Download ABOVE-THRESHOLD IONIZATION: FROM CLASSICAL FEATURES

ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 48
ABOVE-THRESHOLD IONIZATION:
FROM CLASSICAL FEATURES TO
QUANTUM EFFECTS
W. BECKER 1 , F. GRASBON 2 , R. KOPOLD 1 , D.B. MILOŠEVIĆ 3 ,
G.G. PAULUS 2 and H. WALTHER 2,4
1 Max-Born-Institut,
Max-Born-Str. 2A, 12489 Berlin, Germany; 2 Max-Planck-Institut für
Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany; 3 Faculty of Science,
University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Hercegovina;
4 Ludwig-Maximilians-Universität München, Germany
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . .
B. Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
II. Direct Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. The Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Quantum-mechanical Description of Direct Electrons . . . .
C. Interferences of Direct Electrons . . . . . . . . . . . . . . . . . .
III. Rescattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. The Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Quantum-mechanical Description . . . . . . . . . . . . . . . . . .
IV. ATI in the Relativistic Regime . . . . . . . . . . . . . . . . . . . . . .
A. Basic Relativistic Kinematics . . . . . . . . . . . . . . . . . . . .
B. Rescattering in the Relativistic Regime . . . . . . . . . . . . . .
V. Quantum Orbits in High-order Harmonic Generation . . . . . . .
A. The Lewenstein Model of High-order Harmonic Generation
B. Elliptically Polarized Fields . . . . . . . . . . . . . . . . . . . . . .
C. HHG by a Two-color Bicircular Field . . . . . . . . . . . . . . .
D. HHG in the Relativistic Regime . . . . . . . . . . . . . . . . . . .
VI. Applications of ATI . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Characterization of High Harmonics . . . . . . . . . . . . . . . .
B. The “Absolute Phase” of Few-cycle Laser Pulses . . . . . . .
VII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction
With the discovery of above-threshold ionization (ATI) by Agostini et al.
(1979) intense-laser atom physics entered the nonperturbative regime. These
experiments recorded the photoelectron kinetic-energy spectra generated by laser
irradiation of atoms. Earlier experiments had measured total ionization rates
35
Copyright © 2002 Elsevier Science (USA)
All rights reserved
ISBN 0-12-003848-X/ISSN 1049-250X/01 $35.00
36
W. Becker et al.
[I
Fig. 1. Photoelectron spectrum in the above-threshold-ionization (ATI) intensity regime. The series
of peaks corresponds to the absorption of photons in excess of the minimum required for ionization.
The figure shows the result of a numerical solution of the Schrödinger equation (Paulus, 1996).
by way of counting ions, and the data were well described by lowest-order
perturbation theory (LOPT) with respect to the electron–field interaction. This
LOPT regime was already highly nonlinear (see, e.g., Mainfray and Manus,
1991), the lowest order being the minimal number N of photons necessary for
ionization. An ATI spectrum consists of a series of peaks separated by the photon
energy, see Fig. 1. They reveal that an atom may absorb many more photons than
the minimum number N , which corresponds to LOPT.
In the 1980s, the photon spectra emitted by laser-irradiated gaseous media
were investigated at comparable laser intensities and were found to exhibit peaks
at odd harmonics of the laser frequency (McPherson et al., 1987; Wildenauer,
1987). The spectra of this high-order harmonic generation (HHG) display a
plateau (Ferray et al., 1988), i.e., the initial decrease of the harmonic yield
with increasing harmonic order is followed by a flat region where the harmonic
intensity is more or less independent of its order. This plateau region terminates
at some well-defined order, the so-called cutoff.
A simple semiclassical model of HHG was furnished by Kulander et al. (1993)
and by Corkum (1993): At some time, an electron enters the continuum by
ionization. Thereafter, the laser’s linearly polarized electric field accelerates the
electron away from the atom. However, when the field changes direction, then,
depending on the initial time of ionization, it may drive the electron back to
its parent ion, where it may recombine into the ground state, emitting its entire
energy – the sum of the kinetic energy that it acquired along its orbit plus the
I]
ATI: CLASSICAL TO QUANTUM
37
binding energy – in the form of one single photon. This simple model beautifully
explains the cutoff energy of the plateau, as well as the fact that the yield of HHG
strongly decreases when the laser field is elliptically polarized. In this event, the
electron misses the ion. This model is often referred to as the simple-man model.
The model suggests (Corkum, 1993) that the electron, when it recollides with
the ion, may very well scatter off it, either elastically or inelastically. Elastic
scattering should contribute to ATI. Indeed, the corresponding characteristic
features in the angular distributions were observed by Yang et al. (1993), and
an extended plateau in the energy spectra due to this mechanism, much like
the plateau of HHG, was identified by Paulus et al. (1994c). Under the same
conditions, a surprisingly large yield of doubly charged ions was recorded
(l’Huillier et al., 1983; Fittinghoff et al., 1992) that was incompatible with a
sequential ionization process. A potential mechanism causing this nonsequential
ionization (NSDI) is inelastic scattering. It was only recently, however, that
this inelastic-scattering scenario emerged as the dominant mechanism of NSDI,
through analysis of measurements of the momentum distribution of the doubly
charged ions (Weber et al., 2000a,b; Moshammer et al., 2000).
The semiclassical rescattering model sketched above has proved invaluable
in providing intuitive understanding and predictive power. It was embedded in
fully quantum-mechanical descriptions of HHG (Lewenstein et al., 1994; Becker
et al., 1994b) and ATI (Becker et al. 1994a; Lewenstein et al., 1995a). This
work has led to the concept of “quantum orbits,” a fully quantum-mechanical
generalization of the classical orbits of the simple-man model that retains the
intuitive appeal of the former, but allows for interference and incorporates
quantum-mechanical tunneling. The quantum orbits arise naturally in the context
of Feynman’s path integral (Salières et al., 2001).
This review will concentrate on ATI and the various formulations of the
rescattering model, from the simplest classical model to the quantum orbits for
elliptical polarization. Alongside with theory, we will provide a review of the
experimental status of ATI. We also give a brief survey of recent applications of
ATI. High-order harmonic generation is considered only insofar as it provides
further illustrations of the concept and application of quantum orbits. We do not
deal with the important collective aspects of HHG, and no attempt is made to
represent the vast literature on HHG. For this purpose, we refer to the recent
reviews by Salières et al. (1999) and Brabec and Krausz (2000). Earlier reviews
pertinent to ATI have been given by Mainfray and Manus (1991), DiMauro and
Agostini (1995), and Protopapas et al. (1997). The entire field of laser–atom
physics has been compactly surveyed by Kulander and Lewenstein (1996) and,
recently, by Joachain et al. (2000). Both of these reviews concentrate on the
theory. Nonsequential double ionization is well covered in a recent focus issue
of Optics Express, Vol. 8.
38
W. Becker et al.
[I
A. Experimental Methods
ATI is observed in the intensity regime 1012 W/cm2 to 1016 W/cm2 . At such
intensities, atoms may ionize so quickly that complete ionization has taken
place before the laser pulse has reached its maximum. This calls, on the
one hand, for atoms with high ionization potential (i.e. the rare gases) and,
on the other, for ultrashort laser pulses. Owing to the rapid progress in
femtosecond laser technology, in particular since the invention of titanium–
sapphire (Ti:Sa) femtosecond lasers (Spence et al., 1991), generation of laser
fields with strengths comparable to inner atomic fields has become routine. The
prerequisite of detailed investigations of ATI, however, has been the development
of femtosecond laser systems with high repetition rate. Owing to the latter,
the detection of faint but qualitatively important features of ATI spectra with
low statistical noise has become possible. This holds, in particular, if multiply
differential ATI spectra are to be studied, such as angle-resolved energy spectra,
or spectra that are very weak, such as for elliptical polarization or outside the
classically allowed regions. State-of-the-art pulses are as short as 5 fs (Nisoli
et al., 1997) and repetition rates reach 100 kHz (Lindner et al., 2001).
The most widespread method of analyzing ATI electrons is time-of-flight
spectroscopy. When the laser pulse creates a photoelectron, it simultaneously
triggers a high-resolution clock. The electrons drift in a field-free flight tube
of known length towards an electron detector, which then gives the respective
stop pulses to the clock. Now, their kinetic energy can easily be calculated from
their time of flight. This approach has by far the highest energy resolution and
is comparatively simple. However, the higher the laser repetition rate, the more
demanding becomes the data aquisition.
Other approaches include photoelectron imaging spectroscopy (Bordas et al.,
1996), which is able to record angle-resolved ATI spectra, and so-called coldtarget recoil-ion-momentum spectroscopy (COLTRIMS) technology (Dörner
et al., 2000), which is capable of providing complete kinematic determination of
the fragments of photoionization, i.e. the electrons and ions. It requires, however,
conditions such that no more than one atom is ionized per laser shot. Therefore,
it can take particular advantage of high laser repetition rates. The disadvantage
of COLTRIMS is the poor energy resolution for the electrons and the exacting
technology.
B. Theoretical Methods
The single-active-electron approximation (SAE) replaces the atom in the laser
field by a single electron that interacts with the laser field and is bound by an
effective potential so optimized as to reproduce the ground state and singly
excited states. Up to now, in single ionization no qualitative effect has been
identified that would reveal electron–electron correlation. The SAE has found
I]
ATI: CLASSICAL TO QUANTUM
39
Fig. 2. (a) Measured and (b) calculated photoelectron spectrum in argon for 800 nm, 120 fs pulses
at the intensities given in TW/cm2 in the figure (10UP = 39 eV). From Nandor et al. (1999).
its most impressive support in the comparison of experimental ATI spectra in
argon with spectra calculated by numerical solution of the three-dimensional
time-dependent Schrödinger equation (TDSE) (Nandor et al., 1999); see Fig. 2.
The agreement between theory and experiment is as remarkable as it has been
achieved for low-order ATI in hydrogen; cf. Dörr et al. (1990) for the Sturmian–
Floquet calculation and Rottke et al. (1990) for the experiment. For helium, a
comparison of total ionization rates with and without the SAE in the abovebarrier regime has lent further support to the SAE (Scrinzi et al., 1999).
Numerical solution of the one-particle TDSE in one dimension was instrumental for the understanding of ATI in its early days; for a review, see Eberly
et al. (1992). For the various methods of solving the TDSE in more than one
dimension we refer to Joachain et al. (2000). Comparatively few papers have
dealt with high-order ATI in three (that is, in effect, two) dimensions. This is
particularly challenging since the emission of plateau electrons is caused by
very small changes in the wave function, and the large excursion amplitudes
of free-electron motion in high-intensity low-frequency fields necessitate a large
spatial grid. This is exacerbated for energies above the cutoff and for elliptical
polarization. Expansion of the radial wave function in terms of a set of B-spline
functions was used by Paulus (1996), by Cormier and Lambropoulos (1997),
and by Lambropoulos et al. (1998). Matrix-iterative methods were employed by
Nurhuda and Faisal (1999). The most detailed calculations have been carried
out by Nandor et al. (1999) and by Muller (1999a,b, 2001a,b). The techniques
are detailed by Muller (1999c). To our knowledge, no results for high-order ATI
for elliptical polarization based on numerical solution of the TDSE have been
published to this day.
Recently, numerical solution of the TDSE for a two-dimensional model
atom by means of the split-operator method has been widely used in order to
40
W. Becker et al.
[II
investigate various problems such as elliptical polarization (Protopapas et al.,
1997), stabilization (Patel et al., 1998; Kylstra et al., 2000), magnetic-drift effects
(Vázquez de Aldana and Roso, 1999; Vázquez de Aldana et al., 2001) and
various low-order relativistic effects (Hu and Keitel, 2001).
Efforts to deal with the two-electron TDSE and, in particular, to compute
double-electron ATI spectra are under way (Smyth et al., 1998; Parker et al.,
2001; Muller, 2001c). In one dimension for each electron, such spectra have
been obtained by Lein et al. (2001).
An approach that is almost complementary to the solution of the TDSE starts
from the analytic solution for a free electron in a plane-wave laser field, the socalled Volkov solution (Volkov, 1935), which is available for the Schrödinger
equation as well as for relativistic wave equations, and considers the binding
potential as a perturbation. The stronger the laser field, the lower its frequency,
and the longer the pulse becomes, the more demanding is the solution of the
TDSE, and the more the Volkov-based methods play out their strengths.
This review concentrates on methods of the latter variety.
II. Direct Ionization
A. The Classical Model
The classical model of strong-field effects divides the ionization process into
several steps (van Linden van den Heuvell and Muller, 1988; Kulander et al.,
1993; Corkum, 1993; Paulus et al., 1994a, 1995). In a first step, an electron
enters the continuum at some time t0 . If this is caused by tunneling (Chin et al.,
1985; Yergeau et al., 1987; Walsh et al., 1994), the corresponding rate is a highly
nonlinear function of the laser electric field E(t0 ). For example, the quasistatic
Ammosov–Delone–Krainov (ADK) tunneling rate (Perelomov et al., 1966a,b;
Ammosov et al., 1986) is given by (in atomic units)
2n∗ − |m| − 1
3
3
4 2EIP
4 2EIP
,
G(t) = AEIP
exp −
|E(t)|
3|E(t)|
(1)
where E(t) is the instantaneous
√ electric field, EIP > 0 is the ionization
potential of the atom, n∗ = Z/ 2EIP is the effective principal quantum number,
Z is the charge of the nucleus, and m is the projection of the angular momentum
on the direction of the laser polarization. The constant A depends on the actual
and the effective quantum numbers. The rate G(t) was derived on the assumption
that the laser √
frequency is low, excited states play no role, and the Keldysh
parameter g = EIP / 2UP is small compared with unity [UP is the ponderomotive
potential of the laser field; see Eq. (3) below]. Instantaneous rates that hold
for arbitrary values of the Keldysh parameter have been presented by Yudin
II]
ATI: CLASSICAL TO QUANTUM
41
and Ivanov (2001b). For the discussion below, the important feature of the
instantaneous ionization rate G(t) is that it develops a sharp maximum at times
when the field E(t) reaches a maximum.
The classical model considers the orbits of electrons that are released into the
laser-field environment at some time t0 . The contribution of such an orbit will
be weighted according to the value of the rate G(t0 ). Classically, an electron born
by tunneling will start its orbit with a velocity of zero at the classical “exit of the
tunnel” at r ~ EIP / |eE|, which, for strong fields, is a few atomic units away from
the position of the ion. We will, usually, ignore this small offset and have the
electronic orbit start at x(t = t0 ) = 0 (the position of the ion) with ẋ(t = t0 ) = 0.
If, after the ionization process, the interaction of the electron with the ion
is negligible, we speak of a “direct” electron, in contrast to the case, to be
considered below in Sect. III, where the electron is driven back to the ion and
rescatters. An unambiguous distinction between direct and rescattered electrons,
in particular for low energy, is possible only in theoretical models.
A.1. Basic kinematics
The second step of the classical model is the evolution of the electron trajectory
in the strong laser field. During this step, the influence of the atomic potential
is neglected. For an intense laser field, the electron’s oscillation amplitude is
much larger than the atomic diameter, and so this is well justified. For a vector
potential A(t) that is chosen so that its cycle average A(t)T is zero, the
electron’s velocity is
mv(t) = e(A(t0 ) − A(t)) ≡ p − eA(t),
(2)
where e = −|e| is the electron’s charge. The velocity consists of a constant term
p ≡ eA(t0 ), which is the drift momentum measured at the detector, and a term
that oscillates in phase with the vector potential A(t). The kinetic energy of this
electron, averaged over a cycle T of the laser field, is
m
p2
e2
v(t)2 T =
+
A(t)2 T ≡ Edrift + UP .
2
2m 2m
The ponderomotive energy
E=
(3)
e2
(4)
A(t)2 T ,
2m
viz. the cycle-averaged kinetic energy of the electron’s wiggling motion, is
frequently employed to characterize the laser intensity. A useful formula is
UP =
Up [eV] = 0.09337I [W/ cm2 ] l2 [m]
for a laser with intensity I and wavelength l. If the electron is to have a nonzero
velocity v0 at time t0 , one has to replace eA(t0 ) by eA(t0 ) + mv0 ≡ p in the
velocity (2).
42
W. Becker et al.
[II
Most of the time, we will be concerned with the monochromatic elliptically
polarized laser field (−1 x 1)
wA
(x̂ sin wt − x ŷ cos wt)
E(t) = 1 + x2
(5)
with vector potential
A(t) = A
1 + x2
(x̂ cos wt + x ŷ sin wt)
(6)
and ponderomotive energy UP = (eA)2 / 4m. The drift energy Edrift = (eA(t0 ))2 / 2m
is restricted to the interval
2x 2
2
UP Edrift UP .
2
1+x
1 + x2
(7)
For linear polarization, it can acquire any value between 0 and 2UP , while
for circular polarization it is restricted to the value UP . Quantum mechanics
considerably softens these classical bounds. However, these bounds are useful as
benchmarks in the analysis of experimental spectra (Bucksbaum et al., 1986),
in particular for high intensity (Mohideen et al., 1993).
In general, it is important to recall that the ionization probability depends
on the electric field, while the drift momentum p = eA(t0 ) is proportional
to the vector potential, both at the time t0 of ionization. The probability of
a certain drift momentum is weighted with the ionization rate at time t0 . The
electron is preferably ionized when the absolute value of the electric field is
near its maximum. Then, for linear polarization, the vector potential and, hence,
the drift momentum are near zero. In order to reach the maximal drift energy
of 2UP , the electron must be ionized when the electric field is zero and, hence,
the ionization rate is very low. This explains the pronounced drop of the ATI
electron spectrum for increasing energy, see Fig. 1. Sometimes, this interplay
between the instantaneous ionization rate and the drift momentum has surprising
consequences, notably for fields where the connection between the two is less
straightforward than for a linearly polarized sinusoidal field, e.g. for a two-color
field (Paulus et al., 1995; Chelkowski and Bandrauk, 2000; Ehlotzky, 2001).
Another illustration is the dodging phenomenon for the direct ATI electrons
in an elliptically polarized laser field (Paulus et al., 1998; Goreslavskii and
Popruzhenko, 1996; Mur et al., 2001), see Fig. 3.
We have tacitly assumed that pulses are short enough to pass over the electron
before it has a chance to experience the transverse spatial gradient of the
focused pulse. In this event, the spatial dependence of the vector potential A(t)
can truly be neglected. Hence the drift momentum p is conserved and is
II]
ATI: CLASSICAL TO QUANTUM
43
Fig. 3. Dependence of the photoelectron yield as a function of the ellipticity x of the elliptically
polarized laser field (5) for electrons with an energy of 16.1 eV. Only electrons emitted parallel to
the major axis of the polarization ellipse are recorded. The ATI spectrum corresponding to linear
polarization (x = 0) is shown in the inset. The laser intensity was 0.8×1014 W/cm2 at a wavelength
of 630 nm. The figure illustrates the dodging effect mentioned in Sect. II.A.1: ionization primarily
takes place when the electric field is near an extremum. For elliptical polarization, the electric field
then points in the direction of the major axis of the polarization ellipse, and the vector potential in
the direction of the minor axis. Hence, the electron’s drift momentum p = eA(t0 ) is in the direction of
the minor axis. It is the larger, the larger the ellipticity x is. Consequently, emission in the direction
of the large component of the field decreases with increasing ellipticity: the electron dodges the
strong component of the field. The effect vanishes when circular polarization is approached and the
distinction between the major and minor axes disappears. From Paulus et al. (1998).
indeed the momentum recorded at the detector outside the field (Kibble, 1966;
Becker et al., 1987). The wiggling energy UP is lost or, in a self-consistent
description, returned to the field when the electron is left behind by the trailing
edge of the pulse. In the opposite case, where the electron escapes from the
pulse perpendicularly to its direction of propagation, the wiggling energy is
converted into drift energy (Muller et al., 1983). The effects of a space-dependent
ponderomotive potential UP were observed in the “surfing” experiments of
Bucksbaum et al. (1987).
B. Quantum-mechanical Description of Direct Electrons
There is an enormous body of work on the quantum-mechanical description of
laser-induced ionization. For reviews, we refer to Delone and Krainov (1994,
1998). Here we want to concentrate on the analytical approach dating back to
Keldysh (1964) and Perelomov, Popov and Terent’ev (Perelomov et al. 1966a,b;
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[II
Perelomov and Popov, 1967). The goal is to find a suitable approximation to the
probability amplitude for detecting an ATI electron with drift momentum p that
originates from laser irradiation of an atom that was in its ground state |y0 before the laser pulse arrived:
Mp =
lim
t → ∞, t → −∞
yp (t) |U (t, t )| y0 (t ) .
(8)
Here, U (t, t ) is the time-evolution operator of the Hamiltonian (à = 1)
1
(9)
H (t) = − ∇2 − er · E(t) + V (r),
2m
which includes the atomic binding potential V (r) and the interaction −er · E(t)
with the laser field. Furthermore, we introduce the Hamiltonians for the atom
without the field and for a free electron in the laser field without the atom,
1
(10)
Ha ≡ Hatom = − ∇2 + V (r),
2m
1
(11)
Hf (t) ≡ Hfield (t) = − ∇2 − er · E(t).
2m
The corresponding time-evolution operators are denoted by Ua and Uf , respectively. In Eq. (8), |yp and |y0 are a scattering state with asymptotic
momentum p and the ground state, respectively, of the atomic Hamiltonian Ha .
The eigenstates of the time-dependent Schrödinger equation with the Hamiltonian Hf (t) are known as the Volkov states and are of compact analytical form.
In the length gauge, one has
|yp(Vv) (t) = |p − eA(t)e−iSp (t) ,
(12)
exp i[(p − eA(t)) · r]
with |p − eA(t) a plane-wave state [r|p − eA(t) = (2p )
and
t
1
Sp (t) =
dt [p − eA(t)]2 .
(13)
2m
The lower limit of the integral is immaterial. It introduces a phase that does not
contribute to any observable.
The time-evolution operator U (t, t ) satisfies integral equations (Dyson equations) that are convenient if one wants to generate perturbation expansions with
respect to either the interaction HI (t) = −er · E(t) with the laser field,
t
(14)
U (t, t ) = Ua (t, t ) − i dt U (t, t) HI (t) Ua (t, t ),
−3/ 2
t
or the binding potential V (r),
t
U (t, t ) = Uf (t, t ) − i
dt Uf (t, t) V U (t, t ).
(15)
t
Equation (14) also holds if U and Ua in the second term on the right-hand side
are interchanged. The equivalent is true of Eq. (15). With the help of the integral
II]
ATI: CLASSICAL TO QUANTUM
45
equation (14), using the orthogonality of the eigenstates of Ha , we rewrite Eq. (8)
in the form
t
dt yp (t) |U (t, t)HI (t)| y0 (t) ,
Mp = −i lim
t→∞
(16)
−∞
which is still exact. A crucial simplification occurs if we now introduce the
strong-field approximation. That is, we make the substitutions |yp → |yp(Vv) and U → Uf , with the result
∞
Mp = −i
dt0 yp(Vv) (t0 ) |HI (t0 )| y0 (t0 ) .
(17)
−∞
The physical content of this substitution is that, after the electron has been
promoted into the continuum at time t0 due to the interaction HI (t0 ) = −er · E(t0 )
with the laser field, it no longer feels the atomic potential. This satisfies the above
definition of a “direct electron.” Amplitudes of the type (17) are called Keldysh–
Faisal–Reiss (KFR) amplitudes (Keldysh, 1964; Perelomov et al., 1966a,b;
Faisal, 1973; Reiss, 1980); for a comparison of the various forms that exist see
Reiss (1992).
In the amplitude (17), one may write
− er · E(t0 ) = Hf (t0 ) − Ha + V (r) = −i
←−− −−→
ð
ð
−i
+ V (r).
ðt0
ðt0
(18)
Via integration by parts, the amplitude (17) can then be rewritten as
∞
Mp = −i
dt0 yp(Vv) (t0 ) |V (r)| y0 (t0 ) .
(19)
−∞
This form is particularly useful for a short-range or zero-range potential, since
these restrict the range of the spatial integration in the matrix element.
Further evaluation of the amplitudes (17) or (19) leads to expansions
in terms of Bessel functions. For sufficiently high intensity (small Keldysh
parameter g), the saddle-point method (method of steepest descent) can be
invoked (Dykhne, 1960). This consists in expanding the phase of the integrand
about the points where the phase is stationary. Given the form of the Volkov
wave functions (12) and the time dependence of the ground-state wave function,
|y0 (t) = exp(iEIP t)|y0 , this amounts to determining the solutions of
d
[EIP t + Sp (t)] = EIP + 12 [p − eA(t)]2 = 0.
dt
(20)
46
W. Becker et al.
[II
Let us consider a periodic (not necessarily monochromatic) vector potential with
period T = 2p / w. In terms of the solutions ts of Eq. (20), the amplitude (19)
can then be written as
p2
Mp ∝
d
+ EIP + UP − nw
2m
n
1/ 2
(21)
2p i
×
ei[EIP ts + Sp (ts )] p − eA(ts )|V |y0 ,
(t )
S
s
p
s
where Sp denotes the second derivative of the action (13) with respect to time.
The sum over s extends over those solutions of Eq. (20) within one period of
the field (e.g. such that 0 Re ts < T ) that have a positive imaginary part.
Obviously, the saddle points are complex unless EIP = 0. For EIP = 0, we retrieve
the classical drift momentum (2) provided p is such that p = eA(t) at some time t.
The imaginary part of t0 can be related to a tunneling time (Hauge and Støvneng,
1989).
In Eq. (21), the ionization amplitude is represented as the coherent sum over all
saddle points within one period of the field. The fact that the spectrum consists
of the discrete energies
Ep ≡
p2
= nw − UP − EIP
2m
(22)
can be attributed to interference of the contributions from different periods.
This interference is destructive, unless the energy corresponds to one of the
discrete peaks (22). Depending on the shape of the vector potential A(t) and
its symmetries, there will be several solutions ts (for a sinusoidal field, there are
two in the upper half plane and two in the lower, which are complex conjugate
to the former) within one period of the field. Their interference creates a beat
pattern in the calculated spectrum. This is, however, difficult to observe due to
its sensitive dependence on the laser intensity, which is not very well controlled
in an experiment so that the interference effects are usually washed out.
Interferences also exist for an elliptically polarized laser field for fixed electron
momentum as a function of the ellipticity. Since, in experiments, the ellipticity is
better defined than the intensity, these interferences have been observed (Paulus
et al., 1998); see next subsection.
The amplitude (19) admits a vector potential A(t) of arbitrary shape; it is
by no means restricted to a monochromatic field of infinite extent. For a pulse
of finite extent, the saddle points are still determined by Eq. (20). They have,
however, no longer any periodicity. Hence, the discreteness of the spectrum is
lost. Interference from different parts of the pulse may lead to unexpected effects
(Raczyński and Zaremba, 1997).
II]
ATI: CLASSICAL TO QUANTUM
47
While for an infinitely long monochromatic pulse the spectrum is symmetric
upon p → −p, this forward–backward symmetry no longer holds for a
finite pulse. Analysis of the spatial asymmetry of the spectrum may aid in
determination of the pulse length or the absolute carrier phase (Dietrich et al.,
2000; Hansen et al., 2001; Paulus et al., 2001b); see Sec. VI.B.
C. Interferences of Direct Electrons
For linear polarization and a drift momentum p = px̂ with |p| eA, there
are two possible ionization times wt01 = p / 2 + d and wt02 = 3p / 2 − d. The
corresponding classical orbits are illustrated in Fig. 4. As discussed above, while
A(t01 ) = A(t02 ), the field satisfies E(t01 ) = −E(t02 ). Hence, electrons ionized at
t01 and t02 depart in opposite directions right after the instant of ionization. As
illustrated in Fig. 4, the electric field changes sign soon after t01 . Hence, the
electron ionized at this time turns around at a later time and acquires the same
drift momentum as the electron ionized at time t02 , which keeps its original
direction. We expect quantum-mechanical interference of the contribution of
these two ionization channels.
For elliptical polarization, classically, there is at most one ionization time
for given drift momentum. However, Eq. (20) for the complex saddle points of
the quantum-mechanical
amplitude always has more than one solution. For the
√
field (6) and p = 2mE x̂, the solutions are
cos wts = z
1 + x2
√
− û ± ûx 2 − ûIP (1 − x 2 ) − x 2 z ,
(23)
where z = (1 − x 2 )/ (1 + x 2 ), û = E/ 2UP , and ûIP = EIP / 2UP . Obviously, the
solutions ts come in complex conjugate pairs. Those in the upper half-plane
Fig. 4. Classical trajectories (dashed lines) of electrons having the same drift momentum. The solid
line is the effective potential V (x) − exE(t) at times t01 (left) and t02 (right). The electron ionized
at t01 is turned around by the field shortly after ionization. In contrast, the electron ionized at t02
maintains its original direction. This is a strongly simplified picture of the physics underlying the
interferences of direct electrons.
48
W. Becker et al.
[II
Fig. 5. (a) Positions of the saddle points wts in the upper half of the complex wt plane in the interval
1 p Re wt 3 p , calculated from Eq. (23) for E = 17w − E − U , where E
IP
p
IP = 15.76 eV,
2
2
UP = 3.68 eV, and àw = 1.96 eV. The arrows indicate the motion of the saddle points for increasing
ellipticity x. The two branches meet at x0 = 0.755. For several values of the ellipticity, insets depict
the ellipse traced out by the electric-field vector, and the positions of the latter at the emission
times Re ts are marked by solid dots. (b) The function Re F, which determines the magnitude of the
amplitude Mp . The existence of the valley near the ellipticity x0 is related to the effect of dodging,
illustrated in Fig. 3. (c) The function cos2 (Im F + y), whose oscillations are caused by constructive
and destructive interference. The essential physics behind this interference is sketched in Fig. 4.
From Paulus et al. (1998).
enter the amplitude (21). The solutions are plotted in Fig. 5. For −x0 < x x0
[with x0 given by the zero of the square root in Eq. (23)], the second square
root on the right-hand side of Eq. (23) is imaginary, and the solutions are
symmetric with respect to Re wts = p . For |x| x0 , this square root is
real, and all solutions have Re wts = p . This has important consequences for
the saddle-point amplitude (21). In the first case, both solutions contribute to
the amplitude and an interference pattern results. This corresponds to the case
of linear polarization discussed above. In the second case, inspection of the
integration contour in the complex plane shows that it has to be routed only
through the one solution that is closest to the real axis (Leubner, 1981). Hence,
there is no interference. In the first case, the amplitude can be written in the
form Mp ~ exp(Re F) cos(Im F + y), in the second case the cosine is absent.
The two arguments are also plotted in Fig. 5.
The corresponding interferences have been observed by Paulus et al. (1998);
see Fig. 6. They are responsible for the undulating pattern in the ellipticity
distribution, which moves to smaller ellipticity for increasing energy. The same
tendency can be observed in the numerical evaluation of the amplitude (21); see
Paulus et al. (1998) for an example.
For elliptical polarization, the KFR amplitude (19) must be applied with
due caution: it predicts fourfold symmetry of the angular distribution, while
II]
ATI: CLASSICAL TO QUANTUM
49
Fig. 6. ATI spectra in xenon for an intensity of 1.2×1014 W/cm2 for various energies in the direction
of the large component of the elliptically polarized field as a function of the ellipticity. The inset
shows the energy spectrum for linear polarization. The three traces for the lower energies display
the interference phenomenon of the direct electrons discussed in Sect. II.C; the one for the highest
energy belongs to a plateau electron. The interference dips are related to Fig. 5c. From Paulus et al.
(1998).
the experimental distributions only display inversion symmetry (Bashkansky
et al., 1988). Mending this deficiency requires improved treatment of the binding
potential (Krstić and Mittleman, 1991). More discussion of this point has been
provided elsewhere by Becker et al., 1998, who also give further references.
The spatial dependence introduced by Coulomb–Volkov solutions in place of the
usual Volkov solutions (12) already suffices to destroy the fourfold symmetry,
and angular distributions have been calculated with their help by Jaroń et al.
(1999). However, even for a zero-range potential the fourfold symmetry is
broken provided the effects of the finite binding energy are treated beyond the
KFR approximation (Borca et al., 2001).
Very similar interferences have been seen by Bryant et al. (1987) in the
photodetachment of H− in a constant electric field. Here the electron, once
detached, has the choice of starting its subsequent travel either against or with
the direction of the electric field, by close analogy with the opposite directions
of initial travel for the ionization times t01 and t02 in the present case; see Fig. 4.
A spatial resolution of the same effect is observed by the photodetachment
microscope of Blondel et al. (1999). The theoretical description reproduces the
50
W. Becker et al.
[III
observed patterns. Additional bottle-neck structures develop when a magnetic
field is applied parallel to the electric field (Kramer et al., 2001).
III. Rescattering
Thus far, we have dealt with “direct” electrons, which after the first step of
ionization leave the laser focus without any additional interaction with the ion.
Next, we will consider the consequences of one such additional encounter.
A. The Classical Theory
The classical model becomes much richer if rescattering effects are taken into
account. To this end, we integrate the electron’s velocity (2) to obtain its
t
trajectory
e
x(t) =
(t − t0 ) A(t0 ) −
dt A(t) .
(24)
m
t0
The condition that the electron return to the ion at some time t1 > t0 is
x(t1 ) = 0. For linear polarization in the x-direction, this implies x(t1 ) = 0, and
y(t) ≡ z(t) ≡ 0. This yields t1 as a function of t0 . We defer discussion of elliptical
polarization to a later time.
When the electron returns, one of the following can happen (Corkum, 1993):
(1) The electron may recombine with the ion, emitting its energy plus the
ionization energy in the form of one photon. This process is responsible
for the plateau of high-order harmonic generation.
(2) The electron may scatter inelastically off the ion. In particular, it may dislodge a second electron (or more) from the ionic ground state. This process is
now believed to constitute the dominant contribution to nonsequential double
ionization.
(3) The electron may scatter elastically. In this process, it can acquire drift
energies much higher than otherwise.
In the following, we will concentrate on this high-order above-threshold
ionization (HATI). We will, however, also briefly discuss high-order harmonic
generation.
From Eq. (2), the kinetic energy of the electron at the time of its return is
Eret =
e2
[A(t1 ) − A(t0 )]2 .
2m
(25)
Maximizing this energy with respect to t0 under the condition that x(t1 ) = 0 yields
Eret, max = 3.17UP for wt0 = 108◦ and wt1 = 342◦ (Corkum, 1993; Kulander
et al., 1993). It is easy to see that after rescattering the electron can attain a
III]
ATI: CLASSICAL TO QUANTUM
51
Fig. 7. Graphical solution of the return time t1 for given start time t0 ; cf. Paulus et al. (1995): The
return condition x(t1 ) = 0 can be written in the form F(t1 ) = F(t0 ) + (t1 − t0 ) F (t0 ), where the
function F(t) = dt A(t) ~ sin wt (solid curve) is an integral of the vector potential A(t) ~ cos wt
(dotted curve). The thick solid straight line, which is the tangent to F(t) at t = t0 , intersects F(t)
for the first time at t = t1 . The start (ionization) time t0 was chosen such that the kinetic energy Eret
(Eq. 25) at the return time t1 is maximal and equal to Eret,max = 3.17UP . The two adjacent straight
lines both yield the same kinetic energy Eret < Eret,max . The figure shows that one starts earlier and
returns later while the other one starts later and returns earlier. Obviously, there can be many more
intersections with larger values of t1 provided the start times are near the extrema of F(t). They
correspond to the orbits with longer travel times.
much higher energy: Suppose that at t = t1 the electron backscatters by 180◦ , so
that mv(t1 − 0) = e[A(t0 ) − A(t1 )] just before and mv(t1 + 0) = −e[A(t0 ) − A(t1 )]
just after the event of backscattering. Then, for t > t1 , the electron’s velocity is
again given by Eq. (2), but with px = e[2A(t1 ) − A(t0 )] so that
Ebs =
e2
[2A(t1 ) − A(t0 )]2 .
2m
(26)
Maximizing Ebs under the same condition as above yields Ebs, max = 10.007UP
(Paulus et al., 1994a) for wt0 = 105◦ and wt1 = 352◦ . These values are very
close to those that afford the maximal return energy.
It is important to keep in mind that for maximal return energy or backscattering energy, the electron has to start its orbit shortly after a maximum of the
electric field strength. As a consequence, it returns or rescatters near a zero of
the field, see Fig. 7. This also provides an intuitive explanation of the energy
52
W. Becker et al.
[III
Fig. 8. Maximum drift energy after rescattering (ATI plateau cutoff) upon the mth return to the ion
core during the ionization process. Electrons with the shortest orbits (m = 1) can acquire the highest
energy, whereas electrons that pass the ion core once before rescattering at the second return (m = 2)
have a rather low energy. Each return corresponds to two quantum orbits: the mth return corresponds
to the quantum orbits 2m + 1 and 2m + 2.
gain through backscattering: if the electron returns near a zero of the field and
backscatters by 180◦ , then it will be accelerated by another half-cycle of the
field.
In general, the equation x(t1 ) = 0 for fixed t0 may have any number of
solutions. This becomes evident from the graphical solution presented in Fig. 7.
If the electron starts at a time t0 just past an extremum of the field, it returns
to the ion many times. These solutions having long “travel times” t1 − t0 are
very important for the intensity-dependent quantum-mechanical enhancements
of the ATI plateau to be discussed in Sect. III.B.7. Here we will be satisfied
with mentioning another property of the classical orbits: obviously, the return
energy will have extrema, e.g. the maximum of Ebs, max = 10.007UP mentioned
above, which is assumed for a certain time t0,max (t0,max = 108◦ in the example).
If we are interested in a fixed energy Ebs < Ebs, max , there are two start times
that will lead to this energy: one earlier than t0, max , the other one later. From
the graphical construction of Fig. 7 it is easy to see that the former has a longer
travel time than the latter. In the closely related case of HHG, these correspond
to the “long” and the “short” orbit (Lewenstein et al., 1995b). The cutoffs of the
solutions with longer and longer travel times are depicted in Fig. 8.
If we consider rescattering into an arbitrary angle q with respect to the
direction of the linearly polarized laser field, we expect a lower maximal energy
since part of the maximal energy 3.17UP of the returning electron will go into the
III]
ATI: CLASSICAL TO QUANTUM
53
transverse motion. This implies that, for fixed energy Ebs , there is a cutoff in the
angular distribution; in other words, rescattering events will only be recorded for
angles such that 0 q qmax (Ebs ). This is a manifestation of rainbow scattering
(Lewenstein et al., 1995a). All of this kinematics is contained in the following
equations (Paulus et al., 1994a):
(27)
Ebs = 12 A(t0 )2 + 2A(t1 ) [A(t1 ) − A(t0 )] (1 ± cos q0 ) ,
A(t1 )
cot q = cot q0 −
.
(28)
sin q0 |A(t0 ) − A(t1 )|
Here q0 is the scattering angle at the instant of rescattering, which may have
any value between 0 and p , as opposed to the observed scattering angle q at
the detector (outside the field). In Eq. (27), the upper (lower) sign holds for
A(t0 ) > A(t1 ) (A(t0 ) < A(t1 )).
Pronounced lobes in the angular distributions about the polarization direction
were first observed by Yang et al. (1993), while the rescattering plateau in the
energy spectrum with its cutoff at 10UP was identified by Paulus et al. (1994b,c).
These spectra prominently display the classical cutoffs at qmax and Ebs,max .
The classical features become the better developed the higher the intensity is.
Hence, they are particularly conspicuous in the strong-field tunneling limit.
This has been shown theoretically by comparison with numerical solutions of
the Schrödinger equation (Paulus et al., 1995) and experimentally for He at
intensities around 1015 W/cm2 . Indeed, the latter spectra show an extended
plateau for energies between 2UP and 10UP (Walker et al., 1996; Sheehy
et al., 1998). For comparatively low intensities, angular distributions have been
recorded in xenon with very high precision by Nandor et al. (1998). They also
show the effects just discussed, but with much additional structure that appears to
be attributable to quantum-mechanical interference and to multiphoton resonance
with ponderomotively upshifted Rydberg states (Freeman resonances; Freeman
et al., 1987).
B. Quantum-mechanical Description
In order to incorporate the possibility of rescattering into the quantummechanical description, we have to allow the freed electron once again to interact
with the ion (Lohr et al., 1997). To this end, we return to the exact equation (16)
and insert the Dyson integral equation (15). This yields two terms. Next, as we
did in Sect. II.B, we replace the exact scattering state |yp by a Volkov state and
the exact time-evolution operator U by the Volkov-time evolution operator Uf .
In other words, we disregard the interaction with the binding potential V (r),
except for the one single interaction that is explicit in the Dyson equation. This
procedure corresponds to adopting the Born approximation for the rescattering
process.
54
W. Becker et al.
[III
Of the two terms, the first is identical with the “direct” amplitude (17) or (19).
The second describes rescattering. Via integration by parts similar to that
explained in Eq. (18) the two terms can be combined into one,
∞
Mp = −i
t1
dt1
−∞
dt0 yp(Vv) (t1 ) |VUf (t1 , t0 )V | y0 (t0 ) ,
(29)
−∞
which now describes both the direct and the rescattered electrons. The physical
content of the amplitude (29) corresponds to the recollision scenario: The
electron is promoted into the continuum at some time t0 ; it propagates in the
continuum subject to the laser field until at the later time t1 it returns to within
the range of the binding potential, whereupon it scatters into its final Volkov
state.
Exact numerical evaluation of the amplitude (29) for a finite-range binding
potential is very cumbersome. For a zero-range potential, however, the spatial
integrations in the matrix element become trivial, and the computation is rather
straightforward. If the field dependence of the Volkov wave function and the
Volkov time-evolution operator is expanded in terms of Bessel functions, one of
the temporal integrations in the amplitude (29) can be carried out analytically
and yields the same d function as in Eq. (21), specifying the peak energies. The
remaining quadrature with respect to the travel time t1 − t0 has to be carried
out numerically; see Lohr et al. (1997) and Milošević and Ehlotzky (1998a),
where explicit formulas can be found; for elliptical polarization see Becker et al.
(1995) and Kopold (2001). Alternatively, the integral over the travel time may
be done first, and the integral over the return time t1 is then evaluated by Fourier
transformation (Milošević and Ehlotzky, 1998b).
The relevance of the rescattering mechanism to ATI and multiple ionization
was suggested early by Kuchiev (1987) and by Beigman and Chichkov (1987).
Improvements of the customary KFR theory by including further interactions
with the binding potential were already discussed by Reiss (1980). The first
explicit calculations of angular-resolved energy spectra were carried out by
Becker et al. (1994a, 1995) and by Bao et al. (1996). Closely related rescattering
models were presented by Smirnov and Krainov (1998) and by Goreslavskii and
Popruzhenko (1998, 2000).
The physics of high-order ATI is related to electron scattering at atoms in the
presence of a strong laser field. In the former case, the initial state of the electron
is a wave packet created by tunneling, while in the latter it is a plane-wave state.
This latter problem was studied theoretically by Bunkin and Fedorov (1966)
and by Kroll and Watson (1973). Corresponding experiments were done by
Weingartshofer et al. (1977, 1983). Some quantum features of electron scattering
in intense laser fields are remarkably similar to HATI; see Kull et al. (2000) and
Görlinger et al. (2000).
III]
ATI: CLASSICAL TO QUANTUM
55
B.1. Saddle-point methods
For sufficiently high intensity, the temporal integrations in the amplitude (29)
can be carried out by the saddle-point method, as in the case of the direct
amplitude (19). This procedure provides much more physical insight than
Bessel-function expansions, and establishes the connection with Feynman’s path
integral, to be discussed below.
In this context, rather than taking advantage of the explicit form of the Volkov
time-evolution operator, we expand it in terms of the Volkov states (12),
Uf (t1 , t0 ) = d 3 k yk(Vv) (t1 ) yk(Vv) (t0 ) ,
(30)
so that the amplitude Mp is represented by the five-dimensional integral
t1
∞
Mp ~
dt1
dt0 d 3 k exp[iSp (t1 , t0 , k)] mp (t1 , t0 , k)
−∞
(31)
−∞
with the function
mp (t1 , t0 , k) = p − eA(t1 ) |V | k − eA(t1 ) k − eA(t0 ) |V | y0 .
(32)
For ATI, the action
Sp (t1 , t0 , k) = −
1
2m
1
−
2m
∞
dt [p − eA(t)]2
t1
t1
dt [k − eA(t)] +
t0
(33)
t0
2
dt EIP
−∞
in the exponent consists of three parts, according to the three stages discussed
above.
As above in Eq. (20), we approximate the amplitude (31) by expanding the
phase (33) of the integrand about its stationary points. In this process, we assume
that the function mp (t1 , t0 , k) depends only weakly on its arguments. Indeed, for
a zero-range potential, it is a constant. We now have to determine the stationary
points with respect to the five variables t1 , t0 and k. They are given by the
solutions of the three conditions (Lewenstein et al., 1995a)
[k − eA(t0 )]2 = −2mEIP ,
t0
(t1 − t0 ) k =
dt eA(t),
(34)
(35)
t1
[k − eA(t1 )]2 = [p − eA(t1 )]2 ,
(36)
respectively. The first condition (34) attempts to enforce energy conservation
at the time of tunneling. The second condition (35) ensures that the electron
56
W. Becker et al.
[III
returns to its parent ion, and the third one (36) expresses that, on this occasion,
it rescatters elastically into its final state. In general, the saddle-point equations
have several solutions (t1s , t0s , ks ), (s = 1, 2, . . .), of which only those are relevant
for which Re t1s > Re t0s , such that the recollision is later than ionization. The
matrix element can be written as
Mp ~
s
(2p ià)5
det(ð 2 Sp /ðqj(s) ðqk(s) )j, k = 1, ..., 5
1/ 2
eiSp (t1s ,t0s ,ks ) mp (t1s , t0s , ks ),
(37)
where qi(s) (i = 1, . . . , 5) runs over the five variables t1s , t0s and ks . As we
noted already for the direct electrons in the context of Eq. (21), the sum has
to be extended only over a subset of the solutions of the saddle-point equations
(34)−(36). However, in the present case, determining this subset may be tricky
(Kopold et al., 2000a). For a periodic field, the sum over the periods in Eq. (37)
can be carried out by Poisson’s formula. This leaves a sum over the saddle points
within one period and produces a d function as in Eq. (21).
The computation of ATI now consists of two separate tasks. First, the solutions
of the saddle-point equations (34)−(36) have to be determined and, second,
the appropriate subset has to be inserted into expression (37). Note that we
apply the saddle-point approximation to the probability amplitude for given final
momentum p, and not to the complete wave function of the final state. This is the
reason why only few solutions contribute, while a semiclassical computation of
the wave function, which contains all possible outcomes, requires consideration
of a very large number of trajectories (van de Sand and Rost, 2000).
Since EIP > 0, the condition (34) of “energy conservation” at the time of
ionization cannot be satisfied for any real time t0 . As a consequence, all solutions
(t1s , t0s , ks ) become complex. If the ionization potential EIP is zero, then, for
a linearly polarized field, the first saddle-point equation (34) implies that the
electron starts on its orbit with a speed of zero. Provided the final momentum p
is classically accessible, the resulting solutions are entirely real. They correspond
to the so-called “simple-man model” (van Linden van den Heuvell and Muller,
1988; Kulander et al., 1993; Corkum, 1993). For EIP Ñ 0, so long as the Keldysh
parameter g 2 = EIP / (2UP ) is small compared with unity, the imaginary parts of
the solutions of Eqs. (34)–(36) are still not too large, and the real parts are still
close to these simple-man solutions. In this case, approximate analytical solutions
to the saddle-point equations can be written down, which yield an analytical
approximation to the amplitude (31) (Goreslavskii and Popruzhenko, 2000). On
the other hand, for elliptical polarization, the solutions are always complex, even
when EIP = 0. This reflects the fact that, for any polarization other than linear,
an electron set free at any time during the optical cycle with velocity zero will
never return to the point where it was released. Equation (34) then only implies
that k − eA(t0 ) is a complex null vector.
III]
ATI: CLASSICAL TO QUANTUM
57
With the solutions (t1s , t0s , ks ) (s = 1, 2, . . .) of Eqs. (34)–(36), the sth quantum
orbit has the form
t
(t − t0s )ks − t0s dt eA(t) (Re t0s t Re t1s ),
(38)
mx(t) =
t
(t − t1s )p − t1s dt eA(t) (t Re t1s ).
We regard the orbit as a function of the real time t. The conditions x(t0 ) = 0
and x(t1 ) = 0, however, are satisfied for the complex times t0 and t1 . As a
consequence, the quantum orbit (38) as a function of real time does not depart
from the origin but, rather, from the “exit of the tunnel.” This is clearly visible in
Figs. 14, 15, 17 and 20 below. In contrast to the start time t0 , the return time t1
is real to a good approximation, see Fig. 10 (below). In consequence, the orbits
return almost exactly to the origin.
B.2. Connection with Feynman’s path integral
Any quantum-mechanical transition amplitude, such as the ionization amplitude (8), can also be represented in terms of Feynman’s path integral. To this
end, we recall the path-integral representation of the complete time-evolution
operator of the system atom + field,
U (rt, r t ) =
D[r(t)]eiS(t,t ) ,
(39)
(rt) ↔ (r t )
t
where S(t, t ) = t dt L[r(t)], t] is the action calculated along a system path, and
the integral measure D[r(t)] mandates summation over all paths that connect
(rt) and (r’ t ) (see, e.g., Schulman, 1977).
The path integral (39) sums over the functional set of all continuous
paths. In the quasi-classical limit, this can be reduced to a sum over all
classical paths, which are those for which the action S(t, t ) is stationary.
For quadratic Hamiltonians, this WKB approximation is exact. In our case,
motivated by the success of the classical three-step model of Sect. III.A, we
have reduced the exact transition amplitude to the form (31). In implementing
the strong-field approximation, we have approximated the exact action of the
system appropriately at the various stages of the process: before the initial
ionization, in between ionization and rescattering, and after rescattering, as in
the decomposition (33) of the action. This still left us with a five-dimensional
variety of paths. Out of those, finally, the saddle-point approximation (37) selects
the handful of “relevant paths” (Antoine et al., 1997; Kopold et al., 2000a;
Salières et al., 2001). These are essentially the orbits of the classical model, yet
quantum mechanics is fully present: Their coherent superposition as expressed
in the form (37) allows for interference of the contributions of different orbits,
and the fact that they are complex accounts for their origin via tunneling.
58
W. Becker et al.
[III
B.3. Connection with closed-orbit theory
There appears to be a close similarity to the concepts of periodic-closed-orbit
theory, see, e.g., Du and Delos (1988), Gutzwiller (1990), and Delande and
Buchleitner (1994). The photoabsorption cross section s (E) of an atom in the
state |yi with energy Ei can be expressed in the form (Du and Delos, 1998)
∞
e2
iEt
s (E) = 4p Re
dt e yi |DU (t, 0)D| yi ,
(40)
àc
0
where D = r · û is the dipole operator responsible for photoabsorption of the
field with polarization û and E ≈ Ei + àw. The quantity U (t, 0) is the timeevolution operator in the presence of the binding potential as well as additional
static external electric and magnetic fields that may be present. In effect, the timeevolution operator propagates wave packets at constant energy that emanate from
the atom and are reflected by the caustics of the potential back to the atom where
they interfere with each other and with the starting wave packets. This leads to
oscillations in the photoabsorption spectrum. In a semiclassical approximation,
the time-evolution operator can be expanded in terms of classical closed orbits
that start from and return to the vicinity of the atom, defined by the spatial range
of the wave function |yi . Since the classical problem is chaotic, there are more
and more such orbits when the energy nears zero. Fourier transformation of the
photoabsorption spectrum reveals the recurrence times of the classical orbits.
There are several differences to the quantum orbits we are considering here.
In our case, the role of the binding potential is, in effect, reduced to acting as
a coherent source of electrons and to causing rescattering, while in closed-orbit
theory the interplay of its spatial shape with the external static fields generates
the rich structure of the closed orbits. In our case, closed orbits are entirely due
to the time dependence of the laser field. The most important difference is that
closed-orbit theory is concerned with total photoabsorption rates as a function
of frequency, while we consider differential electron spectra for a laser field
with fixed frequency. In other words, our orbits depend on the final state of
the electron.
From Eq. (19), in view of the completeness of the Volkov states, the total
ionization probability due to direct electrons is
d 3 p |Mp |2
∞
t1
2
(Vv)
= 2e Re
dt1
dt0 y0 (t1 ) r · E(t1 )U
(t1 , t0 )r · E(t0 ) y0 (t0 ) .
−∞
−∞
(41)
This differs from the photoabsorption cross section (40) only by the presence
of the Volkov time-evolution operator U (Vv) (t1 , t0 ), which reflects the strongfield approximation, instead of the exact time-evolution operator U (t, 0) of the
III]
ATI: CLASSICAL TO QUANTUM
59
time-independent problem [for which the time-evolution operator U (t, t ) only
depends on the time difference t − t ]. In the total ionization probability (41),
via the same partial integration (18) as above, the electron–field interaction
r · E(t) can be replaced by V (r). The result then looks like the differential
HATI amplitude (29) except that it is sandwiched by the ground state. This
correspondence is a manifestation of the optical theorem.
B.4. The role of the binding potential
The improved Keldysh approximation (29) has been written down for an
arbitrary binding potential V (r). The expansion in terms of the binding potential,
introduced via the Dyson equation (15), is a strong-field approximation (SFA),
which is valid when the electron’s quiver amplitude is so large that most of its
orbit is outside the range of the binding potential. This is trivially guaranteed
for the three-dimensional binding potential of zero range,
V (r) =
ð
2p
d(r) r.
mú
ðr
(42)
This potential supports a single (s-wave) bound state at the energy −ú 2 / 2m
and a continuum that is undistorted from the free continuum except for the
s wave, as required by completeness (Demkov and Ostrovskii, 1989). Without the
regularization operator (ð/ðr) r, which acts on the subsequent state, the potential
does not admit any bound state. There are several possibilities to adjust the one
parameter ú to an individual atom or ion. In most cases one will determine it
so as to reproduce the ionization potential; see, however, Sect. III.B.7.
The zero-range potential (42) underlies many of the explicit results exhibited
in this chapter. However, we emphasize that the amplitude (29), as well as its
saddle-point approximation (37), hold for a much wider class of potentials.
Regardless of the potential, the saddle-point equations (34)−(36) have the
electron start from and return to the center of the binding potential, which is
the origin, and do not depend on its shape. The potential only enters via the
form factors in Eq. (32). For the SFA to be applicable, they must depend on
time only weakly. The procedure corresponds to the Born approximation. It will
be the better justified, the shorter the range of the potential is, so that the form
factor depends only weakly on the momenta. Excited bound states do not enter
the amplitude (29) regardless of the potential used.
For a comparison of a high-order ATI spectrum calculated for the zero-range
potential (42) with the same spectrum extracted from a solution of the threedimensional TDSE for hydrogen, see Cormier and Lambropoulos (1997) for the
latter and Kopold and Becker (1999) for the former. There is good qualitative
agreement within the ATI plateau; in particular, the positions of the dips in the
spectrum that are due to destructive interference agree within a few percent.
The comparison confirms that the detailed shape of the potential has only
60
W. Becker et al.
[III
minor significance for the HATI spectrum. Clearly, however, the real physical
systems best described by a zero-range potential are negative ions with a s-wave
ground state.
B.5. A homogeneous integral equation
An alternative route to the standard KFR matrix element (19) and its improved
version (29) starts from the homogeneous integral equation
t
|Y(t) = −i
dt Uf (t, t) V |Y(t),
(43)
−∞
which holds for the state that develops out of the unperturbed ground state
due to its interaction with the laser field. This integral equation can be derived
immediately from the Dyson equation (15) if one applies both sides of the latter
to the atomic ground state |y0 (t ) in the limit where t → −∞. By inspection,
one may convince oneself that the term Uf (t, t )|y0 t ) makes no contribution
for t − t → ∞ so that Eq. (43) is left. This equation was first introduced in the
context of the quasi-energy formalism by Berson (1975) and by Manakov and
Rapoport (1975) for circular polarization and Manakov and Fainshtein (1980)
for arbitrary polarization.
Inserting on the right-hand side of the integral equation (43) the expansion (30)
of Uf in terms of Volkov states and replacing |Y(t) by the unperturbed
atomic ground state |y0 (t), one can read off the matrix element (19) for direct
ionization. Iterating Eq. (43) one gets
t
t
|Y(t) = −
dt
dt Uf (t, t) VUf (t, t ) V |Y(t ),
(44)
−∞
−∞
which yields the improved KFR amplitude (29) in the same fashion.
The integral equation (43) is particularly useful for the zero-range potential (42), since in this case it allows one to calculate the wave function in all space
provided it is known at the origin. For the latter, to a first approximation, one
may employ the unperturbed wave function. Better approximations are obtained
by using more accurate expressions. These incorporate the possibility that the
ionized electron revisits the core, as illustrated by Eq. (44).
For the zero-range potential and a monochromatic plane wave with circular
polarization, it can be shown that the wave function near the origin exactly obeys
Y(r, t) ∝ (1/r − ú) exp(−iEt) for all times. The complex quasi-energy E has
to be determined as the eigenvalue of a nonlinear integral equation (Berson,
1975; Manakov and Rapoport, 1975). For any polarization other than circular,
the time dependence at the origin is given by a Floquet expansion (Manakov
and Fainshtein, 1980; Manakov et al., 2000). The interaction with a laser field
for a finite period of time was considered along similar lines by Faisal et al.
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ATI: CLASSICAL TO QUANTUM
61
(1990) and Filipowicz et al. (1991); see also Gottlieb et al. (1991) and Robustelli
et al. (1997). The integral equation (43) was also used for two-center potentials
in order to model negative molecular ions (Krstić et al., 1991; Kopold et al.,
1998).
B.6. Quantum orbits for linear polarization
For linear polarization, Fig. 9 presents a calculated ATI spectrum that is typical
of a high laser intensity, cf. the data of Walker et al. (1996). The solid circles
that make up the topmost curve of the upper panel were calculated from the
amplitude (29) by means of a zero-range potential, while the other curves
give the results of including an increasing number of quantum orbits in the
saddle-point approximation (37). The spectra that result from just the sth pair
(which comprises the orbits 2s − 1 and 2s) are displayed in the lower panel.
Quantitatively, the first pair dominates the entire spectrum, but the contribution
of the second pair comes close, in particular near its cutoff around 7UP . The
contribution of the third pair is already weaker by almost one order of magnitude,
and the subsequent pairs hardly play a role anymore. Indeed, in the upper panel,
already the third curve from bottom virtually agrees with the result of the exact
calculation.
The dependence of the parameters t1s , t1s − t0s (the travel time), and ks on the
electron energy Ep is illustrated in Fig. 10 for the two orbits (s = 1, 2) having the
shortest travel times. These parameters uniquely specify the quantum orbits in
space and time. Their behavior is very different for energies below and above the
classical cutoff at 10UP . Below the cutoff, the imaginary parts of the parameters
are only weakly dependent on the energy. Both orbits have to be included in
the sum (37), and their interference leads to the beat pattern, which is visible
in the spectrum of Fig. 9. Notice that the imaginary parts of both the return
times t1s and the momenta kxs are small. In contrast, the imaginary part of the
travel times t1s − t0s , which are related via t0s to the tunneling rate, is substantial;
see Fig. 5, where the ionization time ts is plotted for the direct electrons. Hence,
after rescattering, the orbits are real for all practical purposes: the electron has
forgotten its origin via tunneling. The parameter values of the two orbits (s = 1, 2)
approach each other closely near the cutoff. At some point, one of the two orbits
(drawn dashed in the figure) has to be dropped from the sum (37). This causes the
artifact of the small spikes visible in Fig. 9. For energies above the cutoff, just one
orbit contributes and, as a consequence, the spectrum smoothly decreases without
any trace of interferences. The real part of the parameters stays approximately
constant, while the imaginary part increases strongly with increasing energy. This
is responsible for the steep drop of the spectrum after the cutoff. Similar behavior,
as a function of ellipticity, occurs in Fig. 5.
The procedure of dropping one of the orbits of each pair after its cutoff
can be replaced by a more rigorous method. In the vicinity of the cutoffs,
62
W. Becker et al.
[III
Fig. 9. Upper panel: ATI spectrum in the direction of the laser field for linear polarization for
1015 W/cm2 , àw = 0.0584 a.u., and a binding energy of EIP = 0.9 a.u. The electron energy is given
in multiples of UP . The curve at the top (solid circles) is the exact result from Eq. (29). The other
curves were calculated from the saddle-point approximation (37). From bottom to top, more and
more quantum orbits are taken into account; the results are displaced with respect to each other for
visual convenience. The curve at the bottom incorporates just the pair of orbits with the shortest
travel times, the next one up includes in addition the pair with the next-to-shortest travel times, and
so on. The occasional small spikes are artifacts of the saddle-point approximation, cf. Goreslavskii
and Popruzhenko (1999) and Kopold et al. (2000a). Lower panel: The envelopes of the contributions
of the individual pairs are shown all on the same scale so that the quantitative relevance of the various
pairs is put in perspective. The cutoffs of the various orbits agree with those displayed in Fig. 8.
From Kopold et al. (2000a).
III]
ATI: CLASSICAL TO QUANTUM
63
Fig. 10. Saddle points (ts , ts , ks ) for the orbits (s = 1, 2) having the two shortest travel times. In
this figure, ts is the return time (elsewhere denoted by t1s ), and ts the start time (elsewhere denoted
by t0s ). The figure shows a comparison of elliptical polarization (x = 0.5, solid circles) and linear
polarization (open squares). The values of the other parameters are those of Fig. 9 (eA = 2.04 a.u.).
The symbols identify electron energies of 11.5, 10.4, 8.92, 6.01, and 2.49, all in multiples of UP .
The dashed orbits have to be dropped from the sum (37) after the cutoff. With thescaling of k
given on the ordinate, the saddle points depend only on the Keldysh parameter g = |E0 |/ 2UP .
an approximation in terms of Airy functions was used by Goreslavskii and
Popruzhenko (2000). A uniform approximation was described in a different
context by Schomerus and Sieber (1997). It reproduces the spectra of Fig. 9
without the spikes (Schomerus and Faria, 2002).
B.7. Enhancements in ATI spectra
In several experiments, pronounced enhancements of groups of ATI peaks in
the plateau region (by up to an order of magnitude) have been observed upon
a change of the laser intensity by just a few percent (Hertlein et al., 1997;
Hansch et al., 1997; Nandor et al., 1999). This behavior suggests a resonant
process. Near the resonances, the contrast of the spectra is remarkably reduced
(Cormier et al., 2001). For the experiments reported so far, the effect is most
pronounced for argon. This holds not only for a laser wavelength of 800 nm
but also for 630 nm (Paulus et al., 1994c). The enhancements are so strong
that in experiments implying significant focal averaging the observed spectral
64
W. Becker et al.
[III
Fig. 11. ATI spectra in argon at 800 nm recorded in the direction of the linearly polarized field for
various intensities rising by increments of 0.1 I0 from 0.5 I0 (bottom curve) to 1.0 I0 (top curve). The
horizontal lines mark the maxima of the ATI plateaus for each intensity. For intensities I > 0.8 I0
a group of ATI peaks between 15 eV and 25 eV quickly grows. (The spectra shown here represent
only a fraction of those actually measured.) From Paulus et al. (2001a).
intensity may well be dominated by these enhancements, regardless of the actual
peak intensity. In this sense, ATI in toto has been called a resonant process
(Muller, 1999b). A big step towards understanding the physical origin of the
enhancements was made in theoretical studies that reproduced the enhancements
in the single-active-electron approximation by numerical solution of the oneparticle time-dependent Schrödinger equation in three dimensions (Muller and
Kooiman, 1998; Muller, 1999a,b; Nandor et al., 1999), thereby ruling out any
mechanism that invokes electron–electron correlation.
In Fig. 11 we show results of a measurement of the same effect, but for
a shorter pulse length of 50 fs (Paulus et al., 2001a). Spectra in an intensity
interval of 0.3 to 1.0 × I0 in steps of 0.1 × I0 are displayed. The maximum
intensity I0 was calibrated by using the cutoff energy of 10UP . This leads to
I0 ≈ 8×1013 W/cm2 . There is a striking difference between the spectra for
I 0.8I0 and those for higher intensity: within a small intensity interval a group
of ATI peaks corresponding to energies between about 15 eV and 25 eV grows
very quickly. In the figure this is emphasized by horizontal lines drawn at the
maximal heights of the plateaus. Increasing the intensity above 0.9I0 leads to a
smaller growth rate of these peaks. The plateau, however, preserves its shape.
For an interpretation, it should be kept in mind that a measured ATI spectrum is
made up of contributions from all intensities I I0 that are contained within the
spatio-temporal pulse profile. This means that a spectrum for a fixed intensity
would show the enhanced group of ATI peaks only at that intensity where it first
III]
ATI: CLASSICAL TO QUANTUM
65
Fig. 12. Comparison of the intensity dependence of ATI electrons with different energies. For visual
convenience, the overall increase in yield with increasing intensity has been subtracted. The electrons
at 6.4 eV and 7.3 eV are due to the strongest Freeman resonances, i.e. resonance with atomic states.
Those labeled “plateau” are electrons in the plateau region of the spectra. As a consequence of
the subtraction of the overall increase, the resonance-like behavior corresponds to those intensities
where the respective curves start rising. It is evident that for the plateau electrons this does not
happen at those intensities where the atomic states shift into resonance. Quite to the contrary, the
intensity at which the yield of the plateau electrons starts its rise is reflected in the yield of the
low-energy electrons by a brief halt in their rise. This is indicated by the dashed circles.
appears in our measurement, namely at I ≈ 0.85I0 = 7×1013 W/cm2 . In other
words, the enhancement happens at a well-defined intensity or at least within a
very narrow intensity interval.
Analyzing the wave function of the atom in the laser field, Muller (1999a)
suggested that the enhancements are related to multiphoton resonances with
ponderomotively upshifted Rydberg states. In some cases, in particular for
electrons with rather low energy, one particular Rydberg state could be definitely
identified as responsible. In others, notably for the strong enhancement that for
appropriate intensities dominates the middle of the plateau, this was not possible
(Muller, 2001a).
A closer look at the data of the measurement shown in Fig. 11 reveals
that under the conditions of this experiment (i.e. a pulse duration of 50 fs
as compared with more than 100 fs in the other measurements mentioned)
resonantly enhanced multiphoton ionization does not play an essential role.
This can be deduced from the different intensity dependence of the enhanced
ATI peaks in the plateau and of the low-energy ATI peaks, see Fig. 12. The
latter are known to originate from atomic resonances (Freeman et al., 1987).
In Muller’s numerical simulations, the existence of excited bound states
appears to be instrumental for the enhancements. Yet, the modified KFR
66
W. Becker et al.
[III
matrix element (29), which does not incorporate any excited states, produces
much the same enhancements (Paulus et al., 2001a; Kopold et al., 2001). An
example is shown in Fig. 13. In these calculations, the enhancements occur for
intensities for which an ATI channel closes. This is the case when
EIP + Up = kàw.
(45)
For an intensity slightly higher than specified by this condition, k + 1 is the
minimum number of photons required for ionization in place of k. Such channel
closings are very visible in the multiphoton-detachment yields of negative ions
(Tang et al., 1991) and have been shown to produce a separate comb of peaks in
the low-energy ATI spectrum (Faisal and Scanzano, 1992). Comparison of the
channel-closing condition (45) with the ATI energy spectrum (22) shows that at
a channel closing electrons may be produced with zero drift momentum p. In
this event, the energy of the k photons is entirely used to overcome the binding
potential raised by the ponderomotive energy, and no energy is left for a drift
motion. An electron having a drift momentum near zero has many recurring
opportunities to rescatter. Indeed, the quantum-orbit analysis of the spectra of
Fig. 13 shows that at the channel closings, and only there, an exceptionally large
number of orbits are required to reproduce the exact result. All of these orbits
conspire to interfere constructively to produce the observed enhancements. In the
tunneling regime, this can be proved analytically (Popruzhenko et al., 2002).
In Muller’s numerical simulations, inspection of the temporal evolution reveals
that at the intensities that produce the enhancements electrons linger about
the ion for many cycles of the field before the final act of rescattering.
A detailed comparison between Muller’s numerical simulations and results based
on Eq. (29) has been made by Kopold et al. (2001). This paper also includes
an assessment of the consequences of focal averaging. It is noteworthy that both
approaches predict ATI enhancements also for helium deeply in the tunneling
regime, in spite of the obvious multiphoton character of the channel-closing
condition (45). Unfortunately, the helium data of Walker et al. (1996) and Sheehy
et al. (1998) do not allow one to draw conclusions about the presence or absence
of enhancements.
The interference interpretation just given requires the existence of a sufficient
number of orbits to contribute to the energy considered. The lower panel of
Fig. 9 shows that too few orbits contribute for energies above about 8UP . Indeed,
the enhancements observed experimentally are restricted to the lower two-thirds
of the plateau. The interpretation also implies that the enhancements should
disappear for ultrashort pulses, where late returns do not occur. This has been
observed in experiments by Paulus et al. (2002). In numerical simulations of
HHG based on the three-dimensional TDSE, the same effect has noticed by
de Bohan et al. (1998).
When the modified KFR matrix element (29) is used to describe data for real
atoms, the ionization potential EIP has to be replaced by an effective (lower)
III]
ATI: CLASSICAL TO QUANTUM
67
Fig. 13. ATI spectra for EIP = 14.7 eV, w = 1.55 a.u., and three intensities: at a channel closing
(h = UP / w = 2.526, middle panel), below the channel closing (h = 2.326, lower panel), and above
(h = 2.626, upper panel). In each panel, the exact result calculated from Eq. (29) is shown (solid
symbols) and approximations involving the first 2 (dashed line), 6 (dot-dashed line), and 40 (solid
line) quantum orbits in Eq. (37). From Kopold et al. (2001).
value that corresponds to the de facto onset of the continuum (Paulus et al.,
2001a; Kopold et al., 2001). It is a fact that, for a Coulomb potential, the actual
onset of the continuum is hard to see and may better be replaced by an effective
68
W. Becker et al.
[III
value. This is illustrated, for example, by the photoabsorption spectra of Garton
and Tomkins (1967).
Numerical simulations predict very similar enhancements in high-order
harmonic spectra (Toma et al., 1999) and in nonsequential double ionization
(NSDI) of helium (Muller, 2001c). In HHG in one dimension, the dependence
of the enhancements on the shape of the potential and the presence or absence
of excited bound states has been investigated (Faria et al., 2002). The results are
largely compatible with the quantum-orbit picture.
In a semiclassical framework, the binding potential can be incorporated into
the orbits. This leads to Coulomb refocusing (Ivanov et al., 1996; Yudin and
Ivanov, 2001a): orbits that would miss the ion in the absence of the binding
potential are refocused to the ion in its presence. This emphasizes the importance
of late returns and leads to a substantial increase of rescattering effects without,
however, resonant behavior. If late returns are cut off due to an ultrashort laser
pulse, the rate of NSDI should decrease. Indeed, this has been experimentally
confirmed by comparison of 12-fs and 50-fs pulses (Bhardwaj et al., 2001).
B.8. Quantum orbits for elliptical polarization
Formulation of a classical model of the simple-man variety to describe
rescattering for an elliptically polarized laser field meets with difficulties. The
problem is that an electron that starts with zero velocity almost never returns
exactly to its starting point if the laser field has elliptical polarization. Formally,
this shows in the saddle-point equations (34)−(36) as follows. For EIP = 0,
Eq. (34) yields k = eA(t0 ) if real solutions are sought. For linear polarization,
this leaves two equations to be solved for t0 and t1 : Eq. (36) and the x-projection
of Eq. (35). Real solutions are obtained, provided the final momentum p is
classically accessible. In contrast, for elliptical polarization, three equations are
left since now both the x-projection and the y-projection of Eq. (35) have to
be considered. Hence, there is no simple-man model for elliptical polarization,
even when EIP = 0. The same situation occurs for HHG. This does not mean
that there is no HATI or HHG for elliptical polarization: quantum-mechanical
wave-function spreading assures overlapping of the wave packet of the returning
electron with the ion (Dietrich et al., 1994; Gottlieb et al., 1996). The complete
absence of HHG for a circularly polarized laser field is sometimes taken as
confirmation of the rescattering mechanism. This conclusion is not rigorous since
there is still sufficient overlapping. Rather, the absence of HHG is due to angularmomentum selection rules or, equivalently, destructive interference.
One might try to formulate a simple-man model for elliptical polarization by
relaxing the requirement that the electron return exactly to the position of the
ion or by admitting a nonzero initial velocity, but in doing so a large amount of
arbitrariness is unavoidable. Instead, we will just solve the saddle-point equations
(34)−(36) and accept and interpret the complex solutions.
III]
ATI: CLASSICAL TO QUANTUM
69
Fig. 14. ATI spectrum in the direction of the large component of the elliptically polarized driving
laser field (5) for x = 0.5 (see the field ellipse in the upper right corner of the figure) and electron
energies between 2.5 and 10.5UP . The other parameters are w = 1.59 eV, EIP = 24.5 eV, and
I = 5×1014 W/cm2 . The open circles give the yields of the individual ATI peaks calculated from
the integral (29) for the zero-range potential. The other curves represent the contributions to the
quantum-path approximation (37) of the shortest trajectories 1 and 2 (dot-dashed line), 3 and 4
(long-dashed line), and 5 and 6 (short-dashed line), as well as the sum of all six (solid line). Please,
note that some of these curves overlap partly or entirely. The orbits responsible for each part of the
spectrum, viz. 1 and 2, 3 and 4, and 5 and 6, are presented near the margins of the figures. The
position of the ion is marked by a cross; notice that the orbits do not depart from there, but rather
from a point several atomic units away from it. This is the point where the electron tunnels into the
continuum. The electron travels the orbits in the direction of the arrows. Experimental data for a
similar situation are shown in Fig. 15. From Kopold et al. (2000b).
The results of such a calculation are presented in Fig. 14. What used to
be the rescattering plateau for linear polarization has turned into a staircase
for elliptical polarization. Each step can be attributed to one particular pair
of orbits, and for each step the real parts of such orbits are displayed in the
figure. The orbits are closely related to their analogs in the case of linear
polarization, exhibited in Fig. 9. In particular, their cutoffs oscillate with
increasing travel time as illustrated in Figs. 8 and 9. The main difference is that
for elliptical polarization the orbits are two-dimensional and encircle the ion. The
pair of orbits with the shortest travel times generates the part of the spectrum
preceding the final (highest-energy) cutoff. However, this part is very weak in
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W. Becker et al.
[III
relation to the yields at lower energies. The latter are generated by orbits with
longer travel times, whose contributions for linear polarization are marginal, see
Fig. 9.
Intuitively, this staircase structure can be understood as follows. Return of the
electron to the ion is possible if the electron has a nonzero initial velocity. This
velocity is largely in the direction of the small component of the elliptically
polarized field. The larger this velocity is, the smaller is the contribution that
the associated orbit makes to the spectrum. [This can be compared with the
distribution of transverse momenta in a Gaussian wave packet (Dietrich et al.,
1994; Gottlieb et al., 1996).] For the shortest orbit, while the large component
of the field changes sign so that the electron is driven back to the core in this
direction, the small component has the same sign for the entire duration of the
orbit. Hence, a particularly large initial velocity in this direction is required in
order to compensate the drift induced by the small field component. For the
longer orbits, the small component changes direction, too, during the travel time
and, consequently, a smaller initial transverse velocity suffices to allow the
electron to return to the ion. Support for these qualitative statements can be
found in the orbits depicted in Fig. 14.
The parameters of the two shortest quantum orbits can be read from Fig. 10
and compared with the case of linear polarization. For elliptical polarization, the
momentum ks has two nonzero components, kxs and kys . Both have substantial
imaginary parts, in particular kys . This is a consequence of the lack of a
classical simple-man model for elliptical polarization, as discussed above. For the
orbits (s = 3, 4) (not shown), the imaginary parts are much smaller, in keeping
with the fact that they make a larger contribution to the spectrum.
Figure 15 presents a corresponding measurement of an ATI spectrum and
displays the staircase structure predicted by the theory. The first step (the one
corresponding to energies below 10 eV) is due to direct electrons and does not
concern us here. The other ones correspond to the steps of Fig. 14. The real
parts of representative orbits, calculated from Eqs. (34)−(36), are shown in the
figure. In order to reach a maximum contrast for the steps, the spectrum was
recorded at 30º to the major axis of the polarization ellipse.
B.9. Interference between direct and rescattered electrons
In the lower part of the plateau, the electron can reach a given energy either
directly or after rescattering so that one expects interference of these two paths.
However, Fig. 9 shows that, for linear polarization and high intensity, the
transition region where both paths make a contribution of comparable magnitude
is very narrow. The situation is more favorable for elliptical polarization:
since the plateau turns into a staircase (Fig.14), the yields of the two paths
remain comparable over a larger energy region. This has permitted experimental
III]
ATI: CLASSICAL TO QUANTUM
71
Fig. 15. ATI spectrum in xenon for an elliptically polarized laser field with ellipticity x = 0.36
and intensity 0.77×1014 W/cm2 for emission at an angle with respect to the polarization axis as
indicated in the upper right. The spectrum has a staircase-like appearance. The respective steps are
shaded differently. For each step, the real parts of the responsible quantum orbits are displayed. The
dots with the crosses mark the position of the atom, and the length scale is given in the upper left
of the figure. From Salières et al. (2001).
observation of this interference effect in the energy-resolved angular distribution
(EAD) (Paulus et al., 2000).
Figure 16 shows a comparison of the EAD’s for linear and for elliptical
polarization at the same intensity. For linear polarization, the standard plateau
in the direction of the laser polarization is very noticeable. The side lobes
corresponding to rainbow scattering, mentioned in Sect. III.A, are also visible.
For elliptical polarization, the plateau has split into two, one to the left of the
direction of the major axis of the field and another weaker one to its right. The
lower panel of Fig. 16 exhibits (on the right) EAD’s of a sequence of ATI peaks
where the interference is best developed and (on the left) compares them with
theoretical calculations from the amplitude (29). The parameters underlying the
calculation do not exactly match the experiment. This is mostly attributable
to the insufficient description of the direct electrons for elliptical polarization.
The theoretical results, however, show the same interference pattern. In order
to make sure that this pattern is really due to interference between direct and
rescattered electrons, the two contributions have been displayed separately for
one of the peaks (s = 17): neither one shows a pronounced dip, only their
coherent superposition does. For more details of the theory we refer to Kopold
(2001).
72
W. Becker et al.
[III
Fig. 16. Upper panels: density plots of measurements of the energy-resolved angular distributions
for Xe at an intensity of 7.7×1013 W/cm2 and a wavelength of 800 nm for (a) linear polarization and
(b) elliptical polarization with ellipticity x = 0.36. The direction of the major axis of the polarization
ellipse is at 0º. Dark means high electron yield. Yields can only be compared horizontally, not
vertically, since the data were normalized separately for each ATI peak. For linear polarization,
the cutoff is at 10UP = 46 eV. Lower panels: (a) theoretical calculation from Eq. (29) of the
angular distribution for the ATI peaks s = 11, . . . , 21 for elliptical polarization (x = 0.48). The
other parameters are EIP = 0.436 a.u. (just below the binding energy of xenon in order to stay away
from a channel closing) and I = 5.7×1013 W/cm2 . For the ATI peak s = 17, the contributions of the
direct electrons (dashed line) and the rescattered electrons (dotted line) are displayed separately. The
slight variation in the former is unrelated to the interference pattern of the total yield (solid line),
which results from the coherent sum of the two contributions. (b) Experimental angular distribution
extracted from the upper panel (b) of the ATI peaks s = 15, . . . , 25. From Paulus et al. (2000).
IV]
ATI: CLASSICAL TO QUANTUM
73
IV. ATI in the Relativistic Regime
A sufficiently intense laser field accelerates an electron from rest to relativistic
velocities |v| ~ c within one cycle. Such intensities are characterized by the
ponderomotive energy UP becoming comparable with or exceeding the electron’s
rest energy mc2 . We will briefly summarize the kinematics of an otherwise free
electron in the presence of such a field. In other words, we will generalize
the simple-man model of Sect. II.A.1 to the case of “relativistic intensity.” The
changes are surprisingly few.
A. Basic Relativistic Kinematics
For a four-vector potential Am = (A0 , A), the electron’s four-vector velocity is
(Jackson, 1999)
mũm = pm − eAm ,
(46)
where ũm = g(c, v) with v = v(t) the ordinary velocity dx/dt, and the usual
relativistic factor g = [1 − (v/c)2 ]−1/ 2 (not to be confused with the Keldysh
parameter). The four-velocity satisfies ũ2 ≡ ũ · ũ ≡ ũm ũm = c2 so that
the four-vector mũ is on the mass shell. Equation (46) is the analog of the
nonrelativistic Eq. (2).
We will consider a plane-wave field of arbitrary polarization,
Am =
2
m
ai (k · x)ei
(47)
i=1
with the four-dimensional wave vector k m = (w/c, k) so that k 2 = 0 and k · ei = 0.
The field (47) differs from the field (6) by the fact that the wave fronts are now
given by k · x = const. in place of t = const., that is, we do no longer make
the dipole approximation. We will assume that the laser field propagates in the
z-direction so that k = |k|ez .
The four-vector pm = (E/c, p) is the canonical momentum. For a vector
potential whose cycle average vanishes, its spatial components p have the
physical meaning of the drift momentum as in the nonrelativistic case. Since
the electron–field interaction depends only on
u ≡ k · x / w = t − z/c,
(48)
the canonical momentum pT ≡ ( px , py , 0) transverse to the propagation direction
as well as p · k = w( p0 − pz )/c are constants of the motion inside the field (47). If
we assume that the laser field (47) is turned on and off as a function of u = t − z/c,
then pT and p · k are also conserved when the electron enters and leaves the field.
74
W. Becker et al.
[IV
As we did for the nonrelativistic simple-man model in Sect. II.A.1, we assume
that the electron is initially, at some space–time instant u0 , at rest. Then
pT = eA(u0 )
and
p0 − pz = mc
for all times.
(49)
From the condition that ũ2 = c2 , using the conditions (49), we obtain the energy
as a function of u,
g=
E
e2
(A(u) − A(u0 ))2 .
=
1
+
mc2
2m2 c2
(50)
This yields the cycle-averaged kinetic energy
Ekin = E − mc2 =
p2T
+ UP .
2m
(51)
This is exactly the same decomposition into drift energy and ponderomotive
energy as in the nonrelativistic case, Eq. (3). The ponderomotive energy is
still defined by Eq. (4), and the classical bounds of the spectrum discussed
in Sect. II.A.1 are unchanged. However, velocity and canonical momentum are
connected by the relativistic expresion mgv = pT − eA, and the cycle average
was performed with respect to u rather than the time t. Since it can be shown
that u is proportional to the electron’s proper time, this was, actually, the proper
thing to do (Kibble, 1966).
The fact that p · k is a conserved quantity implies that the electron’s velocity
in the propagation direction of the laser field is given by
pz = mgvz = mc(g − 1) = Ekin /c.
(52)
The presence of this momentum reflects the fact that a laser photon has a
momentum in the direction of its propagation or, alternatively, that the magnetic
field via the Lorentz force causes a drift in the propagation direction or,
alternatively, that the laser field exerts radiation pressure. All three statements
are essentially equivalent. As a consequence, electrons born with zero velocity in
a relativistic laser field are no longer emitted in the direction of its polarization,
but acquire a component in the propagation direction of the laser so that, for
circular polarization, they are emitted in a cone given by the angle
|vT (t)| 2
2m
tan q =
=
(53)
=
|vz (t)| ∞ |pT |
g∞ − 1
with respect to the propagation direction. The subscript ∞ characterizes
quantities outside of the laser field. In the derivation, Eqs. (49)–(52) were used.
The angle q has been observed by Moore et al. (1995, 1999) for intensities
IV]
ATI: CLASSICAL TO QUANTUM
75
of several 1018 W/cm2 and l = 1.053 mm and was used to draw conclusions
regarding the actual (nonzero) value of the initial velocity (McKnaught et al.,
1997), which can be introduced as discussed in the nonrelativistic case in
Sect. II.A.1. There are, however, still some unresolved issues in the interpretation
of these experiments (Taı̈eb et al., 2001).
The cycle average of Eq. (50) can be written in the covariant form
p2 = m2 c2 − (eA)2 ≡ m2∗ c2 > m2 c2 ,
(54)
where p2 and (eA)2 < 0 are invariant four-dimensional scalar products. This
relation is often used to introduce the so-called “relativistic effective mass” m∗ .
It occurs very naturally in the context of the Klein–Gordon equation
(iðm − eAm )2 − m2 c2 Y = 0,
(55)
which explicitly displays the effective mass. However, one has to keep in
mind that this apparently increased mass is just due to the transverse wiggling
motion of the electron, viz. the ponderomotive energy, and that there is nothing
especially relativistic about it. All the same, envisioning the ponderomotive
energy as a mass increase makes sense since, like the rest mass, it is an energy
reservoir that is not easily tapped.
The classical kinematics just discussed are embedded in quantum-mechanical
calculations, which can be carried out along the lines of the strong-field
approximation (17), taking the relativistic instead of the nonrelativistic Volkov
wave function (Reiss, 1990; Faisal and Radożycki, 1993; Crawford and Reiss,
1997). In particular, the stationary-phase approximation is well justified, leading
to a form similar to Eq. (21) (Krainov and Shokri, 1995; Popov et al., 1997; Mur
et al., 1998; Krainov, 1999; Ortner and Rylyuk, 2000).
B. Rescattering in the Relativistic Regime
With increasing laser intensity, the first relativistic effect to become significant –
before the ponderomotive potential becomes comparable with the electronic rest
mass – is the drift momentum (52) in the direction of propagation of the laser
field, which can be traced to the Lorentz force. This has virtually no effect on
the initial process of ionization where the electron’s velocity is low, but since it
is always positive it prevents the electron from returning to the ion. Therefore,
with increasing intensity it gradually eliminates the significance of rescattering
processes. This effect can be estimated by calculating the distance by which
the electron misses the ion in the z-direction when it returns to the ion in the
76
W. Becker et al.
[V
x−y plane (approximately at the time tret ≈ T/ 2). From Eqs. (52) and (51) (where,
for simplicity, we only kept UP ), we obtain
vz T/ 2 ≈
UP
l
2mc2
(56)
with l the wavelength of the laser field. Obviously, this distance can exceed the
width of the wave packet of the returning electron to the point where it does not
overlap anymore with the ion, even when UP /mc2 1.
In HHG the consequences have been investigated in a number of recent
theoretical works (see Sect. V.D) and were found to cause a dramatic drop of the
plateau. The same should be expected for high-order ATI, but to our knowledge,
this has not been explored in detail. However, in the analysis of multiplenonsequential-ionization experiments of neon at 2 × 1018 W/cm2 a conspicuous
suppression of the highest charge state has been attributed to the magnetic-fieldinduced drift (Dammasch et al., 2001).
V. Quantum Orbits in High-order Harmonic Generation
According to the rescattering model, the physics of high-order ATI and highorder harmonic generation differ only in the third step: elastic scattering versus
recombination. Correspondingly, the description in terms of quantum orbits can
be applied to HHG as well; in fact, quantum orbits were introduced for the first
time in the analysis of HHG by Lewenstein et al. (1994). It is from the practical
point of view that the two processes differ greatly: HHG by one single atom has
never been observed, only HHG by an ensemble of atoms. This introduces phase
matching as an additional consideration, equal in significance to the single-atom
behavior (Salières et al., 1999; Brabec and Krausz, 2000).
Below we will consider examples of a quantum-orbit analysis of HHG for
several nonstandard situations. The first example is an elliptically polarized laser
field. A bichromatic elliptically polarized laser field was considered by Milošević
et al. (2000), and in Sect. V.C we concentrated on a special case of such a field:
a two-color bicircular field. Finally, in Sect. V.D the quantum-orbits formalism is
extended into the relativistic regime. A bichromatic linearly polarized laser field
was investigated by Faria et al. (2000), and a simplified version of the quantumorbits formalism was used to deal with problems in the presence of a laser field
and an additional static electric field (Milošević and Starace, 1998, 1999c) or
a laser field and an additional magnetic field (Milošević and Starace, 1999a,b,
2000).
V]
ATI: CLASSICAL TO QUANTUM
77
A. The Lewenstein Model of High-order Harmonic Generation
The matrix element for emission of a photon with frequency W and polarization û
in the HHG process in the context of the strong-field approximation (Lewenstein
et al., 1994),
Mû (W) ~
∞
dt1
−∞
t1
dt0
−∞
d 3 k exp [iSW (t1 , t0 , k)] mû (t1 , t0 , k),
(57)
has the same structure as the corresponding expression (31) for ATI. The
function
mû (t1 , t0 , k) = y0 |er · û| k − eA(t1 ) k − eA(t0 ) |er · E(t0 )| y0 (58)
is the product of two matrix elements: one that describes the ionization at time t0
due to interaction with the laser field, and another one at time t1 that corresponds
to recombination into the ground state followed by emission of the high-order
harmonic photon having the polarization û. The difference to ATI is mostly in
the first term of the action:
∞
SW (t1 , t0 , k) =
t1
dt (EIP − W) −
1
2m
t1
t0
dt [k − eA(t)]2 +
t0
dt EIP ,
(59)
−∞
which now refers to the emitted photon. The corresponding saddle-point
approximation of Eq. (57) is like the HATI approximation (37), except that the
summation is now over saddle points that are solutions of the system of equations
(34), (35) and (Lewenstein et al., 1995b, Kopold et al., 2000b)
[k − eA(t1 )]2 = 2m(W − EIP ).
(60)
The last equation corresponds to the condition of energy conservation at the time
of recombination and replaces the condition (36) of elastic rescattering in HATI.
For a linearly polarized monochromatic field, quantum orbits were employed
from the very beginning for the evaluation of HHG in the Lewenstein model
(Lewenstein et al., 1994, 1995b) and routinely applied in the theoretical analysis
and interpretation (Salières et al., 1999). Conversely, numerical solutions of the
TDSE were analyzed in terms of the short (t1 ) and the long (t2 ) quantum orbit,
and the dominance of these two orbits was corroborated (Gaarde et al., 1999;
Kim et al., 2001). The contributions of the long and the short orbit could be
spatially resolved in an experiment by Bellini et al. (1998). Spectral resolution
was achieved by exploiting the dependence of phase matching on the position of
the atomic jet with respect to the laser focus by Lee et al. (2001) and by Salières
et al. (2001).
78
W. Becker et al.
[V
At the end of Sect. II.B we remarked that the quantum-orbit formalism
is not restricted to periodic fields, but can equally well be applied to finite
pulses. For a periodic field, interference of contributions from different cycles
generates a discrete spectrum. For a finite pulse, it enhances or suppresses
particular frequency intervals. This was dubbed “intra-atomic phase matching”
by Christov et al. (2001) and has been calculated in terms of quantum orbits; in
the context of the TDSE, see Watson et al. (1997). This mechanism underlies
the engineering of a HHG spectrum by tailoring the pulse shape in a feedbackcontrolled experiment (Bartels et al., 2000, 2001). Individual HHG peaks could
be enhanced by up to an order of magnitude.
A description of HHG that is practically equivalent to the Lewenstein
model is based on the integral equation (43) and the zero-range potential (42)
(Becker et al., 1990, 1994b). The equivalence implies that the contribution of
“continuum–continuum terms” is insignificant (Becker et al., 1997). The threestep nature of HHG – direct ATI followed by continuum propagation followed
by laser-assisted recombination – is particularly emphasized in the approach of
Kuchiev and Ostrovsky (1999, 2001), where the integration over the intermediate
momentum k is replaced by a discrete summation over ATI channels. The latter
is carried out by a variant of the saddle-point approach, which is reminiscent of
Regge poles and leads to a complex effective channel number.
B. Elliptically Polarized Fields
High-order-harmonic generation by an elliptically polarized field is of great
interest for applications such as the generation of sub-femtosecond pulses
(Corkum et al., 1994). For theoretical calculations in the context of the SFA, see
Becker et al. (1994, 1997) and Antoine et al. (1996); for a fairly comprehensive
list of references, see Milošević (2000). Fields having polarization other than
linear generate particularly appealing quantum orbits since they allow them to
unfold in a plane. As an example, Fig. 17 shows a HHG spectrum for the
elliptically polarized laser field (5) (Kopold et al., 2000b; Milošević, 2000).
The figure confirms that the “exact results” are well approximated by the
contributions of only the six shortest orbits. This figure is the analog of Fig. 14
for HATI. The spectrum exhibits the same staircase structure, and everything
said there applies here as well.
C. HHG by a Two-color Bicircular Field
The bichromatic w–2w laser field
E(t) = 12 i E1 e+ e−iwt + E2 e− e−2iwt + c.c.,
(61)
whose two components
√ are circularly polarized and counter-rotating in the same
plane (e± = (x̂ ± iŷ)/ 2), is known to generate high harmonics very efficiently;
V]
ATI: CLASSICAL TO QUANTUM
79
Fig. 17. High-order harmonic spectrum for an elliptically polarized laser field with the same
parameters as in Fig. 14 and harmonic orders between 25 and 77. The open circles are calculated
from the integral (57), and the curves labeled 1 through 6 represent the individual contributions to
the quantum-orbit approximation of the six shortest quantum orbits, numbered as in Fig. 14. The
contributions from quantum orbits 2, 4 and 6 have to be dropped above their intersections with
curves 1, 3 and 6, respectively. The coherent sum of all six orbits is represented by the solid line.
Typical orbits responsible for each part of the spectrum are depicted as in Fig. 14. From Kopold
et al. (2000b).
see Eichmann et al. (1995) for experimental results and Long et al. (1995) for
a theoretical description. We will call this field “bicircular.” This high efficiency
was surprising because, for a monochromatic field, the harmonic emission
rate decreases with increasing ellipticity (cf. the preceding subsection) and a
circularly polarized laser field does not produce any harmonics at all. A more
detailed analysis, based on the quantum-orbits formalism, gives an explanation
of this effect (Milošević et al., 2000, 2001a,b). The harmonics produced this way
can be of a practical importance because of their high intensity (Milošević and
Sandner, 2000) and temporal characteristics (attosecond pulse trains; Milošević
and Becker, 2000). The more general case of an rw−sw (with r and s integers)
bicircular field was considered by Milošević et al. (2001a).
For the laser field (61), selection rules only permit emission of circularly
polarized harmonics with frequencies W = (3n ± 1) w and helicities ±1.
Similar selection rules govern harmonic generation by a ring-shaped molecule
(Ceccherini and Bauer, 2001) or a carbon nanotube (Alon et al., 2000).
80
W. Becker et al.
[V
Fig. 18. Harmonic-emission rate as a function of the harmonic order for the bicircular laser field (61)
with w = 1.6 eV and intensities I1 = I2 = 4×1014 W/ cm2 . The ionization potential is EIP = 15.76 eV
(argon). The inset shows the laser electric-field vector in the x−y plane for times − 12 T t 12 T ,
with T = 2p / w being the period of the field (61). The arrows indicate the time evolution of the field.
The ionization time t0 and the recombination time t1 of the three harmonics W = 19w, 31w and
43w are marked by asterisks and solid circles, respectively. These times and harmonics correspond
to the dominant saddle-point solution 2 in Fig. 19. In between the ionization time (asterisks) and the
recombination time (solid circles) the x-component of the electric field changes from its negative
maximum to its positive maximum, whereas its y-component remains small and does not change
sign. From Milošević et al. (2000).
Figure 18 presents an example of the harmonic spectrum for the bicircular
field (61). The results are obtained by numerical integration from Eq. (57).
Compared with the spectrum of a monochromatic linearly polarized field (see,
for example, the nonrelativistic curve in Fig. 22), the spectrum is comparably
smooth. Furthermore, the cutoff is less pronounced and there are small oscillations after the cutoff. These features can be explained in terms of the quantum
orbits. Figure 19a shows the first eleven solutions (those having the shortest travel
times) of the system of the saddle-point equations (34), (35) and (60), while
Fig. 19b shows the individual contributions to the harmonic emission rate of the
first eight of these solutions (Milošević et al., 2000). Obviously, in the plateau
region the contribution of a single orbit, corresponding to solution 2, is dominant
by one order of magnitude, while in the cutoff region more solutions are relevant
(in particular solution 5). This is just the opposite of the standard situation of the
monochromatic linearly polarized field (Lewenstein et al., 1995b) where essentially two orbits contribute in the plateau and just one in the cutoff region. Figure 19a tells which solutions are dominant. The probability of harmonic emission
V]
ATI: CLASSICAL TO QUANTUM
81
Fig. 19. Saddle-point analysis of the results of Fig. 18. (a) The imaginary part of the recombination
time t1 as a function of the real part of the travel time t1 − t0 , obtained from the solutions of the
saddle-point equations (34), (35) and (60). Each point on the curves corresponds to a specific value
of the harmonic frequency W, which is treated as a continuous variable. For the interval of Re(t1 − t0 )
covered in the figure, eleven solutions were found, which are labeled with the numbers in boldface
italics. Values of the harmonic order that approximately determine the cutoffs for each particular
solution are marked by stars with the corresponding harmonic numbers next to them. Those values of
the harmonic order for which | Im t1 | is minimal are identified as well. (b) The partial contributions
to the harmonic-emission rate of each of the first eight solutions of the saddle-point equations. From
Milošević et al. (2000).
82
W. Becker et al.
[V
Fig. 20. Real parts of the quantum orbits for the same parameters as in Fig. 18 and for the harmonic
W = 43w. Five orbits are shown that correspond to the saddle-point solutions 2, 3, 4, 5 and 8 in
Fig. 19. The direction of the electron’s travel is given by the arrows. In each case, the electron
is “born” a few atomic units away from the position of the ion (at the origin), where its orbit
almost exactly terminates. The dominant contribution to the 43rd harmonic intensity comes from
the shortest orbit number 2, whose shape closely resembles the orbit in the case of a linearly polarized
monochromatic field. From Milošević et al. (2000).
decreases with increasing absolute value of the imaginary part of the recombination time t1 . The possible cutoff of the harmonic spectrum can be defined as the
value of the harmonic order after which | Im t1 | becomes larger than (say) 0.01T .
The probability of HHG is maximal when | Im t1 | is minimal. For each solution in
Fig. 19a, these points are marked by asterisks and by the corresponding harmonic
order. As a consequence of wave-function spreading, the emission rate decreases
with increasing travel time t1 − t0 . This gives an additional reason why the
contribution of solution 2 is dominant in the plateau region.
Let us now consider the quantum orbits. In Fig. 20 for the fixed harmonic
W = 43w, we present the five orbits that correspond to saddle-point solutions
2, 3, 4, 5 and 8 in Fig. 19. The dominant contribution comes from the shortest
orbit 2 (thick line). It starts at the point (4.06, 0.66) by setting off in the negative
y-direction, but soon turns until it travels at an angle of 68º to the negative y-axis.
Thereafter, it is essentially linear, as would be the case for a linearly polarized
field. This behavior can be understood by inspection of the driving bicircular field
depicted in the inset of Fig. 18, where the start time and the recombination time
of the orbit are marked. During the entire length of the orbit, the field exerts
a force in the positive y-direction. The effect of this force is canceled by the
electron’s initial velocity in the negative y-direction. The force in the x-direction
is much like that in the case of a linearly polarized driving field. Since HHG by a
V]
ATI: CLASSICAL TO QUANTUM
83
linearly polarized field is most efficient, this makes plausible the high efficiency
of HHG by the bicircular field.
The orbit that corresponds to solution 3 has a shape similar to that of
orbit 2, but is much longer. The corresponding travel time is longer, too,
and, consequently, the contribution of solution 3 to the emission rate of the
43rd harmonic is smaller. The other orbits are still longer and more complicated
so that their contribution is negligible.
The electric field of a group of plateau harmonics is displayed in Fig. 18. It
shows interesting behavior, which again reflects the threefold symmetry of the
field (61), see the inset of Fig. 18. If the group of harmonics includes harmonics
of either parity, then the field consists of a sequence of essentially linearly
polarized, strongly chirped attosecond pulses, each rotated by 120º with respect
to the previous one. If, on the other hand, one were able to select harmonics of
definite helicity, i. e. either W = (3n + 1) w or W = (3n − 1) w, then one would
obtain a sequence of attosecond pulses with approximately circular polarization.
Both cases are illustrated in Fig. 21.
Fig. 21. Parametric polar plot of the electric-field vector of a group of harmonics during one cycle
of the bicircular field (61) on an arbitrary isotropic scale. The position of the origin is indicated in
the upper and the left margin. The parameters are I1 = I2 = 9.36×1014 W/cm2 , àw = 1.6 eV, and
EIP = 24.6 eV. The plot displays two traces: The circular trace is generated by the ten harmonics
W = (3n + 1)w with n = 10, . . . , 19, all having positive helicity. The starlike trace is generated by all
harmonics W = (3n ± 1)w between the orders 31 and 59, regardless of their helicity. The curve at
the bottom represents the x-component of the field of the latter group over one cycle, the time scale
being given on the horizontal axis. It shows that the field is strongly chirped. The black blob at the
center is due to the fact that the field is near zero throughout most of the cycle, cf. the trace of the
x-component. From Milošević and Becker (2000).
84
W. Becker et al.
[V
D. HHG in the Relativistic Regime
Quantum orbits can also be employed in the relativistic regime starting from
the Klein–Gordon equation (55). Milošević et al. (2001c, 2002) found that
the relativistic harmonic-emission matrix element has a form similar to that in
Eq. (57), but with the relativistic action (à = c = 1)
∞
t1
t0
SW (t1 , t0 , k) =
du (EIP − m − W) −
du ek (u) +
du (EIP − m),
(62)
t1
t0
−∞
where
ek (u) = Ek + eA(u) ·
k + 2e A(u)
Ek − ẑ · k
(63)
and Ek = (k 2 + m2 )1/ 2 , u = (t − z)/ w. Solving the classical Hamilton–Jacobi
equation for Hamilton’s principal function it can be shown that ek (u) is the
classical relativistic electron energy in the laser field. In the relativistic case,
the function mû (t1 , t0 , k) in Eq. (57) consists of two parts: the dominant part
is responsible for the emission of odd harmonics W = (2n + 1) w, while
the other one originates from the intensity-dependent drift momentum of the
electron in the field and allows for emission of even harmonics W = 2nw.
Similarly to the nonrelativistic case, the integral over the intermediate electron
momentum k can be calculated by the saddle-point method. The stationarity
t
condition t01 du ðek (u)/ðk = 0, with ðek /ðk = dr/dt, implies r(t0 ) = r(t1 ), so
that the stationary relativistic electron orbit is such that the electron starts from
and returns to the nucleus. As above, the start time and, to a lesser degree, the
recombination time are complex.
In the relativistic case, the stationary momentum k = ks is introduced in
the following way. For fixed t0 and t1 , its component ks⊥ perpendicular to the
photon’s direction of propagation ẑ is given by
t0
(t1 − t0 )ks⊥ =
du eA(u).
(64)
Introducing M = e
2
2
t1
t0
t1
2
2
du A (u)/ (t1 − t0 ) − ks⊥
> 0, one has
2
ks2 = ks⊥
+
2 2
)
(M2 − ks⊥
,
2
2
4(m + M )
(65)
which yields eks as a function of t0 and t1 . The two stationarity equations
connected with the integrals over t0 and t1 are
eks (t0 ) = m − EIP ,
W = eks (t1 ) + EIP − m.
(66)
(67)
As in the nonrelativistic case, they express energy conservation at the time of
tunneling t0 and at the time of recombination t1 , respectively. The final expression
V]
ATI: CLASSICAL TO QUANTUM
85
Fig. 22. Harmonic-emission rate as a function of the harmonic order for ultrahigh-order harmonic
generation by an Ar8+ ion (EIP = 422 eV) in the presence of an 800-nm Ti:Sa laser having the
intensity 1.5×1018 W/ cm2 . Both the nonrelativistic and the relativistic results are shown. The
corresponding relativistic electron orbit with the shortest travel time that is responsible for the
emission of the harmonic W = 100 000w is shown in the inset. The arrows indicate which way the
electron travels the orbit. The laser field is linearly polarized in the x-direction and the v × B electron
drift is in the z-direction. From Milošević et al. (2002).
for the relativistic harmonic-emission matrix element has the form (37) with (62),
where the summation is now over the appropriate subset of the relativistic saddle
points (t1s , t0s , ks ) that are the solutions of the system of equations (64)−(67).
In the relativistic case it is very difficult to evaluate the harmonic-emission
rates by numerical integration. For very high laser-field intensities and ultra-high
harmonic orders, this is practically impossible, so that the saddle-point method
is the only way to produce reasonable results. Figure 22 presents an example.
The nonrelativistic result is obtained from Eq. (37) where the summation is
over the solutions of the system of the nonrelativistic saddle-point equations
(34), (35) and (60). It is, of course, inapplicable for the high intensity of
1.5×1018 W/cm2 at 800 nm and is only shown to demonstrate the dramatic
impact of relativistic kinematics. For the relativistic result, the summation in
Eq. (37) is over the relativistic solutions of Eqs. (64)−(67). The relativistic
harmonic-emission rate assumes a convex shape, and the difference between
the relativistic and nonrelativistic results reaches several hundred orders of
magnitude in the upper part of the nonrelativistic plateau. The origin of this
dramatic suppression is the magnetic-field-induced v × B drift. The significance
86
W. Becker et al.
[VI
of this drift for the rescattering mechanism was emphasized early by Kulyagin
et al. (1996). This is illustrated in the inset of Fig. 22, which shows the real
part of the dominant shortest orbit for the harmonic W = 100 000w. In order to
counteract this drift so that the electron is able to return to the ion, the electron
has to take off with a very substantial initial velocity in the direction opposite
to the laser propagation. The probability of such a large initial velocity is low,
and this is the reason for the strong suppression. As in the nonrelativistic case,
the electron is “born” at a distance of 7.5 a.u. from the nucleus.
The nonrelativistic harmonic yield shows a pronounced multiplateau structure.
While this is an artifact of the nonrelativistic approximation for the intensity
of Fig. 22, it is a real effect for lower laser-field intensities where relativistic
effects are still small (Walser et al., 2000; Kylstra et al., 2001; Milošević et al.,
2001c, 2002). In this case, the three plateaus visible in the nonrelativistic curve of
Fig. 22 are related to the three pairs of orbits, whose contribution to the harmonic
emission rate is dominant in the particular spectral region (see Figs. 2 and 3
of Milošević et al., 2001c). These are very similar to the pairs of orbits that
we have discussed for the elliptically polarized laser field in Fig. 17. However,
for the very high intensity of Fig. 22, the contribution of the shortest of these
orbits becomes so dominant that the multiplateau and the interference-related
oscillatory structure disappear completely. The reason is that the effect of the
v × B drift increases with increasing travel time; see Eqs. (52) and (56) in
Sect. IV.A. This is in contrast to the nonrelativistic case of elliptical polarization,
where longer orbits may be favored because the minor component of the field
oscillates and, therefore, for a longer orbit a smaller initial velocity may be
sufficient to allow the electron to return.
VI. Applications of ATI
Experimental and theoretical advances in understanding ATI – some of which
have been treated in this review – permit its application to the investigation of
other effects. One obvious idea is to exploit the nonlinear properties of ATI.
This is particularly relevant to characterization of high-order harmonics and
measurement of attosecond pulses in the soft-X-ray regime. In this spectral
region (vacuum UV) virtually all bulk non-linear media are opaque. ATI,
in contrast, is usually studied under high- or ultra-high-vacuum conditions.
Another advantage over conventional nonlinear optics is that the nonlinear effect
of photoelectron emission can be observed from more or less any direction,
whereby different properties of the effect can be exploited.
A. Characterization of High Harmonics
The most straightforward approach to characterize high-order harmonics is a
cross-correlation scheme: An (isolated) harmonic of frequency qw, where q is
VI]
ATI: CLASSICAL TO QUANTUM
87
an odd integer, produces electrons by single-photon ionization with a kinetic
energy Eq = qàw − EIP . Simultaneous presence of a fraction of the fundamental
laser beam in the near infrared (NIR) produces sidebands, i.e. electrons with
energies qàw − EIP ± màw (m q). The strength of the sidebands can be
changed by temporally delaying the fundamental with respect to the harmonic
by a time t. Optimal overlapping of the pulses (t = 0) leads to a maximum in
the strength of the sidebands, whereas complete separation entirely eliminates
them. The strength of the sidebands as a function of t can be used to determine
the duration of the harmonic pulse.
For theoretical modeling, the simple ansatz of Becker et al. (1986) can be used,
which assumes that an electron is born in the presence of the laser field with a
positive initial energy Ei , which will be identified with Eq . For Up àw, which
is well satisfied for the weak field we will consider, the differential ionization
rate in the field direction is given by (in atomic units)
∞
ð 2G
Jm2
∝ |p| ·
ðEðW
m = −∞
Ef
2(Ei + mw)
d(E − mw − Ei ).
w2
(68)
m is the order
Here, Ef is the amplitude of the electric field of the fundamental,
of the sideband, p is the momentum of the photoelectron (|p| = 2(Ei + mw)),
and Jm is the Bessel function of the first kind. The intensities of the side bands
are not, in general, symmetric. However, for sufficiently weak fields, both fields
can be treated by lowest-order perturbation theory. It follows that a sideband of
2|m|
order m is proportional to Eh2 Ef , where Eh is the field strength of the harmonic
radiation. In this case, the cross correlation for a sideband of order m can be
calculated as
∞
2|m|
Eh2 (t) · Ef (t − t) dt.
(69)
Cm (t) =
−∞
Figure 23 (overleaf) shows the result of a corresponding calculation, which
is compared with results from a numerical solution of the appropriate onedimensional Schrödinger equation. The agreement is nearly perfect.
Hence, if the NIR pulse is precisely known, the pulse duration of the harmonic
(and even its shape) can be determined by deconvolution of the cross-correlation
functions. Numerical and experimental investigations of this problem were made
by Véniard et al. (1995) and Schins et al. (1996), respectively.
A.1. Measurement of attosecond pulses
Apparently, an experiment as discussed above will not be able to determine
harmonic-pulse durations significantly shorter than that of the fundamental
in the NIR spectral region. In 1996 already, Véniard et al. pointed out that
88
W. Becker et al.
[VI
Fig. 23. Cross-correlation of near-infrared and soft-X-ray pulses. A harmonic of order q creates
photoelectrons at the kinetic energy qàw − EIP . Sidebands are created by simultaneous irradiation
with the fundamental of frequency w. Plotted are the heights of the sidebands for various side-band
orders m versus the delay t between the fundamental and the harmonic. The solid line represents
the analytical approximation (69), whereas the points were calculated by numerically solving the
appropriate (one-dimensional) Schrödinger equation. In each case, the analytical approximation was
normalized to the maximum of the numerical result.
the cross correlation of harmonic and NIR radiation provides access to the
relative phase of neighboring harmonics. This is an extremely important insight
because the phase dependence of the harmonics as a function of their order
(or frequency) determines whether they are mode-locked and whether the
corresponding pulses – which would constitute attosecond pulses in the softX-ray region if bandwidth-limited – are chirped. In fact, Paul et al. (2001) used
this scheme for the first observation of a train of attosecond pulses.
In order to achieve phase measurement of adjacent harmonics, the conditions
have to be chosen such that only sidebands of order m = ±1 are generated
with appreciable amplitude. This calls for intensities of the NIR beam below 1012 W/cm2 . Along with the fact that the NIR field generates only oddorder harmonics this ensures that only two adjacent harmonics contribute to each
sideband. An electron with energy Eq = qàw − EIP , with q an even integer,
can be generated by absorption of the lower harmonic plus one NIR photon
(Eq = (q − 1)àw + àw) or by absorption of the upper harmonic and emission
of one NIR photon (Eq = (q + 1)àw − àw). Each of these two channels receives
contributions from two different quantum paths, which are related to the temporal
order of the interaction with the harmonic and the NIR field. (In contrast to
the quantum orbits we considered elsewhere in this chapter, the quantum paths
here are defined in state space rather than position space.) The photoelectron
VI]
ATI: CLASSICAL TO QUANTUM
89
Fig. 24. Reconstruction of a train of attosecond pulses synthesized from the five harmonics
q = 11, . . . , 19. The attosecond pulses are separated by 1.35 fs, which is half the cycle time of
the driving laser. The latter is represented by the dashed cosine function. Reprinted with permission
from Paul et al. (2001), Science 292, 1689, fig. 4. © 2001 American Association for the Advancement
of Science.
yield at energy Eq is proportional to the square of the (coherent) sum of the
amplitudes of all four quantum paths. Due to the fact that two paths represent
absorption from the NIR field whereas the other two represent emission into it,
the interference term between these two contributions is essentially proportional
to cos(÷q−1 − ÷q+1 + 2àwt). By varying the delay t between the harmonic and
the NIR radiation, the difference ÷q−1 − ÷q+1 of the phases of the two harmonics
can be recorded. The result of the corresponding experiment (Paul et al., 2001)
is that the phase of the harmonics depends almost linearly on their frequency.
Hence, the harmonics considered in the experiment (q = 11 to 19) are modelocked and make up a train of attosecond pulses of 250 as FWHM duration, see
Fig. 24.
A.2. Isolated attosecond pulses
With respect to applications, isolated attosecond pulses appear more useful than
a train of pulses separated by half the period of the fundamental. Isolated
attosecond pulses could be generated by sufficiently short fundamental pulses,
i.e. pulses of about 5 fs, which consist of less than two optical cycles (few-cycle
regime). Then, however, the spectral width of the harmonics will be so broad
that it is no longer possible to identify individual sidebands as necessary for the
method of Paul et al. (2001).
Nevertheless, Drescher et al. (2001) and Hentschel et al. (2002) succeeded
in performing measurements of the harmonic-pulse length with a resolution of
1.8 fs and 150 as, respectively. The experimental setup, in principle, resembles
that of Paul et al. with the difference that higher intensities of the NIR ra-
90
W. Becker et al.
[VI
diation are used for the photoionization cross correlation. In addition, only
photoelectrons ejected perpendicularly to the laser polarization are detected.
The motivation for choosing these conditions can be deduced from a classical
analysis of trajectories of electrons that were injected into the electric field of
the few-cycle NIR pulse by absorption of a harmonic photon. If the duration of
the X-ray pulse is shorter than the optical period T in the NIR, then the final
kinetic energy of the photoelectrons depends on the phase wt0 when the injection
took place, i. e. it exhibits a modulation with a period of T/ 2. By delaying
the fundamental with respect to the harmonic, the modulation can be recorded.
This was done in the experiment of Drescher et al. (2001). Hentschel et al.
(2002) relized that the width of the photoelectrons’ kinetic energy distribution
also exhibits such a modulation, and is measureable with much higher precision
than the center of mass of the distribution. For the two approaches, it is not the
envelope of the fundamental that enters the correlation function, but rather the
optical period. The restriction to photoelectrons emitted perpendicularly to the
laser polarization suppresses the influence of effects related to the emission and
absorption of photons from the laser field, i.e. the sidebands which were crucial
for the experiment of Paul et al. (2001).
B. The “Absolute Phase” of Few-cycle Laser Pulses
The need for highest intensities and extremely broad bandwidths in several areas
of the natural sciences is driving the development to shorter and shorter laser
pulses. At a FWHM duration shorter than a few optical cycles the time variation
of the pulse’s electric field depends on the phase f of the carrier frequency with
respect to the center of the envelope, the so-called “absolute phase.” The electric
field should be written as
E(t) = E0 (t) ex cos(wt + f),
(70)
where the function E0 (t) is maximal at t = 0. Clearly, for a long pulse the phase f
can be practically eliminated by resetting the clock. For a short pulse, however,
the shape of the field (70) strongly depends on this phase, which, therefore, will
influence various effects of the laser–atom interaction. This is one reason for the
significance of this new parameter of laser pulses. The precise knowledge and
control of the absolute phase will pave the way to new regimes in coherent X-ray
generation and attosecond generation; for an overview see Krausz (2001). In addition, such extremely well-defined laser pulses are likely to have applications for
the coherent control of chemical reactions and other processes. Another reason is
that phase control of femtosecond laser pulses has already had a huge impact on
frequency metrology. This is because phase-stabilized femtosecond lasers can be
viewed as ultra-broadband frequency combs that can be used to measure optical
frequencies with atomic-clock precision; see, e.g., Jones et al. (2000).
VIII]
ATI: CLASSICAL TO QUANTUM
91
With current laser technology, only femtosecond laser oscillators can be phasestabilized (Reichert et al., 1999; Apolonski et al., 2000), which is sufficient for
frequency metrology. Strong-field effects require amplified laser pulses. Nisoli
et al. (1997) demonstrated that it is possible to generate powerful (>500 mJ)
laser pulses in the few-cycle regime. However, these are not stabilized and,
accordingly, the absolute phase changes in a random fashion from pulse to pulse.
In a recent experiment, Paulus et al. (2001) were able to detect effects due
to the absolute phase by performing a shot-to-shot analysis of the number of
photoelectrons emitted in opposite directions. To this end, a field-free drift tube is
placed symmetrically around the target gas. Each end of the tube is equipped with
an electron detector. Because of its characteristic appearance, this was dubbed a
stereo-ATI spectrometer.
A characteristic feature of few-cycle pulses such as (70) is that, depending
on the absolute phase, the peak electric-field strength (and thus also the vector
potential) is different in the positive and negative x-directions. Recall from
Eq. (2) that the electron’s drift momentum depends on the vector potential at its
time of birth. Therefore, depending on the value of the absolute phase, such a
laser pulse creates more electrons in one direction than in the other. A theoretical
analysis of the photoelectrons’ angular distribution was given by Dietrich et al.
(2000) and Hansen et al. (2001). An equivalent statement is that the number
of electrons emitted to the left vs. those emitted to the right is anticorrelated:
A laser shot for which many electrons are seen at the right detector is likely to
produce only a few that go left, and vice versa. This can be proved by correlation
analysis. Each laser shot is sorted into a contingency map according to the
number of electrons recorded at both detectors. Anticorrelations can then be seen
in structures perpendicular to the diagonal, see Fig. 25 (overleaf).
VII. Acknowledgments
We learned a lot in discussions with S.L. Chin, M. Dörr, C. Faria, S.P.
Goreslavskii, C.J. Joachain, M. Kleber, V.P. Krainov, M. Lewenstein, A. Lohr,
H.G. Muller, S.V. Popruzhenko, and W. Sandner. This work was supported in
part by Deutsche Forschungsgemeinschaft and Volkswagen Stiftung.
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