Laser-driven acceleration with Bessel beams
B. Hafizi,
1
E. Esarey,
2
and P. Sprangle
2
1
Icarus Research, Inc., P.O. Box 30780, Bethesda, Maryland 20824-0780
2
Beam Physics Branch, Plasma Physics Division, Naval Research Laboratory, Washington, D.C. 20375-5346
Received 30 September 1996
The possibility of enhancing the energy gain in laser-driven accelerators by using Bessel laser beams is
examined. A formalism based on Huygens’ principle is developed to describe the diffraction of finite power
bounded Bessel beams. An analytical expression for the maximum propagation distance is derived and found
to be in excellent agreement with numerical calculations. Scaling laws are derived for the propagation length,
acceleration gradient, and energy gain in various accelerators. Assuming that the energy gain is limited only by
diffraction i.e., in the absence of phase velocity slippage, a comparison is made between Gaussian and Bessel
beam drivers. For equal beam powers, the energy gain can be increased by a factor of N
1/2
by utilizing a Bessel
beam with N lobes, provided that the acceleration gradient is linearly proportional to the laser field. This is the
case in the inverse free electron laser and the inverse Cherenkov accelerators. If the acceleration gradient is
proportional to the square of the laser field e.g., the laser wakefield, plasma beat wave, and vacuum beat wave
accelerators, the energy gain is comparable with either beam profile. S1063-651X9713503-6
PACS numbers: 41.75.Cn
I. INTRODUCTION
Laser-driven accelerators rely on the large intensities that
can be achieved when laser beams are focused down to spot
sizes on the order of several wavelengths 1–17. The asso-
ciated gradients are typically much larger than the 100
MV/m in proposed next-generation X -band linacs. However,
a shortcoming of many of these schemes is that the interac-
tion length over which the high intensity can be sustained is
relatively short due to transverse spreading diffraction. Ra-
diation from a laser cavity is usually in the form of the fun-
damental and higher-order Gaussian modes. For such a beam
the Rayleigh length, i.e., the free-space scale length for dif-
fraction, is given by
Z
RG
=kr
0
2
/2, 1
where r
0
is the minimum spot size of the beam at the focal
point and =2/k is the free-space wavelength 18. In
vacuum or in a gas 4–13 acceleration can be achieved by
direct interaction of the axial component of the laser field E
z
with the particles, where z is the propagation direction. Us-
ing “•E=0, the axial electric field is related to the domi-
nant transverse field E
by E
z
/ z =-“
•E
. For a Gauss-
ian beam, E
z
=O ( E
0
/ kr
0
), where E
0
is the transverse field
amplitude. The product E
z
Z
RG
provides an estimate of the
energy gain, assuming that the interaction is synchronous,
i.e., neglecting phase velocity slippage.
This paper addresses the scaling of and the maximization
of the energy gain in various accelerators driven by lasers
with two different transverse mode profiles. In particular,
laser accelerators driven by Gaussian beams will be com-
pared to those driven by Bessel beams 19–22. The diffrac-
tion of Bessel beams is examined and an analytical expres-
sion for the maximum propagation distance is derived and
compared to numerical calculations. It is shown that a Bessel
beam can enhance the energy gain by a factor of N
1/2
com-
pared to a Gaussian beam of the same power, provided that
i the acceleration gradient is linearly proportional to the
laser field, and ii the acceleration distance is limited by
diffraction and not by phase detuning or some other mecha-
nism, where N is the number of transverse rings lobes in
the Bessel beam. The specific example of the inverse Cher-
enkov accelerator is examined in detail.
The mode structure of a laser beam can be altered using
common optical elements, including holographically gener-
ated zone plates 23 and axicons 11,12,24–29. Notable
examples of such beams are the Bessel beam of order n ,
J
n
( k
r ), where k
is the transverse wave number and r is
the radial coordinate. The J
0
beam has been the subject of
much theoretical and experimental analysis as a paradigm of
what are referred to as ‘‘diffraction-free’’ beams 19. In
reality any beam with finite transverse extent is subject to
spreading and the designation diffraction free is a misnomer.
Indeed, careful comparison of a Bessel beam with a Gaussian
beam reveals that the latter has a better energy transfer ca-
pability 20–22.
However, since Bessel beams are sharply peaked and
have a large depth of field they may be more useful than the
familiar Gaussian beams in certain applications. For ex-
ample, direct laser acceleration relies on the interaction of a
particle with the axial electric field of the laser. The funda-
mental Gaussian and the J
0
beams are not efficient for direct
acceleration since there is no on-axis electric field associated
with either. However, E
z
( r =0) 0 for higher-order Gauss-
ian and Bessel beams. Bessel beams have been created using
axicon lenses and, in particular, a zeroth-order Bessel beam
was used in channel guiding experiments 29 and a radially
polarized, first-order Bessel beam has been used in experi-
ments on the inverse Cherenkov accelerator 12.
II. BESSEL AND GAUSSIAN LASER BEAMS
A. Ideal Bessel beams
In vacuum, the Cartesian components of the laser electric
field, E
i
( i =x , y , z ), satisfy the scalar wave equation
PHYSICAL REVIEW E MARCH 1997 VOLUME 55, NUMBER 3
55 1063-651X/97/553/35397/$10.00 3539 © 1997 The American Physical Society