# The future of X-ray polarimetry towards the 3-Dimensional photoelectron
track reconstruction

Dawoon E. Kim  Alessandro Di Marco  Paolo Soffitta  Enrico Costa  Sergio
Fabiani  Fabio Muleri  Ajay Ratheesh  Fabio La Monaca  John Rankin  Ettore Del
Monte  Alda Rubini

###### Abstract

The development of the first X-ray polarimeter, based on the photoelectric
effect 20 years ago and implemented thanks to advances in gas amplification
structures and readout techniques, had a significant impact in opening a new
window for X-ray polarimetry. This system measures the X-ray polarization by
reconstructing the initial direction of the photoelectron, emitted by the
interaction of an incident photon with an atomic electron, in a gas mixture
from an ionization track collected on a two-dimensional plane. However, actual
X-ray polarimeters, are still requiring relatively long exposure time and
cannot coupled with high effective area mirrors or concentrators. In this
context, _the high yield polarimetry experiment in X-rays_ (Hype-X) project is
currently underway, aiming to improve the sensitivity of the next generation
X-ray polarimetry detectors taking advantage of the recent advancements in
imaging techniques for high-resolution time projection chambers. In
particular, we are evaluating the use of TIMEPIX3 to be applied for the read-
out of a gas detector, which will allow us to obtain a three-dimensional image
of the photoelectron track. To evaluate the improvement achievable by using a
3D track reconstruction, in this paper, we have reproduced a three-dimensional
photoelectron track from a ‘Geant4’ Monte Carlo simulation and examined the
sensitivity of X-ray polarimetry using a new three-dimensional track
reconstruction algorithm. We report the improvement of the modulation factor
with three-dimensional track reconstruction as $\sim 5\%$ (relative) in the
2-8 keV range and $\sim 17\%$ (relative) in the 2-4 keV range compared to the
current two-dimensional polarimetry system. This is equivalent to add a
further telescope to the three-telescope systems now employed in space on
board the IXPE mission.

##  1 Introduction

X-ray polarimetry has been considered for a long time as a prominent tool for
exploring the hidden properties of light emanating from the energetic and
innermost regions of astrophysical sources in the universe. Nevertheless, due
to the lack of suitable sensitive instrumentation, polarimetry had remained a
veiled field in X-ray Astronomy compared to other wavelengths. In this
context, the development of the first photoelectron-based X-ray polarimeter
[1] had a big impact to open a new window for X-ray astrophysics. This system
measures the X-ray polarization by reconstructing the initial direction of the
photoelectron produced by the interaction of an incident photon with an atomic
electron in a gas mixture (i.e., DME; Dimethyl ether) from the collected two-
dimensional ionization track. The first polarimeter built following this
concept, the Gas Pixel Detectors (GPD) [1, 2, 3], has been installed in the
CubeSat mission polarimeter light (PolarLight) [4] and is utilized as the
focal plane of the Imaging X-ray Polarimetry Explorer (IXPE) [5] that
significantly expanding this field [see e.g., 6].

Nonetheless, one of the major limitation of X-ray polarimeters is the dead
time, which restricts the effective area of the mirrors or concentrators to be
coupled with the detector. Therefore, in order to expand our perspective and
delve deeper into the universe with X-ray polarimetry, enhancing the
performance of the GPD is essential, as it will advance the sensitivity of
X-ray polarimetry. In this context, _the high yield polarimetry experiment in
X-rays_ (Hype-X) project aims to improve the sensitivity of the X-ray
polarimeter by combining the capabilities of the GPD thanks to recent
advancements in detectors with imaging and timing properties, and ultimately
developing the next generation of X-ray polarimetry detectors.

The application of the TIMEPIX3 [7] as the ASIC of a gas detector, which can
provide single electron sensitivity and 3D tracking, will allow to perform a
time-resolved X-ray polarimetry with virtually no dead time. The TIMEPIX3 can
measure simultaneously _time of arrival_ (TOA) and _time-over-threshold_ (TOT)
for each pixel, effectively providing charge and time measurement for three-
dimensional tracking reconstruction. This capability will not only improve
sensitivity by creating a three-dimensional image of the photoelectron track
but also they will allow it to sustain a high counting rate, enabling the use
of large effective areas for collecting a large number of photons thanks to
its parallel read-out system. In this regard, it is crucial to develop a new
three-dimensional track reconstruction algorithm for determining the initial
direction of the photoelectron based on the three-dimensional track image.
Therefore, in this paper, we aim to present our initial attempt at this new
algorithm for three-dimensional track reconstruction. The reduction of the
dead time can also be very effective on the sensitivity but this is case
dependent and is not the object of the present study.

In this work, we present a three-dimensional track reconstruction algorithm
and its application to ‘Geant4’ Monte Carlo simulation [8] to compare the
sensitivity for X-ray polarimetry transitioning from 2D to 3D track
reconstruction. In §2, we provide a brief introduction to the basic concept of
the current two-dimensional photoelectric X-ray polarimeter. Next, we
introduce our three-dimensional charge distribution reconstruction process and
initial attempt to determine polarization properties by establishing a new
three-dimensional moments track reconstruction algorithms in §3 and §4. In §5,
we present a comparison of X-ray polarization properties between two different
methods, aiming to quantify the prospective polarization improvement with the
new method by simulating a number of photoelectron track events in different
energy bands.

##  2 Basic concept of photoelectric X-ray polarimeter

Figure 1: Left: schematic view of a photoelectric X-ray polarimeter. The
incoming X-ray photon is absorbed in the gas mixture inside the detector and
creates a photoelectron track. The initial direction of this track, which is
expressed by $\theta$ and $\varphi$, is correlated with the electric field
direction of the absorbed photon. Right: histogram of emission angles of the
photoelectrons (_modulation curve_). In the case of unpolarized sources, the
distribution is flat, while polarized sources produce cosine square
modulation.

The Gas Pixel Detector analyzes the polarization information based on the
initial direction of the photoelectron produced by the interaction between the
X-ray photon and the gas mixture inside the detector. Figure 1 illustrates the
concept of a photoelectric polarimeter. In detail, when the X-ray photon
interacts with absorption material in the detector, a photoelectron is ejected
from the interacting atom. The angular distribution of the ejection direction
follows the photoelectric differential cross-section:

$\small{{\rm{d}\sigma_{Ph}}\over{\rm{d}\Omega}}\propto\cos^{2}\varphi{{{\sin}^{2}\theta}\over{\left(1-\beta{\cos\theta}\right)^{4}}}$
(2.1)

Where $\varphi$ and $\theta$ represent the azimuthal and polar angles of the
photoelectron, respectively, and $\beta$ is the photoelectron velocity in
units of the speed of light. The emission direction of the photoelectron is
most probably aligned along the direction of the incoming beam electric field.
The modest bending forward given by the term at the denominator at our
energies is almost negligible. In the subsequent step, the photoelectron
interacts in the gas mixture and induces an ionization track inside the
detector system until it loses all the energy and stops this process. Then,
the drift field from the window to the multiplication stage close to the ASIC
readout plane forces the track to move to the Gas Electron Multiplier and is
eventually collected by the ASIC CMOS chip, resulting in a two-dimensional
projected photoelectron track. However, the photoelectron interacting in the
gas mixture loses information about its initial emission during the subsequent
processes in the detector. Thereby, the current GPD system employs a
reconstruction algorithm based on the moment analysis proposed by [2, 9] to
measure the emission direction of the photoelectron and the absorption
position.

In order to examine the source polarization properties, it requires collecting
enough photoelectrons to build the so-called _modulation curve_ , that is, the
histogram of the photoelectron directions of emission (see left panel of
Figure 1). The modulation curve shows different shapes according to the
presence or absence of polarization of the incoming photons. For instance, a
polarized photon beam presents an anisotropic distribution according to the
photoelectron emission direction, and an unpolarized one shows a uniform
distribution. Hence, by the amplitude and phase of the modulation shown in the
curve, we can evaluate the source polarization properties. We first fit the
modulation curve with the function:

$\small{\mathcal{M}}\left(\varphi\right)={A}+B{\cos^{2}}\left(\varphi-\varphi_{0}\right),$
(2.2)

where $A$ express constant term and $B$ is the amplitude of the modulation
curve. Then, we calculate the polarization degree of the X-ray source (PD)
according to:

$\small{\rm PD}={{1}\over{\mu}}{{B}\over{2A+B}},$ (2.3)

where $\mu$ represents the modulation amplitude measured by the detector for
100 % polarization photon beam. In this regard, $\mu$ can be considered as a
scale factor to evaluate the polarization of the source. The polarization
angle (PA) is derived as the phase of a $\cos^{2}$ fit of the modulation
curve.

##  3 Reconstruction of 3-dimensional charge distribution from the projected
two-dimensional track image

Figure 2: Concept of three-dimensional charge distribution reproduction. The
yellow and blue dots indicate the original interaction points of the
photoelectron track. The blue conical shapes illustrate the Gaussian diffusion
process and the red hexagons depict the activated charged pixel on the ASIC in
two dimension space. The rightmost red track describes the final three-
dimensional charge distribution.

The outcomes of the next generation polarimeter will be a charge distribution
of the photoelectron track in three-dimensional space (X, Y, Z, $Q$). In this
work, we created the three-dimensional charge distribution by extending the
‘Geant4’ simulation [8]. The Geant4 simulation provides the original
interaction points coordinate of the track $(x_{i},y_{i},z_{i})$ and the two-
dimensional charge map (X, Y, $Q$ in Figure 2) following a Gaussian diffusion
process with a transverse diffusion coefficient ($\sigma_{t}=68\,{\rm\mu
m\,cm^{-1/2}}$; calculated with Garfield based on pure DME at 800 mbar)
expressed as:

$\small
f(x,y)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(x-\mu_{x})^{2}+(y-\mu_{y})^{2}}{2\sigma^{2}}\right],$
(3.1)

where $\sigma$ is $\sigma_{t}\sqrt{z_{i}}$. Hence, we reproduce the three-
dimensional charge distribution by tracing the contributions of each
interaction point on the two-dimensional charge map and assigning the averaged
$z_{i}$ to each pixel. Figure 2 illustrates the concept of this reproduction
process. In this process, we also calculated the standard deviation of $Z$
direction on each charged pixel ($\sigma_{i}$). This quantity is exploited in
the track reconstruction process described in §4.

##  4 3-dimensional moments track reconstruction algorithm

In order to determine the initial direction of a three-dimensional track, we
developed a new reconstruction algorithm, based on a two-step analysis
procedure, using a regression method and three-dimensional rotation. The basic
concept of this algorithm is to apply the standard two-dimensional moment
analysis method which was developed and applied by the IXPE collaboration [2,
9].

As a first step, we determine the regression plane of the track
($\texttt{Z}=A\texttt{X}+B\texttt{Y}+C$; shown as a red plane in the left
panel of Figure 3), which represents the averaged three-dimensional positions
of the track, by the least chi-square method according to:

$\small\chi^{2}=\sum_{i=1}^{n}{\left({\tt{Z}-A\tt{X}-B\tt{Y}-C\over\sigma_{i}}\right)^{2},\qquad\frac{\partial\chi^{2}}{\partial
A}=\frac{\partial\chi^{2}}{\partial B}=\frac{\partial\chi^{2}}{\partial
C}=0,}$ (4.1)

where X, Y, and Z are three dimension coordinates of each charged pixel.
$\sigma_{i}$ indicates the standard deviation into the Z direction of each
charged pixel (described in §3).

When the regression plane is determined, we projected the track onto it by
replacing Z of each charged pixel with $\texttt{Z}^{\prime}$. Then, we rotate
the XY plane for applying the two-dimensional moment analysis. The rotation
was conducted by constraining the rotation matrices expressed:

$\small R_{x}=\begin{bmatrix}1&0&0\\\ 0&\cos{\alpha}&-\sin{\alpha}\\\
0&\sin{\alpha}&\cos{\alpha}\\\ \end{bmatrix},\quad
R_{y}=\begin{bmatrix}\cos{\beta}&0&-\sin{\beta}\\\ 0&1&0\\\
\sin{\beta}&0&\cos{\beta}\\\ \end{bmatrix},\quad
R_{z}=\begin{bmatrix}\cos{\gamma}&\sin{\gamma}&0\\\
-\sin{\gamma}&\cos{\gamma}&0\\\ 0&0&1\\\ \end{bmatrix},$ (4.2)

where each rotation matrix ($R_{x}$, $R_{y}$, $R_{z}$) represents $\alpha$,
$\beta$, and $\gamma$ angle rotation around the corresponding axes $x$, $y$,
and $z$. These rotation matrices allow to convert the track properties into
the rotated from the (T′) and original frame (T) and vice-versa:

$\small{\bf R}=R_{x}\cdot R_{y}\cdot R_{z},\qquad T^{\prime}={\bf R}\cdot
T\leftrightarrow{\bf R}^{-1}T^{\prime}=T$ (4.3)

The angle for each rotation matrix was calculated with respect to each X and Y
axis ($\alpha$ and $\beta$ in Figure 3), however, here in the case of $R_{z}$
was constrained as an identity matrix because the rotation is performed to the
two-dimensional plane. Next, based on the two-dimensional moment analysis
method [2, 9], we find the principal axis of the initial direction on the
$\texttt{X}^{\prime}\texttt{Y}^{\prime}$ plane:

  1. 1.

Finding a barycenter $(x_{b},y_{b})$ of charge distribution based on the
measured $\texttt{X}^{\prime}$ and $\texttt{Y}^{\prime}$ position and each
charge density $Q$.

  2. 2.

Identifying the first principal axis $(\phi_{max})$ of the charge distribution
through a second moment analysis centered on the barycentric position.

  3. 3.

By analyzing the third moment, to distinguish the head and tail area of the
track from the Bragg peak at the end of the photoelectron path, where a large
amount of energy is collected (yellow region in Figure 3).

  4. 4.

Only considering the track starting area (white half-ring area opposite to the
yellow region in Figure 3), repeat steps 1 and 2 to estimate the absorbed
point $(x^{\prime}_{b},y^{\prime}_{b})$ of the incident X-ray photon (yellow
dot with red outline in right panel of Figure 3) and find the final principal
axis (red arrow line in right panel of Figure 3) of the initial direction
based on the absorbed point in three dimension space.

In step 3, the initial portion of the track is empirically determined as a
region that is $1.5-3.0$ times the length of the major axis of the ellipse
($\sqrt{M_{2}(\phi_{max})}$), so that we remove the Bragg peak region. In
addition, the absorption points $(x^{\prime}_{b},y^{\prime}_{b})$ measured by
the reconstruction process allow to reproduce the image of the observed
source. At this stage, the total charge density of the single track is
converted into the energy of the absorbed photon.

Lastly, we rotate the principal axis of the initial direction (red arrow line
in the right panel on Figure 3) back into the original frame (red solid line
in the left panel on Figure 3) and measure the $\theta$ and $\varphi$ of the
photoelectron in the three-dimensional space.

Figure 3: The three-dimensional photoelectron track reconstruction process.
Left: the three-dimensional charge distribution with the determined regression
plane. The yellow dot indicates the absorption point, and the red line
represents the principal axis of the initial direction. Right: a detailed view
of the two-dimensional moment analysis.

##  5 Sensitivity enhancement by the new 3-dimensional moments track
reconstruction

In order to quantify the improvement in the sensitivity to X-ray polarization
that can be obtained thanks to the 3D track reconstruction, we simulated 50000
events using the Geant4 simulation. For the simulation input, we considered
monochromatic sources, having the energies reported in Table 1, in a 1.5-8 keV
setting, with a PD of 100% and a PA of 45 degrees. We estimated the modulation
factor and determined the PA based on modulation curves for each case. The
obtained values are summarized in Table 1 and Figure 4 shows a comparison of
these values as a function of energy.

Table 1: The modulation factor and polarization angle are estimated at different energy levels in two-dimensional (2D) and three-dimensional (3D) reconstruction.  Energy ($\rm{keV}$) |  Modulation factor ($\mu$) |  PA (∘)
---|---|---
| 2D 3D | 2D 3D
1.5 |  0.022 $\pm$ 0.007 0.060 $\pm$ 0.008 |  55.00 $\pm$ 9.59 59.01 $\pm$ 3.63
2.0 |  0.112 $\pm$ 0.007 0.172 $\pm$ 0.010 |  44.69 $\pm$ 1.87 49.27 $\pm$ 1.67
3.0 |  0.291 $\pm$ 0.006 0.339 $\pm$ 0.008 |  45.26 $\pm$ 0.62 45.84 $\pm$ 0.64
4.0 |  0.368 $\pm$ 0.007 0.389 $\pm$ 0.007 |  45.26 $\pm$ 0.55 45.84 $\pm$ 0.52
5.0 |  0.411 $\pm$ 0.007 0.425 $\pm$ 0.006 |  44.69 $\pm$ 0.46 45.26 $\pm$ 0.39
6.0 |  0.451 $\pm$ 0.006 0.459 $\pm$ 0.007 |  45.26 $\pm$ 0.35 45.84 $\pm$ 0.39
7.0 |  0.488 $\pm$ 0.007 0.492 $\pm$ 0.009 |  44.69 $\pm$ 0.40 45.26 $\pm$ 0.47
8.0 |  0.518 $\pm$ 0.007 0.510 $\pm$ 0.007 |  45.26 $\pm$ 0.36 45.84 $\pm$ 0.37

Figure 4: Comparison of modulation factors (left, middle) and polarization
angles (right) between the two-dimensional (blue) and three-dimensional (red)
reconstruction processes as functions of energy. In the left panel, the black
solid line represents the measured modulation factor for the photoelectric
polarimeters on-board IXPE [10, 9, 11], while blue and red lines are obtained
by simulating either a 2D or a 3D photoelectron reconstruction. Middle panel
shows the modulation factor difference ($\Delta\mu$) between $\mu_{3D}$ and
$\mu_{2D}$ in percentage unit. In the right panel, the dashed line denotes the
simulation input value of the PA, which is 45 degree.

As a result, we found that the three-dimensional reconstruction exhibited
better performance compared to the current two-dimensional algorithm, with an
improved modulation factor of $\sim$5% (relative) in the 2-8 keV range and
$\sim$17% (relative) in the 2-4 keV range. This improvement, being the quantum
efficiency equal in the two methods, corresponds to the overall improvement of
the performances. This means, for instance, that we could reduce the exposure
time as $\sim$9.0 days in 2-8 keV and $\sim$7.3 days in 2-4 $\rm{keV}$ for 10
days observation achievement with the 2D algorithm. In particular, the
$\sim$17% improvement in the 2-4 keV range is physically equivalent to add an
additional telescope (a four-telescope system) to a hypothetical three-
telescope system like IXPE. However, we also note that the final angle
measurement deviates in the soft energy band, mainly outside the IXPE band,
when using the current three-dimensional reconstruction. This issue should be
addressed in the enhanced reconstruction algorithm in the future.

##  6 Conclusions

The three-dimensional reconstruction process of the photoelectron track allows
for an improvement of the performances, especially at low energies, where the
tracks are shorter and appear almost circular when projected onto 2D planes.
The three-dimensional reconstruction requires the knowledge of the z
coordinate, this can be obtained by timing information provided from the
TIMEPIX3 [7], thanks to its high time resolution. Currently, a collaboration
between INAF/IAPS and the University of Bonn is developing this next-
generation detector system based on this technology. Additionally, a new
Geant4 Monte Carlo simulation software is in development at INAF/IAPS to
contribute to modeling the response of this new detector.

## Acknowledgments

Acknowledgments. The authors acknowledge funding from the Minister of
University and Research PRIN 2020MZ884C “Hype-X: High Yield Polarimetry
Experiment in X-rays” and the INAF MiniGrant “New generation of 3D detectors
for X-ray polarimetry: simulation of performances”

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