L. Shao, Y. K. Ho*, P. X. Wang, Q. Kong, Y. Q. Yuan, N. Cao, J. Pang,
Y. J. Xie
Institute of Modern Physics, Fudan University,Shanghai 200433, China; * Corresponding author. Eemail: hoyk@fudan.ac.cn ; Fax: 86-21-65643815
Abstract. In this paper we present a comparison between the effects on electron dynamics by a linearly polarized and a circularly polarized field of an intense laser beam. Special attention is given to the vacuum laser acceleration scheme, also known as capture and acceleration scenario
(CAS). It has been found that CAS phenomenon can occur in both polarized fields but there are differences. The CAS of a circularly polarized field exhibits axisymmetric feature, whereas CAS of a linearly polarized field can only be observed when electrons are injected nearly along the polarization plane. Given the same laser parameters (intensity, beam width), the maximum energy gained by the electrons from linearly polarized field is higher than that from circularly polarized field. Physical explanations based on the space distributions of the field amplitudes are described. We hope this study provides significant help to those designing experimental setup to verify CAS as well as to those developing laser accelerators of the future.
The rapid development of intense laser technology in recent years has stimulated many frontier research areas in both applied and fundamental physics
[1
"
3]
.
Among these new areas we have studied the feasibility of laser-driven electron acceleration in vacuum and have obtained meaningful results
[4]
. We found a free electron can be captured by the laser field and accelerated violently with an acceleration gradient in excess of IGeV/cm . We call this process 'capture and acceleration scenario' (CAS), and present a physical explanation to this acceleration scheme
[]
.
In all our previous work the laser beam was assumed to be linearly polarized and the electrons were injected along the electric polarization plane of the electromagnetic field. The main tasks of the present investigation are to study the effect of the field polarization state on the CAS.
This study has great interest to experimentalists designing setup to verify CAS or implementing the laser accelerator.
CP634, Science of Superstrong Field Interactions, edited by K. Nakajima and M. Deguchi
© 2002 American Institute of Physics 0-7354-0089-X/02/$ 19.00
337

The schematic geometry of the electron scattering by laser beam used in the calculation is shown in Fig.l. The laser beam adopted is in the lowest-order Hermit-
Gaussian (0, 0) mode
[6]
expressed analytically in the paraxial approximation and
incident direction of the electron and the propagation direction of the laser is defined as 6 . Four-dimensional vector configurations of energy-momentum (7.
:
,P
xi;
,P
yi:
,P zi
,) and (j f9
P jf9
P jf9
P vf
) are used to specify the incoming and outgoing states of electrons respectively. For a linearly polarized (LP) field, we assume that the electric component of the laser field is polarized along the x-axis. The angle between the projection on the x-y plane of the electron's incident direction and the x-axis is defined as the polarization azimuth angle a .
.,P ,P ,P )
7
yi
7
zr
FIGURE 1. Schematic geometry of electron scattering by a laser beam. The laser propagates along the z-axis, WQ is the laser beam width at the waist. The free electron enters from the minus-x side. The four-dimensional vectors ( 7 . , P x
. , P .,P z
.) and ( y ,p> , p , p ) denote the incident and outgoing state of the electron, respectively. The Lorentz factor J designates the electron energy and the momentum
P is normalized in units of m e
c. Angle 6 is the electron incident angle, with tan6
=
P t i/P
Z i (P t i denotes the transverse component of P.). Angle Ot denotes the polarization azimuth angle. Style for
Figure Captions. Center this text if it doesn't run for more than one line.
338

As for the circularly polarized (CP) field, the electromagnetic field components of the laser in the normal paraxial approximation are expressed in the following equations:
£,=£^^-±-4-1^, (1)
x
x
(3)
2K(z)
Where w
0
is the radius of the beam width at focus, w(z) = w
0
[1 + (z/Z
R
)
2
j ,
R(z) = z[l + (Z
R
/zJ\, (j)(z) = tan'
1
(z/Z
R
), Z
R
= kw^/2 is the Rayleigh length.
And if an LP field and a CP field have the same field strength #
0
, (a
0
= eE
Q
/m e ccn is a dimensionless parameter measuring the laser intensity, with -e and m e
being the electron charge and mass respectively, en the angular frequency of the electromagnetic wave and c the speed of light in vacuum), the energy flow densities
I
L
= 1.366xl0
18
(^)
2
, and I c
=2.13x10
18
(^)
2
. (6)
A, A, for the LP and CP field respectively. In equation (6) A is the wave length of the laser of unit jum and the unit of / is W/cm
2
azimuth angle Ot for LP fields. Figure 2 shows the electron final energy y f
versus the initial phase ^
0
of the field at different a values. Various values of ^
0
correspond electrons impinge on the laser beam at different delay-time. It can be seen from this figure that with a being increased y f
decreases quickly and the capture plateau becomes narrower. This means for LP field we have to inject electrons into an azimuthal angle as smaller as possible to achieve the CAS mode. Figure 3 presents a
339

comparison of y^ (the maximum value of J
f
as the initial laser phase 0
0
is varied
intensities for the both fields are a
0
= 100. This figure illustrates two properties. One is that the electron dynamics in LP fields are strongly axis-asymmetric, whereas in CP fields they are axisymmetric. The other is that the maximum energy gained by electrons from the CP field is a bit lower than that from the LP field.
8000
6000
4000
2000 oc=0
50 100 150 200 250 300 350 d>
Y
FIGURE 2. The final electron energy y, versus the initial phase 0 o
of the field at different Ot values for LP fields with a
0
=100. Other parameters used are kw
Q
= 170, tan 6 = 0.1 and P t
. = 3.
340

8000
6000
4000
2000 o -180
2000
4000
6000
8000
210
120
240
270
90
60
Q
0
=100
30
300
330
FIGURE 3. A comparison between the effects of LP and CP field on the maximum final energy J ^ vs. Ot. Other parameters used are the same as in Fig. 2. The solid line and the dashed line denote the
LP and CP field respectively.
The physical explanation to the first point is according to the distribution of the longitudinal component of the electric field, which plays a major role in accelerating electron in longitudinal direction. Figure 4(a) and (b) show the amplitude distributions
the longitudinal component of the electric field distributes only along the polarization plane of the transverse component of the electric field for LP field, but it has a same distribution both along the polarization plane and normal to the polarization plane for
CP field. Since the amplitude of E z
is not axisymmetric in LP field, it could be wellunderstood that the electron dynamics in LP field exhibit axis-asymmetric characteristics. In contrast to that, the whole CP field amplitudes, including the
341

longitudinal component amplitude, are symmetric with respect to the azimuth angle a .
Therefore, the electron dynamics in CP field should be axisymmetric.
-500
-500
-500
FIGURE 4. The distribution of the longitudinal electric field amplitude at the z = 0 plane with a
0
= 100, (a) is for LP field and (b) is for CP field. The bold circle on the bottom plane represents the laser beam width on the focused plane.
To the second property of the Fig. 3, it can be explained as follows. In the CP field, in addition to the - eE x
force, which is the same for both fields with the same a
0
,
342

there is another transverse force -eE y
for CP only. This force will pull the electrons away from the laser axis which is the strong region of the electric longitudinal
force is decreased. Accordingly the maximum energy gained by electrons from the LP field is larger than that from the CP field.
1. CAS can emerge in both the LP field and CP field provided the laser intensities are high enough.
2. CAS of the LP field exhibits highly axis-asymmetric feature. Only in a small range of the polarization azimuth angle will the incident electrons be substantially
characteristics is the rule.
The authors would like to thank A. M. Sessler and E. Esarey for helpful discussion, and C. M. Fou for carefully reading of this manuscript. This work is partly supported by the National Natural Science Foundation of China under Contracts No. 19984001 and No. 10076002, National High-Tech ICF Committee in China, Engineering-
Physics Research Institute Foundation of China, National Key Basic Research Special
Foundation (NKBRSF) under grant No. G1999075200, and Zhonglu fellowship.
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