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Grating-Assisted Phase Matching in Extreme Nonlinear Optics
Grating-Assisted Phase Matching in Extreme Nonlinear Optics
Oren Cohen,
*
Xiaoshi Zhang, Amy L. Lytle, Tenio Popmintchev, Margaret M. Murnane, and Henry C. Kapteyn
Department of Physics and JILA, University of Colorado at Boulder and NIST, Boulder, Colorado 80309, USA
(Received 19 April 2007; published 31 July 2007)
We propose a new technique for phase matching high harmonic generation that can be used for
generating bright, tabletop, tunable, and coherent x-ray sources at keV photon energies. A weak quasi-cw
counterpropagating eld induces a sinusoidal modulation in the phase of the emitted harmonics that
can be used for correcting the large plasma-induced phase mismatch. We develop an analytical model
that describes this grating-assisted x-ray phase matching and predicts that very modest intensities
( < 10
10
W=cm
2
) of quasi-cw counterpropagating elds are required for implementation.
DOI:
10.1103/PhysRevLett.99.053902
PACS numbers: 42.65.Ky
Phase matching techniques such as quasi-phase match-
ing [
1
,
2
] (QPM) and grating-assisted phase matching [
2
,
3
]
(GAPM) play an important role in converting light from
one frequency to another, expanding the wavelength range
of coherent light sources. Recently, the generation of co-
herent laser-like soft x-ray beams using the process of
high-order harmonic generation (HHG) has received con-
siderable attention [
4
]. However, efcient up-conversion
into the soft x-ray region of the spectrum is challenging,
because ionizing radiation is strongly absorbed by all
matter. Hence, traditional phase matching techniques that
rely on anisotropic crystalline solids or periodically poled
materials cannot be used and novel techniques must be
devised.
In high harmonic generation, an electron is rst ionized
by the driving laser. Once free, the electron oscillates in the
continuum in response to the laser eld. A small fraction of
the ionized electrons recombine with the parent ion and
liberate their excess energy as a short-wavelength photon.
This process represents an extreme limit of nonlinear
optics, where dozens, hundreds, or even thousands of
visible photons, each with energy 1 2 eV, are combined
together, resulting in coherent beams at photon energies up
to a few keV [
5
]. A major limitation to date, however, is the
relatively low conversion efciency from laser light to
harmonics, particularly to photon energies >130 eV.
This low efciency is not due to the effective nonlinearity
of the process, which is nonperturbative and thus scales
relatively slowly with increasing photon energy [
6
].
Rather, the problem to date has been the inability to
efciently phase-match this high-order conversion process
[
4
]. In the ideal case of phase matched nonlinear conver-
sion, the high-order polarization and the generated har-
monic propagate with the same phase velocity, so that the
harmonic signal builds up coherently over the entire length
of the medium. If the phase velocities of the two waves
differ, the signal buildup is limited to the coherence length
L
c
, which corresponds to that distance over which a phase
slip of
accumulates between the two waves. Medium
lengths longer than L
c
result only in oscillation of the
generated signal due to repeated destructive and construc-
tive interference. In HHG, optimum phase matched con-
version can only be obtained for photon energies below
130 eV. In this case, use of a hollow waveguide [
7
,
8
] or
a shallow-focus geometry [
9
] makes it possible to balance
geometrical dispersion with material and plasma disper-
sion. This phase matching technique relies on the presence
of neutral atoms in the medium, and is therefore limited to
the case of weak ionization (i.e. <0:5%5%, depending on
the gas). However, higher photon energies are generated at
higher laser intensities where the medium is more highly
ionized and where this type of phase matching is
impossible.
Few approaches have been discussed for partial phase
matching at high photon energies. Modulated waveguides
partially readjust the phase slip between the driving laser
and harmonic waves by periodically modulating the laser
intensity [
10 12
]. More recently, all-optical QPM was
implemented using a train of counterpropagating pulses
[
13
]. In that work, the counterpropagating light suppresses
the HHG process in regions where it intersects with the
driving pulse [
14 16
]. A train of counterpropagating
pulses can be used for implementing QPM by suppressing
emission from out-of-phase regions [
13
]. However, these
QPM implementations are limited to the case where the
coherence length is larger than
10
m. At keV energies,
however, the coherence length is typically in the micron
range [
17
,
18
]. An approach that can increase the coherence
length at high energies is nonadiabatic self-phase matching
(NSPM) [
17
,
18
]. In NSPM, L
c
is extended somewhat due
to the subcycle evolution of an intense, 5 fs, pulse that
results from rapid ionization of the medium. However, the
increase in L
c
is maintained for a short propagation dis-
tance (few microns) because of the inevitable phase slip
resulting from absorption and defocusing of the driving
laser. NSPM is thus not a general technique for phase
matching over an extended distance. Finally, high-order
difference frequency mixing has been proposed as a way to
compensate for the effect of plasma on phase matching
conditions [
19 21
].
PRL 99, 053902 (2007)
P H Y S I C A L
R E V I E W
L E T T E R S
week ending
3 AUGUST 2007
0031-9007= 07=99(5)=053902(4)
053902-1
© 2007 The American Physical Society In this Letter, we propose a new technique for phase
matched frequency conversion into the x-ray region of the
spectrum. A weak quasi-cw counterpropagating eld in-
duces a sinusoidal modulation on the phase of the gener-
ated harmonics, which is formally equivalent to a
modulation in the refractive index for the driving laser.
Exploiting this correspondence, we show that phase match-
ing of HHG with a quasi-cw counterpropagating eld is
equivalent to conventional low-order harmonic generation
under grating-assisted phase matching conditions [
2
,
3
]. A
simple analytical model predicts the optimal conditions,
such as the intensities and wavelengths of the forward and
backward propagating waves, for implementing GAPM in
HHG. This is the rst technique that appears to be feasible
for phase matching very high-order harmonics over ex-
tended distances in plasma waveguides, to generate bright,
tunable, and narrow bandwidth x-ray beams at keV photon
energies.
A distinctive property of HHG is that the emitted har-
monics are phase shifted relative to the driving laser. This
extra phase, which is primarily acquired by the electron
along its femtosecond boomerang path under the inu-
ence of the laser eld, is very large, reaching tens or
hundreds of radians. It is also proportional to the intensity
of the driving laser [
22
]. Thus, by inducing a shallow
modulation in the laser intensity along the propagation
direction, for example by interfering the driving laser pulse
with a weak counterpropagating beam, leads to modulated
phase change that can be used to correct the phase mis-
match of the HHG process.
We develop a simple analytical model of the induced
phase matching process. Consider a driving pulse,
E
F
t; z
E
0
A
1
z; t cos !
1
t
2 n
1
z=
1
, at wavelength
1
that propagates along z and interferes with a weak
counterpropagating
beam,
E
B
z; t
E
0
r cos !
2
t
2 n
2
z=
2
, at wavelength
2
where E
0
is the peak eld,
A
1
z; t
is a normalized envelope, r
1 is a eld ratio
parameter, !
1;2
2 c=
1;2
are the angular frequencies,
and n
1;2
1 are the refractive indices [Fig.
1(a)
].
Transforming into a frame moving in the forward direction
at phase velocity of the driving eld c=n
1
, where c is the
velocity of light at vacuum gives E
F
; z
E
0
A
1
; z
cos !
1
and E
B
; z
E
0
r cos !
2
2 z=
, where
2
= n
1
n
2
2
=2 and
t
n
1
z=c
. The joint
intensity I ; z / E
F
; z
E
B
; z
2
has a component
that is proportional to r and is modulated along the propa-
gation direction with periodicity
(we neglect the very
week term that is proportional to r
2
). For example, the
intensity modulation at
0 is proportional to
I z /
2E
02
r cos 2 z=
. Thus, the phase change of the emitted
harmonics is given by

HHG
z
z=L
c
A cos 2 z=
(1)
where the rst term is the linear growth in the phase change
due to the phase mismatch of the conversion process and
the second term describes the induced phase modulation
with amplitude A / r and periodicity
by the counter-
propagating light.
To study the effect of the induced modulations on the
phase matching conditions, we calculate the coherent
buildup of the high-order harmonic eld by

E
HHG
Z
L
0
E
0
HHG
exp i
HHG
dz
(2)
where E
0
HHG
is the harmonic eld generate