# Using the LHeC ERL to Generate High-Energy Photons

N.S. Mirian<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY
Germany
E. Salehi<EMAIL_ADDRESS>UVSOR  Institute for Molecular Science  Okazaki
Japan
F. Zimmermann  CERN  Meyrin  Switzerland

###### Abstract

The Large Hadron electron Collider (LHeC) is a proposed future particle
physics project colliding 60 GeV electrons from a six-pass recirculating
energy-recovery linac (ERL) with 7 TeV protons stored in the LHC. The ERL
technology allows for much higher beam current and, therefore, higher
luminosity than a traditional linac. The high-current, high-energy electron
beam can also be used to drive a free electron laser (FEL). In this
contribution, we examine how the LHeC ERL can serve as a source of high-energy
photons for studies in nuclear physics, high-energy physics, Axion detection,
dark energy, and protein crystallography. In the first section, we discuss the
performance of the LHeC-based FEL, operated in the SASE mode for generating
photon pulses at wavelengths ranging from 200 keV to 600 keV. In the second
section, we investigate photon production via Laser Compton scattering (LCS).

##  1 Introduction

Photon beams with high energies ($>$10 keV) offer novel possibilities for
research and application. An intense high-energy photon beam with multiple
outstanding characteristics, such as energy tunability, good directivity,
quasi-monochromaticity, etc., can be employed for numerous applications in
nuclear physics, high energy physics, and non-destructive material analysis.
Potential applications of the high photon energy include protein
crystallography, coronary angiography, nanotechnology, along with searches for
Dark Photons and Axion-like Particles (ALPs).

Recently, in Ref. [1] we have demonstrated the potential generation and
amplification of photons with a few hundred-kilo electron-volt energy by using
the high-energy electron beam available at the Large Hadron electron Collider
(LHeC). The LHeC is a proposed future particle-physics project, colliding 60
GeV electrons from a six-pass recirculating energy-recovery linac (ERL) with 7
TeV protons circulating in the existing LHC [2, 3]. The ERL technology allows
for much higher beam current and, therefore, higher luminosity than a
traditional linac. We considered using the high-current, high-energy electron
beam of he LHeC to drive a free electron laser (FEL). Figure 1 sketches the
reconfigured LHeC-FEL. We demonstrated that such ERL-based high-energy FEL
would have the potential to provide orders of magnitude higher average
brilliance at sub Å wavelengths than any other FEL either existing or
proposed. We investigated the performance of an LHeC-based FEL operation in
the self-amplified spontaneous emission (SASE) mode, using the LHeC electron
beam after one or two turns of acceleration, with beam energies of, e.g., 10,
20, 30, and 40 GeV, to produce X-ray pulses at wavelengths ranging from 1.5
keV to 24 keV. In addition, we explored a possible path to use the 40 GeV
electron beam for generating photon pulses at even much higher photon energy
up to 100 keV, which would also allow a pioneering step into the picometer
wavelength regime [1].

Figure 1:  LHeC recirculating linac reconfigured for FEL operation [1].

However, photonuclear experiments as well as quantum electrodynamics (QED) and
quantum chromodynamics (QCD) experiments for high-energy physics require
electromagnetic interactions in at energies extending from several MeV to GeV.
Such gamma rays can be generated by Laser Compton scattering (LCS) of an
intense laser beam off a high-energy electron beam. The LCS technique
generates intense gamma-rays exhibiting various remarkable properties, e.g.,
quasi-monochromaticity, energy tunability, good directivity, and high
polarization, which can allow many applications in nuclear physics and non-
destructive analysis.

In this article, we demonstrate the potential of LHeC-FEL for generating high
photon energy via the FEL and LCS processes.

##  2 Reconfigured LHeC-FEL

To generate a high photon energy beam, we can utilize the 40 and 30 GeV
electron beams of the LHeC. As indicated in Fig. 1, the beam energy of 40 GeV
can be attained after two passes through the two 10 GeV linacs, instead of the
three passes reaching 60 GeV, in the standard LHeC mode of operation. A beam
energy of 30 GeV is also readily obtained after either 1.5 turns or after two
turns with appropriate linac voltages and phasing.

In this study, we used the ELEGANT particle tracking simulation of Ref. [1].
Table 1 summarizes the 40 GeV electron beam parameters optimized for LHeC FEL
operation. The bunch compression using three linac passages and three arcs
increases the peak bunch current by more than an order of magnitude while
preserving a reasonable transverse emittance and energy spread suitable for
FEL operation.

Table 1:  The main LHeC-ERL electron beam parameters. Peak current, bunch length, and transverse emittance were obtained from a tracking simulation with the code ELEGANT [1, 5]. Parameters | Unit | Value
---|---|---
injection energy | GeV | 0.5
final energy | GeV | 40/30
electrons per bunch |  | 3$\times 10^{9}$
initial FWHM bunch length |  $\mu$m | 234
final FWHM bunch length |  $\mu$m | 35
initial peak beam current | kA | 0.6
final peak beam current | kA | 4
final hor. normalized emittance |  $\mu$m | 0.9
final vert. normalized emittance |  $\mu$m | 0.4
bunch spacing | ns | 25
final rms energy spread | $\%$ | 0.01

##  3 Generation of a few hundred keV photon energy by FEL

To produce the high-energy SASE photons with the 40 GeV and 30 GeV electron
beams from the second linac of the LHeC, consider our FEL simulation with a
“Delta undulator” featuring an 18 mm period and 5 mm minimum gap. The Delta
undulator [4] can shape the FEL photon polarization.

Figure 2:  Simulated power growth for cases of helical (H) and linear (L)
polarization for photon energies higher than 206 keV. The simulations studied
the SASE amplification for an electron beam of either 30 GeV (black lines) or
40 GeV (red and blue lines) passing through a helical or planar Delta
undulator FEL line. The black lines refer to 206 keV, the red ones to 412 keV,
and the blue lines to 618 keV.

Figure 2 displays GENESIS code [6] simulations for the helical and planar sets
up of the Delta undulator FEL line, generating photons with helical and linear
polarization, respectively. The simulations at a wavelength of 206 keV were
performed for a 30 GeV electron beam passing through the Delta undulator, with
a gap of $\sim 6.5$ mm, considering either helical or linear polarization,
with results illustrated by the black solid and dashed lines, respectively.
The figure also presents the radiation amplification at photon energies of 412
and 618 keV, represented by the red and blue lines, respectively, for a 40 GeV
beam passing through the helical or planar Delta undulator line.

Predictions from the 1D and 3D FEL theories of Refs. [7, 8, 9] are in good
agreement with the GENESIS simulation results for wavelengths of a few hundred
keV photon energy, presented in Fig. 2 [1].

##  4 Generation of a few hundred MeV to few GeV photon energy by LCS

The field of high-energy and nuclear physics requires moving into the realm of
photon energies even greater than MeV for some specific experiments. to this
end, Laser-Compton backscattering technique is a unique technique to generate
photons of energy much higher than a few MeV. The schematic of laser Compton
scattering is illustrated in Fig. 3.

Figure 3: Scattering process in the laboratory frame.

A laser photon of energy $E_{p}$ and an electron with kinetic energy $E_{e}$
collide at an angle ${\phi}$. After the collision, the photon and the electron
are scattered with energy $E_{\gamma}$ and $E_{se}$, respectively. Thanks to
energy conservation, the energy of the scattered photon is directly related to
its scattering angle $\theta_{\gamma}$, according to

$E_{\gamma}=\frac{E_{p}(1-\beta\cos\phi)}{(1-\beta\cos\theta_{\gamma})+\frac{E_{p}}{m_{0}c^{2}}\sqrt{1-\beta^{2}}(1-\cos(\theta_{\gamma}-\phi))},$
(1)

where $m_{0}$ denotes the rest mass of the electron, $\beta$ the ratio of the
electron velocity and light velocity $c$, and $\theta_{\gamma}$ the angle of
the scattered gamma-ray photon. In the case of a head-on collision, $\phi=\pi$
Eq. (1) can be simplified to

$E_{\gamma}=\frac{(1+\beta)E_{p}}{1-\beta\cos\theta_{\gamma}+(E_{p}/m_{0}c^{2})\sqrt{1-\beta^{2}}(1+\cos\theta_{\gamma})}\;.$
(2)

The maximum scattered photon energy, which can be observed in the backward
direction of the incident photon, namely at $\theta_{\gamma}=0$, is given by
$E_{\gamma}=4\gamma^{2}E_{p}$ with $\gamma$ the Lorentz factor,
$\gamma=\sqrt{1-\beta^{2}}$, and $E_{p}$ the energy of the primary laser
photon. As revealed by this equation, the energy of the backward scattered
photons is estimated to be $4\gamma^{2}$ times higher than the energy of the
initial laser photons. For LHeC, in the case of a CO2 laser with laser photon
energy $E_{p}=0.117\leavevmode\nobreak\ $eV, power of $0.5\leavevmode\nobreak\
W$, and beam size $\sigma_{x,y}=5\leavevmode\nobreak\ mm$ the maximum energy
of the scattered photons is estimated to be a few GeV, that is in the high
energy of gamma-ray photons.

Figure 4: Left: Energy distribution of the scattered photons. Right: Rate of
gamma-ray photons per second and per radian versus the scattering angle
$\theta$, from a head-on collision of a 10.6 $\mu$m laser beam with an
electron energy of 30 GeV (blue) and 40 GeV (red).

The rate of gamma-ray scattered photons per unit time, $\dot{N}_{\gamma}$ can
be evaluated by a numerical calculation of the product of luminosity $L$ and
scattering cross-section $\sigma(\theta_{\gamma})$ integrated over the
scattering angle up tp $\theta_{\gamma}$ [10]:
$\dot{N}_{\gamma}=L\sigma(\theta_{\gamma})$. The differential cross section
for an electron in the laboratory frame can be derived from the Klein–Nishina
formula [11] in the electron rest frame followed by a Lorentz transformation.
In the electron rest frame, the laser photon always collides “head-on”.
Therefore, the differential cross-section is independent of the incident angle
of the laser $\phi$. The differential cross section can be written as [12] :

$\frac{d\sigma}{d\Omega}=\frac{1}{\gamma^{2}(1-\beta\cos\theta)^{2}}\frac{r_{0}^{2}}{2R^{2}}\left(R+\frac{1}{R}-1+\left(\frac{\cos\theta-\beta}{1-\beta\cos\theta}\right)^{2}\right)\;,$
(3)

where
$R=\left(1+\frac{\gamma(1+\beta)E_{p}}{m_{0}c^{2}}\left(1+\frac{\cos\theta-\beta}{1-\beta\cos\theta}\right)\right)$
and $r_{0}$ designates the classical electron radius. The luminosity is [9]

$L=\frac{fN_{e}N_{p}}{2\pi}\int_{\frac{-L_{\rm int}}{2}}^{\frac{-L_{\rm
int}}{2}}\frac{1}{\sqrt{\sigma_{ex}^{2}+\sigma_{Lx}^{2}}\sqrt{\sigma_{ey}^{2}+\sigma_{Ly}^{2}}}\,dz.$
(4)

Here, $N_{e}$ is the number of electrons per bunch, $N_{p}$ is the number of
laser photons, and $f$ is the collision frequency, while $\sigma_{x}$ and
$\sigma_{y}$ are the rms horizontal and perpendicular beam sizes,
respectively. The subscripts $e$ and $L$ refer to the electron beam and to the
laser, respectively. From the above equations, the total number of scattered
photons per unit of time and per scattered photon energy unit can be written
as:

$\frac{dN_{\gamma}}{dt\;dE_{\gamma}}=\frac{fN_{e}N_{p}}{2\pi}\frac{L_{\rm
int}}{\sqrt{\sigma_{ex}^{2}+\sigma_{Lx}^{2}}\sqrt{\sigma_{ey}^{2}+\sigma_{Ly}^{2}}}\frac{d\sigma}{dE_{\gamma}}\;,$
(5)

where $L_{\rm int}$ is the effective interaction length.

Figure 5:  Energy distribution of the scattered gamma-ray photons versus
photon energy produced by the head-on collision of a $\lambda$=10.6 $\mu$m
laser beam with a 40 GeV electron beam.

Based on Eq. (2), the energy distribution of the scattered photons can be
evaluated as a function of scattering angle, as is shown in Fig. 4 (left).
Figure 4 shows the energy distribution of scattered gamma-ray photons (left)
and the rate of gamma-ray scattered photons per solid angle as a function of
the scattering angle (right) for the collision of $0.117$ eV CO2 laser photons
with the LHeC electron beam at an energy of 30 GeV (blue) and 40 GeV (red).
The figure indicates that most of the high energy scattered photons and most
of the photons fall within a very narrow scattering angle. As can also be seen
in Fig. 4, for higher electron energy 40 GeV, the scattered photon energy is
larger. Using equations (3) and (4), we can calculate the energy spectrum
inside an aperture of radius $R=5$ mm at a distance $l=8$ m from the center of
the collision region extending over the length $L_{\rm int}=7$ m. The result
is shown in Fig. 5. The spectrum exhibits an edge corresponding to a high-
energy cutoff, which is determined by the incident electron and photon
energies. The total gamma-ray flux normalized to laser power and beam current
has been evaluated at approximately $30.541$ photons/W/mA/s.

##  5 Conclusions

The generation of high-energy photon beams opens the door to novel research
and application opportunities. Many applications in nuclear physics, high-
energy physics, and non-destructive analysis call for an intense high-energy
photon beam with energy tunability, good directivity, quasi-monochromaticity,
and high polarization. In this article, we examined the generation of high-
quality, high-energy photon beams at the proposed CERN-based LHeC collider,
via two different approaches, promising the generation of intense x- or gamma-
rays with photon energies above 200 keV. We assumed the simulated electron
beam of Ref. [1] after two turns behind the second linac, with a beam energy
of either 30 or 40 GeV. Simulation results presented in the first part of this
article suggest that, by employing the ’Delta undulator’ with 18 mm period
length, SASE FEL radiation with photon energies in the range of 200-620 keV
can be produced. In the second part, we described the laser Compton scattering
concept, enabling the generation of even much higher photon energies in the
range of 400 MeV to 2.5 GeV, and we discussed the energy-angle correlations in
the resulting photon beams.

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