# Influence of catastrophes and hidden dynamical symmetries on ultrafast
backscattered photoelectrons

T. Rook  L. Cruz Rodriguez  C. Figueira de Morisson Faria  Department of
Physics and Astronomy, University College London
Gower Street, London WC1E 6BT, UK

###### Abstract

We discuss the effect of using potentials with a Coulomb tail and different
degrees of softening in the photoelectron momentum distributions (PMDs) using
the recently implemented hybrid forward-boundary CQSFA (H-CQSFA). We show that
introducing a softening in the Coulomb interaction influences the ridges
observed in the PMDs associated with backscattered electron trajectories. In
the limit of a hard-core Coulomb interaction, the re-scattering ridges close
along the polarization axis, while for a soft-core potential, they are
interrupted at ridge-specific angles. We analyze the momentum mapping of the
different orbits leading to the ridges. For the hard-core potential, there
exist two types of saddle-point solutions that coalesce at the ridge. By
increasing the softening, we show that two additional solutions emerge as the
result of breaking a hidden dynamical symmetry associated exclusively with the
Coulomb potential. Further signatures of this symmetry breaking are
encountered in subsets of momentum-space trajectories. Finally, we use
scattering theory to show how the softening affects the maximal scattering
angle and provide estimates that agree with our observations from the CQSFA.
This implies that, in the presence of residual binding potentials in the
electron’s continuum propagation, the distinction between purely kinematic and
dynamic caustics becomes blurred.

††preprint: APS/123-QED

##  I Introduction

Caustics and catastrophes describe divergent phenomena, in which gradual
changes in the input parameters lead to abrupt changes in a dynamical system’s
response. They have been observed and studied in a wide range of areas, such
as optics [1, 2], mechanics [3], and even biological sciences [4]. Within
strong-field laser-matter interaction, catastrophe theory has been used in
high-order harmonic generation, to identify regions of exceptional brightness
in the high-order harmonic spectra, which can be tuned in order to enhance
specific groups of harmonics [5, 6], and employed to probe multielectron
dynamics at a giant resonance [7]. In strong-field ionization, catastrophe
theory has been used to explain a series of sharply peaked low-energy
structures observed in photoelectron spectra with low frequency, typically
mid-IR driving fields [8, 9, 10, 11, 12, 13]. Furthermore, caustic-type
(Glory) trajectories have also been identified in ultrafast photoelectron
holography [14, 15, 16].

Thereby, a key aspect is that, similarly to the geometrical optics scenario in
which rays of light coalesce, in strong-field and attosecond physics caustics
are associated with coalescing electron trajectories. This picture builds up
on the paradigm that describes strong-field phenomena as the result of the
laser-induced recollision of a previously freed electron with its parent ion
[17, 18]. Coalescing trajectories are observed near the high-order harmonic
(HHG) or high-order above-threshold ionization (ATI) cutoff [19], as well as
in laser-induced nonsequential double ionization [20, 21, 22]. Cutoffs mark
the maximal kinetic energy a rescattering photoelectron may have. For lower
energies, electron trajectories occur in pairs. For each pair, there is a
short and a long trajectory, arriving before and after a field zero crossing,
respectively. Coalescence of more than two trajectories has also been observed
in specific scenarios, such as Glory scattering [16].

Possibly the best-known caustics in high-order ATI are the rescattering
ridges, which occur when pairs of backscattered trajectories coalesce. These
ridges may extend up to a high photoelectron energy, which, for the shortest
trajectory pair and linearly polarized monochromatic fields, is around
$10U_{p}$, where $U_{p}$ is the ponderomotive energy. The prevalent argument,
backed by classical models [23, 24], the strong-field approximation (SFA) [25]
and the adiabatic approximation [26], is that the rescattering ridge is
primarily a kinematic caustic. One should note, however, that, for such
methods, the acts of rescattering are spatially localized and well defined, as
they are either constructed for short-range potentials [26], or as a Born-type
expansion around field-dressed plane waves [25]. The latter assumption
restricts the influence of the rescattering potential to a single point, to
which the electron returns.

If, on the other hand, the residual binding potential is incorporated in the
electron’s continuum propagation, can one really say whether the electron is
rescattered, merely deflected, or minimally influenced by it? In the SFA, an
electron can be unambiguously classified as “direct” or “rescattered”,
depending on whether it reaches the detector without or with further
interaction with the core. However, orbit-based methods beyond the strong-
field approximation have revealed that this distinction is often blurred.
There are, for instance, hybrid orbits, which fit neither classification [8,
27, 28, 29], and, even for orbits that can be neatly classified, the binding
potential may lead to considerable differences from their Coulomb-free
counterparts [30, 31, 32].

A wide range of examples can be found in ultrafast photoelectron holography,
in which well-known holographic structures such as the fan [33, 34] and the
spider [35, 36] can only be reproduced accurately if the binding potential and
the driving field are incorporated on equal footing (see [37] for a review).
Important breakthroughs in the interfering orbits leading to these structures
have been achieved in our previous work using the Coulomb Quantum-Orbit
Strong-Field Approximation (CQSFA). We have shown that, by introducing angle-
dependent distortions in direct ATI orbits, the long-range tail of the Coulomb
potential leads to fan-shaped fringes [30, 31]. Furthermore, obtaining spider-
like structures required hybrid types of orbits, which do not fall into the
binary direct-rescattered description [38, 28, 29]. Other important examples
are carpet-like structures, which, in high photoelectron energy ranges, have
been interpreted as caused by direct SFA orbits [39]. Comparisons of the CQSFA
with experiments [40] and probing the interference carpets with filtering
techniques [41] have shown that this was a case of mistaken identity and that,
in reality, the carpets are formed by the interference of hybrid and
rescattered orbits. It is also noteworthy that only a subset of Coulomb-
distorted rescattered orbits has a clear rescattered SFA counterpart. These
orbits have been identified recently in [32] and lead to the caustics
associated with forward-scattering and backscattering orbits. Thereby, a
legitimate question is whether the rescattering ridges are purely kinematic,
or if they are affected by the shape of the scattering potential. For
instance, soft-core potentials are widely used due to their Coulomb-like tail,
but lack the Coulomb singularity. Remnants of ridges and the role of softening
have been found elsewhere, and discussed employing classical-trajectory
arguments [42]. However, the focus was placed on holographic structures
instead of the ridges.

In the present paper, we study the rescattering ridge due to backscattered
electron orbits considering binding potentials with different degrees of
softening using the hybrid forward-boundary CQSFA (H-CQSFA) recently developed
in [43]. We perform a comprehensive study of the role of the Coulomb
singularity in obtaining closed backscattered ridges and show how using a
soft-core Coulomb potential affects the maximum rescattering angle leading to
open ridges. We perform the initial to final momentum mapping to separate the
different sets of solutions contributing to the ridges. Using elements of
catastrophe theory, we find that softening the potential gives rise to two
additional sets of solutions, which are associated with breaking a dynamical
symmetry that exists for the Coulomb potential but not for its soft-core
counterpart. This symmetry is associated with the conservation of the Laplace-
Runge-Lenz (LRL) vector. We also analyze electron trajectories corresponding
to each of these branches and find that, for a hard-core potential, they are
approximately circular in momentum space, which are signatures of the LRL
vector being conserved. In contrast, for the soft-core potential non-Coulomb
dynamics alters these features and some of these orbits become elliptical.
Finally, we use scattering theory to understand how the softening parameter
influences the maximum scattering angle in a field-free scenario. We show that
the maximum angle observed in the rescattering ridges agrees well with the
predictions of the scattering model.

The article is organized as follows. In Sec. IIA we introduce the general
theoretical framework of the H-CQSFA. Subsquently, we discuss the concept of
focal points and caustics in Sec. IIB. Section III contains the main results
showing photoelectron momentum distributions and momentum mapping, and in Sec.
IV, we present a trajectory and scattering angle analysis. Then, in Sec. V, we
draw the main conclusions of our work. Atomic units are used throughout unless
otherwise stated.

##  II Background

Throughout, we employ the Coulomb Quantum Orbit Strong Field Approximation
(CQSFA) and saddle-point methods. Here we will only provide a brief outline of
this method with only the equations that are essential for understanding the
results in the subsequent sections. For extensive discussions and more details
we refer to our previous publications [27, 30, 31, 38, 44, 43].

###  II.1 General CQSFA expressions

Within this framework, the CQSFA transition amplitude from a bound state
$|\psi_{0}\rangle$ to a continuum state with final momentum $\mathbf{p}_{f}$
reads as

$\displaystyle M(\mathbf{p}_{f})$ $\displaystyle\propto$
$\displaystyle\lim_{t\rightarrow\infty}\sum_{s}D^{-1/2}\mathcal{C}(t^{\prime}_{s})e^{iS(\mathbf{p}_{s},\mathbf{r}_{s},t,t^{\prime}_{s})-i\pi\nu_{s}/2},$
(1)

where
$D=\text{det}\left[\partial\mathbf{p}_{s}(t)/\mathbf{p}_{s}(t^{\prime}_{s})\right]$,
and $\nu_{s}$ is the Maslov phase associated with a solution $s$ as calculated
by the prescription given in [45] and [46].

Here, the semi-classical action is

$S(\mathbf{p},\mathbf{r},t,t^{\prime})=I_{p}t^{\prime}-\int^{t}_{t^{\prime}}[\dot{\mathbf{p}}(\tau)\cdot\mathbf{r}(\tau)+H(\mathbf{r}(\tau),\mathbf{p}(\tau),\tau)]d\tau,$
(2)

where $t^{\prime}$ denotes the ionization time, $t$ is the time at which the
electron reaches the detector, $I_{p}$ gives the target’s ionization
potential, $\mathbf{r}$ and $\mathbf{p}$ are the electron’s intermediate
coordinate and momentum, respectively, parametrized as functions of the
intermediate time $\tau$.

The electronic Hamiltonian is given by

$H(\mathbf{r}(\tau),\mathbf{p}(\tau),\tau)=\frac{1}{2}\left[\mathbf{p}(\tau)+\mathbf{A}(\tau)\right]^{2}+V(\mathbf{r}(\tau)),$
(3)

where $\mathbf{A}$ is the vector potential. The atomic potential is taken as

$V(\mathbf{r})=-1/\sqrt{\mathbf{r}^{2}+\alpha^{2}}$ (4)

with a parameter $\alpha$ to soften the Coulomb singularity. In this work, we
vary this parameter to assess its influence on the dynamics.

The variables $t^{\prime}_{s}$, $\mathbf{r}_{s}$ and $\mathbf{p}_{s}$ in Eq.
(1) are the solutions of the saddle-point Eqs.

$[\mathbf{p}(t^{\prime})+\mathbf{A}(t^{\prime})]^{2}=-2I_{p},$ (5)
$\displaystyle\mathbf{\dot{r}}(\tau)=\mathbf{p}(\tau)+\mathbf{A}(\tau),$ (6)
$\displaystyle\mathbf{\dot{p}}(\tau)=-\nabla_{r}V(\mathbf{r}(\tau)),$ (7)

which have been derived by taking the action to be stationary as indicated by
the subscripts in Eq. (1).

Eqs. (6) and (7) are coupled and give the electron’s propagation in the
continuum, and, in the limit of vanishing ionization potential, can be
associated with classical equations of motion. Eq. (5) gives the conservation
of energy upon tunnel ionization. It is convenient to use a two-pronged
contour, whose first part starts at $t^{\prime}$ and extends vertically along
the imaginary axis up to $\mathrm{Re}[t^{\prime}]$, and whose second part
starts in $\mathrm{Re}[t^{\prime}]$ and extends along the real axis up to $t$.
This choice allows a neat visualization of the sub-barrier and the continuum
dynamics, and to define a tunnel exit as the point in space for which the
electron reaches the continuum. One should bear in mind that this concept is
somewhat arbitrary as discussed in [47, 44]. The sum in Eq. (1) is over the
distinct saddle point trajectories which have final momentum $\mathbf{p}_{f}$.

In principle, one may solve the full complex problem, which, however, brings
the difficulty of branch cuts [48, 44, 37] and does not change the PMDs
significantly. For that reason, we perform a further approximation, namely
take the tunnel exit $z_{0}$ to be real. Explicitly, this gives

$z_{0}=\mathrm{Re}[r_{0||}(\mathrm{Re}[t^{\prime}])],$ (8)

where $\mathbf{r}_{0}(\mathrm{Re}[t^{\prime}])$ is the tunnel trajectory

$\textbf{r}_{0}(\tau)=\displaystyle\int_{t^{\prime}}^{\tau}(\textbf{p}_{0}+\textbf{A}(\tau^{\prime}))d\tau^{\prime}.$
(9)

integrated along the imaginary axis up to $\tau=\mathrm{Re}[t^{\prime}]$ and
the subscript indicates its component along the driving-field polarization
direction. In the tunnel trajectory equation, $\textbf{p}_{0}$ is the
electron’s under-the-barrier momentum, which has been taken as constant from
$t^{\prime}$ to $\mathrm{Re}[t^{\prime}]$.

The prefactor
$\det[{\partial\mathbf{p}_{s}(t)}/{\partial\mathbf{p}_{s}(t^{\prime}_{s})}]$
comes from the quadratic fluctuations around the saddle points and gives
information about the stability of specific orbits. The prefactor

$\mathcal{C}(t^{\prime}_{s})=\sqrt{\frac{2\pi
i}{\partial^{2}S(\mathbf{\tilde{p}}_{s},\textbf{r}_{s},t,t^{\prime}_{s})/\partial
t^{\prime
2}_{s}}}\langle\mathbf{p}+\mathbf{A}(t^{\prime}_{s})|\mathbf{r}\cdot\mathbf{E}(t^{\prime}_{s})|\psi_{0}\rangle,$
(10)

where $\mathbf{E}(t)=-d\mathbf{A}(t)/dt$ is the electric field, stems from the
saddle-point equation (5). The matrix element contains information about the
initial bound-state geometry and also occurs in the standard strong-field
approximation.

###  II.2 Model and orbit types

Here we consider a monochromatic, linearly polarized driving field so that the
vector potential reads

$\mathbf{A}(t)=A_{0}\cos(\omega t)\hat{z}=2\sqrt{U_{p}}\cos(\omega t)\hat{z},$
(11)

where $\hat{z}$ is the polarization direction of the electric field, and
$U_{p}$ is the ponderomotive energy. We assume the target to be hydrogen
($I_{p}=0.5$ a.u.) and take the electron to be initially in a $1s$ state.
Information about how the prefactor is set up in this case is provided in
[27]. For other targets and initial states, see, for instance, our
publications [49, 40, 29, 50, 41]. Instead of using the original
implementation of the CQSFA in [27, 31], which solved a purely boundary
problem using as initial guesses orbits with pre-determined dynamics, in this
work we use the hybrid forward-boundary version of the CQSFA (H-CQSFA)
discussed in [43], in which an initial forward ensemble of orbits is launched,
and they are subsequently used as guesses for a boundary problem.

The main focus of this work will be the backscattered trajectories leading to
the ridges. Below we explain how they can be singled out in the CQSFA
framework. The standard classification of trajectories introduced for the
CQSFA [27] uses the product $\Pi_{\perp}=p_{f,\perp}p_{0,\perp}$ of the
initial and final momentum components perpendicular to the driving-field
polarization, and $\Pi_{\parallel}=z_{0}p_{f,\parallel}$ of the tunnel exit
and the final parallel momentum component.

For orbits type 1, $\Pi_{\perp}>0$ and $\Pi_{\parallel}>0$, for orbits type 2,
$\Pi_{\perp}>0$ and $\Pi_{\parallel}<0$, for orbits type 3, $\Pi_{\perp}<0$
and $\Pi_{\parallel}<0$ and for orbit 4, $\Pi_{\perp}<0$ and
$\Pi_{\parallel}>0$. This classification was very appropriate for the original
boundary problem implementation of the CQSFA, as it essentially started
recursively from the standard SFA without rescattering (for details see [28]).
It happens that, for such a restricted manifold of solutions, the
aforementioned classification separates nicely trajectories which have
qualitatively different characters. Orbit 1 reaches the detector without
further interaction, orbits 2 and 3 are field-dressed hyperbole and orbit 4
essentially goes around the core in a slingshot-type motion. However, this
classification is not sufficient if it is used in conjunction with the
solutions found using the forward-boundary implementation of the CQSFA [43].
To rectify this, it is necessary to consider in the classification more
information about the trajectory than merely its endpoints.

In [32], we have shown that a subset of these trajectories compares very
closely with the SFA backscattered trajectories, which are known to lead to
the ridge [25]. One of the parameters used in the classification is the
rescattering time. Structurally, the SFA is a Born-type expansion that will
lead to, along with the ionization time, a series of times where instantaneous
interactions with the core occur: the rescattering times. In the CQSFA, the
only time which is given by the saddle-point equations is the ionization time.
However, the lack of a pre-defined rescattering time does not mean that there
are no trajectories which behave similarly to the backscattered trajectories
of the SFA. Their dynamics are such that after ionization they, travel for
some time under the combined influence of the electric field and the central
potential, return to a close vicinity of the core, change momentum over a very
short time interval and then travel away from the core. We define an analog of
the rescattering time, $t_{\textrm{resc}}$, in the CQSFA to be the time at
which the position of the photoelectron is closest to the core.

In order to choose this subset of CQSFA orbits, we must first select orbits 3
and 4 according to the condition dictated by $\Pi_{\perp}$ and
$\Pi_{\parallel}$. Subsequently, we filter out those that cross the
polarization axis more than once. Finally, one compares the range of
rescattering times to those obtained for the SFA backscattered orbits. For a
more detailed discussion see our previous paper [32].

###  II.3 Focal points and caustics

Next, we will briefly introduce the concept of caustics and focal points and
explain their importance to our studies. Caustics separate different sheets of
solutions and, in their vicinity, large deviations from the expected amplitude
arise. The focal behavior of solutions in separate sheets will be different.
Thus, focal points provide a road map for identifying them in the parameter
space.

The trajectories defined by the saddle point equations Eq. [(5)-(7)] exhibit
focal points at times $t$ such that the Jacobian
$J_{s}(t)={\partial\mathbf{p}_{s}(t)}/{\partial\mathbf{p}_{s}(t^{\prime}_{s})}$
has zero determinant. The Maslov phase $\nu_{s}$, included in Eq. (1),
associated with a specific semiclassical trajectory, is the asymptotic value
of a time-dependent phase factor which takes integer values and changes only
at focal points of the trajectory [51]. Recently, prescriptions for computing
this phase have been provided for semiclassical treatments of strong field
ionization [46, 45]. Additionally, since this formulation of the CQSFA
considers only trajectories confined to the plane containing the polarization
vector and an additional perpendicular vector chosen without loss of
generality due to the cylindrical symmetry of the Hamiltonian, additional
phases may have to be inserted by hand to account for focal points which exist
in the full 3-dimensional ensemble of trajectories [46, 50].

The dependence of the amplitude in Eq. (1) on the negative square root of
$J_{s}(t)$ leads to a divergence at focal points of the trajectory. When a
focal point occurs as $t\rightarrow\infty$, this divergence manifests itself
in the photoelectron momentum distributions of the amplitude
$M(\mathbf{p}_{f})$ calculated from Eq. (1) as bright regions known as
caustics. This is due to the theory only being accurate under the assumption
that solutions of the saddle point equations Eq. [(5)-(7)] are well separated.
This assumption fails, if a focal point occurs at the end of the saddle point
trajectory, due to the coalescence of two or more saddle point solutions. This
can be understood by associating singularities in the momentum mapping
$\mathbf{p}_{0}\rightarrow\mathbf{p}_{f}$ with folds and cusps in the manifold
of saddle point solutions. These folds can occur at classical high-energy
cutoffs analogous to the high-energy cutoffs which are seen in the strong
field approximation (SFA) for the backward-scattered, high-order above
threshold ionized photoelectrons [52, 32]. Additionally, caustics occur in the
low-energy regions and have been associated with low-energy structures
observed in theory [53, 12, 13] and experiment [54, 55].

To gain an intuitive understanding of what exactly is happening at a focal
point of a given semiclassical trajectory, it is necessary to consider the
trajectories in its immediate neighborhood as well. In Fig. 1, the momentum
path of a trajectory containing a focal point is shown. For a selection of
times $t$ during the trajectories’ propagation, the direction of the vectors
$J_{s}(t)\hat{\mathbf{u}}$ and $J_{s}(t)\hat{\mathbf{v}}$ has been illustrated
at the points $\mathbf{p}(t)$ along the momentum path where
$\hat{\mathbf{u}}=[1~{}0]^{\textrm{T}}$ and
$\hat{\mathbf{v}}=[0~{}1]^{\textrm{T}}$, where T indicates the transpose. For
the start time, the Jacobian is equal to the identity matrix so the vectors
define a square area element. For all other times which do not happen to be at
focal points of the trajectory, the vectors $J_{s}(t)\hat{\mathbf{u}}$ and
$J_{s}(t)\hat{\mathbf{v}}$ define a parallelogram and illustrate the linear
deformation of the trajectories in the immediate neighborhood of the one
illustrated in Fig. 1 under the momentum mapping at time $t$. At a focal
point, the dimension of the vector space spanned by the vectors
$J_{s}(t)\hat{\mathbf{u}}$ and $J_{s}(t)\hat{\mathbf{v}}$ is less than $2$.
This means that the collection of trajectories which come from a neighborhood
of a trajectory with a focal point as shown in Fig. 1, are focused down into a
1-dimensional line or to a single point. This picture clarifies why the
existence of a focal point as $t\rightarrow\infty$ leads to a caustic, since
the finite probability associated with trajectories coming from a specific
area in initial momentum is squeezed into a subset of the final momentum which
has zero measure.

Figure 1: Schematic representation of the dynamical behavior around focal
points. Panel (a) contains a momentum space path (yellow solid line) for a
typical backscattered trajectory in a Coulomb potential. The capital letter I
indicates the initial momentum of trajectory at the time of ionization. The
red and black arrows along this path are a schematic representation of the
vectors which define the deformation in time of a square area element,
centered around the initial momentum of the trajectory represented by the
yellow line, as the trajectories are propagated. When these vectors are
linearly dependent and hence the area defined by the two vectors is zero, the
trajectory has a focal point. The two focal points along this example
trajectory have been indicated using a green scatter point. In panel (b), the
focusing of the path around a focal point has been illustrated. An ensemble of
trajectories, displaced in the direction of the eigenvector associated with
the zero eigenvalue of Jacobian at the time the focal point occurs, starting
from the initial momentum of the trajectory displayed in panel (a), have been
plotted (scale adjusted for visibility). At the time of the focal point, the
ensemble of trajectories all focus at a single point in momentum space and
this bunching is illustrated more clearly in the enlarged region shown within
the black dashed line.

##  III Photoelectron momentum distributions and momentum mapping

In this section, we address how specific contributions to photoelectron
momentum distributions (PMDs) relate to caustics and different sheets of
solutions, populating different regions in momentum space. In Fig. 2, we plot
the full photoelectron momentum distributions (PMDs) computed using the
H-CQSFA for three specific values of the softening parameter $\alpha$ (see Eq.
(4)). A small value, $\alpha=10^{-6}$ (Fig. 2(a)), that gives a good
approximation for the exact Coulomb potential, and hence we call it hard-core
Coulomb potential, and the much larger values $\alpha=10^{-4}$ (Fig. 2(b)) and
$10^{-2}$ (Fig. 2(c)) to obtain a soft-core Coulomb potential. Throughout, we
consider ionization times within a single field cycle in order to avoid
prominent above-threshold ionization (ATI) rings arising from inter-cycle
interference. These rings are irrelevant to the present work and have been
studied elsewhere [56, 37, 30, 31, 28, 44].

All panels exhibit the key holographic patterns, such as the fan, the spider,
and the spiral, and a caustic whose apex is around
$(p_{f\parallel},p_{f\perp})=(0,1.3)$. For simplicity, we have considered a
unit cell from $\omega t=0$ to $\omega t=2\pi$. This is an arbitrary choice of
endpoints, which leads to the asymmetries in holographic patterns seen in the
figure. This artifact can be eliminated by setting $\omega t\rightarrow\omega
t+\phi$ therein, with $0\leq\phi\leq 2\pi$, in Eq. (11) and performing an
incoherent sum of the resulting PMDs [50, 41], but this has not been done as
it is not essential to the present discussion. Furthermore, it has practically
no influence in the orbits leading to the ridges.

Moreover, one may identify at least three ridges, which intersect at
$p_{f\parallel}=0$, and annular-like interference fringes that follow them.
The ridges are classical features related to the electron’s maximal energy
upon reaching the detector, with the outermost ridge corresponding to the
shortest pair of backscattered orbits. This specific pair gives the kinetic
energy of approximately $10U_{p}$, which is the well-known rescattered ATI
cutoff. At least two other ridges at lower energies, associated with longer
returns, are also present. In the CQSFA framework, these ridges have been
identified in [43] and directly compared with the SFA in [32].

A noteworthy feature is that, depending on the potential softening, the ridges
may or may not close. Indeed, the minimal angle, with regard to the
$p_{f\parallel}$ axis, will vary according to this parameter. For a hard-core
potential [Fig. 2(a)], the rescattering ridges will close, and this angle will
be approximately $0^{\circ}$, while for the soft-core potentials [Figs. 2(b)
and (c)], they are interrupted by caustics at ridge-specific angles with
regard to the polarization axis. These angles increase with the softening
parameter. For clarity, in Table 1 we provide the electron’s excursion times
in the continuum and the ridges’ energies for the shortest three pairs of
returning backscattered trajectories, calculated for the parameters of Fig. 2.
This information will facilitate the subsequent discussion.

Pair |  $\mathbf{Re}[t_{\mathrm{resc}}-t^{\prime}$] | $E_{ridge}$
---|---|---
1st shortest | 0.72$T$ | 10.9$U_{p}$
2nd shortest | 1.25$T$ | 6.8$U_{p}$
3rd shortest | 1.77$T$ | 9.8$U_{p}$
Table 1: Classical excursion times and approximate maximum kinetic energies
$E_{ridge}$ at the rescattering ridges for the three shortest pairs of
backscattered electron orbits, calculated for the parameters in Fig. 2.
According to the H-CQSFA classification based on $\Pi_{\parallel}$ and
$\Pi_{\perp}$, the first and third shortest pairs are type 4 orbits, and the
second shortest pair are type 3 orbits. The rescattered times in the H-CQSFA
have been determined according to our previous publication [32].  Figure 2:
Photoelectron momentum distributions for atomic Hydrogen ($I_{p}=0.5$ a.u.) in
the presence of an $800$ nm wavelength, $1.5\times
10^{14}~{}\textrm{Wcm}^{-2}$ intensity, linearly polarized and monochromatic
field, computed using the H-CQSFA, as functions of the final momentum
components $p_{f\parallel}$ and $p_{f\perp}$ parallel and perpendicular to the
laser-field polarization. Panels (a), (b) and (c) correspond to Coulomb
softening parameters of $\alpha=1\times 10^{-6}$, $\alpha=1\times 10^{-4}$ and
$\alpha=1\times 10^{-2}$, respectively, given in red on the right top corner
of the figure.

Next, we will focus on the three specific pairs of orbits leading to the
rescattering ridges observed in Fig. 2. In the CQSFA framework, these are
orbits type 3 and 4 which have been selected according to the additional
criteria in Sec. II.2 to single out backscattered orbits [32]. They lead not
only to the ridges but also to ring-shaped patterns following them. These
rings stem from the quantum interference of the short and the long orbit in
each pair.

In Fig. 3(a), we show the PMD constructed using only the first shortest
backscattered orbit pair for the hard-core Coulomb potential. This leads to
the primary rescattering ridge that goes up to over $10U_{p}$ and extends up
to the laser polarization axis, as indicated by the red dotted arrow in the
figure. We have omitted the PMD for the next two returns as they will present
similar behavior. For the soft-core Coulomb potential shown in Fig 2(c), we
isolate the contributions from the three shortest pairs of orbits in Figs.
3(b)-(d). We observe that the ridges seem to fold around a minimal
rescattering angle $\theta^{(-2)}_{i}$, as represented by the red arrows in
the figure. Both the ridges’ energies and the folding angles vary according to
the specific orbit pair, as indicated by the subindex $i$ in the angle. The
superscript indicates the exponent of the softening parameter. Furthermore,
the annular interference patterns caused by the interference within specific
backscattered orbit pairs have been singled out and are now very clear.
Nonetheless, there is a key difference: while for a hard-core potential [Fig.
3(a)] the fringes follow the ridge up to the polarization axis, for a soft-
core potential [Fig. 3(b)-(d)] interference patterns are only present up to
the region for which the ridges start to fold. This is a consequence of both
interfering orbits only extending up to a specific angle.

Figure 3: Photoelectron momentum distributions as functions of the final
momentum components $p_{f\parallel}$ and $p_{f\perp}$ parallel and
perpendicular to the laser-field polarization, produced for the same atomic
and field parameters as Fig. 2 using the H-CQSFA, but considering only the
specific contributions of the different pairs of backscattering trajectories.
In Panel (a), we employ the same softening as in Fig. 2(a), and in Panel (b)
we use the same softening as in Fig. 2(c). Panels (b), (c), and (d) have been
calculated for the first, second, and third shortest pair of backscattered
orbits as given in Table 1. The white arrows indicate the minimal scattering
angles $\theta^{(i)}_{j}$ observed directly, while the red arrows indicate
those obtained from the estimates in Sec. IV. Thereby, the superscript gives
the exponent of the softening parameter, while the subscript denotes the orbit
pair according to Table 1.

The PMDs in Fig. 3 can be better understood by looking at the final-to-initial
momentum mapping. This mapping relates the final momentum region occupied by
the electron at the detector to specific regions in momentum space from which
the orbits are launched. Momentum maps have been used by us in [43] to
identify different types of electron orbits and their signatures. Here, we
will employ them as a tool to determine momentum constraints, as the different
ridges for hard- and soft-core potentials are classical features.

For simplicity, in the figures that follow, we will focus on the orbits
leading to the primary rescattering ridge whose energy is around $10U_{p}$,
but the arguments employed in the subsequent analysis are general and can be
applied to the remaining pairs of orbits. Furthermore, we have considered the
initial momenta in the upper half-plane. For the orbits associated with the
primary rescattering ridge, this leads to final momenta in the lower half
plane, of which we just show a single quadrant (see upper panels in Figs. 4
and 6). In Fig. 3, we have used the upper half plane for the final momenta to
construct the PMD, which means that the initial momenta arein the lower half-
plane. However, this does not alter the subsequent conclusions as the PMDs are
symmetric with regard to a reflection upon the $p_{f\parallel}$ axis. For the
specific case of a monochromatic field, we also expect a reflection symmetry
with regard to the $p_{f\perp}$ axis for the momentum mapping. Therefore, we
have only included ionization times in a single half-cycle without loss of
generality.

Fig. 4 displays this mapping for the hard-core potential. Because the mapping
is multivalued, it is useful to separate the different types of orbits leading
to the ridge. It is also useful to initiate the discussion from the final
momenta, as the region they occupy can be traced more readily to the PMDs. The
final momenta associated with the two types of interfering orbits, short and
long, are given by the yellow [Fig. 4(a)(i)] and black [Fig. 4(a)(ii)] scatter
points in the upper panels of the figure, respectively. An electron following
the long orbit is freed shortly after the maximum of the field and returns
after a field zero crossing, while an electron along the short orbit is
released comparatively later and returns before the zero crossing. For the
hard-core potential, this pair of orbits coalesce at the rescattering ridge,
as indicated by the dashed lines at the edges of the scatter plots. Because,
in the H-CQSFA framework, they correspond to two types of orbit 4, we will
refer to them as Orbits $4^{(h)}_{s}$ and $4^{(h)}_{l}$, respectively, where
the superscript $(h)$ indicates a hard-core potential and the subscripts $l,s$
specify the type of orbit. The short and long trajectories are separated by a
caustic in the momentum space.

We have observed that, for this particular pair of orbits and potential, the
short (long) orbits have an odd (even) number of focal points, each of which
occurs at a time when the Jacobian is vanishing. Not all focal points will
cause relevant dynamical changes in the system. Some focal points occur in
pairs, so that the Maslov phases associated with them will cancel each other.
They are not directly related to rescattering. However, the remaining focal
points will appear at the same time as a spike in the Jacobian, which is
associated with rescatering and is important for the dynamics. Both orbits
undergo one act of rescattering, and exhibit a single spike in the Jacobian.
Physically, a spike indicates sudden changes in the electron momentum. A
summary of these findings is provided in Table 2.

The corresponding initial momentum maps are given in Fig. 4(b), where the same
color code as for the final momenta was used. The overall momenta for the long
orbit is much smaller, which is consistent with ionization times being closer
to the peak of the field. Moreover, the density of points is highest close to
the polarization axis and is lowest near and at the boundary between the black
and yellow regions. The high density is in line with the expected dynamics for
a linearly polarized field, as ionization will happen predominantly close to
the axis. Nonetheless, these regions are distorted by the presence of the
Coulomb potential.

In summary, for a hard-core potential, there is a fold in the initial to final
momentum mapping. The two different sheets of solutions obtained for the final
momentum are stitched together via the rescattering ridge and go all the way
to the polarization axis. In Fig. 5(a), we present a three-dimensional
representation of this mapping. The figure shows that the yellow and the black
branches determined by the short and long orbits fold around the caustic
associated with the cutoff, in agreement with the dashed lines in Fig. 4(a).
The two branches on the surface reflect the fact that two returning
backscattered trajectories lead to the same energy.

Figure 4: Final [panels (a)] and initial [panel (b)] momentum grid points that
are reached by trajectories in the two branches of solutions which contribute
to the rescattering ridge for a softening parameter $\alpha=1\times 10^{-6}$
and all other parameters as described in Fig. 2, shown by yellow (short orbit)
and black (long orbit) scatter points. The initial momentum of trajectories
for the field parameters described in Fig. 2 which have softening parameter
$\alpha=1\times 10^{-6}$ and final momenta as shown in panel (a)(i) and
(a)(ii) are displayed in panel (b). The dashed lines in the upper panels of
the figure indicate the rescattering ridge, for which the long and short orbit
coalesce. The types of orbits resulting in the final momenta in panel (a)(i)
and (a)(ii) are indicated by the red labels in the lower right corners.

The remaining panels in Fig. 5, obtained with the soft-core potential employed
in Fig. 3(c), show that softening adds complexity to the problem, and that two
additional solutions have emerged. Indeed, Figs. 5(b) to (d) depict four
distinct surfaces, which fold at constraints defined either by the maximal
rescattering angle or by the backscattering ridges. The four surfaces
intercept at a single point. Similarly to the hard-core potential case, we
will employ projections of this surface to better understand the electron’s
dynamics and, whenever possible, link the projections to the actual surface.
These projections are given by the momentum maps in Fig. 6.

For the soft-core potential, the number of focal points along a trajectory is
now insufficient to completely classify the solutions. As specified in Table
2, now there are short orbits with an even number of focal points, and long
orbits with an odd number of focal points. Two of these solutions, displayed
in black and red in Fig. 6, are long trajectories, but the number of focal
points associated with them has different parity. Both types of orbits are
being focused by the potential, but the trajectories starting in the red
region are subsequently being defocused due to the softening. The solutions
displayed in yellow and green are classified as short orbits, and behave
similarly, with the trajectories starting in the green region undergoing
subsequent defocusing. For the red and green solutions, there exist two spikes
in the Jacobian, while a single spike exists for the remaining solutions. A
spike in the Jacobian happens when the rate of change in the momentum is high,
and this can be associated with rescattering.

In the H-CQSFA framework, the soft-core solutions correspond to four types of
orbit 4. Therefore, similarly to what has been done for the hard-core case, we
will refer to them as Orbits $4^{(s)}_{s}$,$4^{(s)}_{s^{\prime}}$,
$4^{(s)}_{l}$ and $4^{(s)}_{l^{\prime}}$, respectively, where the superscript
$(s)$ denotes a soft-core potential and the subscripts
$l,l^{\prime},s,s^{\prime}$ specify the type of orbit (long or short). The
primes have been introduced to refer to the two additional solutions that
exist in the soft-core case. For clarity, a summary of the types of orbits 4
investigated in this work is also given in Table 2.

Softening | Orbit | Focal points | Spikes (Jacobian)
---|---|---|---
$\alpha=10^{-6}$ | $4^{(h)}_{s}$ | odd | 1
$4^{(h)}_{l}$ | even | 1
$\alpha=10^{-2}$ | $4^{(s)}_{s}$ | odd | 1
$4^{(s)}_{l}$ | even | 1
$4^{(s)}_{s^{\prime}}$ | even | 2
$4^{(s)}_{l^{\prime}}$ | odd | 2
Table 2: Orbit classification for the first shortest pair of H-CQSFA
backscattered orbits (second column), computed using residual binding
potentials of different softening (first column), together with the number of
focal points and spikes in the Jacobian (third and fourth column,
respectively). The potential with $\alpha=10^{-6}$ behaves practically like a
hard-core Coulomb potential, while for $\alpha=10^{-2}$ the dynamics
associated with the soft-core potential prevail. The classification
$4^{(i)}_{j}$, with $i=h,s$ and $j=s,l,s^{\prime},l^{\prime}$ indicates that
we are dealing with H-CQSFA orbits associated with potentials of different
softening (hard- or soft-core) and length (long or short).

Figs. 6(a)(i)-(iv) show the final momentum maps associated with each of these
solutions. A noteworthy feature is that there are no solutions close to the
polarization axis. This confirms that the behavior observed for the PMD in
Fig. 3 is due to a change in kinematic constraints. The figure also shows
that, for the soft-core case, different sheets of solutions obtained for the
final momenta merge in several places, not only the rescattering ridge. The
boundaries at which different solutions are glued together are represented by
the dashed lines in each panel, and the color indicates to which solution it
is connected. For instance, in Fig. 6(a)(i), we see that the solution
$4^{(s)}_{s}$, whose final momenta are represented by the yellow scatter plot,
still merges at the ridge with the long orbit $4^{(s)}_{l}$, whose final
momentum mapping is given by the black scatter plot in Fig. 6(a)(ii). This
behavior resembles that of their hard-core counterparts. However, in addition
to that, $4^{(s)}_{s}$ is also connected to the solution
$4^{(s)}_{s^{\prime}}$ [Fig. 6(a)(iii)] as the green dashed line indicates.
The solution $4^{(s)}_{s^{\prime}}$, on its turn, is also connected to the
$4^{(s)}_{l^{\prime}}$ solution [shown in Fig. 6(a)(iv)] via the rescattering
ridge. Solutions $4^{(s)}_{l^{\prime}}$ and $4^{(s)}_{l}$ coalesce at the
boundary defined near the minimal rescattering angle. All four solutions merge
at a single point, where the ridges separated by half a cycle intersect. This
point also marks the apex of a caustic associated with type 3 orbit [31, 43].

Figure 5: Three dimensional representation of the initial perpendicular
momentum map as a function of the electron’s final momentum components,
calculated for the same parameters as in the previous figures, and a softening
of $\alpha=1\times 10^{-6}$ [panel (a)] and $\alpha=1\times 10^{-2}$
[panels(b), (c) and (d)]. The colors of the surfaces match those used in the
two-dimensional projections in Figs. 4 and 6, and the arrows indicate the
merging of two or more sheets.

This merging is better seen in the full three-dimensional surface, displayed
in Figs. 5(b) to (d). The arrows in Fig. 5(b), indicate the coalescence of two
solutions along the fold introduced by the softening. The left arrow shows the
merging of the red and black branches of solutions, closest to the
$p_{f,\parallel}$ axes, while the right arrow indicates the fusion of the
yellow and green branches. The coalescence of these sheets is determined by
the minimal rescattering angle. In Fig. 5(c), we emphasize the merging of two
sheets of solutions along the rescattering ridges: the red and green sheets,
in agreement with Figs. Fig. 6(a)(iii) and (iv) [lower arrow], as well as the
yellow and black solution in agreement with Figs. 6(a)(i) and (ii)[upper
arrow]. Finally, the arrow in Fig. 5(d) shows the point at which all four
sheets merge.

Figure 6: Final [panels(a)] and initial [panel (b)] momentum grid points
reached by trajectories in the four branches of solutions which contribute to
the rescattering ridge for a softened potential ($\alpha=1\times 10^{-2}$) and
all other parameters as described in Fig. 2. The momentum regions associated
with the trajectories $4_{s}^{(s)}$ $4_{s^{\prime}}^{(s)}$, $4_{l}^{(s)}$,
$4_{l^{\prime}}^{(s)}$ are shown by yellow, green (short), black and red
(long) scatter points, respectively. In panels (a) the red labels in the top
right corners indicate the orbit types. The dashed lines at the boundaries of
the scatter graphs indicate the merging of two or more sheets of solutions,
and the colors indicate with which sheet the merging occur. In panel (a)(i),
the upper (lower) dashed lines indicate a merging with the solution
$4_{s^{\prime}}^{(s)}$ ($4_{l}^{(s)}$). In panel (a)(ii) the upper (lower)
dashed lines indicates a merging with the solution $4_{l^{\prime}}^{(s)}$
($4_{s}^{(s)}$). In panel (a)(iii) the upper (lower) dashed lines indicate a
merging with the solution $4_{s}^{(s)}$ ($4_{l^{\prime}}l^{(s)}$). In panel
(a)(iv) the upper (lower) dashed lines indicate a merging with the solution
$4_{l}^{(s)}$ ($4_{s^{\prime}}l^{(s)}$). Panel (b) displays the initial
momenta corresponding to the four sheets of solutions, with the additional
solutions occupying the regions of very small perpendicular momenta
$p_{0\perp}$. Both yellow and red points correspond to trajectories with an
odd number of focal points while the black and green points correspond to
trajectories with an even number of focal points.

The initial momentum map, shown in Fig. 6(b) for the soft-core potential, is
practically identical to its hard-core counterpart in the regions of large
parallel momenta. However, close to the laser-polarization axis, there are
significant differences, with the appearance of two more branches. This is
expected from the discussion of Fig. 6(a). An interesting feature is that the
red and green regions are much more localized in the
$p_{0\parallel}p_{0\perp}$ plane than the black and yellow ones and approach
them from below. This suggests a much stronger interaction with the core, with
lower perpendicular momentum components. As the softening is increased, the
region occupied by the green and red solutions widens. Furthermore, in
contrast to what happens for the hard-core potential, they never touch the
$p_{0\parallel}$ axis. These are precisely the momentum regions that would
close the ridge in Figs. 6(a)(i)-(iv) and Fig. 3(b).

##  IV  Trajectory analysis and rescattering angles

Next, we will discuss why the softening of the potential influences the
rescattering angle. We will start by looking at the electron trajectories in
position and momentum space for the soft-core and the hard-core Coulomb
potentials. These trajectories are displayed in Figs. 7(a) to (c), together
with the Jacobian associated with each solution [Fig. 7(d)]. Although the
hard-core potential exhibits only two branches of solutions, whose initial
conditions are indicated in Figs. 4 and 6 by black and yellow scatter plots,
in order to facilitate a comparison with the soft-core potential, in the
figure we have slightly changed this convention and employed the same colors
for orbits with the same initial conditions.

Figure 7: Position (panel (a)) and momentum (panel (b)) space trajectories and
the inverse arcsinh of the Jacobian determinant along these trajectories
(panel(d)) are plotted for a softened potential with $\alpha=1\times 10^{-2}$
(hard potential with $\alpha=1\times 10^{-6}$) using solid (dashed) lines with
field parameters as described in Fig. 2. For the softened potential, the
colors of the trajectories correspond to the color of the scatter point in
Fig. 6 representing the boundary values of the trajectories momentum and the
final momentum of each trajectory shown for the softened potential is
$(p_{f\parallel},p_{f\perp})=(1,-1.2)$ atomic units. The initial momenta for
the trajectories shown with the yellow, black, green and red lines are
$(p_{0\parallel},p_{0\perp})=(-0.3949,0.01005)$,
$(p_{0\parallel},p_{0\perp})=(0.09192,0.008766)$,
$(p_{0\parallel},p_{0\perp})=(-0.3903,0.003019)$ and
$(p_{0\parallel},p_{0\perp})=(0.09549,0.001135)$ atomic units, respectively
for both the softened and hard potential. In panel (c), a contour plot of the
difference between the soft and Coulomb potential (labeled as the Coulomb
deviation) is shown with a more detailed view of some of the position space
paths shown in panel (a) for which the same color and line-style scheme has
been used. The parameters of the field are the same as those used in Fig. 2.

Fig. 7(a) shows that the trajectories corresponding to the green and the red
branches of solutions interact more strongly with the core than those given by
the black and yellow curves. For the soft-core potential, this behavior is
easily understood in conjunction with the initial-momentum mapping in Fig.
6(b), which indicates that they are released in the continuum with much lower
transverse momenta. Therefore, we expect that they will be more influenced by
the softening, while the remaining orbits will be more affected by the
potential tail.

For the hard-core potential, the trajectories shown have the same initial
momentum as the trajectories in the softened potential, but the final momenta
are all different. The differences between the $4_{s}^{(h)}$ ($4_{l}^{(h)}$)
and the $4_{s}^{(s)}$ ($4_{l}^{(s)}$) trajectories with the same initial
momenta are small but there is a significant difference for the $4_{s}^{(h)}$
($4_{l}^{(h)}$) and the $4_{s^{\prime}}^{(s)}$ ($4_{l^{\prime}}^{(s)}$)
trajectories. This is expected as the trajectories associated with the black
and yellow branches in Fig. 6 do not deviate strongly from their Coulomb-type
counterparts. For the softened potential, although the position space
trajectories in Fig. 7(a) begin on very different paths, eventually the red
(green) trajectory coincides precisely with the black (yellow) trajectory. The
pairwise convergence of trajectories in different branches to the same
asymptotic coordinates is a consequence of their long-distance behavior only
having two possible branches of solutions, as it is determined by the Coulomb
tail.

The non-Coulomb influence is also visible in momentum space, displayed in Fig.
7(b). The black and the yellow soft-core trajectories are nearly circular, and
behave like their Coulomb counterparts. In contrast, the green and red
trajectories are elliptical for the soft-core potential and nearly circular
for the Coulomb potential. The circular shape observed for the Coulomb orbits
stems from the conserved quantity associated with the dynamical symmetry of
the Coulomb potential, which is the Laplace-Runge-Lenz (LRL) vector. In the
pure Coulomb dynamics, LRL vector conservation implies that paths in momentum
space are circular and they have a radius given by $1/L$, where $L$ is the
angular momentum, and a center defined by the LRL vector [57, 58, 59]. Because
rescattering occurs around a field zero crossing, this symmetry is
approximately observed for scenarios in which the Coulomb-type dynamics are
prevalent. This renders the momentum-space orbit approximately circular and
happens if, during the rescattering process, the trajectory remains at a
distance from the core greater than the length scale comparable with the
softening parameter $\alpha$. Examples are the yellow and black orbits, which
remain very similar and nearly circular for both potentials. However, this
symmetry is unambiguously broken once the non-Coulomb dynamics determined by
the softening becomes more important. This is observed for the green and red
orbits, which reach closer to the core than $\alpha$, and whose momentum paths
vary dramatically from being circular.

Although the difference between the momentum paths for the hard- and soft-core
potential is very visible, the difference between the position paths is less
obvious as they differ on a length scale much shorter than the electron
excursion amplitude 111The excursion amplitude is given by
$\sqrt{2U_{p}}/\omega$, and, for the parameters used in this work is around
14.2 a.u.. To show explicitly these differences, in Fig. 7(c), the position-
space paths close to the core, for the two new types of trajectory that arise
for a softened potential, are displayed with solid lines.

Fig. 7(c) shows that the green and red trajectories behave in very distinct
ways, depending on whether the potential is hard- or soft-core. The
trajectories associated with the Coulomb potential, illustrated by dashed
lines, are hyperbolic and undergo a much stronger deflection than those
related to the soft-core potential. The difference between the Coulomb and the
soft-core forces, indicated by the shaded area in the figure, is also
significant. This is in agreement with the maximal angle obtained for the
momentum maps, and also explains the different asymptotic behavior for the
hard- and soft-core potential. In contrast, the yellow and black orbits are
very close in asymptotic momentum.

Finally, in Fig. 7(d) we present the Jacobian determinant as a function of
time for specific orbits around the instance of rescattering. This is a very
short time interval but it contains a large amount of the electrons’ dynamics.
For trajectories that pass close enough to the core to be altered
significantly by the absence of the Coulomb singularity (green and red
orbits), there is a large spike in $\textrm{det}(J)$ towards $-\infty$
followed by a spike towards $+\infty$. Conversely, for trajectories that do
not pass as close to the core, so that they are similar to those of electrons
in an actual Coulomb potential (black and yellow orbits), there is just a
single large spike in $\textrm{det}(J)$ towards $-\infty$.

Figure 8: Field free trajectories with impact parameter represented by the
colour of the line are shown for the hard (softened) potential in panel (a)
(panel(b)). The black arrows indicate the direction of flow of the trajectory.
In panel (c), the scattering angle $\phi$ is shown for a range of values of
the impact parameter for the softened (hard) [Coulomb] potential with a solid
red (dashed orange) [dot-dashed black] line. In Panel (d), the minimum value
of the angle $\theta$ across all values of the impact parameter, is plotted
against the energy of Eq. (14) for three distinct values of the softening
parameter $a$. The vertical dashed black lines indicate the energy which
corresponds to the rescattering ridge associated with the 1st, 2nd and 3rd act
of rescattering. The red circles, blue squares and orange triangles indicate
the observed minimum angle at each of these rescattering ridges for a
softening parameter of $1\times 10^{-2}$, $1\times 10^{-4}$ and $1\times
10^{-6}$ respectively.

The role of the softening in the potential can be further explored using
scattering theory. Figs. 8(a) and (b) provide a schematic representation of a
scattering process by a hard-core ($\alpha=10^{-6}$) and a soft-core
($\alpha=10^{-2}$) Coulomb potential. The figures show that, for a hard-core
Coulomb potential, an incoming electron can be rescattered up to an angle
close to $\pi$, while, for a soft-core potential, the particle will not feel
the potential and will pass through it for angles larger than a certain
critical value. Explicit expressions for the scattering angles of a Coulomb
and soft-core potential have been provided in [60] for the field-free case. A
schematic representation of the scattering process, as well as the relevant
parameters, is shown in Fig. 9. In the center of mass frame, the scattering
angle $\phi$ can be expressed as,

$\phi(b)=\displaystyle\int_{r_{min}}^{\infty}\frac{(b/r^{2})dr}{\sqrt{1-(b/r)^{2}+1/(k_{e}\sqrt{r^{2}+\alpha^{2}})}}$
(12)

where $r_{min}$ is the largest root of the denominator, $b$ is the impact
parameter, $k_{e}$ is the kinetic energy of the electron before the collision
and $\alpha$ is the softening parameter as defined in Eq. (4). As shown in
[60], for a pure Coulomb interaction, it is possible to determine the value of
the angle analytically, and the electron reaches a maximum scattering angle of
$\pi$ when the impact parameter goes to zero. When a softening parameter is
included, we need to perform numerical estimates of the angle as a function of
the different parameters.

To evaluate the angle numerically, we start by first finding the roots of the
denominator in Eq. (12) to determine $r_{min}$ and then evaluate the integral
numerically. We need to introduce a few approximations to apply the scattering
theory to our particular problem. We consider that the perpendicular momentum
component remains constant from ionization until the rescattering event,
$p_{0\perp}\approx p_{\perp}(t_{\mathrm{resc}})$, where the rescattering time
has been determined as in [32]. Therefore, the impact parameter is
approximated as a linear function of the initial transverse momentum and the
travel time $\Delta t=\mathrm{Re}[t_{\mathrm{resc}}-t^{\prime}]$. This gives

$b\propto p_{0\perp}\Delta t.$ (13)

To approximate the initial kinetic energy, we need to consider the vector
potential contribution in the final parallel momentum. Considering that
rescattering mainly occurs at the maximum of the vector potential (minimum of
the electric field), we will approach the vector potential at the time of
rescattering by its maximum value, $A_{0}=2\sqrt{U_{p}}$. For each ridge,
whose maximum energy is given by $E_{ridge}$ according to Table 1, we
approximate the initial kinetic energy required to evaluate the angle in Eq.
(12) using

$k_{e}=\frac{(\sqrt{2E_{ridge}}-A_{0})^{2}}{2}.$ (14)

In Fig. 8(c), we plot the scattering angle as a function of the impact
parameter for initial kinetic energy corresponding to the $10.9U_{p}$ ridge.
For both the hard-core and the soft-core Coulomb potential, the angle exhibits
a maximum and then decays as the impact parameter approaches zero. The maximum
is sharper and closer to $\pi$ for the hard-core Coulomb potential. As the
softening increases, the maximum occurs at a larger impact parameter, and as
the impact parameter is reduced further, the scattering angle decreases. In
the limit of vanishing softening, as we approach the exact Coulomb potential,
the angle decreases monotonously as the impact parameter increases, reaching a
maximum value of $\pi$ as the impact parameter approaches zero [shown with the
dotted brown line in Fig. 8(c)]. This sheds light on the behavior observed in
the upper panels of the figure. For the soft-core potential, as the impact
parameter approaches zero, the trajectories pass by without feeling the
potential.

In Fig. 8(d), the dots represent the maximum angle $\theta$ defined by the red
arrows in Fig. 2 and represented in the schematic model in Fig. 9, for the
three shortest pairs of backscattered trajectories as a function of the
initial kinetic energy for each value of the softening parameter. For the
hard-core Coulomb potential, we have set the value to zero. With lines, we
plot the results obtained from our scattering model, where $\theta$ can be
approximately defined in terms of the scattering angle $\phi$, the peak of the
vector potential $\mathbf{A}_{0}$ and the scattering kinetic energy $k_{e}$ as

$\theta=\tan^{-1}\left[\frac{\sqrt{2k_{e}}\sin{(2\pi-2\phi)}}{\sqrt{2k_{e}}\cos{(2\pi-2\phi)}+A_{0}}\right].$
(15)

The predictions of the scattering model follow the same trends as the angle
estimated from the PMDs in Fig. 2, although the agreement is better for the
first return leading to the high-energy ridge. Intuitively this could be
explained if we consider that the trajectories contributing to the first
return are closer to the scattering model, while for later returns the
influence of the residual potential for a longer time prior to rescattering
precludes determining a clear $t_{\mathrm{resc}}$. This issue also implies
that Born-type theories such as the SFA exhibit a worse agreement with fully
Coulomb-distorted approaches for longer orbits [32]. The values obtained agree
reasonably well with the white arrows in Fig. 2($c^{\prime}$).

Figure 9: A schematic showing the simplified model used to understand the
behavior of trajectories in the presence of a softened potential. It shows the
process of converting from the scattering angle $\phi$ computed from the
field-free spherically symmetric scattering problem, to the angle $\theta$
which we observe in our photoelectron momentum distributions. The inset panel
shows the type of shift which occurs to the rescattering ridges by
approximately $\mathbf{A}_{0}$ due to the field dressing of the scattered
electron.

##  V Conclusions

In this work, we employ the hybrid version of the Coulomb Quantum Orbit
Strong-Field Approximation (H-CQSFA) [43, 61] to assess how different binding
potentials influence the rescattering ridge in strong-field ionization. The
ridge is a well-known structure that occurs in the context of strong-field
ionization, which, in the existing literature, is interpreted as a kinematic
constraint associated with backscattered electron trajectories [23, 25].
Because the H-CQSFA is a fully Coulomb-distorted trajectory-based approach
that considers the binding potential and the external field on equal footing,
the full dynamics are incorporated. Therefore, the time interval and spatial
ranges for which rescattering occurs may or may not be well defined. This is
in stark contrast to Born-type approaches such as the Strong-Field
Approximation (SFA), for which rescattering is temporally and spatially
confined. The backscattered trajectories are singled out from the manifold of
existing trajectories by following the prescription in [32].

A key result of the present paper is that the maximal scattering angle is
strongly influenced by the potential softening. While, for a hard-core Coulomb
potential, the maximum rescattering angle is $\pi$ and the ridges close, for a
soft-core potential the maximum rescattering angle is smaller and the ridges
fold. This angle differs for specific pairs or orbits, and softening
parameters, which suggests that the rescattering ridge is not purely
kinematic. These findings are interpreted using catastrophe theory, which
shows that the long and short backscattered orbits within a pair, as functions
of the electron’s initial conditions, form a surface that folds where the two
orbits coalesce, that is, at the ridge. Introducing a softening parameter
creates two additional sets of solutions, which are qualitatively different.
These solutions never touch the polarization axis, but have relatively low
initial transverse momentum components. This implies that they will interact
strongly with the core.

Their appearance can be explained by the fact that the softening breaks a
hidden dynamical symmetry of the Coulomb potential, which is not present for
the soft-core potential. The Coulomb potential is not merely spherically
symmetric but is hyperspherically symmetric. This means that it is a basis of
the special orthogonal group in four dimensions, $SO(4)$, while the soft-core
potential has the expected $SO(3)$ symmetry [57, 58, 59]. The conserved
quantity associated with this additional symmetry is the Laplace-Runge-Lenz
(LRL) vector.

These symmetry arguments are exact in the field-free case, and, for a field-
dressed scenario, approximately hold during rescattering. This can be
intuitively understood by the fact that rescattering occurs near a field zero
crossing, so that the dynamics are dominated by the Coulomb field. At a zero
crossing, the laser-driving forces will be close to zero, while the vector
potential will be at its maximum. Therefore, it is reasonable to assume that
the influence of the laser field will be merely to shift the momentum
constraints and dress the Coulomb interaction via the vector potential. This
premise has been widely employed to construct approaches such as the Coulomb-
Volkov Approximation [62, 63, 64, 65, 66, 67, 68], or the Quantitative
Rescattering Theory [69, 70], which dress Coulomb-scattering waves and neglect
the residual binding potential when computing the electron orbits in the
continuum, or to include Coulomb effects in electron-electron interaction [21,
71].

Signatures of the Coulomb hidden symmetry are circular orbits in momentum
space, which are caused by LRL vector conservation. If the hard-core Coulomb
potential is taken, this holds for all orbits, while for the soft-core
potential, this only holds for orbits whose perihelium is in the region for
which the non-Coulomb effects are negligible. This will not hold for two
branches of solutions with very small initial transverse momenta. These
solutions are strongly interacting with the core, and very sensitive to non-
Coulomb effects. The non-Coulomb orbits exhibit elliptical shapes in momentum
space. In position space, they are similar to their Coulomb counterparts but
are much less deflected upon return. For those orbits, we have also identified
an additional spike in the Jacobian determinant, which is absent for their
Coulomb counterparts. The spikes in $\textrm{det}(J)$ are caused by the very
rapid change in momentum at the instant of rescattering meaning that
neighboring trajectories that lag behind or rush ahead slightly will have a
relatively large separation at a given time during rescattering (even though
they follow extremely similar paths).

Furthermore, analytical estimates of the scattering angles show that, while
Coulomb-type orbits behave like hyperbolae and can be backscattered up to an
angle $\pi$, orbits scattered by a soft-core potential will be scattered up to
a maximal angle. Beyond that, they will just pass through. The outcome of the
estimates follows the same trend as the angles read directly from the PMDs,
although the agreement is only good for the primary rescattering ridge. This
possibly occurs because, in this case, the joint influence of the field and
the potential is harder to disentangle before and during rescattering. This
assumption is supported by previous results, which show a worse agreement
between the SFA and H-CQSFA for longer orbits or lower photoelectron energies
[32].

Finally, our results show that, in the soft-core case, the surface defined by
the minimal angle in the soft-core case acts as some kind of cutoff, although
it depends on the interplay between the binding potential and the laser field.
This indicates once more that, in a Coulomb-distorted scenario, it is hard to
distinguish between dynamic and kinematic constraints. The present work
revisits a well-known feature in laser-induced rescattering and brings a fresh
perspective on the limitations of soft-core models. This is important due to
the widespread use of soft-core potentials in the modeling of strong-field
phenomena, and their resulting from laser-induced rescattering.

###### Acknowledgements.

We thank A.S. Maxwell and B.B. Augstein for useful discussions. This work was
funded by grants No. EP/J019143/1 and EP/T517793/1, from the UK Engineering
and Physical Sciences Research Council (EPSRC).

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