High-harmonic generation

16 Pages·2006·1.75 MB·English

High Harmonic Generation
Basic Techniques & Challenges
C. Michael R. Greaves
Regimes of Nonlinear Optics
564 T. Brabec and F. Krausz: Intense few-cycle laser fields
lomb field. As a consequence, the laser field only slightly
perturbs the atomic quantum states under nonresonant
excitation conditions. The energy levels suffer only a
2
faint shift proportional to E , which is referred to as the
a
ac Stark shift (Landau and Lifshitz, 1977). The atoms
remain, with a high probability, in their ground state and
the extension of the wave function of the ground state
remains on the order of the Bohr radius a . Nonlinear
B
interactions taking place under these conditions can be
well described by a perturbative approach, and hence we
refer to this parameter range as the regime of perturba-
tive nonlinear optics.
If the electric-field strength becomes comparable to
(or higher than) the binding atomic Coulomb field expe-
rienced by the outer-shell electrons, an electron can es-
cape with a substantial probability from its bound state
FIG. 24. Regimes of nonlinear optics. Only nonresonant inter-
(via tunneling or above-barrier detachment) before the
actions have been considered by assuming the detuning &) to
laser electric field reverses its sign. The electron wave
be larger than the photon energy (*1 eV). The boundaries
packet liberated by optical-field ionization subsequently
between different regimes are not sharply defined. In the
wiggles in the linearly polarized electric field. The am-
strong-field regime, the intensity scale applies to the visible
plitude of the wiggle exceeds the Bohr radius by several
and near-infrared spectral range.
orders of magnitude and the cycle-averaged kinetic en-
ergy of the electron exceeds the binding energy W .
b
introduce field-strength scale parameters that roughly
The parameter range giving rise to these processes is
define the borders of the above regimes. The relation of
referred to as the strong-field regime of nonlinear optics.
the induced medium polarization to the incident fields is
The atomic polarization response is dominated by the
indispensable for describing the propagation of intense
ionization process and the contribution from the bound
radiation in atomic media, i.e., for the interaction of in-
electrons is negligible.
tense light with matter in a macroscopic volume.
Nonlinear polarization induced by optical-field ioniza-
tion emerges only as long as the electron remains in
close proximity to its parent ion. Once the electron is set
A. Perturbative nonlinear optics
free, its trajectory is governed by the Newtonian equa-
tions of motion, resulting in a linear response (with a At low and moderate intensities the polarization, P
2
small remaining nonlinearity originating from the parent " As/m # (where As represents Ampere seconds), of an
ion’s reduced polarizibility). A strong nonlinearity arises atomic ensemble can be expanded into a Taylor series
only at intensities order(s) of magnitude higher due to with respect to the electric field and can be given as the
optical-field ionization stripping the next electron and/or superposition of the linear and nonlinear responses P
(1)
to the wiggle energy of free electrons becoming compa- !$ % E"P , where
0 nl
2
rable to their rest energy mc , indicating the onset of
(2) 2 (3) 3 (4) 4
P !$ % E "$ % E "$ % E "•••, (19)
nl 0 0 0
relativistic nonlinear optics.
$12
Figure 24 assigns the relevant intensity regions to the $ !8.85#10 As/V m is the vacuum permeability, and
0
(k) k$1
above regimes of nonlinear optics for visible and near- % " (m/V) # is the kth-order susceptibility (Bloem-
infrared radiation. Here we focus on nonrelativistic bergen, 1965). It is implicit in Eq. (19) that the atomic
light-matter interactions. We shall (i) review how exploi- polarization instantly follows the change of the field,
tation of processes in the perturbative regime allows which is usually a good approximation even on a time
generation of intense light pulses in the few-cycle regime scale of a few femtoseconds. This is because the induced
and (ii) demonstrate that these pulses are capable of atomic dipole moment is of purely electronic origin with
significantly extending the frontiers of strong-field non- a response time on the order of 1/&, where &!!’
ik
linear optics. Whereas in the perturbative regime the $’ !, ’ represents the transition frequency from the
0 ik
intensity envelope governs the evolution of nonlinear initial (usually ground) quantum state i into some ex-
optical processes, in the strong-field regime the electric cited state k for which !’ $’! is the minimum and the
ik
(and at relativistic intensities also the magnetic) fields dipole transition matrix element ( is nonzero, and ’
ik 0
take control. Because the generation of ultrashort light is the laser carrier frequency [see Eq. (14)]. Since the
pulses relies on perturbative processes, the phase ! [see typical transition frequency from the atomic ground
0
Eq. (14)] is unknown, and the electric field of these state to the lowest excited state significantly exceeds the
pulses is indefinite. The light fields are expected to be- laser frequency in the visible and near-infrared range,
come accessible when strong-field processes with few- 1/& is typically less than 1 fs. In molecules and con-
cycle wave packets are induced (see Sec. VIII). In Secs. densed matter, nuclear motion may also provide a sig-
IV. A-IV.C, we analyze the polarization response of an nificant contribution to the induced dipole moment. This
atomic medium irradiated with strong laser fields and contribution has a response time of hundreds of femto-
Rev. Mod. Phys., Vol. 72, No. 2, April 2000
Outline
• Optical Field Atomic Ionization
• High Harmonic Generation (HHG)
• Microscopic Description
• Macroscopic Description
• Techniques for Phase Matched Generation of
Coherent Soft X-rays
• References
Optical Field Atomic
Ionization
570 T. Brabec and F. Krausz: Intense few-cycle laser fields
energy loss suffered by the laser pulse according to Eq.
• Harmonics are generated by
(24) is in good agreement with experimental observa-
the tunneling, transport and
tions (Fig. 29).
Equations (31) and (23) take proper account of non-
recombination of electrons
linear interactions in the strong-field regime where the
use of intense few-cycle pulses pushes the nonlinear re-
• Atomic ionization proceeds
sponse of (ionizing) matter to unprecedented extremes.
by perturbation of the
As a result, coherently generated harmonics of the driv-
ing laser radiation now exceed the 300th order. Model-
Coulomb potential in the
ing even predicts significant further improvement (in
presence of the electric field
terms of both photon energy and conversion efficiency)
in previously unaccessed parameter ranges and provides of the driving laser pulse
relevant guidelines for the experimenters (Sec. VII).
This same theoretical framework also accounts for the • Ionization rate by the
fascinating temporal behavior of x-ray harmonic radia-
Quasistatic Approximation
tion (Sec. VII) as well as the dependence of strong-field
processes on the carrier phase in few-cycle wave packets
FIG. 30. Regimes of atomic ionization. Exposing an atom to
(Sec. VIII).
an intense laser field will result in a modified potential (solid
curve) composed of the Coulomb potential (dashed curve) and
VI. OPTICAL-FIELD IONIZATION OF ATOMS
the time-dependent effective potential of the optical pulse. (a)
At moderate intensities the resulting potential is close to the
When matter is exposed to intense laser fields, a
unperturbed Coulomb potential and an electron can be liber-
wealth of exciting phenomena can be observed, includ-
ated only upon simultaneous absorption of N photons, result-
ing high harmonic generation (Corkum, 1993), above- ing in multiphoton ionization. The multiphoton ionization rate
threshold ionization (Corkum et al., 1989), atomic stabi- scales with the Nth power of the intensity of the optical pulse.
lization (Pont and Gavrila, 1990), x-ray lasing (Lemoff (b) At sufficiently high field strengths the Coulomb barrier
et al., 1995; Hooker et al., 1995), laser-induced damage becomes narrow, allowing optical tunneling ionization to take
over and resulting in a tunneling current that follows adiabati-
of dielectrics (Du et al., 1994; Stuart et al., 1995; Lenzner
cally the variation of the resultant potential. (c) At very high
et al., 1998; Tien et al., 1999) and molecular dissociation
field strengths, the electric field amplitude reaches values suf-
(Seidemann et al., 1995; Chelkowski et al., 1996). The
ficient to suppress the Coulomb barrier below the energy level
key process triggering all of these strong-field phenom-
of the ground state, opening the way to above-barrier ioniza-
ena is ionization. The theoretical investigation of these
tion.
processes is significantly simplified by the analytic de-
scription of ionization that is available in the limiting
cases characterized by !!1 and !"1 [see Eq. (21)]. The static field ionization rates, namely, the Keldysh theory
two limiting cases of multiphoton ionization and optical- (Keldysh, 1965; Reiss, 1980) and the Ammosov-Delone-
field ionization are illustrated schematically in Figs. Krainov theory (Oppenheimer, 1928; Perelomov et al.,
30(a)–(c), respectively. 1996; Ammosov et al., 1986). In Fig. 31, the Ammosov-
In what follows we shall restrict our discussion to the Delone-Krainov and Keldysh ionization rates for hydro-
strong-field limit !"#1, for which significant ionization gen and helium atoms exposed to a static electric field
takes place. In this parameter range the quasistatic ap- are compared with the rates obtained from an exact nu-
¨
proximation is valid, which relies on the assumption that merical solution of the time-independent Schrodinger
the perturbed electron wave function reaches a quasi-
static state before the electric field changes significantly
(Shakeshaft et al., 1990). Then the fraction of electrons
ionized in the laser field E(t) as a function of time t may
be calculated by Eq. (23). At the threshold for the de-
tachment of the second (or higher) electrons, Eq. (23)
must be generalized by calculating a sum over the indi-
vidual ionization processes, (1/n ) # n (t), determined
a i ei
by the ionization rates w for the ith electron. Further-
i
more, possible nonsequential ionization channels must
be taken into account (Fittinghoff et al., 1992; Corkum,
1993; Walker et al., 1994; Augst et al., 1995; Lablanquie
et al., 1995; Brabec et al., 1996; Dorner et al., 1998;
FIG. 31. Static field ionization rates in hydrogen and helium
Rosen et al., 1999).
atoms vs the electric-field strength in atomic units: solid curve,
The power of the quasistatic approximation rests on
numerical result; dashed curve, Ammosov-Delone-Krainov
the fact that ionization in time-varying laser fields may
formula; dash-dotted curve, Keldysh theory for hydrogen. The
be calculated by using the static field ionization rate w.
dotted line denotes the barrier suppression field strength for H
There exist two approaches for analytic calculation of and He.
Rev. Mod. Phys., Vol. 72, No. 2, April 2000
Strong Field Regime
• Measure of field strength is given by Keldysh
scale parameter:
1 eE
a
=
γ ω √2mW
0 b
1
• > 1 defines the strong field regime
γ
• Coulomb potential is suppressed
• Electron can tunnel through the potential barrier
• Corkum model: Classical description of the
electron evolution after ionization
Outline
• Optical Field Atomic Ionization
• High Harmonic Generation (HHG)
• Microscopic Description
• Macroscopic Description
• Techniques for Phase Matched Generation of
Coherent Soft X-rays
• References
HHG Microscopic Analysis
• HHG photon radiation dependent on the
acceleration of the electron
2
∂
< ψ r ψ >
e e
∂t2 | |
• Two simplifying assumptions:
1. Electron is a free particle in continuum states
2. Only ground state interactions are considered
• Under these assumptions:
1
d (τ ) = a (τ )a (τ , τ )a (τ )
n ion b propogation b recombination
√
i
τ
b
!
Microscopic II
• Model valid for linearly polarized light
• Elliptically polarized light is less efficient
• a time is on the order of the oscillation
propogation
period of the electric field
• Photon emission occurs upon recombination
with the ground state
Microscopic III
τ
propogation
free electron
E(t)
t
τ τ
b recombination
E-field potential high KE electron
suppresses Coulomb recombination yields
field harmonic photon
Microscopic IV
( ω) = I + 3.2U
!
max p p
2πca I
0 L
U =
p
ω2
• Periodic process yields odd harmonics of the
laser field