# Imaging reconstruction method on X-ray data of CMOS polarimeter combined
with coded aperture

Tsubasa Tamba  Hirokazu Odaka  Taihei Watanabe  Toshiya Iwata  Tomoaki Kasuga
Atsushi Tanimoto  Satoshi Takashima  Masahiro Ichihashi  Hiromasa Suzuki  Aya
Bamba  Japan Aerospace Exploration Agency, Institute of Space and
Astronautical Science, 3-1-1, Yoshino-dai, Chuo-ku, Sagamihara, Kanagawa
252-5210, Japan  Department of Physics, Faculty of Science, The University of
Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Department of Earth and
Space Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka
560-0043, Japan  Kavli IPMU, The University of Tokyo, Kashiwa 113-0033, Japan
Graduate School of Science and Engineering, Kagoshima University, 1-21-24,
Korimoto, Kagoshima, Kagoshima 890-0065, Japan  Research Center for Early
Universe, Faculty of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-
ku, Tokyo 113-0033, Japan  Trans-Scale Quantum Science Institute, The
University of Tokyo, Tokyo 113-0033, Japan

###### Abstract

X-ray polarization is a powerful tool for unveiling the anisotropic
characteristics of high-energy celestial objects. We present a novel imaging
reconstruction method designed for hard X-ray polarimeters employing a Si CMOS
sensor and coded apertures, which function as a photoelectron tracker and
imaging optics, respectively. Faced with challenges posed by substantial
artifacts and background noise in the coded aperture imaging associated with
the conventional balanced correlation method, we adopt the Expectation-
Maximization (EM) algorithm as the foundation of our imaging reconstruction
method. The newly developed imaging reconstruction method is validated with
imaging polarimetry and a series of X-ray beam experiments. The method
demonstrates the capability to accurately reproduce an extended source
comprising multiple segments with distinct polarization degrees. Comparative
analysis exhibits a significant enhancement in imaging reconstruction accuracy
compared to the balanced correlation method, with the background noise levels
reduced to 17%. The outcomes of this study enhance the feasibility of Cube-Sat
imaging polarimetry missions in the hard X-ray band, as the combination of Si
CMOS sensors and coded apertures is a promising approach for realizing it.

###### keywords:

polarimetry , hard X-rays , CMOS imaging sensor , coded aperture imaging ,
CubeSat , EM algorithm

##  1 Introduction

X-ray polarization is a distinctive method for capturing unprecedented images
of high-energy celestial objects. It carries essential information about
anisotropic physical phenomena, such as the magnetic field structure
responsible for synchrotron radiation, the emission geometry of Compton
scattering, and the properties of the strong gravitational field around
compact objects. Recent advancements in X-ray polarization studies have been
marked by the launch of the Imaging X-ray Polarimeter Explorer (IXPE; [1, 2])
in 2021. This mission, which covers the soft X-ray band of 2–8 keV, has
substantially broadened our capabilities in this field (e.g., Crab Nebula
observation: [3]). While IXPE focuses on the soft X-ray band through
photoabsorption, the Soft Gamma-ray Detector (SGD) onboard Hitomi [4] and the
Polarized Gamma-ray Observer+ (PoGO+;[5]) utilize Compton scattering to detect
X-ray polarization in the hard X-ray band. Notably, both missions successfully
detected polarization from the Crab Nebula in the higher end of the hard X-ray
band: 60–160 keV for SGD [6] and 20–160 keV for PoGO+ [7]. However, a
significant observational gap exists in the lower end of the hard X-ray band
due to the absence of established observational techniques, specifically in
the energy range of 10–30 keV. This energy range is particularly important
because non-thermal components, carrying anisotropic information, become
dominant over unpolarized thermal emission above $\sim 10\;{\rm keV}$. In
addition, the abundance of photon counts in this band, compared to higher
energy bands, provides a rich dataset for detailed analysis.

To address this observational gap, a 6U CubeSat mission named the Coded
Imaging Polarimeter of High Energy Radiation (cipher; [8]) is under
development. It aims to conduct imaging polarimetry in the 10–30 keV band and
obtain the polarization map for bright extended sources such as the Crab
Nebula. This imaging polarimeter is characterized by a micro-pixel
($\sim{\rm\mu m}$ pixel pitch) Si CMOS sensor and coded apertures. The micro-
pixel CMOS sensor serves as a sophisticated photoelectron tracker, enabling
the determination of the polarization angle of incident photons. Given the
spatial constraints of the CubeSat mission, coded apertures provide the
solution for imaging capability, eliminating the need for X-ray mirrors. The
CMOS sensor has demonstrated polarization detection capabilities in the 6–30
keV range [9, 10], meeting the fundamental requirements of cipher. The imaging
reconstruction of the coded apertures is also established [11], employing a
conventional reconstruction technique called the “balanced correlation method”
[12]. However, this method introduces significant artifacts and noise levels,
which hinder effective imaging polarimetry.

Recognizing the limitation of the balanced correlation method, this paper
presents a novel imaging reconstruction method for polarimetry utilizing the
Expectation-Maximization (EM) algorithm [13]. It is a statistical approach to
efficiently derive maximum likelihood estimates, and has been applied to
imaging reconstruction techniques extensively (e.g., [14, 15, 16, 17, 18]).
This method holds the potential to achieve imaging polarimetry with enhanced
precision, with reduced artifacts and noise levels. The remainder of this
paper is organized as follows. We first provide the formulation of the EM
algorithm that is applied to imaging polarimetry, and describe the
experimental setup to examine the new method in Section 2. The data processing
and results of imaging reconstruction are presented in Section 3. We then
discuss the precision of our new imaging reconstruction method by comparing it
with the conventional balanced correlation method in Section 4. Conclusions
are presented in Section 5.

##  2 Imaging reconstruction method

###  2.1 Application of EM algorithm to imaging polarimetry

We developed a novel reconstruction method for the coded aperture imaging
based on the EM algorithm [13]. This statistical approach involves estimating
the maximum likelihood from incomplete experimental data through iterative
cycles of Expectation (E-) and Maximization (M-) steps, continuously updating
the estimation. The $l$-th E-step and M-step in the imaging reconstruction
analysis are defined as follows:

$\displaystyle{\rm(E-step)}\;\;\;\;\;\tilde{D}_{v}^{(l)}$ $\displaystyle=$
$\displaystyle\sum_{u}p(v\,|\,u)\tilde{S}_{u}^{(l)},$ (1)
$\displaystyle{\rm(M-step)}\;\;\;\;\;\tilde{S}_{u}^{(l+1)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sum_{v^{\prime}}D_{v^{\prime}}}\sum_{v}D_{v}\frac{p(v\,|\,u)\tilde{S}_{u}^{(l)}}{\tilde{D}_{v}^{(l)}},$
(2)

where $l$ increments by 1 at each iteration [14]. In these equations,
$\tilde{S}_{u}^{(l)}$, $\tilde{D_{v}^{(l)}}$, and $D_{v}$ represent the
estimated incident source distribution, the estimated event distribution on
the detector, and the actual event distribution derived from experimental
data, respectively. The term $p(v\,|\,u)$ denotes the posterior probability
that a photon from the $u$-th sky region is detected in the $v$-th detector
region.

The method is applied to the imaging polarimeter with a CMOS sensor and coded
apertures. The sky region is defined as a two-dimensional angular
distribution, $\bm{u}=(\theta_{x},\;\theta_{y})$, with respect to the optical
axis, while the detector region is defined as a two-dimensional pixel
distribution, $\bm{v}=(d_{x},\;d_{y})$. The polarization angle and
polarization degree need to be assigned to each sky region. Since the CMOS
sensor is sensitive only to the horizontal and vertical directions, we focused
on two Stokes parameters, $I$ and $Q$, ignoring the effect of $U$. These
Stokes parameters are linked to the experimental data through the branching
ratio in double-pixel events, which is, the ratio between horizontally-
elongated (H-type) and vertically-elongated (V-type) events [8]. Equations (1)
and (2) can be expressed as

$\displaystyle\tilde{D}_{\bm{v},\,t}^{(l)}$ $\displaystyle=$
$\displaystyle\sum_{\bm{u}}p(\bm{v}\,|\,\bm{u})\tilde{S}_{\bm{u},\,t}^{(l)}$
(3) $\displaystyle\tilde{S}_{\bm{u},\,t}^{(l+1)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sum_{\bm{v}^{\prime}}D_{\bm{v}^{\prime},\,t}}\sum_{\bm{v}}D_{\bm{v},\,t}\frac{p(\bm{v}\,|\,\bm{u})\tilde{S}_{\bm{u},\,t}^{(l)}}{\tilde{D}_{\bm{v},\,t}^{(l)}},$
(4)

where $t$ is a two-valued variable representing $t={\rm H-type}$ or $t={\rm
V-type}$. The Stokes parameters $I$ and $Q$ can be calculated using

$\displaystyle\begin{pmatrix}I_{\bm{u}}^{(l)}\\\ Q_{\bm{u}}^{(l)}\\\
\end{pmatrix}=\begin{pmatrix}1&1\\\ 1/m&-1/m\\\
\end{pmatrix}\begin{pmatrix}S_{\bm{u},\,{\rm H-type}}^{(l)}\\\
S_{\bm{u},\,{\rm V-type}}^{(l)}\\\ \end{pmatrix},$ (5)

where $m$ denotes the modulation factor of the CMOS sensor. The posterior
probability $p(\bm{v}\,|\,\bm{u})$ is calculated from the coded aperture
pattern:

$\displaystyle p(\bm{v}\,|\,\bm{u})\propto\begin{cases}1&(\bm{u}\;{\rm
to}\;\bm{v}\;{\rm pass\;through\;aperture})\\\ \tau&(\bm{u}\;{\rm
to}\;\bm{v}\;{\rm pass\;through\;mask})\\\ 0&{\rm(otherwise)},\end{cases}$ (6)

where $\tau$ is the transmittance of the coded aperture.

###  2.2 Beam experiment

X-ray beam experiments were conducted at BL20B2 in SPring-8, a synchrotron
radiation facility in Hyogo, Japan, to evaluate the imaging-polarimetry
capability of the system. The beamline emits polarized monochromatic X-rays
with a polarization degree of nearly 100% (e.g., [9]). The left panel of
Figure 1 shows the experimental setup at the beamline. We utilized the
GMAX0505RF sensor, manufactured by GPixel Inc., as the polarization detector.
This Si CMOS sensor, with a pixel pitch of $2.5\;{\rm\mu m}$, was originally
designed for visible light and near-infrared imaging but also has sensitivity
for the X-ray polarization [10]. The sensor was positioned at the farthest
part from the beam source. Upstream of it, a $25\;{\rm cm}$ collimator was
mounted to collimate incident X-rays. On the upstream side of the collimator,
a $0.1\;{\rm mm}$ thick board made of SUS304 was attached. This board features
numerous holes, functioning as eight independent coded apertures with distinct
random patterns. The collimator ensures that the shadows of these patterns do
not overlap at the detector plane. Each coded aperture comprises of a
$64\times 64$ pattern with a pitch size of $35\;{\rm\mu m}$, covering
$896\times 896$ pixels on the detector plane (Figure 1 middle). The system has
a field of view and angular resolution of $0.5^{\circ}\times 0.5^{\circ}$ and
$29^{\prime\prime}$, respectively. The orientation of the collimator
($\theta_{x}$ and $\theta_{y}$) is adjustable using a rotation stage,
facilitating the control of the incident X-ray direction. Moreover, the entire
system can rotate around the beam axis, allowing for the adjustment of the
polarization angle detected by the CMOS sensor.

A comprehensive sky scan was conducted by manipulating the rotation stage, and
an image for an extended source was created by merging the acquired data. The
data points for the imaging scan experiments are illustrated in the right
panel of Figure 1, covering a $420^{\prime\prime}\times 420^{\prime\prime}$
plane with a pitch size of $15^{\prime\prime}$. A total of $29\times 29=841$
scan points were utilized. Table 1 provides the details on the datasets
employed in this paper. To verify the hard X-ray imaging polarimetry, we
exposed the system to $16.0\;{\rm keV}$ polarized X-rays with polarization
angles of both $0^{\circ}$ and $90^{\circ}$. This beam energy was selected by
balancing between the polarization detection capability and the quantum
efficiency of the sensor. Lower-energy photons would result in low sensitivity
for polarization detection because they are more likely to be recorded as
single-pixel events. Conversely, higher-energy photons lead to poor
statistical quantity due to their smaller cross section with the Si CMOS
sensor. The incident beam was attenuated by a $120\;{\rm\mu m}$ Cu filter to
avoid pile-up. Each dataset has 10 frames per scan point, resulting in a total
of 8410 frames. The initial four datasets in Table 1 were merged to generate
the “$0^{\circ}$ polarization data”, while the latter four were merged to
generate the “$90^{\circ}$ polarization data”.

Figure 1:  (left) Imaging polarimetry setup at Spring-8 beam line. (middle) Picture of coded apertures. (right) Sky region subject to the imaging scan experiment.  Table 1: List of datasets.  | Start time
---
(JST)
| Beam energy
---
(keV)
| Polarization angle
---
(degree)
| Frame exposure
---
(ms)
Number of frames
2021-11-03 18:50 | 16.0 | 0 | 600 | 8410
2021-11-03 20:40 | 16.0 | 0 | 600 | 8410
2021-11-03 22:42 | 16.0 | 0 | 600 | 8410
2021-11-04 00:22 | 16.0 | 0 | 600 | 8410
2021-11-04 03:42 | 16.0 | 90 | 600 | 8410
2021-11-04 05:23 | 16.0 | 90 | 600 | 8410
2021-11-04 07:00 | 16.0 | 90 | 600 | 8410
2021-11-04 08:54 | 16.0 | 90 | 600 | 8410

##  3 Results

###  3.1 Data processing

We performed standard data processing on the acquired data using ComptonSoft
[19, 20, 21], which included pedestal subtraction, bad pixel exclusion, event
extraction, and event classification. Subsequent to the data processing,
double-pixel events exhibiting either horizontal (H-type) or vertical (V-type)
elongation were specifically selected. The polarization angle and polarization
degree can be determined by measuring the ratio between the two types of
events. The numbers of H-type/V-type events were 9485530/8008517 for
$0^{\circ}$ polarization and 7622987/9741000 for $90^{\circ}$ polarization.
The modulation factor was calculated from these values to be $m=0.1033\pm
0.0001$ for $16.0\;{\rm keV}$ polarized photons (for more details of the
modulation factor, see [10]).

In addition to the $0^{\circ}$ and $90^{\circ}$ polarization data, a new
dataset named “imaging polarimetry test data” with various polarization
degrees depending on incident directions was generated. This dataset can
easily be generated by appropriately blending the $0^{\circ}$ and $90^{\circ}$
polarization data; for instance, an unpolarized source can be simulated by
mixing equal proportions of the $0^{\circ}$ and $90^{\circ}$ polarization
data. To evaluate the imaging polarimetry capability, we divided the entire
scan region into four sub-regions and assigned distinct polarization degrees
($Q/I$) to them. Figure 2 illustrates the definition of the segmentation. We
assigned $Q/I=0.4$, $-1.0$ ($90^{\circ}$ polarization), $1.0$ ($0^{\circ}$
polarization), and $0.0$ (unpolarized) to Regions 1, 2, 3, and 4,
respectively, where the intensity $I$ was spatially uniform. The first value
($Q/I=0.4$) was assigned to examine a “realistic” polarization degree for a
celestial object, while the latter three values ($Q/I=-1.0$, $1.0$, $0.0$)
were included to examine extreme cases.

Figure 2: Definition of the imaging polarimetry test data (see Section 3.1).

###  3.2 Imaging polarimetry

The EM algorithm was applied to the reconstruction of the polarization map
from the encoded images on the detector plane. We set
$\tilde{S}_{\bm{u},\,t}^{(0)}$ as a spatially uniform distribution, and
iterated through the expectation and maximization steps using equations (3)
and (4). The reconstruction was simultaneously applied to all eight coded
apertures, with the transmittance of the apertures set to $\tau=0.02$
(corresponding to $0.1\;{\rm mm}$ SUS304 for $16\;{\rm keV}$ photons). The
encoded image on the detector plane was binned in every $4\times 4$ pixels due
to the limited memory resources, but this did not affect the results as the
angular resolution is primarily dominated by the aperture pitch. The
reconstructed images were generated with an image pixel size of
$20^{\prime\prime}\times 20^{\prime\prime}$, which is slightly smaller than
the angular resolution of the system. The EM steps were iterated until
$l=1500$, ensuring a sufficient convergence. The reconstructed images were
obtained after the vignetting correction was applied.

As a simple case, we first conducted the imaging reconstruction on the
$0^{\circ}$ polarization data, where $I$ and $Q$ distributions should ideally
be identical. The upper panels of Figure 3 display the reconstructed images of
$I$, $Q$, and $Q/I$. The uniformly distributed images of $I$ and $Q$ were
successfully reconstructed, consistent with the actual scan region of
$420^{\prime\prime}\times 420^{\prime\prime}$. The polarization degrees also
yield a convincing value of $Q/I=1$ throughout the entire image. It is
important to note that the $Q/I$ image indicates horizontal polarization of
incident photons even beyond the scan region. This artifact arises because a
portion of the detected photons are inaccurately projected to the outer
region. Consequently, careful examination of both $I$ and $Q/I$ images is
necessary, and polarization degrees should not be relied on in regions where
the photon intensity $I$ is faint. We also tried various patterns of uniformly
distributed $Q/I$ values (0.5, 0, -0.5, -1.0) and confirmed that the
reconstructed images successfully reproduced the uniformly distributed
polarization degrees.

Subsequently, we applied the imaging reconstruction to the imaging polarimetry
test data, which is described in Section 3.1 and defined in Figure 2. The
lower panels of Figure 3 display the reconstructed images. A uniform
distribution of $I$ extending over $420^{\prime\prime}\times
420^{\prime\prime}$ similar to the $0^{\circ}$ polarization data was observed.
But a totally different image of $Q$, clearly divided into four segments, was
observed. Each of the four regions corresponds to the definition in Figure 2.
The reconstructed image of $Q/I$ successfully reflects the distinct
polarization degrees of the four regions, displaying $\sim 0.4$, $\sim-1.0$,
$\sim 1.0$, and $\sim 0.0$ for Regions 1, 2, 3, and 4, respectively.

In Figure 4, a detailed evaluation of the imaging reconstruction results is
presented. The left panel shows the distribution of $I/I_{\rm max}$ for the
four regions, where $I_{\rm max}$ corresponds to the maximum value of the
entire image. The four regions display comparable distributions, indicating
that the spatially uniform distribution of $I$ is successfully reproduced. The
right panel exhibits the distribution of the polarization degree $Q/I$ for the
four regions. The four regions display peaks at the predetermined polarization
degrees and are significantly distinguishable from each other. Still, non-
negligible uncertainties of $\Delta(Q/I)\sim 0.2$ remain due to systematic
uncertainties stemming from potentially non-uniform modulation factors across
the detector plane and biased projections caused by randomized patterns of the
coded apertures. While a poorer statistical quantity would result in worse
reproducibility, this imaging analysis benefits from an average of $\sim
40000$ photon counts contributing each imaging pixel.

Figure 3: The upper panels show the reconstructed images from $0^{\circ}$
polarization data, while the lower panels show those from imaging polarimetry
test data (see text). Reconstructed images include
$I(\theta_{x},\;\theta_{y})$ (left), $Q(\theta_{x},\;\theta_{y})$ (middle),
and polarization degree $Q/I$ (right).  Figure 4: Evaluation of imaging
reconstruction on the imaging polarimetry test data. The left panel shows the
histogram of source intensity plotted for the Regions 1–4. The right panel
displays the histogram of polarization degree plotted for Regions 1–4.

##  4 Discussion

In Section 3, our formulation of the imaging reconstruction method using the
EM algorithm was examined, and the polarization map was successfully
reconstructed from the imaging polarimetry test data. In this section, the
accuracy of the imaging reconstruction was evaluated by comparing our method
with the conventional balanced correlation method [12]. The balanced
correlation method is a prominent decoding method for coded apertures,
especially suitable for Uniformly Redundant Array (URA; [22, 23]), but it also
works for random pattern arrays used in our system (for details of the
balanced correlation method in our system, see Section 2 of [11]).

The left panel of Figure 5 shows the reconstructed $I$ image of the
$0^{\circ}$ polarization data using the balanced correlation method. Despite
the incident flux being spatially uniform along the scan region, it exhibits a
significant imbalance with unpredictably intense signals in the upper left
region. The non-source region also displays large fluctuations with a noisy
background in the upper left region. These artifacts and noisy background
signals arise from the spatially non-uniform distribution of the apertures,
which could back-project more signals to the upper left region of the sky. In
the right panel of Figure 5, the intensity distributions of the source and
background regions are presented, comparing the results between the EM
algorithm and the balanced correlation method. The EM algorithm exhibits sharp
distributions in both source and background regions, with the two
distributions clearly separated. On the other hand, the balanced correlation
method has flat distributions for the source and background regions, and these
two distributions are largely overlapped. This difference indicates fewer
artifacts and lower noise levels in the EM algorithm compared to the balanced
correlation method.

The noise levels can be evaluated by

$\displaystyle L_{\rm noise}=\frac{\sigma_{\rm bkg}}{\mu_{\rm src}},$ (7)

where $\sigma_{\rm bkg}$ denotes the standard deviation in the background
region, and $\mu_{\rm src}$ is the averaged intensity in the source region.
Here, the source region represents a collection of the image pixels within the
central $420^{\prime\prime}\times 420^{\prime\prime}$ area while the
background region denotes a collection of the image pixels outside this area.
We obtain $L_{\rm noise}=0.13$ for the EM algorithm and $L_{\rm noise}=0.75$
for the balanced correlation method, which means that our new imaging
reconstruction method reduces the noise level to $17\%$ of the conventional
method. The significant artifacts and noise levels in the balanced correlation
method had prevented us from creating any efficient polarization map, but the
improved method using the EM algorithm has successfully realized it with
sufficiently reduced artifacts and noise levels, as depicted in Figures 3 and
4.

Several aspects still require refinement for the application of this system
and analysis method to imaging polarimetry. Firstly, while we assumed a
uniform modulation factor within the CMOS sensor, this assumption needs
careful examination. Deviations of $\sim 0.01$ from the adopted original value
($m=0.1033$) could occur when focusing on specific limited areas of the
sensor. A better understanding of the dependence of the modulation factor on
the sensor pixels would mitigate systematic uncertainties in imagin
polarimetry. Secondly, the biased random patterns of the coded apertures also
require scrutiny. As observed in the left panel of Figure 5, generated by the
balanced correlation method, these coded apertures tend to project more
photons to the upper left region of the reconstructed image, potentiallly
imacting the accuracy of the imaging reconstruction in the case of the EM
algorithm. It is still underway to develop more sophisticated coded aperture
patterns. Furthermore, the performance of imaging polarimetry is influenced by
the shape of the polarization distribution. While in this study, the
boundaries between different polarization profiles are located around the
center of the field of view, we also confirmed that the reconstruction
accuracy diminishes when these boundaries are located farther from the center.
This is due to reduced sensitivity of the coded apertures in these regions.
These considerations are beyond the scope of this paper and will be addressed
in future discussions.

Figure 5:  (left) Reconstructed $I$ image of $0^{\circ}$ polarization data
using the balanced correlation method. (right) Intensity distributions of
source (src) and background (bkg) regions compared between the EM algorithm
(labeled as EM) and the balanced correlation method (labeled as BC).

##  5 Conclusions

We developed a new imaging reconstruction method for hard X-ray imaging
polarimetry employing a combination of a CMOS sensor and coded apertures.
Motivated by the significant artifacts and background noise levels associated
with the conventional balanced correlation method, we introduced the EM
algorithm as the foundation of our new imaging reconstruction method. The
effectiveness of the newly developed method was confirmed through X-ray beam
experiments at Spring-8, where the imaging polarimeter was exposed to the
X-ray beam and captured an extended source by a comprehensive scan of the sky.
The method exhibited remarkable capabilities in accurately reproducing an
extended source comprising multiple segments characterized by distinct
polarization degrees. Specifically, it successfully reproduced four regions
with $Q/I=-1.0$, 0.0, 0.4, 1.0. Our developed imaging reconstruction method
achieved a significant reduction in artifacts and noise levels compared to the
balanced correlation method. The background noise level experienced a
significant reduction to $17\%$. The outcomes of this study demonstrate
sufficient feasibility for the hard X-ray imaging polarimetry utilizing the
combination of CMOS sensors and coded apertures. This approach represents one
of the most promising ways to achieve CubeSat missions on hard X-ray
polarimetry.

## Acknowledgment

We thank the anonymous referees, who improved this work with their valuable
comments. We appreciate the helpful technical support by M.Hoshino and
K.Uesugi at SPring-8. This research is supported by the Japan Society for the
Promotion of Science (JSPS) KAKENHI Grant No. 20J20050, 23KJ2214 (TT),
18H05861, 19H01906, 19H05185, 22H00128, 22K18277 (HO), 20J00119 (AT), and
23KJ0780 (TI). The synchrotron radiation experiments were performed at BL20B2
in SPring-8 with the approval of the Japan Synchrotron Radiation Research
Institute (JASRI) (Proposal No. 2019B1369, 2020A1343, 2021B1542, and
2022B1477). This research is also supported by Society for Promotion of Space
Science (TT).

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