −1 (see, e.g., Falceta-Gonçalves et al. 2008;Poidevin et al. 2013;King et al. 2018;Hensley et al. 2019).
In the case where the large-scale magnetic field has an inclination of i = 35 • corresponding to the pitch angle of the Sagittarius arm with respect to the POS ( §3.4), it should be noted that the estimated B turb /B unif values derived from the observed σPA shown in Table 5 may be overestimated by a factor of 1.22 compared to the true value.
In the following, no correction for i will be applied, and we will proceed with the discussion assuming B turb /B unif ≃ σPA .
The obtained B turb /B unif ratios range from 0.13 to 0.14 (≃ 7 • -8 • ) for the two more distant clouds, indicating that the magnetic fields associated with these clouds remain undisturbed by the random motions of the surrounding gas.In contrast, the closer three clouds exhibit B turb /B unif ratios of approximately 0.37 to 0.45 (≃ 21 • -26 • ), suggesting a high degree of perturbation in their magnetic fields.
The observation covers a broader range for more distant clouds, potentially measuring a wider range of magnetic field orientations.However, the fact that we observe more ordered magnetic field orientations for more distant clouds is contrary to this expectation.This indicates that the observed difference in the degree of magnetic field perturbation is not due to the fact that the observation probes different spatial scales of the magnetic field for clouds at varying distances.Figure 13 presents the dependence of σPA (which corresponds to B turb /B unif ) on N H .The two more distant clouds, where the magnetic fields remain undisturbed, exhibit relatively large column densities of N H = 2.1-2.8 × 10 21 cm −2 .In contrast, the three closer clouds, where the magnetic fields show a high degree of perturbation, have relatively small column densities of N H = 0.5-1.5 × 10 21 cm −2 (Table 4).As a result, the correlation between N H and σPA demonstrates a rough anticorrelation between the two variables.
Assuming equipartition between gas kinetic energy and the magnetic field energy, we can convert B turb /B unif (= σPA ) to the magnetic field intensity.Based on the discussions in Skalidis & Tassis (2021); Skalidis et al. (2021), we estimate the magnetic field strength in the POS using the following equation: where σ v,N T is the gas non-thermal velocity dispersion, σ θ is the position angle dispersion of the polarization due to the turbulent magnetic field (= σPA ), ρ is the gas density, µ is the average particle mass (including hydrogen and helium), m H is the hydrogen atomic mass, and n H is the gas number density.Since the gas is primarily atomic (Section 3), we adopt µ = 1.4 (Pattle et al. 2022).As a result, we obtain the following values for the magnetic field strength in each cloud.
12.1 +2.6 −2.2 , 13.5 +∞ −3.4 , 12.5 +1.5 −1.2 , 21.7 +10.6 −3.3 , 22.8 +8.4 −3.9 × n H 10 2 cm −3 1/2 ∆V FWHM 5 km s −1 µG, for the foreground, 1.23 kpc, 1.47 kpc, 1.63 kpc, and 2.23 kpc clouds, respectively.Here we normalize the results by the reference values of the gas number density n H = 10 2 cm −3 and the full width at half maximum value of gas velocity ∆V FWHM = 5 km s −1 , since no estimated values are available (Section 3.4).The assumption of n H = 10 2 cm −3 is equivalent to assuming the LOS thickness of about 1.5 -8.9 pc for the five clouds in the LOS, whose column densities are 0.47 -2.76×10 21 cm −2 (Table 4).We apply ∆V FWHM = 5 km s −1 as a proxy for the HI gas velocity dispersion around CO cloud cores since Nishimura et al. (2015) observed the Orion A and Orion B molecular clouds in 12 CO(J = 2-1) emission line and estimated that the linewidths (≃ ∆V FWHM ) are generally 2 -5 km s −1 .If we adopt a velocity dispersion of ∆V FWHM = 2 km s −1 , it should be noted that the estimated magnetic field strength will proportionally decrease.On the other hand, if the uniform magnetic field is inclined i = 35 • from the POS, the estimated magnetic field strength will be approximately 10% larger (by a factor of √ 1.22, see Section 4.2.2).Heiles & Troland (1980) measured the global magnetic field strength associated with the HI gas in the Sagittarius arm by the Zeeman splitting of 21 cm HI emission line in the tangential direction (l = 51 • , b = 0 • ) of the Sagittarius arm.They obtained a value of ∼ 16 µG.Our estimated magnetic field strengths for the clouds in the Sagittarius arm are roughly consistent with their estimation, although the magnetic field strengths estimate given here will change depending on the gas velocity dispersion and density.
Using OH Zeeman effect measurements, Crutcher (2012) found that the magnetic field strength in low column density interstellar clouds is typically ∼ 10 µG, which makes these clouds magnetically subcritical.In contrast, clouds with higher column densities are often magnetically supercritical; the gas contracts gravitationally and drags the magnetic field lines inward.The column density threshold at which the cloud transition from subcritical to supercritical is N H ∼ 2 × 10 21 cm −2 (also see Pattle et al. 2022, for a review).
The column density of each cloud we observe corresponds to this threshold value, as indicated in Figure 13.Clouds with higher column densities tend to exhibit more ordered magnetic fields (Figure 13), which can be indicative of stronger magnetic field strengths (∼ 20 µG).The higher degree of orderliness may be the result of gas contraction, indicating that these clouds may be magnetically supercritical and the gravitational forces within the cloud are comparable to or greater than the magnetic pressure and the thermal pressure.The presence of a stronger and ordered magnetic field can have significant implications for the dynamics and evolution of the cloud, influencing processes such as star formation and gas kinematics.Thus, our method of separately estimating the magnetic field turbulence and polarization efficiency of each cloud along the LOS can be a promising means of observing interstellar clouds that are magnetically near-critical.
As shown in Figures 6 and 9, we have successfully detected multiple flips in PA along the LOS in the Sagittarius arm.These flips may be due to the local deformation of the magnetic field around individual clouds.However, each cloud's magnetic field is smooth within the observed region (Figures 6, 9).The physical size of the observed region is 3.8 pc × 6.1 pc for the 1.23 kpc cloud and 6.8 pc × 11 pc for the 2.23 kpc cloud.Thus, the results indicate the possibility of a global magnetic field flipping on a scale sufficiently larger than the observed area and possibly even larger than the size of individual clouds.This possibility should be investigated by observing several neighboring regions.

SUMMARY
We completed an R C -band polarimetric survey around l = 14 • .15,b = −1 • .47 in a direction that threads the Sagittarius spiral arm using HONIR, an imaging polarimeter on the Kanata Telescope, Hiroshima University.We selected a region where a large number of Gaia stars were measured with sufficient precision.We found that the polarization position angles (= position angles of the magnetic field projected onto the POS) in the LOS vary significantly at each of four locations at distances of 1.23 kpc, 1.47 kpc, 1.63 kpc, and 2.23 kpc.Based on these data, we found four isolated clouds at these locations in the LOS and a foreground cloud at d < 200 pc, which is possibly an outskirt of the Aquila Rift, for a total of five clouds, producing the observed polarization.
The column density of each cloud is ≲ 2.8×10 21 cm −2 .No corresponding CO molecular clouds are found in the literature, suggesting that these clouds are primarily atomic and may be the surroundings of denser molecular clouds.Thanks to the distinct change in the magnetic fields'position angles, we have detected tenuous dust clouds along the LOS with high sensitivity.
We successfully extracted the magnetic field characteristics of each cloud by differencing the polarimetry data and the Gaia stellar extinction data by distance.Individual clouds' estimated magnetic field structure is smooth within the 17 ′ × 10 ′ observed region.The scale length of the structure is thus expected to be ≳ 10 ′ , corresponding to ≳ 10 pc in physical scales.
Individual clouds' magnetic field position angles are 134.5  -60 • ), in contrast to the current understanding that the large-scale magnetic field in the Galactic disk is parallel to the Galactic plane.
The polarization efficiency of each dust cloud is P/A G = 0.4% mag −1 for the foreground cloud and 1.0 -1.4% mag −1 for each dust cloud in the Sagittarius arm.These values are comparable to or lower than those of Taurus and Perseus (1.5% mag −1 ; Doi et al. 2021b).Besides the angular offset from the Galactic plane mentioned above, the magnetic field of individual clouds may be inclined to the POS at different angles, causing slightly lower polarization efficiency of clouds and their variation.
The turbulent amplitude of the magnetic field associated with each cloud (σ PA ), which can be used as indicators of the turbulent-to-uniform magnetic field intensity ratio B turb /B unif , is weakly correlated with the column density of each dust cloud, ranging from σPA = 21 • .0-25 • .9(B turb /B unif = 0.37 -0.45) for three clouds with relatively low column density N H = 0.47 -1.48 × 10 21 cm −2 and σPA = 7 • .3-8 • .1 (B turb /B unif = 0.13 -0.14) for two clouds with relatively high column density N H = 2.09 -2.76 × 10 21 cm −2 .Assuming the general values n H = 10 2 cm −3 and ∆V FWHM = 5 km s −1 as the gas density and gas velocity dispersion, we estimated the magnetic field strength 12 -13 µG for the low column density clouds and ∼ 20 µG for the high column density ones.
Our observations show an anti-correlation between polarization angular dispersion σ PA and polarization fraction P that was found by Planck observations (Planck Collaboration et al. 2020b).We showed that this anticorrelation can be obtained from the shift of data points in the q-u plane while keeping σ q,u in each region constant due to geometrical depolarization.The magnetic field structure of each region we observed is smooth, even at scales below Planck's spatial resolution.Therefore, the difference between our observations and Planck's is likely due to the difference in distances probed instead of differences in beam sizes.
As demonstrated above, by combining optical polarimetry data with Gaia catalog distances and interstellar extinction, we could estimate each cloud's magnetic turbulence and polarization efficiency along the LOS separately.We argue that this method is a functional tool for investigating the turbulent nature of the magnetic field at the periphery of interstellar molecular clouds in 3D.
The authors are grateful to an anonymous referee, who provided thorough and thoughtful suggestions for improving various aspects of the paper.This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.This research has been supported by JSPS KAKENHI grants 25247016, 18H01250, 18H03720, 20K03276, and 20K04013.M.  Facilities: Kanata:1.5m(HONIR), Gaia, Planck Software: strucchange (Zeileis et al. 2002(Zeileis et al. , 2003)), astropy (Astropy Collaboration et al. 2013, 2018, 2022), Source Extractor (Bertin & Arnouts 1996) APPENDIX A. COORDINATE CONVERSION OF THE NORMALIZED STOKES PARAMETERS Q AND U The measured normalized Stokes parameters, q and u, are defined in equatorial coordinates.We convert these values into Galactic coordinates, q Gal and u Gal , to align the polarization measurements with the Galactic coordinate system for the discussion in this paper.

D. GALACTIC SURFACE DENSITY OF DUST CLOUDS
We estimate the surface density of dust clouds in the Galaxy by calculating their all-sky 3D distribution.Breakpoint analyses are conducted on the celestial plane at intervals of 6 ′ .87,using stars with 'geometric' distances (Bailer-Jones et al. 2021, with uncertainties ⩽ 20%) and estimations of A G (DR3 catalog).The spatial resolution of our estimation is set at 15 ′ .Along each line of sight, the distance to the dust clouds is estimated.Median A G values between breakpoints are computed and differentiated to assess the increase in A G value at each breakpoint position, representing the column density of dust clouds at those positions.To convert A G to hydrogen column density, we assume N H = A G • 2.21 × 10 21 /0.789 (Güver & Özel 2009;Wang & Chen 2019).Finally, we estimate the surface density of dust clouds by integrating the column density within a range of ±100 pc from the Galactic plane, using 20 pc × 20 pc bins on the Galactic plane.Notes.
The standard deviation of q and u measured in the direction perpendicular to the mean q-u vector (σ q,u⊥ ) within each distance range, along with their associated observational uncertainties, is presented.

E. AVERAGE OBSERVED VALUES FOR EACH DISTANCE RANGE
To estimate the intrinsic physical parameters of the magnetic field associated with each dust cloud, we follow a two-step process.First, we calculate the average observed values for each distance range that corresponds to each dust cloud along the LOS.These average values represent the measured properties of the cloud.Then, to obtain the intrinsic values, we subtract the average value of the immediately preceding cloud from the average value of the specific cloud.This difference provides an estimation of the intrinsic physical parameters specific to each dust cloud, which helps us understand the properties of the magnetic field associated with each cloud in a more accurate and meaningful way.
Table 7 displays the average polarization fraction (P ), average A G , and the average N H values.The N H values are derived from A G using the conversion N H = A G • 2.21 × 10 21 /0.789 (Güver & Özel 2009;Wang & Chen 2019).These average observed values serve as the basis for estimating the intrinsic values, which are presented in Table 4. Table 8 displays the standard deviation of q and u measured in the direction perpendicular to the mean q-u vector (σ q,u⊥ ) and their observational uncertainties within each distance range.These values serve as the basis for estimating the intrinsic σ q,u⊥ values, which are presented in Table 5.
F. ANISOTROPY OBSERVED IN σ Q AND σ U DISTRIBUTIONS As described in Section 4.1.2,the dependence of σPA on P/σ q,u deviates from the theoretical value σG(PA) shown in Equation (2), when σ q and σ u are not isotropically distributed with respect to their phase angles in the q-u plane.In addition to the phase angle dependence of the observed σ q and σ u values, the non-Gaussian distribution of observed q and u values on the q-u plane (a distribution with non-zero skewness) produces deviations from Equation (2).By Figure 1.Observed stellar polarization pseudo-vectors (white line segments).The data of 105 stars with errors on PA below δPA ⩽ 10 • are shown, out of 184 stars with significant polarization detection and accurate distance estimation.A reference scale of P is shown in the lower left corner of the figure.The background is from the Second Generation Digitized Sky Survey red image (McLean et al. 2000).The orange line segments are magnetic field position angles obtained from Planck data at 353 GHz (Planck Collaboration et al. 2020a, resolution set to 10 ′ ).The Planck's line segments only show the orientation of the magnetic field, estimated by rotating the polarization PA by 90 • , and their length is not related to the Planck-measured polarization degree.

Figure 2 .
Figure 2. Histogram of PA.The bin width is set as 20 • .We show 105 stars with δPA ⩽ 10 • , the same as in Figure1.The two black arrows indicate the PA of Planck's magnetic field inside the observed region (see Figure1, spatial resolution is set to 10 ′ ).The vertical dotted line represents the position angle for the Galactic Plane (PA = 90 • ).

Figure 3 .
Figure 3. Distance dependence of polarimetry data (PA and P ; our observed 184 values), and AG (the Gaia DR3 cataloged 259 values).An observed PA of 90 • indicates that the magnetic field is parallel to the Galactic plane.The vertical dashed lines indicate the breakpoints of polarimetry data and AG estimated by the breakpoint analysis.Shaded areas correspond to 68% confidence intervals (C.I.) of the estimation.See text and Doi et al. (2021b) for the details of the breakpoint analysis.

Figure 3
Figure 3 also shows interstellar extinction (A G ) values taken from the Gaia DR3 catalog.We find an apparent increase of about two mag in A G at distances beyond ∼ 1.2 kpc.It further becomes A G ⩾ 2.5 mag beyond ∼ 2 kpc.We can attribute this increase in interstellar extinction at distances of about 1.2 -2 kpc to the dust in the Sagittarius arm.The foreground component of A G < 1 mag (0.53 mag or 1.5 × 10 21 H-atom cm −2 for d < 1.23 kpc, see Table4) can be attributed to the cloud(s) in the outskirt of the Aquila Rift at d < 200 pc (Section 1), and it is likely related to the Local Bubble shell(Lallement et al. 2019;Pelgrims et al. 2020, also see Figure5).
Figure 5.The red dashed line represents the sightline of the observation, while the positions of the four identified dust clouds (indicated by their distances) are shown on the Galactic plane.The coordinates are heliocentric Galactic Cartesian coordinates, with the Sun located at the coordinate origin.The X-axis points towards the Galactic center, the Y-axis points in the direction of Galactic rotation (Galactic plane at l = 90 • ), and the Z-axis points towards the Galactic North pole (not depicted in the figure).The color scale represents the surface density of the dust cloud within Z = ±100 pc (see Appendix D for the surface density estimation).The regions of high dust surface density surrounding the 1.23 kpc, 1.47 kpc, and 1.63 kpc clouds correspond to the Sagittarius spiral arm.

Figure 6 .
Figure 6.Spatial distribution of the observed position angles (PA) for each distance range.Here we plot 105 data with uncertainties δPA ⩽ 10 • .
The raw observed values for each distance range are shown in the 'Observed' column.The mean intrinsic PA for each cloud, calculated by subtracting the foreground contributions of all components in front of the respective cloud, is presented in the 'Intrinsic' column.The circular standard deviation (SD) values subtracting the foreground contribution are shown in the 'Differential' column.Note that they are biased by the variation caused by the observational error and not showing the intrinsic magnetic field angular variation of each cloud.See text andDoi et al. (2020) for the definitions of the circular mean and the circular standard deviation.a The 'Intrinsic' and 'Differential' values of the foreground cloud and the 'All' values are the same as those of the observed values because there is no foreground component to be subtracted.

Figure 7 .
Figure 7. Histogram of the polarization angles (PA) for each distance range.We plot data with uncertainties δPA ⩽ 10 • .The bin width of the histograms is 20 • .The vertical dotted line indicates PA = 90 • that is the position angle parallel to the Galactic plane.
Figure 8.Distribution of q Gal -u Gal by distance range.Colored contours are the 1-σ contours of the q Gal -u Gal data scatter for each distance range.The black line segments connect the average q Gal -u Gal values of individual distance ranges and indicate the intrinsic polarization of each cloud.

Figure 9 .
Figure 9. Same as shown in Figure 6, but with the polarization pseudo-vectors of each cloud adjusted by subtracting the average foreground contribution and correcting for their average PA and P values.It is important to note that the individual pseudo-vectors in the figure are not corrected for the contribution of the foreground component to the variance of PA and P ; this correction can only be made statistically.Therefore, the vectors depicted in the figure are corrected only for their average values.
Chen et al. (2020) identified dust clouds by referring to Gaia DR2 interstellar extinction data.Their clouds No. 505 at l = 14 • .821,b = −1 • .107and No. 506 at l = 13 • .370,b = −0 • .212may correspond to our observed cloud(s) because of their spatial proximity to our FOV (l = 14 • .15,b = −1 • .47).The angular distances between the outer edges of their clouds and our FOV are ∼ 0.5 • (for the spatial extent of the clouds, see their Figures 505 and 506 available online 2 ).The distance estimate of cloud No. 505 is 1815.2± 42.8 pc and cloud No. 506 is 1793.0 ± 42.3 pc.The average distance of our detected four clouds (1.23 kpc, 1.47 kpc, 1.63 kpc, Anti-Correlation between Position Angle Dispersion and Polarization FractionPlanck Collaboration et al. ( Figure11.Relationship between the observed values of the polarization fraction (P ) and the polarization angle dispersion (σPA) in our samples.The dashed lines represent the relationship between the position angle dispersion and the P value, based on the marginal distribution of PA described by Equation (2).These lines are plotted for σq,u values of 0.8%, 0.6%, 0.4%, and 0.2%, representing different levels of dispersion.The diagonal dotted line shows the approximation of the theoretical P -dependence, which corresponds to the σPA ∝ P −1 correlation pointed out by PlanckCollaboration et al. (2020b), in the case of σq,u = 0.4% (see also Figure12).

Figure 13 .
Figure13.Correlation between the gas column density NH and the turbulent magnetic field amplitude σPA, which serves as an indicator of the turbulent-to-uniform magnetic field intensity ratio B turb /B unif .The error bars of σPA indicate the 15.9% and 84.1% quantiles obtained from the 10,000 Monte Carlo simulations described in Section 3.3.
α, δ) represents the equatorial coordinate (J2000) position of the source, and (α NGP , δ NGP ) represents the equatorial coordinate (J2000) position of the north Galactic pole(Cox 2000): Gaia source ID in DR 3. b Gaia DR 3 coordinate positions.The reference epoch is 2016.c Distance to the stars (r med geo) and their 16th and 84th percentiles (r lo geo and r hi geo) estimated by Bailer-Jones et al. (2021).dThe debiased values of P as determined by Equation (B5) are shown.Due to the debiasing process, it is possible for the P 2 value in Equation (B5) to be negative.In the online data files, P values for data points with negative P 2 values are indicated as '< 0'. e Extinction in G band (AG) values (ag gspphot) and their 16th and 84th percentiles (ag gspphot lower and ag gspphot upper) taken from the Gaia DR3 catalog(Gaia Collaboration et al. 2022).

Table 1 .
Breakpoints estimated in q Gal , u Gal and AG.

Table 2 .
Fitted parameters within each distance range.Statistical deviation of the maximum likelihood slope value from 0. b Statistical p-value for the null hypothesis that the slope of the distribution is equal to 0. a

Table 3 .
Angular mean and standard deviation of PA within each distance range.
789 (mag; Wang & Chen 2019) and N H /A V = 2.21 × 10 21 (H-atoms cm −2 mag −1 ; Güver & Özel 2009).The estimated A G and N H values are presented in Table 4.The average A G and N H values of the raw observed data used for evaluating these intrinsic A G and N H values can be found in Appendix E.
a NH = b Average of all the observed data.

Table 5 .
The turbulent magnetic field's angular amplitude.
Tahani is supported by the Banting Fellowship (Natural Sciences and Engineering Research Council Canada) hosted at Stanford University and the Kavli Institute for Particle Astrophysics and Cosmology (KIPAC) Fellowship.CVR thanks the Brazilian Conselho Nacional de Desenvolvimento Científico e Tecnológico -CNPq (Proc: 310930/2021-9).AMM's work and optical/NIR polarimetry at IAG have been supported over the years by several grants from the São Paulo state funding agency FAPESP, especially 01/12589-1 and 10/19694-4.AMM has also been partially supported by the Brazilian agency CNPq (grant 310506/2015-8).AMM graduate students have received grants over the years from the Brazilian agency CAPES.

Table 6 .
Data Identification Gaia ID a R.A. b Dec. b l b r med geo c r lo geo c r hi geo c Gal δPA Gal ag gspphot e ag gspphot lower e ag gspphot upper e
a NH =

Table 8 .
Standard deviation and uncertainties in observed q and u.