Chapter 8
Chaos due to Relativistic Effect
8.1 Basic Relation of an Electron in Relativistic Laser Field
When plane strong laser with linearly polarized in the y-direction propagates in the
x-direction in vacuum, the basic equations for an electron in the plane of x and y
under the wave field by the laser are given as follows with inclusion of the force by a
longitudinal electric field E x in the x-direction:
dp x
dt
¼ Àev y B z À eE x
ð8:1:1Þ
dp y
dt
¼ ÀeE y þ ev x B z
ð8:1:2Þ
mc
2 dγ
dt
¼ Àev y E y À ev x E x
ð8:1:3Þ
where E y and B z are due to the laser field and given with the vector potential in the
form:
E y ¼ À
∂A
∂t
B z ¼
∂A
∂z
ð8:1:4Þ
The vector potential of the plane wave propagating in the x-direction is given as a
vector in the y-direction (0, A, 0) in the form:
A ¼ A 0 cos ξ
ð Þ
ð8:1:5Þ
© Springer Nature Switzerland AG 2020
H. Takabe, The Physics of Laser Plasmas and Applications - Volume 1, Springer
Series in Plasma Science and Technology,
https://doi.org/10.1007/978-3-030-49613-5_8
287