Proceedings of International Conference on Physics in ... - KEK

with C (2)
µ = Γ (2)
µ − ˆ L (1) Γ (1)
µ + ˆ L (1)2 Γ (0)
µ /2. Then, we have
V (0)µ C (2)
µ = mca2 0x0
4kz
mc
p ′′
[ ( ) (
1 1
+
η L2 R
+ 1
L 2
3 a0x0
l
16 kz
˜X ′′2 + l2
4
)
mc
p ′′
]
η
. (11)
Here, R ≡ ([ ∂2 xa0x(x0) ] ) −1
/a0x0 is the scale length <strong>of</strong>
the field curvature. Since the new Hamiltonian, −Γ ′′
0, does
not contain the variable Z ′′ , the corresponding component
p ′′
η is found to be constant. Then, the equations <strong>of</strong> motion
in the x-direction are reduced to
dX ′′
dη
dP ′′
x
dη
′′ P x
= − , (12)
kzpη0
= −mca0x0 l
2
[
1
L +
( )
1 1
+
L2 R
˜X ′′
]
. (13)
These equations determine the particle motion up to the
second order <strong>of</strong> ϵ, which varies slowly compared with the
period <strong>of</strong> the fast oscillation appeared in the zeroth-order
orbit.
In the case 1/L 2 +1/R ≥ 0, we obtain a slow oscillatory
motion given by
P ′′
x = α sin θη + P (2)
x0
X ′′ = − α 1
(cos θη − 1) +
mc θζ0kz
cos θη, (14)
P (2)
x0
mca0x0
l
sin θη + X′′ 0 ,
θ
(15)
where θ = l √ (1/L 2 + 1/R) /2, α is a con-
stant determined by the initial condition as α =
mca0x0θ ( 1 − l/ ( 2Lθ 2) − 7l/ (8L) ) , X ′′
0 is the initial
particle position and P (2)
x0 is the second-order initial value
<strong>of</strong> P ′′
x calculated by the second-order backward Lie transformation.
It is remarkably noted that the unbounded secular
motion originating from the first-order ponderomotive
force given in Eq. (10) is changed to the bounded solution,
Eqs. (14) and (15), by taking into account the second order
curvature terms. This motion corresponds to a betatron
oscillation by which the particle is confined in the finite
radial region. Note that since the amplitude factor α and
the period θ are defined by the local gradient and curvature
at the initial particle position, they are valid only in the
region where the variation <strong>of</strong> the curvature is sufficiently
small during one cycle <strong>of</strong> the long period oscillations.
In the case 1/L2 + 1/R < 0, Eqs. (12) and (13) yield to
the solution
P ′′
x =
(2)
α + P x0
e
2
θη +
(2)
−α + P x0
e
2
−θη . (16)
This solution indicates that the particle is rapidly ejected
from the region <strong>of</strong> large laser field amplitude. Taking the
expansion <strong>of</strong> the right-hand side <strong>of</strong> Eq. (16) assuming θη ∼
O (ϵ), Eq. (16) leads to
P ′′
x = αθη + P (2)
x0
X ′′ = α
mc
1
kzζ0
, (17)
θ
2 η2 +
P (2)
x0
mc
1
kzζ0
η + X ′′
0 . (18)
This solution is consistent with that obtained in Eq. (10)
up to the first order, though the second order collection is
included in Eqs. (17) and (18).
Here, we have neglected the z-component <strong>of</strong> the vector
potential, az, for simplicity. The inclusion <strong>of</strong> az may cause
modulation to the amplitude factor α and/or the period θ,
which will be discussed separately.
SUMMARY
We derived a equation system describing the relativistic
ponderomotive force and the related particle dynamics
in a transversely-focused linearly-polarized laser field
up to the second order with respect to ϵ. In the first order,
we obtained the ponderomotive force proportional to
the field gradient in the x- and y-directions that is essentially
the same as the result in Ref. [6]. In the second order,
we found that the particle can exhibit a slow period
betatron-like oscillatory motion characterized by the curvature
<strong>of</strong> the laser field amplitude. This suggests that the
control <strong>of</strong> the curvature is important in confining the particle
and keeping the laser-particle interaction in transversely
localized high-intensity laser fields. The betatron-like oscillation
may cause intense radiation that will be discussed
in a future paper. The present result up to the first order
and the expansion form, Eqs. (17) and (18), up to the second
order are consistent with those obtained by performing
the perturbation expansion directly to the equation <strong>of</strong> motion.
However, in the present analysis, the nonlocal solutions,
Eqs. (14), (15) and (16) are obtained for the first time
through the Lie perturbation approach.
REFERENCES
[1] E. A. Startsev and C. J. McKinstrie, Phys. Rev. E 55 (1996)
7527.
[2] P. Gibbon, ”Short Pulse Laser Interactions with Matter”, Imperial
College Press, London, p. 36 (2005).
[3] J. R. Cary and R. G. Littlejohn, Ann. Phys. 151 (1983) 1.
[4] Y. Kishimoto, S. Tokuda and K. Sakamoto, Phys. Plasmas 2
(1995) 1316.
[5] K. Imadera and Y. Kishimoto, accepted for publication in
Plasma Fusion Res..
[6] N. Iwata, K. Imadera and Y. Kishimoto, Plasma Fusion Res.
5 (2010) 028.
[7] E. S. Sarachik and G. T. Schappert, Phys. Rev. D 1 (1970)
2738.