Research Papers in Physics and Astronomy
Anthony F. Starace Publications
University of Nebraska - LincolnYear
Analytic Description of the High-Energy
Plateau in Harmonic Generation by
Atoms: Can the Harmonic Power
Increase with Increasing Laser
M. V. Frolov∗
M. Yu. Emelin∗∗
N. L. Manakov†
M. Yu. Ryabikin††
T. S. Sarantseva‡
Anthony F. Starace‡‡
∗Voronezh State University
†Voronezh State University
‡Voronezh State University
∗∗Institute of Applied Physics, Russian Academy of Sciences
††Institute of Applied Physics, Russian Academy of Sciences
‡‡University of Nebraska-Lincoln, email@example.com
This paper is posted at DigitalCommons@University of Nebraska - Lincoln.
Analytic Description of the High-Energy Plateau in Harmonic Generation by Atoms:
Can the Harmonic Power Increase with Increasing Laser Wavelengths?
M.V. Frolov,1N.L. Manakov,1T.S. Sarantseva,1M.Yu. Emelin,2M.Yu. Ryabikin,2and Anthony F. Starace3
1Department of Physics, Voronezh State University, Voronezh 394006, Russia
2Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia
3Department of Physics and Astronomy, The University of Nebraska, Lincoln, Nebraska 68588-0111, USA
(Received 27 February 2009; published 16 June 2009; publisher error corrected 18 June 2009)
A closed-form analytic formula for high-order harmonic generation (HHG) rates for atoms (that
generalizes an HHG formula for negative ions [M.V. Frolov et al., J. Phys. B 42, 035601 (2009)]) is used
to study laser wavelength scaling of the HHG yield for harmonic energies in the cutoff region of the HHG
plateau. We predict increases of the harmonic power for HHG by Ar, Kr, and Xe with increasing
wavelength ? over atom-specific intervals of ? in the infrared region, ? ? ð0:8 ? 2:0Þ ?m.
DOI: 10.1103/PhysRevLett.102.243901PACS numbers: 42.65.Ky, 32.80.Rm, 32.80.Wr
For over two decades, the generation of harmonics of
intense femtosecond laser radiation by atoms and mole-
cules has been one of the most studied processes in intense
laser physics. More recently, high-order harmonic genera-
tion (HHG) has become fundamentally important for a
wide range of diverse applications. In particular, it is a
key component of attosecond science, providing a means
to produce attosecond XUV pulses . It also underlies
tabletop sources of coherent soft x-ray radiation in impor-
tant energy regions, such as the ‘‘water window’’ (cf., e.g.,
Ref. , which reports the generation of 300 and 450 eV
harmonics of the driving laser wavelength, ? ¼ 1:6 ?m).
In this regard, investigation of the scaling of the HHG yield
with increasing ? is of great interest (cf. Refs. [3–7]).
Finally, very promising are recent applications of HHG-
based methods for extracting field-free atomic and molecu-
lar data, as in tomographic imaging of molecular orbitals
[8,9] and in the extraction of atomic photorecombination
cross sections (PRCSs) from HHG experimental data .
These latter applications are based on an intuitive parame-
trization of HHG rates (based on thewell-known three-step
HHG scenario [11,12]) in terms of the PRCS and an
‘‘electron wave packet’’ (EWP) resulting from the ioniza-
tion and laser-acceleration steps of the active electron
within the three-step scenario. As shown in Refs. [13,14],
this parametrization is well supported by direct numerical
solutions of the time-dependent Schro ¨dinger equation
(TDSE) for a single active electron. However, the analytic
structure of the EWP remains a ‘‘black box.’’
In this Letter we present a closed form analytic formula
for the HHG rate for harmonics at the high-energy end of
the HHG plateau. Included is an analytic formula for the
EWP, which is largely independent of the atomic target (in
agreement with numerical results of Refs. [13,14]). We use
our analytic results to analyze the wavelength scaling of
the HHG yield in the region of the HHG plateau cutoff. We
find that the scaling law for the yield of harmonics near the
cutoff in rare gases is different from that predicted in
Ref.  (and partly supported experimentally ) for
harmonics below the plateau cutoff. Moreover, we show
that in some cases the HHG efficiency increases with
increasing ? in the long-wavelength domain.
To deduce our analytic results for the emission of high-
harmonic photons by an electron bound in an atomic
(Coulomb) potential that interacts with a laser electric field
FðtÞ ¼ ^ zFcos!t (where F and ! are the field amplitude
and frequency), we discuss first our recent analytic result
c?lmðrÞ, with energy E0¼ ?@2?2=ð2meÞ and angular mo-
mentum l . This latter result was derived quantum
mechanically [in the tunneling limit, ? ? 1, where ? ¼
for HHG rates, RðE?Þ (E?¼ @? ¼ n@!), for the case of
an electron bound by a short-range potential in the state
@!=ðeF??1Þ is the Keldysh parameter] based on a general,
ab initio formulation for the HHG amplitude  that was
applied to the case of HHG by an electron in a short-range
potential using time-dependent effective range (TDER)
theory . To generalize these short-range potential re-
sults to the case of a long-range (Coulomb) potential, we
represent Eq. (28) for RðE?Þ in Ref.  as a product of
RðE?Þ ¼ Ið~ F;!ÞWðEÞ?ðrÞðEÞ;E ¼ E?? jE0j;
and interpret each of them within the three-step scenario.
(i) The dimensionless ‘‘ionization factor,’’ Ið~ F;!Þ, is
I ð~ F;!Þ ¼
ð2l þ 1Þ??vat
?stð~ FÞ;vat¼ e2=@;
where ~ F ¼ Fjcos~ ?0j ¼ 0:951F is an ‘‘effective’’ static
electric field, where ~ ?0defines the moment of ionization,
ti¼ ~ ?0!?1¼ ?0:45T (T ¼ 2?=!), within the classical
three-step HHG scenario. ?stð~ FÞ is the decay rate for a
weakly bound state c?lm¼0ðrÞ in a static field~F :
PRL 102, 243901 (2009)
19 JUNE 2009
? 2009 The American Physical Society
?stð~ FÞ ¼jE0j
ð2l þ 1ÞC2
=ðe@Þ ¼ ð?a0Þ3Fat, a0is the Bohr
radius, Fat¼ 5:14 ? 109V=cm, and C?lis the dimension-
less coefficient in the asymptotic form of c?lmðrÞ:
(ii) The ‘‘propagation factor,’’ WðEÞ (measured in
cm?2s?1), describes the propagation of an ionized electron
in the laser field FðtÞ from the time of its ionization, ti, to
the time of its recombination, tr¼ ~ ?!?1¼ 0:2T:
W ðEÞ ¼p
1016W=cm2. The Airy function’s argument,
, ?t ? tr? ti¼ 4:086!?1¼ 0:65 T,
where p ¼
? ¼ 1:866, I is the laser intensity, and Iat¼ 3:51 ?
? ¼ ð?Iat=IÞ1=3ðE ? EmaxÞ=Eat;
involves the difference between the electron energy E,
corresponding to a given harmonic energy E?, and the
maximum energy, Emax¼ 3:17upþ 0:32jE0j [where up¼
e2F2=ð4me!2Þ is the ponderomotive energy], gained from
the laser field by the ionized electron along the shortest
closed trajectory for the return time ?t [15,19]. [Note that
the HHG amplitude in terms of Airy functions (with argu-
ments different from ?) and their first derivatives was
obtained earlier in Ref.  for the bound s state in a
zero-range potential model. However, neither the argu-
ments of these Airy functions nor other factors in the
HHG amplitude were presented explicitly.]
(iii) The factor ?ðrÞðEÞ in Eq. (1) is the TDER result for
the differential PRCS for an electron with momentum p ¼
p^ z recombining to the bound state c?lm¼0ðrÞ with emis-
sion of a harmonic photon (of energy E?), whose polar-
ization is the same as that of the laser field FðtÞ :
where ? ¼ e2=ð@cÞ. Note that the PRCS in (7) is equiva-
lent to that in the Born approximation. This is because in
the TDER theory the electron interacts with the binding
potential only through the l-wave scattering phase ?lðEÞ,
while the (l ? 1)-wave continuum channels (which con-
tribute to the PRCS according to dipole selection rules)
remain undistorted by the short-range potential.
The analytic result for RðE?Þ provides HHG rates for
the high-energy end of the HHG plateau that are in ex-
cellent agreement with exact TDER results for negative
ions (cf. Ref. ). As each of the three factors in Eq. (1)
has a transparent physical meaning within the three-step
HHG scenario [which, as is commonly accepted, does not
depend on the atomic species], one can expect that an
?ðrÞðEÞ ¼ ?3C2
0;q ¼ p=ð@?Þ;
appropriategeneralizationof Eq.(1)should give an accept-
able description of HHG for atoms. This generalization
consists in the replacement of two of the three factors in
Eq. (1) by their corresponding atomic counterparts:
RðE?Þ ¼ Iað~ F;!ÞWðEÞ?ðrÞ
where the propagation factor WðEÞ is unchanged, because
Eq. (5) is essentially independent of the atomic structure,
describing free-electron motion in the laser field FðtÞ. The
TDER PRCS (7) should be replaced by the PRCS ?ðrÞ
specific atom. For instance, for the ground state hydrogen
a is given by 
a for a
a ðEÞ ¼ 32??3
q2ðq2þ 1Þ2ð1 ? e?2?=qÞa2
where q ¼ pa0=@. Finally, the ‘‘Coulomb-modified’’ ion-
ization factor Iað~ F;!Þ is given by [cf. Eq. (2)]
Iað~ F;!Þ ¼
ð2l þ 1Þ??avat
c?alm¼0ðrÞ of an active atomic electron with energy Ea¼
is the tunneling rate for a bound state
?ð@?aÞ2=ð2meÞ in a static electric field~F :
ð2l þ 1ÞC2
stð~ FÞ ¼jEaj
where ? ¼ Z=ð?aa0Þ, Z is the charge of the atomic core
(Z ¼ 1 for neutral atoms), Fa¼ ð?aa0Þ3Fat, and C?alis
given by the asymptotic form of c?almðrÞ [cf. Eq. (4)]:
We stress that only the PRCS in Eq. (8) is sensitive to the
energy-dependent atomic dynamics, while the EWP,
WðEÞ ¼ Iað~ F;!ÞWðEÞ;
contains only two ‘‘static’’ atomic parameters, jEaj and
C?al. Equations (5) and (10) show clearly that WðEÞ rep-
resents the flux of recombining electrons with velocity v ¼
p=me. Thus Eqs. (1) and (8) justify analytically the ad hoc
parametrization of the HHG yield in Refs. [13,14].
To test the accuracy of Eq. (8), we compare our analytic
results with results of numerical solutions of the TDSE for
a hydrogen atom subjected to a trapezoidal laser pulsewith
a 16-cycle flat top of intensity I ¼ 2 ? 1014W=cm2and
two-cycle ramps for turn-on or -off. The TDSE was solved
by using the FFT split-operator spectral method  and
the imaginary potential method in order to absorb reflec-
tion of waves from the grid boundary . The accuracy of
the numerical solution of the TDSE was monitored by
increasing the grid size and by decreasing the grid cell.
[The final results were obtained for a grid cell of ð0:4a0Þ3
over a box of 100a0in both the x and y axes and up to z *
6eF=ðme!2Þ in the direction of laser polarization. A fixed
mesh width in t was used: ?t ¼ 0:48 a:u:] As an additional
PRL 102, 243901 (2009)
19 JUNE 2009
test, we have checked that the harmonic power, P ¼
E?RðE?Þ, integrated over the interval 20 ? E??
50 eV decreases with increasing ? over the interval
0:8–1:6 ?m as ??xwith x ? 5:2. This result is close to
the result x ? 4:8 of Ref.  for an eight-cycle flat-top sine
pulse with a half-cycle turn-on and turn-off and I ¼ 1:6 ?
Figure 1 shows good agreement of the TDSE and ana-
lytic results at the high-energy ends of the HHG spectra.
The accuracy of the analytic result (8) is better for larger ?
[since Eq. (1) was derived in  in the low-frequency
(long-wavelength) limit]. Since the PRCS (9) varies
smoothly with energy, the oscillatory pattern along the
high-energy plateau in Fig. 1 maps interference oscilla-
tions of the EWP, WðEÞ, as a function of E?(cf. Ref. 
for details). Both our analytic results and the TDSE results
in Fig. 1 show that the harmonic power at the cutoff scales
differently from the widely accepted law Pð?Þ ? ??3
[19,24]. According to Eqs. (2) and (5), the EWP, WðEÞ,
decreases for large ? as ??4, whereas the PRCS at the
HHG cutoff energy (E?¼ Ec? jEaj þ 3:17up? ?2)
gives an extra dependence on ?. In the limit ? ! 1, the
argument of the PRCS in Eq. (8) goes to infinity, so that the
Born approximation may be used for ?ðrÞ
E?5=2? ??5for a Coulomb potential, while ?ðrÞ? ?2l?1
for a short-range potential [cf. Eq. (7)]. Hence, Pð?Þ scales
as ??7for a Coulomb field and ?2l?3for a short-range
potential. Thus, the ??3-scaling is realized only for an s
state in a short-range potential (to which, in fact, the
analyses in Refs. [19,24] correspond). In our analysis
above we assume ? ! 1 (or ? ! 0), while for ? in the
interval from 1.0 to 1:6 ?m, the Keldysh parameter varies
from ? ¼ 0:60 to 0.38 for I ¼ 2 ? 1014W=cm2. Thus, the
scaling law ??6:1for ? ¼ 1:0 ?m and 1:6 ?m, observed
inFig. 1,is slightlylesssteep than theasymptoticone, ??7.
The situation is different for nonhydrogenic atoms. For
these atoms, the photoionization cross section (PICS),
which is related to the PRCS by the equation of detailed
a ðEÞ. Thus, ?ðrÞ
balance , exhibits irregularities caused by Cooper min-
ima, potential barrier and electron correlation effects, etc.
. Owing to these irregularities, a universal scaling law
for Pð?Þ is not possible. Moreover, they can affect the
interference structures in the HHG spectra shown in Fig. 1.
As the oscillations in Pð?Þ [considered as a function of
harmonic energy E?(cf. Fig. 1)] are smoothed when
integrated over E?, the energy-integrated HHG power
(i.e., that of a group of near-cutoff harmonics) should
elucidate more definitively the atomic dynamic effects
caused by irregularities of ?ðrÞ
grated over E?in an energy interval ?E below the cutoff
(from 20 to 50 eV), it was found numerically to satisfy the
following scaling law : P?Eð?Þ ? ??ð5?6Þ. We consider
here the harmonic power integrated over a fixed energy
interval ?E centered at the cutoff energy Ecð? jEaj þ
P?Eð?Þ ¼ @!X
where n?¼ ½ðEc? ?E=2Þ=ð@!Þ?, and [x] is the integer
part of x. Using Eq. (8), the integral in Eq. (14) may be
approximated forthe case
2Eatð?Iat=IÞ?1=3] by taking into account that the argument
of the Airy function Aið?Þ in Eq. (5) for this case varies
over the interval ?1 < ? < þ1 for E?in the interval Ec?
?E=2 [cf. Eq. (6)]. Since Aið?Þ is smooth in the interval
j?j < 1, it may be replaced by a constant, e.g., Aið0Þ. As a
result, Eq. (14) (after changing the integration variable to
E ¼ E?? jEaj) gives
a ðEÞ. When Pð?Þ was inte-
Harmonic energy (eV)
Harmonic power (10-16Js-1)
FIG. 1 (color online).
2 ? 1014W=cm2and two laser-field wavelengths, ? ¼ 1:0 ?m
(left) and ? ¼ 1:6 ?m (right). Thin lines: numerical TDSE
results; thick lines: analytic results obtained using Eq. (8).
HHG spectra for the H atom for I ¼
FIG. 2 (color online).
vs ? for H and the rare gases with I ¼ 2 ? 1014W=cm2and two
energy intervals ?E: 20 (solid lines) and 30 eV (dotted lines).
Circles and squares in panel (a) mark the TDSE results for ?E ¼
20 and 30 eV, respectively.
Harmonic power, P?Eð?Þ [cf. Eq. (14)],
PRL 102, 243901 (2009)
19 JUNE 2009
This expression shows clearly that the ?-scaling of P?Eð?Þ Download full-text
for harmonic energies near the cutoff (for a given ?) is
For both short-range and Coulomb potentials, the PICSs
decrease smoothly with increasing E, giving a smooth
dependence of P?Eð?Þ on ?. Indeed, in the long-
wavelength limit, the Born result for ?ðrÞ
may be used in Eq. (15). That gives P?Eð?Þ ? ??6for the
case of a Coulomb potential, while the PRCS (7) gives
P?Eð?Þ ? ?2l?2for a short-range potential. For rare gases,
theresult isdifferent.In Fig.2wepresentP?Eð?ÞforHand
the rare gases obtained by using Eq. (8) and summing over
n in Eq. (14) numerically. We present also the exact results
for H [using TDSE results for RðE?Þ], which show the
accuracy of using the analytic result (8) for RðE?Þ in
Eq. (14). The PRCS data for recombination to the outer s
shell of He and outer p-shells of other rare gases were
deduced from the PICSs data for these shells found in
Refs.  (He),  (Ne),  (Ar),  (Kr), and 
(Xe). For H, He, and Ne, the PRCSs are smooth in the
considered interval of energy E, leading to a smooth de-
crease of P?Eð?Þ with increasing ?, approximately as
??5:3, ??4:6, and ??3:9, respectively. For Ar and Kr, the
PICSs have Cooper minima, which lead to minima in
P?Eð?Þ near ? ¼ 0:9 and 1:1 ?m. Finally, the result for
Xe shows a broad maximum. This maximum is a ‘‘multi-
electron replica’’ of the known ‘‘giant’’ resonance in the
PICS from the inner 4d shell in Xe that appears in the PICS
from the outer 5p shell due to interchannel couplings .
Therefore, for the heavy rare gases, the dependence of
P?Eð?Þ on ? is irregular and can increase with increasing
? in the long-wavelength region.
To conclude, Eqs. (1) and (8) for HHG rates provide
closed-form, analytic quantum expressions of the famous
semiclassical three-step scenario for the HHG process,
having the same level of transparency and simplicity as
the Keldysh result for tunnel ionization. The three factors
in Eq. (8) describe, respectively, the ‘‘atomic’’ processes of
ionization and recombination [Iað~ F;!Þ and ?ðrÞ
the laser-driven propagation of the ionized electron
[WðEÞ]. The EWP, WðEÞ, in Eq. (13) is not sensitive to
the energy-dependent atomic dynamics; it describes all
oscillatory structures in the HHG spectrum, which origi-
nate from interference of two (short and long) classical
electron trajectories (cf. Ref. ). Using Eq. (8), the
?-scaling law for the integrated power of near-cutoff har-
monics is shown to be sensitive to the energy dependence
of the atom’s PRCS. This dependence provides a means to
increase the frequency of harmonics without significant
loss, or even a possible gain, in the HHG efficiency by
increasing the laser wavelength into an optimum interval
for a particular atomic target.
This work was supported in part by RFBR Grant No. 07-
02-00574, by NSF Grant PHY-0601196, by Russian
a ðEÞ in Eq. (9)
a ðEÞ] and
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