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form e + nγL → e ′ + γ where a photon γ is emitted upon
absorption <strong>of</strong> n laser photons γL by the incoming electron.
Figure 5: Feynman diagrams for NLC.
The process has been analysed in [5, 8] where formulae
for cross sections or emission rates can be found. The main
features <strong>of</strong> NLC may be summarised as follows. There is
no threshold to overcome which implies that there is a classical
limit (nonlinear Thomson scattering) corresponding to
ω ≪ mc2 . In the linear regime (a0 ≪ 1) one finds the
usual Compton upshift for the emitted photon frequency,
ω ′ 4γ2 eω where γe is the electron gamma factor. This is
nowadays being exploited for the generation <strong>of</strong> monoenergetic
gamma rays <strong>of</strong> high peak brillance [12]. The nonlinear
regime (a0 > 1) is characterised by an intensity dependent
cross section, σ(a0), determining the number <strong>of</strong> produced
photons, Nγ ∼ σ(a0)NeNγL . This very fact alters
the Compton upshift the maximum <strong>of</strong> which now becomes
ω ′ n,max 4γ 2 enω/(1 + a 2 0) , n = 1, 2, . . . . (3)
In particular, we note the appearance <strong>of</strong> higher harmonics
(n > 1) and the appearance <strong>of</strong> a 2 0 in the denominator. Thus,
there is a reduction (red-shift) <strong>of</strong> the kinematic Compton
edge which for the first harmonic amounts to
ω ′ max 4γ 2 eω −→ 4γ 2 eω/a 2 0 , (a0 ≫ 1) . (4)
Figure 6: Emission spectrum for NLC.
This redshift w.r.t. linear Compton scattering should be a
clear experimental signal and is highlighted in Fig. 6 (dis-
playing the γ emission spectrum) by a red arrow. The
higher harmonics are visible as side bands to the right <strong>of</strong>
the main (n = 1) spectral peak [13].
Effects due to finite transverse pulse extension are easily
understood qualitatively. The previous situation is typical
for laser beams that are not too strongly focussed such that
the electron beam (radius rb) does not feel the decrease
<strong>of</strong> intensity in transverse direction. Clearly, this requires
rb ≪ w0 with w0 denoting the laser beam waist size (see
Fig. 7, right panel). On the other hand, when the laser beam
is tightly focussed (w0 < rb, Fig. 7, left panel) the electrons
will also probe the boundaries <strong>of</strong> the beam which in
turn will modify the spectra <strong>of</strong> Fig. 6. It is somewhat unfortunate
that the highly nonlinear situation (a0 ≫ 1) corresponds
to a tight focus. This suggests that the experimental
detection <strong>of</strong> the redshift will require a fine tuning compromise
and in particular a very narrow electron beam. For a
detailed study the reader is referred to [14].
Figure 7: Left: Tight laser focus. Right: Wide laser focus.
Laser pair production (PP)
The strong-field QED Feynman diagram for laser induced
PP is obtained from the NLC diagram <strong>of</strong> Fig. 5 via
crossing, i.e. by swapping the outgoing gamma with the
incoming electron which turns into an outgoing positron
(Fig. 8).
Figure 8: Feynman diagram for laser PP obtained from
NLC via crossing.
Expanding the diagram on the right-hand side corresponds
to pair creation stimulated by n laser photons,
γ + nγL → e + e − . Both processes <strong>of</strong> Fig. 8 have been
employed in the experiment SLAC E-144. First, high energy
gammas (30 GeV) have been obtained through nonlinear
Compton upshifting whereupon these were brought
into collision with the laser again. The centre-<strong>of</strong>-mass energy
<strong>of</strong> the colliding photons (for the second harmonic)
was just about enough to produce pairs <strong>of</strong> electrons and