Lecture Notes in Physics

12 H. Schwoerer
field along the direction of the E-field with a velocity vx and a mean kinetic
energy, also called quiver energy Uosc,
vx = eE
m0ω sin(ωt) and Uosc = e2E2 0
. (2.4)
4m0ω2 If the light pulse has passed through the electron, the electron is again at rest
at the original position, no energy is transferred between light and electron.
When we increase the electric field strength, finally the quiver velocity of
the electron approaches the speed of light c and the quiver energy gets in the
range of the rest energy m0c 2 of the electron and higher. In that regime, the
magnetic field of the light wave cannot be neglected anymore in the interaction
with the electron. The equation of motion of the electron has to be solved with
the full Lorentz force
F L = dp
dt = −e · E + v × B . (2.5)
Since the velocity v due to the electric field is along ˆx and the magnetic field
B directs into ˆy (see Fig. 2.3), the v × B-term introduces an electron motion
in ˆz-direction. Solving the relativistic equation of motion of the electron in
the plane electromagnetic wave results in the momenta
px = − eE0
ωm0c sin(ωt − kz) =−a0 sin(ωt − kz),
eE0 2
pz =
sin
ωm0c
2 (ωt − kz) = a20 2 sin2 (ωt − kz) . (2.6)
Here we have introduced the relativistic parameter a0 =eE0/ωm0c, which
is the ratio between the classical momentum eE0/ω as it results from (2.4)
and the rest momentum m0c. We see from (2.6) that the electron oscillates in
transverse direction ˆx with the light frequency ω. The longitudinal velocity is
always positive (in laser propagation direction) and oscillates with twice the
light frequency. Overall, the electron moves on a zig-zag–shaped trajectory as
displayed in Fig. 2.3. In a frame moving forward with the averaged longitudinal
electron velocity 〈vz〉t = (eE0/2cω) 2 , the electron undergoes a trajectory
resembling an 8 (see inset in Fig. 2.3). The higher the relativistic parameter
a0, the thicker the eight.
The energy of the electron is can also be expressed with help of a0:
E = γm0c 2
= 1+ a20 sin 2 (ωt − kz)
· m0c
2
2 . (2.7)
As we see from (2.6), the longitudinal momentum scales with the square of the
intensity, whereas the transverse scales only linearly with I. Therefore, at high
electric field strength the forward motion of the electron becomes dominant
over the transverse oscillation.