Lecture Notes in Physics

16 H. Schwoerer
Emax =4γ 2 wm0c 2 , lmax ≈ γ 2 w ·c/ωp , with γw =
1
. (2.10)
1 − (ωp/ω) 2
Remark that the Lorentz factor γw of the plasma wave does depend only on
its phase velocity, which is determined by the dispersion relation of the plasma
or, in other words, by the ratio of the plasma density to the critical density.
In particular, the maximum energy gain of an electron in a wakefield does not
depend on the light intensity. The light intensity basically has to provide the
plasma density modulation over a long distance. As in the case of the ponderomotive
acceleration, electrons enter the wakefield with different velocities
and on different phases with respect to the plasma wave. Consequently, the
energy gain and the corresponding spectrum of the emitted electrons is broad
and even Boltzmann like, so that again a temperature can be attributed to
the wakefield accelerated electrons. Typical experimentally achieved energy
gains are in the MeV range (see [8, 10] and references therein).
2.3.4 Self-Focussing and Relativistic Channeling
In the discussion so far we have described only the effect of the light field
onto the plasma. Because of its dispersion relation, the plasma modifies vice
versa the propagation of the laser pulse. We will see that even though complex
in detail, the overall effect simplifies and optimizes the electron acceleration
process.
The index of refraction of a plasma was given by (2.1):
npl =
1 − ω2 pl
=
ω2 1 −
nee2 . (2.11)
γm0ɛ0ω2 In the current context we have to discuss the influence of the electron density
ne on the light propagation. We know that the ponderomotive force of
the laser beam pushes electrons in radial direction out of the optical axis: a
hollow channel is generated along the laser propagation (see Fig. 2.6 [left]).
From the numerator in (2.1) we see that the speed of light in the plasma
(determined by vp = c/npl) increases with increasing electron density. Therefore,
the ponderomotively induced plasma channel acts as a positive lens on
the laser beam. From the denominator in (2.1) we see that the same effect is
caused by the relativistic mass increase of the electrons, which is larger on the
optical axis of the beam than in its wings. From their origin these mechanisms
are called ponderomotive and relativistic self-focussing, respectively. In competition
with the natural diffraction of the beam and further ionization, both
effects lead to a filamentation of the laser beam over a distance, which can
be much longer than the confocal length (Rayleigh length) of the focussing
geometry. This channeling can be beautifully monitored through the nonlinear
Thomson scattering of the relativistic electrons in the channel, which is
the emission of the figure-of-eight electron motion at harmonics of the laser
frequency (see Fig. 2.6 and [11, 12]).