Generation of higher-order harmonics upon the addition of high-frequency XUV radiation
to IR radiation: Generalization of the three-step model
Avner Fleischer
*
National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada
and Schulich Faculty of Chemistry and Minerva Center for Nonlinear Physics of Complex Systems,
Technion–Israel Institute of Technology, Haifa 32000, Israel
Received 26 February 2008; published 18 November 2008
The irradiation of atoms by a strong, quasicontinuous IR laser field of frequency results in the emission of
odd harmonics of “IR harmonics”, by high-harmonic generation. It has been recently shown A. Fleischer
and N. Moiseyev, Phys. Rev. A 77, 010102R2008 that the addition of a weak XUV field of harmonic
frequency q to the IR driver field leads to the appearance of a new set of higher-order harmonics “XUV
harmonics” q 2K q , K integers which were absent in the spectra in the presence of the IR field alone. Here
we generalize these results by studying the generation of XUV harmonics upon the addition of an arbitrary
high-frequency XUV field q ˜
, with frequency not necessarily a harmonic of , and amplitude
q ˜
in
, which might
be large. We have found that as the intensity of the XUV field increases, higher sets of XUV harmonics
q ˜ 2K ,2q ˜ 2K -1 ,3q ˜ 2K , .... gradually appear, where each XUV harmonics set n
q ˜
q ˜ 2K -1
+ modn
q ˜
,2 n
q ˜
=1,2,3,... is
q ˜
in
q ˜
2
2
2n
q ˜
times weaker than the set of IR harmonics. The mechanism respon-
sible for the appearance of the XUV harmonics is analyzed analytically using a generalization of the semi-
classical recollision three-step model of high-harmonic generation HHG. It is shown that the emitted HHG
radiation field can be written as a series of terms, with the zeroth-order term representing the three-step
recollision mechanism in its most familiar context P. B. Corkum, Phys. Rev. Lett. 71, 1994 1993 and giving
rise to IR harmonics in the spectra. The higher-order terms, which are responsible for the appearance of the
new sets of XUV harmonics in the spectra, are shown to originate from the polarization ac-Stark oscillations
which is induced by the XUV field on the returning electronic wave packet in the recollision process. The XUV
harmonics are formed by the same electron trajectories which form the IR harmonics and have the same
emission times as the IR harmonics. Due to the small quiver amplitude of the ac-Stark oscillation, they are,
however, much weaker than the IR harmonics. Nevertheless, this mechanism allows an extension of the cutoff
in the harmonic generation spectra without the necessity of increasing the IR field intensity, as is verified
numerically by a quantum-mechanical simulation of a Xe atom irradiated by a combination of IR and XUV
classical fields.
DOI: 10.1103/PhysRevA.78.053413 PACS numbers: 03.65.-w, 42.65.Ky
I. INTRODUCTION
Focusing intense linearly polarized monochromatic infra-
red IR laser pulses into a gas jet of atoms can lead to the
emission of high-energy photons with frequencies extending
into the extreme ultraviolet XUV and X-ray region by
high-harmonic generation HHG. All major features of
HHG, such as its comblike spectrum of odd-integer harmon-
ics to be called “IR harmonics”, its photons’ maximal en-
ergy to be called “IR cutoff”, and the emission times of
each harmonic, could be well reproduced using a semiclas-
sical three-step recollision model 1–3: under the influence
of an intense laser field the electron of an atom tunnels out of
the modified Coulomb potential, gains kinetic energy as a
free particle in the field, and finally may recombine with the
parent ion to release the sum of its kinetic energy and the
ionization potential as a high-energy photon. The emission
times of different harmonics are perfectly synchronized with
the driver field, making the HHG process a promising
method for the production of an adjustable coherent X-ray
source. The current method of achieving the state-of-the-art
IR cutoff positions in the harmonic generation spectra HGS
makes use of high-intensity few-femtosecond IR laser
pulses. The main drawback of this method is that the elec-
tronic plasma, which is inevitably formed at such high inten-
sities, disperses the propagating harmonics and severely lim-
its their phase matching. The method to achieve higher-
energy harmonics which will be presented here does not
suffer from this limitation, since it allows the use of an IR
source of moderate intensity which produces a small amount
of plasma. It uses an XUV driving field which is shined on
the atom simultaneously with the IR one. Taking the fre-
quency of the XUV field larger than the IR cutoff frequency
and inside some spectral window of the HHG generating gas,
new higher-order harmonics to be called “XUV harmon-
ics”, which were absent in the spectra in the presence of the
IR field alone, could be produced with frequencies well
above the IR cutoff frequency, while the XUV photoioniza-
tion could be suppressed, thus unaltering the amount of elec-
tronic plasma. As will be shown, the main drawback of this
method is that the new XUV harmonics have a relatively low
intensity.
In the context of strong laser-matter interactions, the idea
of contaminating the strong IR field with a second or more
higher-frequency usually ultraviolet UV fields is not a
*
Avner.Fleischer@nrc.ca
PHYSICAL REVIEW A 78, 053413 2008
1050-2947/2008/785/05341313 ©2008 The American Physical Society 053413-1