# Response of the first POLAR-2 Prototype to Polarized Beams

Merlin Kole  Nicolas De Angelis  Ana Bacelj  Franck Cadoux  Agnieszka
Elwertowska  Johannes Hulsman  Hancheng Li  Grzegorz Łubian  Tomasz Kowalski
Gilles Koziol  Agnieszka Pollo  Nicolas Produit  Dominik Rybka  Adrien Stil
Jianchao Sun  Xin Wu  Kacper Zezuliński  Shuang-Nan Zhang

###### Abstract

POLAR-2 is a dedicated gamma-ray polarimeter currently foreseen to be launched
towards the China Space Station around 2027. The design of the detector is
based on the legacy of its predecessor mission POLAR which was launched in
2016. POLAR-2 aims to measure the polarization of the Gamma-ray Burst prompt
emission within the $30-800\,\mathrm{keV}$ energy range. Thanks to its high
sensitivity to gamma-ray polarization, as well as its large effective area,
POLAR-2 will provide the most precise measurements of this type to date. Such
measurements are key to improve our understanding of the astrophysical
processes responsible for Gamma-Ray Bursts. The detector consists of a
segmented array of plastic scintillator bars, each one of which is read out by
a Silicon PhotoMultiplier channel. The flight model of POLAR-2 will contain a
total of 6400 scintillators. These are divided into 100 groups of 64 bars
each, in so-called polarimeter modules. In recent years, the collaboration has
designed and produced the first prototypes of these polarimeter modules and
subjected these to space qualification tests. In addition, in April 2023, the
first of these modules were calibrated using fully polarized gamma-ray beams
at the European Synchrotron Radiation Facility (ESRF) in France. In this work,
we will present the results of this calibration campaign and compare these to
the simulated performance of the POLAR-2 modules. Potential improvements to
the design are also discussed. Finally, the measurements are used, in
combination with the verified simulation framework, to estimate the scientific
performance of the full POLAR-2 detector and compare it to its predecessor.

##  1 Introduction

Gamma–ray bursts (GRBs) were first discovered in 1967 and remain one of the
most researched topics in astrophysics. A GRB consists of a gamma–ray
component which lasts from seconds to minutes, called the prompt emission,
followed by a longer–lasting afterglow which can be observed over a wide range
of the electromagnetic spectrum. Thanks to over 50 years of research and the
detection of over 10’000 GRBs, much has been learned about these phenomena.
Insights into their origin were gained through precise measurements of their
location in the sky which indicate that they have an extra–galactic origin [1,
2, 3, 4]. Measurements of a second parameter of the gamma-rays, their time of
arrival, has led to measurements of the duration of GRBs. These in turn
allowed GRBs to be divided into two classes [5]: long and short GRBs. The
class of long GRBs is defined as those with a prompt emission lasting over 2
seconds. Strong evidence exists that such GRBs are the result of the death of
massive stars [6, 7]. During such events the prompt emission is theorized to
be emitted from within two highly relativistic jets powered by a central
engine. Subsequent interactions of the jets with the interstellar medium later
result in the afterglow emission. Short GRBs are defined as those GRBs which
have a prompt emission time below 2 seconds. Such GRBs, which typically also
have a harder energy spectrum compared to their longer counterparts, were
theorized to be the result of the merger of two compact objects such as binary
neutron stars [8]. Strong evidence supporting this theory was found in 2017,
through the joint detection of GRB 170817A with the gravitational wave (GW)
event GW170817 [9, 10]. Although all these measurements have provided
important insights into the progenitor system of GRBs, much still remains
unknown about these extreme phenomena. Especially questions regarding the
nature of the relativistic jets, such as their structure, the presence of
magnetic fields and how the gamma–rays are produced within them, remain poorly
understood. While significant efforts have been made to gain a deeper
understanding using measurements of the energy spectra, arrival time and
direction of the photons, little progress has been made in recent decades. To
break this stalemate, measurements of the gamma–ray polarization have been
proposed in the last decades. Such measurements are theorized to be able to
answer many open questions regarding GRBs [11].

In this paper we will first provide a brief overview of the field of gamma–ray
polarimetry. This is followed in section 3 by an overview of the POLAR-2
detector design with a particular focus on the polarimeter modules and how
these are used to measure the polarization of the incoming photons. The
POLAR-2 simulation framework will be introduced in section 4. Section 5 will
cover the calibration campaign performed at ESRF including an overview of the
measurement and simulation results. Based on this outcome, suggestions for
improvements to the detector will be presented in section 6 and finally, the
foreseen performance of the full detector will be presented in section 7.

##  2 Gamma-Ray Polarimetry

The prompt emission of GRBs peaks in the $10-1000\,\mathrm{keV}$ energy range.
Within this energy range, the polarization of the photons can be deduced by
letting the photons Compton scatter in the detector and measuring their
azimuthal Compton scattering angle. This is possible as a dependence of the
azimuthal Compton scattering angle on the polarization vector of the incoming
photon exists. This can be seen in the Klein-Nishina equation:

$\frac{d\sigma}{d\Omega}=\frac{r_{o}^{2}}{2}\frac{E^{\prime
2}}{E^{2}}\left(\frac{E^{\prime}}{E}+\frac{E}{E^{\prime}}-2\sin^{2}\theta\cos^{2}\phi\right).$
(2.1)

where $r_{0}=e^{2}/m_{e}c^{2}$ is the classical electron radius with $e$ the
elementary charge, $E$ is the initial photon energy and $E^{\prime}$ the
photon energy of the outgoing photon. $\theta$ is the polar scattering angle
and finally $\phi$ is the azimuthal angle between the polarization vector of
the incoming photon and the velocity vector of the outgoing photon in the
plane normal to the momentum vector of the photon. This is illustrated in
figure 1 which also indicates how the dependency between the polarization and
the azimuthal scattering angle can be used in a segmented detector array. In
such a detector, an incoming photon can scatter in one detector element and
then undergo a second interaction, which can be photo-absorption or Compton
scattering, in a second detector segment. The relative position between the
two elements can be used to measure the scattering angle. It should be noted
that the $\cos^{2}$ term in the Compton scattering cross-section results in a
$180^{\circ}$ symmetry. This means that it is not important to identify which
one of the interactions was the first one. However, in case of more than 2
interactions in the detector, the order of the interactions is not
known111unless one has extremely high timing precision but this is not
realistic for space missions. As a result it is not clear which interaction
locations should be used to calculate the scattering angle. In addition, any
photon which is directly absorbed can also not be used for polarimetry, nor
can any photon which leaves the detector without a second interaction. As a
result, the efficiency of gamma–ray polarimetry is typically low, thereby
giving rise to the first difficulty in this field.

For photons where the angle is measured, the inferred scattering angle can be
entered in a histogram to produce a scattering angle distribution, often
referred to as a modulation curve. As can be seen from the Klein-Nishina
equation, when using a perfect detector, the modulation curve will show a
$180^{\circ}$ degree period for a $100\%$ polarized photon flux and will be
flat when the incoming flux is unpolarized. The relative amplitude of this
modulation, $\mu$, is therefore linearly related to the polarization degree
(PD). The polarization angle (PA) can be extracted from the phase of the
modulation as the location of the minimum in the curve.

Extracting the PD from the scattering angle distribution also requires the
amplitude of the $180^{\circ}$ modulation for a $100\%$ polarized beam, here
called $\mu_{100}$. Using this, the PD can be extracted using:

$PD=\frac{\mu}{\mu_{100}}$ (2.2)

It should be noted that $\mu_{100}$ typically has a strong dependence on the
incoming photon energy as well as on their incoming angle. Therefore the value
of $\mu_{100}$ needs to be known for each energy and incoming angle. As it is
not realistic to measure the value of $\mu_{100}$ for the full parameter
space, this is typically derived through Monte Carlo simulations. Thereby
making polarimetry highly reliant on simulations and making detailed
validation of the used simulation framework vital for a reliable analysis.

It is important to note that the above description is only valid when one uses
a perfect detector. In a real detector, the size, shape and uniformity of the
sensitivity of the various detector elements, will induce systematic effects
in the scattering angle distribution. As a result, a real scattering angle
distribution will show a $180^{\circ}$ modulation induced by the polarization,
as well as instrument-induced non-harmonic functions and harmonic functions
which can have periods of $180^{\circ}$ as well. The latter can be mistaken
for polarization, while the other effects will also affect the accuracy of
deducing the polarization from the scattering angle distribution using a
simple fit.

As is discussed in detail in [12] there are two analysis methods to get around
this. In the first method, the instrumental effects are removed by dividing
the measured scattering angle distribution by the one produced using an
unpolarized flux. In the second, the instrumental effects are included in the
instrument response and the measured distribution is fitted using forward
folding. One can, in theory, produce such an instrument response, or the
scattering angle distribution for an unpolarized flux, from calibration
measurements. However, realistically, one cannot produce this for each
possible incoming photon energy and incoming angle. As a result, the
production of the response of the polarimeter needs to be produced using Monte
Carlo (MC) simulations. As the analysis of gamma–ray polarimetry depends
strongly on these MC simulations, any issues in these simulations will result
in systematic errors in the measured PD. Systematic errors resulting from
imperfections in the MC simulations are a second source of difficulty in the
field of gamma–ray polarimetry [11, 12].

Figure 1: An illustration of how the polarization of an incoming photon can be
measured using a segmented detector array. Left: Schematic of a photon with
incoming energy E, scatters with a polar angle $\theta$ and azimuthal angle
$\phi$ before continuing with energy E’. The angle $\eta$, defined as the
angle between the two detector segments with an energy deposition can be
measured. Middle: A photon with energy E interacting in the detector segment
located at the origin. The colors of the surrounding segments indicate the
probability for a second interaction to take place there based on the
polarization vector $\vec{p}$ (darker color indicates higher probability).
Below this the typical distribution of $\eta$ is shown for this scenario.
Right: The same as the middle but with $\vec{p}$ along the y-axis.

Although highly complex, polarization measurements of the GRB prompt emission
have been performed. A detailed overview of all the measurements published up
to 2022 can be found in [11]. The most precise measurements to date were
produced by the POLAR mission. POLAR was a Chinese-European mission dedicated
to measuring the polarization of the GRB prompt emission in the
$50-500\,\mathrm{keV}$ energy range [13]. This instrument was launched as part
of the second Chinese spacelab, the Tiangong-2 in September 2016. It operated
successfully from October 2016 until April 2017 at which point a problem in
its high-voltage power supply did not allow for further scientific data
taking. During the 6 months it operated POLAR detected a total of at least 55
GRBs, of which 14 were bright enough to allow for polarization measurements.
The results of GRB polarimetry of POLAR, published in [14, 15], indicate that
the polarization is typically below $40\%$ when integrating over the full
prompt emission period. In addition, time-resolved analysis was performed
during which hints were found that the PA varies rapidly during the emission,
thereby smearing out the true PD value [16]. Finally, energy-resolved
polarization studies were performed, however, no significant dependency on the
energy was found [17]. Apart from studying GRBs, the data from POLAR was also
used to perform spectral polarimetric measurements of the Crab pulsar [18].

The measurement results from POLAR agree with the majority of the GRB emission
models as these predict low polarization levels. Only models containing
synchrotron emission from a toroidal magentic fields predict polarization
degrees up to $50\%$ and therefore appear to disagree with the POLAR data
[11]. The low levels of PD currently predicted by the majority of the models,
as well as the potentially varying PA, imply that significantly more precise
measurements are required for GRBs. In addition, although the polarization
measurements of the Crab pulsar did agree with previously published results,
the large statistical errors did not allow to fully probe the emission
mechanism [18]. For these purposes the POLAR-2 project was initiated in 2018.

##  3 The POLAR-2 Detector

Figure 2: An exploded view of the POLAR-2 payload [19]. The sensitive part of
the payload consists of 100 polarimeter modules which can be seen below the
carbon fiber cover. Below the 100 polarimeter modules, which are placed in an
aluminum grid, two additional aluminum grids that house the low-voltage power
supply and the back-end electronics can be seen.

###  3.1 Polarimeter Module: Mechanical Design

The design of the POLAR-2 instrument is shown in figure 2 while that of a
polarimeter module is shown in figure 3. The photo-detection area of the
module consists of 64 plastic scintillator bars with dimensions of $5.9\times
5.9\times 125\,\mathrm{mm^{3}}$. The material of the final design will be
EJ-248M by Eljen [20], however, for the polarimeter module tested at ESRF the
plastic was EJ-200 [21]. The EJ-200 material used here was selected based on
its high light yield compared to EJ-248M with 10 optical photons/keV for
EJ-200 compared to 9.2 optical photons/keV for EJ-248M. This higher efficiency
remains when convolving the emission spectrum of these scintillators, which
both peak at 425 nm, with the photo-detection efficiency vs wavelength of the
Silicon PhotoMultiplier (SiPM) which reads it out [19]. However, recent
studies, which are discussed in detail in [19, 22], indicate that, due to the
smoother surface quality achievable for EJ-248M, the number of photons which
reach the bottom of the scintillator is higher for EJ-248M, despite the
difference in light yield. This effect was additionally verified during the
tests described here by also irradiating a polarimeter module containing
EJ-248M. Detailed comparisons of the performance, apart from the light yield
comparison, were not possible as the electronics connected to the second
polarimeter module were damaged prior to the test, resulting in several
malfunctioning channels.

The scintillators are held in place using a plastic, 3D-printed grid, which
ensures each scintillator is placed directly on top of a SiPM channel with a
precision of 50 $\mu m$. Each scintillator is individually wrapped in
reflective foils. To accomplish this, the first 4 sheets of Vikuiti by 3M with
a thickness of 65 $\mu$m, also called Enhanced Specular Reflector film (ESR),
[23] are used to cover the 4 sides of the scintillator, the combination is
then wrapped in a layer of Claryl of 12 $\mu m$ [24] which provides a second
optical insulation layer. An additional layer of Vikuiti, is placed at the top
of the scintillators to increase the light collection further. Above this
layer there is a 1 mm thick layer of sorbothane, a material which combines the
properties of rubber, silicone and elastic polymers, purchased from a
commercial vendor (IsolateIT), and a carbon fiber plate of 0.75 mm thick which
together serve to provide pressure to the top of the scintillators to improve
their coupling to the SiPMs. The SiPM readout consists of 4 arrays of
S13361-6075NE-04 produced by Hamamatsu [25], each of which contains 16
channels with a size of $6\times 6\,\mathrm{mm^{2}}$. The SiPMs were selected
based on their high photo-detection efficiency (PDE), as high as $50\%$, and
the possibility to get these in large area arrays. It should be noted that the
newer, S14 family of Hamamatsu is also available. However, the S13 was
selected based on the shorter decay time of the pulses of $\approx
100\,\mathrm{ns}$ compared to $\approx 600\,\mathrm{ns}$ for the S14. More
details on the SiPM selection are presented in [19]. Between the SiPM and the
scintillators there is a 250 $\mu m$ thick layer of MAPSIL [26, 27] which
serves both as an optical light guide and, together with the sorbothane layer
at the top, to protect the SiPM from damage during the vibration and shock
sustained during the launch. The resistance of this design against vibrations
and shocks was successfully tested during a dedicated test in 2022 [19]. The
SiPM array is mounted on an aluminium and plastic frame which holds the
electronics, as shown in figure 3. This mechanics is designed to thermally
insulate the SiPM arrays from the electronics, thereby keeping their
temperature lower. During the tests discussed here, the SiPMs were operating
at room temperature (measured in the middle of the array) while the
electronics is around 35∘C as measured using a temperature sensor embedded in
the electronics.

The combination of the electronics and the scintillator array and its
surrounding components are all placed inside a carbon fiber socket with a
thickness of 0.75 mm. This socket provides the mechanical support for all the
components inside and ensures the system is light tight. The electronics frame
is finally supported from the bottom by an aluminium end-cap which provides
pressure to the electronics frame and couples to the surrounding mechanics.
For these tests, this consisted of an aluminium frame which is mounted on a
translation table as described in section 5.

Figure 3: An exploded view of a POLAR-2 polarimeter module [19].

###  3.2 Electronics Design

The 64 detector channels of each polarimeter module are individually read out
using a front-end electronics board placed below the SiPM array and
scintillator array. The readout consists of 1 FPGA, 2 CITIROC-1A ASICs (each
with 32 channels), 4 ADCs, and a DC/DC converter (LT3482 by Linear Technology)
used for providing the bias voltage to the SiPMs.

The LT3482 is controlled from the FPGA, an IGLOOv2 by Microsemi, and supplies
a common high voltage to the 64 SiPM channels. It is however known that the
breakdown voltages of each channel vary. Therefore, the breakdown of each
channel was measured prior to the detector assembly at different temperatures.
It was found that the breakdown voltages vary within 0.1 V within one detector
module. The CITIROC ASICs used for the SiPM readout can be used to correct for
these variations using a DAC value which can be applied to the input from each
channel on the ASIC. Using this DAC value the voltage over each channel can be
modified by up to 2.5 V, thereby ensuring each channel is provided with an
equal over-voltage. During the tests discussed here this over-voltage was set
to be 3.0 V.

The signal from each SiPM channel are directly fed into one of the two ASICs
which handle the amplification and discrimination. In order to avoid a clear
$180^{\circ}$ asymmetry in the detector, it was decided to alternate the ASIC
used for each subsequent channel, as a result one ASIC reads out all odd
numbered channels and the other all even numbered ones. The channel numbering
convention is shown in figure 4. The ASICs are capable of analyzing the
incoming signal either using a peak sensing algorithm as well as through a
sampling and hold. Based on preliminary tests it was found that for the
typically short signals from the combination of plastic scintillators and
SiPMs, the peak sensing algorithm works well and was therefore selected.
Within the ASIC, the peak sensing is applied to the signal produced by two
amplifiers, one with a high gain and one with a low gain. The integration time
is $\approx 100\,\mathrm{ns}$ for each channel. In the case of POLAR-2 the
high gain will be used to measure the deposited energy in the
$0-100\,\mathrm{keV}$ range, while that of the low gain is used to measure
from approximately $50-800\,\mathrm{keV}$. In the case of POLAR-2 the signal
from the high gain output is used in the ASIC trigger. Two separate
discriminators are available for each of the 32 channels to give additional
flexibility. Following the convention of the
supplier222https://www.weeroc.com/my-weeroc/download-
center/citiroc-1a/89-citiroc1a-datasheet-v2-53/file the first discriminator is
referred to as the charge discriminator, the second the time discriminator.
While the time trigger can be used to provide detailed timing information for
each of the individual 32 channels. The output from the 32 charge
discriminators are coupled in an OR output only. For POLAR-2 the two available
discriminators are used to provide two separate trigger signals. In the final
POLAR-2 logic the ’charge threshold’ is set higher than that of the ’time
trigger’ to give flexibility in the trigger logic.

Figure 4: The channel numbering convention of the POLAR-2 polarimeter module
as seen from the top. The scattering angle definition is also indicated on the
left. The red dotted line indicates the path scanned by the synchrotron beam
during an irradiation scan with the beam starting below channel 0 and ending
below channel 7.

The trigger patterns from the ASICs are provided to the FPGA which uses this
to make a trigger decision. In the case of the ESRF test, only the charge
trigger was used and all channels were read out if at least one channel is
above the charge trigger threshold. In the logic of the final polarimeter the
time trigger will be set at a lower voltage, and a readout of the module is
performed when at least 2 channels are above this threshold level within a 100
ns time period. This short period ensures that contamination of the Compton
scattering signal by chance coincidence events within the detector stays
negligible. Although this functionality was present at the ESRF test, the two
thresholds were set at equal values in order to simplify the readout during
this first irradiation campaign.

In case the FPGA starts a readout, the output from both the high gain and low
gain amplifier of each of the 64 channels is provided to the ADCs. Each ASIC
has 2 ADCs, one for the high gain output and one for the low gain. Therefore,
a total of 128 ADC values are provided to the FPGA for each trigger. In
addition, $2\times 64$ booleans are provided containing the charge and the
time trigger information of each channel.

It should be noted that, although both the high gain and low gain output from
each channel are read out for each triggered event, the ADC values for all
channels which did not issue a charge trigger is typically 0. This is due to a
setting on the ASIC which, by default, only starts the peak sensing on the
input signal if a charge trigger is issued by this channel. If not, a baseline
value is simply provided. It was found during the analysis phase of this
campaign that this causes significant issues for the detector calibration and
therefore this logic has since been updated in the POLAR-2 firmware, as will
be discussed in section 6.

During this irradiation campaign no prototype of the POLAR-2 back-end
electronics was used. Instead, a General Purpose Input Output (GPIO) board
produced by the electronics group of the DPNC department of the University of
Geneva was used to communicate with the front-end electronics. The GPIO allows
to configure the FPGA on the front-end and handles the data transmission from
LVDS lines. The GPIO in turn provides the data to a desktop which, during the
ESRF campaign, was placed outside of the hutch (the irradiation room with the
beam at ESRF).

###  3.3 Analysis Pipeline

The data provided by the polarimeter module has to undergo several steps in
order to convert the ADC output provided by the ASICs into keV. In this
section the various steps of the analysis pipeline which is applied to the
measured, as well as the simulation data, are presented.

####  3.3.1 Pedestal Subtraction

The left side of figure 5 shows the typical 12-bit ADC spectrum from the high
gain output taken with a $60\,\mathrm{keV}$ beam, while the right side shows
that produced by the low gain. A clear shift in the baseline can be seen for
both; a result of both the pedestal of the ADC as well as that from the ASIC
output. The first step in the analysis procedure consists of subtracting this
pedestal value from each of the 64 channels both for the high gain and the low
gain.

In order to subtract this value the exact pedestal values need to be measured.
For the data analyzed here these measurements were provided by forcing a
readout of each channel while no high voltage was provided to the SiPM. This
results in 2 distributions of ADC values for each channel (one for the high
gain and one for the low gain output) which are well described by a normal
distribution. The mean of this represents the pedestal value and the width,
typically around 10 ADC, the electronic noise on this channel. The fit results
of the mean are used for pedestal subtraction, while both the mean and the
width are used in the simulations which will be described in the next section.
The spectrum of figure 5 after pedestal subtraction is shown in figure 6.

Figure 5: Left: The raw ADC spectrum from the high gain amplifier as measured
for a 60 keV photon beam. The photo-peak can be seen around 2500 ADC while the
trigger threshold (charge threshold) can be seen around 800 ADC. Right:. The
raw ADC spectrum from the low gain amplifier as measured for a 60 keV photon
beam. The same features as in 5 can be observed however, due to the lower gain
the ADC values are compressed. Figure 6: The raw ADC spectrum from the high
gain amplifier as measured for a 60 keV photon beam after pedestal
subtraction. The triggered events are shown in red, while the non triggering
events are shown in blue.

####  3.3.2 Non-linearity Correction

In order to understand the correlation between the ADC values from the high
gain and the low gain the two can be plotted against one another for each
channel. The result can be seen in figure 7. Although the correlation between
the two is linear at low ADC values, when the ADC value in the high gain
output exceeds $\approx\,2500$ ADC a non-linear behaviour is observed, this is
likely a result of saturation of the high gain amplifier. This behaviour has
been reported by other users of the CITIROC ASIC [28], and was also seen with
the POLAR-2 front-end when injecting external charges directly into the ASIC.

Figure 7: The pedestal subtracted output of the high gain vs the pedestal
subtracted output of the low gain. A clear non-linear behaviour can be
observed when the high gain ADC value starts to exceed 2500 ADC. The fit of a
non-linear function is shown in red, while a blue dotted line indicates above
which high gain ADC value the linear correction is applied.

To ensure a better linearity of the high gain ADC value, its value is
corrected in the analysis using the low gain ADC value. For this purpose, the
linear relation was measured at low ADC values and parameterized using a
first-order polynomial. For high gain ADC values which exceed 2000 ADC the
corrected value is calculated based on the linear parameters from the fit and
the low gain ADC value. In addition, in order to include this effect in the
simulations, the correlation at values exceeding $\approx\,2000$ ADC was
fitted using an error function. The linear parameters are used in the data
analysis part to correct the high gain values when these exceed 2500 ADC. In
case the correction exceeds 4095 ADC, it is fixed at this level. An example of
the correlation between the two ADC values after correction is shown in figure
8.

Figure 8: Equal to figure 7 but after applying a correction to the high gain
ADC.

####  3.3.3 Optical Crosstalk

One of the major improvements in POLAR-2, compared to POLAR, regards to the
reduction of optical crosstalk. In the design of POLAR optical crosstalk
between different detector channels within a detector module was primarily a
result of the use of the Multi-Anode Photo-Multiplier Tube (MAPMT) for the
readout. In the MAPMT the 64 channels share one borosilicate window. This
implies that optical photons from one scintillator bar can enter the anodes of
a neighbouring channel, thereby producing a significant amount of crosstalk. A
second issue of the use of a MAPMT is that the window is fragile, which for
space instrumentation prompts the use of a damper material between the
scintillators and the window. This damper material, which in the case of POLAR
was a 3 mm thick layer of MAPSIL [13], induces additional optical crosstalk.
Finally, the optical insulation between scintillators was achieved using a
single layer of Vikuiti by 3M, which although highly efficient, will result in
a small amount of additional crosstalk. This, however, in the case of POLAR is
negligible compared to the optical crosstalk induced by the window and the
damper. As a result, the typical optical crosstalk from one channel to its
direct neighbor (for example, channel 0 to 1) was between $10$ and $20\%$
[29]. Crosstalk to the diagonal neighbors (for example from channel 0 to 9)
was around $5\%$ while that to the second neighbors (for example from channel
0 to 2) was of the order of $1\%$.

For POLAR-2 the use of a SiPM significantly reduces the amount of expected
optical crosstalk. The issue of a thick shared entrance window is removed,
while the less fragile nature of the SiPM significantly reduces the need for a
damper material. Although no thick damper is required, some kind of material
which serves both as a damper and an optical coupling between the
scintillators and the SiPM is required. For this purpose a $250\,\mu m$ thick
MAPSIL layer is molded directly on top of the SiPM [19]. It should be noted
that the MAPSIL layer is not optically insulated on the outside, as this is
mechanically complex to do for an elastic material. This means some light,
approximately $1\%$ will be lost on the outside of the module. Finally, as the
two main sources of optical crosstalk are significantly reduced the third,
related to photons penetrating the optical insulator materials, was further
addressed in this design. Compared to POLAR, where one layer of Vikuiti was
placed between scintillators, for POLAR-2 each scintillator bar is wrapped
covered on all sides with a sheet of Vikuiti, this is then further wrapped in
a layer of Claryl. As a result there are now 4 reflective layers between each
scintillator compared to 1 in POLAR.

The expected level of optical crosstalk with this setup was simulated and is
addressed in detail in a separate paper [22]. The simulations predict only
significant crosstalk to the direct neighbors which is of the level of
$2-3\%$.

Due to the aforementioned issue where peak sensing is not enabled for channels
which do not issue a charge trigger, only energy depositions above this
threshold, at $\approx 8\,\mathrm{keV}$, are recorded correctly. As a result
of this, along with the low levels of crosstalk, the ESRF data could not be
used to perform accurate measurements of the crosstalk. Instead, the crosstalk
was measured after a firmware update, which will be described in section 6,
using the same polarimeter module and the same settings as employed during the
ESRF beam test.

To measure the optical crosstalk the polarimeter module was irradiated with a
${}^{137}Cs$ source which emits 662 keV photons. The data was then processed
using the above steps with pedestal and non-linearity correction. Using the
data in this format, the ADC value in channel x was plotted against that of
channel y, with the only cut that channel y must issue a charge trigger. The
resulting 2d histograms show, in case of neighboring bars, a linear
distribution, which can be fitted with a first order polynomial which is
forced to have an offset of 0.

It should be noted at this point that the correlation found using the above
method is not purely the optical crosstalk as it is measured in ADC and the
two channels do not have the same gain. As is described in detail in [30], the
conversion of energy depositions to electrical signals in the polarimeter
module can be described using several matrices. We start with a vector
$\vec{E}_{\rm dep}$ which contains the energy depositions in the 64
scintillators. The number of optical photons which reach the bottom of a
scintillator for a channel can be described as $N_{\rm bar}=s\,c\,E_{\rm dep}$
with $s$ the scintillation efficiency and $c$ the photon collection
efficiency. As the scintillators here are all produced by the same
manufacturer we can assume that $s$ is equal for all channels. For the full
polarimeter module we can therefore write:

$\vec{N}_{\rm bar}=B\vec{E}_{\rm dep}$ (3.1)

with
$B=sDiag(c1,c2,\textperiodcentered\textperiodcentered\textperiodcentered,c64)$.
We now have to additionally take into account that due to optical crosstalk,
an energy deposition in channel x can also result in optical photons reaching
the bottom of bar y. For this we introduce matrix $X$ which contains the
crosstalk between all the channels. Here $x_{ij}$ is the crosstalk from
channel i to channel j, with $x_{ij}$ being a value between 0 and 1, and
$x_{ij}=x_{ji}$. The number of optical photons reaching the SiPM channel j
from an energy deposition in channel i is then $N_{\rm SiPM}=x_{ij}N_{\rm
bar}$, so:

$\vec{N}_{\rm SiPM}=X\vec{N}_{\rm bar}$ (3.2)

Finally, the number of optical photons which reach a channel x is transformed
into an ADC value in this channel through the gain of the SiPM and the ASIC
channel. The total gain can be described as $g$ and we can calculate the
measured energy $\vec{E}_{\rm meas}$ as:

$\vec{E}_{\rm meas}=G\vec{N}_{\rm SiPM}$ (3.3)

Where $G$ is a diagonal matrix consisting of elements $g_{ii}$. The complete
path to convert the deposited energy to the measured energy can now be written
as:

$\vec{E}_{\rm meas}=(GXB)\vec{E}_{\rm dep}=R\vec{E}_{\rm dep}$ (3.4)

The crosstalk measured using the method above is the ratio between the
energies measured in ADC, so $\frac{E_{j}^{\rm meas}}{E_{i}^{\rm meas}}$. It
contains contributions from the matrices $B$, $G$ and $X$. We can assume that
the variations in the elements in $B$ are minimal, as each scintillator is
produced in the same way. We then state that what is measured using our linear
fit is:

$f_{ij}=\frac{E_{j}^{\rm meas}}{E_{i}^{\rm meas}}=\frac{r_{ji}E_{i}^{\rm
dep}}{r_{ii}E_{i}^{\rm dep}}=\frac{g_{j}x_{ji}}{g_{i}x_{ii}}$ (3.5)

So, for example, we can see that whereas we would expect the true optical
crosstalk to be symmetric, so $x_{ij}=x_{ji}$, this is not the case for the
ratio in the measured energies contained in the elements $f_{ij}$ which make
up the matrix $F$.

Finally, what is required for the analysis here is the matrix $R$, such that
one can perform the conversion:

$\vec{E}_{\rm dep}=R^{-1}\vec{E}_{\rm meas}$ (3.6)

As is again described in detail in [30], we can retrieve matrix $R$ by
measuring the matrix $F$, through our linear fits which produce elements
$f_{ij}$ as well as a diagonal matrix $M$ which contains, the elements
$r_{ii}$ of matrix R. The diagonal elements can be understood as the ADC to
keV conversion for a single channel. These elements can be measured using an
${}^{241}{\rm Am}$ source which emits 59.5 keV photons resulting in a photo-
peak in each channel. An example of this is shown later in this paper in
figure 33 which shows a clear photo-peak around 2000 ADC. By fitting this peak
the value of $r_{ii}$ (as measured in ADC / keV) is retrieved for each
channel. Finally, using the equations above, we can now retrieve matrix R
using:

$R=F^{T}M$ (3.7)

We can then also calculate the matrix $X$ containing the pure optical cross
talk. An example of this is presented on the left in figure 9. A histogram
showing the optical crosstalk to the direct neighbors for each channel is
shown in figure 9. The histogram can be seen to peak between $2$ and $3\%$,
the results therefore match the values predicted in [22].

Figure 9: Left: The optical crosstalk matrix (in which we have set elements
$x_{ii}$ to 0 for visualization purposes) produced using the above described
method. It can be seen that, apart from a few small outliers, the cross talk
is prominent from channels x to channels $x\pm 1$ and $x\pm 8$ which can be
understood when looking at the channel numbering convention. Right: A
histogram showing the amount of corrected optical crosstalk as measured to the
direct neighbors of each channel (so typically 4 entries per channel).

As we now have the correct crosstalk as well as the gain for each channel, we
can produce the matrix $R$. Then finally, we can apply the correction for the
gain and the crosstalk to convert the pedestal subtracted ADC value to the
energy in keV using:

$\vec{E}_{\rm dep}=R^{-1}\vec{E}_{\rm meas}$ (3.8)

An example of the spectrum from irradiation by a 60 keV beam after this
correction is shown in figure 10. The photo–peak can be seen to be correctly
placed at 60 keV.

Figure 10: The energy spectrum as measured when irradiated by a 60 keV beam
for a typical detector channel after pedestal subtraction, non-linearity
correction, as well as the crosstalk and gain corrections.

##  4 Simulations

The simulation software developed for POLAR-2 consists of 2 steps. The
physical interactions of the gamma–rays in the detector are simulated using
Geant4 [31]. The output of Geant4 is then passed on to a digitization Monte
Carlo (MC) which handles the conversion of the deposited energies to an ADC
signal. The final result is a data file which has the same format as the real
data prior to pedestal subtraction.

The decision to divide the simulation into two steps was taken to simplify the
production of detector responses in the future. This is because the detector
response changes due to factors such as bias voltage settings and detector
temperature. As such effects are only taken into account in the digitization,
the Geant4 part will only have to be performed once, while this output can
then be used to produce the detector responses for various instrument
configurations. Both steps are presented in detail below.

###  4.1 Geant4

The Geant4 part of the simulation includes a detailed model of the POLAR-2
detector. For the purpose of the ESRF analysis only a single polarimeter
module was simulated placed inside of an aluminium grid which can hold 9
polarimeter modules. The module as simulated is shown in figure 11. This mass
model was implemented in Geant4 using standard Geant4 objects rather than
using a GDML input. Although this approach is more time consuming to develop,
it makes for a human readable code which can be debugged. The simulated module
includes all the components present in figure 3 including the plastic
scintillators, the damping materials and carbon fiber plate above it, the
carbon fiber sockets, the plastic grids which hold the scintillators in place,
the optical pad, and the SiPMs. A simplified version of the front-end
electronics and their support structure is also present, although, for the
purpose of these simulations these will not have a significant effect on the
results. It was decided not to include the reflective foils (Vikuiti and
Claryl) in the simulations after initial tests showed these have no
significant effect on the results. As they do slow down the simulations
significantly, it was decided to remove these from the simulation geometry.

Figure 11: The mass model of a POLAR-2 polarimeter module as implemented in
Geant4. The scintillators are shown in blue held by a plastic grid at the
bottom. The carbon fiber housing is made transparent here to show the
contents. Below the scintillators the thin SiPM array is shown in brown with
below this the mechanical support structure of the electronics. Finally the
brown cylinder is a copper bar which serves to extract heat from the SiPM
array.

The incoming particles are simulated using the General Particle Source (GPS)
function which allows for a large flexibility. The physical interactions of
the photons inside of the detector is handled using the
G4EmLivermorePolarizedPhysics list. This physics list was previously found to
handle the Compton scattering, including the polarization, correctly [32]. As
the conversion of the energy deposition to the electrical signal is handled in
the digitization, only the amount of energy deposited, the detector channel
where this occurred and the position of the interaction within the
scintillator are stored. The latter is stored in case a dependency of the
light yield on this position is found in the analysis. It should be noted that
one correction to the deposited energy is already applied within Geant4. This
regards the Birks’ effect [33] where a Birks’ constant of
$0.143\,\mathrm{mm/MeV}$ is used, as taken from [34].

For each primary photon the Birks’ corrected energy depositions, the channel
number and the position within the scintillator are stored within a ROOT file
which is passed to the digitizer.

###  4.2 Digitization

The ROOT file produced by Geant4 is read-in by the digitizer along with
several configuration files which contain information on the various
properties of each channel. For each primary photon the following steps are
performed:

  * •

For each energy deposition ($E$) the deposited energy is converted to a number
of optical photons. This conversion is done using the light yield (LY)
provided in the data sheet for the EJ-200 material which is 10.0 optical
photons / keV. It should be noted that in the operational temperature range
applicable for these tests (room temperature) and its operation in space
($-0^{\circ}C$) the light yield does not change for this material [21]. To
take into account the statistical fluctuations the value of E/LY is used as
the $\lambda$ parameter in a Poisson distribution.

  * •

The light collection efficiency, defined as the fraction of the produced
optical photons which reach the SiPM entrance window, is taken into account by
multiplying the total number of produced photons by a factor of $0.30$ which
is based on optical simulations performed of the system [19]. In the case of
POLAR the number of photons which reached the PMT was dependent on where these
photons are produced in the bar. However, for POLAR-2, optical simulations did
not show such a dependence, which was also verified by measurements which will
be presented in section 5.3. This difference can be attributed to the uniform
shape of the scintillator bars in POLAR-2 compared to those used in POLAR
which were tapered. The tapering causes a dependence on the light yield for
interactions in these tapered areas. The uniform shape of the bars used in
POLAR-2, along with the uniform wrapping and their small size compared to the
optical absorption length of the scintillator material (380 cm) [21], allows
for their position independent light yield. A detailed discussion can be found
in [19].

  * •

Before converting the optical photons into photo-electrons in the SiPM the
optical crosstalk has to be applied. For this purpose the true optical
crosstalk, called $x_{ij}$ before, is required. In order to calculate this,
the matrix $F$, as presented in section 3.3.3, is used along with the gain
matrix. The matrix $F$ is provided along with the gain for each channel as an
input configuration file. The matrix $X$ is calculated once, at the start of
the simulations, and subsequently applied to distribute the number of produced
optical photons to the 64 scintillators. The number of optical photons which
reaches channel j from channel i is calculated by multiplying the total number
of optical photons by $x_{ij}$ and using this as the $\lambda$ in a Poisson
distribution.

  * •

The optical photons which reach the SiPM are now converted into photo-
electrons. For this purpose the number of photo-electrons is picked from a
Poisson distribution centered around the number of optical photons multiplied
by the photo-detection efficiency of the SiPM. For the Hamamatsu SiPMs used
here, for the applied bias voltage, this value is 0.5.

  * •

Subsequently, the intrinsic SiPM crosstalk is added to the number of photo-
electrons for each channel. The intrinsic crosstalk is defined as the
probability for a photo-electron to induce a second photo-electron. For the
type of SiPMs used here this is approximately $15\%$. The probability for $N$
photo-electrons to be added to the signal is $0.15^{N}$. To take this into
account a random number is taken between 0 and 1 and compared to $0.15^{N}$ to
pick the value of $N$.

  * •

The number of photo-electrons can now be converted to an ADC value. For this
purpose a configuration file containing, for each channel, the distance
between the photo-electron peaks ($\rm P.E._{dist}$) and the width of each
peak, is read in to provide these values. This value allows to calculate the
mean ADC value for the given number photo-electrons which is used as the mean
of a normal distribution. The width of the photo-electron peak,
$\sigma_{N_{\rm P.E.}}$, is described as:

$\sigma_{N_{\rm P.E.}}=\sqrt{\sigma_{1}^{2}N_{\rm P.E.}+\sigma_{e}^{2}}$ (4.1)

where $N_{\rm P.E.}$ is the number of photo-electrons, $\sigma_{e}$ is the
electronic contribution to the width and, $\sigma_{1}$ the gain contribution.
More details on this parameterization can be found in [35]. The ADC value is
calculated by picking a random number from a normal distribution centred
around the number of photo-electrons multiplied by the $\rm P.E._{dist}$ and
with a width of $\sigma_{N_{\rm P.E.}}$. The values of $\rm P.E._{dist}$,
$\sigma_{1}$ and $\sigma_{e}$ are produced by fitting, in the data from each
channel, the first 4 p.e. peaks using the sum of 4 normal distributions with a
fixed distance $\rm P.E._{dist}$ and a width of $\sqrt{\sigma_{1}^{2}N_{\rm
P.E.}+\sigma_{e}^{2}}$. The typical values of $\rm P.E._{dist}$, $\sigma_{1}$
and $\sigma_{e}$ are $\approx 40,\,2$ and $4\,\mathrm{ADC}$ respectively. It
should be noted that the approach used here simulates a fully linear
correlation between the number of p.e. and the ADC value. While the
measurements presented here are within the linear response part of the SiPM,
some non-linear behaviour is seen in the ASIC. This is taken in account by one
of the subsequent steps in the digitization.

  * •

The ADC value calculated in the previous step is that for the high gain
output. Data from a configuration file which contains the gain non-linearity
parameters mentioned in section 3.3.2 are then used to calculate the low gain
output. For this the linear parameters which correlate the two ADC values are
used.

  * •

The high gain ADC value is now corrected to account for the non-linear gain by
using the non-linear parameters (those which describe an error function).

  * •

Electronic noise is now added to each channel. The level of this noise is
based on a measurement of the width of the baseline for each channel. The
noise added is taken from a normal distribution centred around 0 with a width
taken from this configuration file.

  * •

The ADC values (in the high gain) are now used to produce the charge triggers
(as well as the time triggers although these are not relevant in the ESRF
analysis). The threshold values are again measured from the real data. For
this purpose the measured threshold position along with the width of this
threshold is read-in from another configuration file. These values are
measured by taking the ratio of the spectrum with only triggered events over
the spectrum with all events. Such a distribution can be fitted using an error
function, from which the mean position and the width of the threshold can be
extracted. This procedure was previously applied successfully in [32].

  * •

The ADC values undergo one final modification by adding the pedestal, which,
again is taken from a configuration file based on measurement data.

  * •

Finally, the trigger logic is applied where the event is kept only if at least
one channel exceeds the charge threshold or 2 channels for the time threshold.

  * •

To reproduce the ESRF data accurately one additional step is applied, where
the ADC values for all channels which do not exceed the charge trigger
threshold are picked from a normal distribution centred around 0 and a width
equal to the electronic noise. This is done to reproduce the issue found with
the peak sensing mentioned earlier. For future simulations this step can be
removed.

After the above steps ADC values (both high gain and low gain) as well as
trigger information are available for each channel. These are stored in ROOT
files which follow the data format of the measured data, thereby allowing this
data to pass through the same analysis chain. An example of a normalized
measured and simulated spectrum (from a 60 keV beam) from one channel is shown
in figure 12. This figure shows the simulated data (in blue) directly after
the digitization, as a result the baseline is not yet subtracted and the
output is in ADC. It should be noted that the measured spectrum has some
energy depositions at high ADC values. These are a result of cosmic rays
interacting in the detector during the data taking (which took several
minutes). Figure 13 shows the same spectrum, again for measurements and
simulations, but now after both the data sets have passed through the analysis
chain described in the previous section. Finally, figure 14 shows the
simulated and measured spectrum after passing through the analysis chain but
for a 40 keV beam. It can be observed that the simulation results match the
measurements well. For example the energy resolution at both 40 and 60 keV
match the data while, unlike a commonly used approach, the energy resolution
is not given as a parameter in the simulations. Rather this energy resolution
results naturally by simulating the various steps, such as the production of
optical photons and their conversion into photo-electrons, in the digitization
process. Due to the lack of data below the threshold the photo-electron peaks
are not clearly visible in these spectra. However, as will be discussed in
section 6, updates to the firmware implemented after the ESRF campaign have
allowed to make these more clearly visible. As will also be discussed in that
section through comparison with measurement data, the photo-electron peaks and
their widths are accurately simulated.

Figure 12: A simulated spectrum (blue) resulting from the digitization
software compared to the measured raw data from the same detector channel
(red) as resulting from a 60 keV beam. Figure 13: The same as figure 12 but
after non-linearity, crosstalk and gain corrections. Figure 14: The same as
figure 13 but for a 40 keV beam.

##  5 ESRF Campaign

This section will start with an overview of the setup used during the ESRF
campaign. The data taken there [36] will be described in detail in the
subsequent subsections.

###  5.1 Irradiation Setup

For the purpose of this calibration campaign the polarimeter module was placed
in an aluminium frame which can support a total of 9 modules. This frame is a
scaled down version of the final grid which will be used in the full detector,
capable of holding 100 modules. This grid, which can be seen in figure 15, was
connected to a custom made aluminium piece which connects it to a translation
table. This translation table is capable of moving the detector within the
plane perpendicular to the incoming beam direction. In addition, it can be
used to rotate the detector around the x-axis (the definition of which is
indicated in figure 15). The data from the polarimeter module is provided by a
dedicated board developed by the University of Geneva which can be seen at the
bottom of figure 15 which also provides the power to the polarimeter module
electronics. This board in turn provides the data to a PC which is placed
outside of the experimental hutch through a single ethernet cable. The
operation of the translation table, as well as the control of the beam is
performed using a Python based interface provided by ESRF.

The beamline ID15A of ESRF was used for this test, details on the beam can be
found in [37]. The synchrotron beam is collimated to a size of 1 mm diameter,
and can be adjusted in the energy range of 40 to 140 keV. During the tests
performed here, irradiations were performed at beam energies of 40 and 60 keV
as well as 100 and 120 keV. As will be discussed in section 6, issues induced
by the high rate experienced at 100 and 120 keV were found during the analysis
phase which means only the data from 40 and 60 keV is analyzed here. It should
be noted here that although the energy resolution of the beam is below $1\%$
[37], third level harmonics resulting from the beam manipulation are present
for both 40 and 60 keV irradiation. As these harmonics are of the level of
$1-2\%$ they were measured prior to the irradiation campaign using a Germanium
detector provided by ESRF. In addition, the polarization of the beam is, based
on simulations provided by the ID15A beamline team, above $99\%$ and is along
the y-axis in figure 15.

Figure 15: The setup as used at the ESRF beamline along with the definitions
used for the x, y and z axis. The polarization of the beam is always along the
y-axis. A single polarimeter module is placed within a scaled down version of
the module grid (a $3\times 3$ grid instead of $10\times 10$ as will be used
in the full detector). The readout electronics are placed behind the
mechanical grid along with a cold plate used to keep the setup at around room
temperature. All this is placed on an x-y translator capable of moving the
detector with respect to the beam position.

As a uniform irradiation of the full detector surface cannot be achieved with
the beam directly, the detector is moved with respect to the beam during
irradiation. Each detector scan consists of moving the beam over the 8 rows of
scintillator as indicated in figure 4. In the setup shown in figure 15,
channel 0 is located at the bottom (x=0) and in the positive y direction,
channel 56 is located in the same x location in the negative y direction.

To ensure the beam irradiates each scintillator for an equal amount of time,
the movement in the y direction starts 10 mm outside of the module and ends 10
mm away from the opposite side. When reaching this point a movement in the x
position is initiated to the centre of the scintillators in the row above,
before moving backwards in the negative y direction. This is repeated until
all 8 rows are scanned. The detector is moved at 1 mm per second, as a result
an on-axis scan takes approximately 10 minutes. As the beam intensity, which
is linearly proportional to the electron flux in the synchrotron ring varies
by several $\%$ during this period, the electron flux is stored in the data.
This allows to correct for fluctuations in the beam intensity during the scan.
Finally, it should be noted that the synchrotron beam has to be refilled every
few hours, thereby producing a sudden increase of several percent in beam
intensity. Although one can correct for this, it was decided to avoid taking
data during these sudden changes in beam intensity. To achieve this, the time
before a refill is automatically monitored in our scanning software which
operates the table movement. During each translation in the x-direction (so in
between the scans of the rows), it is checked if a refill will occur during
the scan of the next row. If so, the start of the y-translation is paused
until after the refill has occurred.

Off-axis scans are performed by rotating the detector around the x-axis, where
the rotation angle is referred to as $\theta$. To accommodate for the larger
detector surface the translation in the y-direction is increased for such
scans. Finally, in order to change the polarization angle, the polarimeter
module is rotated by 90 degrees around the z-axis.

###  5.2 Light Yield

Using the data analysis pipeline discussed in section 3.3 the spectra of all
64 channels inside the polarimeter module are processed. The spectrum, after
pedestal and non-linearity correction, from a 60 keV irradiation can be fitted
using a Gaussian function. This provides the gain as measured in ADC/keV. In
addition, the photo-electron peaks, or fingers, can be fitted in order to
extract the distance, as measured in ADC, between them. This second fit
provides the ADC/photo-electron ratio. Using the combination of these two
ratios one can finally calculate the photo-electrons/keV ratio, which gives an
indication of the detection efficiency or light yield detection. It should
finally be noted that, as explained for the optical crosstalk, the true gain
cannot directly be measured as it is affected by the crosstalk. We therefore
finally correct for the optical crosstalk to produce the true light yield for
each channel. The results for this measurement are shown below in figure 16.
While the uncertainty from the fit of the photo-peak is of the order of $1\%$,
the fit on the distance between the fingers has a more significant error of
$\approx 10\%$. Overall, the systematic error on each entry in the
distribution is therefore around $10\%$, indicating that some intrinsic
differences between the channels do exist. Such differences are, for example,
the differences in the quality of the scintillators and their optical coupling
to the SiPM.

The light yield detection presented here is measured using a 60 keV beam. It
is important to note that the Birks effect remains non-negligible at these
energies. At higher energies, the light yield detection will therefore be
approximately $10\%$ higher.

Using optical simulations presented in [19] the light yield detection,
including the optical crosstalk, was predicted both for a polarimeter module
with EJ-200 and for one with EJ-248M. The results in figure 16 are for one
with EJ-200, which based on the predictions from [19] is expected to be333The
values provided in [19] are for higher energies and therefore need to be
corrected for the Birks effect, resulting in decrease of $\approx 10\%$. They
also already include the optical crosstalk which is corrected for in this work
and result in an opposite correction of $10\%$. $1.31\pm 0.16$ photo-
electrons/keV. Measurements with a second polarimeter module (which contained
several damaged SiPM channels and was therefore not used for polarization
analysis) which uses EJ-248M as the scintillating material are shown in figure
17. The light yield can be seen to be $\approx 1.5$ photo-electrons/keV which
again matches the simulations presented in [19] that expects $1.43\pm 0.15$
photo-electrons/keV. The larger spread in these results compared to that in
figure 16 can be attributed to the lower quality of this polarimeter module.
As mentioned several SiPM channels were damaged and could therefore not be
used, while others contained minor scratches. As a result, both the intrinsic
differences between the channels are larger, while in addition, the typical
errors on the various fits required to produce the light yield are larger.
Overall, we do however, see an improvement in the light yield consistent with
simulations, although it is clear that more detailed tests are required in the
future.

To put this number into comparison, for POLAR, this value was only $0.3$
photo-electrons/keV. The large difference between these two can mainly be
attributed to the difference in the scintillator shape, where the POLAR
scintillators were tapered towards the bottom to a size of $5\times
5\,\mathrm{mm^{2}}$ in order to reduce optical crosstalk. Despite this, a
second improvement comes still from the significant reduction in the optical
crosstalk, which in case of a POLAR detector, induced a loss which could
exceed $50\%$ of the light to neighboring channels. For POLAR-2 this has been
reduced to $\approx 10\%$. Finally the increase in the photo-detection
efficiency of the detector of $50\%$ against $20\%$ for POLAR, further
improves this.

The importance of this value, in the case of a SiPM based detector, can be
explained in the form of the dark noise. As SiPMs suffer from significant dark
noise, the trigger threshold for a channel should exceed a certain number of
photo-electrons, or fingers, in order to reduce the dark noise induced trigger
rate. The number of photo-electrons produced for a dark noise event scales
with the intrinsic crosstalk of the SiPM, which in the case of the SiPMs used
here is of the order of $10\%$. As a result, $10\%$ of the dark noise induced
events have more than 1 photo-electron, while a fraction of $0.10^{N+1}$
exceed N photo-electrons. With a typical dark noise of 1 MHz per channel,
reducing this to 1 Hz correlates to a trigger threshold exceeding 7 photo-
electrons. Using the typical light yield of a POLAR-2 channel, this translates
to approximately444The value of 5 keV used here is calculated using a linear
correlation between p.e. and keV. However, at low energies the Birks’ effect
should be taken into account. The real threshold for energy deposition is
therefore $\approx 6\,\mathrm{keV}$. 5 keV. For the measurements performed
during the tests here, the threshold was conservatively set to 8 keV, thereby
making the dark noise contribution negligible during these measurements.

Figure 16: The distribution of the light yield as measured for the polarimeter
module which uses EJ-200 as the scintillating material. The distribution is
fitted with a Gaussian and the mean and width are presented. Figure 17: The
distribution of the light yield as measured for the polarimeter module which
uses EJ-248M as the scintillating material. The distribution is fitted with a
Gaussian and the mean and width are presented.

###  5.3 Position Dependent Light Yield

The results presented in section 5.2 were acquired by using a 60 keV beam and
irradiating the scintillator bars from the top. In the case of an off-axis
irradiation the distribution of the photo-absorption position in the bar would
be different from that of the on-axis case. In case a dependence between the
interaction position and the light yield exists, the results presented in
section 5.2 would only be valid for an on-axis beam. In order to understand
the response of the POLAR-2 instrument, where GRBs are very unlikely to come
directly from the zenith, it is therefore vital to know whether the light
yield depends on the interaction position within a scintillator bar.

While optical simulations presented in [22] indicate that the dependence is
minimal, this needs to be verified using measurements. For this purpose we use
the data from one of the off-axis irradiation runs where the polarimeter
module was tilted with an angle of $30^{\circ}$ with respect to the 60 keV
beam. During such a scan, the beam irradiates the majority of the bars for a
short period, while the bars on one of the outer edges are all irradiated for
a total of $\approx 65$ seconds. This is illustrated in figure 18, which shows
the scintillators, as in figure 4 but tilted along the x-axis. During the 65
seconds the beam also scans the side of scintillator channels 56 to 63.
Therefore producing a high probability for photo-absorption to occur at the
top during one part of the scan, and a high probability for this to occur at
the bottom during a different part. A dependence of the light yield on the
interaction position would result in a shift in the photo-absorption peak.

Figure 18: The same as figure 4 but now tilted along the x-axis to indicate an
off-axis scan. The black dotted start and stop lines correspond to when the
beam moves in and out of the scintillators used for studying the position
dependent light yield.

To study this we use the data from channel 56 from a scan where the beam
enters the top of the bar at around 60 seconds (indicated as the black start
line in figure 18) and leaves the bottom of the bar at around 125 seconds
(indicated as the Stop line in figure 18). The mean position of the photopeak
is shown as a function of time in figure 19. The two red lines indicate when
the beam enters and leaves the bar. It can be observed that while the beam is
not pointing at the bar the photo-peak is slightly below 60 keV. This can be
explained by the fact that an incoming photon cannot deposit its full 60 keV
photon in this bar as it first needs to scatter in a neighboring bar, thereby
losing some of its energy. While the beam is irradiating the scintillator bar
directly we see that all measurements are compatible with 60 keV and we do not
see any trend in the position of the peak position. Similar results were found
for all the 8 bars studied in this way for this scan. The results therefore
support the simulations presented in [22] and we will continue to assume that
the light yield in a bar does not depend on the interaction position.

Figure 19: The mean position of the photo-absorption peak as a function of
time during an off-axis scan of one channel with 60 keV photons. The time at
which the beam enters the bar is indicated with the red line around 60
seconds, while the moment it leaves is indicated with the second red line.

###  5.4 On-Axis Modulation

Using both the scanned data, as well as the simulated one, scattering angle
distributions can be produced for various energies. As not every triggered
event contains energy depositions in two or more scintillator bars an event
selection has to be applied before. In the study performed here this event
selection is kept simple. The scattering angle, measured as the angle from the
x-axis as defined in figure 4, is calculated if at least two energy
depositions are found where both have issued a charge trigger. The requirement
for a charge trigger is needed as, as discussed before, the peak sensing on
the ASIC was not initiated for channels in which no charge trigger is issued.
As a result, the ADC values in these channels are not correlated to actual
energy depositions. The scattering angle is calculated from the bar with the
second highest energy deposition to the bar with the highest energy
deposition. It should be noted that the order used here is arbitrary as
inverting it will only produce a phase shift of $180^{\circ}$. Furthermore, to
calculate the angle, a random position is selected within the two scintillator
bars. This is required to account for the lack of information on the
interaction position within the bars and to avoid a discrete set of scattering
angles to appear in the scattering angle distribution.

The selection applied here is the most simple one, while future studies can be
used to optimize the selection to increase, for example, the $\mu_{100}$ or
the number of selected events. Such an optimization study will need to be
applied on the full detector, so with 100 instead of 1 module. We therefore
restrict ourselves here to the simple analysis with the main purpose of
validating that the instrument is indeed sensitive to polarization and, more
importantly, to validate the MC simulation software.

It should be stressed that, before applying the event selection on both the
simulated and measured data, both types of data go through the same analysis
pipeline which includes pedestal subtraction, non-linearity correction, cross
talk and gain correction. Two minor corrections are however applied to the
measured data which are not applied to the simulated data. The first of these
is the application of a cut on the data to only include events recorded while
the beam was pointing at the detector module. Although, in normal
circumstances only a fraction of the data is recorded when the beam is not
irradiating the detector, during some scans the synchrotron electrons were
refilled during a scan. The motor operating algorithm was written such that
during such a refill of the synchrotron ring, which typically lasts several
minutes, the beam was moved out of the detector while this happened. As a
result, some scans include several minutes of additional background data which
is removed from the analysis through this cut.

The second is a minor correction to account for variations in the beam
intensity during irradiation. During a typical scan of the detector the beam
intensity varies by a few $\%$ as the synchrotron electron beam slowly
decreases in intensity. The intensity of the beam was recorded in the house-
keeping data and each event entered in the scattering angle distribution is
weighted by the relative beam intensity, thereby ensuring a fully uniform
irradiation of the detector surface.

####  5.4.1 Modulation for a 60 keV beam

Figures 20 and 21 show the normalized measured and simulated scattering angle
distributions resulting from a 60 keV beam with orthogonal polarization
angles. The polarization of the beam is, in both cases along the Y-axis of
figure 15. As the module is rotated in between scan (along the z-axis), for
figure 20 the x-axis, as defined in the polarimeter module, is along the
y-axis of the setup, while in figure 21 the polarimeter module is rotated such
that the x-axes of the two systems are aligned. Both of the scattering angle
distributions show a clear $180^{\circ}$ modulation where the phase is shifted
by $90^{\circ}$ as expected. In addition to the polarization-induced effect,
clear instrumental effects can be seen. These become easier to spot when we
combine the two curves. By combining two polarized beams with an orthogonal
polarization an unpolarized beam is created. This can be illustrated through
Stokes parameters for a polarized beam of intensity ’I’ added to one with
equal intensity but orthogonal polarization:

${\displaystyle{\begin{pmatrix}I\\\ I\\\ 0\\\
0\end{pmatrix}}+{\begin{pmatrix}I\\\ -I\\\ 0\\\
0\end{pmatrix}}={\begin{pmatrix}I\\\ 0\\\ 0\\\ 0\end{pmatrix}}}$ (5.1)

Therefore by summing and using a weighting correction for the beam intensities
a fully unpolarized beam response can be produced. The result, both from
simulations (blue) and measurements (red) can be seen in figure 22. The figure
shows clear peaks at every $90^{\circ}$ resulting from the square geometry of
the detector module and the scintillator bars. In addition, it can be observed
that not all of the peaks have the same height, resulting from a non-
uniformity in the sensitivity of the various detector channels. These come,
for example, from the difference in the gain and threshold of each channel.
Although these effects indicate that the detector does not have a fully
uniform response, it is important to note that the non-uniformity is also seen
in the simulated distribution, indicating that these are modeled correctly in
the simulations.

Figure 20: The normalized scattering angle distributions produced using
simulations (blue) and measurements (red) for a 60 keV beam with PD = $100\%$
and PA=$0^{\circ}$ (polarization vector along the x-axis of the polarimeter
module). Figure 21: The normalized scattering angle distributions produced
using simulations (blue) and measurements (red) for a 60 keV beam with PD =
$100\%$ and PA=$90^{\circ}$ (the polarization vector aligned with the y-axis
of the module). Figure 22: The normalized scattering angle distributions
produced using simulations (blue) and measurements (red) for a 60 keV beam
with PD = $0\%$.

The unpolarized response can now also be used to remove the instrumental
effects thereby leaving only the polarization-induced effects. An example from
measurements is shown in figure 23. The resulting distributions, fitted with a
$180^{\circ}$ harmonic function, represent the response of the detector to a
$100\%$ polarized beam and can therefore be used to extract the $\mu_{100}$.
The values of $\mu_{100}$ from measurement were found to be $(23.9\pm
0.9+0.2)\%$ for PA = $90^{\circ}$ and $(24.6\pm 1.2+0.2)\%$ for PA =
$0^{\circ}$. The second, asymmetric error here stems from the uncertainty on
the beam polarization which is known to be > $99\%$. From simulations, we find
$(24.3\pm 2.0)\%$ and $(26.0\pm 2.4)\%$ respectively. The simulations
therefore match the measurements within the error.

Figure 23: The normalized corrected scattering angle distributions from
measurements for a 60 keV beam with PD = $100\%$ and PA = $90^{\circ}$. The
$\mu_{100}$ extracted from here is $(23.9\pm 0.9)\%$.

While the data taken for this campaign can be corrected directly using other
measured data to extract the value of $\mu_{100}$, this will not be the case
for science data. The scattering angle distributions measured from, for
example a GRB, will be the result of a complex spectral shape which, for
practical reasons, cannot be reproduced in the lab for each measured GRB. The
correction using the unpolarized scattering angle distribution therefore
relies on the production of such a curve using simulations. Inaccurate
simulations will firstly not remove all the instrumental effects from the
modulation curve, therefore not producing a $180^{\circ}$ modulation. Secondly
the amplitude of this modulation can be wrong, therefore producing an
inaccurate measurement of the polarization of the GRB. We therefore
additionally tested the correction of the measured data using an unpolarized
scattering angle distribution produced using the simulations. The results for
$E=60\,\mathrm{keV}$, PD = $100\%$ and PA = $90^{\circ}$ are shown in figure
24. It can be seen that the result accurately reproduces the $\mu_{100}$ with
a value of $(23.0\pm{0.9})\%$.

It should finally be noted that although the beam is calibrated to produce a
$60\,\mathrm{keV}$ beam with a FWHM of $\approx 0.5\,\mathrm{keV}$, some
contamination from third order harmonics is present in the beam. For the 60
keV beam the relative strength of this $180\,\mathrm{keV}$ line was about
$3\%$ and was included in the simulations.

Figure 24: The normalized measured scattering angle distributions corrected
using an unpolarized scattering angle distribution produced using simulations
for measurements for a 60 keV beam with PD = $100\%$ and PA = $0^{\circ}$. The
$\mu_{100}$ extracted from here is $23.0\pm 0.9\%$.

####  5.4.2 Modulation for a 40 keV beam

Figures 25 and 26 show the measured and simulated scattering angle
distributions resulting from a 40 keV beam with orthogonal polarization
angles. The unpolarized response, produced again by summing the orthogonal
polarization results with a weighting factor to account for beam intensities,
is shown in figure 27. Again the simulated responses can be seen to match the
measured ones well. The presence of a third order harmonic was taken into
account in the simulations. From the figures we can clearly see that the
relative effect of the polarization is significantly smaller than at
$60\,\mathrm{keV}$. This is reflected in the $\mu_{100}$ which were calculated
to be $(14.7\pm 0.9+0.1)\%$ and $(13.9\pm 1.1+0.1)\%$ for PA = $0^{\circ}$ and
PA = $90^{\circ}$ from measurements. From the simulations we find $(15.9\pm
1.3)\%$ and $(15.0\pm 1.2)\%$ for these values of PA. The results are
therefore again in agreement. The results are therefore again in agreement. An
overview of all the results is presented in table 1.

Energy (keV) |  $\mu_{100}$ Meas. PA$=0^{\circ}$ |  $\mu_{100}$ Sim. PA$=0^{\circ}$ |  $\mu_{100}$ Meas. PA$=90^{\circ}$ |  $\mu_{100}$ Sim. PA$=90^{\circ}$
---|---|---|---|---
40 | $(14.7\pm 0.9+0.1)\%$ | $(15.9\pm 1.3)\%$ | $(13.9\pm 1.1+0.1)\%$ | $(15.0\pm 1.2)\%$
60 | $(24.6\pm 1.2+0.2)\%$ | $(24.3\pm 2.0)\%$ | $(23.9\pm 0.9+0.2)\%$ | $(26.0\pm 2.4)\%$
Table 1: Overview table of the on-axis results. Figure 25: The normalized
scattering angle distributions produced using simulations (blue) and
measurements (red) for a 40 keV beam with PD = $100\%$ and PA = $0^{\circ}$ .
Figure 26: The normalized scattering angle distributions produced using
simulations (blue) and measurements (red) for a 40 keV beam with PD = $100\%$
and PA = $90^{\circ}$ . Figure 27: The normalized scattering angle
distributions produced using simulations (blue) and measurements (red) for a
40 keV beam with PD = $0\%$.

###  5.5 Off-Axis Modulation

Due to the transient nature of GRBs, POLAR-2 needs to observe a large portion
of the sky in order to have a larger chance of detecting GRBs. As the detector
response changes for GRBs entering the detector off-axis it is important to
also study the scattering angle distribution for such scenarios.

It should be noted here that it is not possible to produce an unpolarized beam
for off-axis measurements. This, as changing the PA requires rotating the
detector around the beam axis. As for an off-axis measurement the detector is
additionally rotated along the x-axis, reproducing an unpolarized beam by
summing two orthogonally polarized beams is not possible here. Therefore, to
reproduce the $\mu_{100}$ here one relies on the simulations, which, was shown
in section 5.4.1 to be accurate.

Both for 40 and 60 keV, off-axis scans were performed at angles of
$15^{\circ}$ and $30^{\circ}$ where the detector was rotated along the x-axis.
As a result, the scanning distance in the y direction was increased to
compensate for the larger area seen by the beam. Channels 56 to 63 were
therefore irradiated directly for a longer time, allowing to produce the
results presented in section 5.3.

For the POLAR detector, both in calibration data and from measurements in-
orbit and simulations, see for example [12], it was found that the modulation
curve changed significantly when changing the incoming angle of the photons.
Here we only observe minor differences which do not result in a significant
change in $\mu_{100}$. The scattering angle distributions as measured using a
60 keV beam for on-axis, $15^{\circ}$ and $30^{\circ}$ off-axis are shown
together from measurements in figure 28. When studying the ratio between the
on-axis observation and the $30^{\circ}$ off-axis case the differences can be
seen more clearly, see figure 29. It can be observed that when moving off-
axis, scattering at angles of $0^{\circ}$ and $180^{\circ}$ becomes less
likely. In addition, scattering at $-90^{\circ}$, meaning for example from bar
56 in the direction of 0, becomes more likely as can be expected with this
rotation. Using simulations we can simulate a system where the detector is
instead rotated around the y-axis and with a perpendicular PA. This is shown
in figure 30, the same effects can be seen here.

For 40 keV irradiation the difference was found to be even less significant
and as a result the values of $\mu_{100}$ did not change significantly. Only
when increasing the energy of the incoming photons does the off-axis effect
become significant, this was tested with simulations of a $120\,\mathrm{keV}$
beam for 0 and $30^{\circ}$ off-axis. The ratio of the modulation curves for
both is shown in figure 31, where the same trend as in figure 29 can be seen
but significantly more pronounced.

The small difference between the off-axis scattering angle distributions at
low energies can be explained by the small energies deposited for photons
scattering at small polar angles. For an on-axis beam photons scattering at
small polar angles will have a large probability of interacting a second time
in the same scintillator, thereby not allowing to measure their scattering
angle and thus not contributing to the scattering angle distribution. When
moving off-axis such photons will likely scatter to a different scintillator,
and thereby contribute to the scattering angle distribution. It is such
photons which would result in a difference in the scattering angle
distribution. However, for low incoming photon energies, such at at 40 keV,
the energy depositions for such interactions are, mostly, below the trigger
threshold so they remain undetected. Therefore, the scattering angle
distributions for on-axis and off-axis remain largely similar at 40 keV. When
increasing the energy to 120 keV, small polar scattering angles are instead
detected and therefore contribute to the off-axis scattering angle
distribution while, for the on-axis case the large probability to interact in
the same scintillator still does not allow for the calculation of a scattering
angle. As a result, the scattering angle distributions at larger energies do
start to differ more significantly.

Figure 28: The measured scattering angle distributions produced for a 60 keV
beam with PD = $100\%$ and PA = $90^{\circ}$ using an on-axis beam (blue) and
$\theta=15^{\circ}$ off-axis beam (red) and $\theta=30^{\circ}$ off-axis beam
(purple). Figure 29: The ratio of the measured scattering angle distributions
for on-axis and $30^{\circ}$ off-axis beam with an energy of 60 keV. A clear
increase at both $90^{\circ}$ (for example from channel 0 to 56) and
$-90^{\circ}$ (for example from channel 56 to 0) can be observed when moving
off-axis. Figure 30: The simulated scattering angle distributions produced for
a 60 keV beam with PD = $100\%$ and PA = $0^{\circ}$ using an on-axis beam
(blue), $15^{\circ}$ off-axis beam (red) and a $30^{\circ}$ off-axis beam
(purple). Figure 31: The ratio of the simulated scattering angle distributions
measured for on-axis and off-axis ($30^{\circ}$) for a 120 keV beam.

##  6 Required Improvements

As discussed before, the DAQ used in the detector module during this
irradiation campaign is not yet finalized and was found to not be optimal yet.
Three main issues with the DAQ were identified:

  1. 1.

Peak sensing was only enabled for detector channels which issues a charge
trigger.

  2. 2.

Too high data rates caused timing issues in the data readout.

  3. 3.

Inaccurate measurement of the dead time.

The last of these was due to issues in the firmware where the dead time of the
electronics for each trigger was not accurately measured. This means that,
with the high data rates experienced at ESRF, no correction for variations in
the dead time of the detector could be applied to the analysis. Although the
changes in the beam intensity were included in the analysis, variations in the
dead time had to be ignored. Such variations could occur due to small
differences in the threshold of the various channels. Due to such issues
additional systematic effects could appear in the modulation curves. However,
as no significant deviations from the simulated curves were seen this issue is
likely negligible. For future measurements this issue should be resolved to
not only confirm that no systematic errors are induced in the modulation
curve, but also to accurately measure the fluences of GRBs. In future beam
tests of the detector, the flux of the incoming beam should be measured
accurately to verify an accurate calculation of the dead time. This will, as
could not be done here, allow to calibrate the effective area for polarization
of the detector as well.

The second of these issues was apparent during data taking with beams of 100
keV and 120 keV. The issue can be seen in figure 32. Although this histogram
only shows events where a charge trigger was issued, many events can be seen
below the trigger threshold. This is, potentially, a result of data packages
being corrupted during the high data taking rate ($\approx 1800$ triggers /
second) experienced with the beam. During the 60 keV irradiation this did not
cause significant issues, as the corrupted packets make up only a very minor
fraction of the data as can be seen in figure 10. Although the spectra taken
at the higher energies looked good upon initial inspection, a more detailed
study of this data found that a significant amount of the data was corrupted
leading to a significant amount of events which appear to trigger below the
threshold. This data was therefore not analyzed in greater detail.

Figure 32: A spectrum as measured by one channel during a scan with a
$120\,\mathrm{keV}$ beam. Only events which issued a charge trigger are shown,
it can be observed that a significant amount of events can be seen far below
the trigger threshold which is around 280 ADC.

The issues found at high trigger rates have now been mitigated through several
improvements in the front-end firmware which allow for a faster data through
put. This can be seen in figure 33 where data was taken with $\approx 2000$
triggers per second using an ${}^{241}Am$ source which produces primarily 59.5
keV photons. The spectrum shows a clear photo–peak, but more importantly shows
no significant amount of triggers below the threshold.

Figure 33: A spectrum from one detector channel taken with a polarimeter
module using an ${}^{241}Am$ source after firmware updates which mitigate the
issues found at ESRF. The inset shows more details of the 0 to 500 ADC range.
The triggered events are shown in red while the non triggered are shown in
blue. A clear separation is seen here. In addition, clear photo-electron peaks
can be seen below the threshold, thereby indicating that the peak sensing is
performed properly here as well.

The final issue in the data from ESRF was that peak sensing was not enabled in
the ASIC for channels which did not issue a charge trigger. As a result, the
ADC values for such channels were typically noise and were not correlated to
the real energy deposition. As such, they could not be used in the
polarization analysis. This caused several secondary issues. The first is that
during the event selection for polarization, as discussed in section 5.4.1,
channels without a charge trigger were not taken into account. This thereby
directly resulted in a lower energy threshold of $\approx$ 8 keV for both the
channels selected in the polarization analysis. As a 60 keV photon has a large
probability of depositing energies below 8 keV when Compton scattering all
such events are lost here.

The second result of the lack of peak sensing is related to the measurement of
the trigger thresholds. The implementation of the threshold in the
simulations, as discussed in section 4, requires both the threshold mean and
its width, which is a result of noise in the threshold DAC. Although
determining the mean position of the threshold is possible, the width of this
threshold is distorted in the final data because the ADC value assigned to all
energy depositions which do not issue a trigger are wrong. As a result the
width used in the simulations had to be fine tuned such that the results
matched the real data, rather than using the measured threshold width.

Since the beam campaign the firmware of the polarimeter module has been
updated such that peak sensing is enabled for all channels as soon as one
channel exceeds the charge threshold (or when the time threshold trigger is
issued). This is achieved by forcing the ASIC to start peak sensing on all
channels using an external line from the FPGA. The decision from the FPGA is
taken within nanoseconds after the trigger is issued by the ASIC. This does
not significantly affect the spectra. The final result of this fix can be seen
again in figure 33 where clear photo-electron peaks, or fingers, can be seen
also below the threshold.

Finally, as was discussed in section 5.2, the scintillator material used in
the polarimeter module which was used for the polarization studies in these
tests is not yet the optimum. Switching this to EJ-248M, as was illustrated in
figure 17, will increase the light yield and thereby allow to decrease the
threshold by approximately $10\%$. This material will therefore be used in the
final version of the POLAR-2 polarimeter module.

##  7 Extrapolated POLAR-2 Performance

After verifying the performance of the POLAR-2 MC simulations framework using
measured data, we can use this to simulate the full 100-modules POLAR-2
detector to study its expected performance. Although the simulation framework
used for this is overall equal to that used to simulate the detector response
for the irradiation campaign, a few modifications were made:

  * •

The number of detector modules was increased to 100 and additional detector
mechanics were added to simulate the full POLAR-2 detector as it is shown in
figure 2. It should be noted that the majority of the components, such as the
aluminium structure below the polarimeter modules, does not affect the
results. Rather, such structures are only important for future simulations of
the background.

  * •

The time trigger logic is added. This implies that in case two channels within
a polarimeter module are above the time trigger the full module is read out.
The time trigger threshold was set at 4 keV. The charge threshold was left at
approximately 8 keV (equal to the values used at ESRF). It is likely, that
this threshold can be lowered in the future, for example, thanks to the use of
EJ-248M as the scintillator material, as well as the lower operating
temperature foreseen to be used in orbit. This has to be confirmed through
detailed tests. It should be noted that this is significantly lower than
typical threshold of spectrometers which use SiPMs. This is possible thanks to
the requirement that two detector channels trigger within a 100 ns window,
thereby significantly reducing the dark noise induced triggers.

  * •

Peak sensing is now enabled for all channels in a polarimeter module when
either a valid time or charge trigger is issued. As a result we now see photo-
electron peaks also below the trigger threshold as in figure 33.

###  7.1 Effective Area

The effective area of POLAR-2 is simulated by irradiating the full detector
using mono-energetic photons emitted from a plane with dimensions of
$800\times 800\,\mathrm{mm^{2}}$ placed above the detector surface. The
simulations were performed using mono-energetic beams with energies ranging
from 10 to 800 keV. The output from the simulations is processed in the data
analysis pipeline identical to that used for the single polarimeter module
setup. The total effective area of the detector is then simply defined as the
number of valid triggers in the detector divided by the simulated photon flux
which for these simulations corresponds to $781\,\mathrm{cm^{-2}}$ for
simulations below 100 keV and $312\,\mathrm{cm^{-2}}$ above this. These
simulations were repeated for photons with an incoming polar angle of
$30^{\circ}$, while keeping the $\phi$ angle at $0^{\circ}$, to understand the
angular dependence.

The results are shown in figure 34 where the effective areas are compared to
that of POLAR. We can see that while above 200 keV the effective area of
POLAR-2 is approximately 4 times that of POLAR, matching the increase in
geometrical area, at lower energies the increase is more significant. This is
thanks mainly to the higher light yield which allows for a lower detection
threshold.

The effect of the lower detection threshold ($\approx 8\,\mathrm{keV}$ for
POLAR-2 compared to $\approx 15\,\mathrm{keV}$ for POLAR) also affects the dip
in the effective area which can be seen for both detectors below 100 keV. This
local minimum is a result of the increasing probability for photons to undergo
Compton scattering with increasing energy. As photons of $\approx
20-30\,\mathrm{keV}$ still have a significant probability to undergo photo-
absorption in the plastic, the energy they deposit is typically above the low
energy threshold. With increasing energies the probability to deposit all the
energy through photo-absorption decreases, resulting in a significant fraction
of the photons to Compton scatter in one scintillator, before leaving the
detector. The maximum energy deposition a 50 keV photon can deposit in the
case where it backscatters out of the detector, is 8 keV. As a result, POLAR-2
will be able to measure such interactions, however, for photons with 40 keV,
which can deposit only 5.5 keV, it is less likely555Due to Poisson
fluctuations in the number of photo-electrons produced by a 5.5 keV
deposition, as well as noise, such interactions can still trigger the
detector.. For POLAR-2 a local minimum is therefore visible around 45 keV. For
POLAR, which has a higher detection threshold this minimum is at a higher
energies and is also more pronounced.

Figure 34: Left: The effective area for all events as a function of energy
from simulations. The effective area for an on-axis flux for POLAR-2 is shown
along with that from POLAR. In addition the effective area for an off-axis
flux with an incoming angle of $\theta=30^{\circ},\phi=0^{\circ}$. Right: the
ratio of the total effective area of POLAR-2 over that of POLAR. The ration
which can be directly attributed to the increase in geometrical area is
indicated with the blue dotted line.

While figure 34 shows the effective area which can be used for measuring the
GRB flux, position and spectrum, figure 35 shows the effective area for all
triggers which, using the simple event selection discussed prior, are used for
polarization measurements. To produce this the total number of entries in the
scattering angle distribution is used rather than the total number of triggers
in the detector. As a result the effective area is lower. Again we can see
that at higher energies the ratio between the two matches the increase in the
geometrical area of the detector while below 200 keV the increase is much more
significant.

Also here, the effective area for a flux with an incoming angle polar of
$30^{\circ}$ and $\phi=0$ is presented which, as expected, is slightly higher
than that for the on-axis case. Finally, here we also show the effective area
for polarization events when we impose an additional, offline, threshold for
the two energy depositions of 8 keV. We can see that this significantly
decreases the effective area for polarization events, specifically at low
incoming photons energies. Such a cut, however, will increase the $\mu_{100}$
as will be discussed next.

Figure 35: Left: The effective area for polarization events as a function of
energy from simulations. The effective area for an on-axis flux for POLAR-2 is
shown along with that from POLAR. In addition the effective area for an off-
axis flux with an incoming angle of $\theta=30^{\circ},\phi=0^{\circ}$.
Finally, the effective area is shown for an on-axis flux where an additional
lower energy threshold of 8 keV is set. Right: The ratio of the effective area
of POLAR-2 over that of POLAR. The ratio which can be directly attributed to
the increase in geometrical area is indicated with the blue dotted line.

###  7.2 Modulation Factor

The simulations used to produce the effective area predictions were also used
to simulate the $\mu_{100}$ for POLAR-2. For this purpose the scattering angle
distributions were produced for a polarized flux, corrected using the
unpolarized distribution and finally fitted using a harmonic function with a
period of $180^{\circ}$ to extract the relative amplitude of the modulation.
The results of this are shown for the on-axis case and a $30^{\circ}$ off-axis
case in figure 36. As was also shown with the data from the ESRF tests in
section 5, the difference in the $\mu_{100}$ between the on-axis and off-axis
case is minimal at lower energies. Only when exceeding $\approx
80\,\mathrm{keV}$ do we really start to see a significant difference.

In addition, the $\mu_{100}$ was also calculated when applying the additional
offline low energy threshold on the data of 8 keV. We can see here that this
significantly increases the $\mu_{100}$ at higher energies with a peak at
$41\%$ compared to $37\%$ without this cut. However, as was shown in figure
35, such a cut does result in a significant decrease in the effective area. A
commonly used figure of merit for polarization measurements is the Minimal
Detectable Polarization (MDP) [38]:

$\mathrm{MDP}=\frac{2\sqrt{\mathrm{-ln}(1-{\rm C.L.})}}{\mu_{100}C_{\rm
s}}\sqrt{C_{\rm s}+C_{\rm b}}\,.$ (7.1)

where $\mathrm{C.L.}$ is the confidence level, $C_{\rm s}$ is the number of
signal events and $C_{\rm b}$ the number of background events. This figure of
merit gives the minimal level of polarization of the source one can
distinguish from being unpolarized using the measurement with a confidence
level C.L. It should be noted that, as discussed in detail in [12], the MDP is
not a fully accurate figure of merit to use for wide field of view
polarimeters like POLAR-2. Although it does provide a good approximation on
how to optimize things, it should not be over interpreted.

Although the MDP scales linearly with $\mu_{100}$ and only through
$\sqrt{C_{\rm s}}$, and therefore the effective area, fully optimizing the
ratio between the $\mu_{100}$ and the effective area requires assumptions on
the spectrum of the incoming flux. In addition, cuts like the 8 keV lower
threshold applied here, will also affect the background rate $C_{\rm b}$,
however, the dependency here can only be studied accurately using real data.

The application of the additional lower energy cut therefore only shows that
an optimization using such cuts, as well as for example cuts on the distance
between the two energy interactions in the detector, should be studied.
However, such optimization studies require also a good understanding of the
background, and therefore will be the subject of dedicated studies in the
future.

Finally, it is important to address concerns regarding the increase in dark
noise in the SiPMs as a result of radiation damage during operation in space.
Such an incrase in dark noise would require an increase in the thresholds to
limit the number of noise induced triggers. For details on this the reader is
referred to the following studies performed by the POLAR-2 team [39, 40].
These studies indicate that, through the use of annealing the required
increase in threshold would amount to approximately $0.75\,\mathrm{keV}$ after
2 years.

Figure 36: The modulation factor as a function of energy from simulations. The
$\mu_{100}$ for an on-axis flux for POLAR-2 is shown along with that for an
off-axis flux with an incoming angle of $\theta=30^{\circ},\phi=0^{\circ}$.
Finally the $\mu_{100}$ is shown for an on-axis flux where an additional lower
energy threshold of 8 keV is set.

##  8 Conclusions

The first POLAR-2 polarimeter module was irradiated using fully polarized
beams at the ID15A ESRF beam facility. The performance of the polarimeter
module indicates that the detector is significantly more sensitive than the
detector modules previously used in the POLAR detector. This is mainly
attributable to the increase in the light yield of the detector achieved
through the use of SiPMs.

During the analysis phase some shortcomings of the current design were
additionally found, all of which have since been solved using upgrades of the
firmware.

The performance of the detector module furthermore matches that predicted
through the POLAR-2 MC simulation framework, thereby verifying its validity.
As a result, this could be used to produce predictions on the performance of
the full POLAR-2 detector. Through this it was found that POLAR-2 will have an
effective area for polarization which already exceeds $100\,\mathrm{cm^{2}}$
at 40 keV, while the data taken at ESRF additionally proved that the
instrument is sensitive to polarization at these energies with a $\mu_{100}$
of $\approx 16\%$. For its predecessor POLAR the effective are only started to
exceed $100\,\mathrm{cm^{2}}$ at 120 keV, while it had no significant
sensitivity to polarization below $50\,\mathrm{keV}$.

Although the simulation framework already shows that the detector should be
sensitive at energies of 30 keV, dedicated tests need to be performed to
confirm this. Further improvements to the POLAR-2 design compared to that used
here at ESRF, for example through changing the plastic scintillator material,
will allow to further decrease the detection threshold. A future test at ESRF
will therefore aim to measure its sensitivity at energies of 30 keV and
potentially 25 keV.

## Acknowledgments

We acknowledge the European Synchrotron Radiation Facility who provided the
beam time and facilities which were crucial for this work [36]. We would in
particular like to thank the team of ID15A who assisted us during the 1 week
campaign.

We gratefully acknowledge the Swiss Space Office of the State Secretariat for
Education, Research and Innovation (ESA PRODEX Programme) which supported the
development and production of the POLAR-2 detector. M.K. and N.D.A.
acknowledge the support of the Swiss National Science Foundation. National
Centre for Nuclear Research acknowledges support from Polish National Science
Center under the grant UMO-2018/30/M/ST9/00757. We gratefully acknowledge the
support from the National Natural Science Foundation of China (Grant No.
11961141013, 11503028), the Xie Jialin Foundation of the Institute of High
Energy Physics, Chinese Academy of Sciences (Grant No. 2019IHEPZZBS111), the
Joint Research Fund in Astronomy under the cooperative agreement between the
National Natural Science Foundation of China and the Chinese Academy of
Sciences (Grant No. U1631242), the National Basic Research Program (973
Program) of China (Grant No. 2014CB845800), the Strategic Priority Research
Program of the Chinese Academy of Sciences (Grant No. XDB23040400), and the
Youth Innovation Promotion Association of Chinese Academy of Sciences.

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