Simulation of Photon Acceleration Using Laser
Wake Fields
H. J. Lee*, C. Kirn*, G. H. Kirn*, J. U. Kirn* and H. Suk*
* Center for Advanced Accelerator, Korea Electrotechnology Research Institute, Seongju-Dong
28-1, Changwon, 641-120, S. Korea
Abstract. Numerical simulations of photon acceleration using laser wake fields are presented with
one- and two-dimensional electromagnetic particle-in-cell codes. Up to 150% of frequency upshift
was observed in the one-dimensional simulation, but around 10% in the two-dimensional simulation
because of laser diffraction as well as transverse plasma motion. The saturation mechanisms of
photon acceleration and the effects of slippage, dispersion, diffraction, and pump depletion are
discussed for parameter optimization.
INTRODUCTION
When a laser pulse is located on a downward density gradient of a plasma wave,
the frequency of the laser pulse is blue-shifted and the group velocity increases. This
phenomenon is called photon acceleration [1]. The interaction length between the laser
pulse and the density gradient can be extended significantly by using a wake field
generated by a short laser pulse. A probe pulse with a frequency co0 is launched with
a time delay following a high intensity drive pulse propagating through a plasma with
density n0 and a plasma frequency (Op0. By the drive pulse, a wake field is generated with
a plasma wavelength A 0 and density perturbation Sn. When the driven plasma motion
is not highly relativistic, the density perturbation is linear and shows sinusoidal form as
8n(£) = 5/70sin(A; 0 £),
(1)
where £ = k0z — co0r, k0 = (cofi — (Qp0)l/2/c, and kp0 = 2n/hp0. From the dispersion relation of an electromagnetic wave propagating through plasmas, the upshifted frequency
is calculated as [2]
Aco
_
K^drco^cosf/; n
(2)
where vg and i are the group velocity of the probe pulse and the propagation time, respectively. With this mechanism, it is possible to vary the laser wavelength continuously
within the frequency upshift limit by changing the interaction length, vgr. Moreover, the
mechanism is also applicable to plasma diagnostics [3].
Solodov et al. [4] reported simulation of photon acceleration in a plasma wake with
a approximation for the electron motion and with laser pulse dynamics simulated in
a spectral paraxial approximation. However, their result shows only a few percent (less
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
187
3.5
0.5
x(mm)
FIGURE 1. Diagram of photon acceleration. The FWHM of the drive (probe) pulse is 50 (8.2) fs and
the plasma density is n0 = 1018 cm~3, of which plasma wavelength kp0 is 33 jiim. The normalized vector
potential of the drive pulse is a0 = eE/mca>0 = 1.0. The induced density perturbation is about 40%.
than 4%) of frequency shift. In this study, we present simulation results of the photon acceleration by one-dimensional (Id) and two-dimensional (2d) electromagnetic particlein-cell codes, Id-XOOPIC [5] and OSIRIS [6], without any approximation for the field
and the plasma motion. We investigate the saturation mechanisms of photon acceleration
such as diffraction, dispersion, slippage by dephasing, and pump depletion.
SIMULATION RESULTS
Figure 1 shows a one-dimensional schematic diagram of photon acceleration. The drive
pulse propagating to the right generates plasma wake fields of which wavelength is A 0,
and the probe pulse is launched with a time delay so that it should be located at the place
where the density gradient is negative, around 20 jum in Fig. 1. The time delay, id, is
associated with the plasma wavelength of the wake field, as id ~ 1.75A 0/c. In order to
generate plasma wake field, the drive pulse intensity should be very high, but the probe
pulse intensity needs to be small enough that the electric field should be lower than the
wave breaking limit, mcco 0/e. The used wavelength of the laser pulses is 1 jum.
Figure 2 shows a i d simulation result in which about 40% of frequency upshift was
observed after 2.0 cm of pulse propagation. The used parameters are described in Fig.
2. Moreover, 150% of frequency upshift was observed with an increased intensity of the
drive pulse, a0 = 2.0. In Id simulation, however, the diffraction of the laser pulses is not
considered and thus the interaction length is much longer than that of real situation.
Figure 3 presents 2d simulation results of plasma density and electric field profiles at
three different times, at t = 1.33,2.65, and 3.98 ps (corresponding to about 400, 800, and
188
(a) 0
(a)o
x (mm)
x(mm)
15
0
(b) 00
Ey
Ey
Ey
Frequency
Frequency
2ω 0
Length
Length (cm)
(cm)
2.0
2.0
(c)
(c)00
xx(m(mm)
m)
1515
15
18
18 cm 33, a 10, and 8.2 fs of the
FIGURE 2.
2. One-dimensional
One-dimensional simulation
FIGURE
results with
with nnn000 1018
cm~
= 11.0,
8.2 fs
fs of
of the
the
FIGURE
simulation results
= 1010
cm , a000 0, and 8.2
FWHMof
FWHM
pulse shape
(b)frequency
vs.interaction
interaction
FWHM
of the
the probe
probe pulse.
pulse. Shown are (a) the
the pulse
shape at
at the
theinitial
initialtime,
time,(b)
(b)
frequencyvs.
vs.
interaction
length,and
and(c)
(c)the
the pulse
pulse shape
shape
when the
the interaction
length,
and
(c)
the
pulse
shape when
cm.
length,
interaction length
length isis 22 cm.
3 , a 2. Shown are the profiles
FIGURE 3. Two-dimensional simulation results with n 0 1018
18 cm
FIGURE 3.
3. Two-dimensional simulation
simulation results
results with
with nn00 10
cm 33,, a
a0 2.
2. Shown
are the
profiles
FIGURE
= fields
1018 cm~
Shown
profiles
of the plasma Two-dimensional
densities (upper) and the probe
pulse electric
(lower)00 at= 1.33,
2.65,are
andthe
3.98
ps,
of the
the plasma
plasma densities
densities (upper)
(upper) and
and the
the probe
probe pulse
pulse electric
electric fields
(lower)
at
1.33,
2.65,
and
3.98
ps,
of
fields
(lower)
at
1.33,
2.65,
and
3.98
ps,
respectively. The displayed simulation domain of each box is 25 µ m by 80 µ m.
respectively.The
The displayed
displayed simulation
simulation domain
domain of
of each
each box
box is
is 25
µ m by
m.
respectively.
25 jiim
by 80
80 µ
jiim.
1200 µ m, respectively). The spot sizes of the drive and the probe pulses are 24 and 12
1200 /im,
µ m, respectively).
respectively). The
The spot
spot sizes
sizes of
of the
the drive
drive and
and the
the probe pulses
pulses are
24
12
1200
are The
24 and
and
µ m, of which
Rayleigh lengths
correspond
to 1810 and
451 µprobe
m, respectively.
focal12
µ m, of
of which
which Rayleigh
Rayleigh lengths
lengths correspond
correspond to
to 1810
and
451
µ
m, respectively.
The
focal
/im,
1810
and
451
/im,
respectively.
The
focal
plane is located at 400 µ m ahead at t=0. As the probe pulse propagates, the wavelength
plane isis located
located at
at 400
400 jum
µ m ahead
ahead at
at t=0.
As
probe
propagates,
the wavelength
plane
t=Q.more
As the
thedispersive.
probe pulse
pulse
propagates,
wavelength
becomes shorten,
but the
pulse becomes
In addition,
the the
pulse
shape is
becomes
shorten,
but
the
pulse
becomes
more
dispersive.
In
addition,
the
pulse shape
is
becomes
but the pulse
dispersive.
In addition,
deformedshorten,
corresponding
to the becomes
wake fieldmore
profile.
The transverse
size ofthe
thepulse
probeshape
pulseis
deformed
corresponding
to
the
wake
field
profile.
The
transverse
size
of
the
probe
pulse
deformed
corresponding
to the
wake field
The transverse
size of the probe pulse
becomes comparable
to the
transverse
sizeprofile.
of the wake
field.
becomes
comparable
to the
theof
transverse
size
of the
the
wake
field.
becomes
comparable
to
transverse
of
wake
field.
The frequency
spectrum
the pulsesize
electric
field
at the
center line is shown in Fig. 4,
The frequency
frequency spectrum
spectrum of
of the
the pulse
pulse electric
electric field
The
field at
at the
the center
center line
line is
is shown
shown in
in Fig.
Fig. 4,
4,
189
0.01
0.01
t=0
t=5000 ω0−1
t=7750 ω0−1
Spectrum
0.001
0.0001
1e-05
0
0.5
1
1.5
2
ω/ω0
2.5
3
3.5
4
FIGURE
65 ps (dashed), and
FIGURE 4.
4. Fourier transformed spectrum
spectrum of
of the
the probe pulse at
at tt = 00 (solid),
(solid), tt = 22.65
and
pss(dotted).
tt = 44.11
lip
(dotted).
where almost 10%
10% of frequency upshift is presented. It is much smaller than that of the
1d
Id result (150%)
(150%) because of shortened
shortened interaction length due to laser pulse diffraction
diffraction
as well
well as
as the
the transverse
transverse plasma motion. The frequency
frequency upshift is almost
almost saturated
saturated after
after
the pulse propagates 1200
1200 µ
j m.
DISCUSSION
Diffraction, wave dispersion, drive pulse depletion, and dephasing between the probe
pulse and the plasma wake field play important roles in the saturation of photon acceleration. As can be seen in Eq. (2), the frequency
upshift, Aco,
∆ω , increases as the plasma
frequency upshift,
density, density perturbation ((5ft
δ n 0), or the interaction length (v
(vggτr)) increases. However,
we cannot increase each of the parameters simultaneously because the saturation mechanisms are related to one another.
∆
ω increases as the plasma density increases, but the plasma wavelength decreases
Aco
and therefore the probe pulse becomes more dispersive when the pulse length of the
probe pulse is comparable to aa half of the plasma wavelength. For this case, the front
and the tail of the probe pulse have different wavelengths and different group velocities.
Moreover, if the pulse length is longer than a half of the plasma wavelength, frequency
downshift can happen at positions where density gradient is positive. It is recommended
to make the plasma wavelength larger than four times the probe pulse length, and thus
there is limitation on increasing the plasma density.
δ8n
n00/nn00 increases as we
we increase the drive pulse intensity, a 0, but the density gradient
becomes sharp
sharp and the probe pulse becomes more dispersive with the same reason
mentioned above. Moreover, if we decrease the laser spot size to increase a0, the
190
Rayleigh length decreases and the interaction length is limited by diffraction.
The interaction length, vgr, is restricted to a few Rayleigh lengths by diffraction, and
also restricted by the drive pulse depletion which gives its energy to plasma wake fields.
The energy depletion is proportional to the interaction length, a\, and CO^O/COQ . Usually,
the length scale of energy depletion is longer than that of diffraction for the considered
plasma density and laser intensity.
As the group velocity of the probe pulse increases by photon acceleration, the difference increases between the group velocity of the probe pulse and the phase velocity of
the wake field. Finally, the probe pulse is to be located out of the acceleration phase.
Dephasing can be controlled by changing the plasma density gradually as the pulse
propagates.
In conclusion, the frequency upshift is more effective for higher nQ and a0, but the
plasma density and the laser intensity of the drive pulse should be chosen carefully so
that the length of negative density gradient region should be longer than the probe pulse
length. Further study is necessary for parameter optimization of the plasma density, the
laser spot size, and interaction length.
ACKNOWLEDGMENTS
This work is supported by the Creative Research Initiatives of the Korea Ministry of
Science and Technology.
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