11institutetext:  Carolyn Kierans 22institutetext: NASA Goddard Space Flight
Center, 8800 Greenbelt Way, Maryland 20771, USA
22email<EMAIL_ADDRESS>33institutetext: Tadayuki Takahashi
44institutetext: Kavli Institute for the Physics and Mathematics of the
Universe (WPI), The University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa, Chiba,
277-8583, Japan 55institutetext: Gottfried Kanbach 66institutetext: Max Planck
Institute for Extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching,
Bayern, Germany

# Compton Telescopes for Gamma-ray Astrophysics

Carolyn Kierans corresponding author Tadayuki Takahashi  and Gottfried Kanbach

###### Abstract

Compton telescopes rely on the dominant interaction mechanism in the MeV
gamma-ray energy range: Compton scattering. By precisely recording the
position and energy of multiple Compton scatter interactions in a detector
volume, a photon’s original direction and energy can be recovered. These
powerful survey instruments can have wide fields of view, good spectroscopy,
and polarization capabilities, and can address many of the open science
questions in the MeV range, and in particular, from multimessenger
astrophysics. The first space-based Compton telescope was launched in 1991 and
progress in the field continues with advancements in detector technology. This
chapter will give an overview of the physics of Compton scattering and the
basic principles of operation of Compton telescopes; electron tracking and
polarization capabilities will be discussed. A brief introduction to Compton
event reconstruction and imaging reconstruction is given. The point spread
function for Compton telescopes and standard performance parameters are
described, and notable instrument designs are introduced.

Keywords Compton scattering; gamma-ray astrophysics; Compton telescopes; MeV
gamma-rays; polarimetry; image reconstruction; event reconstruction;
spectroscopy

##  1 Introduction

Megaelectronvolt (MeV) gamma rays are an excellent tool to explore the cosmos:
they travel long distances without deviation or absorption, they probe deeper
into dense regions than radiation at other wavelengths, and they are emitted
by the most extreme and energetic processes in the universe. In particular,
the soft and medium energy regime, from $\sim$100 keV to 100 MeV, is ripe with
scientific opportunities. With recent advances in multimessenger astrophysics,
the community’s demand for sensitive medium energy gamma-ray telescopes has
grown as the sources which generate gravitational waves, neutrinos, and cosmic
rays shine bright in the low-end of the gamma-ray spectrum.

The MeV range is rich with scientific potential. Neutron stars, black holes,
magnetars with extreme magnetic fields, and active galactic nuclei with
relativistic jets are natural accelerators that emit gamma rays through
bremsstrahlung, inverse Compton, and synchrotron radiation. Emissions in the
MeV range constrain the thermal and non-thermal energy budget for high-energy
Galactic and extra-galactic sources. There are also compelling reasons to
believe that indirect signatures of dark matter will be found at these
energies. And finally, the MeV range is the natural scale to probe nuclear
processes in the Galaxy and beyond. Many radioactive isotopes decay by
emitting gamma rays up to the nuclear binding energies $\lesssim 8$ MeV per
nucleon. Radioactive isotopes generated in stellar and explosive
nucleosynthesis are proton rich and often undergo $\beta^{+}$ decay, emitting
a positron which later annihilates and generates signature 511 keV photons.

However, the soft to medium energy gamma-ray regime remains one of the least
explored energy ranges in multi-wavelength astrophysics. This gap in
sensitivity, often referred to as the MeV Gap, has greatly hindered progress
of science due to limited observations in this thermal to non-thermal
transition regime. Figure 1 shows the continuum sensitivity of current and
past telescopes in the X-ray and high-energy gamma-ray regime, and it is clear
that the sensitivity in the MeV Gap is orders of magnitude less than
neighboring bands.

Figure 1:  The differential sensitivity for current and past X-ray and gamma-
ray missions shows the limited performance achieved in the MeV regime. The
reduced sensitivity in the range from 100 keV to 100 MeV is referred to as the
MeV Gap. The XMM sensitivity is for a 1.6 Ms observation at 4$\sigma$
detection significance Hasinger et al. (2001); XMM-Newton Science Operations
Centre . The NuSTAR Koglin et al. (2005), Suzaku HXD Fukazawa et al. (2009),
INTEGRAL ISGRI Lebrun (2005), PICsIT Di Cocco et al. (2003), and SPI Roques et
al. (2003) sensitivities are shown for 3$\sigma$ detections with 100 ks
exposure and $\Delta E/E=0.5$, assuming statistical error only. The
sensitivity of COMPTEL corresponds to the 9 yr all-sky survey of CGRO
Schönfelder et al. (2000). CGRO/EGRET sensitivity (3$\sigma$) is taken from
Kamae (2005) for a 300 hr exposure, which is consistent with the numbers given
in Thompson et al. (1993). The sensitivity of Fermi/LAT is for the 10 yr
survey Ajello et al. (2021); Fer . The 5$\sigma$ sensitivities for MAGIC,
H.E.S.S, and CTA (simulated) correspond to 50 hr observations Zanin et al.
(2021); CTA . The dashed grey line representing the Crab flux is calculated
from Naima Zabalza (2015).

The lack of sensitive telescopes in the MeV range is not due to a lack of
scientific interest, as demonstrated throughout Volume 3 and 4 of this
handbook, but instead due to physical, technological, and environmental
challenges (i.e., background radiation). The interaction cross-section between
photons and matter is lowest in this range, and the dominant mechanism is
Compton scattering, which produces long-range secondaries. Large detector
volumes are needed to stop and contain photons with interaction depths on the
order of $\sim$10 g/cm2. And with no focusing element, these large detector
volumes result in high backgrounds, not only from atmospheric albedo emission,
but also from activation within and around the instrument. While this high
background does not significantly impact the detection sensitivity to high-
energy transients due to the short bright signals, studies of steady-state
sources in the MeV range require sophisticated background rejection
techniques.

To build a sensitive telescope to explore the MeV Gap, it must make use of the
dominant interaction mechanism in this range: Compton scattering. In Compton
scattering, gamma rays will partially transfer their energy to bound
electrons, and by doing so, the electron recoils and the gamma ray is
scattered at an angle relative to its initial direction with lower energy. The
Compton-scattered photon will then interact a second time, and potentially a
third and fourth, before finally absorbing all its energy in the detector.
Only the complete measurement of the secondary products, i.e. the energized
target electron and the de-energized scattered photon, allow for the
determination of the initial energy and direction of the incident photon.

The pioneering telescope that opened the MeV range as a new window to
astronomy was the Imaging Compton Telescope (COMPTEL Schönfelder et al.
(1993)) aboard the Compton Gamma Ray Observatory (CGRO). COMPTEL covered the
energy range 0.8–30 MeV from its launch in 1991 until its termination in 2000.
It produced the first all-sky MeV survey, mapped the diffuse Galactic emission
in the continuum and in the light of radioactive gamma-ray lines (e.g. 26Al,
44Ti), and detected more than 30 gamma-ray sources (pulsars, pulsar wind
nebulae, black holes, and active galaxies). COMPTEL revolutionized gamma-ray
astrophysics, and many of the techniques and tools developed for the mission
are still being used in modern Compton telescopes. As the last instrument to
observe much of the MeV Gap, there are continuing efforts to build upon
COMPTEL’s legacy.

The power of Compton telescopes come from the single photon detection and
imaging capabilities. With a large field of view and modest angular
resolutions ($\sim$1∘), Compton telescopes make powerful survey instruments.
The single photon detections and necessary event reconstruction allow for
effective background rejection capabilities. Additionally, Compton scattering
is inherently sensitive to the linear polarization of incoming gamma rays, and
there is a class of Compton telescopes designed specifically for polarization
measurements of transients, such as gamma-ray bursts. Recent advances in
technology have shown the breadth and power of Compton telescopes, and looking
forward the future is bright with the Compton Spectrometer and Imager (COSI)
mission Tomsick and COSI Collaboration (2022) selected by NASA to launch in
2026.

This chapter aims to provide a concise, comprehensive, and up-to date overview
of Compton telescopes for astrophysics. The physics of Compton scattering,
which is the basis of telescope operation, will be introduced in Section 2. An
overview of the operating principles of Compton telescopes including the
COMPTEL-like double-scattering approach, modern compact Compton telescopes,
and electron tracking capabilities, will be given in Section 3. Section 4
provides a discussion of event reconstruction techniques, and Section 5 covers
the standard performance parameters for a Compton telescope. And finally,
notable Compton telescope instruments will be briefly introduced in Section 6.

##  2 Physics of Compton Scattering

A major debate was conducted after the discovery of X-rays (also called
Röntgen-rays in the early days) in the 1900’s on the nature of this new,
penetrating radiation. Arthur H. Compton, who received the Nobel prize in
Physics in 1927 for the discovery of Compton scattering, summarized the
profound findings in his Nobel lecture Compton (1927):

> All phenomena in the realm of light are found in parallel in the realm of
> X-rays. Reflection, refraction, diffuse scattering, polarization,
> diffraction, emission and absorption spectra, photoelectric effect, all of
> the essential characteristics of light have been found also to be
> characteristic of X-rays. At the same time it has been found that some of
> these phenomena undergo a gradual change as we proceed to the extreme
> frequencies of X-rays.

For high-energy X-rays, a modified component of scattered radiation was found
at large angles ($\varphi$) offset from the primary beam ($\lambda_{0}$) and
shifted to longer wavelengths ($\lambda_{scat}$), i.e. lower energies. This
experimental result could not be explained by coherent scattering of
electromagnetic waves on free or bound electrons (Thomson or Rayleigh
scattering). Arthur Compton introduced the new concept that quanta of light,
photons, scattering off electrons can be described by the law of mechanics
Compton (1923).

Figure 2: A photon with initial momentum $\vec{p}_{0}=h\nu_{0}/c$ scatters off
an electron assumed to be unbound and at rest. The photon scatters at an angle
$\varphi$ relative to the initial direction, and has a momentum
$\vec{p}_{scat}=h\nu_{scat}/c$. The electron recoils from the scatter at an
angle $\Theta$ and with momentum $\vec{p_{e}}=m_{e}\nu/\sqrt{1-\beta^{2}}$.
Through the conservation of energy and momentum, the relation between the
scattering angle and the final photon energy can be determined.

Compton showed that scattering photons conserve energy and momentum, as shown
schematically in Figure 2. For a photon with initial energy $E_{0}$, energy
conservation gives

$E_{0}=E_{scat}+E_{e},$ (1)

where $E_{scat}$ is the scattered photon energy, and the electron, with final
energy $E_{e}$, is assumed to initially be free and at rest (this limitation
is discussed in Section 5.1). Conservation of momentum gives

$\vec{p_{0}}=\vec{p}_{scat}+\vec{p_{e}},$ (2)

where the initial momentum of the photon is
$\left|\vec{p_{0}}\right|=h\nu_{0}/c$, and the final reduced photon momentum
is $\left|\vec{p}_{scat}\right|={h\nu_{scat}/c}$. The recoil electron is
accelerated and the final momentum is given by
$\left|\vec{p_{e}}\right|=m_{e}v\gamma$, with $\
\gamma={1/{\sqrt{1-\beta^{2}}}}$ and $\beta=v/c$. This leads to the well-known
Compton equation:

$\lambda_{scat}-\lambda_{0}=\frac{h}{m_{e}c}(1-\cos\varphi),$ (3)

where $h$, $m_{e}$, and $c$ are Planck’s constant, the electron rest mass, and
the speed of light, respectively. The fraction ${h/{m_{e}c}}=2.426\ 10^{-12}$
m is often called the Compton wavelength, which is the wavelength shift for a
$90^{\circ}$ scattering. It is important to note that a photon can never lose
all its energy in a Compton scattering process, even if it is completely
backscattered.

Equation 3 can be solved for $\varphi$ and with wavelength converted to
energy:

$\cos\varphi=1-m_{e}c^{2}\left(\frac{1}{E_{scat}}-\frac{1}{E_{0}}\right),$ (4)

where the energy of the scattered photon is

$E_{scat}=h\nu_{scat}=\frac{E_{0}}{1+\frac{E_{0}}{m_{e}c^{2}}{(1-\cos\varphi)}}.$
(5)

As one can see in Equation 4 and 5, the final energy of the Compton-scattered
photon can be uniquely determined with the initial energy and Compton scatter
angle information.

Figure 3: The spectral shape of the recoil electron energy for a photon with
initial energy $E_{0}=h\nu$. There is a continuum of electron energies
observed due to the range of allowable Compton scatter angles. The maximum
recoil electron energy is achieved for backscatter interactions when
$\varphi=180^{\circ}$; this defines what is referred to as the Compton edge in
a measured spectrum. Modified from Knoll (2000).

As detectors are sensitive to the ionization from the recoil electron, it is
important to consider the kinetic energy of the Compton-scattered electron:

$E_{e}=E_{0}-E_{scat}=E_{0}\left(\frac{E_{0}(1-\cos\varphi)}{{m_{e}c^{2}}+E_{0}(1-\cos\varphi)}\right).$
(6)

The spectral shape of the electron energy is shown in Figure 3. To better
understand this distribution, we consider the extreme cases of grazing-angle
scattering and backscattering. For $\varphi\approx 0$, Equation 6 gives the
recoil electron energy $E_{e}\approx 0$, and Equation 5 gives $E_{scat}\approx
E_{0}$. For a head-on collision where the photon backscatters at
$\varphi=180^{\circ}$, the maximum amount of energy is transferred to the
electron and Equation 6 reduces to

$\left.E_{e}\right\rvert_{\varphi=180^{\circ}}=E_{0}\left(\frac{2E_{0}}{m_{e}c^{2}+2E_{0}}\right)=E_{0}\left(1-\frac{1}{1+\frac{2E_{0}}{m_{e}c^{2}}}\right).$
(7)

The difference between the initial photon energy and the maximum recoil
electron energy is given by

$E_{c}=E_{0}-\left.E_{e}\right\rvert_{\varphi=180^{\circ}}=\frac{E_{0}}{1+\frac{2E_{0}}{m_{e}c^{2}}},$
(8)

which is often called the Compton edge and is depicted in Figure 3.

The Compton-scattered electron energy $E_{e}$ can also be expressed in terms
of the angle $\Theta$ between the incident photon and the direction of the
recoil electron (see Figure 2), making use of the relation

$\cot(\Theta)=(1+\alpha)\tan(\varphi/2),$

where $\alpha={E_{0}/{m_{e}c^{2}}}$

$E_{e}={{2E_{0}\alpha\cos^{2}\Theta}\over{(1+\alpha)^{2}-\alpha^{2}\cos^{2}\Theta}}.$
(9)

The full quantum-mechanical treatment of the scattering of unpolarized
energetic photons on single, free target electrons
$\gamma+e^{-}\rightarrow\gamma+e^{-}$ was derived by Oskar Klein and Yoshio
Nishina in 1928. The differential cross-section for Compton scattering is
given by the Klein-Nishina formula Klein and Nishina (1929):

$\frac{d\sigma}{d\Omega}=\frac{{r_{0}}^{2}}{2}\left(\frac{E_{scat}}{E_{0}}\right)^{2}\left(\frac{E_{scat}}{E_{0}}+\frac{E_{0}}{E_{scat}}-\sin^{2}\varphi\right),\\\
$ (10)

where $r_{0}=2.818\times 10^{-15}{\rm\ m}$ is the classical electron radius,
$E_{0}$ is the initial energy of the incident photon, $E_{scat}$ is the energy
of the scattered photon, $\varphi$ is the scattering polar angle, as shown in
Figure 2. From this equation it can be seen that higher energy photons will,
in general, have smaller Compton scatter angles, and lower energy photons will
result in larger scatter angles. The Klein-Nishina differential cross-section
is shown graphically in Figure 4 for a number of different energies.

Figure 4: The Klein-Nishina differential cross-section (m2/sr) as a function
of Compton scatter angle shows the energy dependence of the scattering
direction. For lower energy photons, such as 10 keV shown with the blue curve,
the magnitude of the differential cross-section only changes by a factor of
two for all Compton scatter angles. For higher energy photons, such as 2 MeV
shown in red, the cross-section is maximized for small scattering angles.

Integration of Equation 10 over $d\Omega$ gives the total cross-section for
Compton scattering. In terms of Thomson units ($\sigma_{T}={8\pi
e^{2}}/{3m_{e}c^{2}}$ = $0.665\times 10^{-28}$ m2), and using dimensionless
units for the photon energy ($\epsilon=E_{0}/{m_{e}c^{2}}$), the total cross-
section is:

$\sigma_{C}={3\over
4}\sigma_{T}\left[{{1+\epsilon}\over{\epsilon^{3}}}\left({{2\epsilon(1+\epsilon)}\over{1+2\epsilon}}-\ln(1+2\epsilon)\right)+{\ln(1+2\epsilon)\over{2\epsilon}}-{{1+3\epsilon}\over{(1+2\epsilon)^{2}}}\right]$
(11)

If the gamma-ray energy is much smaller than electron mass, the total cross-
section is reduced to the Thomson cross-section, $\sigma_{T}$.

The differential cross-section for linearly polarized photons was derived by
Heitler (1954) in 1954:

$\frac{d\sigma}{d\Omega}=\frac{{r_{0}}^{2}}{2}\left(\frac{E_{scat}}{E_{0}}\right)^{2}\left(\frac{E_{scat}}{E_{0}}+\frac{E_{0}}{E_{scat}}-2\sin^{2}\varphi\cos^{2}\eta\right),\\\
$ (12)

where the polar Compton scatter angle $\varphi$ and the azimuthal scattering
angle $\eta$ are explicitly written Evans (1955). There is a similar
dependence on the Compton scatter angle $\varphi$ in Equation 10; however, the
polarized differential cross-section is maximized when the $\cos^{2}\eta$ term
equals zero. In other words, photons will predominantly scatter at
$90^{\circ}$ relative to the the initial photon’s electric field vector,
defined as $\eta=0$. This results in an azimuthal-dependence of the cross-
section, and therefore, Compton telescopes can inherently detect polarization
if a measurement of $\eta$ is made Lei et al. (1997). The definition of the
scattering angles relative to the initial photon electric field vector
$\vec{\xi}$ is shown in Figure 5.

Figure 5: The Klein-Nishina differential cross-section for polarized photons
(Equation 12) is maximized when the azimuthal scattering angle $\eta$ is
perpendicular to the photon’s initial electric field vector $\vec{\xi}$. By
measuring the azimuthal scattering angle $\eta$, an instrument can be
sensitive to the linear polarization of incoming photons.

Equation 10 not only describes the energy dependence of the Compton scattering
cross-section on single electrons, but also implicitly the total cross-section
for scattering in material of atomic number ($Z$) and density $\rho$. Since
each single scattering event is independent, the total cross-section is the
sum of all events and depends directly on the electron density in the target
material. Detailed compilations of gamma-ray cross-sections for various
materials can be accessed with the internet based application $XCOM$ Berger et
al. (2017).

It is convenient to describe the total cross-section in terms of the mass
attenuation coefficient $\mu$, particularly when comparing Compton scattering
to the dominant photon interactions at lower and higher energies, i.e.,
photoelectric absorption and pair production. These coefficients, given in
units of cm2/g, are most useful to estimate what fraction of photons from an
incoming beam with intensity $I_{0}$ are lost by interactions, either by
absorption or scattering. The intensity $I$ of a beam after traversing matter
with thickness $x$ (g/cm2) is

$I=I_{0}e^{-\mu x},$ (13)

where $x$ is the product of the geometrical path length $l$ and density $\rho$
of the transversed material.

As an illustrative example, we can calculate the amount of silicon needed to
attenuate 50% of incident gamma rays with initial energy of 511 keV. From
$XCOM$, the $\mu$ for silicon at 511 keV is $8.67\times 10^{-2}$ cm2/g, and
the density $\rho$ is 2.33 g/cm3. Rearranging Equation 13 for the path length
gives

$l=\frac{1}{\rho\mu}\ln\left(\frac{I}{I_{0}}\right)=\frac{\ln(0.5)}{(2.33~{}\text{g/cm}^{3})(8.67\times
10^{-2}~{}\text{cm${}^{2}$/g})}=3.4~{}\text{cm}.$ (14)

Therefore, over 3 cm of silicon are needed to scatter or absorb only half of
incident 511 keV photons. This exemplifies the challenges of low interaction
cross-sections for Compton telescopes where large detector volumes are
required.

(a) Plastic Scintillator

(b) Silicon

(c) Sodium Iodide (thallium doped)

Figure 6: The mass attenuation coefficients for three common gamma-ray
detector materials: (a) plastic scintillator (hydrocarbon-based polymers), (b)
silicon, and (c) sodium iodide NaI(Tl). Compton scattering dominates over a
much larger energy range from $\sim$30 keV to 30 MeV for low-Z materials, as
shown here for plastic scintillator. For high-Z materials, such as NaI,
Compton scattering only is dominant from $\sim$300 keV to 7 MeV. Calculations
are from $XCOM$ Berger et al. (2017).

Figure 6 depicts the mass attenuation coefficient $\mu$ as a function of
energy for three common detector materials: a low-Z plastic scintillator
(hydrocarbon-based material of density $\rho\approx 1$ g/cm3), the ubiquitous
semiconductor silicon ($\rho=2.33$ g/cm3), and the common high-Z sodium iodide
doped with thallium NaI(Tl) ($\rho=3.67$ g/cm3). Compton scattering dominates
in the range 30 keV to 30 MeV for the plastic scintillator, while for Na(Tl)
this range is reduced to 300 keV to 8 MeV. These differences are the basis for
many Compton telescope designs that separate scattering and absorption
processes in multiple interactions, and Figure 7 shows the interplay between
the three major interaction processes as a function of the target material
atomic number and initial photon energy. It is evident that materials with low
to medium Z, e.g. hydrocarbon-based scintillators or solid state detectors
like silicon ($Z=14$), are the most efficient Compton scatterers over a wide
energy range. Photoabsorption of the Compton-scattered photon is most
efficient in a high-Z and high density detector, such as NaI(Tl) or cesium
iodide CsI(Tl) scintillators, and solid state detectors made of cadmium
telluride (CdTe) or cadmium zinc telluride (CZT).

Figure 7: The regions of dominant interaction cross-sections for the
photoelectric effect, Compton scattering, and pair production are shown as a
function of photon energy and for detector materials of atomic number Z.
Photoelectric effect and Compton interactions have the same cross-section
along the boundary $\sigma=\tau$, and pair creation dominates beyond
$\sigma=\kappa$. Figure From Evans (1955).

##  3 Basic Operating Principles of Compton Telescopes

In the 1960’s and early 1970’s, the imaging and spectroscopy of celestial
sources in the energy range from 100 keV to 10’s of MeV was considered a very
difficult task. Instruments relying primarily on photoabsorption and pair
production were developed, but were too inefficient at these energies. The
only physical process with reasonable interaction probability in this energy
range is Compton scattering (see Figure 7). However, it is impossible to
derive the initial gamma ray arrival direction and energy (i.e. for imaging or
spectroscopy) from only the first Compton-scattering interaction; the photon
must be fully absorbed and the energy and position of each interaction must be
recorded to reconstruct a Compton-scattering event.

Various work-arounds were attempted to obtain a coarse angular resolution
through the use of massive collimators and anti-coincidence shields
surrounding detectors large enough to contain the full energy of a Compton
event and its secondaries. For sources characterized by a clear temporal
signal (e.g. pulsars, solar flares, gamma-ray bursts), these instruments could
provide important scientific results; however, a more general astronomical
imaging instrument at MeV energies was elusive. Progress towards a sensitive
MeV telescope was only made with a dedicated effort to measure the locations
of energy deposits from Compton interactions.

Figure 8: A Compton telescope works by measuring the energy and position of
one or more Compton scattering interactions. For a two-site event, the initial
energy of the photon is determined from the total absorbed energy
$E_{0}=E_{1}+E_{2}$, where as the direction of the incoming photon is
constrained to a circle on the sky defined by the Compton scatter angle
$\varphi$ of the first interaction. When multiple photons are detected
originating from the same source, the location of the source is determined by
the overlap of each back-projected event circle, as is visualized for 10
events on the right. Modified from Kierans (2018).

Compton telescopes rely on the precise measurement of the energy and location
of Compton scattering interactions which allow for the arrival direction of
the primary gamma ray to be constrained. For example, let us consider a two-
site event, that is, a photon event that consists of a single Compton scatter
followed by photoelectric absorption. The basic operating principle is shown
in Figure 8, where a detector volume measures the two interactions, fully
containing the event. The Compton scatter angle $\varphi$ of the first
interaction can be derived from the energy measurements. For a two-site event,
Equation 4 gives

$\cos\varphi=1-\frac{m_{e}c^{2}}{E_{2}}+\frac{m_{e}c^{2}}{E_{1}+E_{2}},$ (15)

where $E_{1}+E_{2}$ is the total deposited energy, which is equal to the
initial photon energy $E_{0}$ from Equation 4, $E_{scat}=E_{2}$, and
$E_{e}=E_{1}$. The direction of the scattered gamma ray between $\vec{r}_{1}$
and $\vec{r}_{2}$ defines a cone on the sky with opening angle given by
$\varphi$. The back-projection of this cone on the celestial sphere is
referred to as the event circle. If the recoil electron direction is not
measured (see electron-tracking Compton telescopes in Section 3.3), one cannot
determine where on the event circle the single gamma-ray photon originates
from. Multiple photons from the same source will have overlapping event
circles in image space, revealing the location of the source (see Section 5.3
for an overview of imaging reconstruction techniques).

###  3.1 The Classic Double Scattering Compton Telescope

The first Compton telescopes for astrophysics were developed in the 1970’s
Herzo et al. (1975); Schönfelder et al. (1973) and were described as “double
scatter” telescopes. These classic Compton telescopes consisted of two
detectors, referred to as D1 and D2, to measure a Compton scattering
interaction and subsequent photoabsorption in two separated planes, as shown
in Figure 9. The trajectory of the scattered photon is known by recording the
energy and location ($E_{1},\ \vec{r}_{1},\ E_{2},\ \vec{r}_{2}$) of the two
interaction sites. Equation 15 then affords an estimate for the scattering
angle of the primary photon.

(a)

(b)

Figure 9: (a) A classic Compton telescope uses a scattering plane (D1) and an
absorption plane (D2) to measure the position and energy of two interactions
in a Compton event. From these measurements, the original photon direction can
be constrained to a circle on the sky called the “event circle.” Adapted from
Schönfelder and Kanbach (2013). (b) The first Compton telescope flown was
COMPTEL onboard CGRO. COMPTEL used the classic Compton telescope configuration
with two planes to map the MeV sky in the 1990’s. Figure from Schönfelder et
al. (1993).

The first successful Compton telescope operating in space was COMPTEL
Schönfelder et al. (1993) on NASA’s Compton Gamma Ray Observatory (CGRO,
1991-2000). COMPTEL consisted of two detector arrays: D1 contained a low-Z
liquid scintillator (NE 213A) to enhance the Compton scattering cross-section,
and D2 contains a high-Z NaI(Tl) scintillator to enhance photoabsorption (see
Figure 6). COMPTEL was a very large instrument, with D1 and D2 being $\sim$1.5
m in diameter with a separation of 1.5 m between the two planes, and together
weighing almost 600 kg. The instrument is shown in Figure 9.

COMPTEL opened the MeV range with 9 years of groundbreaking observations and
provided the first all-sky survey in the 0.75 to 30 MeV energy range. The
COMPTEL source catalog Schönfelder et al. (2000) includes 63 sources, 32 of
which are steady state, from spin-down pulsars and stellar black-hole
candidates to supernova remnants and active galactic nuclei. Potentially the
most notable science achievement from COMPTEL observations was the all-sky map
of the 1.8 MeV line from 26Al decay Diehl et al. (1993); Plüschke et al.
(2001), which showed structure and hot spots consistent with star forming
regions. While significant in its contributions, COMPTEL’s achieved
sensitivity was still modest Schönfelder (2004).

While the instrument was massive, COMPTEL’s efficiency was low. The effective
area, which is a measure of the equivalent area of an ideal detector, ranged
from only 10 to 50 cm2 depending on energy and event selections, i.e. the
efficiency of detecting incident photons was less than 1%. COMPTEL’s angular
resolution of a few degrees was limited at low energies by the energy
resolution of the scintillators, and at higher energies by the position
resolution of D1 and D2 (see Section 5.1). COMPTEL’s field of view (FOV;
$\sim$1 sr) was limited by the lower-energy thresholds of the two detector
planes.

COMPTEL employed two essential background reduction features. Due to the large
separation between D1 and D2, a time-of-flight measurement was possible to
determine the direction of the scattered photons. A selection of downward
scattered photons was performed with a time-delay window of 4.5 ns. Secondly,
the D1 liquid scintillator allowed for a discrimination of photon and neutron
interactions based on differing pulse shapes. This further reduced the
background observed in flight. These powerful background-rejection techniques
allowed COMPTEL to achieve reasonable sensitivity despite its low effective
area, and ultimately led to COMPTEL’s unprecedented mapping of the MeV sky.

###  3.2 Modern Compton Telescopes

COMPTEL’s double-scattering concept with large separated planes was a limiting
factor for the FOV and measurable Compton scatter angles, and thus led to the
low efficiency of the instrument. Additionally, COMPTEL was only able to
measure two-site events, i.e. a single Compton scatter in D1 and a
photoabsorption in D2. However, depending on the atomic number of the detector
material, a photon can scatter multiple times before finally stopping in a
photoabsorption event. Ideally, one would want to track the photon trajectory
by measuring all secondary and tertiary (and beyond) interactions after the
initial Compton scatter. While COMPTEL effectively measured interactions in
two planes, most modern Compton telescopes are based around the design
philosophy of building a 3D position-sensitive detector volume that acts both
as the scatterer and absorber. With modern advancements in detector technology
achieving (sub)millimetre spatial resolution, the efficiency and performance
of Compton telescopes have significantly improved in the past few decades
since COMPTEL’s development.

Figure 10: Most modern Compton telescopes use high position-resolution
detectors with good energy resolution to measure the position and energy of
multiple interactions, with no geometrical limit on the Compton scatter angle.
As in Figure 8, the direction of the photon is determined by the Compton
scattering angle of the first interaction. Modified from Kierans (2018).

Combining the scattering and absorbing detector capabilities without large
separation allows for the detection of multiple Compton scatters with no
geometric limitation on the scattering angle, as long as the detector volume
contains the interactions. Both of these features increase the efficiency of
the telescope and improve upon the polarization capabilities (discussed in
Section 3.4 and 5.4) compared to the classic Compton telescope design. An
example of this is shown in Figure 10, where a photon Compton scatters twice
before fully depositing its energy with a photoabsorption event. All three
interactions are measured within a large 3D position-sensitive detector
volume, indicated by the blue area.

For a three-site event, as in Figure 10, the Compton scattering angle of the
first interaction can be determined in the same way as for two interactions;
however, the total absorbed energy is now $E_{0}=E_{1}+E_{2}+E_{3}$, and the
energy deposited after the second interaction must be accounted for in
$E_{scat}$ in Equation 4:

$\cos\varphi=1-\frac{m_{e}c^{2}}{E_{2}+E_{3}}+\frac{m_{e}c^{2}}{E_{1}+E_{2}+E_{3}}.$
(16)

This formula can be scaled to a large number of interactions as long as the
initial energy includes all energy deposits from the fully-absorbed photon
$E_{0}=E_{1}+E_{2}+...+E_{n}$, and the scattered photon energy is equal to the
total deposited energy excluding $E_{1}$: $E_{scat}=E_{2}+...+E_{n}$. The
additional event pattern recognition achievable with these multi-scatter
Compton telescopes enables sophisticated background rejection techniques, as
discussed further in Section 4.

After COMPTEL, there were multiple competing technologies that were trying to
achieve the next-generation MeV telescope design with position-sensitive
detectors Kurfess et al. (2000). One such instrument developed in the 1990’s
was the Liquid Xenon Gamma-Ray Imaging Telescope (LXeGRIT) Aprile et al.
(2000a, b). LXeGRIT used a time projection chamber (TPC) filled with liquid
xenon to achieve the event-by-event Compton imaging within one large detector
volume. Gamma rays interact with xenon via ionization and a release of
ultraviolet scintillation photons. When a high voltage is applied, the charges
drift to the anode and cathode, and allow for the total energy deposit to be
recorded. The X-Y interaction location is measured by orthogonal sensing wires
embedded in the liquid xenon, and the Z coordinate is determined via the drift
time. The homogeneous detector volume allowed for a much more efficient design
than the classic double-scattering Compton telescope. Similar techniques are
used in other modern Compton telescopes SMILE Tanimori et al. (2015); Takada
et al. (2022) and GRAMS Aramaki et al. (2020).

The large instrumented detector volume realized by liquid or gaseous TPCs have
their advantages, but are generally limited in stopping power and energy
resolution. Other instrument designs in the 1990’s and early 2000’s started
employing semiconductor detectors, which have the advantage of superior energy
resolution. However, it is not possible to fabricate or instrument arbitrarily
large active volumes of semiconductors; the detection volume must be segmented
into smaller detectors each with energy and internal position sensitivity.
This was achieved in the early days through the use of the ubiquitous
semiconductor silicon Kamae et al. (1987). The telescope volume was broken up
into thin planes of silicon with 2-D resolution, where the Z-dimension of the
interaction was defined as the location of the hit detector plane with no
internal depth resolution. A multilayered design was developed by both the
Medium Energy Gamma-ray Astronomy (MEGA) telescope Bloser et al. (2002);
Kanbach et al. (2005) and the Tracking and Imaging Gamma-Ray Telescope (TIGRE)
Tumer et al. (1994) and proved to be a viable detection technique for Compton
telescopes. The added advantage of the thin silicon detector planes was the
potential to measure the direction of the Compton-scattered electron in more
than one of the silicon planes, which can then be used to further constrain
the kinematics of the scatter, as will be discussed further in Section 3.3.

Other methods of building a 3D position-sensitive detector volume with good
energy resolution have been explored since COMPTEL showed the power of Compton
imaging. Scintillators, where the internal interaction position within a
crystal can be inferred from the light intensity measured at the ends of long
scintillating bars, have been used when energy resolution is not a priority,
but a large effective area or stopping power is a driver. Semiconductor
detectors, with their high spectral resolution, are ideal for Compton
telescopes, since the better energy resolution provides a more accurate
measure of the Compton scattering angle $\varphi$. Other than the thin multi-
layer silicon detector approach, both germanium Amman and Luke (2000); Coburn
et al. (2003) and CZT Xu (2006); Du et al. (2001); Lee et al. (2016) with a
large form factor have been demonstrated. High-purity double-sided strip
germanium detectors will be used in COSI Tomsick and COSI Collaboration
(2022). With an individual crystal size of $8\times 8\times 1.5$ cm3, the
total telescope consists of 16 tightly packed germanium crystals with
independent readout. The COSI detectors achieve an internal position
resolution of $\lesssim 1.5$ mm3: the X-Y position is determined by the charge
collection on orthogonal strips, and the Z dimension is determined from the
charge collection timing Amman and Luke (2000). Other semiconductor-based
Compton telescopes are discussed in Section 6.

Figure 11: There are three common configurations for Compton telescopes. (a)
The classic Compton telescope uses two detector planes as a scatterer and
absorber to measure the interaction of double-scattering events. (b) The
compact Compton telescope uses a single 3-D position-sensitive volume to
measure multiple Compton scatter interactions. (c) The multi-layer design uses
many thin layers of detectors, with the added advantage of electron tracking.
Modified from Yabu (2022).

As long as a precise measurement of the energy and 3D location of each
interaction is achieved, the actual detector implementation for a Compton
telescope can be quite diverse. A summary of three common configurations for
modern Compton telescopes are shown Figure 11. The classic Compton telescope
uses two detector planes, as demonstrated with COMPTEL. Alternatively, a large
3D position-sensitive volume can measure the position and energy of multiple
Compton scatter interactions, increasing the detection efficiency, as
demonstrated with LXeGRIT. The third configuration achieves more precise
energy and/or position information from each scatter using a detection volume
segmented into multiple smaller position-sensitive detectors. One standard
design uses thin detectors with 2D resolution, as demonstrated with MEGA.

Due to the increased efficiency and smaller resolution elements, most modern
Compton telescope are compact in design. One important point to note is that
compact designs no longer have time-of-flight capabilities to determine the
temporal order of interactions. While there are still efforts to modernize
time-of-flight Compton capabilities Bloser et al. (2018), most modern Compton
telescopes require an extra step for event reconstruction; the relative order
of interactions can be determined by the Klein-Nishina cross section and by
redundant information in the energy deposits and relative positions of the
interactions. The process for determining the most probable sequence of
interactions is referred to as Compton sequencing or event reconstruction and
is discussed further in Section 4.

###  3.3 Electron Tracking

In previous discussions, the direction of the incident gamma ray has been
restricted to the Compton event circle where the uncertainty in the azimuthal
direction is due to a lack of knowledge of the Compton-scattered electron’s
direction. Now, if the direction of the recoil electron is known, momentum
conservation can further restrict the direction of each incident gamma ray.
With a more accurate determination of the origin of the gamma ray, background
rejection can be improved Akyüz et al. (2004).

Figure 12: By measuring the direction of the Compton-scattered electron of the
first interaction, the incident photon direction can be constrained within a
reduced arc segment of the original Compton event circle. The measured recoil
electron direction is depicted with blue dots representing energy deposits in
multiple detector layers, and the length of the event arc is dictated by the
precision of the electron trajectory measurement. Modified from Brewer et al.
(2021).

The kinematics of the recoil electron from a Compton interaction can be
calculated in an analogous way to the scattered photon as derived in Section
2. However, since the direction of the Compton-scattered electron $\Theta$ is
less precise than the direction of the Compton-scattered photon due to a short
penetration depth and Molièere scattering Bethe (1953), it is preferable to
define the additional kinematic constraint relative to the original photon
event circle. With the recoil electron direction, the initial event circle can
be reduced to a small segment, defined as the event arc. Figure 12 shows this
concept, where a multi-layer Compton telescope is used to measure energy
deposited from the Compton-scattered electron through multiple detector
layers. The length of the event arc is related to the uncertainty in the
electron’s recoil direction. Figure 13 shows the back-projected Compton event
circles in image space with and without electron tracking; by using the recoil
electron trajectory, the source position is easily discernible with fewer
photons. It is important to note that electron-tracking is a powerful
background reduction technique and allows for a more clearly defined point
spread function in image space, but does not in general improve the angular
resolution of a telescope due to the large uncertainties in the recoil
electron direction. The details on how electron tracking affects the point
spread function of the instrument will be discussed in detail in Section 5.1.

(a) 10 untracked Compton events.

(b) 10 Compton events with electron tracking.

Figure 13: The back-projection of event circles from untracked Compton events
compared with tracked events highlights the benefit of the additional
kinematic information. While the angular resolution does not change, the point
source is more easily discernible, and background contamination can be
significantly reduced. Therefore, electron tracking can enhance the
sensitivity of a telescope. Figure from Mizumura et al. (2014).

The TIGRE and MEGA groups were the first to demonstrate electron tracking
capabilities for Compton imaging; with multiple layers of thin silicon
detectors, they were able to measure the momentum direction of the Compton-
scattered electrons based on the ionization tracks through multiple silicon
layers. Furthermore, electron-tracking Compton imaging has also been
demonstrated in large gaseous time-projection chambers Tanimori et al. (2004);
Kabuki et al. (2007); Orito et al. (2005).

Any telescope that is capable of detecting the track of the Compton-scattered
electron also has the enhanced capability of being sensitive to pair-
production events. As the gamma-ray energy increases and pair-production
become dominant, the direction and energy of an incident gamma rays can be
constrained by measuring the electron and positron trajectories to reconstruct
the vertex and opening angle of the pair conversion. A combination electron-
tracking Compton and pair telescope is an enticing technology that can span
the majority of the MeV Gap with one telescope. This has been explored by a
number of different telescope concepts, as discussed further in Section 6.

###  3.4 Dedicated Polarimeter

There also exists a class of Compton telescopes that is not optimized for
imaging, but for polarimetery. As Equation 12 shows, there is an asymmetry in
the cross-section for photons that scatter parallel and perpendicular to the
initial photon’s electric field vector. This asymmetry, or the amplitude of
the polarization response, is maximized when the $\sin^{2}\varphi$ term is
maximized, or at polar Compton scattering angles $\varphi=90^{\circ}$. A
polarized source will result in a sinusoidal distribution in the azimuthal
scattering angle ($\eta$ in Figure 5). Therefore, instruments that measure the
azimuthal scattering angle, especially at large Compton scatter angles, can
ultimately detect the polarization of a gamma-ray source.

(c)

(d)

Figure 14: (a) Photons will predominately scatter at 90∘ relative to their
initial electric field vector, according to Equation 12. Therefore, a gamma-
ray polarimeter can determine the degree of polarization of a source by
measuring the azimuthal scattering direction $\eta$ of photons. Modified from
Lei et al. (1997). (b) The azimuthal scattering direction for a polarized
source gives a sinusoidal response, referred to as the ASAD (azimuthal
scattering angle distribution), which is maximized at 90∘ relative to the
direction of the initial electric field vector. Modified from Bellm (2011).

The simplest version of a Compton polarimeter is shown in Figure 14, where the
initial photon is scattered in a central detector element and absorbed in a
detector placed at $90^{\circ}$. If this secondary detector is rotated to
sample the counts in the azimuthal (Figure 14), the direction and degree of
polarization can be measured, as described in detail in Section 5.4.

Most dedicated polarimeters are compact in design to be sensitive to 90∘
Compton scatter angles. As the azimuthal scatter distribution is sinusoidal,
the distribution can be coarsely sampled without a significant loss of
sensitivity. In addition, the measured energy resolution for dedicated
polarimeters is not as crucial as for imaging Compton telescopes as the energy
information is not used in the reconstruction (but can provide spectral
information for the incoming photons). Therefore, most dedicated polarimeters
take advantage of scintillator detectors, which are simpler than segmented
semiconductor detectors used in most compact and multi-layer Compton
telescopes.

Figure 15: The LargE Area busrt Polarimeter (LEAP) telescope is a dedicated
Compton polarimeter that measures the azimuthal scattering direction with
scintillating bars. LEAP is compact by design and even with fairly coarse
position knowledge of each interaction, the scatter vector can be used to
precisely measure the polarization of incoming photons. Figure from McConnell
et al. (2021).

An example of a dedicated polarimeter telescope is the LargE Area burst
Polarimeter (LEAP) shown in Figure 15. LEAP is a NASA Mission of Opportunity
concept considered in the 2019 selection round, and is similar to other
Compton polarimeter designs. LEAP uses two different types of scintillators to
measure the azimuthal scattering direction: a low-Z plastic scintillator is
optimized for scattering and a high-Z scintillator optimized for absorption.
The use of scintillators allows for a simple, high-efficiency, and large
field-of-view design. However, without the high-resolution 3D position and
energy information for each interaction, Compton imaging is not possible in
dedicated polarimeter. These instruments are therefore optimized for transient
detections when a large signal-to-background is available.

##  4 Event Reconstruction

Event reconstruction for Compton telescopes is the process of translating the
measured positions and energies of detected photon interactions to the initial
gamma-ray direction defined by the event circle (or the event arc for
electron-tracking Compton telescopes). For classic Compton telescopes with a
limit of two interactions and time-of-flight information to determine the
temporal sequence of hits, event reconstruction is a fairly straightforward
process. With the measured energy and Equation 4, the Compton scatter angle of
the first interaction can be determined, and the axis of the event circle can
be defined from the measured positions (see Figure 9).

As modern Compton telescopes become more compact and sophisticated, so too
have the event reconstruction techniques. The timing resolution is not
sufficient to determine the temporal sequence of interactions in compact
Compton telescopes. Therefore, in order to determine the Compton event circle,
one must first determine which energy deposits correspond to the first and
second interactions for each gamma ray. This is done by testing all $N!$
combinations of the interactions for an event with $N$ energy deposits and
determining the most probable sequence.

The process of event reconstruction for modern telescopes is computationally
intensive; however, there are also significant benefits. The first is that
with the added number of allowable interactions compared to a classic double-
scatter Compton telescope, the efficiency of the instrument increases
significantly. The average number of interactions depends on the photon energy
and the detector material, but higher energy photons can often scatter $>$5
times in a detector before being photoabsorbed; sophisticated reconstruction
techniques are needed to properly trace a high-energy gamma-ray path through a
large detector volume. Second, and more importantly, the main advantage of
event reconstruction is background rejection Aprile et al. (1993); Boggs and
Jean (2000). Events that have any missing energy, from incomplete absorption
or interactions in passive material, or events associated with $\beta^{+}$
decays within the instrument are effectively background events, but with a
filtering step in the reconstruction, these unusable events are rejected and
the signal-to-background ratio increases.

The first step of reconstruction is event identification, outlined in Section
4.1, where it must be determined whether or not the measured energy deposits
are consistent with a Compton scattering event (or a charged particle track or
pair event), and, when applicable, whether there are measured electron tracks.
Then the Compton sequence determination can be achieved by taking advantage of
redundant information in the kinematics and geometrical information measured
for each event. The standard approach is referred to as Compton Kinematic
Discrimination Boggs and Jean (2000), or mean squared difference method. This
straightforward and fast sequence reconstruction technique is employed by
various research groups and will be detailed in Section 4.2. It is difficult
to unambiguously determine the correct order for events which only have two
interactions and no discernible temporal separation, and thus two-site event
reconstruction will be considered separately in Section 4.3.

###  4.1 Event Identification and Track Recognition

Before determining the correct sequence of interactions for an event, one must
characterize the measured energy deposits and determine if they are consistent
with a Compton-scattering event, a pair event, a charged-particle track, or
any other potential interaction. Telescopes that have tracking capabilities
have more possible event types to identify and will be the basis of the
presented approach, which follows Zoglauer (2005). Detectors without electron
tracking may use a simplified version of the steps listed below.

Photons which interact with multiple Compton scatters within a detector volume
leave separated, isolated energy deposits as shown in Figure 10, i.e., there
is no easily identifiable straight path or pattern. Therefore, it is easiest
to first confirm if the measured interactions are consistent with event types
that do have easily recognizable patterns (i.e. pair events or charged-
particle tracks); if no clear pattern is found, then compatibility with
Compton interactions can be checked. The general steps for event
identification for a tracking Compton telescope are as follows:

  1. 1.

Search for pair event vertex

  2. 2.

Search for high-energy charged particle straight tracks

  3. 3.

Search for recoil electron tracks from Compton interactions

  4. 4.

Search for Compton interaction sequence

Pair events are recognizable by the characteristic inverted “V” from the
electron and positron (i.e., the pair) ionization tracks in the detector
volume. In a multi-layer Compton telescope, as described in Section 3.2, the
interactions from the electron and positron are measured after conversion in
subsequent layers with increasing separation. If a vertex is found with a
pattern recognition algorithm, the event can be flagged as a pair event and
the initial photon direction can be determined by the trajectory of each
particle, weighted by their relative energies. Pair reconstruction is
described in detail in Charles and Chiang (2021) and has been matured for the
Fermi Large Area Telescope Atwood et al. (2009).

The second easily identifiable event type is the straight tracks left by high-
energy charged particles, such as cosmic rays and muons. With linear
ionization tracks, the reconstruction of high-energy charged particles is
straightforward and can be accomplished by fitting a straight line to the
measured energy deposits.

#### Recoil Electron Track Reconstruction

The third step in event identification is to search for tracks left by Compton
recoil electrons. In Compton scattering, when a photon interacts with an
electron, it will transfer momentum in the scatter, causing the electron to be
ejected from its atom and recoil. The recoil electron will then leave an
ionization track in the detector material, and the length of the track is
based on the initial electron momentum and the density of the material. The
track lengths in high-Z materials, such as germanium, CdTe, and CZT, are too
short to measure the direction of motion and the full electron energy is
deposited within one position resolution element of the detector. For lower Z
material, such as silicon, or the low densities found in gaseous time
projection chambers, the track length can be longer than a few millimeters and
span multiple resolution elements. For example, in multi-layer Compton
telescopes consisting of thin 2D sensitive silicon detectors, the recoil
electron can deposit energy in 3-4 layers with $\sim$1 cm spacing.

There are two main purposes for electron track recognition in event
reconstruction: the initial Compton-scattering interaction location is needed
for Compton sequencing (Section 4.2), and the direction of motion of the
electron can be used to kinematically constrain the incoming photon direction
(Section 3.3 and 5.1). Unfortunately, the initial interaction location and
electron trajectory are not always easy to discern from the measured energy
deposit as ionization tracks left by recoil electrons are rarely straight and
often short, as seen in Figure 16. The paths of recoil electrons can make
U-shaped tracks, and with poor spatial resolution or coarse sampling in a
multi-layer Compton telescope, the discrete sampling of the energy deposits
complicates the reconstruction. Higher energy electrons ($\gtrsim$1 MeV) will
have long and initially straight tracks, but at lower energies, the track
length decreases and the non-linearity increases.

Figure 16: The measured recoil electron track from Plimley et al. (2011) shows
the curving nature of the path and the complication of track reconstruction
(see Section 5.1). This high spatial resolution measurement, with each pixel
being $10.5\times 10.5$ $\mu$m2, is the first demonstration of a reconstructed
recoil electron track within a 650 $\mu$m-thick silicon charge-coupled device
(CCD). The deposited energy from the recoil electron is 227 keV. Based on the
Bragg peak, the larger energy deposit on the bottom right is likely the end of
the track.

There have been multiple approaches to the electron track reconstruction. The
simplest version relies on the assumption that as the electron loses energy,
it will tend to deposit more energy, as defined by the Bethe-Bloch equation
and the Bragg peak Particle Data Group et al. (2020). Therefore, the track end
point with the smaller measured energy is more likely to be the beginning of
the electron’s path. The electron track reconstruction has been performed
using different algorithms: a figure-of-merit approach Zoglauer (2005),
Bayesian approach Zoglauer (2005), momentum analysis Black et al. (2007), and
graph theory Yoneda et al. (2018). Different approaches are used based on the
relative spatial resolution of the track.

Measuring the electron-scatter direction allows one to further constrain the
kinematics of the Compton scattering interaction, reducing the photon
direction from a back-projected circle to an arc. The precision of the
electron recoil direction determines the length of the event arc. Since the
precision of the electron’s direction is often much worse than that achieved
through the measurements of the scattered photons, this additional kinematic
information does not in general improve the angular resolution but can serve
as a method to reduce the observational background.

As described in Section 3.3 and shown in Section 2, the kinematics of the
recoil electron can be treated in the same way as the scattered photon.
Haefner et al. (2014) demonstrated that the source location can be determined
entirely by the recoil electron without needing to measure the Compton-
scattered photon, increasing the measurement efficiency. However, the angular
resolution and sensitivity of this detection technique is limited.

After pair and high-energy charged particle events have been identified, and
Compton recoil electron tracks have been reconstructed to determine the
initial interaction position, the measured energy deposits can be tested for
compatibility with Compton scattering. The processes of determining the most
probable sequence of interactions for a single photon is called Compton
sequencing.

###  4.2 Compton Sequencing

Compton event reconstruction starts with a collection of hits within the
detector volume, each with energy and position information. With no prior
knowledge or assumptions about the first interaction location, all $N$!
possible combinations of the hit order, where $N$ is the number of
interactions, must be analyzed. The first algorithm developed for kinematic
event reconstruction was introduced by Aprile et al. (1993), and later
formalized by Boggs and Jean (2000). It is referred to as Compton Kinematic
Discrimination, the chi-squared approach, or mean squared difference method,
and as it remains one of the more popular approaches for events with 3$+$
interactions, it will be described here.

Figure 17: In Compton Kinematic Discrimination, the Compton scatter angle
$\varphi_{l}$ of a central interaction $l$ is determined in two ways:
kinematically with the energies $E$ and geometrically with the scatter
directions $\vec{g}$. Minimizing the difference between these two measures for
each Compton scatter leads to the most probable sequence of interactions.
Modified from Zoglauer (2005).

For gamma-ray events which have at least 3 interactions (two Compton scatters
and one photoabsorption; see Figure 17), the Compton scatter angle of the
central interaction(s), denoted by $l$, can be determined in two ways. The
first is kinematically with the Compton equation

$\cos\varphi^{kin}_{l}=1-\frac{m_{e}c^{2}}{E_{l_{+}}}+\frac{m_{e}c^{2}}{E_{l}+E_{l_{+}}},$
(17)

where $E_{l_{+}}$ is the total energy of all interactions following $l$;
$E_{l}$ is not included in $E_{l_{+}}$. Second, the Compton scatter angle can
be determined geometrically considering the angles

$\cos\varphi^{geo}_{l}=\frac{\vec{g_{k}}\cdot\vec{g_{l}}}{|\vec{g_{k}}||\vec{g_{l}}|},$
(18)

where $\vec{g_{k}}$ is the incoming gamma-ray direction and $\vec{g_{l}}$ is
the outgoing direction for interaction $l$.

The two measures of $\cos\varphi_{l}$, from Equation 17 and 18, should be
identical for the correct order of interactions in an ideal instrument. In the
classic Compton Kinematic Discrimination approach, a $\chi^{2}$ quality factor
$Q$ is assigned to each permutation of $N$ interactions (Zoglauer, 2005)

$Q=\sum^{N-1}_{i=2}\frac{(\cos\varphi^{kin}_{i}-\cos\varphi^{geo}_{i})^{2}}{(d\cos\varphi^{kin}_{i})^{2}+(d\cos\varphi^{geo}_{i})^{2}},$
(19)

where $d\cos\varphi_{i}$ are the uncertainties determined through error
propagation and $i$ is the interaction index. The sequence with the lowest
quality factor is the best estimate of the correct kinematic ordering of the
event.

Another common method for Compton sequencing uses a deterministic approach
based on the Klein-Nishina formula Xu et al. (2004). Again, all $N$! possible
combinations of the interaction sequence must be checked, but the most
probable sequence of energy deposits is based on the physics of the
interaction. Recent work was done to improve upon the deterministic event
reconstruction techniques by developing a more efficient probabilistic method
that can account for escape gamma-ray events Yoneda et al. (2021). For events
with three interactions or more, it has been shown that the gamma-ray energy
can be estimated even if the event is not fully absorbed Kurfess et al.
(2000); Tashenov and Gerl (2010).

In real measurements, the reconstruction becomes more complicated due to the
finite spectral and spatial resolution of the detectors, sub-threshold
interactions, and energy deposits in passive material. Any missing energy will
result in a mis-reconstructed event and the calculated event circle will not
align with the initial photon direction (see Section 5.1). The level of
accuracy for event reconstruction is therefore a very important performance
parameter for Compton telescopes. The standard Compton Kinematic
Reconstruction algorithm was initially found to properly reconstruct $50-70\%$
of 3$+$ site events Oberlack et al. (2000); Boggs and Jean (2000). The
standard approach has been modified for improved efficiency by a number of
groups, for example, by assuming the first interaction is the largest energy
deposit, the sequencing performance improves by 20% for $\gtrsim$1 MeV gamma
rays Shy and He (2020). An evaluation of different Compton sequencing
approaches can be found in Thrall et al. (2008); Lee et al. (2021). More
recently, advancements in machine learning approaches have shown promise for a
much higher reconstruction efficiency Zoglauer and Boggs (2007); Zoglauer and
team (2021); Cao et al. (2019).

One of the main advantages of Compton sequencing is the capability to reject
events which do not conform with the expected kinematics of Compton
scattering. This is effectively a method of background rejection. Photons that
do not deposit all of their energy in the active detector volume will not be
properly reconstructed and are thus improperly imaged and contribute to the
background during observations. By setting a maximum acceptable level for the
quality of the event (e.g., given by Equation 19 for the classic Compton
Kinematic Discrimination), any improperly reconstructed Compton event can be
rejected. Likewise, pair production or $\beta$-decay events in the instrument
can be rejected with this approach. Only events that are statistically
determined to be “good” Compton events are kept after Compton sequencing. The
challenges of high backgrounds in Compton telescopes are addressed further in
Section 5.5.

###  4.3 Two-Site Event Reconstruction

For two-site events, with one Compton scatter followed by a photoabsorption,
only a small fraction of events can be sequenced unambiguously Boggs and Jean
(2000). A simple comparison test, the Compton Edge Test Xu (2006), can
distinguish the order of some two-site events. We know that the maximum energy
deposited in Compton scatter is from backscattering given by Equation 7; this
energy defines the Compton edge in Figure 3. It is therefore easy to compare
the two measured energies with this relation, where $E_{0}$ is the sum of the
two energies. If either of the measured energies is larger than the maximum
allowable energy of the Compton edge for the given $E_{0}$, then that
interaction is necessarily a photoabsorption and must be the second
interaction in the sequence.

Another simple relation can be used if the initial photon energy is less than
256 keV Lehner et al. (2004); Lee et al. (2021). As will be derived below, if
$E_{0}\lesssim m_{e}c^{2}/2$ is satisfied, the energy deposit in the first
interaction, or in the Compton scatter $E_{S}$, is deduced to be smaller than
the energy deposited through photoabsorption in the absorber $E_{A}$. With
$E_{0}=E_{A}+E_{S}$, this can be derived starting again with the maximum
energy of the first interaction defined by the Compton edge in Equation 7:

$\displaystyle\text{max}(E_{S})$ $\displaystyle=$
$\displaystyle\left.E_{e}\right\rvert_{\varphi=180^{\circ}}=E_{0}\left(1-\frac{1}{1+\frac{2E_{0}}{m_{e}c^{2}}}\right).$

If $2E_{0}<m_{e}c^{2}$, then

$\text{max}(E_{S})<E_{0}/2,\\\ $ (21)

and with $E_{0}=E_{A}+E_{S}$

$\displaystyle\text{max}(E_{S})$ $\displaystyle<\frac{E_{A}+E_{S}}{2}$ (22)
$\displaystyle\text{max}(E_{S})$ $\displaystyle<E_{A}.$ (23)

Therefore, when the initial photon has energy less than $m_{e}c^{2}/2$,
$E_{S}$ must be less than $E_{A}$, and the smaller energy deposit must
correspond to the Compton scatter interaction. For all other two-site events
there is no unique ordering; however, it has been shown through simulations
that Equation 23 generally holds until $\sim$400 keV Boggs and Jean (2000);
Lehner et al. (2004). Conversely, if the incident photon energy is greater
than $\sim$400 keV, the first interaction is likely to have a larger energy
deposit.

For a more sophisticated approach, one can use a deterministic algorithm based
on the Klein-Nishina cross-section to determine which hit order is more
probable Xu et al. (2004). In doing so, one can also take into account the
probability of photoabsorption for a certain distance between interactions to
better understand the most likely order.

A drawback of the two-site event reconstruction is the loss of background
rejection; there is no redundant information available to determine if the two
measured interactions correspond to a fully absorbed photon. With the
additional uncertainties with two-site sequencing, the sensitivity of a
telescope may be improved by excluding two-site events for some observations.

##  5 Compton Telescope Performance Parameters

The capabilities of a Compton telescope are driven by the precision of the
energy and position measurements of each interaction within the detector
volume, and the precision of the event reconstruction. The science question at
hand drives the design of a Compton telescope. For example, gamma-ray burst
polarization measurements can be done with simple scintillators to measure the
azimuthal scatter direction without imposing strict requirements on the energy
resolution or spatial resolution of the detectors. However, achieving a
continuum sensitivity that is 1–2 orders of magnitude better than COMPTEL (see
Figure 1) requires a detector with excellent spectral and spatial resolution
and strong background rejection. In this section, the point spread function
for Compton telescopes will be introduced, and the standard Compton telescope
performance parameters, such as the angular resolution and sensitivity, will
be discussed.

###  5.1 Point Spread Function

The raw data space for Compton telescopes consists of all possible
measurements of the energy and position of Compton-scattering interactions
within the detector volume. However, only the first two interactions are
needed for imaging capabilities, where the key parameters are the Compton-
scattered photon direction and the Compton scattering angle, as described in
Section 3. Therefore, Compton telescope analysis is generally done in a
reduced data space consisting of three-dimensions: the Compton scatter angle
$\varphi$, and the polar and azimuthal angles of the Compton-scattered photon
direction, $\chi$ and $\psi$, respectively. These angles define the three
orthogonal axes of the data space, as shown in Figure 18. The total energy of
the gamma ray is generally considered a fourth dimension of this data space,
though it will not be explicitly written here. This data space, along with the
fundamentals of modern Compton telescope analysis techniques, was pioneered
for the COMPTEL mission Schönfelder et al. (1993); von Ballmoos et al. (1989);
Diehl et al. (1992) and is sometimes referred to as the COMPTEL Data Space, or
simply the Compton Data Space (CDS).

(a)

(b)

Figure 18: (a) Schematic diagram of the first two interactions of a Compton
event showing the definition of the CDS angles ($\chi,\psi,\varphi$) for a
gamma-ray source located at ($\chi_{0},\psi_{0}$). The Compton scatter angle
of the first interaction, $\varphi$, is calculated from the energy deposits.
The polar and azimuthal angles of the Compton-scattered gamma ray direction (
($\chi,\psi$) are measured relative to the detector coordinate system. These
three parameters define the kinematics and geometry of a Compton event.
Modified from Kierans (2018). (b) The point spread function in the CDS is an
easily discernible cone with apex at the source location ($\chi_{0},\psi_{0}$)
and a 90∘ opening angle.

The point spread function in the Compton Data Space is a cone with the apex at
the location of the source (Figure 18). The opening angle of the cone is 90∘
since the Compton scatter angle is equal (within measurement error) to the
deviation of the scattered gamma-ray direction from the known source location,
as seen in Figure 18. Therefore, the CDS contains redundant information about
the Compton scatter angle: the $\varphi$ dimension is calculated based on the
measured energies with Equation 4, and the scatter direction ($\chi,\psi$) is
defined relative to the detector coordinates, but is related to the geometric
measurement of the Compton scatter angle. In contrast to the projected event
circles in image space, each Compton event is a point in the CDS at
($\chi,\psi,\varphi$), and the accumulation of properly reconstructed Compton
events from point-source emission fills the CDS along the walls of the three-
dimensional cone. The CDS cone is connected to the familiar Compton event
circle in image space: the probability of any region of the sky contributing
to one bin in the CDS ($\chi,\psi,\varphi$) is represented by the back-
projected event circle.

The response density along the cone follows the Klein-Nishina cross-section.
For example, the smaller Compton scatter angles near the cone apex are likely
to be populated by higher energy events. Furthermore, the photon polarization
information is encoded in the CDS cone. The modulation in the Compton
azimuthal scatter angle $\eta$, defined in Equation 12 and shown in Figure 14,
can be defined in terms of the scatter direction ($\chi,\psi$) in detector
coordinates ($\eta\equiv\psi$ for on-axis sources). The population of the CDS
cone is also impacted by the detector geometry where some scattering
directions can be suppressed.

(a)

(b)

Figure 19: (a) It can be illustrative to rotate the CDS coordinates so that
the measurement of the scattering direction is made relative to the incident
photon direction from a known source location ($\chi_{0},\psi_{0}$). Then the
scattering angles ($\chi,\psi$) in the CDS equate to the geometrically
measured polar Compton scattering angle ($\varphi^{geo}$) and the azimuthal
Compton scattering angle ($\eta$), as defined in Figure 5. (b) For on-axis
sources, the point spread function in the CDS translates into a two-
dimensional plane at 45∘. Modulation from a polarized source will result in a
sinusoidal distribution along the $\psi$ dimension.

To better understand the CDS, it can be illustrative to rotate into a
reference frame where the scattering direction ($\chi,\psi$) is defined
relative to the initial photon direction of a known source position
($\chi_{0},\psi_{0}$). This transformation, as visualized in Figure 19, means
that the polar scattering angle $\chi$ becomes a geometric measure of the
Compton scattering angle $\varphi^{geo}$, and the azimuthal scattering
direction $\psi$ becomes equivalent to azimuthal scatter angle $\eta$. In the
CDS, the distribution of events fills a two-dimensional plane at 45∘, as shown
in Figure 19. The polarization response and geometric asymmetries will now
show as a sinusoidal modulation along the $\psi$ ($=\eta$) dimension. For
unpolarized sources at a known location, the CDS can be further simplified by
projecting into two dimensions (with some loss of discriminating power); by
integrating over the azimuthal scattering angle $\eta$, a simplified 2D CDS
can be defined by $\varphi^{kin}$ and $\varphi^{geo}$ Schönfelder et al.
(1993). A simulated point spread function in this projected 2D Compton Data
Space is shown in Figure 20, where the deviations from
$\varphi^{kin}=\varphi^{geo}$ are from uncertainties in the measured positions
and energies close to the line, and the width of the CDS cone corresponds to
the angular resolution of the instrument. The events that are far from the
$\varphi^{kin}=\varphi^{geo}$ line are improperly reconstructed: an
incompletely absorbed recoil electron results in a measure of
$\varphi^{kin}<\varphi^{geo}$, while incompletely absorbed gamma rays result
in $\varphi^{kin}>\varphi^{geo}$.

(a)

(b)

Figure 20: When the source position is known, projection of the CDS into two
dimensions allows for a direct relation between the kinematically measured
Compton scatter angle ($\varphi^{kin}$) and the geometrically measured angle
($\varphi^{geo}$). Events that are properly reconstructed lie along the
$\varphi^{kin}=\varphi^{geo}$ line, where the spread is defined by
uncertainties in the energy and positions measurements. (a) A simulation of a
511 keV point source measured in a COSI-like instrument shows this 2D
projection of the CDS point spread function. All of the events located off of
the $\varphi^{kin}=\varphi^{geo}$ line are improperly reconstructed, most
often due to missing energy. Modified from Kierans (2018). (b) The
geometrically measured Compton scattering angle is given by the angle between
the initial photon direction $\vec{g_{0}}$ and the scattered gamma-ray
direction $\vec{g_{1}}$. The difference between the kinematic and geometric
Compton scatter angle gives the Angular Resolution Measure (ARM),
$\Delta\varphi_{ARM}$ for each event.

The 3D CDS point spread function (Figure 18) forms the basis for sensitive
Compton telescope data analysis. Compared to the image space with overlapping
Compton event circles causing source confusion, the CDS provides strong
discriminating power and allows for a better separation of background and
source emission regions Zoglauer et al. (2021).

#### Angular Resolution Measure

(a)

(b)

Figure 21: (a) The smallest angular distance between the known source location
and each Compton event circle defines the ARM, $\Delta\varphi_{ARM}$. (b) A
histogram of the $\Delta\varphi_{\text{ARM}}$ values gives a one-dimensional
projection of the point spread function, and the FWHM of the ARM distribution
defines the angular resolution of a Compton telescope. As an example, the
histogram shown here is from a measurement of a far-field laboratory 511 keV
source with the COSI balloon detector Kierans et al. (2017) and has a 6∘
angular resolution.

The angular resolution of a Compton telescope given by the width of the CDS
cone walls and is generally dominated by the uncertainty in the measured
energy and position of interactions. The most common definition of the angular
resolution is the full width at half maximum (FWHM) of the Angular Resolution
Measure (ARM) distribution, which is a one-dimensional projection of the CDS
cone. The ARM for each event is defined as the minimum distance between the
geometrically calculated Compton scatter angle and the kinematically
calculated Compton scatter angle:

$\Delta\varphi_{\text{ARM}}=\varphi^{geo}-\varphi^{kin}.$ (24)

The kinematic Compton scatter angle is determined from the measured energy
deposits using the Compton equation:

$\varphi^{kin}=\arccos\left[1-m_{e}c^{2}\left(\frac{1}{E_{scat}}-\frac{1}{E_{0}}\right)\right]$
(25)

with $E_{0}$ being the total deposited energy, as detailed in Section 2. The
geometrically determined Compton scatter angle is found by taking the dot
product of the initial photon direction $\vec{g_{0}}$ and the measured
scattered gamma-ray direction $\vec{g_{1}}$:

$\varphi^{geo}=\arccos\left[\frac{\vec{g_{0}}\cdot\vec{g_{1}}}{|\vec{g_{0}}||\vec{g_{1}}|}\right],$
(26)

as defined in Figure 20. Note that this is similar to the redundant
information used in Compton Kinematic Discrimination (Equations 17 and 18);
however, here we must know the initial source location to define
$\vec{g_{0}}$. This definition of $\Delta\varphi_{\text{ARM}}$ is equivalent
to the shortest angular distance between the known source location and the
Compton event circle in image space. This is depicted for three event circles
in image space in Figure 21.

The FWHM of the ARM distribution is the standard definition for the angular
resolution. A histogram of the measured $\Delta\varphi_{\text{ARM}}$ values
from a COSI-like detector is shown in Figure 21. For an ideal detector, one
expects a sharp feature at $\Delta\varphi_{\text{ARM}}=0$ indicating that the
event circle correctly overlaps the known source location. Measurement errors
in the position and energy of interactions within the detector broaden this
response. Events that are improperly reconstructed, often due to incomplete
absorption, end up in the wings of this ARM distribution $\sim 50^{\circ}$.
Sophisticated event reconstruction techniques, as discussed in Section 4, and
a selection on the quality of the Compton event can reduce the prominence of
these off-ARM-peak events Zoglauer (2005); Zoglauer et al. (2021).

#### Scatter Plane Distribution

Compton telescopes that have electron tracking allow for a more defined point
spread function in image space, as shown in Figure 13. By measuring the recoil
electron trajectory from the first Compton interaction in a telescope, the
standard event circle can be reduced to an arc.

The measured direction of the recoil electron is less precise than the
direction of the scattered gamma ray (longer lever arm) and, therefore, the
ARM is still used to describe the angular resolution of an electron-tracking
Compton telescope. It is convenient to define the electron’s kinematics
relative to the center of the photon cone defined by $\vec{g_{1}}$, and this
is commonly done in terms of the Scatter Plane Deviation (SPD). The SPD
describes the angle between the scattered plane defined by the known source
location and the true scattered gamma-ray direction ($\vec{g_{0}}$ and
$\vec{g_{1}}$ in Figure 20), and the measured scatter plane defined by the
scattered gamma-ray direction and the recoil electron direction ($\vec{e}$):

$\Delta\nu_{SPD}=\arccos\left[(\vec{g_{1}}\times\vec{g_{0}})\cdot(\vec{g_{1}}\times\vec{e})\right].$
(27)

This is equivalent to the angular distance between the known source location
and the calculated photon origin for each event circle, as depicted in Figure
22. The SPD is a measure of the length of the Compton event arc.

Figure 22: The SPD $\Delta\nu_{SPD}$ is defined as the angular distance
between the known source location (yellow star) and the measured origin from
the scattered electron and gamma-ray kinematics. Adapted from Zoglauer (2005).

The Compton data space for a tracking telescope includes two more dimensions
to describe the recoil electron direction. While a 5+ dimensional data space
can be used for the most accurate Compton telescope analysis with enough
computation resources, it is currently more common for the SPD to be used as
an event selection in list-mode imaging approaches (see Section 5.3). By only
using events with a small SPD, the photon direction can be more precisely
constrained and the image-space representation of the source is much more
defined. In the presence of background, a selection on the SPD for each gamma-
ray event significantly reduces the number of background photons that overlap
with the source region, increasing the sensitivity of the telescope.

Electron tracking capabilities have been achieved in both gaseous and solid-
state detectors. The low density in gaseous TPCs allows for the recoil
electron to subtend a few centimeters in the detector volume, and thus can
result in a fairly accurate measure of the initial scattering direction. The
SMILE-II instrument demonstrated a FWHM of measured SPDs $\sim 75^{\circ}$ for
events from a 137Cs source that transferred more than 80 keV to the recoil
electron Tanimori et al. (2015). However, low-density gaseous TPCs require a
large detector size and are limited in terms of the energy resolution, which
in turn limits the angular resolution. Semiconductor detectors have advantages
with high density and high spectral resolution; however, the path length for
recoil electrons is generally much shorter. The recoil electron direction can
be measured in multi-layer silicon Compton telescopes O’Neill et al. (2003);
Kanbach et al. (2005), but the effects of Molière scattering are a limitation;
Kanbach et al. (2005) achieved a SPD FWHM of $\sim 80^{\circ}$ for a 88Y
source. To more precisely measure the recoil electron direction in silicon,
the detector requires a spatial resolution around 10 $\mu$m. Track
reconstruction within a single solid-state detector layer was first
demonstrated for Charged Coupled Devices (CCDs) with $10.5\times 10.5$ $\mu$m2
pixel size Vetter et al. (2011); Plimley et al. (2011); see Figure 16. More
recently, a hybrid detector with complementary metal–oxide–semiconductor
(CMOS) 20 $\mu$m pitch readout on one side of a 500 $\mu$m-thick silicon
wafer, and strip electrodes on the other side, was developed to achieve high
spatial resolution tracking information, improved energy resolution, and
$\mu$sec timing information Yoneda et al. (2018); Yabu (2022).

#### Uncertainties in the Angular Resolution

The angular uncertainty for each event can be determined from the
uncertainties in the measured energies, which contribute to the Compton
scatter angle calculation, and the uncertainties in the measured positions of
the first two interactions, which determine the axis of the event circle von
Ballmoos et al. (1989). While there is a fundamental limit to the angular
resolution of Compton telescopes (see Section 5.1), in most modern designs,
the dominant factor is usually either the position resolution or energy
resolution of the detectors.

Figure 23: The error in the angular resolution is affected by both the
uncertainty in the measured energy of the scattered gamma ray and the recoil
electron. These are plotted separately assuming a fraction error: the solid
lines show the error in the angular resolution from only the error from the
scattered gamma-ray energy of $\Delta E/E_{scat}=0.5,2,5,$ and 10% of $E_{g}$,
given by Equation 28. The dashed lines show the contribution from the
uncertainty in the recoil electron energy with the same fractional error. The
total photon energy was assumed to be 2 MeV. Adapted from Zoglauer (2005).

It can be seen from Equation 4 that the measured energy of each interaction
plays a role in calculating the Compton scattering angle for each event, and
thus is a defining factor for the ARM. The precision with which the energy can
be measured, i.e. the energy resolution, depends on the detector properties
and readout electronics. A propagation of energy measurement errors for
Equation 4 gives the following error on the Compton scattering angle Zoglauer
(2005); Lehner et al. (2004):

$d\varphi^{kin}=\frac{E_{0}}{\sin\varphi}\sqrt{\left(\frac{1}{E_{scat}^{2}}-\frac{1}{(E_{e}+E_{scat})^{2}}\right)^{2}dE_{scat}^{2}+\frac{1}{(E_{e}+E_{scat})^{4}}dE_{e}^{2}}$
(28)

where $E_{0}$ is the initial photon energy, $dE_{scat}$ is the uncertainty in
the Compton-scattered photon energy, and $dE_{e}$ is the uncertainty in the
recoil electron energy, using the same notation in Section 2. For events with
more than two interactions, $dE_{scat}$ will include the measured energy error
from each additional interaction added in quadrature. This relation is shown
in Figure 23 where the initial photon energy is assumed to be 2 MeV, and the
two different contributions, $dE_{scat}$ and $dE_{e}$, are plotted separately
for set errors of $\Delta E/E$ = 0.5, 2, 5 and 10%. The angular resolution is
significantly impacted by a poor energy measurement of the scattered gamma
ray, especially at higher Compton scatter angles. It is clear that to achieve
an angular resolution $\sim 1^{\circ}$, the energy resolution of the detector
must be $\lesssim 1-2$%, and by selecting only events which have a smaller
Compton scatter angle, the angular resolution can be improved.

The uncertainty in the position measurements for the first two interactions
translates to an error in the axis of the Compton circle. When the distance
between these two interactions is small, this effect becomes more pronounced.
For example, for interactions that are 0.5 cm apart, a spatial resolution of 1
mm results in a significant error on the scattering direction compared to
events that are separated by $>1$ cm. The size of this effect can be simply
estimated as $d\varphi^{geo}\lesssim\text{tan}(\frac{dx}{D})$, where $dx$ is
the spatial resolution and $D$ is the distance between interactions. For a 1
mm spatial resolution, the potential angular deviation is as much as 11∘ for a
separation of 0.5 cm and only 6∘ at separation of 1 cm. This is only an
approximation, and likely an upper limit since there is a higher probability
of interacting near the center of a spatial resolution element than near the
corner when the scattering direction is off-axis Lehner et al. (2004). For a
detailed calculation of the angular uncertainty from the detector position
resolution considering different resolutions in each dimension, see Xu (2006);
Takeda (2009); Xu et al. (2004). In general, the uncertainty in the angular
resolution from the error in the position measurements is more pronounced at
lower energies since these events will have a smaller distance between the
first and second interactions.

Figure 24: The simulated angular resolution for a silicon and cadmium
telluride instrument is shown broken out into the main sources of error. The
uncertainty from the position resolution ($\sim$2∘) is relatively independent
of energy and starts dominating above 300 keV. The energy resolution and
Doppler broadening is highly energy-dependent and dominates the ARM below 300
keV. Figure from Takeda (2009).

The total angular uncertainty is given by a sum of the squares of the two
independent components:

$d\varphi_{total}=\sqrt{(d\varphi^{kin})^{2}+(d\varphi^{geo})^{2}}.$ (29)

As an example of how the different contributions affect the angular resolution
of a telescope, Figure 24 shows the measured and simulated FWHM of the ARM
distribution for a balloon-borne silicon and cadmium telluride Compton
telescope Takeda et al. (2007). The simulated response is broken down into the
three main contributing factors: the energy resolution, position resolution,
and Doppler broadening (Section 5.1). The angular resolution is dominated by
uncertainties in the energy $<300$ keV, and the position uncertainty, which is
relatively independent of energy, starts to dominate at higher energies.

The effect of the position and energy error on the angular resolution can be
reduced through event selections. If events with larger Compton scatter angle
are rejected, then the contribution of the error measurement on the scattered
gamma-ray energy will be reduced. Similarly, if events with a small separation
distance between the first and second interaction are rejected, then events
with a more precise measurement of the scattering direction will remain. Both
examples can result in improved angular resolution at the cost of using fewer,
higher-quality events in the analysis. There is a constant trade-off between
the angular resolution and the efficiency of the telescope and different
observations will call for different optimizations.

#### Doppler broadening as a Lower Limit to the Angular Resolution

The Klein-Nishina cross section (Equation 10) was derived with the assumption
that the target electron is a free particle at rest. However, in a Compton
telescope the electrons are bound in an atom with finite momentum. Therefore,
there is an additional uncertainty in the interaction energy which
fundamentally limits the angular resolution that can be achieved with a
Compton telescope Zoglauer and Kanbach (2003). This effect is referred to as
Doppler broadening Du Mond (1929).

The Klein-Nishina formula can be modified to include the effects of the bound
electron’s momentum. According to the modified formula Ribberfors (1975);
Tumer et al. (1994), the magnitude of Doppler broadening depends on the
initial photon direction, Compton scattering angle, and the atomic number
Zoglauer and Kanbach (2003); Takeda (2009). The angular uncertainty is
generally small for low-Z material, such as plastic scintillator or silicon,
but can be a dominant factor for high-Z material, such as germanium, CdTe, or
CZT.

Figure 25: Doppler broadening is a fundamental limit to the angular resolution
of a Compton telescope, and the magnitude of the effect is highly dependant on
the material atomic number Z and the initial photon energy. The angular
resolution, as measured by the FWHM of the ARM distribution, is simulated in
Geant4 for common detector materials. Figure from Yabu et al. (2021).

Yabu et al. (2021) performed a study of the Doppler-broadening effects in
silicon and other common detector materials. Figure 25 shows the angular
resolution simulated with the Geant4 Agostinelli et al. (2003) toolkit as a
function of energy for CdTe, CZT, germanium (Ge), NaI, liquid argon (Ar),
gadolinium aluminum gallium garnet crystal (GAGG), and silicon. Silicon, with
its low atomic number, has the lowest achievable angular resolution among the
simulated materials, with the Doppler-limited angular resolution almost a
factor of two lower than CZT. There is also a strong dependence on the shell
structure of the atoms: the worst uncertainty is obtained when the electron
orbitals are completely filled and the best is obtained for alkaline or
alkaline earth metals Zoglauer (2005). Figure 24 shows the Doppler broadening
contribution to the total angular resolution of a silicon and cadmium
telluride instrument, and the relation to other uncertainty factors from the
measured energy and position resolution of the detectors.

###  5.2 Sensitivity

The most impactful performance characteristic of a Compton telescope is its
sensitivity (see Figure 1). It is equivalent to the lowest source flux
detectable and is derived from the standard way of calculating detection
significance. The sensitivity of a telescope is given by Jean et al. (1996);
Staubert (1985):

$F_{min}(E)=\frac{n\sqrt{N_{S}+N_{B}}}{A_{eff}T_{obs}}=\frac{n^{2}+n\sqrt{n^{2}+4N_{B}}}{2A_{eff}T_{obs}}$
(30)

where $n$ is the detection significance (typically chosen to be 3 or
5$\sigma$), $N_{S}$ is the number of source photons detected, $N_{B}$ is the
number of background photons, $A_{eff}$ is the effective area, and $T_{obs}$
is the observation time.

At first glance, the main contributions to the sensitivity are the number of
detected background counts, the effective area, and the time. However, $N_{B}$
is defined by the size of the source resolution element and depends on the
angular resolution measure, the energy bin width, and the SPD (for tracking
detectors), which all in turn depend on the detector energy resolution and/or
position resolution. Only the source and background photons whose event
circles overlap with the source position are included. The standard assumption
is to use a selection on the ARM distance from the source position of
1$\times$FWHM of the ARM distribution. The background counts are calculated by
integrating the expected background spectrum in the measured energy bin
$\Delta$E.

The narrow-line sensitivity can be calculated for gamma rays with known
energy, such as nuclear lines. Equation 30 can be used; however, only the
number of background photons within the energy resolution of the instrument
are included, where $\Delta E$ is usually taken to be 1.5–2$\times$FWHM. The
narrow-line sensitivity, therefore, can be enhanced by having an excellent
energy resolution as fewer background photons will lie beneath the photopeak.

To properly calculate the sensitivity of a Compton telescope, detailed
instrument simulations with an accurate detector response are necessary. The
background count rate in a certain resolution element can only be determined
through simulations since a significant fraction of the background
contribution comes from activation of the instrument material (Section 5.5).
And an accurate measure of the effective area is only possible through
detailed instrument simulations, where event selections can significantly
affect the outcome. Furthermore, calculations performed in the Compton data
space instead of the image space allow for a much lower sensitivity since the
source and background regions are much easier to separate. For examples of
sensitivity calculations for Compton imagers, see Boggs and Jean (2001);
Zoglauer (2005); Aprile et al. (1998).

###  5.3 Imaging Capabilities

The goal of Compton telescope imaging is to recover an accurate measurement of
the spatial distribution of source emission. The simplest imaging approach for
Compton telescopes is the back-projection of event circles onto the sky (see
Figure 8 and 13). One can discover the position of gamma-ray sources by
detecting regions in which multiple event circles overlap one another; the
density of the event circles in each bin can be used to identify sources.
Figure 26(a) shows that the back-projection of 200 simulated gamma-ray events
clearly reveals the source in the center of the field-of-view, but there are
many additional features in the image due to the extended nature of Compton
event circles. Considering the high-background environments in space
observations, more sophisticated techniques are needed to reveal the true
source distribution.

(a) Back-projection of a point source simulation with 200 events.

(b) Image obtained after 5 iterations of the LM-ML-EM algorithm.

Figure 26: (a) The back-projection of a 500 keV point source simulation with a
Compton telescope detecting 200 events. The event circles overlap at the
location of the source in the center of the image, shown with the red hot
spot. (b) After 5 iterations of the LM-ML-EM imaging algorithm on the image in
(a), the original source is reconstructed. From Kierans (2018).

The measured imaging response of a Compton telescope to a source with
intensity distribution $I(l,b)$ can be written von Ballmoos et al. (1989);
Zoglauer et al. (2021):

$D(\chi,\psi,\varphi)=R(\chi,\psi,\varphi;l,b)\times
I(l,b)+B(\chi,\psi,\varphi)$ (31)

where $D$ is the measured data in the Compton Data Space (see Section 5.1),
$R$ is the instrumental imaging response, and $B$ is the measured background
in the data space. The response $R$ defines the probability that a photon
emitted from the Galactic image-space bin $(l,b)$ is detected in a given data
space bin $(\chi,\psi,\varphi)$. The task of image reconstruction is to invert
the measurement process to determine $I(l,b)$. As the problem is not directly
invertible, iterative approaches must be used.

There are two general classes of iterative imaging reconstruction approaches
used for Compton telescopes Frandes et al. (2016): binned mode and list mode.
In binned mode imaging, the imaging response $R$ is determined through
laboratory measurements and supplemented by simulations with enough statistics
to fill all bins of the data space. While Equation 31 shows a five-dimensional
response (three dimensions for the data space, and two for the image space),
each additional parameter used to describe the event (energy, time, recoil
electron direction, etc.) adds another dimension to this data space. List-mode
methods Barrett et al. (1997) use a list of events with all of their measured
parameters to define the data space, and the image response, i.e. the back-
projected circles or arcs, is calculated on an event-by-event basis. The size
of the data space is proportional to the number of events.

One of the main advantages of list-mode imaging is its capacity to save all of
the information from photon interactions with full precision. Event parameters
that are not traditionally in the image data space, like the distance between
the first two interactions or the electron recoil direction, can be saved and
used to further select the quality of events. A common list-mode algorithm
used in Compton telescopes is the List Mode Maximum-Likelihood Expectation-
Maximization (LM-ML-EM) Wilderman et al. (1998); Zoglauer et al. (2011).
Figure 26(b) shows the results of five iterations of the LM-ML-EM algorithm
for the 200 simulated events shown in 26(a); the expected point source is
recovered with very little variation in the field of view outside of the
source location. Other list-mode imaging algorithms include Filtered Back-
Projection Xu et al. (2004); Haefner et al. (2014) and stochastic origin
ensembles Andreyev et al. (2011); Daniel et al. (2020), and many variations of
these approaches are used.

Unfortunately, there are significant limitations to list-mode imaging,
especially for astrophysical observations. Without taking into account the
full response of the telescope, asymmetries in the data space not modeled in
list-mode (for example the peak at large $\Delta\varphi_{ARM}$ in Figure 21)
can cause systematics in the imaging. Furthermore, list-mode imaging does not
allow for the determination of the absolute flux. So while list-mode imaging
is often used for Compton imaging analysis Zoglauer et al. (2011), especially
for terrestrial applications Xu et al. (2004), astrophysical observations
strongly benefit from a binned-mode imaging technique.

In a binned image space, the imaging response describes the probability that
an event emitted from a Galactic location ($l,b$) in image space is detected
in a given bin ($\chi,\psi,\varphi$) in the Compton Data Space. Each bin in
the response matrix $R$ must be filled with sufficient statistics, and
therefore, computationally intensive simulations are required. The simulated
response matrix must however be calibrated and benchmarked with high-
statistics measurements of laboratory sources to understand systematics.
Additional dimensions can be included, such as the gamma-ray energy, time, and
recoil electron scatter direction, to better discern the source emission, but
the number of bins can quickly scale beyond the capabilities of state-of-the-
art personal computers.

A number of different binned-mode algorithms have been developed for Compton
imaging Ikeda et al. (2014). COMPTEL predominantly used the Maximum Entropy
Method (MEM) Strong et al. (1990) and the Richardson-Lucy approach Richardson
(1972); Lucy (1974) for many of their imaging analyses (e.g. see Knödlseder et
al. (1996); Strong (1995); Diehl et al. (1995)). Additionally, the Multi-
resolution Regularized Expectation Maximization (MREM) approach was developed
to suppress noise structures in the imaging Knödlseder et al. (1999). Each
algorithm has different advantages: the Richardson-Lucy approach is better
suited for point sources, the MREM method is optimized for very smooth images,
and MEM is a middle ground Zoglauer et al. (2021). Figure 27 shows a
comparison for the 26Al emissions as measured by COMPTEL using the MEM and
MREM approaches. The underlying data for both of these images are the same,
and while there are similarities in the large-scale emission, they show
different levels of small scale emission features due to the optimizations of
the iterative methods.

(c) MEM image reconstruction

(d) MREM image reconstruction

Figure 27: The 26Al emission measured by COMPTEL over 9 years shows emission
along the Galactic disk with two different imaging techniques. (a) The Maximum
Entropy Method (MEM) creates a lumpy, structured image enhancing point-source
and noise features. (b) The Multi-resolution Regularized Expectation
Maximization (MREM) algorithm was developed to create the smoothest
distribution consistent with the observed data. Both images have the same
underlying data. From Plüschke et al. (2001).

The broad range of Compton imaging techniques all have slightly different
advantages. List-mode approaches give a straightforward way to find bright
point sources, without needing a predefined imaging response, and are
sufficient for terrestrial applications and quick checks in astronomical
observations. The most accurate and sensitive imaging is achieved through
binned-mode algorithms which can account for any measurements systematics in
the imaging response. Different binned-mode techniques are better suited for
enhancing point source emission features, while others are optimized for
smooth extended emission. Additionally, model fitting and forward-folding
approaches are also used for Compton telescope imaging analysis. Depending on
the observation, the best practice is to compare multiple imaging methods to
achieve a clear understanding of the underlying emission.

###  5.4 Polarization Capabilities

As introduced in Section 3.4, Compton telescopes are inherently sensitive to
polarization. The response of a polarized source is defined by a sinusoidal
modulation in the azimuthal Compton scattering angle $\eta$ from Equation 12.

The standard method for polarization analysis is to fit the histogram of
azimuthal scattering angles $\eta$ with a cosine function to determine the
polarzation fraction and polarization angle of the incident photons Lei et al.
(1997); Tatischeff et al. (2019). The modulation amplitude of the azimuthal
scattering angle distribution (ASAD), depicted in Figure 14, is directly
proportional to the polarization fraction of the incident beam. The level of
polarization is quantified by the modulation factor:

$\mu=\frac{A}{B}=\frac{C_{max}-C_{min}}{C_{max}+C_{min}}$ (32)

where $A$ is the amplitude of the sinusoidal response, and $B$ is the offset,
and $C_{max}$ and $C_{min}$ are the maximum and minimum values in count space
as indicated in Figure 14. Since the maximum modulation response $C_{max}$
occurs when $\eta=90^{\circ}$, and the minimum at $\eta=0^{\circ}$, we can
evaluate Equation 12 to find the modulation factor as a function of the energy
and Compton scattering angle:

$\mu(E,\varphi)=\frac{\frac{d\sigma}{d\Omega}(\eta=90^{\circ})-\frac{d\sigma}{d\Omega}(\eta=0^{\circ})}{\frac{d\sigma}{d\Omega}(\eta=90^{\circ})+\frac{d\sigma}{d\Omega}(\eta=0^{\circ})}=\frac{\sin^{2}\varphi}{\frac{E_{scat}}{E_{0}}+\frac{E_{0}}{E_{scat}}-\sin^{2}\varphi}.$
(33)

This relation is shown in Figure 28. The modulation from a polarized source is
largest at lower energies and for Compton scattering angles $\varphi\sim
90^{\circ}$. Geometric effects in the detector and asymmetries in the
instrument response can give rise to false modulation signals in the ASAD;
therefore, it is common to normalize the measured ASAD with a simulated ASAD
from unpolarized emission.

Figure 28: The modulation factor describes the amplitude of the polarization
response. The modulation factor, and therefore the polarization response, is
maximized at Compton scattering angles $\varphi=90^{\circ}$ and for lower
incident photon energies. Adapted from Lei et al. (1997).

In the standard analysis, the degree of linear polarization $\Pi$ is derived
by normalizing $\mu$ in such a way that

$\Pi=\frac{\mu}{\mu_{100}},$ (34)

where $\mu_{100}$ is the modulation factor for 100% linearly polarized
photons.

The polarization sensitivity of a telescope is generally reported in terms of
the minimum detectable polarization (MDP). The MDP defines the polarization
for which there is only a small chance that the measured modulation is due to
statistical fluctuations:

$\text{MDP}=\frac{4.29}{\mu_{100}S}\sqrt{\frac{S+B}{T_{obs}}}$ (35)

The 4.29 factor corresponds to a 99% confidence level Weisskopf et al. (2010),
$\mu_{100}$ is the modulation factor from a 100% polarized source determined
by simulations, $S$ and $B$ are the source and background count rates,
respectively, and $T_{obs}$ is the observation time.

The general steps for the standard Compton polarization analysis method are as
follows:

  1. 1.

Simulate 100% polarized source to find $\mu_{100}$.

  2. 2.

Determine event selections that give the lowest MDP.

  3. 3.

Subtract background ASAD from the measured ASAD.

  4. 4.

Simulate unpolarized source and divide background-subtracted ASAD by the
unpolarized ASAD to correct for geometric systematics.

  5. 5.

Fit background-subtracted and geometry-corrected ASAD with

$C(\eta)=A\cos(2(\eta-\eta_{0}))+B,$ (36)

where $A$, $B$, and $\eta_{0}$ are free parameters. The modulation factor is
given by $\mu=\frac{A}{B}$.

  6. 6.

Scale the modulation factor by $\mu_{100}$ to find the polarization fraction
$\Pi=\frac{\mu}{\mu_{100}}$.

The standard method, and other more sophisticated techniques, rely heavily on
simulations to find the response to an unpolarized and 100% polarized source.
Therefore, instrument simulations need to be benchmarked with real
measurements and pre-flight polarization calibrations are essential Mizuno et
al. (2005); Bloser et al. (2009); Kole et al. (2016). As systematic errors
almost always result in an overestimation of polarization, and off-axis
emissions tend to produce false modulation signals, ASAD measurements of
unpolarized sources are essential to understand systematics.

As the number of gamma-ray polarization measurements is rapidly increasing
Chattopadhyay et al. (2019); Kole et al. (2020), the analysis techniques are
quickly becoming more sensitive and sophisticated. For telescopes with three-
dimensional position resolution that measure the Compton polar angle in
addition to the azimuthal scattering angle, Krawczynski (2011) show that a
significant improvement in the sensitivity is achieved when both the polar and
azimuthal scattering angles are used with a Maximum Likelihood analysis
approach. This has recently been demonstrated for real measured data with the
COSI balloon instrument Lowell et al. (2017a, b). Analysis for the dedicated
GRB polarimeter POLAR has leveraged the existing Multi-Mission Maximum
Likelihood (3ML) framework to simultaneously model the spectral and
polarization data Burgess et al. (2019); Kole et al. (2020). Bernard (2015)
show similar improvements in the sensitivity by using a moments method to
compute a simple estimator of the polarization fraction.

###  5.5 Limitations and Challenges

While the science in the MeV range is enticing and continues to drive the
scientific community, observations in this range have been slow to progress
compared to other wavelengths (see Figure 1). This is due to a number of
reasons that have been addressed throughout this Chapter. The low Compton
scattering cross section (see Figure 6) requires large detector volumes to
fully contain the high-energy photon events, which can be difficult to
instrument with sufficient position resolution while maintaining a good energy
response. The incident photon direction can only be constrained to a large
circle on the sky, or at best an event arc for instruments with tracking
capabilities (Section 3). Therefore, while Compton telescopes provide powerful
single-photon detection, they are not true direct imagers. With each photon
depositing its energy in multiple discrete interactions throughout the
detector volume, sophisticated algorithms are required for event sequencing
and reconstruction (see Section 4) to determine the initial photon parameters.
The high-level analysis for Compton telescopes requires the use of large data
spaces with a heavy reliance on simulations to understand the instrument
response (see Section 5.3). And finally, even an ideal Compton telescope, with
infinite position and energy resolution, as well as perfect event
reconstruction and response determination, has only a modest angular
resolution since there is a fundamental limit based on the unknowable momentum
of bound electrons (see Section 5.1). A further complication for Compton
telescopes is the high background radiation that is prevalent in the MeV
range.

#### Background Radiation

Background radiation is one of the dominant factors that determines a Compton
telescope’s sensitivity (see Section 5.2). The large volume detectors that are
required to fully absorb Compton events, and the large FOV, result in large
background signals von Ballmoos (2005).

Figure 29: Simulated spectra of the reconstructed background Compton events
show the separate background contributions of the eASTROGAM mission concept.
An orbit of $15^{\circ}$ is used, but lower inclinations $\lesssim 5^{\circ}$
have been shown to reduce the SAA induced background. Figure from Cumani et
al. (2019).

The nuclear activation lines that are produced in interesting cosmic sources
can also plague the instrument as background lines. After high-energy cosmic-
ray collisions occur within the instrument mass, which is inevitable at
balloon-altitudes and in satellite orbits, the material around the detector
will become activated and the radiation from these instrumental nuclear decays
contribute a source of significant background. Furthermore, cosmic-ray
interactions with particles in the atmosphere will produce a broadband
spectrum of gamma rays that is orders of magnitude more intense than Galactic
sources. This atmospheric emission, referred to as albedo radiation, is
strongest near the horizon.

Cumani et al. (2019) present an overview of the background components from 10
keV to 100 GeV in low-Earth orbit. Simulations of the response of a proposed
gamma-ray mission (eASTROGAM De Angelis et al. (2017)) are performed to
understand the contributions of each background component, including
activation from charged cosmic rays and the passage through the South Atlantic
Anomaly (SAA). Figure 29 shows the reconstructed background Compton events for
an orbit inclination of 15∘.

A number of lessons about the reduction of background radiation have been
learned from previous MeV telescope designs Schönfelder (2004): anti-
coincidence detectors (ACDs) surrounding the Compton telescope can
significantly reduce the background by vetoing signals caused by cosmic-rays;
low-activation materials near the detector element will reduce instrumental
background; and discrimination of neutrons and cosmic rays, either in
electronics (pulse-shape identifiers), through vetoing, or in the software
analysis chain can lower the dead-time and false identification of gamma-ray
signals. Electron tracking capabilities and selections on the quality of the
events also provide a way to reduce the background. Additionally, detailed
simulations for a proper benchmarking of the instrument prior to launch are
essential to understand the source and background response of the telescope.

##  6 Notable Compton Telescope Designs

The development of Compton telescopes began in the 1960s and 1970s and there
have been significant improvements in design and technologies since then. In
this section, the focus will be on notable, mature instruments with space or
balloon-flight experience, or substantial laboratory measurements and
calibration campaigns. This is not meant to be an exhaustive list of all
instrument development efforts, but a representative sample. Reference
Cavallari and Frontera (2017); Tatischeff et al. (2019); von Ballmoos (2005)
for an overview of past instruments, Engel et al. (2022) for a summary of some
future missions, and other chapters in this handbook for further details on
past and present Compton telescope designs.

The variety of Compton telescopes is extensive, with different detector
materials, geometries, spatial and spectral resolution capabilities, and
general design optimizations. We have attempted to organize the many
development efforts into four main categories: semiconductor-based Compton
imagers (Section 6.1), gaseous/liquid detectors (Section 6.2), dedicated
polarimeters (Section 6.3), and instruments capable of detecting both Compton
scattering and pair creation (Section 6.4). We also include a brief
description of instrument applications in other fields, such as medical
imaging and nuclear defense, in Section 6.5.

###  6.1 Semiconductor-based Compton Imagers

(a) Double-sided strip Ge detector.

(b) Pixelated CZT detector.

(c) Multi-layer double-sided strip Si and CdTe detectors.

Figure 30:  (a) Large volume double-sided strip germanium detectors ($8\times
8\times 1.5$ cm3) have been developed with 0.5 mm strip pitch for the Gamma-
Ray Imager/Polarimeter for Solar flares (GRIPS) instrument Shih et al. (2012).
Modified from Amman (2020). (b) Large volume CZT detectors are an emerging
technology. The ($4\times 4\times 1.5$ cm3) monolithic detector with 1.8 mm
pitch pixelated electrodes has demonstrated excellent position and energy
resolution. Figure from Chen et al. (2018). (c) As silicon detectors cannot be
made thicker than $\sim$1 mm, multi-layer designs are more prevalent. An
instrument which combines layers of $32\times 32$ mm double-sided strip
silicon and double-sided strip CdTe with positional resolution of 250 $\mu$m
has been used in environmental monitoring Takahashi et al. (2012); Takeda et
al. (2015) and medical imaging Yabu et al. (2021); Yabu (2022).

Many of the modern Compton imaging telescopes employ semiconductor detectors.
The desired energy resolution is inherent to these detectors, and the two or
three-dimensional good position resolution within each detector volume is
achieved by segmented electrodes. The most common semiconductor materials used
are germanium, CdTe, CZT, and silicon, and three examples of Compton telescope
detector designs with these materials are shown in Figure 30. Recent work into
artificial diamond detectors has shown excellent energy, position, and most
notably timing resolution, that can potentially enable a compact and efficient
Compton telescope using time-of-flight Bloser et al. (2019). One challenge
with semiconductor detectors is the difficulty to make large volumes, and the
detection efficiency is limited compared to Compton telescopes using
scintillators.

There are two general designs for semiconductor Compton telescopes: multi-
layer silicon detectors with 2D position resolution, and thick, large-volume
Ge or CZT detectors with internal 3D position sensitivity. We highlight
examples of each concept here by focusing on the Soft Gamma-ray Detector on
board Hitomi, which employed a multi-layer design, and the newly selected
Compton Spectrometer and Imager, which utilizes thick germanium detectors.

#### Soft Gamma-ray Detector on Hitomi

The Soft Gamma-ray Detector (SGD) on board the Hitomi satellite was designed
to cover an energy range of $60-600$ keV with a sensitivity 10 times better
than that of the Suzaku Hard X-ray Detector Takahashi et al. (2018); Tajima et
al. (2018). The SGD combined a multi-layer Compton telescope with an active
well-type shield to reduce the FOV and significantly increase the signal-to-
background ratio Ichinohe et al. (2016). Hitomi was launched in February 2016,
and sadly the SGD only had a few days of nominal observations before the
spacecraft ceased to function in March 2016.

Figure 31: (a) The Hitomi SGD combined a multi-layer Compton telescope with an
active BGO well-type shield to achieve excellent background rejection from
60-600 keV. Three identical Compton telescope modules composed the SGD. (b)
The Compton camera was composed of 32 layers of silicon pad detectors and 8
layers of CdTe pad detectors, both with a pixel pitch of 3.2 mm. Additional
CdTe sensors surrounded the silicon detectors to increase the detection
efficiency for events with large Compton scattering angles. Figure from Tajima
et al. (2018).

The Compton telescope for the SGD was made of 32 layers of 0.625 mm thick
silicon pad detectors and 8 layers of 0.75 mm thick CdTe pad detectors, as
shown in Figure 31. The sides of the silicon detectors were also surrounded by
two layers of CdTe detectors to measure the Compton-scattered photons with
large scattering angles. Each silicon detector had a $16\times 16$ array of
$3.2\times 3.2$ mm2 pads (called pads instead of pixels due to their size),
and the CdTe sensors, which are one quarter of the size of the silicon, had an
$8\times 8$ array of $3.2\times 3.2$ mm2 pads. The design combined a low-Z
material (Si) optimized for Compton scattering and a high-Z material (CdTe) to
act as an efficient absorber to obtain high detection efficiency. Photons
which Compton scattered in the silicon detector and were fully absorbed in the
CdTe sensors were used for Compton imaging, but in principle each layer could
act as a scatterer or absorber. An additional advantage of the silicon sensors
was the limited effect of Doppler broadening (see Section 5.1).

The multi-layer silicon and CdTe detector was mounted at the bottom of a well-
type active shield, as shown in Figure 31. The major advantage the narrow FOV
was enhanced background rejection. If the reconstructed event circle for a
specific event did not align with the FOV defined by the active-well shield,
then that event was considered background and rejected. Most of the background
could be rejected by requiring this condition. The opening angle provided by
the bismuth germanate (BGO) shield was $\sim$10∘ at 500 keV. An additional
fine phosphor bronze collimator installed in the opening of the BGO well
restricted the FOV of the telescope to 33′ for photons below 100 keV, to
minimize the flux from the cosmic X-ray background in the FOV.

The SGD was also designed to function as a Compton polarimeter Katsuta et al.
(2016). Fortunately, it observed the Crab nebula for 5 ks during the initial
Hitomi test phase Hitomi Collaboration et al. (2018). From this short
detection, the polarization fraction of the phase-integrated Crab emission
(sum of pulsar and nebula emissions) was found to be $22\pm 11$% with the
polarization angle being $111\pm 13^{\circ}$ in the energy range of 60–160 keV
(errors correspond to the 1$\sigma$ deviation). Figure 32 shows the measured
background-subtracted ASAD for the SGD Crab measurement, where the analysis
procedure was similar to that described in Section 5.4. The polarization angle
measured by SGD is one sigma from the projected spin axis of the pulsar,
$124.0^{\circ}\pm 0.1^{\circ}$. While the observation time was much less than
planned, the results show the capability of polarization measurement with a
multi-layered semiconductor Compton telescope and the functionality of the SGD
in flight.

Figure 32: The SGD observed the Crab nebula for a total of 5 ks during the
testing phase of Hitomi. With this observation, the polarization capabilities
and functionality of the SGD were demonstrated. The background-subtracted ASAD
from the Crab observations shown here corresponds to a polarization fraction
of $22\pm 11$% with a polarization angle of $111\pm 13^{\circ}$ in the energy
range of 60–160 keV. Figure from Hitomi Collaboration et al. (2018).

#### The Compton Spectrometer and Imager

The Compton Spectrometer and Imager (COSI) is a Compton telescope that was
recently selected to be NASA’s next Small Explorer mission with a launch in
2026 Tomsick and COSI Collaboration (2022). COSI is sensitive in the 0.2-5 MeV
range and is designed to study sources of line emissions, such as Galactic
positrons and nucleosynthesis, and perform polarization studies of gamma-ray
bursts and compact objects. The heart of COSI consists of an array of 16 high-
purity double-sided strip germanium detectors which provide excellent energy
resolution and $\sim$1.5 mm3 position resolution. The Compton imaging
capabilities give COSI a narrow line sensitivity that is better than
INTEGRAL/SPI, despite its smaller size. The COSI instrument was advanced
through technology development as a balloon-borne instrument Kierans et al.
(2017); Beechert et al. (2022), and measurements from balloon flights provide
a proof of principle of the telescope capabilities Zoglauer et al. (2021);
Lowell et al. (2017b); Siegert et al. (2020).

Figure 33: COSI uses 16 double-sided strip germanium detectors to measure
Compton interaction with high spectral and spatial resolution. Each detector
has 1.16 mm strip pitch and a $<1.5$ mm3 internal position resolution. The
detector array is housed in an aluminum cryostat and cooled to liquid nitrogen
temperatures with a mechanical cryocooler. The cryostat is surrounded on five
sides with an anti-coincidence shield to effectively constrain the FOV and
reduce the background. Figure from Tomsick and COSI Collaboration (2022).

The COSI instrument is shown in Figure 33. The 16 germanium detectors are each
$8\times 8\times 1.5$ cm3 with a strip pitch of 1.16 mm (see Figure 30). Each
of the COSI germanium detectors can act as a Compton imager alone, but
combined in an array increases the active detector volume and detection
efficiency. While the germanium gives COSI its excellent energy resolution, it
must be cooled to liquid nitrogen temperatures. The COSI detector array is
housed in an evacuated cryostat, with readout electronics mounted externally,
and detectors are cooled by a mechanical cryocooler.

The COSI cryostat is surrounded on five sides by a BGO anti-coincidence shield
read out with photomultiplier tubes (PMTs). The BGO shield effectively
constrains the instantaneous FOV to 25% of the sky, and reduces the
background. Interactions detected in the BGO shields are used to veto photons
from outside the FOV, and reject events that may not be fully absorbed in the
germanium array. With the all-sky FOV and high detection efficiency, the BGO
detectors act as a sensitive transient monitor.

###  6.2 Gaseous and Liquid Time Projection Chambers

The large sensitive detector volumes needed for Compton imaging are easily
achieved with gaseous and liquid time projection chambers (TPCs), and these
designs have been pursued since the early 1990’s alongside semiconductor
concepts. Gaseous and liquid TPCs require significantly fewer electronics
channels, and therefore lower power consumption, compared to the multi-layer
or segmented design of semiconductor-based instruments. Liquid designs provide
high detection efficiency, while the tracks of Compton recoil electrons in low
density gaseous TPCs are long and easy to reconstruct. Therefore, the
additional background rejection capabilities of electron tracking is an
additional design driver. The most common designs use either liquid or gaseous
argon or xenon Tanimori et al. (2004); Aramaki et al. (2020); Bloser et al.
(2019); Hunter et al. (2014).

When a gamma ray interacts with liquid argon or xenon via Compton scattering,
it produces an ionization cloud and simultaneously generates ultraviolet
scintillation light. Liquid argon and liquid xenon TPCs (LArTPC and LXeTPC)
generally measure the scintillation light with PMTs or silicon
photomultipliers (SiPMs) and the ionization tracks of electrons with two
dimensional electrodes, such as wires or segmented electrode pads Aramaki et
al. (2020). These detectors have three-dimensional spatial sensitivity due to
the combination of the horizontal electrodes and the drift time between the
light and the electron signals. The first fully realized TPC Compton imaging
instrument was LXeGRIT.

#### Liquid Xenon Gamma-Ray Imaging Telescope

The Liquid Xenon Gamma-Ray Imaging Telescope (LXeGRIT) was developed at
Columbia University and the University of New Hampshire Tumer et al. (1994) in
the 1990’s and early 2000’s. The instrument was based on a high-purity liquid
xenon time projection chamber, as shown in Figure 34.

Figure 34: The LXeGRIT time projection chamber used UV scintillation light to
provide the event trigger and timing, and the drift of the ionization charge
cloud past orthogonal sensing wires gave a measure of the X-Y position of the
interaction. The drift time gave the interaction depth. Figure from Aprile et
al. (2001).

The detector measured ionization and scintillation from Compton-scattering
interactions. The fast UV scintillation light, measured by PMTs at the bottom
of the chamber, gave the initial event trigger and the time of the
interaction. The position of the interaction was read out on X-Y directional
sets of wires through which the ionization charge drifts; the timing
difference provided the Z-distance of the interaction sites. The demonstrated
position resolution was $\sim$1 mm, and the energy resolution was 14% FWHM at
511 keV Curioni (2004). LXeGRIT pioneered the use of event pattern recognition
to reject background events, and the initial concept of the Compton Kinematic
Discrimination sequencing approach was developed for the LXeGRIT instrument
Aprile et al. (1993); Oberlack et al. (2000). LXeGRIT was tested at balloon
altitudes in 1999 and 2000 Curioni et al. (2007), and while LXeGRIT is no long
in active development, the instrument demonstration has enabled more modern
and capable designs using liquid TPCs Aramaki et al. (2020).

###  6.3 Dedicated Polarimeters

There is a sub-class of Compton telescopes that are optimized for detection
efficiency and polarization sensitivity, particularly for transient events, as
described in Section 3.4. In general, these instruments use scintillator
detectors which have poor position resolution and/or energy resolution and
cannot perform full event reconstruction for imaging. However, to be an
effective Compton polarimeter, only the azimuthal scattering angle needs to be
measured, and scintillator detectors provide excellent detection efficiency in
large volumes with sufficient position resolution.

The first gamma-ray polarization measurements in fact came from instruments
not designed for Compton scattering Coburn and Boggs (2003); Willis et al.
(2005). By measuring coincident interactions in separate detector volumes, an
azimuthal scattering angle distribution can be determined; however, without
careful, dedicated polarization calibrations of the instrument before launch,
systematics are difficult to address.

The Cadmium Zinc Telluride Imager (CZTI) on Astrosat Bhalerao et al. (2017),
India’s first satellite, is designed as a hard X-ray imaging instrument with a
coded-mask, but the mask becomes transparent above 100 keV. While not a
dedicated polarimeter, CZTI was calibrated prior to launch and has been used
to constrain the polarization of GRBs Chattopadhyay et al. (2019) and the Crab
pulsar Vadawale et al. (2018) by measuring the azimuthal scattering direction
between neighboring pixels Vadawale et al. (2015).

More recently, there has been a significant efforts devoted to the development
of next-generation gamma-ray polarimeters McConnell et al. (2021) (e.g. see
Figure 15). The first prolific dedicated Compton polarimeter was the POLAR
instrument, which has reported polarization measurements for a similar number
of GRBs as CZTI.

#### POLAR

POLAR was a dedicated Compton polarimeter launched in 2016 to the Chinese
space laboratory Tiangong-2, and during its 6 month operation it detected 55
GRBs Kole et al. (2020). POLAR was comprised of 25 modules (in a 5$\times$5
layout), each housing 64 thin and elongated plastic scintillator bars