# Numerical calculations of a high brilliance synchrotron source and on issues
with characterizing strong radiation damping effects in non-linear
Thomson/Compton backscattering experiments

A. G. R. Thomas1,2, C. P. Ridgers3, S. S. Bulanov4, B. J. Griffin2, S. P. D.
Mangles5 1Centre for Ultrafast Optical Science, University of Michigan, Ann
Arbor, MI 48109, US.  2Department of Nuclear Engineering and Radiological
Sciences, University of Michigan, Ann Arbor, MI 48109, USA.  3Clarendon
Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK.  4University
California Berkeley, Berkeley, CA 94720 USA.  5Blackett Laboratory, Imperial
College London, London SW7 2AZ, UK

###### Abstract

A number of theoretical calculations have studied the effect of radiation
reaction forces on radiation distributions in strong field counter-propagating
electron beam-laser interactions, but could these effects – including quantum
corrections – be observed in interactions with realistic bunches and focusing
fields, as is hoped in a number of soon to be proposed experiments? We present
numerical calculations of the angularly resolved radiation spectrum from an
electron bunch with parameters similar to those produced in laser wakefield
acceleration experiments, interacting with an intense, ultrashort laser pulse.
For our parameters, the effects of radiation damping on the angular
distribution and energy distribution of _photons_ is not easily discernible
for a “realistic” moderate emittance electron beam. However, experiments using
such a counter-propagating beam-laser geometry should be able to measure such
effects using current laser systems through measurement of the _electron beam_
properties. In addition, the brilliance of this source is very high, with peak
spectral brilliance exceeding $10^{29}$ photons s-1mm-2mrad${}^{-2}(0.1$%
bandwidth$)^{-1}$ with approximately 2% efficiency and with a peak energy of
10 MeV.

###### pacs:

41.75.Jv,52.38.Ph,41.60.-m

##  I Introduction

The recent development of ultra-high intensity laser systems has generated a
great amount of interest in a class of well known theoretical problems
involving the interaction of strong fields with relativistic electron beams
that have not been experimentally demonstrated. Relativistic electron beams
are regularly measured in experiments by laser wakefield acceleration
Tajima_PRL_1979 ; mangles ; geddes ; faure  and are characterized by being of
relatively high current density in short bunches. In laser wakefield
acceleration, oscillations of the electrons in the electromagnetic fields of
electron plasma cavities created by laser driven ponderomotive expulsion have
been shown to result in extremely bright sources of x-rays Rousse_PRL_2004 ;
ISI:000242538700029 ; ISI:000254024500038 ; ISI:000253724500017 ;
Thomas_POP_2009 ; Kneip_NP_2010 .

Another proposed source of radiation using the wakefield accelerated electron
beam is Thomson or Compton backscattering from a second laser Esarey_PRE_1993
; Hartemann_PRL_1996 ; Salamin_JPA_1998 ; Ueshima_LPB_1999 ;
Avetissian_PRE_2002 ; Lee_OE_2003 ; Brown_PRSTAB_2004 ; Hartemann_PRE_2005 ;
Koga_POP_2005 ; Albert_POP_2011 . In this scheme a counter-propagating laser
is used as a short wavelength undulator for producing high brightness,
monochromatic gamma rays. An undulator in a conventional synchrotron is
characterized by a strength parameter $K$ that characterizes the oscillation
amplitude relative to wavelength. For small $K$, the radiation is
monochromatic. For large $K$ the radiation is characterized by a synchrotron-
like spectrum Jacksonbook . In the counter-propagating laser scheme, the field
strength parameter (normalized peak vector potential)
$a_{0}=|eF_{0}|/m_{e}c\omega_{0}$ is analogous to $K$. $F_{0}$ is the peak
electric field strength of a laser with central angular frequency
$\omega_{0}$. For a laser with $a_{0}\ll 1$ ($I\lambda^{2}\ll 10^{18}$ Wcm-2),
the radiation is monochromatic. For $a_{0}>1$ harmonics in the radiation
spectrum start to appear, and for $a_{0}\gg 1$ the spectrum becomes broad. For
linear polarization of the laser there is also longitudinal motion due to the
Lorentz force, and therefore downshifting of the fundamental frequency occurs
He_PRL_2003 ; ISI:000182450200081 . The monochromatic regime using a laser
wakefield accelerated electron bunch has been proposed as a good source for
applications He_PRL_2003 ; Hartemann_PRSTAB_2007 ; Albert_PRSTAB_2010 ;
Albert_PRSTAB_2011 . In addition, experiments using this counter propagating
geometry with a very high intensity laser (Fig 1) should be an interesting
testbed for studying radiation reaction forces and non-linear quantum
electrodynamics Bulanov_NIMPR_2011 , due to the high field strength in the
electron rest frame.

Figure 1: Schematic of counter-propagating laser-beam interaction geometry
using laser wakefield accelerated electrons.

The transverse component of the laser vector potential is Lorentz invariant,
so the radiation emission of an $a_{0}\leq 1$ interaction is very different to
an $a_{0}\gg 1$ interaction independent of the reference frame (and therefore
electron energy, in the colliding geometry). The emission of photons in such
processes clearly indicates that a force should be applied to the electron to
conserve momentum. Conversely, the electric field strength is not a Lorentz
invariant, and hence the electron energy in this geometry may be crucial to
determining whether the field is quantum electrodynamically strong or not.

In this paper, radiation damping effects on the full angular and energy
distribution of photons produced in the counter-propagating geometry
interaction between a tightly focused $\sim 10^{22}$ Wcm-2 ultrashort pulse
with a electron beam is studied by solving modified classical equations of
motion numerically and generating spectra with a numerical radiation
spectrometer Thomas_PRSTAB_2010 . The layout of the manuscript is as follows:
First we parameterize the interplay between the field strength $a_{0}$ and
electron energy $\gamma m_{e}c^{2}$ in the colliding pulse geometry, and
identify the regime relevant to near term experiments where radiation damping
is strong but quantum electrodynamic effects are relatively small. Next we
introduce the numerical model for calculating both the electron dynamics and
radiation spectra. We then proceed to calculate the $\gamma$-ray spectrum with
“realistic” conditions, then examine the effect of radiation reaction on the
photon and electron phase-spaces. Finally, we show that semi-classical
corrections to the radiation reaction force may be observable in experiments.

##  II Parameterizing strong field interactions

###  II.1 Radiation reaction force effects

Although properly described by quantum electrodynamics, the radiation force
has a classical form that is self-consistent within the limits that the
acceleration timescale is much larger than $\tau_{0}=2e^{2}/3mc^{3}=6.4\times
10^{-24}$ s Dirac_PRSA_1938 ; Rohrlich_PRE_2008 . It is principally a damping
of motion due to loss of momentum to the radiation. The Lorentz-Abraham-Dirac
equation is a third order differential equation of motion for a charged
particle in the presence of accelerating forces, and includes the change of
momentum due to the radiation generated by the charge. The force on an
electron is given in covariant form by:

$\frac{d}{d\tau}v^{\mu}=-\frac{e}{m_{e}}\left(F^{\mu\nu}v_{\nu}+\tau_{0}D^{\mu}\right)\;,$
(1)

where $D^{\mu}$ is the radiation reaction (damping) force,
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the
electromagnetic field tensor and $v_{\nu}={dx^{\nu}}/{d\tau}=\\{\gamma
c,-\gamma\mathbf{v}\\}$ is the particle four-velocity.

The radiation reaction force, according to the Lorentz-Abraham-Dirac model, is
a source of much controversy precisely because it is a third order
differential equation, which allows for self-accelerating solutions that do
not conserve energy, for example. Various authors have reformulated the
equation to eliminate the third order term (See Sokolov Sokolov_POP_2009 ,
Hammond Hammond_PRA_2010  and references within). These are generally
identical to first order in $\tau_{0}$ (and are therefore basically all
equivalent to the Landau-Lifshitz form of the radiation reaction force
Landau_Lifshitz ), but are otherwise not identical. The modified force can be
written in the form Rohrlich_PRE_2008 :

$\frac{d}{d\tau}v^{\mu}=-\frac{e}{m_{e}}\left[F^{\mu\nu}v_{\nu}-\tau_{0}P^{\mu\alpha}\frac{d}{d\tau}\left(F_{\alpha}^{\nu}v_{\nu}\right)\right]\;.$
(2)

where $P^{\mu\alpha}=\eta^{\mu\alpha}+{v^{\mu}v^{\alpha}}/{c^{2}}$ and
$\eta^{\mu\nu}$ is the Minkowski metric tensor with trace -2. In Ref.
Bulanov_PRE_2011 , several examples were given, which show that the solutions
of the Lorentz-Abraham-Dirac model and equation 2 are identical in the
classical regime.

One of the interesting phenomena arising from this laser-electron interaction
is that the radiation damping is theoretically predicted to be so extreme that
for a sufficiently intense laser, the electron beam may lose almost all its
energy in the interaction time. In particular, Koga et al. studied the effect
of radiation damping on the radiation spectrum Koga_POP_2005 . Di Piazza et
al. also studied the effect of radiation damping on the angular distribution
of radiation Piazza_PRL_2009 . The effects of ‘real world’ conditions on the
radiation spectrum emitted have also been somewhat previously studied, for
example the effects of higher-order field corrections for tightly focused
pulses Hartemann_AJS_2000 ; Lee_EPL_2010 .

Radiation damping can be parameterized by considering the energy loss of the
electron due to the most significant damping term Koga_POP_2005 ;
Bulanov_NIMPR_2011 . Here we proceed from equation 2, where ignoring terms of
$\tau_{0}^{2}$ and higher and the Schott term, the damping contribution can be
written in the form Rohrlich_PRE_2008 :

$\frac{d}{d\tau}v^{\mu}=-\frac{e}{m_{e}}F_{\alpha}^{\nu}v_{\nu}\left[\eta^{\alpha\mu}-\tau_{0}\frac{e}{m_{e}c^{2}}v_{\nu}v^{\mu}F^{\alpha\nu}\right]\;,$
(3)

The electromagnetic four-force can be written in the form:

$\displaystyle
F^{\alpha\nu}v_{\nu}=-\frac{dA^{\alpha}}{d\tau}+v_{\nu}\partial^{\alpha}A^{\nu}\;.$
(4)

For the case of a linearly polarized plane wave,
$A^{\mu}=\Re\left[\\{A_{0}\\}^{\mu}e^{i\kappa_{\alpha}x^{\alpha}}f(\kappa_{\beta}x^{\beta}/\omega_{0}t_{L})\right]$
where $\kappa_{\alpha}$ is the four-wave-vector
$\kappa_{\alpha}=\omega_{0}\\{1,-\mathbf{\hat{k}}/c\\}$,
$f(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})$ is a function describing the
temporal envelope and $t_{L}$ is the pulse duration, interacting with a
counter-propagating electron with initial Lorentz factor $\gamma_{0}$ obeying
$a_{0}\ll\gamma_{0}\ll(a_{0}\omega_{0}\tau_{0})^{-1/2}$, the zeroth component
is well approximated by:

$\frac{d\gamma}{d\tau}=-\gamma\tau_{0}\frac{da^{\mu}}{d\tau}\frac{da_{\mu}}{d\tau}\;,$
(5)

where $a^{\mu}=eA^{\mu}/m_{e}c$. The condition on $\gamma$ is so that the
longitudinal Lorentz force is minimized but radiation damping does not affect
the transverse oscillations of the electron. For a slowly varying gaussian
envelope, i.e. $(1/f)df/d\tau\ll\kappa_{\mu}v^{\mu}$ with
$f=\exp(-(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})^{2})$, and averaging over
the fast oscillations, we can integrate to obtain the total energy loss by the
particle:

$\frac{\Delta\gamma_{\infty}}{\gamma_{0}}=\frac{\sqrt{\frac{\pi}{2}}\tau_{0}t_{L}\omega_{0}^{2}\gamma_{0}a_{0}^{2}}{1+\sqrt{\frac{\pi}{2}}\tau_{0}t_{L}\omega_{0}^{2}\gamma_{0}a_{0}^{2}}\;,$
(6)

This is similar to the result in Ref. Bulanov_PRE_2011 , but with a different
definition for the pulse duration because here $t_{L}$ is close to the full-
width-at-half-maximum duration commonly used in experiments. From this
expression, we can define a parameter
$\psi=10\sqrt{\frac{\pi}{2}}\tau_{0}t_{L}\omega_{0}^{2}\gamma_{0}a_{0}^{2}t_{rad}$,
for a particular characteristic timescale for radiation damping $t_{rad}$,
such that:

$\frac{\Delta\gamma_{\infty}}{\gamma_{0}}=\frac{0.1\psi(t/t_{rad})}{1+0.1\psi(t/t_{rad})}\;.$
(7)

which clearly defines strong radiation damping for $\psi\geq 1$ and weak
radiation damping for $\psi\ll 1$. Here we choose $t_{rad}=2\pi/\omega_{0}$ –
that is to say a laser period – which is slightly different from the choice of
Koga et al. Koga_POP_2005 , who chose the pulse duration for $t_{rad}$.
However, we have also added a factor of 10 into $\psi$, which is such that
$\psi=1$ corresponds to a 10% energy loss in a single cycle, which therefore
results in a condition similar to that of Koga et al. Koga_POP_2005 , since
they considered an approximately 10 cycle pulse. In addition, a 10% loss in a
single cycle can reasonably be defined as the threshold of “significant”
damping. Hence:

$\psi=10\sqrt{2\pi^{3}}\omega_{0}\tau_{0}\gamma_{0}a_{0}^{2}\;.$ (8)

For a 800 nm laser, $\psi=1.2\times 10^{-6}\gamma_{0}a_{0}^{2}$. The condition
$\psi=1$ leads to the condition for the laser pulse vector potential, i.e. the
strong radiation damping regime is realized for

$a_{0}>a_{rad}=\left(10\sqrt{2\pi^{3}}\omega_{0}\tau_{0}\gamma_{0}\right)^{-1/2}.$
(9)

###  II.2 Quantum electrodynamics effects

Quantum electrodynamically strong interactions are parameterized by a
relativistically and gauge invariant parameters
$\chi_{e}=||F_{\mu\nu}v^{\nu}||/(cE_{cr})$ and
$\chi_{\gamma}=||F_{\mu\nu}\hbar k^{\nu}||/(m_{e}cE_{cr})$ Ritus_1979 , where
$\hbar k^{\nu}$ is the four-momentum of a photon and
$E_{cr}=m_{e}^{2}c^{3}/e\hbar=1.32\times 10^{18}$ Vm-1 is the Schwinger or
critical field of quantum electrodynamics. These parameters determine the
rates of photon creation by an electron or an electron-positron pair creation
by high-energy photon in a strong electromagnetic field, the latter being the
Breit-Wheeler process Breit_PR_1934 . The photon emission probability for
$\chi_{e}\ll 1$ is $\approx\left(5\alpha
m_{e}^{2}/2\sqrt{3}p_{0}\right)\chi_{e}$ and for $\chi_{e}\gg 1$ is
$\approx\left[14\Gamma(2/3)\alpha
m_{e}^{2}/27p_{0}\right]\left(3\chi_{e}\right)^{2/3}$, where $p_{0}$ is the
electron energy and $\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt$ is the Euler
gamma function, and $\alpha=e^{2}/4\pi\epsilon_{0}\hbar c=1/137$ is the fine
structure constant Ritus_1979 . The pair production probability by a photon
for $\chi_{\gamma}\ll 1$ is $\approx\left(3\sqrt{3}\alpha
m_{e}^{2}/16\sqrt{2}k_{0}\right)\chi_{\gamma}\exp\left(-8/3\chi_{\gamma}\right)$
and for $\chi_{\gamma}\gg 1$ is $\approx\left[15\Gamma^{4}(2/3)\alpha
m_{e}^{2}/28\pi k_{0}\right]\left(3\chi_{\gamma}\right)^{2/3}$, where $k_{0}$
is the photon energy Ritus_1979 . Previously it was shown that extremely high
intensity counter propagating laser pulses could lead to prolific pair
production Bell_PRL_2008 ; Fedotov_PRL_2010 ; Bulanov_PRL_2010 .

For multi-100 TW lasers, such as the Hercules Yanovsky:2008  or Astra Gemini
Hooker_JPIV_2006  lasers, with focused field strength $|E|\sim 10^{-3}E_{cr}$,
interaction with GeV energy electron beams should be sufficient to achieve
$\chi_{e}\sim 1$ Schwinger_PR_1951 ; Sokolov_PRL_2010 ; Sokolov_PRE_2010 .
However the conversion of emitted photons into electron-positron pairs will be
suppressed due to the $\exp\left(-8/3\chi_{\gamma}\right)$ in the expression
for the probability for $\chi_{\gamma}\ll 1$.

A notable experiment in a similar geometry, using the 46 GeV electron beam
from the the Stanford Linear Accelerator (SLAC) colliding with a laser with
intensity of $I_{0}\sim 10^{18}$ Wcm-2, was an important demonstration of non-
linear quantum electrodynamics (multi photon Breit-Wheeler pair production)
Burke_PRL_1997 . A simplified version of the parameter $\chi_{e}$ for the
situation of an electron beam with energy $E=\gamma_{0}m_{e}c^{2}$ colliding
with a laser field with field strength parameter $a_{0}$ can be written as
Sokolov_PRE_2010

$\chi_{e}=\frac{2\hbar}{m_{e}c^{2}}\omega_{0}\gamma_{0}a_{0}\;.$ (10)

For an 800 nm laser system, this gives $\chi_{e}=6\times
10^{-6}\gamma_{0}a_{0}$. For the SLAC experiment (using a 527 nm laser), the
small $a_{0}$ ($a_{0}<1$) was compensated by the high beam energy
($\gamma_{0}\sim 10^{5}$), so that $\chi_{e}\approx 0.4$.

Quantum electrodynamics effects may be considered significant when the energy
of the emitted photons becomes of the order of the electron energy,
$\hbar\omega\gtrsim\gamma_{0}m_{e}c^{2}$. For a head-on collision of an
electron and a laser pulse, a characteristic emitted photon energy is
$\hbar\omega\approx\hbar\omega_{0}a_{0}\gamma_{0}^{2}$ Bulanov_NIMPR_2011 ,
which corresponds to the condition $\chi_{e}\sim 1$. Hence, quantum
electrodynamics effects may be considered to be strong for a field strength
of:

$a_{0}>a_{Q}=\frac{m_{e}c^{2}}{2\hbar\omega_{0}\gamma_{0}}.$ (11)

However, quantum effects in the radiation damping of electrons becomes
noticeable for much lower laser field strengths. It is well known Ritus_1979 ;
Erber_RMP_1966 ; Kirk_PPCF_2009 ; Sokolov_PRE_2010 , that the classical
description of an electron radiating in a strong electromagnetic field
overestimates the total emitted power. It is connected with the fact that in
the quantum description the emitted photon energy may not exceed the electron
energy, whereas the classical approach does not have such a restriction. This
effect can be approximately taken into account by introducing a function
$g(\chi_{e})$ into the expression for the total power of emitted radiation
Erber_RMP_1966 ; Kirk_PPCF_2009 ; Sokolov_PRE_2010 . $g(\chi_{e})$ enters the
equation of motion by modifying the expression for the radiation reaction
force as:

$\frac{d}{d\tau}v^{\mu}=-\frac{e}{m_{e}}F_{\alpha}^{\nu}v_{\nu}\left[\eta^{\alpha\mu}-g(\chi_{e})\tau_{0}\frac{e}{m_{e}c^{2}}v_{\nu}v^{\mu}F^{\alpha\nu}\right]\;,$
(12)

The strong damping parameter $\psi$ can be modified to include this quantum
effect to obtain a parameter $\psi_{Q}=\langle g(\chi_{e})\rangle\psi$, where
$\langle g(\chi_{e})\rangle$ is the time average of the $g$ factor. To do
this, we make use of a polynomial fraction fit to data for $g(\chi_{e})$ given
in Ref Kirk_PPCF_2009 :

$\displaystyle
g(\chi_{e})=\left(3.7\chi_{e}^{3}+31\chi_{e}^{2}+12\chi_{e}+1\right)^{-4/9}\;,$
(13)

which for $\chi_{e}\rightarrow 0$, $g\rightarrow 1$. The condition
$\chi_{e}=1$ corresponds to $g(\chi_{e})=0.18$, but even for $\chi_{e}=0.1$
this factor has a value of $g(\chi_{e})=0.66$. The time averaged field
strength parameter $a_{0}/\sqrt{2}$ (for linear polarization) is used to
approximate $\langle g(\chi_{e})\rangle\approx g(\langle\chi_{e}\rangle)$,
which is valid for $\chi_{e}\ll 1$.

Figure 2: The function $\psi_{Q}$ as a function of $a_{0}$ and $\gamma_{0}$
for an 800 nm central wavelength laser. The solid line indicates the threshold
between classical and quantum radiation reaction forces, and the dotted line
indicates the threshold where $g(\chi_{e})$ begins to be significant.

The modified strong damping parameter $\psi_{Q}$ for an 800 nm wavelength
laser is shown as a function of $a_{0}$ and $\gamma_{0}$ in figure 2. As shown
in Refs Kirk_PPCF_2009 ; Sokolov_PRE_2010 , for $\chi_{e}\sim 0.1$, the
spectrum emitted should not be changed significantly in shape, but $\langle
g(\chi_{e})\rangle\approx g(\langle\chi_{e}\rangle)$ indicates that the energy
loss of the electron beam due to radiation damping should be changed by a
measurable amount. This is also consistent with what we observe with our
model.

###  II.3 Parameter regimes involving $\chi_{e}$ and $\psi_{Q}$

The counter-propagating geometry laser-electron beam experiment is an
excellent testbed for studying quantum electrodynamics and strong radiation
damping effects. This is because of the ability to choose between strongly
radiation damped behavior ($\psi_{Q}\gtrsim 1$), or fields that are quantum
electrodynamically strong ($\chi_{e}\gtrsim 1$), _or_ a situation where both
$\psi_{Q}\gtrsim 1$ and $\chi_{e}\gtrsim 1$ simultaneously; conditions where
even more exotic effects may occur. These are controlled through variation of
the laser field strength, $a_{0}$, central frequency, $\omega_{0}$, and the
electron beam energy, $\gamma_{0}m_{e}c^{2}$. We can compare the requirements
for $a_{0}$, $\omega_{0}$ and $\gamma_{0}$ for the interaction to be in the
strong radiation damping regime or quantum electrodynamics dominant regime
relevant to experiments using 30 fs class lasers.

Table 1 shows parameters for different scenarios for strongly radiation damped
($\psi_{Q}\gtrsim 1$) and quantum electrodynamically strong ($\chi_{e}\gtrsim
1$) physics in a non-linear Thomson/Compton scattering geometry for an 800 nm
laser pulse with intensity $I_{L}$ colliding with an electron beam with energy
$E_{b}$. (a) corresponds to the SLAC experiment Burke_PRL_1997 . (b)
corresponds to near term experiments using intense 30 fs lasers such as
Hercules Yanovsky:2008  or Astra Gemini Hooker_JPIV_2006  and a laser
wakefield generated electron beam. (c) corresponds to an ‘ideal’ experiment
using two laser beam lines (as Astra Gemini has) with the current maximum
experimentally demonstrated laser intensity Bahk_OL_2004  and laser-wakefield
accelerated electron beam energy Leemans_NP_2006 . The SLAC experiment is
shown for comparative purposes only since the quasi-static field approximation
is not valid for this case Hu_PRL_2010 .

Table 1: Different scenarios for strong radiation damping ($\psi_{Q}\gtrsim 1$) and QED strong ($\chi_{e}\gtrsim 1$) physics in a non-linear Thomson scattering geometry for 800 nm wavelength laser pulses with intensity $I_{L}$ colliding with an electron beam with energy $E_{b}$. |  $E_{b}$ / GeV |  $I_{L}$ / Wcm-2 | $a_{0}$ | $a_{rad}$ | $a_{Q}$ | $\chi_{e}$ | $\psi$ | $\psi_{Q}$
---|---|---|---|---|---|---|---|---
(a) | 46.6 | $1\times 10^{18}$ | 0.5 | 2.5 | 1.2 | 0.43 | 0.045 | 0.018
(b) | 0.2 | $5\times 10^{21}$ | 50 | 46 | 420 | 0.12 | 1.2 | 0.74
(c) | 1 | $2\times 10^{22}$ | 100 | 21 | 84 | 1.2 | 23 | 3.7

In the first case (a) the laser vector-potential is of the order of both
$a_{rad}$ and $a_{Q}$. Case (b) corresponds to a situation where there will be
strong radiation damping but quantum effects will be weak,
$a_{Q}>a_{0}>a_{rad}$. In case (c) the laser is sufficiently intense for both
radiation damping and quantum recoil to be manifest, $a_{0}>a_{Q}>a_{rad}$.

Since our model is classical – that is to say involving equations of motion
only – it is restricted to the parameter range where $\chi^{2}\ll 1$
Sokolov_PRE_2010 . For the parameters described here, $\chi^{2}=0.014$ and so
the classical approach is reasonable. This also motivates the use of the
description of this process as “non-linear Thomson scattering” rather than
“Compton” scattering. We also calculate the electron spectrum after the
interaction in the presence of radiation damping with and without the $g$
factor, showing that quantum modifications to radiation losses may be
measurable.

##  III The model and numerical methods

The spectral intensity of radiation emitted by a number $N_{P}$ of
accelerating point charges can be expressed, in the far-field, as Jacksonbook
:

$\frac{d^{2}I}{d\omega
d\Omega}=\frac{\mu_{0}e^{2}c}{16\pi^{3}}\omega^{2}\Bigg{|}{\int_{-\infty}^{\infty}\sum_{j=1}^{N_{P}}\mathbf{\hat{s}}\times\mathbf{\beta}_{j}e^{i\omega(t-\mathbf{n}\cdot\mathbf{r}_{j}/c)}}dt\Bigg{|}^{2}\;,$
(14)

where the unit vector $\mathbf{\hat{s}}$ is in the direction of observation,
at a distance far compared with the scale of the emission region. This can be
written alternatively in terms of proper time, $\tau$:

$\frac{d^{2}I}{d\omega
d\Omega}=\frac{\mu_{0}e^{2}c}{16\pi^{3}}\omega^{2}\Bigg{|}\sum_{j=1}^{N_{P}}{\int_{-\infty}^{\infty}\mathbf{\hat{s}}\times\mathbf{v}_{j}e^{i\kappa_{\alpha}x_{j}^{\alpha}}}d\tau\Bigg{|}^{2}\;,$
(15)

where $\mathbf{v}_{j}$ is the momentum part of the $j$th particle’s four-
velocity defined as:

$v_{j}^{\alpha}=\frac{dx_{j}^{\alpha}}{d\tau}\;.$ (16)

To numerically integrate the equations of motion for charged particles, both
$x^{\alpha}$ and $v^{\alpha}$ have to be recorded at a number of discrete
points. To then perform the spectral integration numerically, equation 15 can
be reduced to the summation:

$\frac{d^{2}I}{d\omega
d\Omega}=\frac{\mu_{0}e^{2}c}{16\pi^{3}}\omega^{2}\Bigg{|}{\sum_{j=1}^{N_{P}}\sum_{n=0}^{N_{\tau}}\mathbf{\hat{s}}\times\mathbf{v}_{j}^{n}e^{i\kappa_{\alpha}x_{j}^{\alpha,n}}}\Delta\tau\Bigg{|}^{2}\;,$
(17)

One of the advantages of using proper time rather than ‘laboratory’ time for
numerical calculations is that the time resolution is effectively adaptive; as
the particle gains inertia and is therefore accelerated at a decreased rate
for a similar force, the time step-size increases. Numerically calculating
this integral by ‘brute force’ has the problem that the exponent is a fast
oscillating function, and therefore without resolving $\omega$ the integral
will in general not converge without a numerical timestep of $\Delta\tau\ll
1/\omega$ Filon_PRSE_1928 ; Nyquist ; ISI:A1949um84500004 . Since we are
interested in $\gamma$-ray photons in excess of an MeV energy generated from a
few fs interaction, the ratio of the necessary time step to the integration
timescale is computationally unfeasibly large. Recently, methods for
overcoming this limit by using interpolation have been developed
Martins_SPIE_2009 ; Thomas_PRSTAB_2010 ; Martins_AAC_2010 . Here we use the
method we previously developed Thomas_PRSTAB_2010 , and the reader is directed
towards that paper for further details of the numerical calculation.

The particle trajectories were calculated in the presence of four-potentials,
$A^{\mu}=\\{A^{0}=\phi/c,A^{1},A^{2},A^{3}\\}$, representative of a spatio-
temporally Gaussian laser pulse with no interaction between electrons. The
laser pulse propagated in the $+\mathbf{\hat{x}}_{3}$ direction with four-
potential described by:

$A^{\mu}=\Re\left[\\{A_{0}\\}^{\mu}(x^{\alpha})e^{i\kappa_{\alpha}x^{\alpha}}f(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})\right]\;,$
(18)

where $\\{A_{0}\\}^{\mu}(x^{\alpha})$ is the spatial distribution of four-
potential, in this case $\kappa_{\alpha}=\\{\omega_{0},0,0,\omega_{0}/c\\}$ is
the laser four wavevector, $f(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})$ is a
function of time describing the temporal envelope. The spatial-temporal
distribution of a tightly focused pulse that satisfies the vacuum Maxwell’s
equations is in general very complicated, but is easier to formulate in terms
of potentials than fields. That is because it is possible to have a purely
transverse (to propagation) vector potential and satisfy the vacuum Maxwell’s
equations, something that is not possible with fields. These can be formulated
by using the Lorentz-invariant Lorenz gauge condition
$\partial_{\mu}A^{\mu}=0$. Using a slowly varying envelope approximation, and
using a transverse vector potential linearly polarized in the
$\hat{\mathbf{x}}_{1}$ direction with propagation in the
$\hat{\mathbf{x}}_{3}$ direction, this can be approximated as
$\\{A_{0}\\}^{0}=-(ic/\omega_{0})\partial\\{A_{0}\\}^{1}/\partial x_{1}$
Davis_PRA_1979 . Here, vector and scalar potentials with corrections to the
basic Gaussian optics formulation were introduced up to order
${\theta_{0}}^{2}$, where $\theta_{0}=2c/\omega_{0}w_{0}$ is the asymptotic
divergence angle of a Gaussian laser beam with a waist of $w_{0}$. This yields
potentials:

$\\{A_{0}\\}^{1}=\left[1+\frac{\theta_{0}^{2}}{2}\left(\frac{1-i\zeta}{1+\zeta^{2}}\right)\left(1-\left(\frac{1-\zeta^{2}}{2\left(1+\zeta^{2}\right)}\right)\rho^{2}\right)\right]\Psi_{0}\;,$
(19)
$\\{A_{0}\\}^{0}=i\theta_{0}\xi_{1}e^{-i\tanh^{-1}\zeta}\left[\\{A_{0}\\}^{1}-{\theta_{0}}^{2}\left(\frac{1-\zeta^{2}}{1+\zeta^{2}}\right)\Psi_{0}\right]\;,$
(20)

and $\\{A_{0}\\}^{2}=\\{A_{0}\\}^{3}=0$, where

$\Psi_{0}=\frac{e^{-i\tanh^{-1}\zeta-(1+i\zeta)\rho^{2}}}{\sqrt{1+\zeta^{2}}}\;,$
(21)

$\rho=\sqrt{{x_{1}}^{2}+{x_{2}}^{2}}/w_{0}$, and
$\zeta=x_{3}\theta_{0}/w_{0}$. $w_{0}/\theta_{0}$ is the Rayleigh range of the
laser. Higher order corrections to the field structure could be employed to
account for extremely tight focusing, but here we restrict our numerical
calculations to foci with $w_{0}>\lambda$, where $\lambda$ is the laser
wavelength; these corrections in ${\theta_{0}}^{2}$ are of magnitude
$(1/\pi^{2})\lambda^{2}/w_{0}^{2}$, so these corrections are up to 10 % of the
zero order fields and can’t be considered negligible, but the next order
corrections are ${\theta_{0}}^{4}$ and therefore of less importance.

An electron beam was modeled using $N_{P}$ particles initiated with a momentum
$p_{0}$ in the $-\mathbf{\hat{x}}_{3}$ direction in front of the laser. In
order to simulate a more realistic beam, rejection sampling against a Gaussian
probability distribution function was used to generate a beam with a spread in
momentum, $\mathbf{\sigma_{p}}$, and position, $\mathbf{\sigma_{x}}$ which
statistically approximated the phase space distribution:

$\displaystyle f_{e}(\mathbf{x},\mathbf{p},t)=$
$\displaystyle\exp\left[-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{p^{2}}{2\sigma_{p}^{2}}\right]\;,$
(22)

where
$x^{2}/\sigma_{x}^{2}=x_{1}^{2}/\sigma_{x_{1}}^{2}+x_{2}^{2}/\sigma_{x_{2}}^{2}+x_{3}^{2}/\sigma_{x_{3}}^{2}$
and
$p^{2}/\sigma_{p}^{2}=p_{1}^{2}/\sigma_{p_{1}}^{2}+p_{2}^{2}/\sigma_{p_{2}}^{2}+\left(p_{3}-p_{0}\right)^{2}/\sigma_{p_{3}}^{2}$
The root-mean-square normalized emittance of the bunch is therefore given by
$\epsilon={\sigma_{p_{1}}}{\sigma_{p_{2}}}{\sigma_{p_{3}}}{\sigma_{x_{1}}}{\sigma_{x_{2}}}{\sigma_{x_{3}}}$.
Although the particle tracking routine could easily calculate a much larger
bunch, due to the computational demands of the numerical spectrometer for a
full angular sweep, the number of electrons in the bunch was limited to
$N_{P}=500$. Radiation from individual electrons was summed incoherently.

A gaussian temporal envelope is used in all cases,
$f=e^{-(\kappa_{\alpha}x^{\alpha}/\omega_{0}t_{L})^{2}}$. The pulse duration
is $t_{L}=65\;\omega_{0}$, where $\omega_{0}$ is the laser angular frequency,
which in the case of a typical $0.8$ $\mu$m laser is $2.36\times 10^{15}$ s-1,
yielding $t_{L}=27.5$ fs at $1/e^{2}$ radius, or 32 fs full-width-at-half-
maximum, of intensity. The electron beam parameters were varied, with $p_{0}$
corresponding to a beam energy typically of 204 MeV ($\gamma_{0}=400$). The
linearly polarized laser, with normalized vector potential of $a_{0}=50$
corresponding to a peak laser intensity of $5.3\times 10^{21}$ Wcm-2 was
focused to a spot with waist $w_{0}=2.55$ $\mu$m, or $w_{0}=20c/\omega_{0}$.
The electron beam parameters are comparable those routinely achieved in laser
wakefield acceleration experiments and are summarized in table 2.

Table 2: Emittance parameters of the electron beam used in the numerical model. (a) Finite momentum spread case, (b) Zero momentum spread case. | $\sigma_{x_{1}}$ | $\sigma_{x_{2}}$ | $\sigma_{x_{3}}$ | $\sigma_{p_{1}}$ | $\sigma_{p_{2}}$ | $\sigma_{p_{3}}$
---|---|---|---|---|---|---
A | $3c/\omega_{0}$ | $3c/\omega_{0}$ | $9c/\omega_{0}$ | $m_{e}c$ | $m_{e}c$ | $10m_{e}c$
B | $3c/\omega_{0}$ | $3c/\omega_{0}$ | $9c/\omega_{0}$ | $0$ | $0$ | $0$

##  IV Numerical results

In this section we detail ‘real world’ numerical calculations of a
backscattering experiment applicable to near term experiments using current
laser systems and laser wakefield accelerated electrons. By ‘real world’, we
mean that the calculation of the radiation spectrum includes the effect of a
Gaussian shaped bunch of electrons with normalized emittance (longitudinal and
transverse) comparable to that produced in laser-wakefield acceleration
interacting with a tightly focused laser, that radiation reaction forces are
included, and that the radiation spectrum is calculated directly from the
electron trajectories. However, the self-consistent absorption of laser pulse
photons is not included. The energy radiated by a $10^{9}$ electron beam will
later be shown to be 0.3 J, which is a non-negligible 2% of the pulse energy
of the laser considered here. Including the depletion of laser energy would
modify the spectrum of photons slightly, but is likely to be less important
than the other effects we consider here.

Figure 3: The angularly resolved spectral intensity ($d^{2}I/d\omega d\Omega$)
due to a 500 electron bunch with $\gamma=400$ and emittance given in table 2
scattering from a laser pulse with $a_{0}=50$ with higher order field
contributions included as in equations 19-21. In (a) the radiation reaction
force is not included, and in (b) the radiation reaction force is included.
The contours are taken at identical spectral intensity levels for both cases,
normalized to the peak which is 1.6517$\times 10^{-26}$ Js-1 at 0.2, 0.3, 0.4,
0.5, 0.6, 0.7 and 0.8.

The spectral intensity $d^{2}I/d\omega d\Omega$, where differential solid
angle $d\Omega=\sin\theta d\theta d\phi$, was calculated on a grid consisting
of 150 cells in $\omega$ over the range
$10^{4}\omega_{0}<\omega<10^{8}\omega_{0}$, with $\Delta\omega$ exponentially
increasing with $\omega$, 117 cells in $\theta$ over the range $0<\theta<30$
mrad and 26 cells in $\phi$ over the range $0<\phi<\pi/2$ rad. For clarity in
the figures, symmetry is assumed and therefore the full range $-30<\theta<30$
mrad and $0<\phi<\pi$ is displayed.

###  IV.1 The high brilliance synchrotron source

The properties of radiation from backscattering of an electron with a laser
pulse have been extensively studied, and we can therefore use analytic
formulae to predict that in the interaction of a electron beam with
$\gamma=400$ with a laser with field strength $a_{0}=50$, the synchrotron-like
spectrum will peak in energy at
$\hbar\omega_{peak}=2.56a_{0}\gamma^{2}\hbar\omega_{0}=30$ MeV
ISI:000182450200081 . Using the numerical model, we can more accurately model
the properties of the radiation produced in a high intensity laser interaction
with a laser wakefield accelerated electron beam as a source for applications,
in particular including the effects of radiation damping Koga_POP_2005 , non-
plane wave laser fields Hartemann_AJS_2000  and calculate the full angular
distribution of radiation.

Figure 3 shows the spectral intensity of radiation produced by a 500 electron
bunch with $\gamma=400$ scattering from a laser pulse with $a_{0}=50$. In this
example, the higher order field contributions are included, as in equations
19-21, as well as beam emittance as given in table 2 case A. This represents
reasonably “realistic” modeling of an experiment and results in a well
collimated, smooth, synchrotron-like radiation emission extending up to very
high energies, with a broad peak at approximately $10$ MeV, which is a factor
of 3 smaller than the analytic prediction due to the radiation reaction and
finite spot effects. Because the laser pulse is linearly polarized, as
expected, the radiation is strongly polarized, but also the angular intensity
distribution has a pronounced ellipticity, with the major axis in the
direction of polarization. Linear polarization also leads to higher photon
energies compared with a circularly polarized pulse with the same pulse
energy.

One other notable effect is that of the higher order terms in the laser
fields. These do not significantly change the spectral shape, but do change
the magnitude non-negligibly. Without the field contributions, the peak
spectral intensity is 1.57$\times 10^{-26}$ Js-1, but with them it is
1.65$\times 10^{-26}$ Js-1, which is a 5% difference. Although the order
$\theta^{2}$ pulse potential corrections have been simply added to the first
order potential – so that the energy in the corrected pulse is higher than the
uncorrected – because the additional potential are $\sim 10$% of the first
order potential, adding the corrections only represents a $\sim$1% increase in
pulse energy. The slight increase in pulse energy alone is too small to
account for the increased radiation output alone. Instead it is the additional
longitudinal motion due to these potentials that increases the spectral
output.

Figure 4: The on-axis peak spectral brilliance ($d^{2}I/d\omega
d\Omega/(1000\hbar t_{L}\pi(w_{0}/2)^{2})$ in standard light-source units of
${\rm photons}\cdot{\rm s}^{-1}{\rm mm}^{-2}{\rm mrad}^{-2}/0.1\%\;{\rm
bandwidth}$) due to a 100 pC electron bunch with $\gamma=400$ and momentum
spread given by table 2 scattering from a 30 fs laser pulse with $a_{0}=50$.

To compare this to other synchrotron light sources, it is also useful to plot
the on-axis spectrum in terms of the standard units of the synchrotron
community, ${\rm photons}\cdot{\rm s}^{-1}{\rm mm}^{-2}{\rm
mrad}^{-2}/0.1\%\;{\rm bandwidth}$. To this, the spectral intensity is
multiplied by a numerical factor that assumes the 500 electrons are a
reasonable statistical representation of a 100 pC electron bunch that is
typical of laser wakefield experiments mangles ; geddes ; faure . Also
necessary for this calculation, the source size of the radiation is taken to
be the laser spot area within the radius of half the pulse waist,
$\pi(w_{0}/2)^{2}$. The on-axis radiation spectrum is shown in figure 4. As
well as peaking at high energies, the peak spectral brilliance is also
extremely high, comparable to the FLASH free electron laser but at
significantly higher photon energies Robinson_NJP_2010  and significantly more
brilliant than conventional synchrotrons. The effect of the high intensity
dramatically increases the brilliance of the source, at the expense of the
band-width which at lower intensity can be extremely narrow, which may be of
more utility for some applications Hartemann_PRSTAB_2007 ; Albert_PRSTAB_2010
; Albert_PRSTAB_2011 .

In figure 5 the cumulative photon number,
$\int_{0}^{E}(dN/dE^{\prime})dE^{\prime}$, _per electron_ is shown for this
spectrum, showing that on average each electron interacting with the laser
field emits approximately 200 photons. When integrated numerically, the total
photon energy emitted by each electron is $3.5\times 10^{-10}$ J. This is 10
times more than the energy of a 200 MeV electron. This result may
superficially appear not to conserve energy, however, the radiated photon
energy is predominantly drawn from the laser pulse. For a bunch of $10^{9}$
electrons, which is of the order 100 pC of charge, the total energy output
would be 0.35 J. For a bunch of this size, depletion of the laser fields – if
treated self-consistently – would modify the electron dynamics and radiation
output, but should only be a small perturbation (the pulse energy used here is
19.0 J) and hence would not be expected to modify this output energy
significantly. Ignoring this correction, the conversion efficiency of laser
pulse energy into $\gamma$-rays is 1.8%.

Figure 5: The cumulative photon number
($\int_{0}^{E}dN/dE^{\prime}dE^{\prime}$) per electron due to a 500 electron
bunch with $\gamma=400$ and momentum spread scattering given by table 2 from a
laser pulse with $a_{0}=50$. Figure 6: The angularly resolved spectral
intensity ($d^{2}I/d\omega d\Omega$) due to a zero emittance 500 electron
bunch with $\gamma=400$ and no scattering from a laser pulse with $a_{0}=50$.
In (a) the radiation reaction force is not included, and in (b) the radiation
reaction force is included. The contours are taken at identical spectral
intensity levels for both cases, normalized to the peak which is 2.6872$\times
10^{-26}$ Js-1 at 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8.

###  IV.2 On the observation of radiation reaction effects in the photon
distribution

It has been suggested that signatures of the radiation reaction forces may be
observed in the photon distribution emitted in a counter propagating
experiment Koga_POP_2005 ; Piazza_PRL_2009 . The numerical calculations
performed here suggest this may be difficult due to the momentum spread of the
electron beam. Figure 6 shows the spectral intensity of radiation emitted
under identical conditions to those of figure 3 except that here the electron
beam has zero momentum spread, as in table 2 case B. The distribution has fine
features that are smoothed out when the electron beam has a momentum spread,
as would be expected. To see more clearly the effect of momentum spread on the
radiation distribution, figures 7 and 8 show two dimensional slices through
the radiation intensity distribution, in the planes parallel and perpendicular
to the laser polarization. In addition, the spectral intensity has been
converted into a photon distribution _per electron_ ,
$\omega_{0}d^{2}N/d\omega d\Omega$, which is more likely to be the form of
data obtained in an experiment (i.e. a histogram of photon hits on an array of
single photon counting detectors).

Figure 7: The photon distribution (normalized to the laser frequency,
$\omega_{0}d^{2}N/d\omega d\Omega$) per electron due to a 500 electron bunch
with $\gamma=400$ and zero momentum spread scattering from a laser pulse with
$a_{0}=50$. In (a) and (c) radiation reaction force is not included, and in
(b) and (d) radiation reaction force is included. (a) and (b) show the photon
distribution in the plane perpendicular to the laser polarization and (c) and
(d) show the photon distribution in the plane parallel to the laser
polarization.

Figure 7 shows the photon distribution from a zero momentum spread electron
beam interaction. (a) and (c) _radiation reaction force is not included_ , and
in (b) and (d) _radiation reaction force is included_. (a) and (b) show the
photon distribution in the plane perpendicular to the laser polarization and
(c) and (d) show the photon distribution in the plane parallel to the laser
polarization. The angular distribution of photons shows pronounced differences
with and without radiation reaction forces, and the energy distribution is
also dramatically changed, in particular resulting in a large number of low
energy photons in the damped case compared to no damping. Another feature is
slow oscillations in the spectral intensity with frequency/energy. These may
be due to the short truncated electron bunch and laser pulse in the time
domain, which will result in long wavelength oscillations in the frequency
domain.

Figure 8: The photon distribution (normalized to the laser frequency,
$\omega_{0}d^{2}N/d\omega d\Omega$) per electron due to a 500 electron bunch
with $\gamma=400$ and momentum spread given by table 2 scattering from a laser
pulse with $a_{0}=50$. In (a) and (c) radiation reaction force is not
included, and in (b) and (d) radiation reaction force is included. (a) and (b)
show the photon distribution in the plane perpendicular to the laser
polarization and (c) and (d) show the photon distribution in the plane
parallel to the laser polarization.

When the electron bunch is given the momentum spread of table 2 case A, the
distinction between the cases with and without radiation reaction force
becomes significantly less distinct. Figure 8 shows the photon distribution
from this interaction. There is little difference in the spectral intensity
distribution with and without radiation reaction force effects, except that
the overall magnitude is reduced, and the peak energy is reduced. Differences
in the angular distribution, however, are small and are likely to be much
smaller than expected shot-to-shot fluctuations in electron beam emittance.
Coupling this to the intrinsic difficulty of measuring high energy photons in
a collimated beam, it appears to be unfeasible to suggest that radiation
reaction effects will be discernible in experimental measurements in this
configuration in the near term.

###  IV.3 On the observation of radiation reaction effects in the electron
phase-space distribution

In contrast to the photon measurements, it should be very easy to observe
radiation reaction effects in the electrons as measured using a standard
scintillating screen configuration. It is typical in laser wakefield
accelerator experiments to measure either the electron beam profile using a
scintillating screen, or electron forward momentum spectrum using a deflecting
magnet and a scintillating screen geddes ; faure . These diagnostics
effectively correspond to the $p_{1}$-$p_{2}$ and $p_{1}$-$p_{3}$ electron
phase-space densities respectively – with a spectrometer, the deflection by
the magnetic field disperses the electrons by $p_{3}$, but the projection in
$p_{1}$ is maintained.

Figure 9: Two dimensional histograms of the $p_{1}$-$p_{2}$ phase space
distribution of a 500 electron bunch with momentum spread according to table 2
case B before (top) and after (bottom) interaction with the high intensity
($a_{0}=50$) pulse. In (a) and (c) there is no radiation reaction, and in (b)
and (d) a radiation reaction model included according to equation 2.

Figure 9 shows the $p_{1}$-$p_{2}$ phase-space density for the electron bunch
before and after the interaction as two dimensional histogram plots. The
electrons are deflected by the laser fields so that the transverse momentum
spread is increased in both cases, consistent with a ponderomotive deflection.
However, there is little difference between the cases with and without
radiation reaction forces. This is because the radiation damping effect
reduces both transverse and longitudinal momenta proportionally (to lowest
order the radiation force in equation 2 is
$dp^{\mu}/d\tau|_{fric}=-\tau_{0}\omega_{0}^{2}\gamma^{2}a^{2}p^{\mu}$), and
hence in general the exit angle of a particular electron $\theta_{exit}\simeq
p_{\bot}/p_{3}$ is not expected to change significantly.

Figure 10: Two dimensional histograms of the $p_{3}$-$p_{1}$ phase space
distribution of a 500 electron bunch with momentum spread according to table 2
case B before (top) and after (bottom) interaction with the high intensity
($a_{0}=50$) pulse. In (a) and (c) there is no radiation reaction, and in (b)
and (d) a radiation reaction model included according to equation 2. _Note
that the horizontal momentum scale is negative and not the same for each phase
space_.

The effect on the electron spectrum is dramatic however, as was also
previously shown by Koga et al. Koga_POP_2005 . In figure 10, two dimensional
histogram plots of the $p_{3}$-$p_{1}$ phase space density of the electron
bunch is shown with and without radiation reaction forces. Under the
conditions modeled here, the electron beam loses almost half its energy when
radiation damping is included (and as expected, it experiences little change
in energy without radiation damping).

Figure 11: Two dimensional histograms of the $p_{3}$-$p_{1}$ phase space
distribution of a 500 electron bunch with momentum spread according to table 2
case B before (top) and after (bottom) interaction with the high intensity
($a_{0}=50$) pulse with radiation damping. (a) and (c) use the purely
classical expression 2 and in (b) and (d) the radiation reaction four-force is
modified by multiplying by the instantaneous $g$ factor given by equation 13.
_Note that the horizontal momentum scale is negative and not the same for each
phase space._

Finally, figure 11 shows two dimensional histogram plots of the
$p_{3}$-$p_{1}$ phase space density of the electron bunch similar to figure
10, but this time the radiation reaction is shown with and without the factor
$g(\chi_{e})$ given by equation 13 included. It can be seen by the plot that
under these conditions the _electron_ spectrum after the interaction with the
$g$ factor differs from the purely classical result by $\sim 10$% relative to
the overall energy loss. There is a smaller energy loss due to the fact that
the expected radiation spectrum is less energetic than the purely classical
result would suggest. This difference between classical and quantum corrected
radiation reaction may be sufficiently large to be distinguishable over
experimental fluctuations if well characterized. The effect of the addition of
$g(\chi_{e})$ on the _photon_ spectrum calculated with classical radiation
reaction forces under these conditions is negligible.

##  V Conclusions

The counter-propagating electron beam, ultra-high-intensity laser interaction
is likely to be attempted by numerous experimental groups in the near future.
In addition to ultimately studying quantum electrodynamic effects, initial
experiments with lower electron beam energies and laser intensities are likely
to be concerned with the brilliant high energy photon output and classical
forms of radiation forces. From these numerical calculations, we predict a
large flux of photons with energy in excess of 1 MeV, in a beam collimated
within a 10 mrad divergence angle, and with an elliptical angular distribution
due to the linear polarization of the laser pulse. Each electron should emit
$\sim$100 photons above 1 MeV for a 200 MeV Gaussian electron beam colliding
with a pulse of intensity $5\times 10^{21}$ Wcm-2. For a typical laser
wakefield accelerated electron bunch with 100 pC charge mangles ; geddes ;
faure , this should result in $\sim 10^{11}$ photons in a broad synchrotron-
like spectrum peaking at 10 MeV with approximately 2% conversion efficiency of
laser energy into $\gamma$-rays, in a beam collimated to less than 10 mrad
divergence and with a peak brightness exceeding $10^{29}$ photons
s-1mm-2mrad${}^{-2}(0.1$% bandwidth$)^{-1}$.

In addition we show that measurements of the _radiation_ will be unlikely to
be able to indicate signatures of radiation reaction forces, and in particular
the ability to distinguish between different classical or quantum formulations
of the radiation force, due to the effects of beam emittance and tight laser
focusing. However, it should still be easy to observe radiation reaction
effects in the electron spectrum, where differences compared with a no-
radiation force model is dramatic, even with moderate beam emittance.
Including quantum effects using the $g(\chi_{e})$ factor under these
parameters causes a sufficiently reduced damping effect on the electron energy
spectrum to be measurable.

Whether signatures of different classical radiation reaction force models can
be observed in experiment is not addressed by the results of this paper.
However, it is unlikely, since the primary measurable effect on the electrons
is energy loss, which is to low order similar for all formulations of the
radiation reaction force. It is likely that the differences between models
will be hidden by the effects of beam emittance and laser focusing conditions
also.

##  VI Acknowledgements

This work was funded by the NSF under contracts 1054164 and PHY-0935197, DARPA
under contract N66001-11-1-4208, EPSRC (Grant numbers EP/I014462/1 and
EP/G055165/1) and the Royal Society.

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∎

11institutetext: Walid Hariri 22institutetext: Labged Laboratory, Computer
Science department
Badji Mokhtar Annaba University
22email<EMAIL_ADDRESS>
# Efficient Masked Face Recognition Method during the COVID-19 Pandemic

Walid Hariri

(Received: date / Accepted: date)

###### Abstract

The coronavirus disease (COVID-19) is an unparalleled crisis leading to a huge
number of casualties and security problems. In order to reduce the spread of
coronavirus, people often wear masks to protect themselves. This makes face
recognition a very difficult task since certain parts of the face are hidden.
A primary focus of researchers during the ongoing coronavirus pandemic is to
come up with suggestions to handle this problem through rapid and efficient
solutions. In this paper, we propose a reliable method based on occlusion
removal and deep learning-based features in order to address the problem of
the masked face recognition process. The first step is to remove the masked
face region. Next, we apply three pre-trained deep Convolutional Neural
Networks (CNN) namely, VGG-16, AlexNet, and ResNet-50, and use them to extract
deep features from the obtained regions (mostly eyes and forehead regions).
The Bag-of-features paradigm is then applied to the feature maps of the last
convolutional layer in order to quantize them and to get a slight
representation comparing to the fully connected layer of classical CNN.
Finally, Multilayer Perceptron (MLP) is applied for the classification
process. Experimental results on Real-World-Masked-Face-Dataset show high
recognition performance compared to other state-of-the-art methods.

###### Keywords:

Face recognition COVID-19 Masked face Deep learning.

††journal:

##  1 Introduction

The COVID-19 can be spread through contact and contaminated surfaces,
therefore, the classical biometric systems based on passwords or fingerprints
are not anymore safe. Face recognition is safer without any need to touch any
device. Recent studies on coronavirus have proven that wearing a face mask by
a healthy and infected population reduces considerably the transmission of
this virus. However, wearing the mask face causes the following problems: 1)
fraudsters and thieves take advantage of the mask, stealing and committing
crimes without being identified. 2) community access control and face
authentication have become very difficult tasks when a grand part of the face
is hidden by a mask. 3) existing face recognition methods are not efficient
when wearing a mask which cannot provide the whole face image for description.
4) exposing the nose region is very important in the task of face recognition
since it is used for face normalization peng2011training , pose correction
lu2005matching , and face matching koudelka2005prescreener . Due to these
problems, face masks have significantly challenged existing face recognition
methods.

To tackle these problems, we distinguish two different tasks namely: face mask
recognition and masked face recognition. The first one checks whether the
person is wearing a mask or no. This can be applied in public places where the
mask is compulsory. Masked face recognition, on the other hand, aims to
recognize a face with a mask basing on the eyes and the forehead regions. In
this paper, we handle the second task using a deep learning-based method. We
use a pre-trained deep learning-based model in order to extract features from
the unmasked face regions (out of the mask region). It is worth stating that
the occlusions in our case can occur in only one predictable facial region
(nose and mouth regions), this can be a good guide to handle this problem
efficiently.

The rest of this paper is organized as follows: Section 2 presents the related
works. In Section 3 we present the motivation and contribution of the paper.
The proposed method is detailed in Section 4. Experimental results are
presented in Section 5. Conclusion ends the paper.

##  2 Related works

Occlusion is a key limitation of real-world 2D face recognition methods.
Generally, it comes out from wearing hats, eyeglasses, masks as well as any
other objects that can occlude a part of the face while leaving others
unaffected. Thus, wearing a mask is considered the most difficult facial
occlusion challenge since it occludes a grand part of the face including the
nose. Many approaches have been proposed to handle this problem. We can
classify them into three categories namely: local matching approach,
restoration approach, and occlusion removal approach.

Matching approach: Aims to compare the similarity between images using a
matching process. Generally, the face image is sampled into a number of
patches of the same size. Feature extraction is then applied to each patch.
Finally, a matching process is applied between probe and gallery faces. The
advantage of this approach is that the sampled patches are not overlapped,
which avoids the influence of occluded regions on the other informative parts.
For example, Martinez et aleix. martinez2002recognizing  sampled the face
region into a fixed number of local patches. matching is then applied for
similarity measure.

Other methods detect the keypoints from the face image, instead of local
patches. For instance, Weng et al. weng2016robust  proposed to recognize
persons of interest from their partial faces. To accomplish this task, they
firstly detected keypoints and extract their textural and geometrical
features. Next, point set matching is carried out to match the obtained
features. Finally, the similarity of the two faces is obtained through the
distance between these two aligned feature sets. Keypoint based matching
method is introduced in Duan et al. duan2018topology . SIFT keypoint
descriptor is applied to select the appropriate keypoints. Gabor ternary
pattern and point set matching are then applied to match the local keypoints
for partial face recognition. In contrast to the above-mentioned methods based
on fixed-size patches matching or keypoints detection, McLaughlin et al.
mclaughlin2016largest  applied the largest matching area at each point of the
face image without any sampling.

Restoration approach: Here, the occluded regions in the probe faces are
restored according to the gallery ones. For instance, Bagchi et al.
bagchi2014robust  proposed to restore facial occlusions. The detection of the
occluded regions is carried out by thresholding the depth map values of the 3D
image. Then the restoration is taken on by Principal Component Analysis (PCA)
wold1987principal . There are also several approaches that rely on the
estimation of the occluded parts. Drira et al. drira20133d  applied a
statistical shape model to predict and restore the partial facial curves.
Iterative closest point (ICP) algorithm has been used to remove occluded
regions in gawali20143d . The restoration is applied using a curve, which uses
a statistical estimation of the curves to manage the occluded parts. Partially
observed curves are completed by using the curves model produced through the
PCA technique.

Occlusion removal approach: In order to avoid a bad reconstruction process,
these approaches aim to detect regions found to be occluded in the face image
and discard them completely from the feature extraction and classification
process. Segmentation based approach is one of the best methods that detect
firstly the occluded region part and using only the non-occluded part in the
following steps. For instance, Priya and Banu priya2014occlusion  divided the
face image into small local patches. Next, to discard the occluded region,
they applied the support vector machine classifier to detect them. Finally, a
mean-based weight matrix is used on the non-occluded regions for face
recognition. Alyuz et al. alyuz20133  applied an occlusion removal and
restoration. They used the global masked projection to remove the occluded
regions. Next, the partial Gappy PCA is applied for the restoration using
eigenvectors.

Since the publication of AlexNet architecture in 2012 by Krizhevsky et al.
krizhevsky2012imagenet , deep CNN have become a common approach in face
recognition. It has also been successfully used in face recognition under
occlusion variation almabdy2019deep . It is seen that the deep learning-based
method are founded on the fact that the human visual system automatically
ignores the occluded regions and only focuses on the non-occluded ones. For
example, Song et al. song2019occlusion  proposed a mask learning technique in
order to discard the feature elements of the masked region for the recognition
process.

Inspired by the high performance of CNN based methods that have strong
robustness to illumination, facial expression, and facial occlusion changes,
we propose in this paper an occlusion removal approach and deep CNN based
model to address the problem of masked face recognition during the COVID-19
pandemic. Motivations and more details about the proposed method are presented
in the following sections.

##  3 Motivation and contribution of the paper

Motivated by the efficiency and the facility of the occlusion removal
approaches, we apply this strategy to discard the masked regions. Experimental
results are carried out on Real-world Masked Face Recognition Dataset (RMFRD)
and Simulated Masked Face Recognition Dataset (SMFRD) presented in
wang2020masked . We start by localizing the mask region. To do so, we apply a
cropping filter in order to obtain only the informative regions of the masked
face (i.e. forehead and eyes). Next, we describe the selected regions using a
pre-trained deep learning model as a feature extractor. This strategy is more
suitable in real-world applications comparing to restoration approaches.
Recently, some works have applied supervised learning on the missing region to
restore them such as in din2020novel . This strategy, however, is a difficult
and highly time-consuming process.

Despite the recent breakthroughs of deep learning architectures in pattern
recognition tasks, they need to estimate millions of parameters in the fully
connected layers that require powerful hardware with high processing capacity
and memory. To address this problem, we present in this paper an efficient
quantization based pooling method for face recognition using three pre-trained
models. To do so, we only consider the feature maps at the last convolutional
layers (also called channels) using Bag-of-Features (BoF) paradigm.

The basic idea of the classical BoF paradigm is to represent images as
orderless sets of local features. To get these sets, the first step is to
extract local features from the training images, each feature represents a
region from the image. Next, the whole features are quantized to compute a
codebook. Test image features are then assigned to the nearest code in the
codebook to be represented by a histogram. In the literature, the BoF paradigm
has been largely used for handcrafted feature quantization lobel2013joint  to
accomplish image classification tasks. A comparative study between BoF and
deep learning for image classification has been made in Loussaief and
Abdelkrim loussaief2018deep . To take full advantage of the two techniques, in
this paper we can consider BoF as a pooling layer in our trainable
convolutional layers. This aims to reduce the number of parameters and makes
it possible to classify masked face images.

This deep quantization technique presents many advantages. It ensures a
lightweight representation that makes the real-world masked face recognition
process a feasible task. Moreover, the masked regions vary from one face to
another, which leads to informative images of different sizes. The proposed
deep quantization allows classifying images from different sizes in order to
handle this issue. Besides, the Deep BoF approach uses a differentiable
quantization scheme that enables simultaneous training of both the quantizer
and the rest of the network, instead of using fixed quantization merely to
minimize the model size passalis2017learning . It is worth stating that our
proposed method doesn’t need to be trained on the mission region after
removing the mask. It instead improves the generalization of the face
recognition process in the presence of the mask during the pandemic of
coronavirus.

Figure 1: Overview of the proposed method. Figure 2: 2D Face rotation. Figure
3: (1): Masked face. (2): Sampling the masked face image into 100 regions of
the same size. (3): Cropping filter.

##  4 The proposed method

Fig. 1 presents an overview of the proposed method. It has four steps:

###  4.1 Preprocessing and cropping filter

The images of the used dataset are already cropped around the face, so we
don’t need a face detection stage to localize the face from each image.
However, we need to correct the rotation of the face so that we can remove the
masked region efficiently. To do so, we detect 68 facial landmarks using Dlib-
ml open-source library introduced in king2009dlib . According to the eye
locations, we apply a 2D rotation to make them horizontal as presented in Fig.
2.

The next step is to apply a cropping filter in order to extract only the non-
masked region. To do so, we firstly normalize all face images into 240
$\times$ 240 pixels. Next, we partition a face into blocks. The principle of
this technique is to divide the image into 100 fixed-size square blocks (24
$\times$ 24 pixels in our case). Then we extract only the blocks including the
non-masked region (blocks from number 1 to 50). Finally, we eliminate the rest
of the blocks as presented in Fig. 3.

###  4.2 Feature extraction layer

To extract deep features from the informative regions, we have employed three
pre-trained models as feature extractors:

#### VGG-16:

simonyan2014very  is trained on the ImageNet dataset which has over 14 million
images and 1000 classes. Its name VGG-16 comes from the fact that it has 16
layers. It contains different layers including convolutional layers, Max
Pooling layers, Activation layers, and Fully Connected (fc) layers. There are
13 convolutional layers, 5 Max Pooling layers, and 3 Dense layers which sum up
to 21 layers but only 16 weight layers. In this work, we choose the VGG-16 as
the base network, and we only consider the feature maps (FMs) at the last
convolutional layer, also called channels. This layer is employed as a feature
extractor and will be used for the quantization in the following stage. Fig. 4
presents VGG-16 architecture.

#### AlexNet:

has been successfully employed for image classification tasks
krizhevsky2017imagenet . This deep model is pre-trained on a few millions of
images from the ImageNet database through eight learned layers, five
convolutional layers and three fully-connected layers. The last fully-
connected layer allows to classify one thousand classes. The fifth
convolutional layer is used in this paper to extract deep features (See Fig.
5).

#### ResNet-50:

he2016deep  has been successfully used in various pattern recognition tasks
such as face and pedestrian detection mliki2020improved . It containing 50
layers trained on the ImageNet dataset. This network is a combination of
Residual network integrations and Deep architecture parsing. Training with
ResNet-50 is faster due to the bottleneck blocks. It is composed of five
convolutional blocks with shortcuts added between layers. The last convolution
layer is used to extract Deep Residual Features (DRF). Fig. 6 shows the
architecture of the ResNet-50 model.

Figure 4: VGG-16 network architecture introduced in simonyan2014very . Figure
5: AlexNet network architecture introduced in krizhevsky2017imagenet . Figure
6: ResNet-50 network architecture introduced in he2016deep . The extracted DRF
are shown.

###  4.3 Deep bag of features layer

From the $i^{th}$ image, we extract feature maps using the feature extraction
layer described above. In order to measure the similarity between the
extracted feature vectors and the codewords also called term vector, we
applied the RBF kernel as similarity metric as proposed in
passalis2017learning . Thus, the first sub-layer will be composed of RBF
neurons, each neuron is referred to a codeword.

As presented in Fig. 1, the size of the extracted feature map defines the
number of the feature vectors that will be used in the BoF layer. Here we
refer by $V_{i}$ to the number of feature vectors extracted from the $i^{t}h$
image. For example, if we have 10 $\times$ 10 feature maps from the last
convolutional layer of the chosen pre-trained model, we will have 100 feature
vectors to feed the quantization step using the BoF paradigm. To build the
codebook, the initialization of the RBF neurons can be carried out manually or
automatically using all the extracted feature vectors overall the dataset. The
most used automatic algorithm is of course k-means. Let $F$ the set of all the
feature vectors, defined by: $F=\\{V_{ij},i=1\ldots V,j=1\ldots V_{i}\\}$ and
$V_{k}$ is the number of the RBF neurons centers referred by $c_{k}$. Note
that these RBF centers are learned afterward to get the final codewords. The
quantization is then applied to extract the histogram with a predefined number
of bins, each bin is referred to a codeword. RBF layer is then used as a
similarity measure, it contains 2 sub-layers:

(I) RBF layer: measures the similarity of the input features of the probe
faces to the RBF centers. Formally: the $j^{th}$ RBF neuron $\phi(X_{j})$ is
defined by Eq.(1):

$\phi(X_{j})=\exp(\|x-c_{j}\|_{2}/\sigma_{j}),$ (1)

Where $x$ is a feature vector and $c_{j}$ is the center of the $j^{th}$ RBF
neuron.

(II) Quantization layer: the output of all the RBF neurons is collected in
this layer that contains the histogram of the global quantized feature vector
that will be used for the classification process. The final histogram is
defined by Eq.(2), where $\phi({V})$ is the output vector of the RBF layer
over the $c_{k}$ bins.

$h_{i}={V_{j}}\sum^{N^{k}}_{k}\phi({V_{jk}})$ (2)

###  4.4 Fully connected layer and classification

Once the global histogram is computed, we pass to the classification stage to
assign each test image to its identity. To do so, we apply the Multilayer
perceptron classifier (MLP) where each face is represented by a term vector.
Deep BoF network can be trained using back-propagation and gradient descent.
Note that the 10-fold cross-validation strategy is applied in our experiments
on the RMFRD dataset. We note $V=[v1,\dots,v_{k}]$ the term vector of each
face, where each $v_{i}$ refers to the occurrence of the term $i$ in the given
face. $t$ is the number of attributes, and $m$ is the number of classes (face
identities). Test faces are defined by their codeword $V$. MLP uses a set of
term occurrences as input values ($v_{i}$) and associated weights ($w_{i}$)
and a sigmoid function ($g$) that sums the weights and maps the results to
output ($y$). Note that the number of hidden layers used in our experiments is
given by: $\frac{m+t}{2}$.

##  5 Experimental results

To evaluate the proposed method, we carried out experiments on very
challenging masked face datasets. In the following, we present the datasets’
content and variations, the experimental results using the quantization of
deep features obtained from three pre-trained models, and a comparative study
with other state-of-the-arts.

###  5.1 Dataset description

Real-World-Masked-Face-Dataset wang2020masked  is a masked face dataset
devoted mainly to improve the recognition performance of the existing face
recognition technology on the masked faces during the COVID-19 pandemic. It
contains three types of images namely, Masked Face Detection Dataset (MFDD),
Real-world Masked Face Recognition Dataset (RMFRD), and Simulated Masked Face
Recognition Dataset (SMFRD). In this paper, we focus on the last two datasets
described in the following.

####

a) RMFRD is one of the richest real-world masked face datasets. It contains
5,000 images of 525 subjects with masks, and 90,000 images without masks which
represent 525 subjects. A semi-automatic annotation strategy has been used to
crop the informative face parts. Fig. 7 presents some pairs of face images
from RMFRD dataset.

####

b) SMFRD contains 500,000 simulated masked faces of 10,000 subjects from two
known datasets Labeled Faces in the Wild (LFW) huang2008labeled  and Webface
yi2014learning . The simulation is carried out using Dlib library dlibwebsite
. This dataset is balanced but more challenging since the simulated masks are
not necessarily in the right position. Fig. 8 shows some examples of simulated
masked faces from SMFRD dataset.

Figure 7: Pairs of face images from RMFRD dataset: face images without a mask
(up) and with a mask (down). Figure 8: Masked faces from SMFRD dataset.

###  5.2 Method performance

The face images were firstly preprocessed as described in Section 4.1. In
contrast to SMFRD dataset, RMFRD is imbalanced (5,000 masked faces vs 90,000
non-masked faces). Therefore, we have applied an over-sampling by cropping
some non-masked faces to get an equivalent number of cropped and full faces.
Next, using the normalized 2D faces, we employ the three pre-trained models
(VGG-16, AlexNet and ResNet-50) separately to extract deep features from their
last convolutional layers as presented in Section 4.2. The output features are
($14\times 14\times 512$, $13\times 13\times 256$, $7\times 7\times 2048$)
dimensional, respectively.

The quantization is then applied to extract the histogram of a number of bins
as presented in Section 4.3. Finally, MLP is applied to classify faces as
presented in Section 4.4. In this experiment, the 10-fold cross-validation
strategy is used to evaluate the recognition performance. The experiments are
repeated ten times in RMFRD and SMFRD datasets separately, where 9 samples are
used as the training set and the remaining sample as the testing set, and the
average results are calculated.

Table 1 reports the classification rates on the RMFRD dataset using four
different sizes of the codebook (i.e. number of codewords in RBF layer) by
(i.e. 50, 60, 70, 100 term vectors per image). We can see that the best
recognition rate is obtained using the third FMs in the last convolutional
layer from VGG-16 with 60 codewords by 91.3%. The second FMs achieved 90.8%
with 50 codewords and outperformed the first FMs over the four codeword sizes.
AlexNet, on the other hand, realized 86.6% with 100 codewords where the best
recognition rate achieved by ResNet-50 was 89.5% with 70 codewords. In this
experiment, it is clear that VGG-16 outperformed the AlexNet and ResNet-50
models.

Table 2 reports the classification rates on the SMFRD dataset. The highest
recognition rate is achieved by the ResNet-50 through the quantization of DRF
features by 88.9%. This performance is achieved using 70 codewords that feed
an MLP classifier. AlexNet model realized good recognition rates comparing to
the VGG-16 model (86.0% vs 85.6% as highest rates).

Table 1: Recognition performance on RMFRD dataset using four codebook sizes. Method | Size 1 | Size 2 | Size 3 | Size 4
---|---|---|---|---
term vectors | 50 | 60 | 70 | 100
| VGG-16 | model |  |
Conv5 FM1 14$\times$14$\times$512 | 88.5% | 89.2% | 87.1% | 87.5%
Conv5 FM2 14$\times$14$\times$512 | 90.8% | 87.4% | 87.2% | 88.0%
Conv5 FM3 14$\times$14$\times$512 | 91.0% | 91.3% | 90.1% | 89.8%
| AlexNet | model |  |
Conv5 FM 13 $\times$ 13 $\times$ 256 | 84.3% | 85.7% | 85.9% | 86.6%
| ResNet-50 | model |  |
Conv5 FM 7$\times$7$\times$2048 | 87.4% | 87.9% | 89.5% | 89.3%
Table 2: Recognition performance on SMFRD dataset using four codebook sizes. Method | Size 1 | Size 2 | Size 3 | Size 4
---|---|---|---|---
term vectors | 50 | 60 | 70 | 100
| VGG-16 | model |  |
Conv5 FM1 14$\times$14$\times$512 | 82.4% | 83.7% | 84.5% | 84.7%
Conv5 FM2 14$\times$14$\times$512 | 83.1% | 83.5% | 85.0% | 85.4%
Conv5 FM3 14$\times$14$\times$512 | 81.7% | 81.3% | 84.4% | 85.6
| AlexNet | model |  |
Conv5 FM 13 $\times$ 13 $\times$ 256 | 83.7% | 83.9% | 84.2% | 86.0%
| ResNet-50 | model |  |
Conv5 FM 7$\times$7$\times$2048 | 83.5% | 84.7% | 88.9% | 88.5%

###  5.3 Performance comparison

To further evaluate the performance of our proposed method, we have compared
the obtained experimental results with those of other face recognizers on the
RMFRD and SMFRD datasets as follows:

#### Comparison with transfer learning-based technique:

We have tested the face recognizer presented in luttrell2018deep  that
achieved a good recognition accuracy on two subsets of the FERET database
phillips1998feret . This technique is based on transfer learning (TL) which
employs pre-trained models and fine-tuning them to recognize masked faces from
RMFRD and SMFRD datasets. The reported results in Table 3 show that the
proposed method outperformed the TL-based method on the RMFRD and SMFRD
datasets.

#### Comparison with covariance-based technique:

Covariance-based features have been applied in hariri20163d  and achieved high
recognition performance on 3D datasets in the presence of occluded regions. We
have employed this method using 2D-based features (texture, gray level, LBP)
to extract covariance descriptors. The evaluation on the RMFRD and SMFRD
datasets confirms the superiority of the proposed method as shown in Table 3.

#### Comparison with deep feature extractor:

Another efficient face recognition method using the same pre-trained models
(AlexNet and ResNet-50) is proposed in almabdy2019deep  and achieved a high
recognition rate on various datasets. Nevertheless, the pre-trained models are
employed in a different manner. It consists of applying a TL technique to
fine-tune the pre-trained models to the problem of masked face recognition
using an SVM classifier. We have tested the this strategy on the masked faces.
The results in Table 3 further demonstrate the efficiency of the BoF paradigm
compared to the use of a machine learning-based classifier directly.

Table 3: Performance comparison with state-of-the-art methods. Method | Dataset | Technique | Masks | Accuracy
---|---|---|---|---
Luttrell et al. luttrell2018deep  | RMFRD | TL | yes | 85.7%
Hariri et al. hariri20163d  | RMFRD | Covariance | yes | 84.6%
Almabdy et al. almabdy2019deep  | RMFRD | CNN+SVM | yes | 87.0%
Our | RMFRD | CNN+BoF | yes | 91.3%
Luttrell et al. luttrell2018deep  | SMFRD | TL | yes | 83.3%
Hariri et al. hariri20163d  | SMFRD | Covariance | yes | 83.8%
Almabdy et al. almabdy2019deep  | SMFRD | CNN+SVM | yes | 86.1%
Our | SMFRD | CNN+BoF | yes | 88.9%

###  5.4 Computation and training times comparison

The comparison of the computation times between the proposed method and
Almabdy et al.’s method almabdy2019deep  shows that the use of the BoF
paradigm decreases the time required to extract deep features and to classify
the masked faces (See Table 4). Note that this comparison is performed using
the same pre-trained models (VGG-16 and AlexNet) on the RMFRD dataset. AlexNet
is lowest training and testing time compared to VGG-16 with less GPU memory
usage.

Table 4: Training and testing time on the RMFRD dataset in milliseconds. Method | AlexNet | VGG-16
---|---|---
Almabdy et al. almabdy2019deep  |  train:550 test:34 |  train:930 test:120
Our |  train: 308 test:21 |  Train:605 test:84

###  5.5 Discussion

The obtained high accuracy compared to other face recognizers is achieved due
to the best features extracted from the last convolutional layers of the pre-
trained models, and the high efficiency of the proposed BoF paradigm that
gives a lightweight and more discriminative power comparing to classical CNN
with softmax function. Moreover, dealing with only the unmasked regions, the
high generalization of the proposed method makes it applicable in real-time
applications. Other methods, however, aim to unmask the masked face using
generative networks such as in din2020novel . This strategy is a greedy task
and not preferable for real-world applications.

The efficiency of each pre-trained model depends on its architecture and the
abstraction level of the extracted features. When dealing with real masked
faces, VGG-16 has achieved the best recognition rate, while ResNet-50
outperformed both VGG-16 and AlexNet on the simulated masked faces. This
behavior can be explained by the fact that VGG-16 features fail to ensure a
high discriminative power comparing to the DRF features that are still
relatively steady compared to their results on the real masked faces. When
dealing with other state-of-the-art recognizers, one of them applied the same
pre-trained models with a different strategy. The proposed method outperformed
TL-based method using the same pre-trained models. This performance is
explained by the fact that the fc layers of the pre-trained models are more
dataset-specific features (generally pre-trained on ImageNet dataset) which is
a very different dataset, thus, this strategy is not always suitable for our
task. Moreover, the proposed method outperformed previous methods in terms of
training time. The achieved performance further confirms that the BoF paradigm
is a slight representation that further reinforces the high discrimination
power of the deep features to feed a machine learning-based classifier.

##  6 Conclusion

The proposed method improves the generalization of the face recognition
process in the presence of the mask. To accomplish this task, we proposed a
deep learning-based method and quantization-based technique to deal with the
recognition of the masked faces. We employed three pre-trained models and used
their last convolutional layers as deep features. The Bof paradigm is applied
to quantize the obtained features and to feed an MLP classifier. The proposed
method outperformed other state-of-the-art methods in terms of accuracy and
time complexity. The proposed method can also be extended to richer
applications such as video retrieval and surveillance. In future work, we look
at the application of deep ensemble models with additional pre-trained models
to enhance the accuracy.

##  7 Acknowledgments

The authors would like to thank DGRSDT, Algeria.

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