# Impact of the continuum Coulomb interaction in quantum-orbit-based
treatments of high-order above-threshold ionization

T. Rook1, D. Habibović2, L. Cruz Rodriguez1, D. B. Milošević2,3, C. Figueira
de Morisson Faria1 1Department of Physics and Astronomy, University College
London, Gower Street, London WC1E 6BT, UK
2 University of Sarajevo, Faculty of Science, Zmaja od Bosne 35, 71000
Sarajevo, Bosnia and Herzegovina
3Academy of Sciences and Arts of Bosnia and Herzegovina, Bistrik 7, 71000
Sarajevo, Bosnia and Herzegovina

###### Abstract

We perform a systematic comparison between photoelectron momentum
distributions computed with the rescattered-quantum orbit strong-field
approximation (RQSFA) and the Coulomb-quantum orbit strong-field approximation
(CQSFA). We exclude direct, hybrid, and multiple scattered CQSFA trajectories,
and focus on the contributions of trajectories that undergo a single act of
rescattering. For this orbit subset, one may establish a one-to-one
correspondence between the RQSFA and CQSFA contributions for backscattered and
forward-scattered trajectory pairs. We assess the influence of the Coulomb
potential on the ionization and rescattering times of specific trajectory
pairs, kinematic constraints determined by rescattering, and quantum
interference between specific pairs of trajectories. We analyze how the
Coulomb potential alters their ionization and return times, and their
interference in photoelectron momentum distributions. We show that Coulomb
effects are not significant for high or medium photoelectron energies and
shorter orbits, while, for lower momentum ranges or longer electron excursion
times in the continuum, the residual Coulomb potential is more important.

††preprint: APS/123-QED

##  I Introduction

The laser-induced rescattering or recombination of an electron with its parent
ion plays a huge role in intense-field laser-matter interaction [1]. This
physical picture has been instrumental in explaining a wide range of
phenomena, which could not be interpreted by other means, and also in
dictating the attosecond $(10^{-18}$ s) time scales for which they occurred.
These processes may lead to the generation of high-order harmonics [2, 3, 4,
5], or photoelectrons with energies well above the ionization threshold, in
what is known as above-threshold ionization (ATI) [6, 7, 8].

Besides the myriad applications, such as attosecond light [9, 10] and electron
[11] pulses and photoelectron holography [12, 13], rescattering processes
naturally call for orbit-based approaches, both classical and quantum
mechanical. Quantum mechanically, electron orbits are associated with
transition amplitudes, and describe the several pathways an electron can take
up to the detector. This is the key ingredient in several semi-analytic
approaches, which have been formulated to describe the dynamics of the
liberated electron under the influence of the driving field and the Coulomb
potential of the residual ion [14, 15]. If the process is analyzed in terms of
the Feynman path-integral formalism [16], the ionization amplitude is written
as the sum of the contributions of different quantum orbits [17, 18]. These
quantum orbits have a well-known spacetime evolution, so the method allows us
to investigate different options available to the freed electron.

The most traditional and widespread of such approaches is the strong-field
approximation (SFA). The SFA assumes that the driving field is so strong that
the influence of the parent ion on the liberated electron can be neglected
during the electron propagation upon eventual rescattering [19, 20, 21, 22,
23]. This approximates the continuum by field-dressed plane waves [24, 6, 25],
which are analytically tractable, and rescattering is incorporated by
constructing a Born-type series around these solutions [26, 27]. This leads to
a clear-cut definition of scattering, which has influenced the main way of
thinking in the research area for decades.

Specifically for ATI, besides the scenario in which the liberated electron
goes directly to the detector (the so-called direct electrons), the
oscillatory character of the applied field can return the electron in the
vicinity of its parent ion and the rescattering may occur. If rescattering
happens, the process is denoted as high-order ATI (HATI) and its signature is
a long plateau with comparable peak intensities [28] (for review see, e.g.,
[29, 30, 31]). These electrons are known as rescattered electrons. Quantum-
orbit methods were employed within the SFA to investigate the direct electrons
(see for example Ref. [20]), while the solutions for the rescattered electrons
were systematically presented and classified in Ref. [32] for linearly
polarized and in Ref. [33] for bicircular driving fields.

The SFA theory was applied to the HATI process in Refs. [34, 26], but the role
of the residual Coulomb potential remained an open question. At first, the
influence of the Coulomb potential was accounted for through the Born series
analyzing the influence of the screening parameter [35], while later
approaches incorporate Coulomb distortions in the resulting scattering waves
[36], combined with the Born series [37], until eventually the continuum
propagation is modified [38, 39, 40, 41], although not all groups of orbits
leading to substantial contributions were taken into consideration [38, 39],
or approximations have been made on the scattering angle [42, 40, 41]. For a
review of the main methods beyond the SFA see the review article [13].

A Coulomb-distorted continuum invites a wide range of questions, such as
whether a distinction between “direct” and “rescattered” electrons can still
be made and, if so, what would be its validity range. Many features identified
in photoelectron holography, such as the fan-shaped structure that occurs near
the ionization threshold [43, 44], or the spider-like structure near the
polarization axis [12], were observed in experiments and in the full numerical
solution of the time-dependent Schrödinger equation, but not in the SFA. More
recent examples are spiral-like structures [45], or twisted holographic
patterns that have been predicted for elliptically polarized fields if the
Coulomb potential is included [46].

An important breakthrough occurred in the past decade, with the development of
strong-field path-integral approaches that incorporate the driving field and
the Coulomb potential on equal footing [47, 48, 49, 50, 51, 52]. These
approaches do not resort to Born-type expansions, which means that the
distinction between “direct” and “rescattered” electrons is blurred. This is
expected as, in reality, the interplay between the residual binding potential
and the driving field is highly non-trivial, which invites many questions.
Would an electron, being deflected by the potential, but whose perihelion is
larger than the Bohr radius be direct or rescattered? For instance, in [53,
54] it was shown that a fan-shaped holographic structure could only be created
if the Coulomb tail was accounted for, but for all purposes, the interfering
orbits were viewed as “direct”. Furthermore, in [47], whole sets of orbits
have been encountered which had no SFA counterpart. What about the scenarios
in which an electron is trapped or recaptured by the ionic potential [55, 56,
57]? Above all, how to compare theories whose structure is so different?

In our earlier publications [50, 53, 54, 58, 46] the comparison between the
results obtained using the SFA and the Coulomb quantum-orbit strong-field
approximation (CQSFA) was made without considering the structural differences
between the theories. For example, in the SFA theory, the intermediate
electron momentum changes abruptly at the moment of rescattering which happens
when the electron is at the position of the core. In the CQSFA, this is not
the case, i.e., the trajectory can go very close to zero, but will never be
zero, while the change of the photoelectron momentum is continuous. In the
present paper, we present a systematic comparison of the results obtained
using our theories. We discuss how the CQSFA trajectories which mimic the
behavior of the SFA trajectories can be isolated, and how good is the
agreement between the partial contributions to the probability density. Also,
we discuss how these differences depend on the photoelectron energy by
tracking the phase accumulated due to the Coulomb potential.

The paper is organized as follows. In Section II we present the two versions
of the SP method: one based on the SFA and one based on the Coulomb-corrected
SFA. In Sec. III we discuss how the trajectories obtained in the SFA theory
can be extracted from the CQSFA theory. Moreover, we discuss the behavior of
some of the trajectories which do not have an SFA analog. In addition, we
briefly present the classification of the quantum orbits for both theories. In
Section IV we present our numerical results and discuss the similarities and
differences between the two theories. Finally, Section V contains our main
conclusions. The atomic units are used unless otherwise stated.

##  II Background

The probability amplitude for the transition from the initial bound state
$\psi_{0}$ to the free final state $\psi_{\mathbf{p}_{f}}$ with momentum
$\mathbf{p}_{f}$ is given by [29]

$M_{\mathbf{p}_{f}}=-i\lim_{t\rightarrow\infty}\int_{-\infty}^{t}dt_{0}\langle\psi_{\mathbf{p}_{f}}(t)|U(t,t_{0})\mathbf{r}\cdot\mathbf{E}(t_{0})|\psi_{0}(t_{0})\rangle,$
(1)

where $U(t,t_{0})$ is the evolution operator which corresponds to the total
Hamiltonian $H(t)=H_{0}+H_{I}(t)$ of the system. Here $H_{0}$ is the time-
independent part which describes the electron exposed to the atomic potential,
while the time-dependent part $H_{I}(t)=\mathbf{r}\cdot\mathbf{E}(t)$
corresponds to the interaction of the electron with the applied field
$\mathbf{E}(t)$.

###  II.1 Strong Field Approximation

The evolution operator $U(t,t_{0})$ can also be written as

$U(t,t_{0})=U_{\mathrm{F}}(t,t_{0})-i\int_{t_{0}}^{t}dt^{\prime\prime}U(t,t^{\prime\prime})V(\mathbf{r})U_{\mathrm{F}}(t^{\prime\prime},t_{0}),$
(2)

where $U_{\mathrm{F}}(t,t_{0})$ is the Volkov evolution operator and
$V(\mathbf{r})$ is the atomic potential. Using only the first term of the
previous expansion we obtain the amplitude of the direct electrons

$M_{\mathbf{p}_{f}}^{(0)}=-i\int_{-\infty}^{\infty}dt_{0}\langle\mathbf{p}_{f}+\mathbf{A}(t_{0})|\mathbf{r}\cdot\mathbf{E}(t_{0})|\psi_{0}\rangle
e^{iS_{d}(\mathbf{p}_{f};t_{0})},$ (3)

with $\mathbf{A}(t)$ being the vector potential of the field
$\mathbf{E}(t)=-d\mathbf{A}(t)/dt$ and
$S_{d}(\mathbf{p}_{f};t_{0})=S_{\mathbf{p}_{f}}(t_{0})+I_{p}t_{0}$ being the
modified action for the direct electrons with
$S_{\mathbf{p}_{f}}(t)=\int^{t}dt^{\prime}[\mathbf{p}_{f}+\mathbf{A}(t^{\prime})]^{2}/2$
and $I_{p}$ the ionization potential. The free final state
$\psi_{\mathbf{p}_{f}}$ is approximated by a plane wave. Finally, $t_{0}$ is
the ionization time. On the other hand, the second term from Eq. (2) gives the
amplitude of the rescattered electrons

$\displaystyle
M_{\mathbf{p}_{f}}^{(1)}=-\int_{-\infty}^{\infty}dt_{0}\int_{t_{0}}^{\infty}dt\int
d{\mathbf{k}}\langle\mathbf{p}_{f}|V(\mathbf{r})|{\mathbf{k}}\rangle$
$\displaystyle\times\langle{\mathbf{k}}+\mathbf{A}(t_{0})|\mathbf{r}\cdot\mathbf{E}(t_{0})|\psi_{0}\rangle
e^{iS_{r}(\mathbf{p}_{f},{\mathbf{k}};t,t_{0})}.$ (4)

Here
$S_{r}(\mathbf{p}_{f},{\mathbf{k}};t,t_{0})=S_{\mathbf{p}_{f}}(t)-S_{{\mathbf{k}}}(t)+S_{{\mathbf{k}}}(t_{0})+I_{p}t_{0}$
is the modified action for the rescattered electrons, while $t$ is the
rescattering time. For the remainder of this paper, this contribution shall be
denoted by SFA.

The integral which appears in Eq. (II.1) can be solved numerically or using
the saddle-point (SP) method. The stationary condition which corresponds to
the momentum ${\mathbf{k}}$ is satisfied for
${\mathbf{k}}_{\text{st}}(t_{0},t)=-\int_{t_{0}}^{t}\mathbf{A}(t^{\prime})dt^{\prime}/(t-t_{0})$,
while the conditions which correspond to the ionization $\partial
S_{r}(\mathbf{p}_{f},{\mathbf{k}}_{\text{st}};t,t_{0})/\partial t_{0}=0$ and
rescattering time $\partial
S_{r}(\mathbf{p}_{f},{\mathbf{k}}_{\text{st}};t,t_{0})/\partial t=0$ lead to
the following SP equations

$[{\mathbf{k}}_{\text{st}}+\mathbf{A}(t_{0})]^{2}=-2I_{p},$ (5)
$[{\mathbf{k}}_{\text{st}}+\mathbf{A}(t)]^{2}=[\mathbf{p}_{f}+\mathbf{A}(t)]^{2},$
(6)

which represent the energy-conservation conditions at the ionization and
rescattering times, respectively. The solutions were classified previously in
[59] for the high-order harmonic generation (see Sec. III). The modified SP
method (see Appendix B in Ref. [20]) leads to the following expression for the
rescattering amplitude

$\displaystyle M_{\mathbf{p}_{f}}^{(1),\mathrm{SP}}$ $\displaystyle=$
$\displaystyle
2i\pi^{3/2}\kappa^{5/2}\sum_{\\{t_{0s},t_{s}\\}}\frac{\langle\mathbf{p}_{f}|V|{\mathbf{k}}_{\text{st}}\rangle}{[i(t_{s}-t_{0s})]^{3/2}}$
(7) $\displaystyle\times\frac{1}{S^{\prime\prime}_{rs0}}\left(\frac{2\pi
i}{S^{\prime\prime}_{rs}}\right)^{1/2}e^{iS_{rs}},$

where we introduced
$S_{rs}=S_{r}(\mathbf{p}_{f},{\mathbf{k}}_{\text{st}};t_{s},t_{0s})$,
$S^{\prime\prime}_{rs0}=\partial^{2}S_{r}/\partial t_{0}^{2}|_{t_{0s},t_{s}}$
and $S^{\prime\prime}_{rs}=\partial^{2}S_{r}/\partial t^{2}|_{t_{0s},t_{s}}$,
$\kappa=(2I_{p})^{1/2}$, and the sum is over the solutions $t_{0s}$ and
$t_{s}$ of the SP equations (5) and (6).

###  II.2 Coulomb Quantum-Orbit Strong Field Approximation

For the approach which takes into account the Coulomb potential, the
probability amplitude (1) can be written as

$\displaystyle M_{\mathbf{p}_{f}}$ $\displaystyle=$
$\displaystyle-i\lim_{t\rightarrow\infty}\int_{-\infty}^{t}dt_{0}\int
d\widetilde{\mathbf{p}}_{0}\langle\widetilde{\mathbf{p}}_{f}(t)|U(t,t_{0})|\widetilde{\mathbf{p}}_{0}\rangle$
(8)
$\displaystyle\times\langle\widetilde{\mathbf{p}}_{0}|\mathbf{r}\cdot\mathbf{E}(t_{0})|\psi_{0}(t_{0})\rangle,$

with $|\widetilde{\mathbf{p}}_{f}(t)\rangle=|\psi_{\mathbf{p}}(t)\rangle$, and
$\widetilde{\mathbf{p}}_{0}=\mathbf{p}_{0}+\mathbf{A}(t_{0})$ and
$\widetilde{\mathbf{p}}_{f}(t)=\mathbf{p}_{f}+\mathbf{A}(t)$ being the initial
and final velocity of the electron at the times $t_{0}$ and $t$, respectively.
The matrix element
$\langle\widetilde{\mathbf{p}}_{f}(t)|U(t,t_{0})|\widetilde{\mathbf{p}}_{0}\rangle$
can be calculated using the path integral method, leading to the following
expression [60, 61, 50, 62]

$\displaystyle M_{\mathbf{p}_{f}}$ $\displaystyle=$
$\displaystyle-i\lim_{t\rightarrow\infty}\int_{-\infty}^{t}dt_{0}\int
d\widetilde{\mathbf{p}}_{0}\int_{\widetilde{\mathbf{p}}_{0}}^{\widetilde{\mathbf{p}}_{f}(t)}\mathcal{D}^{\prime}\widetilde{\mathbf{p}}\int\frac{\mathcal{D}\mathbf{r}}{(2\pi)^{3}}$
(9) $\displaystyle\times
e^{iS(\widetilde{\mathbf{p}},\mathbf{r},t,t_{0})}\langle\widetilde{\mathbf{p}}_{0}|\mathbf{r}\cdot\mathbf{E}(t_{0})|\psi_{0}\rangle,$

where $\mathcal{D}^{\prime}\mathbf{p}$ and $\mathcal{D}\mathbf{r}$ are the
integration measures for the path integrals, prime indicates a restriction,
and the action reads

$S(\widetilde{\mathbf{p}},\mathbf{r},t,t_{0})=I_{p}t_{0}-\int_{t_{0}}^{t}[\dot{\mathbf{p}}(t^{\prime})\cdot\mathbf{r}(t^{\prime})+H(t^{\prime})]dt^{\prime},$
(10)

with
$H(t^{\prime})=H[\mathbf{r}(t^{\prime}),\mathbf{p}(t^{\prime}),t^{\prime}]=[\mathbf{p}(t^{\prime})+\mathbf{A}(t^{\prime})]^{2}/2+V[\mathbf{r}(t^{\prime})]$.
The stationary conditions for the variables $t_{0}$, $\mathbf{p}$ and
$\mathbf{r}$ lead to the equation

$[\mathbf{p}(t_{0})+\mathbf{A}(t_{0})]^{2}+2V[\mathbf{r}(t_{0})]=-2I_{p},$
(11)

as well as to the classical equations of motion of the electron

$\dot{\mathbf{p}}=-\nabla_{r}V[\mathbf{r}(t^{\prime})],\quad\dot{\mathbf{r}}=\mathbf{p}+\mathbf{A}(t^{\prime}).$
(12)

A classification of the solutions to the system of equations (11) and (12) was
introduced in [50] based upon the initial conditions and the final momentum of
the photoelectron trajectory. However, this classification was found to be
incomplete [63] due to the fact that more information about the intermediate
momentum and position is necessary for a full characterisation of the
qualitative nature of the trajectory (see Sec. III). Using the solutions
$t_{0s}$, $\mathbf{p}_{s}$, and $\mathbf{r}_{s}$ of these Coulomb-corrected SP
equations, the probability amplitude can be written as

$\displaystyle M_{\mathbf{p}_{f}}^{\text{CSP}}$ $\displaystyle\propto$
$\displaystyle\lim_{t\rightarrow\infty}\sum_{s}D^{-1/2}\mathcal{C}(t_{0s})e^{iS_{s}-i\pi\nu_{s}/2},$
(13)

where $S_{s}=S(\widetilde{\mathbf{p}}_{s},\mathbf{r}_{s},t,t_{0s})$,
$D=\text{det}\left[\partial\mathbf{p}_{s}(t)/\partial\mathbf{p}_{s}(t_{0s})\right]$,

$\mathcal{C}(t_{0s})=\sqrt{\frac{2\pi
i}{\partial^{2}S(\widetilde{\mathbf{p}}_{s},\mathbf{r}_{s},t,t_{0s})/\partial
t_{0s}^{2}}}\langle\mathbf{p}+\mathbf{A}(t_{0s})|\mathbf{r}\cdot\mathbf{E}(t_{0s})|\psi_{0}\rangle,$
(14)

and $\nu_{s}$ is the Maslov phase associated with a solution $s$ as calculated
by the prescription given in [64] and [65].

In the following section, we compare our results obtained using the SFA and
CQSFA theories using the example of the hydrogen atom exposed to a linearly
polarized field. The ionization potential is $0.5$ a.u. Our driving field is
$\mathbf{E}(t)=E_{0}\sin(\omega t)\hat{\mathbf{e}}_{x}$ with $E_{0}$ being the
amplitude, and we define the emission angle $\theta_{e}$ as the angle between
the final photoelectron momentum and the unit vector $\hat{\mathbf{e}}_{x}$.

##  III Momentum Mapping and Orbit Correspondence

In this section, we explain how the two theoretical models are related to each
other. The SFA and CQSFA theories are different in a way that the SFA is a
Born-expansion-based theory, meaning that the electrons which go directly to
the detector, as well as the electrons which experience one or more
rescattering events, are clearly identifiable. The continuum is approximated
by laser-dressed plane waves and the electron rescatters with the potential
when it returns to the site of its release, namely the origin. Spatially, this
assumption locates the interaction at a single point.

On the other hand, the CQSFA approach treats the Coulomb and the laser-field
potentials equally. Because of the spatial range of the Coulomb potential, it
is not straightforward to determine whether the electron can be viewed as
direct, undergoing a mere deflection, a soft collision, or a hard collision.
Furthermore, the potential may trap the outgoing electron, or lead to multiple
collisions. This makes the difference between the direct and rescattered
electrons blurred, and also means that not every orbit in the CQSFA will
exhibit an SFA counterpart. Therefore, we must identify those that do. The
main goal of this section is to understand what types of orbits can be
compared between these methods and the procedure one must follow in order to
do this.

We start with a short reminder about the classification of the SP solutions
for both theories. In the SFA theory, the rescattering solutions occur in
pairs associated with a longer and shorter orbit, which coalesce for maximal
classically allowed rescattering energies. These energies are associated with
sharp decreases in the photoelectron signal, which are known as ‘cutoffs’
[66]. They are classified into two groups: the backward- and forward-
scattering solutions. In the former case, the electron is rescattered in
approximately the opposite direction with respect to the direction of its
motion before the scattering event, while in the latter case, the electron is
only deflected during this event. Therefore, within the SFA, the parameter
used to enforce this distinction is the rescattering angle.

The backward-scattering solutions are classified using the multiindex
$(\alpha,\beta,m)$. For the return times within one optical cycle, there are
infinitely many solutions with different travel times $\tau=t-t_{0}$. The
index $m$ gives the approximate value of the travel time in units of laser-
field period $T=2\pi/\omega$, $\omega$ being the angular frequency. For
$-(m+1/2)T<t_{0}<-(m-1/2)T$, $m=0,1,2,\ldots$, there are two pairs of
solutions distinguished by the index $\beta$ in such a way that $\beta=-1$
($\beta=+1$) corresponds to the longer (shorter) travel time. Furthermore, the
two solutions of one pair are distinguished by the index $\alpha$. The long
(short) solution is denoted with $\alpha=-1$ ($\alpha=+1$).

The forward-scattering solutions were classified in [32] using the multiindex
$(\mu,\nu)$ in such a way that $\mu=0,1,2,\ldots$, measures the travel time,
while index $\nu=\pm 1$ is used to distinguish the two solutions of one pair.
The positions of the cutoff for these solutions were classically found in
[67], while those positions with quantum corrections were given in [68].
Besides the shortest and the second-shortest pair, other forward-scattering
solutions contribute only in a low-energy part of the spectra
($E_{\mathbf{p}_{f}}<0.04U_{p}$, $U_{p}=E_{0}^{2}/(4\omega^{2})$ being the
ponderomotive energy).

For the CQSFA theory, the classification of the solutions was introduced in
[50, 54] based upon the initial conditions and the final momentum of the
photoelectron trajectory. The solutions with $p_{f,y}p_{0,y}>0$ are denoted as
‘orbit 1’ for $z_{0}p_{f,x}>0$ and ‘orbit 2’ for $z_{0}p_{f,x}<0$, while for
$p_{f,y}p_{0,y}<0$ the solutions are denoted as ‘orbit 3’ for $z_{0}p_{f,x}<0$
and ‘orbit 4’ for $z_{0}p_{f,x}>0$. In those early publications, orbit 1 was
identified with a direct SFA orbit in which the electron is freed in the
direction of the detector, orbits 2 and 3 behave as field-dressed hyperbolas
starting half a cycle earlier or later, with the difference that orbit 2
interacts less with the core and can still be viewed as “direct”, while orbit
3 has a hybrid character. Orbit 4 goes around the core before reaching the
detector, and is identified as backscattered. In more recent publications [58,
63], however, we found that these distinctions are blurred and a wide range of
behaviors for each of the four types of orbit can be identified by applying
spatial filters. If the orbit’s distance of closest approach $r_{c}$ is larger
than the radius $r_{T}$ determined by the tunnel exit, the CQSFA orbit is
considered ‘direct’. If the orbit’s perihelion lies between the tunnel exit
and the Bohr radius, that is, $r_{T}\geq r_{c}\geq r_{0}$, we refer to the
orbit as ‘soft scattered’, while if $r_{c}<r_{0}$ the CQSFA orbit is ‘hard
scattered’. A thorough analysis using these filters has been performed in
[63].

Here, instead of using a spatial filter, in order to compare the CQSFA with
the SFA, we eliminate all orbits which do not have analogous SFA solutions by
ruling out multiple scattering events or single scattering events induced
significantly by the long range of the Coulomb potential. The simple criterion
which was used is keeping only the solutions which correspond to the
trajectories that cross the $r_{y}=0$ axis once. It turns out that this
criterion allows one to recover families of solutions which may be compared
with the single-rescattering SFA solutions. Additional constraints have been
imposed by matching the CQSFA rescattering times to their SFA counterparts, as
will be discussed below.

Following the approximations stated above, the rescattering solutions found in
the SFA theory can be reproduced from the orbit 3 and orbit 4 solutions from
the CQSFA theory. Orbit 1 and 2 solutions mainly correspond to the direct
electrons. The relation between the solutions obtained using our theories is
summarized in Table 1.

Orbit | $z_{0}p_{f,x}$ | $p_{f,y}p_{0,y}$ | SFA orbit
---|---|---|---
1 | $+$ | $+$ | direct
2 | $-$ | $+$ | direct
3 | $-$ | $-$ |  $(\beta,m)=(1,1),(1,2)$,
|  |  | forward-scattering
4 | $+$ | $-$ | $(\beta,m)=(-1,0),(-1,1)$
Table 1: Main types of orbits identified using CQSFA approach. The $+$ ($-$)
sign on each cell indicates a positive (negative) value of the product. The
fourth column contains the SFA orbits which can be extracted from the
corresponding CQSFA orbits.

Here, the SFA orbits that can be extracted from the corresponding CQSFA orbits
are given in the fourth column. For the SFA solutions $(\beta,m)=(1,0)$ [32]
we have not found the CQSFA analogue. Finally, it is worth mentioning that the
CQSFA orbits are classified using the boundary conditions of the trajectories
that specify the four types of orbits, while the SFA orbits are classified
using the value of the electron’s velocity just before rescattering.

Now we illustrate how we extracted the earlier-mentioned SFA solutions from
the CQSFA theory. In Fig. 1(a) we present the values of the initial momentum
of semiclassical trajectories calculated using the CQSFA theory whose final
momenta belong to a uniform grid. Different types of orbits (see Table 1) are
coded with different colors as indicated in the legend. For simplicity, we
have considered only solutions with the ionization time in the second half
cycle. The half-cycle symmetry that is inherent to this field [69, 70]
guarantees that there will be no loss of information. Both half cycles have
been considered in [63]. The solutions with the final momentum in the first
(second) quadrant of the final momentum plane correspond to orbit 1 (2). On
the other hand, the solutions with the final momentum in the third (fourth)
quadrant of the final momentum plane, shown in Fig. 1, correspond to orbit 3
(4). The island close to the origin leads to the rescattering ridges, while
the island centered at $(p_{0,x},p_{0,y})=(-2U_{p}^{1/2},0.2U_{p}^{1/2})$
gives rise to caustics [63]. Both of these islands are visible in Fig. 1(a),
while in Fig. 1(b) only the central island, which is responsible for the
ridges shown in Fig. 1(c), has been presented.

Figure 1: Panel(a): The values of the initial momentum of semiclassical
trajectories calculated using the CQSFA whose final momenta belong to a
uniform grid. Different types of orbits are presented with different colors as
indicated in the legend. The results are obtained for the second-half cycle of
the monochromatic field. Panels (b) and (c): The values of the initial [panel
(b)] and final [panel (c)] momentum of semiclassical trajectories for orbits 3
and 4, obtained using the CQSFA theory, which have analogous counterparts in
the SFA backward-scattered solutions. They are coded in different colors as
indicated in the legend in panel (c). The intensity of the applied field is
$I=E_{0}^{2}=2\times 10^{14}$ W/cm2, while the angular frequency $\omega$
corresponds to the wavelength of $800$ nm.

Using specific subsets of the initial momentum points, rescattered
trajectories analogous to those obtained in the SFA can be obtained. In Fig.
1(b) we display these subsets in such a way that the red, blue, orange and
pink regions lead to the backward-scattering SFA solutions listed in order of
the increasing travel time. For example, the solutions represented by the red
points correspond to the backward-scattered SFA solutions with the shortest
travel time. Since these solutions can all be classified as orbit 3 or orbit
4, only the solutions classified as such are displayed in Fig.1(b) and
Fig.1(c). For the CQSFA we choose the rescattering time to be the time at
which the trajectory reaches its perihelion. Physically, this is the most
appropriate assumption one can make to compare with the SFA, which is a Born-
type expansion for which rescattering times are well defined according to Eq.
(6).

The range of these solutions regarding the final momentum is shown in Fig.
1(c), with rescattering ridges marking the maximal classical momenta
associated with specific orbits. The rescattering ridges formed by the first-
and third-shortest solutions intersect the $p_{f,x}=0$ axis, while this is not
the case for the second- and fourth-shortest solutions. The first- and third-
shortest solutions with the final momentum $p_{f,x}<0$ (not shown in the
figure) correspond to the SFA forward-scattering solutions. Similar solutions
do not exist for the second- and fourth-shortest pairs so that the forward-
scattering SFA solutions can only be extracted from the orbit 3 CQSFA
solutions. The solutions represented by the black dots [see Fig. 1(b)] account
for the remaining CQSFA trajectories. These include the forward-scattering
solutions and the solutions which include multiple scattering events.

Despite the fact that the direct SFA orbits can be extracted from orbits 1 and
2, this does not mean that orbits classified as 1 and 2 correspond only to the
direct electrons. To illustrate this, we show a few examples of the
rescattering trajectories corresponding to either orbit 1 or orbit 2 CQSFA
solutions.

Figure 2: The values of the initial [panel (a)] and final [panel (b)]
photoelectron momentum of the semiclassical orbit 1 trajectory calculated
using the CQSFA theory for the situation in which the trajectories make one
loop in the clockwise (blue regions) or counterclockwise (red regions)
direction around the core, as well as for the trajectories which do not make
any loop around the core (orange regions). Panel (c): The examples of orbit 1
trajectories for given values of the initial conditions.

In Fig. 2 we present the values of the initial [panel (a)] and final [panel
(b)] photoelectron momentum of the semiclassical orbit 1 trajectories
calculated using the CQSFA theory. Among all solutions, we extract those which
correspond to the trajectories which make a loop in the clockwise (blue
regions) or counterclockwise (red regions) direction around the core. The
regions which correspond to the trajectories which do not make a loop around
the core are denoted by orange color. The three regions of the final momentum
which correspond to these trajectories partiality overlap [see Fig. 2(b)].
However, the ridges of these trajectories can still be easily noticed. The
ridge associated to the trajectory which makes a loop in the clockwise
direction has the highest energy, while the lowest-energy ridge corresponds to
the trajectory which makes a loop in the counterclockwise direction around the
core. The examples of these trajectories are shown in Fig. 2(c). The values of
the initial-momentum components for the trajectory which makes a loop in the
clockwise (counterclockwise) direction are $p_{0,x}=0.00214$ and
$p_{0,y}=0.212$ ($p_{0,x}=0.3679$ and $p_{0,y}=0.0711$), while for the
trajectory which does not make a loop around the core, these values are
$p_{0,x}=-0.0254$ and $p_{0,y}=0.135$, in units of $U_{p}^{1/2}$. For clarity,
although the field dressing means that initially some of the trajectories
shown in Fig. 2(c) travel into the 2nd quadrant of the plane, they are indeed
orbit 1 trajectories and will eventually move into the 1st quadrant outside of
the position ranges shown in the figure. Clearly, these are rescattering
trajectories even though they correspond to the orbit 1 solutions.
Additionally, the radial vector of the perihelion of the trajectory which
loops around the core in a counter-clockwise direction [see the red line in
Fig. 2(c)] makes an angle of $55^{\circ}$ with the negative $r_{x}$ axis. This
shift of the perihelion is what allows the photoelectron to scatter into the
first quadrant and is an effect which relies entirely upon the presence of a
long-range Coulomb potential. Similar conclusions are valid for the orbit 2
solutions.

$(\beta,m)=$ $(1,2)$ $~{}~{}~{}~{}~{}(-1,1)$ $~{}~{}~{}~{}~{}(1,1)$
$~{}~{}~{}~{}~{}(-1,0)$

Figure 3: Ionization [panels (a) and (c)] and rescattering [panels (b) and
(d)] times for the SP solutions relevant in the medium- and high-energy
regions of the spectrum calculated using the SFA [panels (a) and (b)] and
CQSFA [panels (c) and (d)] theories. The intensity and frequency of the
applied field are the same as in Fig. 1. The emission angle is
$\theta_{e}=0^{\circ}$. The values of the multiindex $(\beta,m)$ are indicated
above panel (a), while the values of index $\alpha$ are indicated within
panels (a) and (b).

##  IV Rescattered electrons - SFA vs CQSFA

In this section, we present numerical results for the ionization and
rescattering times as well as the photoelectron momentum distributions
obtained using the strong-field approximation and the Coulomb quantum-orbit
strong-field approximation theories.

###  IV.1 Times and orbits

First, we investigate and classify the solutions for the ionization and
rescattering times obtained using the SFA and CQSFA theories. The low-energy
region is dominated by the contribution of direct electrons which means that
the rescattered solutions, which we consider here, are most relevant in the
medium- and high-energy regions of the spectra.

In Fig. 3 we present the SP solutions for the ionization [panels (a) and (c)]
and rescattering times [panels (b) and (d)] obtained using the SFA and CQSFA
theories as indicated in the panels. All solutions exhibit arch-like
structures associated with pairs of orbits that coalesce at specific energies.
These energies correspond to maxima of the electron’s classical kinetic
energy, and give kinematic constraints that may be associated with a cutoff in
the photoelectron spectra. The highest cutoff energy, at $10U_{p}$, is
associated with the shortest pair of orbits [71]. The SFA classification of
the solutions is presented in Fig. 3(a). More precisely, the values of the
multiindex $(\beta,m)$ are indicated above panel (a), while the values of
index $\alpha$ are indicated within panels (a) and (b). Nonetheless, at first
glance, there are a few differences between the SFA and CQSFA results. For
example, the ionization and rescattering times calculated using the CQSFA
theory are shifted with respect to those obtained using the SFA. Also, the
arch-like structures, calculated using the CQSFA, are closed for the low-
energy values.

In order to better assess the aforementioned differences, in Fig. 4 we present
the results for the ionization and rescattering times for the shortest
[$(\beta,m)=(-1,0)$] [panels (a) and (b)] and second-shortest
[$(\beta,m)=(1,1)$] [panels (c) and (d)] solutions of the SP equations
calculated using the SFA (dashed lines) and CQSFA (solid lines) theories.

Figure 4: Ionization [panels (a) and (c)] and rescattering [panels (b) and
(d)] times for the shortest [panels (a) and (b)] and the second-shortest
[panels (c) and (d)] SP solutions calculated using the SFA (dashed lines) and
CQSFA (solid lines) theories.

Both, the ionization and rescattering times are shifted when the Coulomb
effects are taken into consideration. For the ionization time, this shift is
approximately the same for both pairs of solutions, while for the rescattering
time, the shift is different for the two pairs. In addition, the effect of the
Coulomb potential is generally different for the two solutions of one pair.
The difference between the rescattering times of the solutions of the shortest
pair [see Fig. 4(b)] is more pronounced than the corresponding difference for
the solutions of the second-shortest pair [see Fig. 4(d)], particularly in the
medium- and high-energy regions of the spectrum. For the first return, the
difference in travel time calculated using the SFA and CQSFA theories is small
for all values of the photoelectron energy. We have checked that this shift
becomes more pronounced for later rescattering occurrences. This happens due
to the fact that, for the later rescattering events, the electron spends more
time close to the core, thus increasing the significance of the Coulomb
potential.

In the low-energy region of the spectrum, the shift of the rescattering time,
due to the Coulomb effect is larger. This can be explained by the fact that
the low-energy electrons spend more time in close proximity to the core than
the high-energy electrons. The Coulomb potential energy $V(\mathbf{r})$ [see
Eq. (12)] of the liberated electron is sensitive to the small oscillations of
the electron subjected to the laser field. In order to illustrate this, in
Fig. 5 we present $V(\mathbf{r})$ as a function of time for the shortest
[panel (a)] and the second shortest [panel (b)] pair of solutions. We present
the potential for the values of the photoelectron energy $E_{\mathbf{p}_{f}}$
as indicated in the legends and for both the short (solid lines) and long
(dashed lines) solutions.

Figure 5: Coulomb potential as a function of time for the shortest [panel (a)]
and the second-shortest [panel (b)] pair of solutions from Fig. 4. The values
of the photoelectron energy are indicated in the panels, while the solid and
dashed lines correspond to the short and long trajectories, respectively.

The rescattering events happen around $0.5T$. By comparing the values of the
potential during the rescattering event for different values of the
photoelectron energy, we see that, for the high and medium values of the
energy, the potential rapidly goes to a large negative value (see the blue and
orange lines in Fig. 5). On the other hand, for the small value of the
photoelectron energy, the behavior of the potential during rescattering is
different for the shortest and second-shortest pairs of solutions. In the case
of the second-shortest pair, the photoelectron does not closely approach the
core and the potential well is approximately as deep as it is at the tunnel
exit [see the red lines in Fig. 5(b)]. In addition, the potential exhibits
oscillations which continue long after rescattering (see the red lines in Fig.
5). This explains the fact that the deviation of the rescattering time
calculated using the CQSFA from the time calculated using the SFA theory is
more pronounced in the low-energy region.

Finally, the two branches of the second-shortest (as well as for the fourth-
shortest) set of solutions approach each other for low-energy in a similar
manner as they do for the high-energy cutoff [see Fig. 4(d)]. This can be
related to the results shown in Fig. 1(c). As we already mentioned in Sec.
III, the second- and fourth-shortest pairs do not intersect with the $p_{f,y}$
axis. Therefore, as the energy of solutions goes to zero in the direction of
emission $\theta_{e}=0^{\circ}$, they approach the caustic and coalesce. This
low-energy caustic also passes through the origin in the SFA theory. However,
we do not observe the same low-energy coalescence of the rescattering and
ionization times. This happens because the presence of a long-range Coulomb
potential implies that the photoelectron requires a certain minimum energy to
escape this interaction.

Figure 6: Photoelectron momentum distributions for the shortest [(panel (a)],
the second shortest [(panel (b)] and the third shortest [(panel (c)] backward-
scattering solutions of the SP equations. The upper-half of the panels
corresponds to the SFA-obtained distributions, while the lower-half of the
panels are calculated using the CQSFA theory. Multiindices $(\beta,m)$ which
correspond to the SFA-obtained results are indicated in the top-left corner of
the panels. Panels (a)(i) and (a)(ii): The linear photoelectron spectra
calculated using the SFA (solid orange line) and the CQSFA (dashed blue line)
for the values of final momentum denoted by the white dashed and red dash-
dotted lines in panel (a). The driving-field parameters are the same as in
Fig. 1.

###  IV.2 Photoelectron momentum distributions

After analyzing the differences between the solutions of the SP equations
obtained using the two methods, we now turn our attention to the differences
in the photoelectron momentum distributions. We devote particular attention to
the three pairs of solutions with the shortest travel time.

In Fig. 6 we present the photoelectron momentum distributions (PMDs) for the
shortest [(panel (a)], the second shortest [(panel (b)] and the third shortest
[(panel (c)] backward-scattering solutions of the SP equations. The upper-half
of the panels corresponds to the SFA distributions, while the lower-half of
the panels are calculated using the CQSFA theory. The results shown in
different panels are normalized independently. The solutions obtained using
the SFA appear in pairs and the divergent contribution beyond the cutoff has
been discarded [59]. The rescattering ridges which appear in the PMDs,
calculated using our two theories, appear at roughly the same place in the
momentum plane. Each ridge is associated with the maximal classical energy
available for a pair of backscattered orbits, and the fringes characterize the
interference between the long and short orbits in a pair. They are well-known
in the SFA [71] and have recently been identified in the CQSFA context [63].
Throughout, we see that the SFA fringes are perfectly circular, while their
CQSFA counterparts exhibit distortions close to $(p_{f,x},p_{f,y})=(0,0)$.
There are also mismatches in the positions of the fringes, which depend on the
pair of orbits taken into consideration.

For the shortest pair of solutions [see Fig. 6(a)], the agreement between the
results obtained using the two earlier mentioned theories is good except for
the regions $p_{f,x}\in[\pm 0.5U_{p}^{1/2},\pm 1.8U_{p}^{1/2}]$,
$|p_{f,y}|<0.7U_{p}^{1/2}$. More specifically, the positions of the minima,
caused by the interference of the two contributions of the pair appear for
approximately the same energy in both cases. In contrast, in the low-energy
region, the CQSFA fringes become tear-shaped and there are mismatches with
regard to the SFA. The main difference appears for the emission in the
direction close to $\theta_{e}=\pm 90^{\circ}$. In this case, the
photoelectron yield calculated using the CQSFA theory is increased with
respect to the yield obtained for other values of the emission angle. This
increase in the photoelectron yield was discussed in [58] and it is not
present in the photoelectron momentum distribution obtained using the SFA
theory. Finally, for energy after the cutoff, the applied SFA theory leads to
the probability density which decreases exponentially as a function of the
photoelectron energy. On the other hand, for the CQSFA, there are no
trajectories with energy beyond the cutoff. In order to include the solutions
beyond the classical cutoff, complex trajectories are required. These
trajectories can be included as shown in [72].

In order to discuss the agreement between the SFA and CQSFA theories as a
function of the photoelectron energy, in Fig. 6(a)(i) we present the
photoelectron momentum spectra for a fixed value of the momentum component
$p_{f,y}=\pm 0.5U_{p}^{1/2}$ as a function of the momentum component $p_{f,x}$
[along the white dashed lines in Fig. 6(a)]. Clearly, the agreement between
the results obtained using our theories is excellent in the medium- and high-
energy parts of the spectrum. As the momentum component $p_{f,x}$ decreases,
the agreement between the positions of the minima and maxima calculated using
our theories becomes slightly worse. On the other hand, the agreement is not
good for $p_{f,x}<U_{p}^{1/2}$, i.e., in the low-energy part of the spectra.
Similar results are presented in Fig. 6(a)(ii) but for a fixed
$p_{f,x}=1.32U_{p}^{1/2}$ [along the red dash-dotted line in Fig. 6(a)] and
for changing $p_{f,y}$. In this case, the photoelectron spectrum is shorter,
but the discrepancies between the results calculated using our two theories
are still not pronounced except for values $p_{f,y}<0.5U_{p}^{1/2}$. The
agreement between the results becomes better with the increase of $p_{f,y}$.
Also, the number of predicted interference minima (and maxima) is the same for
both theories.

For the second shortest pair of solutions [see Fig. 6(b)], the agreement
between the results obtained using SFA and CQSFA theories is much weaker than
for the shortest pair. In particular, the positions of the interference minima
do not coincide very well indicating that the accumulated phase differences
are not the same. Intuitively, this makes sense because the contribution of
this pair of solutions is mostly significant in the low- and medium-energy
parts of the spectra where the Coulomb effects are particularly pronounced.
Furthermore, for $p_{f,x}\approx\pm 1.85U_{p}^{1/2}$ and a small $p_{f,y}$,
caustics appear in the results obtained using the CQSFA theory. These caustics
are associated with orbits clustering, but are overestimated due to an
artefact of the theory, which fails to work well in the vicinity of coalescing
saddle points. Similarly as for the shortest pair of solutions, the increase
of the photoelectron yield is predicted by the CQSFA theory, for the emission
in the direction close to the $\theta_{e}=\pm 90^{\circ}$. In the end, we
mention that the solutions of the second-shortest pair (and all other
backward-scattering solutions with $\beta=1$) have irregular behavior for the
energy close to zero in the SFA approach [68].

Finally, for the third-shortest pair, the agreement between the results
obtained using our two theories is reasonable - not as good as for the
shortest pair, but better than that observed for the second shortest pair.
However, the fringes’ positions are mismatched and the CQSFA exhibits a larger
number of interference minima (6 against the 5 minima encountered for the
SFA). There are also caustics which now appear for $p_{f,x}\approx\pm
1.5U_{p}^{1/2}$ and small $p_{f,y}$. Besides that, the increase of the
photoelectron yield in the close to orthogonal direction exists, but it is not
pronounced as for the second-shortest pairs.

Figure 7: Real part of the difference between the actions $S^{L}$ and $S^{S}$,
corresponding to the long (superscript ’$L$’) and short (superscript ’$S$’)
orbits of the shortest [panel (a) and (b)] and second-shortest [panel (c)]
pair of solutions, modulus $2\pi$ as a function of the momentum component. The
results are calculated using the SFA (dotted red lines) and CQSFA (dotted blue
lines) theories along the lines $p_{f,y}=0.5U_{p}^{1/2}$ and
$p_{f,x}=1.32U_{p}^{1/2}$ for the shortest and $p_{f,x}=2.2U_{p}^{1/2}$ for
the second-shortest pair.

Up to now, we have discussed the rings which appear in the photoelectron
momentum distributions calculated using SFA and CQSFA theories. We have
concluded that, for the shortest pair of solutions, there is close agreement
between the results in the medium- and high-energy parts of the spectra, while
it is the case that for the solutions of the second shortest pair, there is
disagreement across the entirety of the spectra. In order to explain this, in
Fig. 7 we present the real part of the difference between the actions $S^{L}$
and $S^{S}$ modulus $2\pi$ (remainder after dividing by $2\pi$) which
correspond to the long (superscript ’$L$’) and short (superscript ’$S$’)
orbits of the shortest [panels (a) and (b)] and second-shortest [panel (c)]
pair of solutions for the fixed values of one momentum component as indicated
in the panels [along the white dashed, red dash-dotted and orange dotted lines
in Figs. 6(a) and 6(b)]. The results are calculated using the SFA (red
squares) and CQSFA (blue dots) theories along the lines
$p_{f,x}=1.32U_{p}^{1/2}$ and $p_{f,y}=0.5U_{p}^{1/2}$ for the shortest and
$p_{f,x}=2.2U_{p}^{1/2}$ for the second-shortest pair. When the difference
between the real parts of the actions is equal to an even (odd) multiple of
$\pi$, maxima (minima) appear in the spectrum. For the shortest pair of
solutions, the difference $\operatorname{Re}S^{L}-\operatorname{Re}S^{S}$
becomes smaller as the longitudinal component of the photoelectron momentum
$p_{f,x}$ increases [see Fig. 7(a)]. For small values of $p_{f,x}$, the
differences between the results obtained using the SFA and CQSFA theories are
pronounced. This is in agreement with the results shown in Fig. 6(a)(i). The
increased number of minima in the CQSFA spectrum compared to the SFA spectrum
can be understood by analysing the behaviour of the difference
$\operatorname{Re}S^{L}-\operatorname{Re}S^{S}$ in the vicinity of
$p_{f,x}\approx 1.3U_{p}^{1/2}$. In this region, the mentioned difference
reaches a deeper minimum for the CQSFA than for the SFA, such that it extends
to the value $\pi$ and forms an additional interference minimum. On the other
hand, for $p_{f,x}=1.32U_{p}^{1/2}$, the difference
$\operatorname{Re}S^{L}-\operatorname{Re}S^{S}$ remains almost constant as a
function of $p_{f,y}$ except for a low-energy region where the Coulomb effects
are pronounced [see Fig. 7(b)]. The chosen vertical slice is in the energy
region for which the continuous Coulomb interaction is less significant and
the accumulated phase difference is similar for both theories. Finally, for
the second-shortest pair of solutions, the difference
$\operatorname{Re}S^{L}-\operatorname{Re}S^{S}$ remains quite pronounced for
the broad range of values of the photoelectron momentum [see Fig. 7(c); in
accordance with results presented in Fig. 6(b)]. The difference
$\operatorname{Re}S^{L}-\operatorname{Re}S^{S}$ still decreases with the
increase of the photoelectron momentum, but, due to the lower position of the
cutoff for this solution, the Coulomb effects remain significant for most
parts of the photoelectron momentum distribution. In addition, the shape of
the Coulomb potential $V[\mathbf{r}(t)]$ may influence the action [see Eq.
(10)]. In particular, for the second-shortest pair, the Coulomb potential
$V(\mathbf{r})$ exhibits a rich structure prior to the rescattering (see Fig.
5), which is not the case for the shortest solution. This implies that, due to
this effect, the difference between the CQSFA-calculated action and the SFA-
calculated action may be more significant than for the shortest pair of
solutions.

Besides the backward-scattering solutions, which are prevalent in the high-
energy part of the spectra, the forward-scattering solutions should also be
taken into consideration. The classification of these solutions is presented
in Sec. III.

Figure 8: Photoelectron momentum distributions for the shortest [(panel (a)]
and the second-shortest [(panel (b)] forward-scattering solutions of the SP
equations. The index $\mu$ used in the SFA classification is shown in the top
left corner of the panels. The upper-half of the panels corresponds to the
SFA-obtained distributions, while the lower-half panels are calculated using
the CQSFA theory. The driving-field parameters are the same as in Fig. 1.

In Fig. 8 we present the photoelectron momentum distributions for the shortest
[(panel (a)] and the second-shortest [(panel (b)] forward-scattering solutions
of the SP equations. The upper half of the panels corresponds to the SFA
distributions, while the lower-half panels are calculated using the CQSFA
theory. The results shown in different panels are normalized independently.
For the shortest pair of solutions, the photoelectron momentum distributions
calculated using our theories qualitatively agree well. However, the positions
of the minima caused by the interference of two contributions are not at the
same place. This is particularly the case for the minima with a small energy.
The fact that the interference minima do not appear at the same place in the
photoelectron momentum plane can be explained in the similar manner as for the
backward-scattering solutions. In addition, for the emission close to the
polarization direction, the CQSFA calculation leads to an increase in the
photoelectron yield in comparison with the one obtained using the SFA theory.
Finally, similarly to the backward-scattering solutions, the CQSFA does not
give any nonzero probability density after the cutoff because the complex
trajectories are not included.

In Fig. 8(b) we see that, for the second-shortest pair of solutions, the
differences between the presented results are much more pronounced than for
the shortest pair. This is expected due to the fact that the contribution of
this solution is significant only in the low-energy part of the spectra. In
conclusion, due to the fact that the contributions of the forward-scattering
solutions are significant for low- or medium-energy regions, the Coulomb
effects should be taken into consideration.

##  V Conclusions

In this work, we compare two semianalytical theories based on the saddle-point
method which allow one to get a clear insight into the underlying physics of
the strong-field ionization process. The first one is the strong-field
approximation (SFA) which assumes that the driving field is so strong that the
Coulomb attraction between the liberated electron and the residual ion can be
neglected during the electron propagation in the continuum. The second theory
is the Coulomb quantum-orbit strong-field approximation (CQSFA) for which the
Coulomb potential and the driving-field potential are treated equally. These
theories are structurally very different. The former is a Born-type-expansion
theory for which the rescattering times and conditions are well-defined
because the interactions with the parent ion during rescattering are localized
at the single point, namely the origin. On the other hand, the latter has a
Coulomb distorted continuum which consequently makes it impossible to clearly
define when the rescattering starts or ends.

To compare the results obtained using the SFA and CQSFA is not as trivial task
as it sounds at first. In particular, for a systematic comparison, it is not
sufficient to remove the Coulomb coupling or make it short-range due to the
structural differences between the two theories. These differences make it
hard to establish a one-to-one correspondence between the orbits obtained with
each method. In order to make this comparison, we have identified the CQSFA
trajectories which mimic the behavior of the Born-type SFA trajectories with a
single act of rescattering. This means that among all orbits which appear in
the CQSFA theory, we have extracted only those with SFA counterparts.

We have found that the agreement between the results obtained using our
theories is fairly good if the solutions used in the CQSFA theory are
restricted to only those which we expect to have a counterpart in the SFA
theory. This happens because the compared orbits accumulate similar phases in
the continuum, although their dynamics can be very different. For example,
before the rescattering, the SFA trajectories are one-dimensional for a
linearly polarized driving field, while for the CQSFA theory, the
corresponding orbits are always two-dimensional. More generally, the SFA
trajectories follow the field, which is not the case with their CQSFA
counterparts due to the always present Coulomb interaction.

In order to systematically investigate the influence of the Coulomb effects,
we have analyzed the photoelectron momentum distributions for different
solutions of the saddle-point equations. We have found that the positions of
the ridges for different solutions are approximately the same for both
theories, while the positions of the minima caused by the interference of the
contributions may be different, depending on the value of the photoelectron
energy. Whether the positions of these minima are reproduced well in the SFA
or not depends on two counterbalancing effects: the time the electron spends
in the continuum and the electron’s energy. A longer time in the continuum
means that the accumulated Coulomb phase will be larger as well, but a higher
kinetic energy means that the electron will be less sensitive to the Coulomb
potential. These effects are clearly visible in the photoelectron momentum
distributions for the three shortest pairs of orbits. The best agreement is
for the shortest pair because the corresponding trajectories have a short
travel time in the continuum and the corresponding energy is the highest. The
worst agreement is with the second shortest pair, because the travel time is
longer than for the shortest pair, and the corresponding energy is the lowest.
Finally, the third shortest pair has a reasonable agreement, because, although
the orbits spend a longer time in the continuum, their energy is higher than
those of the second-shortest pair, which partly counterbalances this effect.
These conclusions have been confirmed by explicit calculation and comparison
between the accumulated phases in the SFA and CQSFA theories.

In conclusion, even though the SFA theory does not include many possible cases
of the electron trajectories and the included trajectories are treated by
neglecting the existing Coulomb potential, it is still a reasonably good
approximation which accounts for a lot of the key dynamics. The obtained
results are usually sufficiently accurate for various purposes. Finally, one
expects the level of agreement between the theories to depend on the
parameters of the driving laser field and the potential. A systematic study of
this dependence could help precisely establish the range of values in the
parameter space for which the application of the SFA theory is acceptable.
Also, the study of the structural differences between the two theories
presented in the current paper could be utilized as the groundwork for
establishing future hybrid theories.

###### Acknowledgements.

D. H. would like to thank the University College London for its kind
hospitality. This work was partly funded by grants No. EP/J019143/1 and
EP/T517793/1, from the UK Engineering and Physical Sciences Research Council
(EPSRC). D. H. and D. B. M. gratefully acknowledge support by the Ministry for
Science, Higher Education, and Youth, Canton Sarajevo, Bosnia and Herzegovina.

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