Download Theory of relativistic harmonic generation

274 / QELS 2001 / FRIDAY MORNING
tions are in good agreement with the experimental results.
In conclusion, we have performed a detailed
analysis of the harmonic generation process in
the few-optical-cycle regime. In particular the
spectral profile of the harmonics with respect to
the order, their intensity and relative spectral
shift, and their divergence properties were measured as a function of the position of the gas jet.
We have developed a nonadiabatic three-dimensional propagation model, whose results are in
good agreement with the experiments.
-
c
.c
v)
3
4m
v
.0
Photon energy (ev)
C
v)
W
-
.
I
-
C
80
100
120
140
160
Photon energy (eV)
QFF5 Fig. 1. Harmonic spectrum in helium generated by 7-fs excitation pulses. The inset shows on
a logarithmic scale an enlarged portion of the cutoff spectral region
shows discrete and well resolved harmonics up to
160 eV. The number of photons per pump pulse
in the 50-150 eV spectral range is -7 x lo6,
which corresponds to a conversion efficiency of
-3 x
As shown in the inset of Fig. 1, which
displays an enlarged portion of the harmonic
spectrum on a logarithmic scale, harmonic radiation at energies as high as 190 eV can be obtained at lower efficiency. Using the spectrograph
focal-plane images we have measured the far field
divergence of the harmonic beam. Figure 2 shows
a three-dimensional harmonic spectrum, from
which we can obtain the conversion efficiency
and the divergence of the single harmonics.
As displayed in Fig. 3, which shows the harmonic divergence as a function of the order, the
spatial width of the beam broadens with order
from 1.5 mrad to 3.2 mrad ranging from 231dharmonic up to 6Sthharmonic, and then narrows in
the cutoff region. This analysis was performed for
various positions of the gas jet with respect to the
laser beam waist. The harmonic divergence is remarkably affected by the jet position.
The experimental results have been analyzed
in terms of a nonadiabatic three-dimensional numerical model3 The single-atom response is calculated using the strong field model, generalized
References
1. T. Brabec and E Krausz, “Intense few-cycle
laser fields: frontiers of nonlinear optics,”
Rev. Mod. Phys. 72,545-591 (2000).
2. M. Nisoli, S. De Silvestri, 0. Svelto, “Generation of high energy 10-fs pulses by a new
pulse compression technique,” Appl. Phys.
Lett. 68,2793-2795 (1996).
3. P. Villoresi, P. Ceccherini, L. Poletto, G. Tondello, C. Altucci, R. Bruzzese, C. de Lisio, M.
Nisoli, S. Stagira, G. Cerullo, S. De Silvestri,
0.Svelto, “Spectral features and modeling of
high-order harmonics generated by sub-10fs pulses,” Phys. Rev. Lett. 85, 2494-2497
(2000).
QFF6
Harmonic M e r
QFF5 Fig. 3. Divergence of the harmonic
beam vs harmonic order
l L 4 5 am
Theory of Relativistic Harmonic Generation
Q-Han Park, Physics Department, Kyunghee
University, Seoul 130-701, Korea
Robert W. Boyd, Institute ofOptics, University of
Rochester, Rochester, NY 14627 USA
to account for nonadiabatic effects. Ionization ef-
fects and ground state depletion are included
using the Ammosov-Delone-Krainovtheory The
single atom response is inserted into a three-dimensional propagation model, which takes into
account both temporal plasma-induced phase
modulation, and spatial plasma lensing effects on
the fundamental beam. The computer simula-
QFF5 Fig. 2. Three-dimensional harmonic spectrum
John E. Sipe, Department ofphysics, University of
Toronto, Toronto, Canada M5S lA7
Alexander L. Gaeta, School ofApplied and
Engineering Physics, Cornell University, Ithaca,
NY 14853 USA
In a recent publication, Chen et al.’ describe their
observation of phase matched third-harmonic
generation (THG) in a highly ionized gas under
laser intensities sufficiently large (2 x 1017
Wlcm’,) that relativistic effects are expected to be
important. The harmonic radiation is found to be
emitted in a narrow cone (of radius 5.6 degrees)
in the near forward direction, with a reasonably
large (2 x lo-*) conversion efficiency. Perhaps
most surprisingly they found that the intensity of
the THG was comparable for linear and circular
polarization of their incident laser beam. In traditional nonlinear optics the process of THG is
known to vanish identically for an isotropic
medium for a circularly polarized fundamental
laser beam? Nonlinear optical interactions are
often described in terms of a nonlinear optical
susceptibility, which for the case of third-harmonic generation relates the polarization of the
medium to the third power of the electric field
amplitude of the incident laser beam. Harmonic
light is radiated in as a consequence of the timevarying polarization of the medium. This procedure presupposes the validity of the electric dipole approximation, both in the calucation of the
nonlinear susceptibility and in terms of calculating how the response of the medium leads to new
frequency components. Detailed examination of
the tensor nature of the nonlinear susceptibility
FRIDAY MORNING
shows that the nonlinear response vanishes for a
circularly polarized intput beam.
In this paper, we show that the argument
about the vanishing THG breaks down under relativistic conditions. We consider the motion of a
free electron in an intense laser field of wavelen h h For field strengths of the order of
. . for h = Ipm corresponds to an inmc lk which
tensity of the order of 2 x l O I 7 Wlcm', the electron is accelerated to relavistic velocities in a single optical period, and thus the excursion of the
electron from its equilibrium poistion occur over
a distance of the order of h. Under such conditions, the radiation emitted by the electron cannot be described within the electric dipole approximation, and in fact cannot be well described
by a multipole expansion for a finite number of
terms. We instead calculate the intensity of the
emitted radiation through use of the LenardWiechert potentials, as described below.
The motion of a free electron in an intense
electromagnetic field of frequency o is well
known. For circularly polarized light the electon
rotates at frequency w in a circle of radius ro =
pclo where p = eEo/ywand $ = m2c? ie2E2,Io2.
For linearly polarized light the electron moves in
a figure-eight pattern of dimensions comparable
to r,,. Even though the motion is strictly harmonic
for the circularly polafized situation, and contains only frequencies o and 2 o in the linearly
polarized case, the radiation field contains all
harmonics of the fundamental frequency in both
cases because of retardation effects. In particular,
the vector potential of the n-th harmonic component of the radiated wave at distance R, from the
center of the electron orbit is given by
SHG-circular
/ QELS 2001 / 275
THG-circular
P
+
where k,, = olc, k = nolc and k has magnitude k
and points in the direction of the observation
point, and i'gives the instantaneous position of
the electron. The power per unit solid angle emitted by this harmonic component is then given by
dIn c
- IZxdnl2R2,
dQ 811
We have evaluated d, analytically for the circularly polarized case and numerically for the linearly polarized case. Some of the predictions of
this model are shown in Fig. 1-Fig. 3. Figure 1
shows radiation patterns in the polar coordinate
for the sencond and the third harmonic generations in the case of the circularly polarized light.
Figure 2 and figure 3 show radiation patterns for
the linear case at various values of angle $. We
find that this model predicts that linearly and circularly polarized light are roughly equally efficient at exciting harmonic generation. Also, we
note that the model predicts the correct conversion efficiency, at least to order of magnitude.
One important prediction of the present
model is that second-harmonic generation
(SHG) should also be emitted, with an efficiency
no smaller than that of the third-harmonic. Although Chen et al.' make no mention of the observation of SHG in their initial publication, SHG
has been observed in subsequent work.4The efficiency of the emission of the SHG is likely to depend upon the subtleties of the phase matching
process, which wiU be discussed.
QFF6 Fig. 1. Radiation patterns for the second and the third harmonic generations in the case of the
circularlypolarized light.
SHG-phi = 0 deg
SHG-phi = 45 deg
SHG-phi = 90 deg
0.
0.0
-0.0
-0.
QFF6 Fig. 2. Radiation patterns for the second harmonic generations in the case of the linearly polarized light.
THG-phi = 45 deg
THG-phi=gO deg
MG-phi = 0 deg
-4
QFF6 Fig. 3. Radiation patterns for the third harmonic generations in the case of the linearly polarized light.
Reference
3.
1. S.-Y. Chen, A. Maksimchuk, E. Esarey, and D.
Umstadter, Phys. Rev. Lett., 84, 5528,2000.
2. J.F. Ward and G.H.C. New, Phys. Rev. 185,57, 4.
1969.
R.W. Boyd, Nonlinear Optics, Academic
Press, Boston, 1992. See especially section
4.2.
D. Umstadter, personal correspondence.