Scattering asymmetry in in-situ Mie polarimetry diagnostic of nanodust clouds

Imaging Mie polarimetry is key to determining spatially resolved information about the properties, i.e. refractive index and grain size, of particle clouds, such as during the growth process in reactive particle producing plasmas. Asymmetries in the intensity maps of the different Stokes parameters resulting from the anisotropic scattering of polarized laser light complicate the analysis and require the use of radiative transfer (RT) simulations. We use RT simulations to investigate the asymmetric scattering behavior based on a model of a typical reactive argon-acetylene plasma. We address possible misinterpretations and explore the potential for analyzing particle properties. We find that the asymmetric pattern of the intensity distributions is highly dependent on the refractive index, providing the potential to determine the refractive index and grain size at any time during the growth process.


Introduction
For the determination of the properties of dust particles grown in reactive plasmas, the analysis of the polarization state of scattered light has proven to be a powerful tool. The size of the dust particles is the key parameter to determine other measures, like dust number density and dust charge (Greiner et al 2018). In typical setups of reactive argon-silane (Boufendi et al 1992) and argon-acetylene plasmas (Denysenko et al 2006), particle clouds of nearly monodisperse spherical particles are created and the particles grow from nanometers up to micrometers (Berndt et al 2009). Figure 1 shows such a cloud, created in a capacitively coupled, radio frequency driven low pressure argon-acetylene discharge (for details see Petersen et al 2022). When irradiating such a particle cloud with laser light, one can measure the polarization state of the scattered light with a polarimeter, typically a rotating compensator polarimeter (Hauge and Dill 1975).
Provided that the particle cloud has a low dust number density resulting in a low optical depth, the scattered light can be analyzed easily based on Mie theory under the assumption of single scattering. For this purpose, Groth et al (2015) presented the CRAS-Mie method. By fitting the polarization state of the scattered light, i.e. the relation of the polarimetric angles Ψ and ∆, measured during the growth process, this method provides the refractive index and the time-dependent particle radius a(t).
The assumption of single scattering limits the applicability of such methods to particle clouds with low optical depths and thus often to particle sizes ≲200 nm (Groth et al 2015). When multiple scattering becomes relevant, the polarization state of the scattered light can no longer be predicted semi-analytically (van de Hulst 1957). Radiative transfer (RT) simulations which numerically solve the RT equation are needed for the analysis instead (Kirchschlager et al 2017, Kobus et al 2022. Especially in spatially resolved measurements, effects due to multiple scattering can complicate the analysis. This is illustrated in figure 2. The intensity distributions of the different Stokes parameters measured for light scattered within the particle cloud show a vertical asymmetry. As we will show later, this Figure 1. Nanodust cloud created in the electrode gap of two circular (60 mm diameter) electrodes in an argon-acetylene plasma. The 2D cross-section of the cylindrical symmetric dust cloud is visualized by a vertically expanded laser sheath going from right to left through the center of the discharge chamber. Figure 2. Stokes parameters of a nanodusty plasma at high optical depth taken with an imaging polarimeter. In contrast to figure 1 a laser beam is sent through the chamber, illuminating only a small area of the nanodust cloud. The main fraction of the intensity scattered toward the imaging polarimeter is due to single scattering of the laser beam. Multiple scattering events are rare, and can only be found above and below the laser beam in the normalized values q, u, v. (a) Stokes vector (I, q, u, v) T as reconstructed from the polarimeter data. The laser beam is asymmetrically scattered to the upper and lower region of the cloud, which is clearly seen in profiles averaged along the x direction of q, and u (b). asymmetry occurs even in the case of a homogeneous density distribution of the dust cloud and is due to the anisotropic scattering of the initially polarized laser light.
Classical setups like Imaging Rotating Compensator Polarimeters (I-RCP) were used successfully (Groth et al 2019) to investigate the layered, inhomogeneous growth of particles. Thanks to the availability of modern, micro polarizer-equipped cameras such as the Sony Polarsens (Sony 2018) measurements with high spatial and temporal resolution are now easily feasible. Using RT simulations, the effects of multiple scattering in imaging Mie polarimetry can be predicted, helping to optimize analysis strategies.
In this study, we want to raise awareness of the asymmetric scattering behavior of polarized light. The spatial asymmetry in the polarized intensity maps can complicate the analysis even for single-point measurements 5 . As we will see below, at high optical depths, where multiple scattering contributes significantly, errors can occur in the determination of particle properties if this asymmetry is ignored and the polarimetric measurement is not performed accurately. However, we also explore the potential of imaging polarimetry and the analysis of the asymmetric pattern for particle property determination. For this purpose, we use a simple model of a reactive argon-acetylene plasma (described in section 2) and perform RT simulations (details in section 3) to simulate the polarization state of scattered laser light influenced by multiple scattering. The resulting asymmetric (polarized) intensity maps of the scattered light are presented in section 4. We explain the origin of the asymmetry in section 5. In section 6, we discuss implications for the analysis of particle properties. We draw attention to possible misinterpretations when ignoring the effects caused by multiple scattering, but we also show that the analysis of the asymmetry offers great potential for the determination of the refractive index of the particles.

Model set-up
The model set-up is based on a typical argon-acetylene plasma, for which Mie polarimetry is used to determine the particle properties of the nanodust cloud in-situ (Groth et al 2015). It has already been used in Kobus et al (2022), where a detailed description and justification of the model is given. Below, we briefly summarize the model parameters and give an illustration in figure 3.
The particle cloud has a cylindrical shape with a radius of r = 3 cm and a height of h = 3 cm and contains a constant particle number density of n = 10 13 m −3 . We simulate monodisperse particle clouds. First, we focus on the simulation of a particle cloud with an exemplary particle radius a = 200 nm. In section 6, we consider the full growth process by simulating a series of monodisperse particle clouds with particle radii equidistantly distributed between 20 nm and 300 nm with a size step of δa = 5 nm. For the refractive index of the particles, we assume N = 1.8 + 0.05i.
A red (λ = 662.6 nm) laser beam traverses the cloud in x direction with an offset of δy = 2.8 cm from the center. The laser light is linearly polarized by 45 • with respect to the x-y plane, corresponding to the Stokes vector S = (1, 0, 1, 0) T . Perpendicular to the propagation direction of the laser and perpendicular to the symmetry axis of the cylinder, a polarimeter to detect the scattered light is placed in the positive y direction.

RT simulations
The Mie scattering of the laser light in the particle cloud is simulated using the 3D Monte-Carlo RT code POLARIS (Reissl et al 2016). Within these simulations, photon packages, characterized by their wavelength λ, Stokes vector S = (I, Q, U, V) T , and travelling direction, are followed as they propagate through the model space. The path of the photon packages is determined based on probability density distributions. These are functions depending on the optical depth, scattering angles and particle properties, such as refractive index and size. Scattering events modify the polarization state of the photon packages according to the Mueller matrix M(θ) for the scattering angle θ (angle between the propagation direction before and after scattering, see figure 4): (1) Prior to this, the Stokes vector in the reference frame of the photon package must be rotated into the reference frame given by the scattering angle φ. The scattering angle φ is the angle between the negative z p -axis of the reference frame of the photon package and the projection of the new travel direction after scattering on the y p -z p plane (see figure 4). The corresponding rotation matrix is: We determine the optical properties including the Mueller matrix for a given particle size and refractive index based on Mie theory with the code miex (Wolf and Voshchinnikov 2004). The intensities of the different Stokes parameters of photon packages that eventually leave the model space in y direction are added on the detector. To improve the signal-to-noise ratio, we use the enforced-first-scattering method (preventing the lack of scattering events due to low optical depths, Cashwell and Everett 1959) and peel-off (for each scattering event, in accordance with the probability of reaching the detector, weighted fractions of the photon packages are sent directly to the detector; Yusef-Zadeh et al 1984). . Model set-up used for RT simulations to investigate the scattering of laser light in a particle cloud. The cylindrical particle cloud (violet) with homogeneous dust number density has a radius of 3 cm. The red laser is polarized 45 • with respect to the scattering plane (x-y plane) and passes through the cloud in x direction near the outer edge at δy = 2.8 cm. In the z direction the setup is symmetrical as the laser traverses the cloud centrally with respect to the z-dimension of the cylinder. The projection of this path on the yp-zp plane (grey) is shown in cyan. To assist the spatial perception, lines behind the yp-zp plane are dashed. The scattering angle φ is defined in the yp-zp plane perpendicular to the propagation direction between the negative zp-axis and the projection of the new travel direction. The rotation by the scattering angle φ around xp provides the reference frame (magenta) for the multiplication with the Mueller matrix. The scattering angle θ is the angle between the propagation direction before (xp, black dashed) and after the scattering (blue dashed).
Intensity maps of the different Stokes parameters simulated for the model described above are shown in figure 5. In agreement with the experimental setup serving as a basis for the chosen model, we integrate over a circular area of radius 5 mm at the center of the detector images to mimic the field of view of a single point polarimeter.
Traditionally, in the context of Mie polarimetry, ∆(Ψ) curves are used for the analysis. Both polarimetric angles Ψ and ∆ are calculated from the normalized Stokes parameters  as follows: Figure 5 shows detector images of the different Stokes parameters for the described model with a particle radius of 200 nm. In order to have a consistent representation with the setup illustrated in figure 3, the z-axis is oriented downward. In the following, therefore, the part of the intensity map at z < 0 cm is phrased 'above the laser beam' , correspondingly, 'below the laser beam' refers to z > 0 cm. The total intensity I is shown on the left panel of figure 5. The highest intensities (yellow horizontal line) occur at the position of the laser beam and result from single scattering by 90 • directly to the polarimeter. Beyond this line, the light was scattered at least twice. It can be seen that despite the homogeneous and thus vertically symmetric density distribution, the brightness distribution is asymmetric with respect to the z direction. At the left edge of the cloud (x ≲ −1.1 cm), the intensity is increased at above the laser beam, whereas at larger values of x the intensity is higher below the laser beam.

Description of the effect
For the normalized Stokes parameters q and u, indicating the linearly polarized radiation, there are additional asymmetric features in vertical direction regarding changes of the sign. Above the laser beam, q is negative. Below, q is positive at first, but then its sign changes along a line starting from the left, where the laser beam enters the cloud, to increasing values of x and z. The normalized Stokes parameter u is positive above the laser beam towards large x values. Below, u is also positive, but then changes its sign only slightly below the beam still within the field of view of the polarimeter.
The circularly polarized light (normalized Stokes parameter v), on the other hand, shows a slight asymmetry in vertical direction only.
To illustrate the polarization direction, we have overlaid the polarization vectors on the intensity maps of the normalized Stokes parameters in figure 5. These emphasize the vertical asymmetry. In the upper part of the particle cloud (z < 0 cm) there is a smooth transition of the polarization direction from horizontal to vertical again to horizontal, with a slightly curved shape above the laser beam. Below the laser beam, in a nearly vertical line directly below the entrance of the laser beam into the cloud (x ≲ −1.1 cm) in the region where q and u show a change of sign, the polarization vectors exhibit a strong directional change.

Origin of the asymmetry
Different aspects are involved in generating the asymmetric intensity distributions of the individual Stokes parameters. First, the linear polarization of the incoming laser radiation of 45 • with respect to the x-y plane together with the anisotropic scattering regarding the scattering angle φ leads to an asymmetric scattering behavior. Second, the proximity of the laser beam to the edge of the cloud leads to a truncation of symmetry-preserving photon paths. These aspects will be discussed in the following.
We first have a look at the impact of the anisotropic scattering regarding the scattering angle φ. To be able to study this effect in isolation, we look at the intensity distributions that result when the laser beam passes through the center of the cloud (δy = 0 cm) instead of close to the edge of the cloud. The resulting intensity maps, which are based on the same set-up as figure 5 except for the shift of the laser beam, are shown in figure 6(b). Again, a vertical asymmetry of the total intensity I can be seen, with a higher intensity below the laser beam.
To explain this, let us imagine two photon packages that travel on paths mirrored at the x-y plane, as illustrated in figure 7(a). Those photon packages pass the same optical depths and are scattered by the same scattering angles θ before reaching the detector at two points mirrored at the x-axis. Intuitively, at those points, the values of the Stokes parameters of both photon packages should be identical. One would therefore expect vertically symmetric intensity distributions. However, the scattering angles φ (in the plane perpendicular to the original travel direction) by which the photon packages are scattered on the two paths are different (see figure 7(a)).
The probability density function of the scattering angle φ has preferential directions with respect to the polarization direction before the scattering event (Fischer et al 1994): ) .
The scattering functions S 1 (θ) and S 2 (θ) are provided from Mie calculations and P L is the degree of linear polarization of the incident radiation. The angle γ indicates the position of the major axis of the polarization ellipse of the incident radiation: Figure 8(a) illustrates the probability density distribution for the scattering angle φ exemplarily for different scattering angles θ. It can be seen that the scattering regarding the angle φ is anisotropic, where depending on the scattering angle θ, scattering is more likely either in a direction perpendicular to the original polarization plane (θ ≲ 73 • , compare figure 8(b) or within the original polarization plane (θ ≳ 73 • ) This means, at the first scattering event, for small scattering angles θ, photon packages are preferentially scattered downward behind the laser beam (z > 0 cm, y < 0 cm) or upward in front of the laser beam (z < 0 cm, y > 0 cm). For scattering angles θ ≳ 73 • , photon packages are preferentially scattered upward behind the laser beam (z < 0 cm, y < 0 cm) or downward in front of the laser beam (z > 0 cm, y > 0 cm) with an even larger anisotropy. Figure 5. Detector images of the total intensity I and normalized Stokes parameters q, u, and v of laser light scattered by a cylindrical particle cloud with homogeneous density containing particles with radii of 200 nm. The laser propagates through the particle cloud from left to right. The white contour lines indicate intensity levels of 10 −4 , 5 × 10 −5 and 10 −5 with respect to the maximum intensity. The intensity maps of the normalized Stokes parameters q, u and v are overlaid with black polarization vectors indicating the major axis of the polarization ellipse (see equation (6)). The blue circles mark the detector area over which we integrate the radiation to mimic the field of view of a polarimeter. Now, at first glance, one might think that photon packages travelling on paths with scattering angles φ 1 and φ ′ 1 differing by 180 • at the first scattering, as illustrated in figure 7(b), would cancel out this effect. However, this is not the case. Firstly, the lengths of the paths s and s ′ through the cloud differ, resulting in different optical depths on the way to the detector and thus different attenuation of the total intensities. Secondly, the scattering angles θ 2 and θ ′ 2 of the second scattering differ. Thus, the anisotropy with respect to Figure 7. Illustration of photon package paths described in section 5. The laser beam enters the cloud perpendicular to the y-z plane shown above and is scattered for the first time at the origin of the reference frame yp, zp shown in black. Afterwards, two photon packages propagate along paths s (red) and s ′ (red dashed) which feature different symmetries: (a) paths mirror-symmetric with respect to the x-y plane, (b) two paths at which the first scattering is rotationally symmetric with respect to the original travel direction of the photon package (xp), i.e. the scattering angles φ1 and φ ′ 1 = φ1 + 180 • result in the same rotation of the reference frame (equation (2)). In each case, the global coordinate system is indicated in gray and a segment of the cylindrical particle cloud in violet. The relevant scattering angles (cyan and blue for each of the two paths) are shown. the scattering angle θ causes different weights of the photon packages reaching the detector on these paths. Thus, these paths also lead to different intensities at two points mirrored at z = 0 cm of the detector.
In the case of the laser beam passing close to the edge of the cloud, the asymmetry is further increased by the fact that paths with the symmetry illustrated in figure 7(b), which do not fully compensate each other due to the different optical depths and different scattering angles θ 2 and θ ′ 2 , but which contribute similarly, are cut off. Now, let us take another look at two paths that are mirrored at the x-y plane, as illustrated in figure 7(a). In addition to the anisotropy regarding scattering angle φ, the different angles φ 1 and φ ′ 1 result in different rotations according to equation (2). While I and V are invariant to this rotation, the Stokes parameters Q and U are affected by the rotations by different angles φ, leading to different outcomes on the two considered paths. In particular, for the Stokes parameter Q, there is a change of sign for one of the paths. Accordingly, the intensity distribution of the normalized Stokes parameter q in figure 6 shows a vertical asymmetry.

Implications
Once the optical depth is high enough for a significant contribution from multiple scattering, Mie polarimetry is affected by the asymmetry described above, potentially complicating the analysis. A simple solution seems to be to avoid the asymmetry. Keeping the typical setup with the laser as the radiation source unchanged in order to keep existing diagnostic methods applicable and to preserve the advantage of spatially precise information, depolarizing the incident laser light appears appropriate. However, this involves large intensity losses and is therefore technically not feasible. Yet, it would be sufficient to rotate the input polarization of the laser beam symmetrically to the x-y plane with an input Stokes vector of S = (1, 1, 0, 0) T . However, this turns out to be impractical. In the case of single scattering, the measured Stokes parameter U would always remain zero, independent of the particle properties, i.e. the particle radius, so that ∆ = π 2 = const. (see equation (4)). Well established methods, such as the CRAS-Mie method, which are based on the analysis of the relation ∆(Ψ), can then no longer be applied.
As the initial polarization of the laser beam and the asymmetric scattering behavior cannot be avoided, we make use of RT simulations to investigate the impact of the asymmetric intensity distribution on the relation ∆(Ψ) and consequently on the analysis of the particle properties.
In this context, the asymmetry can already have a noticeable impact on spatially unresolved measurements. To illustrate this, in the following, we focus on two models with significant contributions from multiple scattering. The model HO, with n = 10 14 m −3 , has a ten times higher optical depth than the one described in section 2. The model LC has the same particle cloud parameters as described in section 2, but the laser radiates without y-offset directly through the center of the cloud (figure 6), resulting in longer paths with higher optical depths along the path of the laser beam and between the laser beam and the polarimeter. For these two models, we perform a series of RT simulations of monodisperse particle clouds with particle radii a between 20 nm to 300 nm and calculate the polarimetric angles Ψ(a) and ∆(a) for different vertical positions of the polarimeter to check the impact of a displacement in z direction. In figure 9 the corresponding ∆(Ψ) relations are shown.
At about the minimum of the relations ∆(Ψ), from which multiple scattering affects the curves, significant deviations appear. For both models, HO and LC, in the case of a polarimeter offset downward to z = 3 mm, the curves are shifted towards smaller values of Ψ. Conversely, for a polarimeter offset upward to z = −3 mm the curves are shifted toward larger values of Ψ. This is because when the polarimeter is shifted, the contribution of single scattering in the field of view of the polarimeter becomes smaller while multiple scattering becomes more relevant. Due to the asymmetry of the brightness distribution with respect to the z direction, the impact of multiple scattering differs depending on whether the polarimeter is moved upwards or downwards. The shift of the relations ∆(Ψ) is accompanied by a change of the position of the maximum. Since the maximum of the curve is an important indication for the refractive index determination (Kobus et al 2022), a displaced positioning of the polarimeter can lead to the determination of an incorrect refractive index and thus to the determination of erroneous grain radii. The actual deviation from the actual value depends on the direction in which the polarimeter is shifted.
However, the asymmetry also provides an opportunity to derive additional constraints on the particle properties. More precisely, the asymmetric patterns of the intensity distributions of the different Stokes parameters depend on the refractive index. This is illustrated for the laser passing through the center and near the edge of the dust cloud in figures 6 and 10, respectively. It can be seen that the vertical asymmetries vary with varying refractive index. For example, the bounds at which the sign changes differ. In the case of the laser beam passing near the edge (figure 10) the region directly below the laser beam (z > 0 cm) where u is positive is much more extended at the refractive index N = 1.5 + 0.05i than in the case of the two refractive indices with larger real parts N = 1.8 + 0.05i and N = 2.1 + 0.05i. In addition, the occurrence of some regions also changes with the refractive index. For example, only at N = 1.5 + 0.05i, the Stokes parameter q shows a narrow elongated negative region starting from the entrance of the laser beam into the cloud in negative x and z directions.
Thanks to these asymmetric patterns, the analysis of spatially resolved imaging polarimetry using RT simulations can be used to constrain the refractive index of the dust particles. In contrast, single point polarimetry, from which the refractive index is determined using the relation ∆(Ψ) assuming a constant refractive index throughout the course of the growth experiment, imaging polarimetry has the advantage of allowing the refractive index to be determined for each individual time step by fitting the asymmetric intensity distributions of the different Stokes parameters. We would like to point out that the vertical asymmetry of the intensity distributions can be additionally influenced by the spatial particle density distribution. This can be taken into account as a free parameter when fitting the intensity distributions of the various Stokes parameters using RT simulations. However, this can cause ambiguity, so that the refractive index and therefore the particle radius cannot be determined. In this case, complementary measurements of the spatial particle distribution (Killer et al 2014, Pretzier et al 1992 can provide a remedy. Figure 9. Illustration of the impact of the polarimeter position based on the models HO (high optical depth, left) and LC (laser beam through center, right). The relation of the polarimetric angles Ψ(a) and ∆(a) (upper panels) result from radiative transfer simulations of two series of monodisperse particle clouds increasing the particle radius as a parameter from 20 nm to 300 nm. The corresponding polarimeter positions are shown in the intensity distributions in the lower panels.

Summary
RT simulations allow the prediction of the polarization state of scattered light, taking into account multiple scattering. Consequently, they are a well suited tool for the optimization of analysis strategies for the determination of the properties of particles in reactive plasmas. Especially in the case of imaging polarimetry, which is required to study inhomogeneous particle clouds, RT simulations are the prerequisite for analyzing the complex interplay of contributions due to multiple scattering.
In the present study, we made use of RT simulations to investigate the anisotropic scattering behavior of polarized light. For a simple model of an argon-acetylene plasma encompassing a cylindrical particle cloud, a vertically asymmetric brightness distribution of the polarized scattered light is observed even though the spatial density distribution of the dust particles is homogeneous (see section 4). This asymmetry is due to the non-isotropic probability distribution of the scattering angle φ in the plane perpendicular to the direction of propagation of the incident polarized light (see section 5 and figure 4). The asymmetric intensity distribution can complicate the analysis and, if not measured carefully, can lead to the determination of incorrect refractive indices and thus to erroneous grain sizes. On the other hand, due to the asymmetric scattering behavior, imaging Mie polarimetry offers the potential to enhance CRAS-Mie or even to relax the constant refractive index constraint of CRAS-Mie, allowing one to trace the temporal evolution of the refractive index and size of the particles (see section 6).

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.