#  Generation of high circular polarization of interstellar Lyman $\alpha$
radiation triggering biological homochirality

Hajime Fukushima1, Hidenobu Yajima1, Masayuki Umemura1
1Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1
Tsukuba, Ibaraki 305-8577, Japan
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The homochirality of biological molecules on the Earth is a long-standing
mystery regarding the origin of life. Circularly polarized ultraviolet (UV)
light could induce the enantiomeric excess of biological molecules in the
interstellar medium, leading to the homochirality on the earth. By performing
3D radiation transfer simulations with multiple scattering processes in
interstellar dusty slabs, we study the generation of circular polarization
(CP) of ultraviolet light at Lyman $\alpha$ ($\lambda=0.1216~{}{\rm\mu m}$) as
well as in the near-infrared (NIR, $\lambda=2.14~{}{\rm\mu m}$) wavelengths.
Our simulations show that the distributions of CP exhibit a symmetric
quadrupole pattern, regardless of wavelength and viewing angle. The CP degree
of scattered light from a dusty slab composed of aligned grains is $\sim 15$
percent for Ly$\alpha$ and $\sim 3$ percent at NIR wavelengths in the case of
oblate grains with an MRN size distribution. We find that the CP degree of
Ly$\alpha$ is well correlated with that in the NIR regardless of viewing
angles, whilst being a factor of $\sim 5$ higher. Thus, high CP of Ly$\alpha$
is expected in sites where NIR CP is detected. We suggest that such circularly
polarized Ly$\alpha$ may initiate the enantiomeric excess of biological
molecules in space.

###### keywords:

astrobiology – polarization – stars: formation – dust, extinction – ISM:
structure – infrared: stars.

††pubyear: 2019††pagerange:  Generation of high circular polarization of
interstellar Lyman $\alpha$ radiation triggering biological homochirality–C

##  1 Introduction

The homochirality of biological molecules on the Earth is a long-standing
unrevealed issue for the origin of life. Biological molecules including amino
acids are detected in meteorites, such as the Murchison and Mukundpura
meteorites (Engel & Macko, 1997; Pizzarello & Yarnes, 2018). Also, the
Hayabusa2 spacecraft recently collected the samples of Ryugu (Yokoyama et al.,
2022), and amino acids were discovered (Nakamura et al., 2022). Besides, the
enantiomeric excess of the L-amino acids is found in the Murchison and
Mukundpura meteorites. It is proposed that circularly polarized light can play
a crucial role in the enantiomeric excesses of biological molecules (see
Bonner, 1991, for a review). Thereafter, Bailey et al. (1998) demonstrated
that the circularly polarized ultraviolet (UV) light in the space can induce
the enantiomeric excesses of biological molecules discovered on the
meteorites.

For now, the observations of circular polarization (CP) caused by dust grains
have been limited at near-infrared (NIR) wavelengths and various degrees of CP
have been detected around young stellar objects (e.g., Chrysostomou et al.,
2000; Fukue et al., 2010; Kwon et al., 2014; Kwon et al., 2016; Kwon et al.,
2018). The NIR CP degree is correlated with the luminosity of the central
stars, and high CP of more than 10 percent has only been observed around
massive stars (Kwon et al., 2014). The previous works showed that aspherical
dust grains efficiently produce the CP light (Gledhill & McCall, 2000; Wolf et
al., 2002; Whitney & Wolff, 2002; Lucas et al., 2004). In Fukushima et al.
(2020a, hearafter Paper I), we have successfully reproduced the quadrupole
patterns of the observed CP maps at NIR wavelengths and suggested that the CP
degree larger than 10 percent could be generated by micron-sized dust grains.

On the other hand, Lucas et al. (2005) theoretically investigated the UV
circular polarization of scattered light around a massive star. They concluded
that the CP degree of singly scattered light is less than 10 percent, and that
dichroic extinction with multiple scatterings is needed to produce higher CP.
However, they only considered the case with the wavelength range larger than
$\lambda=0.2~{}{\rm\mu m}$. The CP degree at the shorter wavelength is still
unclear.

The absorption of CP light by chiral molecules is caused by the circular
dichroism (CD). If a continuous flux exists in a wide range of wavelengths,
the CD alternates in sign and sums to zero over the whole spectrum due to the
Kuhn-Condon sum rule (Kuhn, 1930; Condon, 1937). Therefore, to initiate the
enantiomeric excess, a circularly polarized strong emission line at a specific
wavelength is requisite. A possibility is Ly$\alpha$ lines from hydrogen since
a lot of distant galaxies with strong Ly$\alpha$ line emissions from hydrogen
have been observed (Ouchi et al., 2020). The solar system, which is located at
the outskirt of the Milky Way, could be exposed to strong Ly$\alpha$ radiation
from central star-forming regions in early galactic phase. Also, high-redshift
Ly$\alpha$ emitting galaxies may produce strong Ly$\alpha$ background
radiation in addition to direct radiation from star-forming regions.
Ly$\alpha$ photons ($\lambda=0.1216~{}{\rm\mu m}$) are produced via the
hydrogen recombination in ionized gas (Yajima et al., 2009, 2014). In the star
cluster formation, photoionization feedback from massive stars is the main
mechanism to disrupt star-forming clouds (e.g., Fukushima et al., 2020b;
Fukushima & Yajima, 2021, 2022). Most stars are born as members of a star
cluster (Lada & Lada, 2003), and sun-like stars are also surrounded by ionized
gas at birth. Therefore, the early solar system could form under the
environment with strong Ly$\alpha$ radiation. Recently, it is suggested that
circularly polarized Ly$\alpha$ is favorable to cause the enantiomeric excess
of the precursors and the amino acids (Shoji et al., 2023) and propylene oxide
(Hori et al., 2022) detected in the Sagittarius B2 star-forming region
(McGuire et al., 2016). However, the generation of Ly$\alpha$ CP has not been
elucidated so far.

In this study, we explore the generation of CP at Lyman $\alpha$
($\lambda=0.1216~{}{\rm\mu m}$). We utilize the 3-dimensional radiation
transfer code developed in Paper I. The simulations follow changes of the
Stokes parameters at each scattering and absorption process. We also
investigate the dependence of the CP distributions on the observational
angles. In Section 2, we describe the numerical method and setups of our
simulations. The main results are presented in Section 3. Section 4 is devoted
to summary and discussion.

##  2 NUMERICAL METHOD

We utilize the radiation transfer code based on the Monte Carlo technique
developed in Paper I that follows a change of the Stokes parameters. We
consider the UV wavelength ($\lambda=0.1216~{}{\rm\mu m}$), which corresponds
to Ly$\alpha$ photons. We calculate scattering and absorption processes in a
dusty slab around a single radiation source.

In the following subsections, we describe the brief summary of radiation
transfer methods in Section 2.1, the models of dust grains in Section 2.2, and
the setup of the dusty slab in Section 2.3. In Appendix A, we describe details
of the simulation method. We also show the results at the NIR wavelength
($\lambda=2.14~{}{\rm\mu m}$) in Appendix B to compare CP at Ly$\alpha$.

###  2.1 Radiation transfer

Photons emitted from stars propagate with absorption and scattering due to
dust grains that change their polarization properties. In our simulations,
each photon packet contains the information of the stokes vectors
${\bf{I}}=(I,Q,U,V)^{T}$. We conduct the Monte Carlo radiation transfer
simulations in the following steps:

Photon injection:

In this study, we assume that radiation from a star is unpolarized initially,
i.e., the Stokes vector is ${\bf I}=(I_{0},0,0,0)$ where $I_{0}$ is the
intensity at the surface of radiation sources. The initial direction of a
photon packet is chosen randomly. Then, the radiation transfer calculation
starts.

Step 1:

We set the optical depth $\tau_{\rm sc,p}$ between the current position and
the next scattering. The optical depth is evaluated as $\tau_{\rm
sc,p}=-\ln(R)$, where $R$ is the random number in the range $[0-1]$.

Step 2:

We calculate the photon propagation until the total optical depth $\tau_{\rm
sc}$ is equal to $\tau_{\rm sc,p}$. In each cell, we estimate the scattering
cross-section of dust grains and add it to the total optical depth $\tau_{\rm
sc}$ by considering the number density of dust grains and the path length over
the cell. If $\tau_{\rm sc}$ is still smaller than $\tau_{\rm sc,p}$, we
consider the dust extinction in the cell as Equation (3) in Paper I.

Step 3:

Once $\tau_{\rm sc}$ reaches $\tau_{\rm sc,p}$, we derive the changes of the
Stokes vector and the propagation angle. The number of angle bins is 1024. The
scattering angle is randomly selected with the weight of the differential
cross-section $dC_{\rm sca}/d\Omega$. Depending on the scattering angle, the
Stokes vector is determined. Finally, we reset $\tau_{\rm sc}$ and $\tau_{\rm
sc,p}$, and we return to Step 1. We consider the radiative transfer to a mock
observer at each scattering process for constructing a two-dimensional image
as Equation (4) in Paper I.

Final step:

When a photon packet reaches the boundary of the computational box, we stop
the radiative transfer. Then we start to calculate the radiative transfer of
the next photon packet.

In Step 3, the Stokes vector after a scattering process is defined with the
Müller matrix as (e.g., Bohren & Huffman, 1983),

$\displaystyle\left(\begin{array}[]{c}I\\\ Q\\\ U\\\ V\end{array}\right)_{\rm
sc}=\frac{1}{k^{2}r^{2}}\left(\begin{matrix}S_{11}&S_{\rm 12}&S_{\rm
13}&S_{\rm 14}\\\ S_{21}&S_{\rm 22}&S_{23}&S_{24}\\\ S_{31}&S_{\rm
32}&S_{33}&S_{34}\\\ S_{41}&S_{\rm
42}&S_{43}&S_{44}\end{matrix}\right)\left(\begin{array}[]{c}I\\\ Q\\\ U\\\
V\end{array}\right)_{\rm i},$ (9)

where $k$ and $r$ are the wave number of photons and the distance from the
scattering point. The Müller matrix depends on the photon propagation and
scattering angles as well as the opening angle between the incident light and
the minor axis of dust grains. As discussed in Section 2.2, we assume that the
shape of dust grains is oblate, that is rotationally symmetric around the
minor axis. More details of Step 3 is described in Appendix A.

In this work, we conduct the simulations with $10^{8}$ photon packets. The
propagation angle of photons packets is discretized by HEALPIX (Górski et al.,
2005) with 1024 angle bins.

###  2.2 Dust grain model

We adopt a dust model with oblate spheroidal grains with axis ratio of 2:1 as
in Paper I. We assume the size distribution of the MRN dust mixture (Mathis et
al., 1977) in this study. The number density of dust grains $(n_{\rm d})$ and
the range of dust radii $(a_{\rm d})$ are given as $n_{\rm d}\propto a_{\rm
d}^{-3.5}$ and $0.005~{}{\rm\mu m}\leqq a_{\rm d}\leqq 0.25~{}{\rm\mu m}$.
Here, the radius of a dust grain is defined as $a_{\rm d}=(3V_{\rm
d}/4\pi)^{1/3}$ where $V_{\rm d}$ is the volume. In Paper I, we found that the
micron-size dust grains mainly produce highly circularly polarized light at
the NIR wavelength. In Appendix B, for viewing angles other than that adopted
in Paper I, we show the results at the NIR wavelength with the MRN dust
mixture and the micron-size dust grains.

For the radiative transfer calculations, we prepare tables of the Müller
matrix (Eq. 9) and the absorption rates (Eq.3 in Paper I) in advance. We use
DDSCAT code (Draine & Flatau, 2013) in which a dust grain is treated as an
array of electric dipoles. The code allows us to calculate the optical
properties at each scattering angle of ($\theta_{\rm s},\phi_{\rm s}$), and
the angle between the minor axis of a dust grain and the direction of incident
light ($\Theta$). The scattering angles are discretized with 1024 angle bins.
The optical properties of dust grains depend not only on the shape but also on
the dielectric function. We adopt the values of dielectric function of
astronomical silicate in Draine (2003), $m=1.9+0.7i$ for $0.1216~{}{\rm\mu m}$
and $m=1.7+0.034i$ for $\lambda=2.14~{}{\rm\mu m}$.

###  2.3 Setup of dusty slab

Figure 1:  The spatial positions of the dusty slab and the observer. The slab
is along the yz-plane. The vector $\vec{a}_{1}$ represents the direction of
the minor axis of dust grains. The angles $(\theta_{\rm obs},\phi_{\rm obs})$
are the position of the observer.

We set a dusty slab with a thickness of $0.2~{}{\rm pc}$ and the radiation
source at the centre, as shown in Fig. 1. In this work, we adopt simple
geometry to understand the origin of CP clearly. We consider a typical scale
of filament induced by turbulence motion as the thickness of the dusty slab
(e.g., McKee & Ostriker, 2007; Arzoumanian et al., 2011; Fukushima et al.,
2020b). Here, we assume that the minor axis direction of the oblate dust
grains ($\vec{a}_{1}$) is aligned along the $z$-axis in the Lab-frame,
$\vec{a}_{1}=(0,0,1)$. The number density of dust grains is set as the
scattering optical depth along the horizontal direction becomes 1.0. The slab
is resolved by $300\times 300\times 60$ cells. The dust column density with
the optical depth is reasonable considering the typical molecular clouds with
the surface gas density of $\sim 100~{}\rm M_{\odot}~{}pc^{-2}$ (e.g., Roman-
Duval et al., 2010).

In Paper I, we only considered the face-on view, which corresponds to the
angle $(\theta_{\rm obs},\phi_{\rm obs})=(90^{\circ},0^{\circ})$. In this
study, we consider observational maps of five viewing angles of $(\theta_{\rm
obs},\phi_{\rm obs})=(90^{\circ},0^{\circ})$, $(90^{\circ},30^{\circ})$, ,
$(90^{\circ},60^{\circ})$$(60^{\circ},0^{\circ})$, and
$(30^{\circ},0^{\circ})$, to investigate dependence of CP degrees on viewing
angles.

Figure 2:  The observational images of Ly$\alpha$ ($\lambda=0.1216~{}{\mu{\rm
m}}$). The angles $(\theta_{\rm obs},\phi_{\rm obs})$ represent the positions
of observers as shown in Fig.1. The top, middle, and bottom columns show the
intensity distribution, linear polarization, and circular polarization. In the
top panels, we plot the ratio of the observed light to stellar light. In the
middle panels, the black bars show the direction of linear polarization.

##  3 Results

###  3.1 Observational maps

We first show the results on the polarization of Ly$\alpha$. Fig. 2 represents
the observational images of the intensity and the degree of the
linear/circular polarization at each viewing angle. The intensity maps are
normalized with the intrinsic intensity $I_{*}$ emitted from the radiation
source. The intensity monotonically decreases as the radial distance
increases. The centre is the brightest because photons directly escape from
the slab without scattering.

The middle panels in Fig. 2 show the linear polarization degree and direction.
The scattered light is linearly polarized along the perpendicular direction of
the scattering plane, resulting in the concentric pattern shown in the figure.
The degree of linear polarization is smaller in the horizontal direction of
the slab as shown in the face-on map (Fig.2-a), and it lessens in the forward
and backward scattered light as shown in Fig.2 (b)-(e). The linear
polarization degree of scattered light by large dust grains is lower than that
of small grains Paper I. At the wavelength of Ly$\alpha$, the Rayleigh
approximation is valid only at $a_{\rm d}<10^{-2}~{}{\rm\mu m}$. The large
dust grains reduce linear polarization degrees.

Figure 3:  The CP degree along the circle of radius $0.25~{}{\rm pc}$ at the
centre of the observation maps. Each line shows the cases with the viewing
angles $(\theta_{\rm obs},\phi_{\rm obs})=(90^{\circ},0^{\circ})$,
$(90^{\circ},30^{\circ})$, and $(90^{\circ},60^{\circ})$, which correspond to
the bottom panels of Fig. 2-a, b, and c. The horizontal axis is the rotational
axis ($\psi$). Each line shows the cases of Ly$\alpha$ (blue) and the NIR
(orange) wavelength. The left (right) vertical axis shows the CP degree of
Ly$\alpha$ (the NIR wavelength). The direction of the horizontal axis
corresponds to $\psi=0^{\circ}$.  Figure 4: Same as Fig. 3, but for the
observational angles $(\theta_{\rm obs},\phi_{\rm
obs})=(90^{\circ},0^{\circ})$, $(60^{\circ},0^{\circ})$, and
$(30^{\circ},0^{\circ})$, which correspond to the bottom panels of Fig. 2-a,
d, and e.

The bottom panels of Fig. 2 show the distributions of the CP degree. We find
that the symmetric quadrupole patterns appear like that at the NIR wavelength
(see Paper I). In Fig. 3 and 4, we show the azimuthal distributions of the CP
degree along the circle of the radius $0.25~{}{\rm pc}$ in Fig. 2. In the
face-on view (Fig. 2-a), the maximum value of the CP degree reaches $\sim 15$
percent. The relation between the CP degree and the inclination angle is not
simple. In the case of $(\theta_{\rm obs},\phi_{\rm
obs})=(60^{\circ},0^{\circ})$, the fraction of the back scattering photons in
the observable flux becomes large, resulting in the high CP degree $\sim 20$
percent. Fig. 3 and 4 also show the azimuthal distributions of the CP at the
NIR wavelength as shown in Fig. 9. The absolute values of CP degree are the
largest ($|V/I|\sim 0.03$) in the diagonal direction of the face-on view (Fig.
9-a). The CP degree decreases as the inclination angle increases because of
the contributions of the forward and backward scattering photons. The CP
degree at the NIR wavelength is correlated with that of Ly$\alpha$ one and it
drops by a factor $\sim 5$.

###  3.2 Physical origin of high CP degree of Ly$\alpha$

Figure 5:  The size dependence of the cross-section and CP degree in the cases
with Ly$\alpha$. Each lines shows the cases of $(\Theta,\theta_{\rm
s},\phi_{\rm s})=(45^{\circ},90^{\circ},270^{\circ})$. The solid line
represents $S_{11}/a^{2.5}$. The dashed line shows $S_{41}/S_{11}$. The black
dashed-line is averaged value of $S_{41}/S_{11}$ with the MRN mixture ($n_{\rm
d}\propto a^{-3.5}$).  Figure 6:  Same as Fig. 5, but for the case with
$\lambda=2.14~{}{\mu\rm m}$. Figure 7:  Upper panel: the dependence of
$S_{41}/S_{11}$ on the wavelength in the case of the scattering angles
$(\Theta,\theta_{\rm s},\phi_{\rm s})=(45^{\circ},90^{\circ},270^{\circ})$.
The circles show the mean absolute values calculated by DDSCAT code. The
filled and non-filled circles represent the positive and negative values. The
solid line shows the values calculated with Equations 78 and 82 in the
Rayleigh approximation. The vertical dashed lines show the wavelength
considered in this study ($\lambda=0.1216~{}{\rm\mu m}$ and $2.14~{}{\rm\mu
m}$). Bottom panel: the dependence of the refractive index of astronomical
silicate (Draine, 2003).

Here we investigate the origin of high CP degree of Ly$\alpha$. The scattering
cross-section for non-polarized photons is determined solely by the $S_{11}$
component of the Müller matrix (see also Eq. 23). The abundance of dust grains
in the logarithmic bin is given as $\propto a_{\rm d}^{-2.5}$. Thus, the
values of $S_{11}/a_{\rm d}^{2.5}$ represent the weight of each logarithmic
bin on the cross-section. Fig. 5 shows the size dependence of $S_{11}/a_{\rm
d}^{2.5}$ in the case with $(\Theta,\theta_{\rm s},\phi_{\rm
s})=(45^{\circ},90^{\circ},270^{\circ})$ at the wavelength of Ly$\alpha$. At
this angle, the CP degree becomes maximum if the Rayleigh approximation is
valid (see also Eq. 82). In the Rayleigh approximation, the cross-section for
scattering is proportional to the dust grain size as $\propto a_{\rm d}^{6}$,
resulting in that $S_{11}/a_{\rm d}^{2.5}\propto a_{\rm d}^{3.5}$. At $a_{\rm
d}<0.025~{}{\rm\mu m}$, the value of $S_{11}/a_{\rm d}^{2.5}$ monotonically
increases as the Rayleigh approximation. At $a_{\rm d}>0.025~{}{\rm\mu m}$,
$S_{11}/a_{\rm d}^{2.5}$ oscillates and damps. Thus, the dust grains of
$a_{\rm d}\sim 0.025~{}{\rm\mu m}$ mainly contribute to the scattering
process. The Rayleigh approximation for $S_{41}/S_{11}$ is valid only if
$a_{\rm d}<10^{-2}~{}{\rm\mu m}$. It slightly increases to $\sim 0.3$ at $\sim
0.02~{}{\rm\mu m}$, but it rapidly decreases to $-0.4$ at $a_{\rm d}\sim
0.04~{}{\rm\mu m}$. At the larger dust grains, $S_{41}/S_{11}$ fluctuates and
converges to zero at $a_{\rm d}\gtrsim 0.2~{}\rm\mu m$. The averaged value of
$S_{41}/S_{11}$ is $\sim 0.2$, which is close to that of the Rayleigh
approximation.

Fig. 6 shows the dust size dependence of $S_{11}/a_{\rm d}^{2.5}$ and
$S_{41}/S_{11}$ at the NIR wavelength to compare that of the wavelength of
Ly$\alpha$. The value of $S_{11}/a_{\rm d}^{2.5}$ monotonically increases as
$\propto a_{\rm d}^{3.5}$ as well as the Rayleigh approximation. Therefore,
the larger dust grains have more contributions to the scattering processes.
The values of $S_{41}/S_{11}$ is constant at $a_{\rm d}<10^{-1}~{}{\rm\mu m}$
where the Rayleigh approximation is valid (also see Appendix C). For the
larger dust grains, $S_{41}/S_{11}$ increases up to 0.05 at $a_{\rm
d}=0.25~{}{\rm\mu m}$. These large dust grains raise the averaged value of
$S_{41}/S_{11}$ in the case of the MRN size distribution model at the NIR
wavelength.

The trend of $S_{41}/S_{11}$ at each wavelength can be understood from the
view of the Rayleigh approximation. Here, we compare our models based on
DDSCAT with the Rayleigh approximation. The solid line in the top panel of
Fig. 7 shows the $S_{41}/S_{11}$ calculated based on the Rayleigh
approximation and the refractive index (see also Appendix C). The value of
$S_{41}/S_{11}$ increases with the ratio of the imaginary part to the real
part of the refractive index ($m=n+ik$). In this figure, we assume that dust
grains are oblate whose axis ratio is $q=0.5$, and the scattering angles are
$(\Theta,\theta_{\rm s},\phi_{\rm s})=(45^{\circ},90^{\circ},270^{\circ})$.
The circle symbols in Fig. 7 show the mean values of $S_{41}/S_{11}$
calculated with DDSCAT code. At $\lambda>3~{}{\rm\mu m}$, the value of
$S_{41}/S_{11}$ nicely match the values derived from the Rayleigh
approximation. At the shorter wavelength, the value of $S_{41}/S_{11}$
deviates from the Rayleigh approximation and increases up to 10 percent due to
the contribution of larger dust grains as shown in Fig. 6. Also, the sign of
$S_{41}/S_{11}$ is reversed at $0.2~{}{\rm\mu m}<\lambda<1~{}{\rm\mu m}$
because the large dust grains have the negative values in these wavelengths.
At $\lambda<0.2~{}{\rm\mu m}$, the imaginary part of the refractive index
increases, and the fluctuation of $S_{41}/S_{11}$ occurs as shown in Fig. 5.
Due to these oscillations of $S_{41}/S_{11}$, the contributions of the large
dust grains are cancelled out. Thus, the mean value of $S_{41}/S_{11}$ reaches
the Rayleigh approximation at $\lambda=0.1216~{}{\rm\mu m}$.

##  4 Summary and Discussion

We have studied the generation of CP via multiple scattering processes on dust
grains by performing three-dimensional radiative transfer simulations. We have
investigated the CP degree of Ly$\alpha$ ($\lambda=0.1216{\rm\mu m}$) and NIR
wavelength ($\lambda=2.14{\rm\mu m}$), assuming a dusty slab composed of
aligned dust grains. The quadrupole patterns of CP appear in both cases,
regardless of the viewing angles. We find that the CP of Ly$\alpha$ can become
much higher than that at the NIR wavelength. The CP degree is $|V|/I\sim 15$
percent for Ly$\alpha$ and $|V|/I\sim 3$ percent for the NIR wavelength.

Lucas et al. (2005) suggested that the high CP degree at the UV wavelengths
($\lambda>0.2~{}\rm\mu m$) was generated by the dichroic extinction process.
In this work, we have focused on the shorter wavelength
$\lambda=0.1216~{}\rm\mu m$, corresponding to Ly$\alpha$ line of hydrogen. The
Ly$\alpha$ photons are emitted from the ionized gas associated with star-
forming regions. The circularly polarized Ly$\alpha$ photons can cause the
enantiomeric excess of amino acids (Shoji et al., 2023) and chiral molecules
(Hori et al., 2022). Therefore, interstellar Ly$\alpha$ radiation with high CP
as shown in our models might induce the enantiomeric excess of amino acids
discovered in the meteorites (Engel & Macko, 1997; Pizzarello & Yarnes, 2018).

In this study, we have supposed a dusty slab composed of oblate dust grains
whose minor axes are aligned with the edge-on angle of the slab. The optical
depth of the slab is assumed to be unity in the perpendicular direction. In
such a configration, single-scattered photons mostly contribute to the CP
generation. Although we have adopted a simplified model of a dusty slab, the
observed quadruple structure of the CP at near-infrared wavelengths is well
reproduced. In optically thick cases, the multiply scattered photons also
contribute to the CP generation via the dichroic extinction process (Lucas et
al., 2005). Actually, the distributions and aligned directions of dust grains
depend on the gas streaming, magnetic, and radiation fields (Andersson et al.,
2015, and reference therein). However, the spatial distributions of gas and
magnetic fields are still under debate because of complicated processes with
stellar feedback (e.g., Kim et al., 2018, 2021; Fukushima et al., 2020b). In a
future work, we will perform simulations using the results of the magneto-
radiation hydrodynamics simulations to investigate the effects of multiple
scattering and the alignment of dust grains in star-forming clouds.

In this work, we have considered the astronomical silicate dust model alone.
On the other hand, the two-component model with silicate and graphite has been
taken into account frequently in previous works (e.g. Mathis et al., 1977;
Draine & Lee, 1984). Since the Stokes parameters depend on the dust component,
the CP degrees in our simulations can change with different dust models.
Although it is difficult to constrain the dust properties only from the
observational data, Draine & Hensley (2021) recently found that the single
component model of large dust grains $(\gtrsim 0.01~{}\mu{\rm m})$ can
reproduce the observed extinction well (see also Hensley & Draine, 2022). In
their model, the dielectric function of grains at $h\nu\gtrsim 8~{}{\rm eV}$
is almost the same as the model used in this study.

We have studied the interaction between photons and dust grains. However,
Ly$\alpha$ photons also interact with neutral hydrogen atoms (Yajima et al.,
2012; Abe et al., 2018) that can induce the linear polarization (Seon et al.,
2022). Then, the linear polarized light can be converted into the circularly
polarized one via scattering with a dust grain or a hydrogen atom. On the
other hand, the coherency of the last scattering angle can get lower by
multiple scattering with hydrogen atoms and dust grains, which likely
decreases the CP degree.

The observation of the polarized light is a powerful tool to investigate
physical properties in the interstellar medium, such as the magnetic fields
(e.g., Hall, 1949; Hiltner, 1949). Recently, the observation of Atacama Large
Millimeter/submillimeter Array (ALMA) discovered the linearly polarized light
in the protoplanetary disks, and it is mainly caused by the scattering on the
dust grains (e.g., Kataoka et al., 2015; Tazaki et al., 2017). The
observations of the circularly polarized light will add more physical
information about the interstellar medium and the protoplanetary disks
(Draine, 2022). CP light has been mainly observed in the near-infrared
($1~{}{\rm\mu m}\lesssim\lambda\lesssim 3~{}{\rm\mu m}$, e.g., Kwon et al.,
2014), but we expect that the degree of CP becomes higher at the mid-infrared
(MIR) wavelength ($10~{}{\rm\mu m}\lesssim\lambda\lesssim 30~{}{\rm\mu m}$) as
shown in Fig. 7. In particular, the emission spectrum of an embedded massive
protostar has a peak around the mid-infrared (Tanaka et al., 2016; Fukushima
et al., 2018). Therefore, highly circularly polarized light in the mid-
infrared wavelength is expected to exist around forming massive stars. Note
that, on the other hand, thermal emission from hot dust can overwhelm the
scattered light at MIR wavelengths in regions where massive stars distribute
densely. If the thermal emission directly escapes the system, the CP degrees
become lower.

Circularly polarized UV can induce the enantiomeric excess of the biological
molecules in meteorites, which fell on to the earth and might trigger the
homochirality. In this paper, we have demonstrated that the Ly$\alpha$ CP is
tightly correlated with that at the infrared wavelength regardless of the
viewing angles and is higher by a factor of $\sim 5$ than that at the infrared
wavelength. We suggest that interstellar Ly$\alpha$ radiation with high CP in
early evolutionary phase of the Milky Way might initialize enantiomeric excess
of the biological molecules.

## Acknowledgements

This work is supported in part by MEXT/JSPS KAKENHI Grant Number
JP23K13139(HF), JP17H04827, JP20H04724, JP21H04489 (HY), JP19H00697(MU), NAOJ
ALMA Scientific Research Grant Numbers 2019-11A, JST FOREST Program, Grant
Number JP-MJFR202Z (HY), and Astro Biology Center Project research AB041008
(HY).

## DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A Calculation Method of Scattering

Figure 8:  The coordinates in the radiation transfer simulations; (1) the
laboratory frame (Lab-frame), (2) the pre-scattering frame (PS-frame) and (3)
the post-scattering frame (S-frame). The vectors $\vec{e}_{\rm k}$,
$\vec{e}_{\perp}$, and $\vec{e}_{\parallel}$ represent the directions of the
photon packet, the parallel and perpendicular vectors defined in each
coordinate. In the PS and S-frames, the vector $\vec{a}_{1}$ represents the
direction of the minor axis of the dust grains, and $\Theta$ is the angle
between the $\rm X_{\rm ps}$ and $\rm X_{\rm s}$ axes. $\theta_{\rm s}$ and
$\phi_{s}$ represent scattering angles.

In Paper I, we briefly summarized the calculation methods of scattering. We
describe the details of them in this Appendix.

The Stokes parameters $Q$ and $U$ depend on the orthonormal basis vectors. In
Equation (9), the basis vectors are defined as the parallel basis vectors
($\vec{e}_{\parallel,i}$ and $\vec{e}_{\parallel,s}$) of the incident, and
scattered light are contained in the same plane as shown in Fig 8-(3). Here,
we consider the three coordinates to simplify the coding as shown in Fig. 8:

(1)

Lab-frame: We calculate the transportation of photon packets in the laboratory
frame (Lab-frame). The parallel and perpendicular vectors of the electric
field are defined as they coincide with the $\vec{e}_{\perp}=-\vec{e}_{\phi}$
and $\vec{e}_{\parallel}=\vec{e}_{\theta}$ where $\vec{e}_{\theta}$ and
$\vec{e}_{\phi}$ are the unit vectors of the polar and azimuthal directions.

(2)

PS-frame: We use the PS-frame (pre-scattering frame) to calculate the
scattering cross-section and determine the scattering direction. In this
frame, the direction of the incident light is the same as the $\rm X_{\rm
ps}$-axis. We set the direction of the minor axis of an oblate dust grain
($\vec{a}_{1}$) in the $\rm X_{\rm ps}$-$\rm Y_{\rm ps}$ plane at $z=0$. The
$\rm Y_{\rm ps}$ and $\rm Z_{\rm ps}$ axes are parallel to the unit vectors of
the incident light ($\vec{e}_{\perp}$ and $\vec{e}_{\parallel}$),
respectively.

(3)

S-frame: In the S-frame, we transform the Stokes parameters with the Müller
matrix $\bf{M}$ defined as Equation (9). In this frame, the polar and
azimuthal angles ($\theta_{\rm s}$ and $\phi_{\rm s}$) of the scattered light
correspond to the scattering angles. The parallel vectors of the incident and
scattering light are defined as coplanar. The direction of the incident light
is the same as the $\rm X_{\rm s}$-axis in this frame.

In each cell, we calculate the cross-section of scattering $C_{\rm sca}$,
which depends on the directions of the minor axis of a dust grain, and the
incident and scattered light. First, we alter the Stokes vector from Lab-frame
$(\bf{I}_{\rm lab})$ to PS-frame $(\bf{I}_{\rm ps})$. The conversion of the
Stokes vector is defined as (Bohren & Huffman, 1983)

$\displaystyle\left(\begin{array}[]{c}I^{{}^{\prime}}\\\ Q^{{}^{\prime}}\\\
U^{{}^{\prime}}\\\
V^{{}^{\prime}}\end{array}\right)=\left(\begin{matrix}1&0&0&0\\\ 0&\cos
2\psi&\sin 2\psi&0\\\ 0&-\sin 2\psi&\cos 2\psi&0\\\ 0&0&0&1\\\
\end{matrix}\right)\left(\begin{array}[]{c}I\\\ Q\\\ U\\\
V\end{array}\right),$ (18)

where the angle $\psi$ is the clockwise rotation angle. The rotation axis
between the Lab and PS-frames ($\psi_{\rm lab\rightarrow ps}$) is obtained as
follows. In the PS-frame, the minor axis of a dust grain $\vec{a}_{1}$ is
coplanar with the ${\rm X}_{\rm ps}$-${\rm Y}_{\rm ps}$ plane. The projection
vector of $\vec{a}_{1}$ to the ${\rm Y}_{\rm ps}$-${\rm Z}_{\rm ps}$ plane
coincides with the perpendicular axis $\vec{e}_{\perp,\rm ps}$. The projection
unit vector of $\vec{a}_{1}$ to the ${\rm Y}_{\rm ps}$-${\rm Z}_{\rm ps}$
plane in the Lab-frame is obtained as

$\displaystyle\vec{b}$ $\displaystyle=$
$\displaystyle\vec{a}_{1}-\left(\vec{a}_{1}\cdot\vec{e}_{\rm
k}\right)\vec{e_{\rm k}}$ (19) $\displaystyle=$
$\displaystyle\vec{a}_{1}-\cos\Theta\vec{e}_{\rm k}$

where $\Theta$ is the angle between the incident light and the minor axis of a
dust grain. The perpendicular axis of the PS-frame in the Lab-frame is given
as $\vec{e}_{\perp,{\rm ps}}=\vec{b}/|\vec{b}|$. The rotation angle is
obtained as

$\displaystyle\psi_{\rm lab\rightarrow
ps}=\begin{cases}\arccos(\vec{e}_{\rm\perp,ps}\cdot\vec{e}_{\rm\perp})&(\vec{e}_{\rm\perp,ps}\cdot\vec{e}_{\rm\parallel}<0)\\\
2\pi-\arccos(\vec{e}_{\rm\perp,ps}\cdot\vec{e}_{\rm\perp})&(\vec{e}_{\rm\perp,ps}\cdot\vec{e}_{\rm\parallel}\geqq
0).\end{cases}$ (20)

We use the Müller matrix $\bf{M}$ to determine the scattering direction in the
PS-frame. Since the matrix is originally defined in the S-frame, we derive the
modified Müller matrix as

$\displaystyle\bf{M}^{{}^{\prime}}=\bf{M}\left(\begin{matrix}1&0&0&0\\\
0&-\cos 2\phi_{\rm s}&\sin 2\phi_{\rm s}&0\\\ 0&-\sin 2\phi_{\rm s}&-\cos
2\phi_{\rm s}&0\\\ 0&0&0&1\\\ \end{matrix}\right),$ (21)

where we adapt the rotational axis from the PS-frame to the S-frame as
$\psi_{\rm ps\rightarrow s}=\pi/2-\phi_{\rm s}$. The differential cross-
section of scattering is estimated with the modified Müller matrix
$\bf{M}^{{}^{\prime}}$ as

$\displaystyle\frac{dC_{\rm
sca}}{d\Omega}=\frac{1}{k^{2}}\frac{(S_{11}^{{}^{\prime}}I_{\rm
ps}+S_{12}^{{}^{\prime}}Q_{\rm ps}+S_{13}^{{}^{\prime}}U_{\rm
ps}+S_{14}^{{}^{\prime}}V_{\rm ps})}{I_{\rm ps}},$ (22)

where ${\bf I}_{\rm ps}=(I_{\rm ps},Q_{\rm ps},U_{\rm ps},V_{\rm ps})$ is the
Stokes vector in the PS-frame. The scattering cross-section is given as

$\displaystyle C_{\rm sca}=\frac{1}{k^{2}}\int
d\Omega\left(\frac{S_{11}^{{}^{\prime}}I_{\rm ps}+S_{12}^{{}^{\prime}}Q_{\rm
ps}+S_{13}^{{}^{\prime}}U_{\rm ps}+S_{14}^{{}^{\prime}}V_{\rm ps}}{I_{\rm
ps}}\right).$ (23)

The scattering angles are randomly selected based on the probability from the
differential cross-section in the PS-frame. Then, we convert the Stokes vector
in the PS-frame ($\bf{I}_{\rm ps}$) to that in the S-frame ($\bf{I}_{\rm s}$)
with the modified Müller matrix $\bf{M}^{{}^{\prime}}$. In our radiative
transfer calculations, the intensity of a photon packet does not change in a
scattering process. We change ratios of $Q$, $U$, and $V$ to $I$. Finally, we
convert the stokes vector $\bf{I}_{\rm s}$ to that in the Lab-frame $\bf{I}$
as in Equation (20).

##  Appendix B The observational map at the NIR wavelength

Figure 9:  Same as Fig. 2, but for the cases at $\lambda=2.14~{}{\mu\rm m}$.

We present the results at the NIR wavelength of $\lambda=2.14~{}{\rm\mu m}$.
In Paper I, we only discussed the face-on view. Here, we simulate the
observational maps of the viewing angles of $(\theta_{\rm obs},\phi_{\rm
obs})=(90^{\circ},0^{\circ})$, $(90^{\circ},30^{\circ})$, ,
$(90^{\circ},60^{\circ})$$(60^{\circ},0^{\circ})$, and
$(30^{\circ},0^{\circ})$. Fig. 9 shows the observational maps from each angle.
Similar to the case of Ly$\alpha$, the intensity decreases outward.

The middle panels in Fig. 9 show the linear polarization. In the same fashion
as seen in Fig. 2, the polarization vector shows the concentric pattern. The
degree of linear polarization mainly depends on the incident and scattered
light angles represented in Fig. 8 as $\theta_{\rm s}$. At $(\theta_{\rm
obs},\phi_{\rm obs})=(90^{\circ},0^{\circ})$, most of observed photons are
perpendicularly scattered, which induces the highly linear polarization. Thus,
the degree of linear polarization reaches $\sim 1$ in the entire region. On
the other angles (b)-(e), the degree of linear polarization decreases in the
horizontal or vertical directions of the slabs because the forward or backward
photons are dominating.

The bottom panels of Fig. 9 show the distributions of the CP degree at the NIR
wavelength. The symmetric quadrupole patterns appear. These patterns are
related to the dependence of the Müller matrix on the angles of $\Theta$,
$\theta_{\rm s}$, and $\phi_{\rm s}$. The value of $S_{41}/S_{11}$ represents
the generation rates of the circular polarization from the non-polarized
light. The Rayleigh approximation is valid only for dust grains of which the
size is much smaller than the wavelength of radiation. It is $<0.1~{}{\mu\rm
m}$ for the NIR wavelength (see also Fig. 6). As discussed in Appendix C, the
$S_{41}$ is estimated with the Rayleigh approximation as (see Eq. 82, Gledhill
& McCall, 2000)

$\displaystyle
S_{41}\propto-(\alpha_{1}\alpha_{3}^{*}-\alpha_{1}^{*}\alpha_{3})\sin
2\Theta\sin\theta\sin\phi.$ (24)

These patterns are consistent with the observed angular distributions of the
CP degrees (e.g., Kwon et al., 2013, 2014; Kwon et al., 2016).

Figure 10:  Same as Fig. 10, but for the size distributions $n_{\rm d}\propto
a_{\rm d}^{-3.5}$ and the size range $0.01~{}\mu{\rm m}\leqq a_{\rm d}\leqq
1~{}\mu{\rm m}$ at $\lambda=2.14~{}{\mu\rm m}$.

In Paper I, we showed that the CP degree exceeds 20 percent even at the NIR
wavelength if the dust slab contains micron-size dust grains. Here, we perform
the additional simulations with the large dust grain size as $0.01~{}{\rm\mu
m}\leqq a_{\rm d}\leqq 1~{}{\rm\mu m}$ and investigate the observational maps
with the different viewing angles. Figure 10 shows the intensity, linear, and
circular polarization maps. In the face-on view, the CP degree exceeds 20
percent in the diagonal direction. The sign of the degree is reversed compared
with the fiducial model of Figure 9, which is caused by the micron-size dust
grains as shown in Paper I. The distributions of CP are the asymmetric
quadrupole patterns in the edge-on view of oblate grains (Fig.10-a, b, and c).
However, in the cases of $(\theta_{\rm obs},\phi_{\rm
obs})=(60^{\circ},0^{\circ})$ and $(30^{\circ},0^{\circ})$, the dipole
patterns of CP light appears unlike the fiducial model. Thus, the CP pattern
of the NIR wavelength is not universal if the micron-size dust grains
contained. The dipole pattern can be a probe of the dust size extending $\sim
1~{}\rm\mu m$.

##  Appendix C Rayleigh approximation

Here, we derive the Müller matrix for an oblate dust grain based on the
Rayleigh approximation. The scattering matrix was calculated in Appendix A of
Gledhill & McCall (2000), but the coordinates adapted in their formula are
different from in Fig. 8. Here, we reconstruct the formula to clarify.

The incident electric field causes the dipole moment as

$\displaystyle\bf{p}=\alpha\bf{E},$ (25)

where $\alpha$ is the polarizability tensor. If the oblate dust grain axis
coincides with the coordinate axis, $\alpha$ only has the diagonal components.
Each diagonal represents as (Bohren & Huffman, 1983)

$\displaystyle\alpha_{i}=4\pi
a_{1}^{2}a_{3}\frac{(m^{2}-1)}{3+3L_{i}(m^{2}-1)},$ (26)

where $m$, $a_{1}$, and $a_{3}$ are the refractive index, the radii of the
major and minor axes. The shape factors $L_{i}$ of oblate satisfy
$L_{1}=L_{2}<L_{3}$ and $L_{1}+L_{2}+L_{3}=1$. The shape factor in the
direction of the major axis $L_{1}$ is given as (Bohren & Huffman, 1983)

$\displaystyle
L_{1}=\frac{g(e)}{2e^{2}}\left[\frac{\pi}{2}-\arctan(g(e))\right]-\frac{g(e)^{2}}{2},$
(27)

where

$\displaystyle g(e)=\left(\frac{1-e^{2}}{e^{2}}\right)^{1/2},$ (28)

and

$\displaystyle e^{2}=1-a_{1}^{2}/a_{3}^{2}.$ (29)

The scattered light is obtained as

$\displaystyle{\bf E}_{\rm s}=\frac{k^{2}e^{ikr}}{4\pi r}({\bf
e}_{{k,s}}\times{\bf p})\times{\bf e}_{{k,s}},$ (30)

where $k$, $r$ and ${\bf e}_{\bf r}$ are the photon wavenumber, the distance
from the scattered point, and the direction of the scattered light. Here, we
adapt the coordinate defined as Fig. 8. The basis vectors of incident light
$({\bf{e}}_{k,i},{\bf e}_{\perp,i},{\bf e}_{\parallel,i})$ and scatted light
$({\bf{e}}_{k,s},{\bf e}_{\perp,s},{\bf e}_{\parallel,s})$ are given as

$\displaystyle{\bf e}_{k,i}=\left(\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right),{\bf e}_{\perp,i}=\left(\begin{array}[]{c}0\\\
\sin\phi_{\rm s}\\\ -\cos\phi_{\rm s}\end{array}\right),{\bf
e}_{\parallel,i}=\left(\begin{array}[]{c}0\\\ \cos\phi_{\rm s}\\\
\sin\phi_{\rm s}\end{array}\right),$ (40)

and

$\displaystyle{\bf e}_{k,s}=\left(\begin{array}[]{c}\cos\theta_{\rm s}\\\
\sin\theta_{\rm s}\cos\phi_{\rm s}\\\ \sin\theta_{\rm s}\sin\phi_{\rm
s}\end{array}\right),{\bf e}_{\perp,s}=\left(\begin{array}[]{c}0\\\
\sin\phi_{\rm s}\\\ -\cos\phi_{\rm s}\end{array}\right),{\bf
e}_{\parallel,s}=\left(\begin{array}[]{c}-\sin\theta_{\rm s}\\\
\cos\theta_{\rm s}\cos\phi_{\rm s}\\\ \cos\theta_{\rm s}\sin\phi_{\rm
s}\end{array}\right).$ (50)

Then, the scattered light is calculated as

$\displaystyle{\bf E}_{\rm s}$ $\displaystyle=F\left(p_{\rm Y}\sin\phi_{\rm
s}-p_{\rm Z}\cos\phi_{\rm s}\right){\bf e}_{\perp,s}$
$\displaystyle+F\left(-p_{\rm X}\sin\theta_{\rm s}+p_{\rm Y}\cos\theta_{\rm
s}\cos\phi_{\rm s}+p_{\rm Z}\cos\theta_{\rm s}\sin\phi_{\rm s}\right){\bf
e}_{\parallel,s}$ (51)

where

$\displaystyle F=\frac{k^{2}e^{ikr}}{4\pi r}.$ (52)

The dipole moment ${\bf p}=(p_{\rm X},p_{\rm Y},p_{\rm Z})$ is obtained by
transforming the diagonal matrix ${\bm{\alpha}}$ into the coordinate system of
(X, Y, Z). The transforming matrix corresponds to rotation by an $\Theta$
angle around the axis Z. The dipole moment is given as

$\displaystyle{\bf p}=R_{z}(\Theta)\mathcal{A}R_{z}(-\Theta){\bf
E}_{i}=\left(\begin{array}[]{c}P_{11}E_{\perp,i}+P_{12}E_{\parallel,i}\\\
P_{21}E_{\perp,i}+P_{22}E_{\parallel,i}\\\
P_{31}E_{\perp,i}+P_{32}E_{\parallel,i}\\\ \end{array}\right),$ (56)

where $R_{z}(\Theta)$ and $\mathcal{A}$ are defined as

$\displaystyle\mathcal{A}=\begin{pmatrix}\alpha_{3}&0&0\\\ 0&\alpha_{1}&0\\\
0&0&\alpha_{1}\end{pmatrix},$ (57)

and

$\displaystyle R_{z}(\Theta)=\left(\begin{matrix}\cos\Theta&-\sin\Theta&0\\\
\sin\Theta&\cos\Theta&0\\\ 0&0&1\end{matrix}\right).$ (58)

The incident electric field is estimated with the basis vectors $({\bf
e}_{\perp,i},{\bf e}_{\parallel,i})$ as

$\displaystyle{\bf E}_{i}=E_{\perp,i}{\bf e}_{\perp,i}+E_{\parallel,i}{\bf
e}_{\parallel,i}=\begin{pmatrix}0\\\ E_{\perp,i}\sin\phi_{\rm
s}+E_{\parallel,i}\cos\phi_{\rm s}\\\ -E_{\perp,i}\cos\phi_{\rm
s}+E_{\parallel,i}\sin\phi_{\rm s}\end{pmatrix}.$ (59)

Substituting equations (57), (58), and (59) into (56), we obtain each factor
for the dipole moment $\bf p$ as

$\displaystyle P_{11}$ $\displaystyle=a\sin\phi_{\rm s},$ (60) $\displaystyle
P_{12}$ $\displaystyle=a\cos\phi_{\rm s},$ (61) $\displaystyle P_{21}$
$\displaystyle=b\sin\phi_{\rm s},$ (62) $\displaystyle P_{22}$
$\displaystyle=b\cos\phi_{\rm s},$ (63) $\displaystyle P_{31}$
$\displaystyle=-c\cos\phi_{\rm s},$ (64) $\displaystyle P_{32}$
$\displaystyle=c\sin\phi_{\rm s},$ (65)

where

$\displaystyle a$ $\displaystyle=(\alpha_{3}-\alpha_{1})\cos\Theta\sin\Theta,$
(66) $\displaystyle b$
$\displaystyle=\alpha_{1}\cos^{2}\Theta+\alpha_{3}\sin^{2}\Theta,$ (67)
$\displaystyle c$ $\displaystyle=\alpha_{1}.$ (68)

The relation between the incident and scattered light is written with the
amplitude scattering matrix as (Bohren & Huffman, 1983),

$\displaystyle\left(\begin{array}[]{c}E_{\parallel}\\\
E_{\perp}\end{array}\right)_{s}=\frac{e^{ik(r-x)}}{-ikr}\left(\begin{matrix}S_{2}&S_{3}\\\
S_{4}&S_{1}\end{matrix}\right)\left(\begin{array}[]{c}E_{\parallel}\\\
E_{\perp}\end{array}\right)_{i},$ (73)

where we ignore the time-dependent term. Substituting equations (51) and (56)
into (73), we obtain the each components of the amplitude scattering matrix as

$\displaystyle S_{1}$ $\displaystyle=\frac{k^{3}}{4\pi
i}\left(b\sin^{2}\phi_{\rm s}+c\cos^{2}\phi_{\rm s}\right),$ (74)
$\displaystyle S_{2}$ $\displaystyle=\frac{k^{3}}{4\pi
i}\left(-a\sin\theta_{\rm s}\cos\phi_{\rm s}+b\cos\theta_{\rm
s}\cos^{2}\phi_{\rm s}+c\cos\theta_{\rm s}\sin^{2}\phi_{\rm s}\right),$ (75)
$\displaystyle S_{3}$ $\displaystyle=\frac{k^{3}}{4\pi
i}\left(-a\sin\theta_{\rm s}\sin\phi_{\rm s}+b\cos\theta_{\rm s}\cos\phi_{\rm
s}\sin\phi_{\rm s}-c\cos\theta_{\rm s}\cos\phi_{\rm s}\sin\phi_{\rm
s}\right),$ (76) $\displaystyle S_{4}$ $\displaystyle=\frac{k^{3}}{4\pi
i}\left(b\cos\phi_{\rm s}\sin\phi_{\rm s}-c\cos\phi_{\rm s}\sin\phi_{\rm
s}\right).$ (77)

The $S_{11}$ and $S_{41}$ components of Müller matrix defined as Equation (9)
are calculated with the amplitude scattering matrix as

$\displaystyle S_{11}$
$\displaystyle=\frac{1}{2}\left(|S_{1}|^{2}+|S_{2}|^{2}+|S_{3}|^{3}+|S_{4}|^{4}\right)$
$\displaystyle=\frac{k^{6}}{32\pi^{2}}\left[f_{11}|\alpha_{1}|^{2}+f_{31}\left(\alpha_{1}\alpha_{3}^{*}+\alpha_{1}^{*}\alpha_{3}\right)+f_{33}|\alpha_{3}|^{2}\right],$
(78)

where

$\displaystyle f_{11}$ $\displaystyle=\cos^{2}\phi_{\rm s}+\cos^{2}\theta_{\rm
s}\sin^{2}\phi_{\rm s}$ $\displaystyle+\cos^{2}\Theta[(\sin\theta_{\rm
s}\sin\Theta+\cos\theta_{\rm s}\cos\phi_{\rm
s}\cos\Theta)^{2}+\sin^{2}\phi_{\rm s}\cos^{2}\Theta],$ (79) $\displaystyle
f_{31}$ $\displaystyle=\cos\Theta\sin\Theta[(\cos^{2}\theta_{\rm
s}-\sin^{2}\theta_{\rm s}\cos^{2}\phi_{\rm s})\cos\Theta\sin\Theta$
$\displaystyle-\cos\theta_{\rm s}\sin\theta_{\rm s}\cos\phi_{\rm
s}(\cos^{2}\Theta-\sin^{2}\Theta)],$ (80) $\displaystyle f_{33}$
$\displaystyle=\sin^{2}\Theta[(\sin\theta_{\rm s}\cos\Theta-\cos\theta_{\rm
s}\cos\phi_{\rm s}\sin\Theta)^{2}+\sin^{2}\phi_{\rm s}\sin^{2}\Theta],$ (81)

and

$\displaystyle S_{41}$ $\displaystyle={\rm
Im}\left(S_{2}^{*}S_{4}+S_{3}^{*}S_{1}\right)$
$\displaystyle=-\frac{k^{6}}{64\pi^{2}}{\rm
Im}(\alpha_{1}\alpha_{3}^{*}-\alpha_{1}^{*}\alpha_{3})\sin\theta_{\rm
s}\sin\phi_{\rm s}\sin 2\Theta.$ (82)

Figure 11:  The relation between $S_{41}/S_{11}$ and the imaginary part of the
refractive index $k$. Each line represents cases with $n=1.5$ (solid), $1.7$
(dashed), and $1.9$ (dot-dashed).

Fig. 11 shows $S_{41}/S_{11}$ with the various refractive index $m=n+ik$ with
the scattering angle and the inclined angle of the dust grain $(\theta_{\rm
s},\phi_{\rm s})=(90^{\circ},270^{\circ})$ and $\Theta=45^{\circ}$. Here, we
adopt the oblate dust grain with the axial ratio of 2:1. The value of
$S_{41}/S_{11}$ only depends on the ratio of the imaginary part to the real
part of the refractive index. At $k/n<0.5$, the value of $S_{41}/S_{11}$
increases with $k/s$. The refractive index of astronomical silicate is
$k/n=0.02$ for $\lambda=2.14~{}{\rm\mu m}$ and $k/n=0.37$ for
$\lambda=0.1216~{}{\rm\mu m}$. Thus, the CP degree of the scattered Ly$\alpha$
photons is more than 5 times higher than that of the NIR light on the Rayleigh
approximation.