Single and Crystalized Ions in
Ultra-Intense Laser Pulses
K a r e n Z. H a t s a g o r t s y a n *, U l r i c h D . J e n t s c h u r a * and C h r i s t o p h H . K e i t e l
*Theoretische Quantendynamik, Fakultat fur Physik, Universitat Freiburg,
Hermann-Herder-Strafie 3, D-79104 Freiburg, Germany
Abstract. Relativistic and quantum mechanical effects in the interaction of atomic systems with a
superintense laser field are investigated. Various atomic sytems are the object of attention, ranging
from highly charged ions to thin crystal layers. A new regime of high-harmonic generation (HHG)
based on multiply charged ions is described, where the dynamics of the tunneled wave packet of
the electron between birth and recombination is strongly modified by the ionic core, producing
a coherent burst of hard X-rays. We also consider relativistic effects in the nonsequential double
ionization of helium. The scheme for phase-matched HHG from a laser-driven thin crystal layer in
the presence of a high background of free electrons is discussed. For a selective range of parameters,
the initial regular structure of the layer is preserved during the strong interaction with the laser
pulse due to the suppression of shock waves in the layer. As a result, short hard x-ray pulses
may be generated with unprecedented coherence, and thus the possibility of applications in x-ray
spectroscopy and holography.
INTRODUCTION
Since its inception, laser radiation has proven to be a powerful tool for both probing
and modifying matter. The increasing laser intensity unveils new regimes of lasermatter interaction and opens new perspectives and applications. These include the laser
acceleration of particles, the fast ignition of nuclear fusion and the creation of shortwavelength coherent radiation sources. The last substantial progress in laser intensities
has been achieved due to the chirped-pulse amplification technique (CPA), which now
enables scientists to deliver focused intensities up to the order of 1022 W/cm 2 [1]. These
laser intensities can impart electrons with tremendous energy. Specifically, in the laser
focus, the electrons can be accelerated up to GeV energies [2]. Meanwhile, a dicusssion
is in agenda to extend the CPA systems capability up to the order of 1028 W/cm 2 using
megajoule laser facilities [3].
For high intensities above 1015 W/cm 2 , the relevant parameter for the description
of the nonlinear optical processes is the so-called Keldysh parameter 7 = y/ p /2C7,
i.e. the ratio of the ionization energy Ip to the electron ponderomotive energy in the
laser field U. Here, U = e2E|/4mo;|, and e and m are the electron charge and mass,
1
Permanent address: Department of Theoretical Physics, Yerevan State University, A. Manoukian Street
1, 375025 Yerevan, Armenia.
2
Email address: keitel@uni-freiburg.de
CP634, Science of Superstrong Field Interactions, edited by K. Nakajima and M. Deguchi
© 2002 American Institute of Physics 0-7354-0089-X/02/$ 19.00
71
respectively. EL and UL represent the laser field amplitude and frequency. In the strongfield regime, j~l > 1, the Coulomb barrier is suppressed by the laser field so that the
electron can tunnel out through the barrier during the laser period, a process known as
tunneling ionization (TI). In the strong-field regime, the relativistic effects, which are the
nondipole interaction, the magnetic-field influence and spin effects, begin to play a role
when the free-electron oscillation velocity becomes comparable to the velocity of light,
i.e. at £ := eEL/mcu}L > 1, where c is the speed of light. The main concern of this report
is the investigation of distinct features of relativistic effects in the strong field regime of
laser-matter interaction, especially in the HHG process. Various atomic sytems will be
the object of attention, particularly, highly charged ions, helium atoms and thin crystal
layers.
QUANTUM RELATIVISTIC INTERACTION OF MULTIPLY
CHARGED IONS WITH SUPER-INTENSE LASER PULSES
The interaction of atomic systems with high-power laser pulses is a very promising
source of high-frequency coherent and short light pulses. The oscillation of small parts
of the electronic wavepacket against the ionic core of such laser-driven atoms was shown
to lead to the generation of extremely high multiples, harmonics of the applied laser
frequency [4]. The shortest highly coherent light generated up to date has the wavelength
2.5 nm [5, 6]. One evident possibility for the increase of the HHG frequency consists in
the increase of the driving laser intensity. The immediate ionization of atomic systems
exposed to intense laser fields above 1015 W/em2 has been an obstacle to this approach.
The use of multiply charged ions circumvents this dilemma, at least to a certain extent.
Multiply charged ions of arbitrary charge states as well as their handling can nowadays
be carried out as a matter of routine. The remaining electrons are then so strongly bound
that even the most powerful laser fields available today can only temporarily remove
electrons from the vicinity of the nucleus until further recombination accompanied by
effective potential
polarization direction
propagation direction
FIGURE 1. Schematic diagram for coherent sub-Angstrom hard x-ray generation in multiply charged
ions. The effective potential originating from the ionic potential and the laser field at maximal strength,
the bound ground state with energy —Ip and the small tunneled part of it are depicted as a function of
the spatial coordinates in the laser polarization and propagation directions. Depending on the velocity
distribution of the tunneled electron wavepacket the arrows indicate that some fraction of the tunneled
wavepacket ionizes in both the laser polarization and propagation directions, while another may return
under the strong influence of the deep ionic potential. From [8] with permission.
72
HHG (see fig. 1).
The dynamics of the electron wavepacket in such intense laser fields are in the
relativistic regime. There has been considerable activity recently in this field. From
recent results we mention novel means of HHG via slow spin flipping [7], In Ref. [8], a
new regime of HHG is found where the quantum relativistic dynamics of the tunneled
electron significantly deviates from the conventional situation. We will give a more
detailed description of this result. Atomic Nitrogen in the sixth ionized state is exposed
to a superintense laser pulse of the order of 1018 W/cm2. The applied laser field still
corresponds to the weakly relativistic regime, and thus the Dirac equation describing
the dynamics of the corresponding wave function may be expanded in powers of the
ratio of electron velocity to the speed of light v/c, including up to fourth-order terms.
The radiation spectrum (see Fig. 2) shows that high harmonics are produced with
o
wavelengths well below the atomic scale of 1 A. The very high harmonic part is well
separated from the remaining part of the spectrum, in contrast to the "usual" mechanism
1
0.632
10
0.634
100
0.636
9
1000
(A)
0.638
10000
0.640
(b)
10
10
FIGURE 2. The radiation signal with sub-^4. harmonics. Calculated is the x-polarized radiation emitted
in the laser magnetic field direction. The laser parameters involve a wavelength of 248nm (KrF laser),
an intensity of 1.2 x 1018 W/cm2, a 10-cycle linear pulse turn-on and a 10-cycle duration with constant
amplitude. The ion is modeled to adapt atomic Nitrogen N+Q with all but one electron removed (for the
o
ion model see [9]). In a) the enlargement emphasizes the region of sub-A hard x rays, b) It shows the
o
o
structure for the spectral regime from 0.632A to 0.64 A From [8] with permission.
73
of HHG. A more intuitive understanding of this new mechanism of HHG can be drawn
from the trajectory of the electronic wave packet. As shown in the left side of Fig. 3,
the trajectory differs significantly from the conventional situation [4]. The tunneled
electron wavepacket between "birth" and recombination is not propagating freely, but
its trajectory is rather significantly modified by the strongly attractive potential of the
nucleus of the multiply charged ion. The wavepacket starting its return to the nucleus
£
200
100
(a)
-100
-200
-0.5
0.8
43
*q
3
0.6
I
u
0.4
0.2
0.5
9X10
6xl
-ISO -120 -60
0
60
120
ISO
<x> (atomic units)
- D.S5T1
CLS356
°"
3X1Q-
0x10
9.5
10.0
10.5
11.0
11.5
Interaction time (unit of laser periods)
FIGURE 3. Left figure: the weakly relativistic motion of the center-of-mass of the tunneled electronic
wavepacket. (x) and (z) denote the center of mass in the laser polarization direction x and the laser
o
propagation direction z, respectively. Right figure: the physical origin of the sub-A harmonics, a) The
center-of-mass motion of the tunneled fraction of the electronic wavepacket along the laser polarization
direction (x-axis) as a function of the interaction time; b) the corresponding ground state population
o
evolution in time and c) the radiation signal R as a function of time at wavelength 0.6571 A (solid line),
o
o
0.6356 A (long dashed line), and 0.6123A (short dashed line) as given by the reverse Fourier transform of
the spectrum in Fig. 2 around the respective frequencies. The parameters are as described in Fig. 2. From
[8] with permission.
A is merely scattered elastically at the nucleus B because of its high velocity (see right
side Fig. 3: (a) is the mean position of the electron in the laser polarization direction
(b) is the ground state population, and (c) is the emitted intensity of the radiation for
o
three selected harmonics with wavelengths below 1 A as a function of time). Significant
interaction with the ionic core takes place during a further recollision, points C-D,
involving sufficiently low velocities, as indicated by the strong peak of the ground-state
population at this time, see right Fig. 3 (b). The highly excited bound and continuum
states relax back to the ground state giving rise to hard x-rays at point D as clearly visible
in right Fig. 3 (c). Meanwhile for lower ion charges and laser intensities harmonics
74
are generated around B. Furthermore, we stress that the wavelength of the generated
coherent radiation may be further reduced with increasing laser intensity and increasing
ionic charge number.
At the end of this section we briefly mention first results concerning relativistic (and
magnetic-field) effects on the laser-induced nonsequential double ionization [11]. As is
well-known, the yield of the laser-induced double ionization of atoms is many orders
of magnitude higher than predicted within a theory using one active electron at a time
[10]. This phenomenon has been explained by a recollision of the first ionized electron.
In [11], a regime is found beyond the dipole approximation where the probability of
double ionization of helium via recollisions is reduced as the recolliding electron may
miss the ionic core with the inner electron due to the Lorentz force (generated by the
magnetic laser field). This effect confirms indirectly that the recollisions are responsible
for the high yield of nonsequential double photoionization. Related effects appear in the
angular distribution with enhanced ionization in the laser propagation direction.
X-RAY GENERATION FROM LASER DRIVEN CRYSTALS
A straightforward extension of the atomic HHG scheme into the ultrarelativistic regime
of laser intensities is barred by serious problems. First of all, one issue is the ionization
and the resulting deterioration of phase matching of the signals of the various ions due
to the presence of the electron background. Many efforts were undertaken to improve
the phase matching [4] with considerable success except for the hard x-ray regime. The
second problem is the radiation pressure in laser propagation direction that prevents
the electrons from a recombination. In this situation, it is therefore reasonable to use
many-body systems with a regular structure. There has been a lot of activity over the
last two decades regarding intense nonrelativistic laser interactions with many-body
systems: molecules [12], clusters [13], and solids [14], and laser interactions on the
surface of overdense plasmas as formed on laser-driven metallic surfaces [15]. In the
latter case, problems arise when approaching the hard x-ray regime due to collective
phenomena during the very intense laser-plasma interaction, these collective phenomena
substantially reduce the spatial coherence of the harmonic radiation. In [16], HHG
via stimulated bremsstrahlung is proposed due to a very short intense laser-crystal
interaction, where HHG is limited due to the rather low efficiency of multiple electron
scattering.
Usually, strong collective effects during laser-solid interaction arise due to the strong
ponderomotive pressure of the laser pulse on the solid surface that launches a shock wave
into the bulk of the medium. We propose a setup for strong laser pulse interaction with a
thin solid layer where collective effects can be effectively suppressed as the transversal
width of the layer, Lx = Ly = 20 //m, with respect to the laser beam propagation direction
is substantially less than the radius of the laser beam RL = 3mm (see Fig.4). Li are
spatial dimensions of the thin solid layer with separations a« between the atoms and
number of atoms Ni in direction i. The subindices i e {x,y,z} refer to the polarization
direction x, magnetic field direction y and propagation direction z of the applied laser
pulse, respectively. In this setup, the ponderomotive forces can be estimated to be
75
Z U/1L, where 1L = R^/LX, and Z is the effective charge number of the ion. The average
ionic displacement is found to be k & Z^c2rlLxm/(4MR\] during the brief lasercrystal interaction, with M being the mass of the identical ions, and TL the interaction
time. For a Tiisapphire laser pulse of 30fs duration, an intensity of / = 5 • 1021 W/cm2
o
and a potassium crystal (Z = 17), we obtain ^ ^ 0.2 A which is far less than the lattice
o
period a = 4.5 A. Thus, in the considered setup, the ponderomotive forces are not
strong enough to move electrons over an appreciable distance during the interaction.
Consequently, no collective instabilities can develop, and the structure of the layer will
be preserved during the process.
We assume that HHG at each atom arises due to the tunneling-recollision-mechanism.
For a potassium crystal, Z =17 electrons will be ionized on average by ththe considered
laser pulse via over-the-barrier-ionization, and it is the last remaining 18 electron that
will generate the harmonics. At these intensities, the medium is highly ionized, and
the refractive index for the laser wave n L is governed by the free electron background.
Nevertheless, for the applied parameters the plasma is yet underdense due to dependence
of the effective electron mass on the laser intensity, which allows the laser wave to
propagate in the plasma, i.e. LJL > UJP. Note, that this setup essentially differs from one
considered in Ref. [16], since in the latter case, the transversal amplitude of the electron
oscillation exceeds the medium (layer) width, and the laser radiation destroyes the layer:
consequently, the above mentioned refractive index becomes irrelevant.
E
laser field
^L
FIGURE 4. Phase matching of HHG from a thin solid layer with periodic ionic structure in an intense
and very short laser pulse. Transverse phase matching, i.e. between ion 1 and 2, requires via Eq. (1)
uti — kl = 27TSi with I = ax sin#, t\ = 0 and integer number si. Longitudinal phase matching, i.e.
between ions 3 and 4, requires analogously wtz — kl* = 2?rs2 with /* = az cos<9, time delay t^ = azriL/c
and integer s2 • 9 is the angle between the propagation direction of the laser pulse z and that of the harmonic
wave k. From [17] with permission of the IOP Publishing Limited.
Maintenance of the periodic structure of the thin crystal layer allows to solve phase
matching problem of HHG in the presence of high free electron background. It is
achieved by imposing the Bragg condition for phase matching between fundamental
76
and harmonic waves generated from various ions:
j = l,2,...
(1)
where 1 is the displacement vector of the ions, o;,k are frequency and wave vector of
the harmonic radiation, respectively, with k = |k| = nu;/c, refractive index n for the
harmonic radiation and t being the time for the laser pulse to propagate the distance 1.
With Eq. (1) and Fig. 4 we find the phase matching condition kaxsin0 = ZTVSI
and azu(nL — ncos#)/c — 27T52, with s\$ — 0,±l,d=2,.... We consider the two most
favorable directions of detection along the laser propagation direction (first scheme:
9 = Q,SI = 0) and at a nonvanishing angle 9 = arccos(nJL/n) (second scheme: 9 ^ 0,
s2 = 0).
(a)
(b)
x1(T
-e05
0.5
-0.5
(0/0),
0.5
FIGURE 5. a) Sketch of a spectrum with periodic phase matching, b) HHG spectral intensity
from a crystal layer Ww = dW/dwdQ scaled by the respective peak value for one ion W& =
x, where x = U/UL — 106- The narrow features (thin lines) correspond to the coherent
HHG from the periodic structure, while the wide features (thick lines) do to the HHG from one ion
multiplied by number of ions and magnified by 3 • 1014, corresponding by intensity to the incoherent
HHG from disordered ions . The solid lines correspond to Nz = 2 • 105 and the dashed lines to Nz = 105 .
The parameters employed are: Lx—Ly — 20 /^m, XL — 800 nm, TL — 30 /s, 0 = Q,ax — ay — az — 4.5
o
A , n - UL — 0.057. Prom [17] with permission of the IOP Publishing Limited.
We calculate the spectral intensity from the regular structure taking advantage of the
fact that the electrons responsible for HHG ionize as well as evolve in the laser field in a
similar manner. Consequently, their trajectories are congruent apart from translations in
space by 1 and in time by t. Thus, to a good approximation, the total HHG spectral intensity of the crystal can be expressed via a coherent sum of the HHG spectral intensity
arising from single ions (see Fig. 5). The peak spectral intensity of harmonic radiation at
the coherence condition (1) is ideally proportional to (NxNyNz)2 and will thus in general
significantly exceed the HHG spectral intensity of irregular systems. Reduced enhancements are still possible within the spectral and angular linewidths: SM/UJ « az/sLz and
60 « /XLX, with A being the harmonic wavelength. The efficiency of the proposed x-ray
radiation source can be estimated by the spectral brightness of the radiation which we
express for convenience in units photons/ (s -
B=
dN
= 3-10
15NxNyNznL
|n x
axayaz(A)
77
(2)
where as usual "0.1% bandwidth" means that the spectral variable B is scaled by 0.1%
of the considered frequency, n = k/k and a^ = (c^/e) jffle~lkrdV, N is the number of
emitted photons, A is the emitter area and O is the solid angle in the emision direction.
For a high laser intensity of / = 5 • 1021 W/cm 2 , hard x-ray HHG is possible in
principle with unprecedented coherence; e.g., for l.SMeV photons, the spectral width
becomes 6u/w w 1.4 • 10~7, and the angular divergence of the emitted radiation 89 «
4-10~ 8 . No reliable estimates are currently available for the single-ion-variable aw
entering into the expression for the total spectral brightness in Eq. (2), because of the
favorable but complex situation that the tunneled wave packet may recombine with
several ions on its journey through the crystal
The suggested scheme of laser-driven thin crystals is attractive because of its high coherence (10"5% bandwidth for MeV photons or 10~3% bandwidth for 3 KeV photons).
In particular, we hope that our scheme could lead to a highly coherent table-top light
source with photon energies above few KeV. Finally the brevity of the hard x-ray pulses
may allow for time-resolved x-ray holography and spectroscopy.
ACKNOWLEDGMENTS
This work was supported by Deutsche Forschungsgemeinschaft (Nachwuchsgruppe
within SFB 276). KZH is also supported by the Humboldt Foundation.
REFERENCES
1. G. Mourou, C.P.J. Barty, and M.D. Periy, Phys. Today 51, No.l, 22 031301 (1998); M. D. Peny et al.,
Opt Lett. 24,160 (1999).
2. Y.I. Salamin and C.H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
3. T. Tajima and G. Mourou, Phys. Rev. ST AB 5, 031301 (2002).
4. C. J. Joachain, M. Dorr, and N. Kylstra, Adv. At. Mol. Opt. Phys. 42, 225 (2000); C. H. Keitel,
Contemp. Phys. 42, 353 (2001).
5. C. Spielmann, et al., Science 278, 661 (1997).
6. Z. H. Chang, et al., Phys. Rev. Lett. 79, 2967 (1997);
7. M. W. Walser and C. H. Keitel, Opt Comm. 199,447 (2001) and Phys. Rev. A 65, 043410 (2002).
8. C.H. Keitel and S. X. Hu, Appl. Phys. Lett., 80, 541 (2002).
9. S. X. Hu and C.H. Keitel, Phys. Rev. Lett., 83, 4709 (1999) and Phys. Rev. A 63, 053402 (2001).
10. J.B. Wateon, etal., Phys. Rjev. Lett. 78,1884 (1997); T. Weber, etal., Phys. Rev. Lett. 84,443 (2000);
R. Moshammer, et al., Phys. Rev. Lett. 84,447 (2000); A. Becker and F.H.M. Faisal, Phys. Rev. Lett 84,
3546 (2000); J.S. Parker, J. Phys. B 34, L69 (2001).
11. J. Prager and C.H. Keitel, J. Phys. B 35, L167 (2002).
12. A.D. Bandrauk, Molecules in Laser Melds (Dekker, New York, 1993).
13. See e.g. A. McPherson et al., Nature (London) 370, 631 (1994); T. Ditmire et al., Nature (London)
398,489 (1999); S. X. Hu and Z. Z. Hu, Appl. Phys. Lett 71, 2605 (1997).
14. L. Plaja and L. Roso-Franco, Phys. Rev. A 45, 8334 (1992); S. Huller and J. Meyer-ter-Vehn, Phys.
Rev. A 48,3906 (1993); F.H.M. Faisal and J.Z. Kaminski, Phys. Rev. A 56, 748 (1997).
15. PA. Norreys et al, Phys. Rev. Lett. 76,1832 (1996); D. von der Linde et al., Phys. Rev. A 52, R25
(1995); P. Gibbon, IEEE J. Quant. El. 33, 11 (1997); K. W. D. Ledinghani, Phys. Rev. Lett. 84, 899
(2000).
16. K.Z. Hateagorteyanand C.H. Keitel, Phys. Rev. Lett. 86, 2277 (2001).
17. K.Z. Hateagortsyanand C.H. Keitel, J. Phys. B 35, L175 (2002).
78