# Dressing effects in laser-assisted ($e,2e$) process in fast electron-
hydrogen atom collisions in an asymmetric coplanar scattering geometry

Gabriela Buică<EMAIL_ADDRESS>Institute of Space Science, P.O. Box
MG-36, Ro 77125, Bucharest-Măgurele, Romania

###### Abstract

We present the theoretical treatment of laser-assisted $(e,2e)$ ionizing
collisions in hydrogen for fast electrons, in the framework of the first-order
Born approximation at moderate laser intensities and photon energies beyond
the soft-photon approximation. The interaction of the laser field with the
incident, scattered, and ejected electrons is treated nonperturbatively by
using Gordon-Volkov wave functions, while the atomic dressing is treated by
using first-order perturbation theory. Within this semi-perturbative formalism
we obtain a new closed formula for the nonlinear triple differential cross
section (TDCS), which is valid for linear as well circular polarizations. New
analytical simple expressions of TDCS are derived in the weak field domain and
low-photon energy limit. It was found that for non-resonant $(e,2e)$ reactions
the analytical formulas obtained for the atomic matrix element in the low-
photon energy limit give a good agreement, qualitative and quantitative, with
the numerical semi-perturbative model calculations. We study the influence of
the photon energy as well of the kinetic energy of the ejected electron on the
TDCS, in the asymmetric coplanar geometry, and show that the dressing of the
atomic target strongly influences the $(e,2e)$ ionization process.

(e,2e) collisions,laser field,scattering,ionization,atomic dressing,triple
differential cross section

###### pacs:

34.80.Qb, 34.50.Rk, 03.65.Nk, 34.80.Dp

##  I INTRODUCTION

It is well known that the study of the atomic ionization process by collisions
with electrons, the so-called $(e,2e)$ reaction, reveals information about the
electronic structure of the atomic target and residual ion b-j89 , and is of
interest in collision theory or in other fields such as plasma physics or
astrophysics, which need reliable scattering cross section data whelan93 .
Camilloni and coworkers camilloni72  were the first to use $(e,2e)$ reaction
as a tool for measuring the momentum distribution of the ejected electrons, in
a coplanar symmetric scattering geometry where the outgoing electrons have
equal energies and polar angles, at high incident and outgoing electron
energies. Since then, an increasing number of $(e,2e)$ experiments have been
performed over the years for different target atoms and for various
kinematical configurations, and the electron momentum spectroscopy (EMS) has
been developed to provide information on the electronic structure of atoms and
molecules weigold99 ; smirnov99 . The symmetric $(e,2e)$ reaction is the basis
of EMS, also known as binary $(e,2e)$ spectroscopy, and is kinematically
characterized by a large momentum transfer of the projectile electron and a
small momentum of the residual ion. Another useful scattering configuration is
the coplanar asymmetric geometry with fast incident electrons (keV) and
ejected electrons of low and moderate energies, where most of the $(e,2e)$
reactions occur coplan94 .

In the past few decades the electron-impact ionization of an atom in the
presence of a laser field has become increasingly interesting and it is often
referred to as the laser-assisted $(e,2e)$ collision ehlotzky98 . Recently,
Hø̈hr and coworkers hohr2007  performed the first kinematically-complete
experiment for laser-assisted ionization in electron–helium collisions at high
incident electron energy ($1$ keV) and showed significant differences of the
triple differential cross section (TDCS) in comparison to the field-free
cross-sections. Very recently, Hiroi and coworkers hiroi2021  reported the
observation the laser-assisted electron-impact ionization of Ar in an
ultrashort intense laser field, and showed that the signal intensity of the
laser-assisted process for one-photon absorption obtained by integrating the
signals over the detection angle ranges is about twice as large as that
estimated by previous theoretical calculations in which the atomic dressing by
the laser field is neglected cavaliere80-81 .

A large number of papers have been published so far and several theoretical
approaches have been proposed, involving ejected electrons of low energies
that are studied under the combined influence of the laser field and the
Coulomb field of the residual ion. The early theoretical works on laser-
assisted $(e,2e)$ scattering have neglected the dressing of the atomic target
by the laser field or have used the closure approximation for laser-atom
interaction. First, Jain and Tzoar introduced the Coulomb-Volkov wave
functions jain78 , which takes into account the influence of the Coulomb field
of the nucleus on the final electron state. Since then, the effect of the
Coulomb interaction in the laser-assisted $(e,2e)$ collisions on hydrogen atom
was studied in several papers by employing different types of final state wave
functions like the Coulomb-Volkov or Coulomb corrected Gordon-Volkov wave
functions. Banerji and Mittleman banerji81  calculated TDCS for ionization of
hydrogen by electron impact, at low photon energies, in which the slow ejected
electron was described by a modified Coulomb wave function, and the laser-
electrons interactions were included in the low-frequency approximation.
Cavaliere and coworkers cavaliere80-81  studied laser-assisted $(e,2e)$
collisions in hydrogen at low photon energies, high incident electron
energies, and ejected electrons with moderate as well as small energies, in
the first-order Born approximation, with the incident and scattered electrons
described by the Gordon-Volkov wave functions gordon ; volkov , while the
ejected electron is represented by a modified Coulomb wave function. Later on,
the dressing of the atomic target by the laser field has been included in the
first-order time-dependent perturbation theory (TDPT), and therefore the
influence of the laser parameters such as intensity, polarization, and photon
energy has attracted a lot of interest from the theoretical point of view.
Joachain and coworkers joachain88  extended the semi-perturbative theory of
Byron and Joachain b-j , and showed the strong influence of a laser field on
the dynamics of laser-assisted $(e,2e)$ collisions in hydrogen, for fast
incident and scattered electrons and slow ejected electrons, in the Ehrhardt
asymmetric coplanar geometry ehrhardt . For $(e,2e)$ collisions in hydrogen
with slow ejected electrons, Martin and coworkers martin89  analyzed the
influence of the laser parameters: photon energy, laser intensity, and
polarization direction on the angular distribution of the ejected electrons.
The influence of laser polarization has also been discussed by Taïeb and
coworkers taieb1991 , who developed a dressed atomic wave functions on a basis
of Sturmian functions, which allowed to take into account accurately the
contribution of the continuum spectrum to the dressing of the atomic states
cionga93 . Very recently, Makhoute and coworkers ajana2019  presented their
numerical results obtained for $(e,2e)$ collision in atomic hydrogen in the
symmetric and asymmetric coplanar scattering geometries, at large photon
energies. For the direct scattering channel the calculation of the specific
radial amplitudes was performed by expanding the atomic wave functions in a
Sturmian basis, whereas the closure approximation was employed for the
exchange channel. As mentioned before most of these previous theoretical works
were focused on scattering geometries involving slow ejected electrons, and
only recently it was shown for ejected electrons of high energies that the
laser field strongly modifies the $(e,2e)$ collisions. New theoretical studies
for laser-assisted EMS at high impact energy and large momentum transfer were
published and it was found that the atomic dressing, calculated in the
closure- and low-frequency approximations, substantially influences the laser-
assisted TDCSs at low kouzakov2010 ; bulychev2012  and large photon energies
khalil2017 .

The purpose of the present paper is to study the laser-assisted $(e,2e)$
reactions in hydrogen, in which the target atom is ionized in collision with
an electron beam in the presence of a laser field, for fast incident and
outgoing electrons, in an asymmetric coplanar scattering geometry, beyond the
soft-photon approximation. We present a new method to derive the relevant
atomic transition amplitude which takes into account the dressing of the
target by the laser field. The laser field alone cannot significantly ionize
the hydrogen atom since the photon energy is considered below the ionization
threshold and the laser intensity is not high enough to allow ionization
through a multiphoton process. We assume fast scattered electrons of
sufficiently high velocity, such that we neglect their interaction with the
Coulomb field of the remaining ion. Similar to the approach used in the
Keldysh-Faisal-Reiss approximation keldysh65 ; faisal73 ; reiss80 , the
influence of the remaining ion on the final state of the fast ejected electron
is neglected, since the residual Coulomb field is weak compared to the laser
field strength. We follow the approach of Ref. joachain88  in which the semi-
perturbative theory b-j  was generalized to laser-assisted fast $(e,2e)$
collisions in atomic hydrogen. In order to simplify the calculations we
introduce several assumptions: (a) It is reasonable to employ a first-order
Born treatment of the projectile-atom interaction, since we consider fast
nonrelativistic collisions such that the velocities of the projectile and
outgoing electrons are much larger than the atomic unit bransden ; joa2012 .
(b) The non-relativistic Gordon-Volkov solutions are used for the incident and
outgoing electrons to describe their interaction with the laser field. (c) The
laser field intensity is considered moderate, but much weaker than the atomic
unit ($3.51\times 10^{16}$ W/cm2), in order to avoid direct one- and
multiphoton ionization. In contrast to other theoretical works we take into
account the atomic dressing effects in the first-order TDPT in the laser
field, going beyond the soft-photon approximation. The photon energy is
considered below the ionization threshold of the hydrogen atom, and one-photon
resonance transitions are allowed between the ground and excited states. (d)
Since the scattered and ejected electrons have high energies and of comparable
order of magnitude, our semi-perturbative formalism takes into account the
exchange effects in the first-order Born approximation.

The manuscript is organized as follows. In Sec. II we present the theoretical
method used in laser-assisted ionization of atomic hydrogen by electron
impact, and derive new analytical formulas for the ionization transition
amplitudes and TDCSs by electron impact. In the low-photon energy limit we
provide simple analytic formulas of TDCSs, in a closed form, for the laser-
assisted $(e,2e)$ ionization process which include the atomic dressing
effects. Numerical results are presented in Sec. III, where the TDCSs for
laser-assisted electron impact ionization of hydrogen are analyzed as a
function of the scattering angle of the ejected electron and as a function of
the photon energy. We have studied the modifications of the angular
distributions of the ejected electrons due to the external laser field at
different ejected electron energies and photon energies. Finally, summary and
conclusions are given in Sec. IV. Atomic units (a.u.) are employed throughout
this manuscript, unless otherwise specified.

##  II Semi-perturbative theory

The laser-assisted scattering of electrons by hydrogen atoms in a laser field
in which the atomic target is ionized, the so-called laser-assisted $(e,2e)$
reaction, can be symbolically represented as:

$\displaystyle e^{-}(E_{i},\mathbf{k}_{i})+{\rm
H}(1s)+N_{i}\,\gamma(\omega,\bm{\varepsilon})\to$ $\displaystyle
e^{-}(E_{f},\mathbf{k}_{f})+e^{-}(E_{e},\mathbf{k}_{e})+{\rm
H}^{+}+N_{f}\,\gamma(\omega,\bm{\varepsilon}),$ (1)

where $E_{i}$ and $E_{f}$, and $\mathbf{k}_{i}\,(\theta_{i},\varphi_{i})$ and
$\mathbf{k}_{f}\,(\theta_{f},\varphi_{f})$ represent the kinetic energy and
the momentum vector of the incident and scattered projectile electrons,
respectively, while $E_{e}$ and $\mathbf{k}_{e}\,(\theta_{e},\varphi_{e})$ are
the kinetic energy and the momentum vector of the ejected electron, as plotted
in Fig. 1. Here $\gamma\,(\omega,\bm{\varepsilon})$ denotes a photon with the
energy $\omega$ and the unit polarization vector $\bm{\varepsilon}$, and
$N=N_{i}-N_{f}$ is the net number of exchanged photons between the projectile-
atom scattering system and the laser field. The laser field is treated
classically, and within the dipole approximation is described as a
monochromatic electric field,

$\bm{{\cal E}}(t)=(i/2){\,\cal E}_{0}\,e^{-i\omega
t}\,\bm{\varepsilon}+\rm{c.c.},$ (2)

where ${\cal E}_{0}$ represents the amplitude of the electric field. The
magnetic vector potential, $\bm{A}(t)$, is simply calculated from $\bm{{\cal
E}}(t)=-\partial_{t}\bm{A}(t)$, as

$\bm{A}(t)=({\cal E}_{0}/\omega)\left[\cos\omega
t\cos(\xi/2)\,\mathbf{e}_{j}+\sin\omega t\sin(\xi/2)\;\mathbf{e}_{l}\right],$
(3)

where
$\bm{\varepsilon}=\cos(\xi/2)\,\mathbf{e}_{j}+i\sin(\xi/2)\,\mathbf{e}_{l}$ is
the polarization vector of the laser beam, with $\mathbf{e}_{j}$ and
$\mathbf{e}_{l}$ two different unit vectors along different orthogonal
directions. $\xi$ represents the degree of ellipticity of the laser field
which varies in the range $-\pi/2\leq\xi\leq\pi/2$, and determines the
ellipticity of the field. The value $\xi=0$ corresponds to a linearly
polarized (LP) laser field, while $\xi=\pi/2$ corresponds to a left-hand
circularly polarized (CP) laser field.

###  II.1 Laser-dressed electronic and atomic wave functions

As mentioned before, we consider that the external laser field has a dominant
influence and neglect the Coulomb interaction between fast outgoing electrons
and residual ion in the scattered and ejected electron wave functions
keldysh65 ; faisal73 ; reiss80 . At sufficiently high projectile kinetic
energies, it is well known that the first-order Born approximation in the
scattering potential can be used to describe the electron impact ionization
process joachain88 ; bransden ; joa2012 . We assume fast incident and outgoing
electrons with kinetic energies much larger than the energy of a bound
electron in the first Bohr orbit ehl1998 , since for the field-free $(e,2e)$
reaction in e-H collisions it is well known that the plane wave approximations
agree well with experiment at kinetic energies above $200$ eV bransden ;
Lohmann81 . Thus, in the non-relativistic regime, as long as both $E_{f}\gg 1$
a.u. and $E_{e}\gg 1$ a.u., we describe the fast scattered and ejected
electrons by Gordon-Volkov wave functions jain78 ; leone89 ; taj2004 . We
should mention that the use of a Coulomb-Volkov wave function provides a more
accurate treatment at small impact kinetic energies, where the effect of the
proton’s potential on the incoming and outgoing electrons is important
whelan93 ; ehl1998 ; zhang2007 . In order to avoid the direct one- and
multiphoton ionization processes, we consider that the electric field
amplitude is weak with respect to the atomic unit of electric field strengths,
${\cal E}_{0}\ll 5.1\times 10^{9}$ V/cm, i.e. the strength of the laser field
is much lower than the Coulomb field strength experienced by an electron in
the first Bohr orbit. Therefore, we describe in a nonperturbative way the
initial and final states of the projectile electron, as well the final state
of the ejected electron interacting with a laser field by non-relativistic
Gordon-Volkov wave functions gordon ; volkov , expressed in the velocity gauge
as

$\chi_{\mathbf{k}}^{V}({\mathbf{r}},t)={(2\pi)^{-3/2}}\exp\left[{i{\mathbf{k}}\cdot{\mathbf{r}}-i{\mathbf{k}}\cdot\bm{\alpha}(t)}-iE_{k}\,t-\frac{i}{2}\int^{t}dt^{\prime}\bm{A}^{2}(t^{\prime})\right],$
(4)

where $\mathbf{r}$ and $\mathbf{k}$ represent the position and momentum
vectors, and $E_{k}=k^{2}/2$ is the kinetic energy of the electron.
$\bm{\alpha}(t)=\int^{t}dt^{\prime}\bm{A}(t^{\prime})$ describes the classical
oscillation motion of a free electron in the electric field defined by Eq.
(2), and by using Eq. (3) we obtain

$\bm{\alpha}(t)=\alpha_{0}\left[\mathbf{e}_{j}\sin\omega
t\cos(\xi/2)+\,\mathbf{e}_{l}\cos\omega t\sin(\xi/2)\right],$ (5)

where ${\alpha_{0}}=\sqrt{I}/\omega^{2}$ is the amplitude of oscillation, and
${I}={\cal E}_{0}^{2}$ denotes the laser intensity. Obviously, as noticed from
Eq. (4), at moderate field strengths the largest effect of the laser field on
the free-electron state is determined by a dimensionless parameter
$k\alpha_{0}$, that depends on the electron and photon energies, and laser
intensity. For example, a laser intensity of $1$ TW/cm2, a photon energy of
$3.1$ eV, and an electron kinetic energy of $200$ eV result in a value of
$k\alpha_{0}\simeq 1.58$, while the ponderomotive energy acquired by an
electron in the electric field $U_{p}={I}/4\omega^{2}$ is about $0.015$ eV,
and, therefore can be safely neglected compared to the photon and unbound
electrons energies employed in the present paper.

The interaction of the hydrogen atom, initially in its ground state, with a
laser field at moderate field strengths is considered within the first-order
TDPT. An approximate solution for the wave function of an electron bound to a
Coulomb potential in the presence of an electric field, also known as the
dressed wave function, is written as

$\Psi_{1s}\left(\mathbf{r}_{1},t\right)=\left[\psi_{1s}^{(0)}(\mathbf{r}_{1},t)+\psi_{1s}^{(1)}(\mathbf{r}_{1},t)\right]\exp\left[-i{E}_{1}t-\frac{i}{2}\int^{t}dt^{\prime}\bm{A}^{2}(t^{\prime})\right],$
(6)

where $\mathbf{r}_{1}$ is the position vector of the bound electron,
$\psi_{1s}^{(0)}$ is the unperturbed wave function of the hydrogen atom ground
state, and $\psi_{1s}^{(1)}$ represents the first-order perturbative
correction to the atomic wave function due to the external laser field. We
employ the following expression of the first-order correction in the velocity
gauge, $\psi_{1s}^{(1)}$, as described by Florescu and Marian in Ref. vf1 ,

$\psi_{1s}^{(1)}(\mathbf{r}_{1},t)=-\frac{\alpha_{0}\omega}{2}\left[\bm{\varepsilon}\cdot\mathbf{w}_{100}(E^{+}_{1};\mathbf{r}_{1})e^{-i\omega
t}+\bm{\varepsilon}^{*}\cdot\mathbf{w}_{100}(E^{-}_{1};\mathbf{r}_{1})e^{i\omega
t}\right],$ (7)

with the linear-response vector, ${\bf{w}}_{100}$, defined by

${\bf{w}}_{100}(E_{1}^{\pm};\mathbf{r}_{1})=-G_{C}(E_{1}^{\pm})\,{\bf{P}}\,\psi_{1s}(\mathbf{r}_{1}),$
(8)

where $\bf{P}$ denotes the momentum operator of the bound electron, and
$G_{C}$ is the Coulomb Green’s function. For the hydrogen atom in its ground
state the linear-response vector was expressed in Ref. vf1  as

${\bf{w}}_{100}(E_{1}^{\pm};\mathbf{r}_{1})=i(4\pi)^{-1/2}{\cal
B}_{101}(E_{1}^{\pm};{r}_{1})\,\hat{\textbf{r}}_{1},$ (9)

where $\hat{\textbf{r}}_{1}=\textbf{r}_{1}/{r}_{1}$, and the energies
$E_{1}^{+}$ and $E_{1}^{-}$ take the following values

$E_{1}^{+}={E}_{1}+\omega+i0,\;\;E_{1}^{-}={E}_{1}-\omega,$ (10)

with $E_{1}=-13.6$ eV representing the energy of the ground state. The radial
function ${\cal B}_{101}$ in Eq. (9) was evaluated vf1  using the Schwinger’s
integral representation of the Coulomb Green’s function in momentum space
including both bound and continuum eigenstates, and can be expressed in terms
of Humbert function, $\Phi_{1}$, as

${\cal
B}_{101}(\tau;{r}_{1})=\frac{2\,\tau}{2-\tau}\left(\frac{2}{1+\tau}\right)^{2+\tau}{r}_{1}\,e^{-r_{1}/\tau}\Phi_{1}(2-\tau,-1-\tau,3-\tau,\xi_{1},\eta_{1}),$
(11)

where the parameter $\tau$ takes two values
$\tau^{\pm}=1/\sqrt{-2E_{1}^{\pm}}$, and the variables of the Humbert function
are $\xi_{1}=(1-\tau)/2$ and $\eta_{1}=(1-\tau){r}_{1}/\tau$.

###  II.2 The nonlinear scattering matrix

We employ a semi-perturbative approach of the scattering process which is
similar to that developed by Byron and Joachain b-j  for free-free
transitions, in which the second-order Born correction is negligible compared
to the laser-dressing effects. The evaluation of the scattering amplitude is
very challenging due to the complex three-body interaction: projectile
electron, bound electron, and laser field. However, since we assume that both
scattered and ejected electrons have large kinetic energies the calculation
simplifies, and we can derive a closed form expression for the TDCS. Thus, the
initial state of the scattering system is calculated as the product of the
initial states of the fast incident electron and atomic target dressed by the
laser field, ${\chi}_{\mathbf{k}_{i}}^{V}(\mathbf{r}_{0},t)$ and
$\Psi_{1s}(\mathbf{r}_{1},t)$, while the final state is calculated as the
product of the final states of the fast scattered and ejected electrons, which
are approximated as Gordon-Volkov wave functions. Our treatment differs from
that of Taïeb and coworkers taieb1991  in the fact that we dress the fast
projectile and ejected electrons to all order in the laser field and we dress
the atomic target by using an atomic wave function corrected to the first
order in the laser field vf1 . As mentioned before, we focus our study at
moderate laser intensities ($I\leq 1$ TW/cm2) and fast projectile electrons
($E_{i}$, $E_{f}\geq 1$ keV) such that the interaction between the projectile
electron and hydrogen atom is well treated within the first-order Born
approximation in the static scattering potential
$V_{d}(r_{0},r_{1})=-1/r_{0}+1/{|\bf{r}_{1}-\bf{r}_{0}|}$ for the direct
channel, and $V_{ex}(r_{0},r_{1})=-1/r_{1}+1/{|\bf{r}_{1}-\bf{r}_{0}|}$ for
the exchange channel.

In order to describe the scattering process (1) we employ the direct and
exchange scattering matrix elements massey , which are calculated at high
kinetic energies of the projectile and ejected electrons as

$\displaystyle S_{fi,d}^{B1}$ $\displaystyle=$
$\displaystyle-i\int_{-\infty}^{+\infty}dt\,\langle\chi_{\mathbf{k}_{f}}^{V}(\mathbf{r}_{0},t)\chi_{\mathbf{k}_{e}}^{V}(\mathbf{r}_{1},t)|V_{d}(r_{0},{r}_{1})|{\chi}_{\mathbf{k}_{i}}^{V}(\mathbf{r}_{0},t)\Psi_{1s}(\mathbf{r}_{1},t)\rangle,$
(12) $\displaystyle S_{fi,ex}^{B1}$ $\displaystyle=$
$\displaystyle-i\int_{-\infty}^{+\infty}dt\,\langle\chi_{\mathbf{k}_{f}}^{V}(\mathbf{r}_{1},t)\chi_{\mathbf{k}_{e}}^{V}(\mathbf{r}_{0},t)|V_{ex}(r_{0},{r}_{1})|{\chi}_{\mathbf{k}_{i}}^{V}(\mathbf{r}_{0},t)\Psi_{1s}(\mathbf{r}_{1},t)\rangle,\,$
(13)

where $\chi_{\mathbf{k}_{i(f)}}^{V}$ and $\chi_{\mathbf{k}_{e}}^{V}$, given by
Eq. (4), represent the Gordon-Volkov wave functions of the projectile and
emitted electrons embedded in the laser field, whereas $\Psi_{1s}$, given by
Eq. (6), represents the wave function of the bound electron interacting with
the laser field. By using the Jacobi-Anger identity Watson , $e^{-ix\sin\omega
t}\equiv\sum_{N=-\infty}^{+\infty}J_{N}(x)e^{-iN\omega t},$ we expand the
oscillating part of the Gordon-Volkov wave functions occurring in the
scattering matrix elements, Eqs. (12) and (13), in terms of the ordinary
Bessel functions of the first kind, $J_{N}$, as

$\exp{[-i\,\mathbf{q}\cdot\bm{\alpha}(t)]}=\sum_{N=-\infty}^{+\infty}J_{N}({\cal
R}_{q})e^{-iN\omega t+iN\phi_{q}},$ (14)

where the argument of the Bessel function is defined by ${\cal
R}_{q}=\alpha_{0}|\bm{\varepsilon}\cdot\mathbf{q}|$, and $\phi_{q}$ represents
the dynamical phase which is calculated as
$e^{i\phi_{q}}=\bm{\varepsilon}\cdot\mathbf{q}/|\bm{\varepsilon}\cdot\mathbf{q}|$,
where $\mathbf{q}=\mathbf{k_{i}}-\mathbf{k_{f}}-\mathbf{k_{e}}$ denotes the
recoil momentum vector of the ionized target, $H^{+}$. Clearly, for a CP laser
field a change of helicity, i.e. $\bm{\varepsilon}\to\bm{\varepsilon}^{*}$,
leads to a change of the sign of the dynamical phase, $\phi_{q}\to-\phi_{q}$
in the TDCS, while for a LP laser field $e^{i\phi_{q}}=\pm 1$, and
$\phi_{q}=n\pi$ with $n$ an integer.

For the direct channel, by replacing Eqs. (4), (6), and (14) into Eq. (12), we
obtain the scattering matrix for electron-hydrogen collisions in a laser
field, after performing the integration with respect to time,

$S_{fi,d}^{B1}=-2\pi
i\sum_{N=N_{min}}^{+\infty}\delta(E_{f}+E_{e}-E_{i}-E_{1}-N\omega)\,T_{N,d}\,,$
(15)

where the Dirac function, $\delta$, assures the energy conservation which
implies that the kinetic energy of the scattered electron is determined by the
relation $E_{f}=E_{i}+E_{1}-E_{e}+N\omega$. Here the kinetic energy of the
residual ion, $E_{q}=q^{2}/2m_{p}$, has been neglected in comparison to any of
the electrons kinetic energies, $E_{j}$, ($j=i,f$, and $e$), since the mass of
the residual ion (proton) is much larger than the electron mass. The energy
spectrum of the scattered electron consists of an elastic line, $N=0$, and a
number of sidebands corresponding to the positive and negative values of $N$.
Obviously, for a given value of the ejected electron energy, $E_{e}$, the net
number of exchanged photons is limited and cannot be smaller than a minimal
value that is the integer of $N_{min}=(E_{e}-E_{i}-E_{1})/\omega$.

The total nonlinear transition amplitude, $T_{N,d}$, for the laser-assisted
$(e,2e)$ ionization process in the direct channel can be split as a sum of two
terms

$T_{N,d}=T^{(0)}_{N,d}+T^{(1)}_{N,d}\,,$ (16)

where $T^{(0)}_{N}$ and $T^{(1)}_{N}$ represent the electronic and atomic
transition amplitudes. The first term on the right-hand side of the total
transition amplitude Eq. (16), $T^{(0)}_{N}$, is the transition amplitude due
to projectile electron contribution, in which the atomic dressing terms are
neglected,

$T^{(0)}_{N,d}=\frac{1}{2^{5/2}\pi^{7/2}}\frac{e^{iN\phi_{q}}}{\Delta^{2}}J_{N}({\cal
R}_{q})\int
d\mathbf{r}_{1}\;e^{-i\mathbf{k}_{e}\cdot\mathbf{r}_{1}}(e^{i\mathbf{\Delta}\cdot\mathbf{r}_{1}}-1)\psi_{1s}(\mathbf{r}_{1}),$
(17)

where the integration over the projectile coordinate, $\textbf{r}_{0}$, was
performed using the Bethe integral, and
$\mathbf{\Delta}=\mathbf{k_{i}}-\mathbf{k_{f}}$ is the vector of momentum
transfer from the incident to the scattered electron. After performing the
radial integration with respect to $\textbf{r}_{1}$ in Eq. (17), the
electronic transition amplitude can be simply expressed as

$T^{(0)}_{N,d}=-\frac{1}{(2\pi)^{2}}J_{N}({\cal
R}_{q})\,f_{ion}^{B_{1}}(\Delta,q,k_{e})\,e^{iN\phi_{q}},$ (18)

where

$f_{ion}^{B_{1}}(\Delta,q,k_{e})=-\frac{2^{5/2}}{\pi\Delta^{2}}\left[\frac{1}{(q^{2}+1)^{2}}-\frac{1}{(k_{e}^{2}+1)^{2}}\right],$
(19)

is the direct scattering amplitude in the first-order plane-wave Born
approximation for ionization of hydrogen atom by electron impact in the
absence of the laser field b-j89 ; taj2004 . In the electronic transition
amplitude, Eq. (18), the interaction between the laser field and the
projectile and ejected electrons is contained in the argument of the Bessel
function ${\cal
R}_{q}=(\sqrt{I}/\omega^{2})|\bm{\varepsilon}\cdot\mathbf{q}|$, and phase
$\phi_{q}$, being decoupled from the kinematic term. This feature is a
characteristic of employing Gordon-Volkov wave functions for fast electrons
and moderate laser intensities ehl1998 ; acgabi2000 ; gabi2015-gabi2017 . The
field-free electronic scattering amplitude $f_{ion}^{B_{1}}$ contains a
factor, $-2/\Delta^{2}$, which is related to the first-order Born amplitude
corresponding to scattering by the Coulomb potential ${-1}/{r_{0}}$, while the
two terms in the squared brackets of Eq. (19) are related to the momentum
transfer to the residual ion, $q$, and the momentum of the ejected electron,
$k_{e}$, respectively. For the field-free $(e,2e)$ collisions in the plane-
wave Born approximation, the first term in the right-side-hand of Eq. (19)
gives rise to the so-called binary encounter peak weigold79 , which occurs at
very low residual ion momentum $q\simeq 0$.

The second term on the right-hand side of Eq. (16), $T^{(1)}_{N}$, represents
the first-order atomic transition amplitude and corresponds to processes in
which the hydrogen atom absorbs or emits one photon and is subsequently
ionized by the projectile electron impact. $T^{(1)}_{N}$ occurs due to
modification of the atomic state by the laser field, the so-called atomic
dressing, which is described by the first-order radiative correction,
$\psi^{(1)}_{1s}(\mathbf{r}_{1},t)$, in Eq. (7). After some straightforward
algebra, integrating over the projectile coordinate, $\textbf{r}_{0}$, the
direct first-order atomic transition amplitude can be written as

$T^{(1)}_{N,d}=-\frac{\alpha_{0}\omega}{2}\left[J_{N-1}({\cal R}_{q})\;{\cal
M}_{at}^{(1)}(\omega)e^{i(N-1)\phi_{q}}+J_{N+1}({\cal R}_{q})\;{\cal
M}_{at}^{(1)}(-\omega)e^{i(N+1)\phi_{q}}\right],$ (20)

where ${\cal M}_{at}^{(1)}(\omega)$ denotes the specific first-order atomic
transition matrix element related to one-photon absorption,

${\cal M}_{at}^{(1)}(\omega)=\frac{1}{2^{5/2}\pi^{7/2}\Delta^{2}}\int
d\mathbf{r}_{1}\;e^{-i\mathbf{k}_{e}\cdot\mathbf{r}_{1}}\;(e^{i\mathbf{\Delta}\cdot\mathbf{r}_{1}}-1)\bm{\varepsilon}\cdot\mathbf{w}_{100}(E^{+}_{1};\mathbf{r}_{1}),$
(21)

whereas the transition matrix element ${\cal M}_{at}^{(1)}(-\omega)$ is
related to one-photon emission

${\cal M}_{at}^{(1)}(-\omega)=\frac{1}{2^{5/2}\pi^{7/2}\Delta^{2}}\int
d\mathbf{r}_{1}\;e^{-i\mathbf{k}_{e}\cdot\mathbf{r}_{1}}(e^{i\mathbf{\Delta}\cdot\mathbf{r}_{1}}-1)\;\bm{\varepsilon}^{*}\cdot\mathbf{w}_{100}(E^{-}_{1};\mathbf{r}_{1}),$
(22)

where the energies $E_{1}^{\pm}$ are given in Eq. (10). Obviously, in Eq. (20)
only one photon is exchanged (emitted or absorbed) between the laser field and
the bound electron, while the remaining $N+1$ or $N-1$ photons are exchanged
between the laser field and the projectile electron. By performing the radial
integral over $\mathbf{r}_{1}$ in Eq. (21) we derive the first-order atomic
matrix element for one-photon absorption as,

${\cal
M}_{at}^{(1)}(\omega)=-\frac{1}{2^{3/2}\pi^{3}\Delta^{2}}\left[(\bm{\varepsilon}\cdot\hat{\mathbf{q}})\;{\cal
J}_{101}(\omega,q)-(\bm{\varepsilon}\cdot\hat{\mathbf{k}}_{e})\;{\cal
J}_{101}(\omega,-k_{e})\right],$ (23)

while the following changes are made $\omega\to-\omega$ and
$\bm{\varepsilon}\to\bm{\varepsilon}^{*}$ in Eq. (23) to obtain the first-
order atomic transition matrix element, ${\cal M}_{at}^{(1)}(-\omega)$, for
one-photon emission. The expression of the atomic radial integral ${\cal
J}_{101}$, is given by

${\cal
J}_{101}(\pm\omega,p)={\int}_{0}^{\infty}dr_{1}\;r_{1}^{2}\;j_{1}(pr_{1})\;{\cal
B}_{101}(E_{1}^{\pm};r_{1}),$ (24)

with ${\cal J}_{101}(\omega,-p)=-{\cal J}_{101}(\omega,p)$, where $p=q$ or
$k_{e}$. After performing some algebra in Eq. (24), by using the expansion of
the spherical Bessel function, $j_{1}$, an analytical form of the radial
integral is obtained in terms of two Appell’s hypergeometric function,
$F_{1}$, as

${\cal
J}_{101}(\omega,p)=\frac{2^{6}\tau}{p\,(2-\tau){(1+\tau)}^{4}}\;Re\left[a^{3}F_{1}(b,1,3,b+1,x,y)-\frac{ia^{2}}{2p}F_{1}(b,2,2,b+1,x,y)\right],$
(25)

in which $a={\tau}/{(1+ip\tau)}$, $b=2-\tau$, and the variables of the
Appell’s hypergeometric function are

$x=\frac{\tau-1}{\tau+1},\;\;y=\frac{(1-\tau)(1-ip\tau)}{(1+\tau)(1+ip\tau)},$
(26)

where the parameter $\tau$ depends on the photon energy, $\omega$, and it
takes two values, $\tau^{-}=1/\sqrt{-2E_{1}^{+}}$ and
$\tau^{+}=1/\sqrt{-2E_{1}^{-}}$, corresponding to the two energies $E_{1}^{+}$
and $E_{1}^{-}$ defined in Eq. (10). The first-order atomic matrix element,
Eq. (23), has a structure that explicitly contains the scalar products
$\bm{\varepsilon}\cdot\hat{\mathbf{q}}$ and
$\bm{\varepsilon}\cdot\hat{\mathbf{k}}_{e}$, depends on the scattering
geometry, being written in a closed form that allows us to analyze the
dependence on the laser field polarization. The last term in the right-hand
side of the electronic scattering amplitude and atomic matrix element, Eqs.
(19) and (23), occurs due to the non-orthogonality of the Gordon-Volkov wave
function of the ejected electron and the initial ground-state wave function of
the hydrogen atom. The structure of Eq. (23) is also similar to other
processes, with the vectors ${\mathbf{q}}$ and $\mathbf{k}_{e}$ replaced by
vectors which are specific to each particular process, such as elastic laser-
assisted scattering of electrons by hydrogen atoms dubois86 ; acgabi2 ,
bremsstrahlung cross sections in the electron-hydrogen atom collisions dubois
, or laser-assisted electron-impact excitation of hydrogen atoms
gabi2015-gabi2017 .

###  II.3 The nonlinear scattering matrix for exchange scattering

Our formalism does not neglect the exchange effects between the scattered and
ejected electrons in both the electronic and atomic terms, since fast incident
and outgoing electrons are involved in the calculation, and, as in the EMS
experiments their kinetic energies could have comparable orders of magnitude.
In the first-order Born approximation in the exchange potential, $V_{ex}$, we
obtain the exchange scattering matrix for the laser-assisted $(e,2e$)
reactions, after performing the integration with respect to time in Eq. (13),

$S_{fi,ex}^{B1}=-2\pi
i\sum_{N=N_{min}}^{+\infty}\delta(E_{f}+E_{e}-E_{i}-E_{1}-N\omega)\,T_{N,ex}\,,$
(27)

where $T_{N,ex}=T^{(0)}_{N,ex}+T^{(1)}_{N,ex}$. The electronic transition
amplitude for the exchange scattering, $T^{(0)}_{N,ex}$, in which the atomic
dressing contribution is neglected, can be expressed as

$T^{(0)}_{N,ex}=-\frac{1}{(2\pi)^{2}}J_{N}({\cal
R}_{q})\,g_{ion,ex}^{B_{1}}(\Delta_{e},q)\;e^{iN\phi_{q}},$ (28)

where

$g_{ion,ex}^{B_{1}}(\Delta_{e},q)=-\frac{2^{5/2}}{\pi\Delta_{e}^{2}(q^{2}+1)^{2}},$
(29)

denotes the electronic exchange amplitude in the absence of the laser field,
that is in agreement to the Born-Ochkur approximation ochkur ; massey , and
$\Delta_{e}$ represents the amplitude of the momentum transfer vector from the
incident to the ejected electron,
$\mathbf{\Delta}_{e}=\mathbf{k}_{i}-\mathbf{k}_{e}$.

Similarly to the direct scattering, the first-order atomic transition
amplitude for the exchange scattering can be expressed as

$T^{(1)}_{N,ex}=-\frac{\alpha_{0}\omega}{2}\left[J_{N-1}({\cal R}_{q})\;{\cal
M}_{at,ex}^{(1)}(\omega)e^{i(N-1)\phi_{q}}+J_{N+1}({\cal R}_{q})\;{\cal
M}_{at,ex}^{(1)}(-\omega)e^{i(N+1)\phi_{q}}\right],$ (30)

where

${\cal
M}_{at,ex}^{(1)}(\pm\omega)=-\frac{\bm{\varepsilon}\cdot\hat{\mathbf{q}}}{2^{3/2}\pi^{3}\Delta_{e}^{2}}\;{\cal
J}_{101}(\pm\omega,q),$ (31)

and ${\cal J}_{101}(\pm\omega,q)$ is calculated from Eq. (25). Obviously, the
exchange effects for both electronic and atomic contributions to the
transition amplitude vary like $\Delta_{e}^{-2}$, and cannot be neglected in
comparison to the contribution of the direct scattering channel if ${k}_{e}$
and ${k}_{f}$ are of comparable order of magnitude. In contrast, for very fast
incident and scattered electrons, with $k_{i}$ and $k_{f}$ much larger than
the atomic unit, and slow ejected electrons, $k_{e}\ll k_{f}$, the electronic
and atomic exchange terms can be neglected compared to the corresponding
direct terms.

###  II.4 The low-photon energy approximation

In the low-photon energy limit where the photon energy is small compared to
the ionization energy of the hydrogen atom (typically in the infrared region),
it is worth presenting some useful simple approximation formulas for the
atomic transition amplitude. In most theoretical works the analytical
calculations cannot be done exactly and, as the photon energy remains small is
expected that only few intermediate bound states to contribute to the atomic
transition amplitude and thus to approximate the complicated analytical
formulas. This is the key of the closure approximation method b-j , which
consists in replacing the difference energy $E_{n}-E_{1}$ by an average
excitation energy $\bar{\omega}\simeq 4/9$ a.u. for the hydrogen atom in
approximating the sum over the intermediate states in the atomic transition
amplitudes. Here we present a different approach based on the low-frequency
approximation (LFA), given by the lowest-order term of the expansion of the
atomic matrix element ${\cal M}_{at}^{(1)}(\omega)$ in powers of the laser
photon energy. After some algebra we derive an approximate formula for the
atomic radial integral, ${\cal J}_{101}$, in the low-photon energy limit
$\omega\ll|E_{1}|$ in Eq. (25), in the first order in $\omega$,

${\cal
J}_{101}(\omega,p)\simeq-\frac{16\,p}{(p^{2}+1)^{3}}\left(1-\frac{\omega}{2}\,\frac{p^{2}-9}{p^{2}+1}\right),$
(32)

where $p=q$ or $k_{e}$, and, therefore the atomic transition amplitude for the
direct process, Eq. (20), in the low-photon energy limit reads as

$\displaystyle T^{(1)}_{N,d}$ $\displaystyle\simeq$
$\displaystyle{\alpha_{0}\omega}\frac{2^{3/2}e^{iN\phi_{q}}}{\pi^{3}\Delta^{2}}\left\\{J_{N-1}({\cal
R}_{q})e^{-i\phi_{q}}\left[\frac{\bm{\varepsilon}\cdot\mathbf{q}}{(q^{2}+1)^{3}}+\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}}{(k_{e}^{2}+1)^{3}}\right]\right.$
(33) $\displaystyle+\,J_{N+1}({\cal
R}_{q})e^{i\phi_{q}}\left[\frac{\bm{\varepsilon}^{*}\cdot\mathbf{q}}{(q^{2}+1)^{3}}+\frac{\bm{\varepsilon}^{*}\cdot\mathbf{k}_{e}}{(k_{e}^{2}+1)^{3}}\right]$
$\displaystyle+\,\frac{\omega}{2}J_{N-1}({\cal
R}_{q})e^{-i\phi_{q}}\left[\frac{\bm{\varepsilon}\cdot{\mathbf{q}}\,(q^{2}-9)}{(q^{2}+1)^{4}}+\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}\,(k_{e}^{2}-9)}{(k_{e}^{2}+1)^{4}}\right]$
$\displaystyle\left.-\,\frac{\omega}{2}J_{N+1}({\cal
R}_{q})e^{i\phi_{q}}\left[\frac{\bm{\varepsilon}^{*}\cdot{\mathbf{q}}\,(q^{2}-9)}{(q^{2}+1)^{4}}+\frac{\bm{\varepsilon}^{*}\cdot\mathbf{k}_{e}\,(k_{e}^{2}-9)}{(k_{e}^{2}+1)^{4}}\right]\right\\}.$

For a LP laser field the following formula holds, $J_{N}({\cal
R}_{q})=J_{N}(\alpha_{0}\,\bm{\varepsilon}\cdot{\mathbf{q}})e^{-iN\phi_{q}}$,
and we obtain from Eqs. (18) and (19) the direct electronic transition
amplitude,

$T^{(0),LP}_{N,d}=\frac{2^{1/2}}{\pi^{3}\Delta^{2}}J_{N}(\alpha_{0}\,\bm{\varepsilon}\cdot{\mathbf{q}})\left[\frac{1}{(q^{2}+1)^{2}}-\frac{1}{(k_{e}^{2}+1)^{2}}\right],$
(34)

and the direct atomic transition amplitude, Eq. (33), simplifies to

$\displaystyle T^{(1),LP}_{N,d}$ $\displaystyle\simeq$
$\displaystyle\frac{2^{5/2}}{\pi^{3}}\frac{\omega}{\Delta^{2}}\left\\{NJ_{N}(\alpha_{0}\,\bm{\varepsilon}\cdot{\mathbf{q}})\left[\frac{1}{(q^{2}+1)^{3}}+\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}}{\bm{\varepsilon}\cdot\mathbf{q}}\frac{1}{(k_{e}^{2}+1)^{3}}\right]\right.$
(35)
$\displaystyle+\left.\,\frac{\alpha_{0}{\omega}}{2}J_{N}^{{}^{\prime}}(\alpha_{0}\,\bm{\varepsilon}\cdot{\mathbf{q}})\,\left[\frac{\bm{\varepsilon}\cdot{\mathbf{q}}\,(q^{2}-9)}{(q^{2}+1)^{4}}+\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}\,(k_{e}^{2}-9)}{(k_{e}^{2}+1)^{4}}\right]\right\\},$

where we have used the recurrence relation
$J_{N-1}(x)+J_{N+1}(x)=J_{N}(x)(2N/x)$, and $J_{N}^{{}^{\prime}}$ is the first
derivative of the Bessel function which satisfies the relation
$J_{N}^{{}^{\prime}}(x)=[J_{N-1}(x)-J_{N+1}(x)]/2$, with
$x=\alpha_{0}(\bm{\varepsilon}\cdot{\mathbf{q}})$, Watson . If we consider the
lowest order in the photon energy $\omega$ in Eq. (35)

$T^{(1),LP}_{N,d}\simeq\frac{2^{5/2}}{\pi^{3}}\frac{N\omega}{\Delta^{2}}J_{N}(\alpha_{0}\,\bm{\varepsilon}\cdot{\mathbf{q}})\left[\frac{1}{(q^{2}+1)^{3}}+\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}}{\bm{\varepsilon}\cdot\mathbf{q}}\frac{1}{(k_{e}^{2}+1)^{3}}\right],$
(36)

we obtain the LFA formula for $N$-photon absorption atomic transition element
in the case of a LP field. However, in the limit $\omega\to 0$ for scattering
parameters such that ${\cal R}_{q}\gg 1$, i.e there is a strong coupling
between the projectile and ejected electrons and the laser field, the
transition amplitudes derived in the semi-perturbative approach do not
diverge, but approach the zero value, due to asymptotic behavior of the Bessel
function of the first kind Watson  at large arguments,
$J_{N}(x)\simeq\sqrt{2/\pi x}\cos(x-N\pi/2-\pi/4)$, for $x\to\infty$.

Furthermore, whenever the condition ${\cal R}_{q}\ll 1$ is satisfied, i.e. the
perturbative regime of low laser intensities where $\alpha_{0}\ll 1$ a.u.
and/or scattering kinematics with $|\bm{\varepsilon}\cdot{\mathbf{q}}|\ll 1$
a.u., we can use the approximate formula for the Bessel function at small
arguments,

$J_{N}({\cal R}_{q})\simeq\frac{1}{N!}\left(\frac{{\cal
R}_{q}}{2}\right)^{N},\quad\text{for \; $N>0$},$ (37)

and $J_{N}({\cal R}_{q})=(-1)^{-N}J_{-N}({\cal R}_{q})$, for $N\leq 0$ Watson
. Thus, in the perturbative region with ${\cal R}_{q}\ll 1$, we obtain from
Eq. (34) a simple formula for the direct electronic transition amplitude for
$N$-photon absorption ($N>0$), as

$T^{(0),LP}_{N,d}\simeq\alpha_{0}^{N}\frac{2^{1/2}}{\pi^{3}\Delta^{2}N!}\left(\frac{\bm{\varepsilon}\cdot\mathbf{q}}{2}\right)^{N}\left[\frac{1}{(q^{2}+1)^{2}}-\frac{1}{(k_{e}^{2}+1)^{2}}\right],$
(38)

while for the direct atomic transition amplitude for $N$-photon absorption in
the LFA we obtain from Eq. (36)

$T^{(1),LP}_{N,d}\simeq\alpha_{0}^{N}\frac{2^{5/2}N\omega}{\pi^{3}\Delta^{2}N!}\left(\frac{\bm{\varepsilon}\cdot\mathbf{q}}{2}\right)^{N}\left[\frac{1}{(q^{2}+1)^{3}}+\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}}{\bm{\varepsilon}\cdot\mathbf{q}}\frac{1}{(k_{e}^{2}+1)^{3}}\right].$
(39)

Expressions similar to Eqs. (34)-(35) and Eqs. (38)-(39) can be easily derived
for the exchange scattering channel. As expected, the electronic and atomic
transition amplitudes are important at scattering and ejected angles where the
momenta $q$, $\Delta$, and $\Delta_{e}$ are small. The ratio of the direct
atomic and electronic transition amplitudes derived in the low-photon energy
limit, Eqs. (39) and (38),

$\frac{T^{(1),LP}_{N,d}}{T^{(0),LP}_{N,d}}\simeq\frac{4N\omega}{q^{2}+1}\left[1+\frac{(q^{2}+1)^{3}}{(k_{e}^{2}+1)^{3}}\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}}{\bm{\varepsilon}\cdot\mathbf{q}}\right]\left[1-\frac{(q^{2}+1)^{2}}{(k_{e}^{2}+1)^{2}}\right]^{-1},$
(40)

shows that, compared to the projectile electron contribution, the first-order
atomic dressing effects for $\omega\ll 1$ a.u. and laser parameters such that
${\cal R}_{q}\ll 1$, are increasing with the net number of exchanged photons,
$N$, and photon energy, $\omega$, and are decreasing with the momenta of the
ejected electron, $k_{e}$, and residual ion, $q$. Obviously, Eq. (40) shows
that differences occur in the TDCSs for absorption or emission of $N$ photons,
which correspond to positive or negatives values of $N$, due to constructive
or destructive interferences of the electronic and atomic terms in TDCS.

In the case of one-photon absorption ($N=1$) in the perturbative regime with
${\cal R}_{q}\ll 1$ and low photon energies we can use the approximate formula
for the Bessel function, Eq. (37), and by keeping only the first order in
laser field intensity, $I$, we obtain simple formulas for the direct
electronic transition amplitude, Eq. (18),

$T^{(0)}_{N=1,d}\simeq\frac{\sqrt{I}}{2^{1/2}\pi^{3}}\frac{\bm{\varepsilon}\cdot\mathbf{q}}{\omega^{2}\Delta^{2}}\left[\frac{1}{(q^{2}+1)^{2}}-\frac{1}{(k_{e}^{2}+1)^{2}}\right],$
(41)

as well for the direct atomic transition amplitude derived in the low-photon
energy limit, Eq. (33),

$T^{(1)}_{N=1,d}\simeq\frac{2^{3/2}}{\pi^{3}}\frac{\sqrt{I}}{\omega\Delta^{2}}\left[\frac{\bm{\varepsilon}\cdot\mathbf{q}}{(q^{2}+1)^{3}}\left(1+\frac{\omega}{2}\;\frac{q^{2}-9}{q^{2}+1}\right)+\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}}{(k_{e}^{2}+1)^{3}}\left(1+\frac{\omega}{2}\;\frac{k_{e}^{2}-9}{k_{e}^{2}+1}\right)\right].$
(42)

Moreover, if we keep the lowest order in the photon energy in Eq. (42) we
obtain a quite simple formula for the direct atomic transition amplitude

$T^{(1)}_{N=1,d}\simeq\frac{2^{3/2}}{\pi^{3}}\frac{\sqrt{I}}{\omega\Delta^{2}}\left[\frac{\bm{\varepsilon}\cdot\mathbf{q}}{(q^{2}+1)^{3}}+\frac{\bm{\varepsilon}\cdot\mathbf{k}_{e}}{(k_{e}^{2}+1)^{3}}\right],$
(43)

in the LFA for one-photon absorption in the perturbative regime.

Similarly, for the exchange scattering we derive simple approximate formulas
for the electronic and atomic transition amplitudes at ${\cal R}_{q}\ll 1$,
Eqs. (28) and (30) in the low-photon energy limit, as

$T^{(0)}_{N=1,ex}\simeq\frac{1}{2^{1/2}\pi^{3}}\frac{\sqrt{I}}{\omega^{2}\Delta_{e}^{2}}\frac{\bm{\varepsilon}\cdot\mathbf{q}}{(q^{2}+1)^{2}},$
(44)
$T^{(1)}_{N=1,ex}\simeq\frac{2^{3/2}}{\pi^{3}}\frac{\sqrt{I}}{\omega\Delta_{e}^{2}}\frac{\bm{\varepsilon}\cdot\mathbf{q}}{(q^{2}+1)^{3}}\left(1+\frac{\omega}{2}\;\frac{q^{2}-9}{q^{2}+1}\right).$
(45)

The infrared divergence in the limit $\omega\to 0$ is evident in all the above
electronic and atomic transition amplitude expressions derived at ${\cal
R}_{q}\ll 1$. Thus, in the perturbative regime and low-photon energy
approximation the electronic transition amplitude varies like $\omega^{-2}$,
while the atomic transition amplitude varies like $\omega^{-1}$, which is
reminiscent of the infrared divergence of quantum electrodynamics Jauch  and
Low theorem Low  in the limit $\omega\to 0$. Clearly, these simple analytical
formulas we have derived for one-photon absorption, as well for nonlinear
atomic transition amplitudes might provide more physical insight into the
laser-assisted $(e,2e)$ reactions.

###  II.5 The triple differential cross section

It is well known that the TDCS can provide useful information about collision
dynamics in the electron-impact ionization process ehlotzky98 . For laser-
assisted $(e,2e)$ collisions accompanied by the transfer of $N$ photons, we
calculate the nonlinear TDCS in the first-order Born approximation in the
scattering potential, for unpolarized incident projectile and hydrogen beams,
and without distinguishing between the final spin states of the electrons,

$\frac{d^{3}{\sigma}_{N}^{B1}}{d\Omega_{f}\,d\Omega_{e}\,dE_{f}}={(2\pi)}^{4}\frac{k_{f}k_{e}}{k_{i}}\left(\frac{1}{4}{\left|T_{N,d}+T_{N,ex}\right|}^{2}+\frac{3}{4}{\left|T_{N,d}-T_{N,ex}\right|}^{2}\right),$
(46)

averaged over the initial spin states and summed over the final spin states.
The projectile electrons are scattered into the solid angle $\Omega_{f}$ and
$\Omega_{f}+d\Omega_{f}$ with the kinetic energy between $E_{f}$ and
$E_{f}+dE_{f}$, and the ejected electrons are emitted within the solid angle
$\Omega_{e}$ and $\Omega_{e}+d\Omega_{e}$. The TDCS is a function of the
electrons momentum vectors ${\mathbf{k}}_{i},{\mathbf{k}}_{f}$, and
${\mathbf{k}}_{e}$, and depends on the laser parameters: intensity $I$, photon
energy $\omega$, and polarization $\bm{\varepsilon}$. The dominant
contribution to TDCS is due to collisions involving small momentum transfers
$\Delta$ and $\Delta_{e}$, small momentum of the residual ion $q$, or near
resonance photon energies. The TDCS for the laser-assisted $(e,2e)$ process is
given by

$\frac{d^{3}{\sigma}^{B1}}{d\Omega_{f}\,d\Omega_{e}\,dE_{f}}=\sum_{N=N_{min}}^{+\infty}\frac{d^{3}{\sigma}_{N}^{B1}}{d\Omega_{f}\,d\Omega_{e}\,dE_{f}}\,.$
(47)

By integrating TDCS over the direction of the scattered electrons,
$\Omega_{f}$, we obtain the double differential cross section of the ejected
electrons, while by integrating TDCS over the direction of the ejected
electrons, $\Omega_{e}$, we derive the double differential cross section of
the scattered electrons. Finally, the total ionization cross section is
deduced by integrating over the angles and energies of the scattered and
ejected electrons.

By neglecting the atomic dressing in Eq. (46), namely $T_{N,d}^{(1)}\simeq 0$
and $T_{N,ex}^{(1)}\simeq 0$, at small momentum of the residual ion, $q\ll
k_{e}$, we obtain a simple formula for the laser-assisted TDCS,

$\frac{d^{3}{\sigma}_{N}^{B1}}{d\Omega_{f}\,d\Omega_{e}\,dE_{f}}\simeq\frac{k_{f}k_{e}}{k_{i}}\,|J_{N}({\cal
R}_{q})|^{2}\frac{4}{\Delta^{4}}\left(1-\frac{\Delta^{2}}{\Delta_{e}^{2}}+\frac{\Delta^{4}}{\Delta_{e}^{4}}\right)|\psi_{1s}^{(0)}(q)|^{2},$
(48)

that “decouples” into a product of three factors: (i) the squared Bessel
function which includes the laser-projectile and ejected electrons
interaction, (ii) the electron-electron collision factor in the first-order
Born approximation

$f_{ee}^{B1}=\frac{1}{4\pi^{4}\Delta^{4}}\left(1-\frac{\Delta^{2}}{\Delta_{e}^{2}}+\frac{\Delta^{4}}{\Delta_{e}^{4}}\right),$

that is the absolute square of the half-off-shell Coulomb-matrix element
summed and averaged over final and initial spin states for fast projectile and
outgoing electrons takahashi2006 , and (iii)
$|\psi_{1s}^{(0)}(q)|^{2}={8}\pi^{-2}(q^{2}+1)^{-4}$ that represents the
squared momentum-space wave function for the ground state of atomic hydrogen
b-j89 . Equation (48) is in agreement to the TDCS derived for EMS by Kouzakov
and coworkers, namely Eq. (26) in Ref. kouzakov2010 . The half-off-shell Mott
scattering TDCS, for fast projectile and outgoing electrons that includes the
exchange terms, massey ; weigold99 , is simply calculated as
$(2\pi)^{4}f_{ee}^{B1}$,

$\left(\frac{d{\sigma}}{d\Omega_{e}}\right)_{ee}=\frac{4}{\Delta^{4}}\left(1-\frac{\Delta^{2}}{\Delta_{e}^{2}}+\frac{\Delta^{4}}{\Delta_{e}^{4}}\right).$
(49)

If we take into account the atomic dressing in Eq. (46) in the low-photon
energy limit $\omega\ll|E_{1}|$, and consider the lowest order in the photon
energy in Eq. (33), at small momentum of the residual ion, $q\ll k_{e}$, we
obtain

$\frac{d^{3}{\sigma}_{N}^{B1}}{d\Omega_{f}\,d\Omega_{e}\,dE_{f}}\simeq\frac{k_{f}k_{e}}{k_{i}}\,|J_{N}({\cal
R}_{q})|^{2}\left(\frac{d{\sigma}}{d\Omega_{e}}\right)_{ee}\left(1+\frac{4N\omega}{q^{2}+1}\right)^{2}|\psi_{1s}^{(0)}(q)|^{2},$
(50)

that is in agreement to the laser-assisted TDCS derived in the low-photon
energy approximation for EMS by Bulychev and coworkers, namely Eqs. (9)-(11)
in Ref. bulychev2012 . In contrast to Eq. (48) in which the atomic dressing
effects are neglected, the TDCS Eq. (50) does not obey the well-known Kroll-
Watson sum rule k-w . Obviously, the TDCS in the laser-assisted $(e,2e)$
collisions provides valuable information about the collision dynamics
weigold99 , electronic structure of the target, and can be used to derive the
momentum density distribution of the target electron, which was first
demonstrated for hydrogen and helium atoms Lohmann81 ; b-j89 .

##  III NUMERICAL EXAMPLES AND DISCUSSION

In this section we present our numerical results for the laser-assisted
electron-impact ionizing collisions in hydrogen, described by Eq. (1), for
fast incident and outgoing electrons, and we apply the semi-perturbative
formulas derived in Sec. II to calculate the nonlinear TDCSs in the presence
of a LP laser field. Obviously, due to the complicated analytical form of the
laser-dressed atomic wave function, the total scattering amplitude has to be
numerically evaluated. It is worth pointing out that the electronic and atomic
transition amplitudes, Eqs. (18), (20), (28), and (30), as well their
approximations derived in Subsec. II.4 are applicable for arbitrary scattering
configurations and laser field polarizations. We study the laser-assisted
$(e,2e)$ process in the coplanar geometry depicted in Fig. 1, in which the
momenta of the electrons, $\mathbf{k}_{i},\mathbf{k}_{f}$, and
$\mathbf{k}_{e}$, lie in the same plane where the two outgoing electrons are
detected in coincidence at the scattering angles $\theta_{f}$ and
$\theta_{e}$, with equal corresponding azimuthal angles
$\varphi_{f}=\varphi_{e}=\varphi_{i}$. The momentum vector of the incident
electron, $\mathbf{k}_{i}$, is taken parallel to the $z$ axis, with
$\theta_{i}=0^{\circ}$ and $\varphi_{i}=0^{\circ}$, and the scattering angle
$\theta_{f}$ of the scattered electron is fixed, while the angle $\theta_{e}$
of the ejected electron is varied. The asymmetric scattering geometry is
considered in which $\theta_{f}\neq\theta_{e}$ and ${k}_{f}\neq{k}_{e}$. At
this point it is useful to recall the differences between the symmetric and
asymmetric scattering geometries, namely the symmetric geometry is defined by
the requirement that the scattering angles and energies of the scattered and
ejected electrons are equal. In a kinematically complete experiment by
measuring the momentum vectors of both ejected electron and ionized target,
$\mathbf{k}_{e}$ and $\mathbf{q}$, we can deduce the momentum of the scattered
electron, $\mathbf{k}_{f}=\mathbf{k}_{i}-\mathbf{k}_{e}-\mathbf{q}$, as well
as the momentum transfer of the scattered electron,
$\mathbf{\Delta}=\mathbf{k}_{i}-\mathbf{k}_{f}$, occurring during the
collision hohr2007 . Thus, from the energy conservation law, the final
momentum of the projectile is given by
${k_{f}}={(k_{i}^{2}-k_{e}^{2}+2E_{1}+2N\omega)}^{1/2}$, while the momentum
transfer of the projectile is simple calculated as
$\Delta=(k_{i}^{2}+k_{f}^{2}-2k_{i}k_{f}\cos\theta_{f})^{1/2}$. The Cartesian
components of the momentum transfer vector, $\mathbf{\Delta}$, are given by
$(-k_{f}\sin\theta_{f},0,k_{i}-k_{f}\cos\theta_{f})$ and the amplitude
$\Delta$ varies in the range $|k_{i}-k_{f}|\leq\Delta\leq k_{i}+k_{f}$, for
forward $\theta_{f}=0^{\circ}$ and backward $\theta_{f}=180^{\circ}$
scattering, respectively. Similarly, the amplitude of the momentum transfer
vector $\mathbf{\Delta}_{e}$ is calculated as
$\Delta_{e}=(k_{i}^{2}+k_{e}^{2}-2k_{i}k_{e}\cos\theta_{e})^{1/2}$. The
amplitude of the recoil momentum vector of the residual ion, ${q}$, is given
by

$q={[\Delta^{2}+k_{e}^{2}-2k_{i}k_{e}\cos\theta_{e}+2k_{f}k_{e}\cos(\theta_{f}-\theta_{e})]}^{1/2}.$
(51)

The argument ${\cal R}_{q}$ of the Bessel functions is calculated as

${\cal R}_{q}^{2}={\cal R}_{i}^{2}+{\cal R}_{f}^{2}+{\cal R}_{e}^{2}-2{\cal
R}_{i}{\cal R}_{f}\cos(\phi_{i}-\phi_{f})-2{\cal R}_{i}{\cal
R}_{e}\cos(\phi_{i}-\phi_{e})+2{\cal R}_{f}{\cal
R}_{e}\cos(\phi_{f}-\phi_{e}),$

where ${\cal R}_{s}=\alpha_{0}|\bm{\varepsilon}\cdot\mathbf{k}_{s}|$ and
$e^{i\phi_{s}}=\bm{\varepsilon}\cdot\mathbf{k}_{s}/|\bm{\varepsilon}\cdot\mathbf{k}_{s}|$,
with $s=i,f$, and $e$. For a LP laser field the dependence of ${\cal R}_{s}$
on the laser polarization is given by ${\cal
R}_{s}=\alpha_{0}|\mathbf{e}_{j}\cdot\mathbf{k}_{s}|$ and $\phi_{s}=n\pi$,
while for a CP field with the polarization unit vector
$\bm{\varepsilon}=(\mathbf{e}_{j}+i\mathbf{e}_{l})/\sqrt{2}$, we obtain ${\cal
R}_{s}=(\alpha_{0}/\sqrt{2})\sqrt{(\mathbf{e}_{j}\cdot\mathbf{k}_{s})^{2}+(\mathbf{e}_{l}\cdot\mathbf{k}_{s})^{2}}$
and
$\phi_{s}=\arctan{(\mathbf{e}_{l}\cdot\mathbf{k}_{s})/(\mathbf{e}_{j}\cdot\mathbf{k}_{s})}+n\pi$,
where $n$ is an integer. In our numerical calculation we consider that the
laser field is linearly polarized in the same direction along the momentum
vector of the incident electron, $\bm{\varepsilon}||\mathbf{k}_{i}$.
Specifically, for a LP laser field and a coplanar scattering geometry with
$\phi_{i}=\phi_{f}=\phi_{e}=0^{\circ}$ the argument of the Bessel function
simplifies to ${\cal
R}_{q}=\alpha_{0}(k_{i}-k_{f}\cos\theta_{f}-k_{e}\cos\theta_{e})^{1/2}$.

To start with, we have checked that the numerical results of TDCSs for the
($e,2e$) scattering of fast electrons by hydrogen atoms in their ground state
are in agreement with earlier numerical data published in the literature. A
very good agreement is obtained with the numerical results of TDCS for one-
and two-photon exchange presented in Fig. 1 of Ref. kouzakov2010  and Figs. 1
and 2 of Ref. bulychev2012 , under the kinematical conditions of EMS (small
momentum of the residual ion $q$ and large momentum transfers $\Delta$ and
$\Delta_{e}$), for incident electrons of kinetic energy $E_{i}=2013.6$ eV, in
a noncoplanar symmetric scattering geometry, and a LP laser of intensity
$4\times 10^{12}$ W/cm2, calculated in the low-frequency approximation at
$\omega=1.17$ eV. At an incident electron kinetic energy $E_{i}=500$ eV, in a
coplanar symmetric geometry, $\bm{\varepsilon}\|\bm{\Delta}$, and a LP laser
of intensities $I=1.3\times 10^{7}$ W/cm2, $10^{2}\times I,10^{4}\times I$,
and $10^{6}\times I$, the behavior of the TDCS calculated from Eq. (46) is in
fair agreement, up to a scaling factor, to the first-order Born calculation of
TDCS for the ionization of hydrogen shown in Figs. 2 and 3 of Ref. khalil2017
. Since the atomic wave function was calculated within the closure
approximation khalil2017 , our numerical results disagree at larger photon
energies $\omega>3$ eV where the atomic dressing effect is more important, and
cannot be accurately described by this approximation. At the resonance photon
energy of $10.2$ eV, laser intensity of $1.3\times 10^{7}$ W/cm2, and
polarization $\bm{\varepsilon}||\bm{k}_{i}$, in the Ehrhardt asymmetric
coplanar geometry, with the incident and ejected electrons kinetic energies
$E_{i}=250$ eV and $E_{e}=5$ eV, and scattering angle $\theta_{f}=3^{\circ}$,
the TDCS given by Eq. (46) is in an satisfactory agreement with the first-Born
TDCS for the ionization of hydrogen plotted in Fig. 3(a) of Ref. taieb1991
where the atomic wave function is calculated using a Coulomb-Sturmian basis.
Obviously, despite the low value of the ejected electron kinetic energy, the
agreement is due to the fact that for one-photon resonance the TDCS is
dominated by the atomic contribution due to $1s-2p$ excitation.

Now, we return our discussion to the scattering geometry depicted in Fig. 1
where the laser polarization, $\bm{\varepsilon}$, is parallel to the incident
electron momentum direction, $\mathbf{k}_{i}$, and the outgoing electrons move
asymmetrically with respect to the direction of the incident electron, with
different scattering and ejected angles, and different kinetic energies. We
have chosen high kinetic energies of the projectile and ejected electrons
(compared to the atomic scale), moderate laser intensities below $1$ TW/cm2
which correspond to electric field strengths lower than $2.7\times 10^{7}$
V/cm, and have considered photon energies below the ionization threshold of
the hydrogen atom. Specifically, a laser intensity of $1$ TW/cm2 and a photon
energy of $1.55$ eV (Ti:sapphire laser) result in a quiver motion amplitude
$\alpha_{0}\simeq 1.64$ a.u. and an argument of the ordinary Bessel function
${\cal R}_{q}\simeq 1.64|\bm{\varepsilon}\cdot{\mathbf{q}}|$, while for a
larger photon energy of $3.1$ eV (Ti:sapphire second harmonic) the
corresponding amplitude $\alpha_{0}$ and the argument ${\cal R}_{q}$ are about
$4$ times smaller. The numerical results obtained for TDSCs in the first-order
Born approximation in the scattering potential, Eq. (46), are compared with
those obtained by considering the atomic contribution in the LFA, Eq. (36),
and those obtained by neglecting the dressing of the target by setting
$T_{N,d}^{(1)}\simeq 0$ and $T_{N,ex}^{(1)}\simeq 0$ in Eq. (46).

In Fig. 2 we present the TDCSs as a function of the angle of the ejected
electron, $\theta_{e}$, with exchange of one photon, $N=1$, at high kinetic
energies of the projectile electron $E_{i}=2$ keV and ejected electron
$E_{e}=200$ eV, and a small scattering angle, $\theta_{f}=5^{\circ}$. The
laser intensity is $I=1$ TW/cm2, while the photon energies we consider are:
$1.55$ eV in Fig. 2(a), $3.1$ eV in Fig. 2(b), $4.65$ eV in Fig. 2(c), and
$9.3$ eV in Fig. 2(d). Figure 3 show similar results to Fig. 2, but for a
larger scattering angle $\theta_{f}=15^{\circ}$. In all figures the solid
lines correspond to the laser-assisted TDCSs calculated from Eq. (46), which
include the laser dressing effects of the projectile and of the hydrogen atom,
the dot-dashed lines correspond to the TDCSs in which the atomic dressing is
considered in the LFA for the direct and as well as exchange scattering, while
the dashed lines correspond to the results in which the atomic dressing is
neglected. As resulted from our theoretical calculations, the TDCS is quite
important at scattering and ejected angles where the recoil momentum $q$ is
small. Thus, at the scattering angle $\theta_{f}=5^{\circ}$ the angular
distribution of the electrons is observed with a highest probability at the
maximum values of TDCSs, which occur at the following detection angles
$\theta_{e}\simeq-61^{\circ}$ in Fig. 2(a), $\theta_{e}\simeq-40^{\circ}$ in
Fig. 2(b), $\theta_{e}\simeq-38^{\circ}$ in Fig. 2(c), and
$\theta_{e}\simeq-39^{\circ}$ in Fig. 2(d). Similar to the free-free
transitions or other laser-assisted processes ehl1998 ; acgabi99 ; joa2012 ,
the net effect of the laser field is to decrease the peak values of the
angular distributions of TDCSs, while the atomic dressing contribution is
increasing with photon energy. The dressing effect of the laser is included in
the argument of the Bessel function through the quiver motion amplitude,
$\alpha_{0}$, in the electronic transition amplitudes, Eqs. (18) and (28), as
well through ${\cal R}_{q}$ and the factor $\alpha_{0}\,\omega$ in the atomic
transition amplitudes, Eqs. (20) and (30). Thus, as the photon energy
increases the atomic dressing effects (included in the full lines) are more
important than the electronic dressing effects (included in the dashed lines),
and the TDCS decreases as suggested by Eqs. (38), (39), and (40) derived in
the low-photon energy limit. At low photon energies which are far from any
atomic resonance, of $1.55$ eV or even $3.1$ eV at $\theta_{f}=5^{\circ}$, the
laser-assisted $(e,2e)$ process depicted in Figs. 2(a)-2(b) and 3(a) is well
described by the LFA (dot-dashed lines), as long as the photon energy is much
smaller than $|E_{1}|$. A clear signature of the nonperturbative effect of the
laser is the oscillatory character of the angular distribution of TDCS, as
shown in Fig. 2(a) compared to Figs. 2(b)-2(d). The nonperturbative behavior,
due to a larger quiver amplitude, is seen at the small photon energy of $1.55$
eV ($\alpha_{0}\simeq 1.64$ a.u. and $U_{p}\simeq 0.03$ eV), and resides in
the occurrence of an increasing number of zeros in the Bessel functions of the
first kind and, therefore, in the TDCS cionga93 . Thus, kinematic minima of
TDCSs appear at ${\cal R}_{q}=0$ if the scalar product
$\bm{\varepsilon}\cdot\mathbf{q}=0$, condition that is fulfilled at ejected
electron angles given by the relation
$\cos\theta_{e}={(k_{i}-k_{f}\cos\theta_{f})}/{k_{e}}$. The first two
kinematical minima which are located on the left- and right-side of the main
maximum in Fig. 2(a) at the photon energy of $1.55$ eV, occur at the angles
$\theta_{e}\simeq-79^{\circ}$ and $80^{\circ}$, while the next minima of TDCS
are due to the zeros of the Bessel function $J_{1}({\cal R}_{q})$ with ${\cal
R}_{q}\neq 0$. At larger photon energies in Figs. 2(b)-2(d) the first two
kinematical minima of the TDCSs occur at the angles
$\theta_{e}\simeq-79^{\circ}$ in Figs. 2(b)-2(d), and at $\theta_{e}\simeq\
80^{\circ}$ in Fig. 2(b), $\theta_{e}\simeq\ 81^{\circ}$ in Fig. 2(c), and
$\theta_{e}\simeq\ 83^{\circ}$ in Fig. 2(d).

In Fig. 4 we show the numerical results for TDCSs plotted in a logarithmic
scale for $N=0,1$ and $2$, at $\omega=3.1$ eV and the scattering angles
$\theta_{f}=5^{\circ}$ in Fig. 4(a) and $\theta_{f}=15^{\circ}$ in Fig. 4(b),
with the same parameters as in Figs. 2(b) and 3(b). The angular distributions
of TDCS at different $N$ present similar features with different magnitudes,
and show that the net effect of the laser field is to decrease the values of
TDCSs and to split the peaks which occur at $N=0$ (full lines) at
$\theta_{e}\simeq-62^{\circ}$ in Fig. 4(a) and $\theta_{e}\simeq-71^{\circ}$
in Fig. 4(b). The splitting of the peaks by the kinematical minima, which is a
well known signature of the laser field on the TDCSs, appears due to
cancellation of the scalar product $\bm{\varepsilon}\cdot\mathbf{q}$ and is
located almost symmetrically with respect to the direction of the incident
electron at the ejected angles $\theta_{e}\simeq-79^{\circ}$ and $80^{\circ}$
in Fig. 4(a) and $\theta_{e}\simeq-74^{\circ}$ and $75^{\circ}$ in Fig. 4(b).

It is well known the projectile electron plays a major role in the scattering
process since it interacts with the atomic target electron (a repulsive
interaction), its nucleus (an attractive interaction), as well as with the
laser field. In Fig. 5 we present the TDCSs for the ionization of hydrogen by
electron impact in the presence of a LP laser field, for absorption of one
photon $N=1$, at a photon energy of $4.65$ eV as a function of the ejected
electron angle. The kinetic energy of the ejected electron is $E_{e}=100$ eV
in Fig. 5(a), $200$ eV in Fig. 5(b), $400$ eV in Fig. 5(c), and $800$ eV in
Fig. 5(d). The other parameters concerning the scattering geometry, incident
projectile energy, angle of the scattered electron, and laser field intensity
are the same as in Fig. 2. Figure 6 shows similar results as in Fig. 5, but
for a larger scattering angle $\theta_{f}=15^{\circ}$. Clearly, the electronic
contribution (dashed lines) underestimates the angular distribution of TDCS.
For kinetic energies of the ejected electron $E_{e}\leq 100$ eV at
$\theta_{f}=5^{\circ}$ and $E_{e}\leq 200$ eV at $\theta_{f}=15^{\circ}$, the
atomic dressing effects are quite important, and the TDCS calculated in the
LFA (dot-dashed lines) fails to describe accurately the laser-assisted
$(e,2e)$ process. As the kinetic energy of the ejected electron increases to
$800$ eV, at small scattering angles the atomic dressing effects are less
important than the electronic dressing effects, as it is shown in Figs. 5(d)
and 6(d). As we approach the symmetric coplanar case of scattered and ejected
electrons of equal energies, $E_{f}\simeq E_{e}\simeq(E_{i}+E_{1}+\omega)/2$,
the minimum of the recoil momentum amplitude $q$ occurs now at larger angles
close to $\theta_{f}\simeq-\theta_{e}\simeq 45^{\circ}$, Eq. (51).

In order clarify the importance of the atomic dressing term we illustrate in
Fig. 7(a) the TDCS, in a logarithmic scale, with respect to the photon energy
for one-photon absorption. The kinetic energies of the projectile and the
ejected electrons are $E_{i}=2$ keV and $E_{e}=200$ eV, while the angles of
the scattered and ejected electrons are chosen $\theta_{f}=15^{\circ}$ and
$\theta_{e}=-55^{\circ}$. The polarization vector of the electric field is
parallel to the momentum of the incident electron, and we consider a moderate
laser intensity, $I=1$ TW/cm2, for which the non-perturbative dressing effects
of the projectile and ejected electrons can be visualized at small photon
energies with $\alpha_{0}>1$. The solid line corresponds to the laser-assisted
TDCS calculated from Eq. (46), which includes the dressing effects of the
projectile and of the atomic target, the dashed line corresponds to TDCS in
which the atomic dressing terms are neglected, while the dot-dashed line
corresponds to the result in which the atomic dressing terms are considered in
the LFA, Eq. (36). The TDCS shows a strongly dependence on the atomic
structure of the target and exhibits a series of resonance peaks which are
associated with one-photon absorption from the initial ground state of the
hydrogen atom, at photon energies that match atomic resonances
$\omega=E_{n}-E_{1}$, and correspond to poles that occur in the atomic radial
integral at $\tau=n$, with $n\geq 2$, as detailed in Fig. 7(b). The noteworthy
feature of the atomic radial integral $J_{101}$, Eq. (25), is that it presents
poles with respect to $\tau$ which arise due to the cancellation of the
$2-\tau$ factor in the denominator, as well as from the poles of the Appell’s
hypergeometric functions $F_{1}$ at $\tau=n^{\prime}$, where $n^{\prime}\geq
3$ is an integer. The origin of these poles resides in the poles of the
Coulomb Green’s functions used for the calculation of the linear-response
vector ${\bf w}_{100}$, vf1 . Clearly, the LFA, which does not take into
account the atomic structure, fails to describe the laser-assisted $(e,2e)$
process at large photon energies (typically in the UV range). Figure 7(c)
shows the energy spectra in the nonperturbative regime at low photon energies,
$\omega\leq 1.6$ eV in Fig. 7(a), domain where at the laser intensity $I=1$
TW/cm2 the quiver amplitude $\alpha_{0}$ is larger than $1$ a.u. and increases
up to $395$ a.u. for $\omega=0.1$ eV. The TDCS presents oscillations due to
the Bessel function $J_{1}({\cal R}_{q})$, and the LFA (dot-dashed line) gives
a good description of the atomic dressing effect. It should be kept in mind
that both the laser intensity and photon energy play an important role, and
obviously, the nonperturbative effects are seen to be important as laser
intensity increases and photon energy decreases due to increasing quiver
motion of the free and bound electrons, and contribute to the oscillatory
behavior of the laser-assisted TDCSs, which resides in the occurrence of
increasing oscillations of the Bessel function.

##  IV Summary and conclusions

We study the electron-impact ionization of hydrogen at large projectile and
ejected electron kinetic energies in the presence of a linearly polarized
laser field, and investigate the laser-assisted $(e,2e)$ reaction at moderate
laser field intensities. We focus our numerical results on the case of the
asymmetric coplanar scattering geometry, where we discuss the importance of
the dressing effects and analyze the influence of the laser field on the TDCS
in several numerical examples. The laser-assisted $(e,2e)$ reaction has a
nonlinear character which consists in multiphoton absorption (emission) of
photons from (to) laser radiation by projectile and ejected electrons and
atomic target. We present a new method to calculate the atomic radial
amplitude in a closed form, which represents the main difficulty in the
evaluation of TDCS. Thus, a semi-perturbative approach is used, in which for
the interaction of the fast incident and outgoing electrons with the laser
field we employ non-perturbative Gordon-Volkov wave functions, while the
interaction of the hydrogen atom with the laser field is considered in first-
order TDPT, and the interaction of the fast incident electron with the
hydrogen atom is treated in the first-order Born approximation. The exchange
between the outgoing electrons can not be ignored when ejected electrons with
large kinetic energy are detected, and is included in the calculation. Our
theoretical formulas and numerical results clearly demonstrate the strong
influence of the photon energy and laser intensity on the dynamics of laser-
assisted $(e,2e)$ process. It was found that the atomic dressing contribution
calculated in first-order TDPT in the laser field substantially modifies the
laser-assisted TDCSs at small momenta $q$, $\Delta$, and $\Delta_{e}$, and for
photon energies close to resonances. The introduction of the laser field in
the $(e,2e)$ reaction changes the profile of TDCS as it is seen in Fig. 4,
where the peaks of TDCSs are reduced in magnitude and splitted by the presence
of the laser, due to appearance of the kinematical minima. We show that the
atomic dressing effects strongly depend on the structure of the atomic target
as is seen in Fig. 7, and cannot be correctly described by the LFA at large
photon energies. At low photon energies we confirm the validity of LFA by
comparing the numerical results obtained for the atomic matrix elements within
LFA with the results obtained by first-order TDPT. Thus, the theoretical
studies remain very useful for understanding essential details of the
scattering signal due to the fact that the derived analytical formulas have
the advantage of giving more physical insight into the laser-assisted $(e,2e)$
process and valuable information in future theoretical and experimental
investigations.

## ACKNOWLEDGMENTS

The work by G. B. was supported by the research program PN 19 15 01 02 through
Contract No. 4N/2019 (Laplas VI) from the UEFISCDI and the Ministry of
Research, Innovation, and Digitization of Romania.

## References

  * (1) F. W. Byron Jr., and C. J. Joachain, Phys. Rep. 179, 211-272 (1989).
  * (2) (e, 2e) & Related Process, edited by C. T. Whelan, H. R. J. Walters, A. Lahmam-Bennani, and H. Ehrhardt, NATO ASI Series, vol 414 (Series C: Mathematical and Physical Sciences), (Springer, Dordrecht, 1993); Coincidence Studies of Electron and Photon Impact Ionization, edited by C. T. Whelan and H. R. J. Walters (Springer Science+Business Media, LLC, New York, 1997).
  * (3) R. Camilloni, A. Giardini Guidoni, R. Tiribelli, and G. Stefani, Phys. Rev. Lett. 29, 618 (1972).
  * (4) E. Weigold and I. E. McCarthy, Electron Momentum Spectroscopy (Kluwer, New York, 1999).
  * (5) V. G. Neudatchin, Yu. V. Popov, and Yu. F. Smirnov, Phys. Usp. 42, 1017 (1999).
  * (6) M. A. Coplan, J. H. Moore, and J. P. Doering, Rev. Mod. Phys. 66, 985 (1994).
  * (7) F. Ehlotzky, A. Jaro, and J. Z. Kaminski, Phys. Rep. 297, 63 (1998).
  * (8) C. Höhr , A. Dorn, B. Najjari, D. Fischer, C.D. Schröter, and J. Ullrich, Phys. Rev. Lett. 94, 153201 (2005); J. Electron Spectrosc. Relat. Phenom. 161, 172 (2007).
  * (9) T. Hiroi, Y. Morimoto, R. Kanya, and K. Yamanouchi, Phys. Rev. A 104, 062812 (2021).
  * (10) M. Jain and N. Tzoar, Phys. Rev. A 18, 538, (1978).
  * (11) J. Banerji and M. H. Mittleman, J. Phys. B 14, 3717 (1981).
  * (12) P. Cavaliere, G. Ferrante, and C. Leone, J. Phys. B 13, 4495 (1980); P. Cavaliere, C. Leone, R. Zangara, and G. Ferrante, Phys. Rev. A 24, 910 (1981).
  * (13) V. Gordon, Z. Physik 40, 117 (1926).
  * (14) D. M. Volkov, Z. Physik 94, 250 (1935).
  * (15) C. J. Joachain, P. Francken, A. Maquet, P. Martin, and V. Véniard, Phys. Rev. Lett. 61, 165 (1988).
  * (16) F. W. Byron Jr. and C. J. Joachain, J. Phys. B 17, L295 (1984).
  * (17) H. Ehrhardt, K. Jung, G. Knoth, and P. Schlemmer, Z. Phys. D 1, 3 (1986).
  * (18) P. Martin, V. Véniard, A. Maquet, P. Francken, and C. J. Joachain, Phys. Rev. A 39, 6178 (1989).
  * (19) R. Taïeb, V. Véniard, A. Maquet, S. Vučić, and R. M. Potvliege, J. Phys. B 24, 3229 (1991).
  * (20) A. Cionga, V. Florescu, A. Maquet, and R. Taïeb, Phys. Rev. A 47, 1830 (1993).
  * (21) A. Makhoute, D. Khalil, and I. Ajana, Atoms 7, 40 (2019).
  * (22) K. A. Kouzakov, Y. V. Popov, and M. Takahashi, Phys. Rev. A 82, 023410 (2010).
  * (23) A. A. Bulychev, K. A. Kouzakov, and Y. V. Popov, Phys. Lett. A 376, 484 (2012).
  * (24) D. Khalil, M. Tlidi, A. Makhoute, and I. Ajana, J. Phys. B: At. Mol. Opt. Phys. 50, 078001 (2017).
  * (25) L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965).
  * (26) F.H.M. Faisal, J. Phys. B: Atom. Molec. Phys. 6, L89 (1973).
  * (27) H. R. Reiss, Phys. Rev. A 22, 1786 (1980).
  * (28) B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules (Longman, London, 1983).
  * (29) C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, Atoms in Intense Laser Fields (Cambridge University Press, Cambridge, 2012), p. 466\.
  * (30) F. Ehlotzky, A. Jaroń, and J. Z. Kamiński, Phys. Rep. 297, 63 (1998).
  * (31) B. Lohmann and E. Weigold, Phys. Lett. A, 86, 139 (1981).
  * (32) C. Leone, S. Bivona, R. Burlon, F. Morales, and G. Ferrante, Phys. Rev. A 40, 1828 (1989).
  * (33) Y. Attaourti and S. Taj, Phys. Rev. A 69, 063411 (2004).
  * (34) J. Zhang and T. Nakajima, Phys. Rev. A 75, 043403 (2007).
  * (35) V. Florescu and T. Marian, Phys. Rev. A 34, 4641 (1986).
  * (36) N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Oxford University Press, London, 1965); C. Joachain, Quantum Collision Theory (North-Holland, Amsterdam, 1987).
  * (37) G. N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1962).
  * (38) A. Cionga, F. Ehlotzky, and G. Zloh, Phys. Rev. A 62, 063406 (2000); J. Phys. B: At. Mol. Opt. Phys. 33, 4939 (2000).
  * (39) G. Buica, Phys. Rev. A 92, 033421 (2015); J. Quant. Spectrosc. Radiat. Transf. 187, 190 (2017).
  * (40) E. Weigold, C. J. Noble, S. T. Hood, and I. Fuss, J. Phys. B: Atom. Molec. Phys. 12, 291, (1979).
  * (41) A. Dubois, A. Maquet, and S. Jetzke, Phys. Rev. A 34, 1888 (1986).
  * (42) A. Cionga, F. Ehlotzky, and G. Zloh, Phys. Rev. A 61, 063417 (2000).
  * (43) A. Dubois and A. Maquet, Phys. Rev. A 40, 4288 (1989).
  * (44) V. I. Ochkur, Sov. Phys. JETP 20, 1175 (1965).
  * (45) M. Jauch and F. Rohrlich, The Theory of Electrons and Photons (Springer, New York, 1976).
  * (46) F. E. Low, Phys. Rev. 110, 974 (1958).
  * (47) Y. Miyake, M. Takahashi, N. Watanabe, Y. Khajuria, Y. Udagawa, Y. Sakai, and T. Mukoyama, Phys. Chem. Chem. Phys. 8, 3022 (2006).
  * (48) N. M. Kroll and K. M. Watson, Phys. Rev. A, 8, 804 (1973).
  * (49) A. Cionga and G. Zloh, Laser Phys. 9(1), 69 (1999).

Figure 1: (Color online) Schematic representation of the scattering geometry
for the $(e,2e)$ reaction. The incident electron has energy $E_{i}$ and
momentum $\mathbf{k_{i}}$, while the scattered electron has energy $E_{f}$ and
momentum $\mathbf{k_{f}}$, and is detected at a fixed scattering angle
$\theta_{f}$. The ejected electron has energy $E_{e}$ and momentum
$\mathbf{k_{e}}$, and its detection angle $\theta_{e}$ is varied.
$\mathbf{\Delta}=\mathbf{k_{i}}-\mathbf{k_{f}}$ denotes the momentum transfer
vector from the incident to the scattered electron, and
$\mathbf{q}=\mathbf{\Delta}-\mathbf{k_{e}}$ is ion recoil momentum vector. For
a coplanar scattering geometry lying in the $(x,z)$-plane the azimuthal angles
$\varphi_{f}$ and $\varphi_{e}$ are equal to $0^{\circ}$ or $180^{\circ}$,
with the incident electron propagating in the $z$-axis direction,
$\varphi_{i}=0^{\circ}$ and $\theta_{i}=0^{\circ}$.  Figure 2: (Color online)
The TDCSs, in atomic units, for the ionization of hydrogen from the ground
state by electron impact in the presence of a LP laser field with
$\bm{\varepsilon}\|\bm{k}_{i}$, as a function of the ejected electron angle
$\theta_{e}$, for one-photon absorption. The kinetic energies of the incident
and ejected electron are $E_{i}=2$ keV and $E_{e}=200$ eV, and the scattering
angle is $\theta_{f}=5^{\circ}$. The laser intensity is $I=1$ TW/cm2, while
the photon energy is $\omega=1.55$ eV in Fig. 2(a), $3.1$ eV in Fig. 2(b),
$4.65$ eV in Fig. 2(c), and $9.3$ eV in Fig. 2(d). The solid lines correspond
to the laser-assisted TDCSs calculated from Eq. (46), the dashed lines
correspond to TDCSs in which the atomic dressing is neglected, and dot-dashed
lines correspond to TDCSs in which the atomic dressing is considered in the
LFA.  Figure 3: (Color online) Similar result as in Figs. 2(a)-2(d), but for a
scattering angle $\theta_{f}=15^{\circ}$. Figure 4: (Color online) The TDCSs
for the ionization of hydrogen by electron impact in the presence of a LP
laser field as a function of the ejected electron angle, $\theta_{e}$, at the
photon energy $\omega=3.1$ eV for no photon exchange (full line), one- and
two-photon absorption (dot-dashed and dashed lines). The scattering angle is
$\theta_{f}=5^{\circ}$ in Fig. 4(a) and $\theta_{f}=15^{\circ}$ in Fig. 4(b).
The other parameters concerning the scattering geometry, incident and ejected
electron energies, and laser intensity are the same as in Fig. 2. Figure 5:
(Color online) The TDCSs for the ionization of hydrogen by electron impact in
the presence of a LP laser field as a function of the ejected electron angle
$\theta_{e}$, for one-photon absorption. The photon energy is $\omega=4.65$
eV, while the kinetic energy of the ejected electron is $E_{e}=100$ eV in Fig.
5(a), $200$ eV in Fig. 5(b), $400$ eV in Fig. 5(c), and $800$ eV in Fig. 5(d).
The other parameters concerning the scattering geometry, kinetic energy of the
incident projectile, angle of the scattered electron, and laser intensity are
the same as in Fig. 2.  Figure 6: (Color online) Similar result as in Figs.
5(a)-5(d), but for a scattering angle $\theta_{f}=15^{\circ}$. Figure 7:
(Color online) The TDCSs for the ionization of hydrogen by electron impact in
the presence of a LP laser field as a function of the photon energy $\omega$,
for one-photon absorption. The solid lines correspond to the laser-assisted
TDCSs calculated from Eq. (46), the dashed lines correspond to TDCSs in which
the atomic dressing is neglected, and dot-dashed lines correspond to TDCSs in
which the atomic dressing is considered in the LFA. The angles of the
scattered and ejected electrons are $\theta_{f}=15^{\circ}$ and
$\theta_{e}=-55^{\circ}$, while the other parameters concerning the scattering
geometry, incident and ejected electron energies, and laser intensity are the
same as in Fig. 2. Figure 7(b) shows the detailed structure of the resonance
peaks which occur at photon energies $\omega=|E_{1}|(1-1/n^{2})$, with $n\geq
2$. Figure 7(c) shows, in a linear scale, the detailed structure of the
oscillatory behavior of TDCS at small photon energies, $\omega\leq 1.6$ eV.