23P179-B
Generation of Energetic Electrons in Preplasma
via Parametric Instability
Alexey V. Arefiev, Boris N. Breizman, and Vladimir N. Khudik
Institute for Fusion Studies, The University of Texas at Austin
1 University Station, C1500, Austin, Texas 78712, USA
A new mechanism is reported that increases electron energy gain from a laser beam with ultra-relativistic
intensity in significantly underdense plasma. The increase occurs when electron oscillations across the ionchannel produced by the laser become parametrically unstable. The instability threshold is determined bythe product of beam intensity and ion density.

1. Introduction
A common element of applications involving
laser-target interactions is the generation ofenergetic electrons. A laser beam incident onto asolid-density target generates an oscillating force atthe target surface. This force injects electrons into
the target with characteristic energy of the order ofthe ponderomotive potential. One way to increaseelectron energy beyond that level is via multiple
interactions with the laser. This requires electron
recirculation in the target during the pulse. Anotherway to increase electron energy is to increaseelectron interaction length with the laser beam. The
conditions for that can naturally occur in a
preplasma in front of the target. A laser prepulse
often creates a transparent preplasma that extends
many wavelength from the target surface along thebeam path. The main pulse propagating throughsuch preplasma can then accelerate electrons toenergies
significantly
higherthan
theponderomotive potential.
We present a mechanism that increases theelectron energy gain in sub-critical preplasma
regardless of the beam polarization, which
distinguishes this mechanism from the previously
discussed betatron resonance [1]. The increase
occurs at ultra-relativistic intensities when electron
oscillations across the channel formed by the laser
become parametrically unstable. The instability
threshold is determined by the product of beam
intensity and ion density. Our findings suggest thatthe population of hot electrons and their energy
could possibly be increased in laser-target
interactions by adjusting the preplasma density andlength in order to trigger the instability and provide
sufficient interaction length for the enhanced energy
gain.
2. ModelWe consider interaction of a plane wave with asingle electron placed in a straight uniform ionchannel. We choose a two-dimensional Cartesian
setup, such that the plane wave propagates along
the z-axis. The static electric field produced by theion space-charge in the channel is directed along they-axis and it is then proportional to y. The ions are
assumed to be immobile.
The plane wave is described by a normalized
vector potential=a a (ξ ) e x cos θ + e y sin θ  ,where θ is the polarization angle and ex and ey areunit vectors. The dimensionless phase variable ξ isdefined asξ ≡ ( ct − z ) λ ,where λ is the laser wave-length, c is the speed oflight, and t is the time in the laboratory (ion) frameof reference. We assume that a(ξ) has a generalform=a (ξ ) a0 F (ξ ) sin ( 2πξ +ψ ) ,where a0 is the maximum amplitude, F(ξ) is a
slowly varying envelope, and ψ is an initial phase.
Starting with the equations of motion for a
relativistic electron in electric and magnetic fields,
we have derived an equation for electron motion
across the channel in the presence of the planewave:
2 du d 2uu
K 2u−+d ξ 2 2 ( C − u 2 2 )  d ξ  2 ( C − u 2 2 )3+K 2u
K 2 a 2 cos 2 θ u
K sin θ da,+=32C − u 2 2 dξ
2 (C − u 2) 2 (C − u 2 2)

whereu ≡ ω p0 y c ,K ≡ ω p 0λ c ,and ω p 0 = 4π n0 e me is the plasma frequency, n0
is the density of the single charged ions in thechannel, and e and me are the electron charge andmass. The constant C for an electron that is initiallyat rest is determined by its initial displacement,2C = 1 + u 2 ( 0 ) 2 . The wave amplitude a0 enters theequation of motion as a product a0K, whichsuggests a convenient dimensionless parameter κdefined asκ ≡ a0 K =ω p 0 a0 λ c .

maximum value achievable without the ions (invacuum), γ vac = 1 + a02 2 .

Note that K  2π in a significantly underdense
plasma, which is the case of our primary interest.
3. Results
In the ultra-relativistic regime (a0>>1), the K2
terms that do not involve a2 can be neglected in theequation of motion. The resulting equation
contains only two parameters (κ and θ ). The caseof an s-polarized wave (θ = 0 ) clearly illustrates
the instability mechanism. The equation of motion
for small oscillations and F=1 reduces tod 2u 1 2 20.
+ κ sin ( 2πξ +ψ ) u =dξ 2 2This is a Mathieu differential equation for a linear
oscillator with a periodically changing frequency.
Such oscillations are susceptible to parametricinstability that causes their amplitude to grow. We
find that the instability threshold is κ* ≈ 10.2 .

Fig. 1. Parametrically unstable electron oscillations inthe s-polarized wave for κ =12 and a0=10. The dashedcurve shows a/a0 as a function of electron axial location.
Note that here umax ≈ 2 and γ vac = 1 + a02 2 .

The transverse oscillations in the channelchange the dephasing rate that determines axial
motion of the electron relatively to the wave. The
dephasing rate decreases significantly as a result of
the instability compared to that in the absence ofions. This allows the electron to stay in phase withthe wave for a longer time and thereby gain moreenergy, with the resulting γ-factor exceeding theFig. 2. Maximum γ-factor in the s-polarized wave
(θ =0) averaged over the initial wave phase.

The solution of the complete equation of
motion and the resulting γ-factor for a set ofparameters corresponding to κ= 12 > κ* are shownin Fig. 1. The oscillations grow significantly due
to the instability. This increase in amplitude causes
axial acceleration, reflected by the increase in γ.
Figure 2 shows the maximum γ-factor as afunction of a0 and κ. At a0>10, the instability
threshold is indeed determined only by parameterκ since the threshold coincides with the dashedvertical line κ =κ*. The peak value of γ is roughly
2.6 times greater than the maximum value that canbe achieved in the absence of ions.
In summary, we have shown that the parametricinstability of an ultra-relativistic electron in the ionchannel provides an opportunity for enhanced
electron energy gain in the laser beam.
Acknowledgments
This work was supported by Sandia National
Laboratory Contract No. PO 990947, National
Nuclear Security Administration Contract No.
DE-FC52-08NA28512, and U.S. Department ofEnergy Contract No. DOE-ER54742. The authors
are grateful to Dr. Marius Schollmeier for
stimulating discussions.
References[1] A. Pukov, Z.-M. Sheng, and J. M. ter Vehn: Phys.
Plasmas 6 (1999) 2847.