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Low-energy electron rescattering in laser-induced ionization
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2014 J. Phys. B: At. Mol. Opt. Phys. 47 204022
(http://iopscience.iop.org/0953-4075/47/20/204022)
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Journal of Physics B: Atomic, Molecular and Optical Physics
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204022 (13pp)
doi:10.1088/0953-4075/47/20/204022
Low-energy electron rescattering in laserinduced ionization
W Becker1, S P Goreslavski2, D B Milošević1,3,4 and G G Paulus5,6
1
Max-Born-Institut, Max-Born-Str. 2a, D-12489 Berlin, Germany
National Research Nuclear University MEPhi, Kashirskoe Shosse 31, 115409 Moscow, Russia
3
Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Hercegovina
4
Academy of Science and Arts of Bosnia and Herzegovina, Bistrik 7, 71000 Sarajevo, Bosnia and
Herzegovina
5
Institute of Optics and Quantum Electronics, Friedrich Schiller University, Max-Wien-Platz 1, D-07743
Jena, Germany
6
Helmholtz Institute Jena, Fröbelstieg 3, D-07743 Jena, Germany
2
E-mail: [email protected], [email protected], [email protected] and [email protected]
Received 15 April 2014, revised 4 July 2014
Accepted for publication 14 August 2014
Published 8 October 2014
Abstract
The low-energy structure (LES) in the energy spectrum of above-threshold ionization of rare-gas
atoms is reinvestigated from three different points of view. First, the role of forward rescattering
in the completely classical simple-man model (SMM) is considered. Then, the corresponding
classical electronic trajectories are retrieved in the quantum-mechanical ionization amplitude
derived in the strong-field approximation augmented to allow for rescattering. Third, classical
trajectories in the presence of both the laser field and the Coulomb field are scrutinized in order
to see how they are related to the LES. It is concluded that the LES is already rooted in the
SMM. The Coulomb field enhances the structure so that it can successfully compete with other
contributions and become visible in the total spectrum.
Keywords: above-threshold ionization, low-energy structure, simple-man model, strong-field
approximation
(Some figures may appear in colour only in the online journal)
laser field, and call it the time of tunneling (not to be confused
with the ‘tunneling time’, viz. the fictitious time interval that
the electron spends inside the classically forbidden barrier).
But tunneling is not essential for the application of the Keldysh theory. The assumption is just that it is meaningful to
think of the electron becoming free at a certain time. This can
also, e.g., be realized in over-the-barrier ionization or in the
absorption of a single UV photon.
In the original Keldysh theory, after this instant of ionization the electron never feels its former binding potential any
more. For most of todayʼs strong-field physics, however, this
assumption is not justified. The electron may once again enter
the range of the potential, and upon that time scatter elastically, recombine into the ground state with emission of one
photon, or liberate a second or more electrons, to mention the
most important scenarios [5–7]. The option for these processes can be straightforwardly incorporated into Keldysh
1. Introduction
Multiphoton ionization of an atom is a very complicated
process, if the atom needs to absorb a large number of photons before the electron becomes free, as is the case for
ionization of a rare-gas atom by an infrared laser field. This is
even more so for ‘above-threshold ionization’ (ATI) where
the electron absorbs many more photons than necessary
[1, 2]. Indeed, early attempts to calculate ionization rates
based on perturbation theory did not yield very useful results
[3]. It was Keldysh who literally cut the Gordian knot into
two simple parts, the bound electron on which the laser field
is a small perturbation, and the freed electron, for which the
laser field is dominant [4]. These two are connected at one
well-defined instant of time. One may envision this as the
time when the electron tunnels to freedom through the barrier
formed by the binding potential and the interaction with the
0953-4075/14/204022+13$33.00
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J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204022
W Becker et al
theory by making allowance for just one additional interaction
with the binding potential [8–11]. Keldysh theory without or
with the option for rescattering is often referred to as the
‘strong-field approximation’ (SFA) [12]. If rescattering is
definitely included, it is sometimes called the ‘improved’
SFA (ISFA).
Keldysh theory is formulated quantum mechanically in
terms of ionization amplitudes, but the physical picture
behind it relies on classical concepts, with the exception of
the process of tunneling. Indeed, under suitable conditions,
classical orbits can be extracted from the transition amplitude
by saddle-point evaluation [8, 13, 14]. This leads to a representation of the amplitude as a superposition of the contributions of classical orbits, much in the fashion of
Feynmanʼs path integral [15]. A lot can be learned about
strong-field physics just by inspecting classical trajectories of
an electron driven by an intense laser field. This approach has
been dubbed the ‘simple-man model’ (SMM) [16].
The ISFA, which accounts for one interaction of the freed
electron with its parent ion after the process of ionization
[8–11], has been especially successful with the description of
processes such as high-order ATI or high-order harmonic
generation where the electron after its liberation is substantially accelerated by the field to higher kinetic energies. It
appeared obvious that the low-energy regime—in intermediate states and especially in the final state—was poorly
described by the SFA owing to its neglect of the Coulomb
potential. On the other hand, in recent years the experimental
interest has focused on just this regime, and qualitatively new
features of low-energy ATI have emerged [17–22]. On the
theory side, various attempts have been made to implement
Coulomb propagation into the SFA phase [23–27].
In this paper, we will be especially concerned with pronounced structures in the electron-energy spectrum in the
direction of the laser polarization at energies of a small
fraction of the ponderomotive energy Up [19–21], which have
become referred to as the ‘low-energy structure’ (LES). Since
the LES unfolds so close to the ionization threshold, it has
been taken for granted that the SFA, which does not at all
account for the Coulomb potential after ionization, or the
ISFA, which does so only to first order in the Born series, are
inadequate for its description [19, 21, 26, 28–31]. Most
researchers expected that a full treatment of the Coulomb
potential, as provided by the time-dependent Schrödinger
equation or by semiclassical methods [21, 22], is required to
generate the LES. However, surprisingly, the LES was
recently reproduced by numerical evaluation of the
ISFA [32].
We will try to shed light on this situation by surveying
the low-energy region of ATI from three different angles. We
will reconsider the standard SMM [16] with the focus on lowenergy forward scattering of the revisiting electron
(section 2). It is well known that the classical orbits of the
SMM are embedded as the so-called ‘quantum orbits’ in the
quantum description of the SFA and ISFA, from which they
can be extracted by saddle-point methods. Hence, we will
consider the manifestation of forward scattering in the ISFA
and retrieve the SMM orbits, classify them, and evaluate the
quantum corrections to the SMM by the ISFA (section 3). In
section 4, we will consider the problem via classical solutions
of the equation of motion in the presence of both the laser and
the Coulomb field, find analytical estimates for the Coulombinduced changes of the electronic trajectories, use them to
derive an upper energy boundary of the LES, and relate the
results to the SFA and the SMM. In the concluding section we
will address the particular role of the Coulomb potential.
2. The SMM and the LES
In the standard SMM [16] an electron starts its orbit at the
time t0 at the exit of the tunnel with zero longitudinal velocity
and with a Gaussian distribution of transverse momenta p⊥.
For a field linearly polarized in the x direction, its longitudinal
motion is governed by (atomic units are used unless noted
otherwise)
x¨ (t ) = −E (t ),
(1)
x˙ (t ) = x˙ ( t0 ) + A (t ) − A ( t0 ),
(2a)
which integrates to
x (t ) = x ( t0 ) + [x˙ ( t0 ) − A ( t0 ) ]
× ( t − t0 ) + F (t ) − F ( t0 ).
(2b)
In the transverse direction, the motion is force free
y (t ) = p⊥ ( t − t0 ).
(3)
We introduced the vector potential A(t) (so that
t
E (t ) = −dA dt ) and its indefinite integral F (t ) = ∫ dτA (τ ).
As mentioned, we assume that at the start time x˙ (t0 ) = 0 and,
for simplicity, also x (t0 ) = 0 . The condition that the electron
revisit at the time tr the longitudinal position of its birth is
x (tr ) = 0 . From (2b) this is
F ( t r ) = F ( t0 ) + ( t r − t0 ) F ′ ( t0 ).
(4)
At the time of the revisit, the kinetic energy of the returning
electron is
E ret =
1
[ A ( tr ) − A ( t0 ) ]2 .
2
(5)
In the recollision at the time tr , the electron may elastically or inelastically rescatter, or recombine into the ground
state emitting a photon, or do something else. We are interested in ATI, i.e., elastic rescattering. If the electron rescatters
by an angle ψ with respect to its incoming velocity (2a),
which is in the (positive or negative) x direction, its velocity
after rescattering will be [33]
vx (t ) = [A (t ) − A ( t r ) ] + cos ψ [A ( t r ) − A ( t0 ) ] ,
(6a)
vy (t ) = sin ψ ⎡⎣A ( t r ) − A ( t0 ) ⎤⎦ .
(6b)
The first term on the right-hand side of (6a) is the additional
velocity imparted by the field after rescattering. For ψ = 0 ,
we have forward scattering (the velocity is unchanged) and
for ψ = π back scattering (scattering into an arbitrary
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J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204022
W Becker et al
indeed, one can convince oneself that equation (12) implies
that Efs = E bs; see the appendix.
The energies E ret, Efs, and E bs are plotted in figure 1 as
functions of τ. One recognizes the well-known back-scattering maximum of 10.01Up and the subsequent local maxima,
which occur for longer travel times. Such classical maxima
give rise to abrupt cutoffs in the classical energy distribution,
which manifest themselves as smooth cutoffs in the quantummechanical spectrum. The same holds true for the maxima of
the forward-scattering energy: they should also be visible as
maxima in the electron spectrum. However, there is a difference: the maxima of 10.01 Up etc are located in the plateau
part of the spectrum, and such energies can only be reached
by backscattering. In contrast, the forward-scattering maxima
lie in the very-low-energy part of the spectrum where they
have to compete with the contribution of the direct electrons.
It is here where the form of the scattering potential comes in.
For a Coulomb potential, the amplitude for forward scattering
is very large—it actually diverges—so it is possible that the
forward-scattering term even dominates the contributions of
the direct electrons. On the other hand, for short-range
potentials the scattering amplitude is small so that the scattered electrons will not be able to compete with the direct
electrons.
Figure 1. The return energy E ret (dot-dashed (blue) line), five times
the forward-scattering energy Efs (dashed (red) line) and the backscattering energy E bs (solid (green) line) as functions of the travel
time τ = ωt .
direction will be considered in section 2.2). The corresponding drift energies are
1 2
A ( t0 ),
2
(7a)
1
[ 2A ( tr ) − A ( t0 ) ]2 .
2
(7b)
E fs =
E bs =
We note that E ret = 0 implies Efs = E bs.
For an arbitrary field E(t), for given start time t0
equation (4) allows for an easy graphical solution [34]. For
the sinusoidal field
E (t ) = E 0 xˆ sin ωt ,
2.1. The soft-recollision model
As just discussed, for some ionization times t0 the electron
will revisit its parent ion with zero longitudinal velocity.
These times are given by the additional condition that
x˙ (tr ) = 0 , which leads to
(8)
equation (4) can be solved for the ‘travel time’ τ ≡ ω (tr − t0 )
in terms of the start time t0 or the return time tr . This yields
A ( t0 ) = 2σ Up f (τ ) (1 − cos τ ),
(9a)
A ( t r ) = −2σ Up f (τ ) (1 − cos τ − τ sin τ ),
(9b)
F ′ ( t r ) = F ′ ( t0 ).
Graphically, these times t0 and tr are found by determining
straight lines that are tangent to F(t) at both t0 and tr . Such a
tangent invariably intersects the curve F(t) at some intermediate time. Hence, before the electron can revisit the ion
with zero longitudinal velocity, it has to revisit at least once
with nonzero velocity.
The soft-recollision model [31, 37] argues that for such
orbits the Coulomb potential will be especially important.
This is because the longitudinal motion occurs on the scale of
the laser wavelength, while the range of the (Coulombic)
binding potential is field independent. Therefore, the longer
the wavelength, the more will the interaction with the Coulomb potential be restricted to those times t when x (t ) ≈ 0,
and the corresponding time interval will be longest for those
orbits where the electron turns around just at x (t ) = 0 , while
for the preceding recollision (a ‘hard recollision’) the interaction time will be brief.
In figure 2 we present a graphical solution of
equations (4) and (13) for the sinusoidal field (8). Symmetry
implies that the afore-mentioned first (brief) encounter with
the ion will happen at times t = (1 + n 2) T with n = 0, 1 ,...
(for 0 ⩽ t0 ⩽ T 2 ) and T = 2π ω the period of the field.
A ( t r ) − A ( t0 ) = −2σ Up f (τ ) (2 − 2 cos τ − τ sin τ ) (9c)
with σ the sign of A (t0 ), A0 = E0 ω, Up = A02 4 , and
f (τ ) = 2 + τ 2 − 2 cos τ − 2τ sin τ .
(10)
Using this in (5) and in (7a) and (7b) we obtain expressions
for the return energy and the final energies for forward and
back scattering, which only depend on the travel time [35, 36]
E ret = 2Up (2 − 2 cos τ − τ sin τ )2 f (τ ),
(11a)
E fs = 2Up (1 − cos τ )2 f (τ ),
(11b)
E bs = 2Up (3 − 3 cos τ − 2τ sin τ )2 f (τ ).
(11c)
The forward-scattering energy has maxima for travel
times such that
tan (τ 2) = τ 2.
(13)
(12)
One can show that in this case E ret = 0 , that is, the electron
returns to its parent ion with zero velocity. Obviously, there is
then no difference between forward and back scattering and,
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J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204022
W Becker et al
2.2. Off-axis rescattering
For rescattering by an arbitrary angle ψ, the final drift
momentum is given by equations (6a) and (6b) with
A (t ) → 0 for t → ∞ and the final kinetic energy is
E kin (ψ ) = A ( t r ) [A ( t r ) − A ( t0 ) ]
1
× (1 − cos ψ ) + A2 ( t0 ).
2
(19)
The lab frame scattering angle θ is defined by
vx (∞) = p (θ ) cos θ,
sinusoidal field (8). The dotted (red) straight line is tangent to F(t) at
both t0 and tr . Further solutions t0 are closer to t T = 0.25 and have
return times tr that lie outside the figure.
p (θ ) = −A ( t r ) cos θ
From figure 2 one can read off the condition
which for the field (8) yields
±
(14)
(
= E ret g (τ ) cos θ ±
(15)
g (τ ) =
(16)
8Up
(2n + 3)2 π 2
(17)
≡  n.
(22)
(23)
A ( t r)
.
sin ψ [ A ( t r ) − A ( t0 ) ]
(24)
Forward and back scattering (ψ = 0 or π) correspond to final
motion along the x axis (θ = 0 or θ = π depending on in
which direction the electron returns), but ψ = π 2 does not
normally correspond to θ = π 2.
The probability of rescattering into a certain angle ψ is
not predicted by this model. It can be adjusted by hand or, by
comparison of experimental data with the model, inferred
from the data. Compare equation (26) for how the rescattering
angle enters the ISFA.
which agree with equation (7) of [31, 37]. The corresponding
energies are
E = p2 2 =
2
),
1 − cos τ − τ sin τ
.
2 − 2 cos τ − τ sin τ
cot θ = cot ψ −
p = − A ( t 0 ) = −F ′ ( t 0 )
2A 0
,
(2n + 3) π
1 − g2 (τ ) sin2 θ
The relation between the angle θ in the lab frame and the
rescattering angle ψ is
The corresponding longitudinal drift momenta are
= A 0 cos ωt0 ≈ A 0 δ =
(21)
with
which is equivalent with equation (12). Solutions have the
form ωt0 = π 2 + δ with δ ≪ 1. To first order, the solutions
are
2
δ ≡ δn =
.
(2n + 3) π
[ A ( tr ) − A ( t0 ) ]2 − sin2 θA2 ( tr ) ,
E kin (θ ) = p2 (θ ) 2
where F (t ) = E (t ) ω2 = F0 sin ωt . This yields
tan ωt0 = (2 + n) π − ωt0,
(20)
We can eliminate the angle ψ from equations (6a) and (6b)
and get
Figure 2. Graphical solution of equations (4) and (13) for the
F ( t 0)
tan ϑ =
= F ′ ( t0 ) ,
(1 + n 2) T − t0
vy (∞) = p (θ ) sin θ.
(18)
3. Quantum orbits of forward scattered electrons
We have n Up = 0.090, 0.032, 0.017 ,... for n = 0, 1, 2 ,....
The exact values from equation (12) (not using (16)) are
0.0944, 0.0330, 0.0167 ,... . A related estimate based on
classical Coulomb trajectories will be presented in
equation (60) and discussed in section 4.1.4.
The imprint of this mechanism in the (p , p⊥ ) velocity
map should be that at the longitudinal momenta (17) the
distribution of transverse momenta narrows, because for these
momenta Coulomb refocusing is especially efficient, and this
effect should increase with increasing laser wavelength. The
resulting enhancement of the spectrum for particular longitudinal momenta matches the recently observed experimental
data of the LES [19–21].
The classical trajectories discussed in the previous section are
embedded in a fully quantum-mechanical solution. There are
various routes towards such a solution. The time-dependent
Schrödinger equation can be solved numerically and the trajectories be extracted [38], but for strong fields and long
wavelengths this is a difficult task though the only one that is
capable of yielding completely accurate results. The most
frequently adopted route is the one opened up by Keldysh,
now often referred to as the SFA or the ISFA, which allows
for one additional interaction of the liberated electron with the
binding potential, corresponding to one recollision in the
SMM. Equivalently, quantum mechanics can be formulated in
terms of Feynmanʼs path integral. In view of the orbits
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J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204022
W Becker et al
discussed in this paper, this is particularly appealing. Indeed,
for processes in strong fields the number of these paths, which
in the full path integral is infinite, can be substantially reduced
to just a handful.
In any case, one obtains an expansion of the ionization
amplitude Mp from the initial ground state to a continuum
state with momentum p with respect to the number of interactions of the liberated electron with the potential so that
[14, 39]
Mp = Mp(0) + Mp(1) + ... .
x q (t )
⎧
( t − t 0s ) k s +
⎪
⎪
=⎨
⎪(t − t )p +
rs
⎪
⎩
∫t
∞
(25)
tr
∫−∞ dtr ∫−∞ dt0 ∫ d3keiS ( t ,t ,k)
p r 0
× p V k m p ( t r , t0, k).
(26)
The plane-wave matrix element of the scattering potential V,
i.e., the field-free scattering amplitude of V, determines the
angular distribution of rescattering and its magnitude relative
to the direct term Mp(0) . The precise form of the remaining
function m p (tr , t0, k) will be of no concern here. The crucial
dependence on the parameters is generated by the action
1
2
1
−
2
Sp ( t r , t0, k) = −
∫t
∫t
∞
dt′ [p + A (t′) ]2
r
tr
2
dt′ [k + A (t′) ] + Ip t0
(27)
with Ip the ionization potential. The integrals in (26) over the
intermediate electron momentum k , the ionization time t0, and
the rescattering time tr can be solved using the saddle-point
method. The resulting stationarity conditions are (see, e.g.,
[14])
1
t r − t0
∫t
parameter [4], δ =
(28)
(29)
1
1
2
2
k + A ( tr ) ] = [ p + A ( tr ) ] .
[
2
2
(30)
∑Mp(1)
{ s˜}
.
Ep (2Up ) and Q = (1 + γ 2 + δ 2 )2 −
pω ( t r − t0 ) A 0 + sin ωt r − sin ωt0 = 0,
(34)
which for the field (8) and for p = k = −A (t0 ) agrees with
the SMM equation (4).
It is possible to introduce a classification of such solutions, similar to that of the backward-scattering solutions of
[45]. In figure 3 we classify these solutions using the (double)
index νμ. The electron kinetic energy Ep (in units of Up ) is
presented as a function of the real part of the forward-scattering time tr (in units of the laser-field optical period T).
Since we are dealing with low energies, the energy is presented on a logarithmic scale. The solutions for the forwardscattering time come in pairs which are denoted by the index
μ = 0, 1, 2, 3, … (the longer the travel time Re (tr − t0 ), the
larger is the value of μ). The members of each pair are
characterized by the index ν = ±1 such that the travel time is
shorter for ν = −1 than for ν = 1. The main difference to the
Physically, these three conditions correspond, respectively, to
the requirement that the electron return to its parent ion, and
to energy conservation at the times t0 and tr .
Equations (28)–(30) have several complex solutions
(t0s, trs, k s ) (s = 1, 2 ,...), and the saddle-point method
requires to select a certain subset {s˜}, from which the result is
built [14]
Mp(1) ≈
(33)
2δ 2 (1 + cos 2θ ). For forward scattering, the emission angle
in the lab frame is θ = 0 or θ = π (cf the relation (24)
between the scattering angles in the lab frame and the particle
frame). Then, the solution for the rescattering time tr is
obtained from equation (28)
0
1
[ k + A ( t0 ) ]2 = −Ip,
2
rs
Here θ is the electron emission angle with respect to
the laser polarization axis, γ = Ip (2Up ) is the Keldysh
tr
dt′A (t′),
A (t′) dt′ , if Re t0s ⩽ t ⩽ Re t rs,
(32)
A (t′)dt′ , if t > Re t rs.
0s
2 cos2 ( Re ωt0 s ) = 1 + γ 2 + δ 2 − Q ,
cos θ
cosh ( Im ωt0 s ) = − δ
.
cos ( Re ωt0 s )
0
k=−
t
t
The real parts of these quantum orbits are virtually identical
with the SMM orbits of the previous section. We shall depict
some quantum orbits below.
In past work, it was possible to describe surprisingly well
not only the plateau and cutoff region of the high-order ATI
spectra, but also the quantum-mechanical phenomenon of
resonancelike enhancement of a group of peaks in the highenergy spectrum, which appears for specific laser-field
intensities; see, e.g., [42–44]. In view of the results of the
SMM, we expect that this formalism can also be applied to
explain the LES. In previous formulations of the quantumorbit formalism, only the back-scattered quantum orbits were
taken into account. We will show in the present paper the
existence of the forward-scattered quantum orbits and argue
that they are responsible for the LES.
The solutions of the saddle-point equations in the case of
backscattering were analyzed in [45]. Let us suppose that
solutions that correspond to forward scattering also exist. For
forward scattering, we have k = p, and equation (30) is
satisfied for all values of tr . The solution of equation (29) for
the time t0 is known in analytical form [35, 46]
We will be concerned with the rescattering amplitude, which
has the form of a five-dimensional integral over the ionization
time t0, the recollision time tr , and the drift momentum k in
between ionization and recollision
Mp(1) =
∫t
(31)
s
Each solution (t0s, trs, k s ) (s = 1, 2 ,...) defines a complex
trajectory in space (a ‘quantum orbit’), which has the form
[40, 41]
5
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204022
W Becker et al
Figure 3. Analysis of the cutoff positions for forward-scattering
saddle-point solutions. The electronʼs kinetic energy Ep in units of
Up (on a logarithmic scale) is presented as a function of the real parts
of the forward scattering times, Re tr T . The five pairs of orbits with
the shortest travel times are given. The orbits are characterized by the
multi-index νμ, where the index μ = 0, 1, 2, … denotes a pair of
solutions, while the members of each pair are distinguished by the
index ν = ±1. There are further solutions that have forward
scattering times beyond the right-hand margin of the figure. Their
maximal (cutoff) energies Ep decrease. The curves have been
calculated for Ar, for emission in the direction θ = 0°, and for a
linearly polarized laser field having the intensity 2 × 1014 W cm−2
and the wavelength 2000 nm. The maximal energies of the curves
denoted by μ = 1, 2, 3, and 4 correspond to the forward-scattering
energy maxima  n with n = 0, 1, 2 , and 3 which have been
discussed in section 2.1; cf equation (18).
Figure 4. Forward-scattered quantum-orbit solutions and trajectories
for the two shortest pairs of orbits (μ = 0, 1) . The ionization time t0
is identical for all panels; it is represented by the blue curve in the
upper left corner of the left panel (a). Left-hand panels: for this t0 in
each optical cycle there is a pair (ν = ±1) of forward-scattering
solutions denoted by μ = 0 and 1, in panels (a) and (b), respectively.
Right-hand panels: real parts of the quantum orbits (32) obtained
using the saddle-point solutions of the corresponding left panel. The
positions where the electron exits from the tunnel and where it
forward scatters on the core are denoted by the same symbols, a
filled circle for ν = 1 and a filled square for ν = −1. The electron
energy is Ep = qω with (a) q = 120 and (b) q = 12. The orbits in
panels (b) correspond to the soft recollision labeled n = 0 in
section 2.1. The red dashed trajectories are shifted up by 5 au (a) and
10 au (b), in order to avoid a visual overlap with the black solid
curves.
back-scattering solutions is that for the forward scattering
solutions the ionization time t0 is the same for all solutions νμ
and is given in analytical form by equation (33). The solution
νμ = 10 is dominant in the region of energies near 1 Up . It
does not have a sharp cutoff as is the case for the solutions
having μ > 0 . The remaining pairs of solutions have sharp
cutoffs at those energies where for given μ the solutions
having ν = −1 and ν =+1 approach each other. These correspond to the maxima of the forward-scattering energies
n (n = μ − 1 = 0, 1 ,...), which were identified within the
soft-recollision model in section 2.1. For the solution μ = 1
this happens near the energy Ep = 0.1 Up . With increasing μ
the cutoff positions decrease.
Let us now analyze in more detail the solutions of
equations (33) and (34) for the times t0 and tr for forward
scattering. In figure 4, in the left panels, we present the two
pairs of such solutions with the shortest travel times. In panel
(a) the ionization time t0 is represented by the blue curve in
the upper left corner. This time is the same for all solutions
for the forward scattering time tr , which is why we only show
it in panel (a). The electron energy, as a continuum parameter,
changes from a small value to the cutoff value along the
curves (Re t0, Im t0 ) and (Re tr , Im tr ) in the complex plane.
For the cutoff values of the energies the curves ν = 1 and
ν = −1 approach each other very closely. Beyond the cutoff
the imaginary part of the time tr has a large positive (for
ν = 1) or negative (for ν = −1) value.
In the right-hand panel of figure 4 we show the electron
trajectories (32), defined as the real part of the quantum
orbit. They are in the direction of the polarization axis x and
start from the tunnel exit, which is indicated by a solid
circle or square. The electron on these real orbits forward
scatters close to the origin at the point Re xq (Re trs ), which
is denoted by the same symbol. With increasing electron
energy the value of Re xq (Re trs ) increases. The real parts of
the quantum orbits after rescattering (for t > Re trs ) are also
exhibited.
Here is a detailed description of the orbits: after the
electron has emerged at the tunnel exit it first moves away
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J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204022
W Becker et al
and ν = ±1 are presented in figure 5(a) (the solution
νμ = −10 is dropped). As expected, the LES below 0.1 Up is
clearly visible. Let us analyse this more carefully. In
figure 5(b) we present the partial differential ionization rates
that correspond to the various forward-scattering saddle-point
solutions, in accordance with the notation explained in
figure 3. One can notice distinct peaks at the energies near
12 ω (solution μ = 1), 4 ω ( μ = 2) and 2 ω (μ = 3). The
contribution of the solution νμ = 10 is smaller but gives a
background (the dash-dotted line) to the other solutions,
which is such that the LES peaks still clearly stick out. This
background extends to energies above 2 Up .
Figure 5 does not include the contribution of the direct
electrons, which is given by the term Mp(0) in equation (25).
The relative magnitudes of the contributions of the direct and
the forward-scattered electrons decide whether or not the LES
is visible in the experimental spectrum. For a short-range
potential forward scattering is weak; for example, for a zerorange potential the matrix element in equation (26) is
〈p | V | k〉 = (4π 2 2I p )−1, which is small. Hence the direct
electrons dominate and the LES is not visible in the total
spectrum. In contrast, for a Coulomb potential we have
〈p | V | k〉 = Z [2π 2 (p − k)2 ], which for forward scattering is
divergent and has to be regularized [32]. In consequence, the
LES can rise above the direct electrons and dominate the lowenergy total spectrum.
Figure 5. Differential ionization rates of Ar as functions of the
electron energy in units of Up for a linearly polarized laser field
having the intensity 2 × 1014 W cm−2 and the wavelength 2000 nm.
The results are obtained using the saddle-point approximation with
forward-scattering saddle-point solutions. The peaks, which are most
clearly seen in panel (b), correspond to the soft-recollision energies
 n with n = μ −1 = 0, 1, …, 7. See the text for explanation.
4. Classical trajectories in the laser field and the
Coulomb field
Both the SMM and the SFA account for the Coulomb field
only in the most perfunctory fashion: the SMM by allowing
for scattering exactly at the position of the ion and the SFA by
the first-order Born approximation for the revisiting electron.
In the SMM, the Coulomb potential can be taken into account
by weighting the scattering event with the angle-dependent
cross section of the scattering potential; in the SFA the firstorder scattering amplitude is automatically included. (It is
important to note that for the Coulomb potential the first-order
Born approximation is exact.) We can get an idea of the
justification or the shortcomings of such approaches by considering exact classical trajectories in the presence of both the
laser field and the attractive Coulomb field. Since the Coulomb field is fully incorporated, there is no analog of the Born
approximation and there is no distinction between direct and
rescattered electrons. The equation of motion then reads, in
place of (1)
from the origin in the negative x direction. After the field has
changed its sign the electron turns around and moves back to
the origin where it forward scatters off the core and continues
in the direction θ = 0°. In the right-hand panel of figure 4 the
corresponding trajectories for fixed energies (a) 120 ω ≈ 1 Up
(a) and 12 ω ≈ 0.1 Up (b) are shown (it should be mentioned
that the used values of q do not have to be integers—they are
chosen only for convenience). The energy in the case (b)
corresponds to the cutoff value shown in figure 3 for μ = 1,
which in the SMM correspond to the soft-recollision energy
 0 of equation (18). Since the trajectories for each of these
pairs of solutions overlap, we shifted up the trajectory
denoted by the red dashed line. The difference between the
red dashed and black solid trajectories in the panel (b) is in
the direction of the electron momentum at the time of forward
scattering. This momentum is in the direction of the negative
(positive) x axis for the red dashed (black solid) curve. In the
right-hand panel of figure 4(a) we see that for ν = −1 the
travel time is extremely short: the symbols that denote the
corresponding ionization and rescattering times almost
overlap.
Let us now calculate the ATI spectra using the forwardscattering saddle-point solutions. The results obtained taking
into account the fifteen solutions μ = 0, 1, 2, 3, 4, 5, 6, 7
x¨ (t ) = −E (t ) − r (t ) r 3 (t ).
(35)
In figure 6 we compare two specific trajectories having
the same initial conditions with (blue solid curves) and
without (red dashed curves) the Coulomb field. The longitudinal motion does not qualitatively change from one to the
other (upper panel). However, the transverse momentum is no
longer conserved when the Coulomb field is taken into
account (lower panel). The figure shows that at ωt ≈ 2π the
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W Becker et al
orbits start in the first and third quarter, long orbits in the
second and fourth. Then in the SMM the first quadrant is
covered by short (3, +) and by long (2, +) trajectories. In
general, in the SMM the contribution from a given quarter
cycle of the laser period with a fixed sign of the initial
transverse velocity fills in exactly one quadrant of the (px , py )
plane, and in each point there are exactly two contributions
(represented by the same color), one from a short and the
other from a long trajectory. With the adopted grid and
parameters, we would have two filled rectangles of size 1 au
by 0.4 au in each quadrant.
With the Coulomb field on, the panels of figure 7 display
a modification of these rectangles under the action of the
Coulomb field. Disregarding strongly (chaotically) scattered
isolated points originating from large-angle scattering off the
ion [29, 48], two gross features are seen. First, the rectangles
are shifted along the field direction since the Coulomb field
increases the drift momentum of long trajectories and
decreases it for short ones. Second, the former rectangles are
compressed in the transverse direction (the lower and upper
rows of points in the panels are closer to the horizontal axis
than 0.4 au). The total population of the (px , py ) plane is
obtained by superimposing all four panels of figure 7. It is
symmetric with respect to reflection about both coordinate
axes so that it is sufficient to investigate in detail the population of only one quadrant of the (px , py ) plane.
In addition to the gross features just discussed, one
clearly sees deviations from the simple SMM picture originating from new types of trajectories, which are qualitatively
distorted by the Coulomb field [26] compared with their
SMM analogs. We discuss this effect taking the first quadrant
as an example. In the first quadrant we have strongly overlapping layers of contributions from short trajectories (3, +)
and from long (2, +) trajectories. However, in addition we
observe two systematic qualitative effects of the Coulomb
field: (i) for small px there are contributions from short (1, +)
trajectories (a narrow strip along the positive py axis; owing to
the Coulomb field the (1, +) orbits have gained positive
momentum and actually have become returning-type(long)
trajectories; (ii) for small py > 0 , there are contributions from
(2, −) orbits (the afore-mentioned trajectories where the
Coulomb field has changed the sign of py; it seems as if the
population of the fourth quadrant has spilled over into the first
quadrant). As mentioned above, the population (ii) is held
responsible for the LES [26].
Figure 6. Classical longitudinal (upper panel) and transverse (lower
panel) trajectory of an electron after its birth by ionization without
(dashed red line, SFA) and with (solid blue line) the Coulomb field.
The simulation is for a 12-cycle pulse with maximal field amplitude
E0 = 0.0655 au, wavelength of 2000 nm, Ip = 0.579 au. The initial
transverse momentum is 0.1 au, and ionization has occurred at a
phase of ωt0 = 98° (8° after a field maximum).
electron goes for the first time through the longitudinal
position x = 0. This ‘hard recollision’ takes place at rather
high velocity so that the effect on both the longitudinal and
the transverse momentum is small. However, this is not
necessarily the case, if the transverse velocity and/or the
impact parameter are small; see equations (46)–(49). The next
two recollisions occur on either side of ωt ≈ 3.5π . This
combined event is close to a soft recollision and, consequently, the effect on the transverse momentum is strong, so
strong that it even changes its sign. Such trajectories have
recently been associated with the LES [26]. Traditionally, the
effect is subsumed under ‘Coulomb refocusing’ [47].
To obtain a general qualitative picture of how the Coulomb field affects the SFA momentum distribution we show
in figure 7 the mapping of the initial conditions for electrons
produced in the four quarters of one laser period
0 < ωt0 < 2π with positive and negative transverse initial
velocity onto the final-momentum momentum plane (px , py )
according to the full equation of motion (35). The grid of
initial conditions is limited to not too large initial transverse
velocities | v0 | ⩽ 0.4 au and start times not too far away from
the field maxima so that | ωδt0 | ⩽ 20°, since trajectories with a
very large drift momentum (transverse or longitudinal) are
only slightly affected by the Coulomb field. The effect of the
Coulomb field becomes clear if we compare with the corresponding SMM mapping. Let us denote by (n , + ) and (n , − )
electrons liberated in the nth quarter cycle of the field
E (t ) ∝ sin ωt with v0 > 0 and v0 < 0 , respectively. Short
orbits do not revisit the plane x = 0, long orbits do. Short
4.1. Semianalytical estimates of Coulomb recollisions
For quantitative estimates of the Coulomb effects we formulate a (semi-) analytical model allowing approximate
evaluation of classical trajectories in the combined laser and
Coulomb field. The basic assumption is that with a low-frequency field the electronʼs oscillation amplitude is very large
so that the liberated electrons are far away from their parent
ions during most of the laser period. Hence, the electron–ion
interaction is most efficient during the small time intervals
when the electron–ion distance is minimal [29–31, 37, 48]. In
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W Becker et al
Figure 7. The four panels each display the (px , py ) plane as it gets populated by electrons liberated in the first half of a sine-shaped field
E (t ) ∝ sin ωt xˆ with transverse momenta v0 such that v0 > 0 (upper left panel), in the second half with v0 > 0 (upper right panel), in the first
half with v0 < 0 (lower left panel), and in the second half with v0 < 0 with the restrictions mentioned in the text. The labels (n, ± ) refer to
electrons liberated in the nth quarter cycle of the field with positive or negative v0. Evolution of the electron momentum after the end of the
laser pulse was not considered in the calculation of the panels [29, 48]. The arrows in each panel identify a rectangular region of the (px , py )
plane outside of which the Coulomb corrections are insignificant. Their positions are determined as explained in sections 4.1.1–4.1.3; see the
text. The parameters are λ = 2000 nm, E0 = 0.0655 au, and Ip = 0.579 au.
essence, the model is the same as [29, 48] but differs from it
in realization. In [29, 48] having in mind the axial symmetry
of the three-dimensional problem, solutions were obtained
using polar coordinates. In contrast, we consider the twodimensional motion in the plane defined by the initial transverse velocity vector and the direction of the linear laser
polarization. Axial symmetry is restored by rotating this plane
around the polarization direction.
Figure 8 establishes the notation used in the following.
Essentially, t0 denotes the time of ionization, t1 the time of the
first return of a long trajectory to the plane x = 0, which by
necessity occurs with nonzero longitudinal velocity (cf
figure 2), and t2 the time of the next U-turn (with a minimum
of | x (t )|), which may be for x (t ) > 0 as depicted in the figure
or for x (t ) < 0. The case when x (t ) = 0 corresponds to a soft
recollision as defined in section 2.1. The Coulomb corrections
to the electron momentum are evaluated by integrating the
Coulomb force over time in the vicinity of the interaction
times tn. These corrections are added to the electron drift
momentum and the electronʼs trajectory after the interaction is
calculated accounting for this modification. The results are
mostly in the form of elliptic integrals, which in many cases
can be replaced by their asymptotic expansions.
Figure 8. Longitudinal position x(t) for a short (upper panel) and a
long (lower panel) trajectory starting in the same laser period and
having positive final momentum px > 0 at the end of the laser pulse.
Dots mark those times when the electron–ion distance and the
Coulomb force reach, respectively, their minimum and their
maximum. The time of ionization is t0; t1 denotes the first revisit of
the plane x = 0, which by necessity occurs with nonzero longitudinal
momentum, and t2 is the time of a U-turn under the action of the
laser field at a distance d (t2 ) from the plane x = 0 (if this happens at
x = 0 it is a ‘soft recollision’ according to [31, 37]).
4.1.1. Coulomb correction to the drift momentum at the tunnel
exit. Calculations begin at the tunnel exit at time t0 [49, 50].
The electron trajectory in the laser field starts at the tunnel exit
x 0 = −Ip E (t0 ) with the initial velocity v(t0 ) = (0, v0 ). Since
the Coulomb force rapidly decreases when the electron moves
away from the tunnel exit it is sufficient to consider only the
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W Becker et al
very initial part of the trajectory
x (t ) = x 0 −
1
E ( t0 )( t − t0 )2 ,
2
expression (40) for the longitudinal coordinate remains valid
for times larger than t1 and can be used to evaluate the time t2
of the next interaction.
For start times close to the field maximum, at
ωt0 = π 2 + δ with δ ≪ 1, the time t1 can be written as
ωt1 = 2π + α with α ≪ 1. Then, the solution to the equation
x C (t1 ) = 0 to a linear approximation in the small parameters
is
y (t ) = v0 ( t − t0 ). (36)
Retaining only the first term in an expansion in the small
velocity v0 2Ip one obtains the corrected drift momentum
px ( t0 ) = A 0 ( − cos ωt0 + ξ sin ωt0 ),
⎛
E ( t0 )
py ( t0 ) = v0 ⎜⎜ 1 −
2Ip2
⎝
⎞
⎟
⎟
⎠
(37)
α = 1 + γ 2 2 − 3π (ξ + δ ) 2,
(38)
where γ = Ip (2Up ) denotes the Keldysh parameter. The
impact parameter and the velocity at time t1 are
with ξ = πω (2Ip )3 2 . Therefore, the electron moves away
from the tunnel exit with the Coulomb-corrected velocity
vxC (t ) = px ( t0 ) + A 0 cos ωt ,
vyC = py ( t0 ).
ρ ( t1) = ( t1 − t0 ) py ( t0 ) =
(39)
=
This modification does not satisfy the initial condition at
t = t0 but correctly describes the drift momentum on large
time scales of the order of the laser period. The correction to
the drift momentum may be the most important and most
ubiquitous modification of the SMM and the SFA by the
Coulomb potential. For circular polarization, where
rescattering does not complicate matters, and in the
quantum (R-matrix) context, it is discussed in reference [27].
Integration of the velocity (39) gives the Coulombcorrected coordinate
x C (t ) = ( t − t0 ) px ( t0 ) + ( A 0 ω)( − sin ωt0 + sin ωt )
Ip
−
.
ωA 0 sin ωt0
(46)
y (t ) = ρ ( t1) + vyC ( t2 )( t − t1).
(47)
(48)
with aosc = E0 ω2 . Obviously, the Coulomb attraction not
only decreases the transverse velocity (Coulomb focusing),
but can even change its sign if
v0 < v* =
4 ( 3πa osc ) ≈ 0.06 au
(49)
for the parameters λ = 2000 nm, E0 = 0.0655 au, and
Ip = 0.579 au, which we use here and below for quantitative
estimates (closer consideration increases this value by
(10–15)% so that v0 < 0.07 au).
On the other hand, if v0 is so large that already at t1 the
transverse distance satisfies ρ (t1 ) > aosc 2 (i.e. v0 > A (3π )≈
0.3 au), then it will further increase during the time interval
t2 − t1. But at such a large transverse distance the Coulomb
force has become so small that its effect can be neglected
regardless of the longitudinal position. If so, then
equations (37) and (46) give the final momentum. The
transverse momenta corresponding to v0 = 0.3 au are marked
by the horizontal arrows in figure 7.
(41)
i.e., an electron with the velocity
crosses the plane x = 0
at a fixed impact parameter ρ (t1 ). The limits of integration in
the rapidly convergent integral of the Coulomb force can be
extended to ±∞ and one has
2
.
ρ ( t1) vxC ( t1)
vyC = py ( t0 ) + pyC ( t1),
⎞
⎛
4
⎟
vyC ( t2 ) ≈ v0 ⎜ 1 −
3πv02 a osc ⎠
⎝
(40)
vxC (t1 )
pyC ( t1) = −
(45)
To zeroth order in the small parameters δ, ξ, and α with
ρ (t1 ) ≈ 3πv0 (2ω) and vxC (t1 ) = A0 , the velocity (46) is
the equation x C (t ) = 0 one finds the time t1. If the electron
trajectory intersects the plane x = 0 with sufficient velocity
(this is what we call a ‘hard recollision’), then near this time
the electron trajectory is approximated by
pxC ( t1) = 0,
(44)
After crossing the x = 0 plane until the next interaction at the
time t2 the electronʼs transverse velocity and coordinate are
4.1.2. Coulomb correction in a hard recollision. By solving
y C (t ) = y C ( t1) ≡ ρ ( t1),
⎞
v0 ⎛
E ⎞⎛
⎜ 1 − 0 ⎟ ⎜ 3π + α − δ⎟ ,
⎜
⎟
2
⎝
⎠
ω⎝
2Ip ⎠ 2
vxC ( t1) = A 0 (1 + δ + ξ ).
Equations (39) and (40) are applicable up to the time of the
next interaction with the ion. It is worth mentioning that in the
absence of subsequent interactions, i.e. if their effect is
negligibly small, equations (39) and (40) will be a fairly good
approximation till the end of a long laser pulse if the
oscillating sin ωt and cos ωt are multiplied by the fieldenvelope function, e.g. by cos2 (ωt 2n ). Equation (38) confirms that Coulomb attraction reduces the initial transverse
momentum compared with the SMM, but only by a small
amount.
x C (t ) = vxC ( t1)( t − t1),
(43)
4.1.3. Coulomb correction in a U turn, soft recollision. At the
time t2, the longitudinal velocity is zero and we have from
equation (39)
(42)
ωt2 = 7π 2 − arcsin ⎡⎣px ( t0 ) A 0 ⎤⎦ ≈ 7π 2 − δ − ξ .
The longitudinal correction is zero since the longitudinal
force is an odd function of t − t1. This means that the
(50)
Substituting this into equation (40) yields the corresponding
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longitudinal position. To linear approximation one finds
d ( t2 ) a osc = 3π (ξ + δ ) − 2 − γ 2 2,
ρ (t1 )
ρ ( t2 ) ≈ ρ ( t1) ≈ 3πv* (2ω) ≈ 24.8 au.
(51)
For start times after tcr with the phase δ = δcr + Δ the
longitudinal distance (51) is
which maybe positive or negative depending on the start time.
At some start time ωtcr = π 2 + δcr , with
(
δcr = 2 + γ 2 2
)
(3π ) − ξ ,
d ( t2 ) = 3πa osc Δ .
(52)
⎛ x 0 ⎞3 2
≈⎜
⎟ .
pxC ( t0) ⎝ r ( t2) ⎠
pxC ( t2)
1
pi = − 2
r ( t2 )
In view of (56) one has
ΔH = γ 2 (6π ) or δ H = − ξ + f (γ )
(59)
with f (γ ) = (2 + γ 2 ) (3π ). Since there was no correction to
p (δ H ) at t1 we arrive at the conclusion that the final
longitudinal momentum for these trajectories coincides with
the drift momentum right after the tunnel exit:
px (δ H ) = A0 f (γ ). The corresponding energy can be interpreted as an upper boundary of the LES,
(53)
ϵH = px2 ( δ H ) 2 = 2Up f 2 (γ ) = Ip f 2 (γ ) γ 2.
2 r ( t2 )
E ( t2 )
× Ji ( d ( t2 ) ρ ( t2 ) ) (i = x , y),
(57)
The phase δ H could be defined by the condition that this ratio
be less than unity meaning that the correction at the time t2
does not change the momentum considerably. But technically
it is more convenient to use the condition
x0
x0
<
⩽ 1.
(58)
r ( t2 )
d ( t2 )
The integral of the Coulomb force over time is readily
calculated with the result
C
(56)
Now, the purpose is to determine a phase δ H = δcr + ΔH
such that the longitudinal Coulomb correction in (54) can be
neglected for δ ⩾ δ H . Such trajectories will be of the first
type. This can be done by comparing the corrections at the
time t2 and at the tunnel exit. The structure of the latter is
similar to (54) (with the replacement r (t2 ) → x 0 ) and, hence,
their ratio is
it turns to zero, d (tcr ) = 0 . This is the case of the soft
recollision investigated in section 2.1. For our parameters, we
have δcr = 9.3°.
If a trajectory starts closer to the field maximum, i.e.
δ < δcr , the average slope of the curve x(t) decreases and the
U-turn at x = 0 changes into a double crossing. In the
quantum-orbit description of section 3 this is reflected by the
fact that a new class of orbits enters the picture. If, on the
other hand, the electron starts after δcr the distance d (t2 )
swiftly increases. For example, for a start just 3° after δcr , we
already have d (t2 ) = 0.5aosc . Coulomb corrections then
quickly become insignificant.
Evaluation of the Coulomb corrections at the time t2
allows us to distinguish between returning trajectories of two
types. The first one consists of trajectories that are affected by
the Coulomb field in a minimum way, i.e. only at the times t0
and t1. The contributions from subsequent returns (at t2 and
possible later times) are negligibly small for these trajectories.
Trajectories with essential contributions from these later
returns are assigned to the second type.
The electron trajectory in the vicinity of a U-turn at t2 is
approximated by
x (t ) = d ( t2 ) − E ( t2 )( t − t2 )2 2,
y (t ) ≈ y ( t2 ) ≡ ρ ( t2 ).
(55)
(60)
In each panel of figure 7, the corresponding momentum is
marked by a vertical arrow at the larger value of | px | (this
corresponds to the long trajectory). The position of the second
vertical arrow at the smaller value of | px | (corresponding to the
short trajectory) is found from the condition that the distance
d (t2 ) (see the upper panel of figure 8) equals half of the large
oscillation amplitude so that the Coulomb correction at the
time t2 can be neglected.
(54)
where r (t2 ) = d 2 (t2 ) + ρ2 (t2 ) and the functions Ji depend
on the ratio d (t2 ) ρ (t2 ).
At the start time ωtcr = π 2 + δcr , we have d (t2 ) = 0
and Jx (0) = 0.85 and Jy (0) = 1.85. The corrections (54) then
depend on the transverse distance ρ (t2 ) and might have a
different value.
As mentioned above, the longitudinal distance d (t2 )
quickly increases when the start time exceeds tcr , and for
d ρ ⩾ 1.5 one can safely use the limits of Ji(x) for large
argument x, which are Jx → πd (t2 ) (2r (t2 )) and
Jy → 3πρ (t2 ) (8r (t2 )). This form makes clear that the change
of momentum is the product of the Coulomb force and an
effective interaction time 2r (t2 ) | E (t2 )| .
Being interested in the momentum distribution along the
polarization direction we consider relatively small v0, which
lead to small final transverse momenta. Such v0 are definitely
less than A0 (3π ) ≈ 0.3 au and rather are of the order of (49).
With such an initial velocity, ρ (t2 ) remains of the order of
4.1.4. Discussion. Obtaining an upper limit ϵH for the LES
was the main purpose of the derivations above. We showed
that for energies larger than ϵH the trajectories suffer only
minimal distortion by the Coulomb field while below ϵH
deformation in one or another way is substantial. It is
interesting to note that in the limit of γ → 0 , we have
ϵH = 8Up (9π 2 ), which agrees with  0 from equation (18).
The term proportional to γ2 in the function f (γ ) reflects a
nonzero length of the tunnel. For the nonzero Keldysh
parameter γ = 0.38, which corresponds to the parameters
specified above (below equation (49)), we find ϵH Up = 0.13.
We also note that according to equation (60) the quantity
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W Becker et al
ϵH Ip depends only on the Keldysh parameter γ. The implied
scaling of the upper boundary of the LES is slower than γ−2,
which agrees with the data [19, 32].
Coulomb potential is unique because the quantum-mechanical
first-order Born approximation yields the exact scattering
cross section, which moreover agrees with the classical
Rutherford result.
In this paper, we focused on on-axis emission. However,
forward scattering (actually, close-to-forward scattering) also
generates off-axis effects in the spectrum. This will be
explored in future work [52]. In addition to the LES, additional structure at low electron energy has recently been
discovered in experiments: a structure well below the LES
energy, which has been called the very-low-energy structure
(VLES) [22] and a substantial enhancement at practically zero
energy, called the zero-energy structure (ZES) [53]. The
VLES appears to be reproduced by semiclassical trajectory
calculations [22], while the origin of the ZES is still under
debate.
5. Conclusions
We have confirmed by three different approaches that the
low-energy regime of above-threshold ionization supports
abundant spectral structure on a scale well below the ponderomotive potential Up ∼ Iλ 2 . Therefore, for the details to
be visible, long laser wavelengths λ are most suitable.
Surprisingly, we found that the LES is already inherent in
the classical SMM, even though the latter only allows for a
single act of rescattering, which at first sight seems to rule
out a proper description of Coulomb propagation effects.
Since the improved SFA incorporates the simple-man orbits
it affords the LES as well and puts it into a quantum framework. Namely, the LES originates from forward rescattering of the liberated electron off its parent ion. Final
energies reached through forward scattering have classical
cutoffs just like those of back scattering. These cutoffs
manifest themselves in the spectrum as substantial
enhancements or smoothed spikes. These exist for any
binding potential, including even the extreme case of a
zero-range potential. Whether or not they are visible in the
total spectrum, which has contributions from the direct
electrons and the rescattered electrons (cf the two terms in
equation (25)), depends on the relative magnitude of their
contributions, which is determined by the matrix element
〈p | V | k〉 in equation (26). For short-range potentials the
enhancements are small compared with the contributions of
the direct electrons, which do not rescatter at all, and on this
background they are not visible [51]. In contrast, for a
Coulomb potential we have 〈p | V | k〉 = Z [2π 2 (p − k)2 ],
which is divergent in the forward direction.
It remains almost a miracle that low-energy laser-atom
effects can be reproduced by a theory conceptually as
straightforward as the SMM and its quantum versions, the
SFA and the ISFA. In order better to understand the connection, we investigated exact classical trajectories in the
presence of both the laser and the Coulomb field. We confirmed that the LES is related to classical trajectories whose
transverse momentum changes its original sign in a close
encounter with the ion [26]. We also derived an analytical
estimate of the upper boundary of the LES, which comes out
in close agreement with the results of the SMM and the ISFA.
The connection between the two approaches, the SMM and
the ISFA on the one hand and the Coulomb trajectories on the
other, is still somewhat opaque, because for rescattering the
SMM and the ISFA only consider orbits with zero initial
transverse momentum while nonzero transverse momentum is
crucial in the Coulomb approach. The orbits of reference [26]
are quantum mechanical, but they are based on Coulomb
correcting the direct-electron orbits (corresponding to the
term Mp(0) in (25)), which have nonzero initial transverse
momentum, too. In all this, one should keep in mind that the
Acknowledgements
We appreciate discussions with Ph Korneev and S Kelvich.
We gratefully acknowledge support by the Alexander von
Humboldt Foundation, the German Federal Ministry of
Education and Research in the framework of the Research
Group Linkage Programme and the German Research Foundation (DFG).
Appendix
Here we prove equation (12) and the subsequent statements.
The condition that Efs be maximal yields
sin τf (τ ) = τ (1 − cos τ )2 ,
(A.1)
which can be factorized as
(τ − sin τ )(τ sin τ − 2 + 2 cos τ ) = 0,
(A.2)
so that the times τn where Efs is maximal are given by the
solutions of
τ sin τ − 2 + 2 cos τ = 0
(A.3)
or, equivalently,
tan
τ
τ
= .
2
2
(A.4)
The remaining statements that at these times E bs = Efs and
E ret = 0 then follow immediately upon using (A.3).
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