Acceleration Channel of Light Beam
and Vacuum Laser Acceleration
YJ. Xie, Y.K. Hoa), Q. Kong, N. Cao, J. Pang, L. Shao
Institute of Modern Physics, Fudan University, Shanghai 200433, China
a) Author to whom correspondence should be addressed,
FAX: +86-21-65643815. Electronic address: hoyk(S)fudan.ac.cn
Abstract. It has been found that for a focused laser beam propagating in vacuum, there exist
"natural" channels for laser driven electron acceleration, which emerge just beyond the beam
width and extend along the diffraction angle for a few Rayleigh lengths. These channels have
the similar characteristics to that of a wave guide tube of conventional accelerators: a
subluminous wave phase velocity in conjunction with a strong longitudinal acceleration electric
field. Relativistic electrons injected into these channels can be captured in the acceleration
phase of the wave for sufficiently long time and gain considerable net energy from the laser
field. The basic conditions for this acceleration scheme to emerge are addressed here.
Due to the recent development of ultra-intense laser with the chirped pulse
amplification (CPA) technique, currently laser intensities have increased to as high as
1019~21W/cm2 [1-4]. Such intense lasers can provide field gradients of 106~107MV/m,
much higher than that of conventional accelerators, which is about l~10MV/m. Thus
laser acceleration of particles, especially electrons, has been the focus of significant
attention. Many schemes have been proposed [5-10] along this line. Among them, the
far-field acceleration has received wide attention [11], because it has the advantage of
a simple experimental setup. Concerning laser acceleration in vacuum, there is a basic
question not answered till this work. Can a free electron gain noticeable net energy
from laser field in vacuum where the interaction length is unlimited? [12] The
so-called Lawson-woodward theorem [13], states that the phase velocity of the laser
field near the focal region is greater than c, thus the inevitable phase slippage would
lead the relativistic electron to experience alternatively acceleration and deceleration
phase as it transverse the laser field, which would result in a cancellation of the
energy gain for an unlimited interaction length.
The crucial issue in above discussion is related to wave phase velocity. In this
paper we study the phase velocity of a focused laser beam. The results show there
exists a region characterized by lower phase velocity (V9 <c). Combined with the
longitudinal electric field in this region, we get natural channels for laser acceleration.
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
286
If fast electrons can be injected into these channels, these electrons can remain
synchronous with the accelerating phase for sufficiently long time, thereby receive
considerable energy from the field. We call this acceleration scheme CAS (Capture
and Acceleration Scenario). Simulations indicate that the electron trajectory is
significantly perturbed by the laser field in the CAS scheme, which in effect limits the
interaction region. Hence no additional optics are required to limit the interaction
distance in the CAS scheme. This allows operation at ultra-high laser intensities and
high energy gains without the restriction of damaging nearby optics. Thus the CAS
scheme shows interesting prospect to become a new principle of realistic laser
accelerators.
For a laser beam of Hermite-Gaussian(0,0) mode polarized in the x-direction and
propagated along the z-axis, the transverse electric field component is [14]
, y, z,0 = E 0 -^exp(-* + y22 ) exp [- iq>]
w(z)
w(z)
(1)
where w0 is the beam width at focus, w(z) = w0[l + / 2 j2 is the beam width,
/ = z /ZR , ZR = kwQ2 1 2 is the Rayleigh length, and phase cp is
<p = kz-CDt- tan-'ffl -<p0 +
*
y
(2)
where cpQ is the initial phase, R(Z)= z[l + /~ 2 J is the radius of the curvature. The
effective phase velocity of the wave along a particle trajectory (Y9)j can be
calculated using the following equation
(3)
where (V^)7 is the gradient of the phase field along the trajectory. From it, we can
obtain the effective phase velocity along different direction, for example, the one
parallel to the z-axis V \
zR(\+ry
287
(4)
ρ=0
ρ=0.5
ρ=1.0
ρ=1.5
1.0006
1.0006
1.0004
1.0004
VϕΖ
1.0002
1.0002
N
1.0000
1.0000
0.9998
0.9996
0.9994
0.9992
-8- 8
-6- 6
-4- 4
-2-
2 -0-
l=z/Z
Z=z/ZRR
0 22
44
6
8
kw00=60
=60
kw
FIGURE1.1.The
Theeffective
effective phase
phase velocity
velocity along
along the
the line
line ρ
p =0, 0.5, 1.0,
1.0, 1.5 paralleling to z-axis
FIGURE
2
2
where f ϕ^= ρ (1 − 2l ) , ρ = ξ 2 + η 2 ， ξ == xx/w
, ηT] == yy/w
/ w00，
/ w00. In particular, the
where
(1 + l )
minimumphase
phase velocity
velocity VV^
is the one along the
to the equal-face
minimum
equal-face
ϕm is the one along the direction normal
thewave.
wave.Generally
Generally we
we call
call itit phase
phase velocity of the wave
ofofthe
Vϕm =
ωCO
=
∇ϕ
ω(0
1 − fϕ
22ρpil
(
) 2 + (k −
)2
2
w0 (1 + l )
Z R (1 + l 2 )
(5)
(5)
Alongthe
thez-axis,
z-axis,the
theeffective
effective phase
phase velocity
velocity equals
equals to
to the
the minimum
minimum phase
Along
phase velocity.
velocity.
The
effective
phase
velocity
along
the
line
p=0,
0.5,
1.0,
1.5
paralleling
to
The effective phase velocity along the line ρ =0, 0.5, 1.0, 1.5 paralleling to
z-axis
aregiven
giveninin Fig.1.
Fig.l. Just
Just as
as what
what LawsonLawson- Woodward
Woodward states,
states, the
the effective
effective phase
z-axis are
phase
velocityalong
alongz-axis
z-axis ((ρp=0)
=0)isis larger
larger than
than c.
c. Thus,
Thus, ifif electrons
electrons move
move along
along this
this line,
line, it
it
velocity
must
suffer
fast
phase
slippage
with
the
laser
field.
But
away
from
z-axis
(p
>W
),
Q
must suffer fast phase slippage with the laser field. But away from z-axis ( ρ > w0 ),
there isis aa region
region (within
(within z<Z
z<ZRR)) where
where VV^ <<cc.. Furthermore
Furthermore as
as for
for the
there
the minimum
minimum
ϕz
phase
velocity,
the
condition
V^
<
c
requires
approximately
p
>
w(z),
which
phase velocity, the condition V < c requires approximately ρ > w(z ) , which
ϕm
extendstotoaaregion
region much
much larger
larger than
than that
that of
of VV <<c.
Fig.2 illustrates
illustrates the
c . Fig.2
the distribution
distribution
extends
ϕz
of
Vyn
on
the
plane
y=0.
It
is
straightforward
to
find
there
exists
a
region
where the
of Vϕm on the plane y=0. It is straightforward to find there exists a region where
the
phase
velocity
is
lower
than
the
light
velocity
in
vacuum
(Fig.2(a)),
which
emerge
phase velocity is lower than the light velocity in vacuum (Fig.2(a)), which emerge
justbeyond
beyondthe
the beam
beam width
width and
and extend
extend along
along the
the diffraction
diffraction angle
angle θ9m ~~ 1l/kw
/ kw00.. In
In the
the
just
m
subluminous
phase
velocity
region,
the
magnitude
of
the
minimum
phase
velocity
is
subluminous phase velocity region,
the
magnitude
of
the
minimum
phase
velocity
is
2
2
of
the
order
V^
~
c[l-l/(kw
)
]
(Fig.2(b)).
This
phenomenon
stems
from
the
Q
of the order Vϕm ~ c[1 − 1 /(kw0 ) ] (Fig.2(b)). This phenomenon stems from the
diffraction effect
effect of
of the
the optical
optical beam,
beam, and
and is
is caused
caused by
by the
diffraction
the phase
phase factor
factor
2
2 + y2 )/2R(z).
k (x + y ) / 2 R( z ) .
288
(a)
(a)
Vϕm
α=
x/
w
β=z/ZR 0
=6
0
kw0
FIGURE 2. The distribution of the minimum phase velocity V
in the
the plane
plane y=0.
y=0.
V(pm
ϕm in
( a)
a) the
the 3D
3D surface
surface plot of
of V
V^
(b) the
the 2D
2D contour of V
V^.
ϕm (b)
ϕm .
Notice that the minimum phase velocity along the z-axis is still superluminous,
i.e., ((V^)^
Vϕm ) ρ =0 =
c /{1 − 1 /[kZR(l
+ ll 22))]}
=c/{l-l/[kZ
] } .. It indicates that the near-axis region of the
R (1 +
beam is not suitable for accelerating charged particles (Also, in this region the
longitudinal component of the accelerating electric field
field is very small, see Fig.3).
This is because the phase velocity in this region is the highest, which leads to fast
phase slippage between electron and laser field. But if the electron can move along
along
(z ) , it is hopeful for electron to be trapped in the laser field and
the beam width w
w(z),
obtain net energy.
For accelerating particles, subluminous phase velocity region alone is not enough.
The acceleration field strength is also an important factor and should be addressed
addressed
here. The amplitude of the longitudinal electric field component is
Ez =
E 0 w0ξ
ρ2
exp(
−
)
1+ l2
Z R (1 + l 2 )
(6)
The variables are defined as before. Fig.3 gives the distribution of Ezz in the plane
y=0.
Just as the distribution of subluminous phase velocity in the plane y=0, the largest
amplitude of the longitudinal electric field emerges not at the beam center, instead, at
the out region of the beam along the polarization axis(x-axis). In order to represent
represent
the ability of the laser field for accelerating charged particles, we introduce a quantity
Q to combine these two factors together. We call it acceleration quality factor, which
is defined as
(
Q = Q 0 (1 − V ϕ m )ξ exp − ρ 2 (1 + l 2 )
) (1 + l )
=
= 00
289
2
Vϕm ≤ c
Vϕm >c
>c
(7)
Unit)
(Relative
Unit)
EzE(Relative
z
α
αy =z/Z
y=
z/ZR
/w
α x=x 0
0
x/w 0
α
x=0=6
kw
R
0
kw 0=6
FIGURE3.3.The
The distribution
distribution of
of the
the normalized
normalized amplitude
FIGURE
amplitude of
of the
the longitudinal
longitudinalelectric
electricfield
fieldEz
Ezininthe
the
FIGURE 3. The distribution of the normalized amplitude of the longitudinal electric field Ez in the
plane
y=0.
plane
planey=0.
y=0.
(Relative
Unit)Unit)
(Relative
Where Q0isis aa normalization
normalization constant chosen
to
Where
to make
make QQ
Q in
in the
the order
order of
ofunit.
unit.
Where go
Q0 is a normalization constant
constant chosen
chosen to
make
in
the
order
of
unit.
the contribution from
from the phase
velocity, and
1-Vϕm/c represents the
1-Vcpm/c
andthe
theremaining
remainingterm
termisisis
represents the contribution
contribution from the
the phase
phase velocity,
velocity, and
the
remaining
term
1-Vϕm/crepresents
proportional tothe
the amplitude of
of the longitudinal
longitudinal electric field,
proportional
field, itititdescribes
describesthe
theeffects
effects
proportional to
to the amplitude
amplitude of the
the longitudinal electric
electric field,
describes
the
effects
of laser intensity on electron’s behavior. For region where Vϕm > c , we simply define
of
region where
where VV^ >>cc,, we
wesimply
simply define
define
of laser
laser intensity
intensity on
on electron's
electron’s behavior.
behavior. For
For region
ϕm
Q ≡ 0 . Because in that case no particles can remain synchronous with the
QQ =
can remain
remain synchronous
synchronous with
with the
the
≡ 00 .. Because
Because in
in that
that case
case no
no particles
particles can
accelerating phase of the laser field.
accelerating
accelerating phase
phase of
of the
the laser
laser field.
field.
The distribution of the acceleration quality factor Q on the plane y=0 is given in
The
factor Q
Q on
on the
the plane
plane y=0
y=0 is
is given
given in
in
The distribution
distribution of
of the
the acceleration
acceleration quality
quality factor
Fig.4.
Fig.4.
Fig.4.
α
y=
α
z/
Z
R
z/
Z
α x=x
y=
/w 0
=60
kw 0/w
0
60
kw 0=
α x=x
R
FIGURE 4. The distribution of the acceleration quality factor Q in the plane y=0.
FIGURE
Q in
in the
the plane
plane y=0.
y=0.
FIGURE 4.
4. The
The distribution
distribution of the acceleration quality factor Q
290
It shows apparently that there are acceleration channels in the field of the focused
laser beam propagating in vacuum. The significant values of Q emerge beyond the
beam width and extend along the diffraction angle of the beam for a few Rayleigh
lengths. This is because both the lower phase velocity region and the large amplitude
of longitudinal electric field locate at the out region of the beam. Along the diffraction
angle of the beam for several Rayleigh lengths, although the effective phase velocity
is still lower than c (Fig.2(b)), the amplitude of longitudinal electric field has already
decreased to very little (Fig.3), thus the factor Q becomes very small. To sum up,
acceleration channels in the field of the focused beam propagating in vacuum have
been found, which shows similar characteristics to that of a wave guide tube of
conventional accelerators: subluminous phase velocity in conjunction with a strong
longitudinal electric field component. Consequently, if one can inject fast electrons
into these channels, then some of them may remain synchronous with the accelerating
phase for sufficiently long time such that they receive considerable energy from the
field.
The acceleration channel can give a reasonable explanation to the mechanism of
CAS. Now we go on to study under what conditions CAS phenomenon can occur. To
study the dynamics of electrons in this acceleration channel, 3D test particle
simulations are utilized that solve the relativistic Newton-Lorentz equations of motion
[15].
Simulation results show that the laser intensity, the electron initial momentum
and the electron incident angle are all very important factors for CAS to occur. There
exists a threshold value of laser intensity (ao)th. Only when ao>(ao)th, the CAS
phenomenon can be observed. Fig.5 shows the electron maximum final energy yfm vs.
the laser intensities ao at different beam width, yfm is the maximum value of yf as 90 is
varied over the range (0,2ji).
____ kw 0 =60
____ kw 0 =80
.......... kw =100
FIGURE 5. The maximum output energy yfm as a function of the laser intensity a0 at different laser
beam width. Solid line for w0=60, Px=0.9, Pz=9; dashed line for w0=80, P x =l.l, Pz=ll;dotted line for
w0=100, Px=1.2, Pz=12; momentums in y-direction in all cases equal zero, we terminate the calculation
at cot=400000.
291
It shows that: (i) (ao)th is strongly dependent on the laser beam width WQ , it
increases quickly with WQ . (ii) The scaling law for net energy gain of CAS electrons
from the laser field (ao>(ao)th) is approximately as yfm <* a0n with n~l as kwo<150.
This feature is consistent with the mechanism underlying CAS, namely the
acceleration occurs primarily in the acceleration channel by the longitudinal electric
field, (iii) With the increasing of beam width wo, the maximum output energy yfm
increase also. This is due to the fact that the work (AW) done by the laser field on the
electron in the longitudinal direction can be approximately estimated by
/7
Zn
2
AW °c Ez -ZR oc —2_.—2_oc E0wQ , where EO is the transverse electric field
kw0
2
amplitude at focus, and ZR the Rayleigh length.
The efficiency of CAS is sensitive to the electron initial incident momentum.
There exists an optimum injection energy range for high CAS efficiency. This range
is also strongly dependent on the laser beam width w 0 , but weakly relied on the laser
intensity. Roughly speaking, when kwQ < 120, the electron incident energy should be
in the range of 5~15MeV. Furthermore, for CAS to appear, electrons should be
injected into the laser field in a small angle (typically tan0=0.1~0.15). According to
the conditions above, the value of (ao)th for a laser beam of kw0 = 50 ~ 60 is 5~6,
corresponding to laser power density around 3xl019W/cm2, which is obtainable by
present laser systems. Thus it is promising for CAS to be tested experimentally.
In conclusion, we find that for a focused laser beam propagating in vacuum, there
exists a subluminous wave phase velocity region, due to the diffraction effect of the
optical beam. Combined with a strong longitudinal electric field component in this
region, it seems phase exist "natural" channels for laser driven electron acceleration,
which emerge just beyond the beam width and extend along the diffraction angle for a
few Rayleigh lengths. Relativistic electrons injected into this channel can be captured
in the acceleration phase for sufficiently long time and gain considerable net energy
from the laser field. It shows a new regime totally different from the other proposed
vacuum electron acceleration schemes. We call this CAS scheme. The
Lawson-Woodward theorem does not apply to this scheme because the particle path
in CAS scheme is not a straight line. In fact, the particle can travel in regions
(primarily away from the axis) where the phase slippage is very small. The basic
conditions for CAS to occur are that the laser intensity should be large enough as
ao>5, and electrons should be injected into the laser field in a small angle (typically
292
tan0=0.1~0.15) with injection energy range be in 5-15MeV. Studies show that CAS
has the potential to become a new principle of realistic laser accelerators.
ACKNOWLEDGMENTS
The authors would like to thank A.M. Sessler and E.H.Esarey for enlightened
discussions. This work is supported partly by the National Natural Science
Foundation of China under contracts No. 19984001 and 10076002, National
High-Tech ICP Committee in China, Engineering-Physics Research Institute
Foundation of China, and National Key Basic Research Special Foundation
(NKBRSF) under Grant No. 1999075200..
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