HIGH-ENERGY LASER PONDEROMOTIVE ACCELERATION.
I. V. Smetanin, P. N. Lebedev Physics Inst., Moscow, Russia,
C. Barnes, SLAC, USA
K. Nakajima, KEK, Tsukuba, Japan
Abstract
A new concept of TeV-range laser ponderomotive
acceleration in a plasma is proposed. Particles areaccelerated in the point-like scattering by the leading
front of the laser pulse, propagating at the group velocity
less than the vacuum speed of light. In this scheme, thegain in particle energy is determined by the groupvelocity and does not depend on laser intensity, which
determines the quantum probability of acceleration. The
quantum and classical analysis of the scheme proposed ispresented. Estimates show that the concept proposed is apromising technique for compact laser acceleration ofTeV energy range.

by the initial energy of particle and pulse group velocity
only, and does not depend on peak intensity. Laser
intensity determines the probability for the particle to bereflected from the leading edge of the pulse, and, thus,
the number of particles captured in the acceleration
regime. To estimate this probability, we use quantum
Klein-Gordon equation. In fact, acceleration in this newconcept is a point-like process, and thus is insensitive tothe phase of injection. Estimates show that accelerationof electrons up to TeV -range of energy is attainable in asingle point-like interaction using current laser
technologies, that makes this novel concept prospective
for the next generation of particle accelerators.

2. CLASSICAL ANALYSIS
1. INTRODUCTION
Ponderomotive motion of charged particles in thefield of non-uniform electromagnetic wave is underinvestigation for a long time, being initiated by P. L.
Kapitza and P. A. M. Dirac [1]. Recent advances in highintensity laser systems [2] has renewed now an interest in
the development of the next generation accelerators
technology [3,4].
In the conventional theory of laser ponderomotive
acceleration (LPA) [5,6], the laser-particle interaction isassumed to proceed in vacuum, i.e., where the laser pulse
group velocity is equal to the vacuum speed of light,
β* = v gr / c = 1 . The acceleration is originated by thetransverse gradient of the ponderomotive potential. As aresult, the accelerated particle is expelled from the focal
region at a definite non-zero scattering angle, and thegain in energy increases with an increase in peakintensity. In a plasma, however, it is more realistic tosuggest that the laser pulse group velocity is less than c,
β * < 1 . In this case, a natural limit for conventional
vacuum LPA mechanism [5,6] arises: with an increase inlaser peak intensity, particle velocity will inevitably
exceed the value of pulse group velocity.
We propose here a new concept of laser
ponderomotive acceleration of particles in a plasma.We
intend to show that the new acceleration regime occurs
when the group velocity of high-power laser pulse is lessthen the vacuum speed of light. The physical mechanismof acceleration is the elastic scattering of particle by the
ponderomotive potential in the reference frame moving
together with the laser pulse. A simple analyticalquantum and classical consideration is presented. We
show that the energy gain is determined for this regime

Let us assume the high-power ultrashort laser pulse
propagating in a medium (a gas or plasma, for example)
with a definite group velocity, β * < 1 . In this newreference frame, the laser pulse ponderomotive potential
becomes static, so the interaction can be considered now
as just the elastic potential scattering. Let the initialenergy and velocity of the particle are γ 0 and β0 ,respectively, and the initial angle of collision, α 0 ,and α 0 = 0 corresponds to the laser pulse co-propagatingwith the accelerated particle. Using Lorentz
transformation, it is easy to find the energy of the particle
in the moving frame

γ ' = (1 − β * β0 cosα 0 )γ 0γ *(1)2 −1/ 2is the characteristic γ where γ * = (1 − β * )factor of the laser pulse. According to the classicalscattering theory, if this energy is less than the field peakponderomotive potential,

γ ' ≤ (1 + a 0 2 / 2)

1/ 2(2)the particle has to be reflected. Here, a0 is thenormalized peak value of the laser field vector potential
amplitude. As an effective Hamiltonian is timeindependent in the moving frame, the final energy γ ofthe particle does not change after scattering (in theinfinity), and, after the Lorentz transform, we have in thelaboratory frameγ (1 − β * β cosα ) = γ '/γ *(3)
In this implicit relation, β is the particle absolutevelocity, and α is the final scattering angle, α=0
corresponds to the co-propagation case.

It is easy to conclude from comparison of Eqs. (1) and(3) that there is the following conservation law for theacceleration mechanism under discussion,γ (1 − βz β *) = const .
(4)
The results of the above simple kinematics approachcoincide with that of electrodynamics consideration. The
integral of motion (4) appears in the form of energy
conservation law in the 1D theory of ultrashort pulse
propagation in the cold underdense plasma [4].
For the particle initially at rest, the maximum energy
gain is at α = 0 , i.e., γ max = (1 + β * 2 )γ * 2 ≈ 2γ *2 ,and the particle velocity exceeds the laser pulse groupvelocity βmax = 2β */(1+ β *2 ) > β * . Thus, the particle
accelerated is in front of the laser pulse during all theinteraction process, and its gain in energy does notdepend on the value of the laser peak intensity, but on
the dielectric properties of the medium only, which isqualitatively different than that for conventional vacuum
LPA [5,6].
The integral of motion (4) coincides in the limit ofvacuum LPA β * = 1 with that derived by Harteman etal. [5], γ (1 − βz ) = const . This integral, however,results in a completely different feature in the laserparticle interaction. Scattering at zero angle α = 0 iscondemned. Expelled particle inevitably has a transversevelocity. At sufficiently high intensities, the quiveramplitude of motion in laser wave becomes comparable
to the beam waist, and electron is effectively expelled out
from the focal region due to the transverse gradient of thelaser field. The laser-particle interaction is thus
terminated within a wavelength, which leads to thestrong dependence on injection conditions [5].
In the new non-vacuum LPA concept which wepropose, there is no strong dependence on the injection
phase. Really, in the moving reference frame, the laser
field vector potential is stationary, and the acceleration
mechanism in this case is more akin to the well-known
potential scattering.
The origin of the difference between conventional
LPA and the regime proposed is that conventional
mechanism utilize the transverse intensity gradient,whereas in the case of our interest the scattering ofparticle by the leading front of the laser pulse occurs.
For the case of injection at definite initial energy, themaximum final energy of particle is for scattering at zeroangle α = 0 and from Eqs. (1) and (3) we get,γ ± = [( β * ± β0 ) 2 + γ 0 −2 ]γ 0γ *2(5)where the signs correspond to the initially counter- (“+”)
or copropagating (“-“) laser pulse and acceleratedparticle. Although the energy gain is maximal forcounter-propagating injection, this interaction geometryis efficient for sufficiently low energies due to thethreshold condition (2). At high injection energies andenergy gains, the co-propagating injection is preferable.

3. QUANTUM ANALYSIS.
As it has been shown above, the most specific featureof the LPA regime under consideration is that the energy
gain for particle accelerated does not depend on the laser
pulse intensity, but on the group velocity only, i.e. on thedielectric properties of the medium. The value of thelaser peak intensity determines only the threshold of thisacceleration regime, i.e. possibility of the accelerationaccording to the classical scattering condition (2). In a
more correct quantum feature, the value of
ponderomotive potential determines the probability ofscattering, which determines the number of accelerated
particles. To estimate its value, one can start with Klein Gordon equation, as the spin effects are evidently notimportant for LPA.
It is convenient to consider the problem in the moving
reference frame, in which the effective potential is timeindependent. As the most interesting case is thescattering at the angle α = 0 , one can use for estimatesthe 1D approximation. As the averaged ponderomotive
potential is time-independent, the energy of the particle
is conserved, and one canseek solution in theasymptoticalform_ϕ ( z , t ) ∝ ex p ( − i ε t / h ) ex p ( ± ik 0 z ) whereε is the particle energy in the moving frame, and_k 0 = ε 2 − m 2 c 4 / hc is the asymptotic wavenumberof electrons in the field-free regions. In fact, we assume
the stationary flow of particles collides with thepotential, and seek the parts of this flow which will bereflected and transmitted. Using the standard methods ofquantum mechanical calculations [], one can easily write
estimates for the reflection R and transmission T=1-R
coefficients in the two following limiting cases .
First, in the case, when the energy of particle isless than the ponderomotive potential, i. e.,_k 0 < eA0 / hc 2 , where A 0 is the peak amplitude ofthe laser vector potential, the approximate solution canbe given in the WKB-approximation : T ≈ exp(−2Γ ),z2(e 2A 2 ( z)2Γ =∫_ 2 2 − k0hcz1)

1/ 2dz(6)Here, the integration is over classically prohibited zone,
between the turning points z1 and z 2 , where theintegrand becomes zero. In this case, most of theparticles, except an exponentially small fraction, arereflected, and consequently, accelerated.
In the opposite case, when the energy is well above_the ponderomotive barrier, k 0 >> eA0 / hc 2 , theresult of the perturbation theory is

( )

(7)

which is, in fact the ratio of the quiver momentum to theinitial electron momentum in the moving frame.
It is interesting to make some estimates for laser
ponderomotive acceleration of electrons in plasma. We
will assume the electron beam with the initial energy 150
MeV ( γ 0 = 300 ) is injected into the interaction region
in the direction of laser pulse propagation. Themaximum energy gain is in the forward direction andgiven by Eq. (9). At given output energy of electrons, therequired group velocity and corresponding Lorentz factor
are

γ*=

1γ +γ0[2 1 + γγ 0 (1 − ββ0 )]

(8)

1/ 2.
× 10 γ * λ [ µm] , anddensity, ne [cm ] ≈ 1115
threshold peak intensity and normalized laser field
amplitude are given by relations (1),(2)−221−2a 0th = 2 (γ * γ (1 − β * β0 ) − 1)
2The

202

corresponding

1/ 2(9)intensityisI [W / cm ] = 137. × 10 a λ [ µm] . In the Fig. 1, thedependencies of the normalized amplitude and laser
peak intensity on the final energy of electrons areshown. The required plasma density and Lorentz factor,corresponding to the pulse group velocity, are shown inthe Fig.2.
The regime proposed makes it possible acceleration ofelectrons up to TeV energy range using modern lasertechnology. For the injectionenergy 275 MeV( γ 0 ~ 550 )
218peak

20−2IL

18IL, 10 W/cm

21000

γ*1000100

ne101101001000

Energy gain, GeVFig. 2. Plasma density and Lorenz -factor ofcorresponding laser pulse group velocity for acceleration
to GeV energies. Injection energy 150 MeV.

ne ~ 1115. × 1012 cm−3
( γ * ~ 3.2 × 10 4 ) the energy gain is ~1TeV( γ ~ 2 × 10 6 ). The thresholdnormalized laser field amplitude required is thena 0 ~ 45 which corresponds to laser peak intensityandThe group velocity determines the required plasma−3-310000

12<< 1ne, 10 cm∫4γ*,2+∞eA0
1 e 2 A 2 ( z)R≈
_ 2 2 exp( 2ik 0 z) dz ~ _k 0 −∞ 2h c
hk 0 cplasmadensity~2.74 × 1021W / cm 2 .
It is interesting to estimate the parameters required
for acceleration of electrons up to PeV -range of energies( γ ~ 2 × 10 9 ). At 700 MeV the injection energy and1024 W / cm2 the peak intensity, we have an estimate forgroup velocity Lorentz-factor γ * ≈ 16. × 10 6 , which−3corresponds to the plasma density ne ≈ 4 × 10 cm .
In conclusion, we have proposed in this paper a new
relativistic ponderomotive acceleration concept, which isbased on the scattering of particles by the leading front of
the laser pulse moving in a plasma at the group velocity
which is less than the speed of light. The gain in particle
energy is determined by the injection energy and pulse
group velocity and does not depend on peak intensity,
which determines the quantum probability ofacceleration. The quantum and classical analysis of theproposed concept is presented. Estimates show that thescheme proposed is prospective for the next generation
accelerator technology.
8a0100REFERENCESa0,

10110100

1000

Energy gain, GeVFig.1. Threshold laser peak intensity and normalized
amplitude corresponding to the GeV -domain energy
gain of electrons. Injection energy – 150 MeV.

[1] P. L. Kapitza and P. A. M. Dirac, Proc. Cambridge Philos.
Soc. 29, 297 (1933)[2] M. D. Perry and G. Mourou, Science 264, 917 (1994)[3] T. Tajima and J.. Dawson, Phys. Rev. Lett. 43, 267 (1979)[4] E. Esarey, P. Sprangle, J. Krall, A. Ting, IEEE Trans.
Plasma Sci. 24, 252 (1996)[5] F. V. Hartemann et al., Phys. Rev. E, 51, 4833 (1995).
[6] B. Quesnel, P. Mora, Phys. Rev. E, 58, 3719 (1998)[7] A. B. Migdal, Qualitative Methods in Quantum Theory,
W.A. Benjamin, Inc., Reading, 1977.