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Unformatted text preview: ty laser pulses favor electron heating
because the hydrodynamic expansion does not have sufﬁcient
time to wash out sharp density gradients nor is it dominated
by the strong ponderomotive force that steepens the density
At this point it is worth mentioning that in most highintensity experiments the main interaction pulse is preceded
by prepulses4 which cause early plasma formation and
expansion, so that the short-pulse interaction does not occur
with a sharp-boundary, solid-density plasma. However,
proﬁle steepening at the critical surface is still effective;
thus one may expect the interactions to still occur with a
sharp density proﬁle, having a lower density jump with
respect to a solid interface. Occasionally, ‘‘preplasma’’ formation may also allow additional electron acceleration
mechanisms to take place in the underdense plasma region
(Esarey, Schroeder, and Leemans, 2009), possibly leading to
electron energies much higher than given by Eq. (6). In more
recent experiments, advanced pulse cleaning techniques
allow minimizing prepulse effects (see Sec. III.E).
1. Heating models A popular electrostatic model of electron heating at a
step-boundary plasma was proposed by Brunel (1987). In
this model, electrons are dragged out of the surface of a
perfect conductor by an oscillating ‘‘capacitor ﬁeld,’’ extending on the vacuum side, and representing the P component of
the incident plus reﬂected laser electric ﬁeld. Electrons are
considered to be ‘‘absorbed’’ when, after having performed
about half of an oscillation on the vacuum side, they reenter
the target, there delivering their energy, which is of the order
of the oscillation energy in the external ﬁeld.5 The model thus
accounts in a simpliﬁed way for the pulsed generation
(once per cycle) of hot electrons directed into the target and
having an energy, roughly speaking, close to the ‘‘vacuum’’
value (6). This simple model is not self-consistent because,
for instance, the capacitor ﬁeld is assumed to vanish inside
the target, implying the presence of a surface charge density.
Nevertheless, following Mulser, Ruhl, and Steinmetz (2001)
it is possible to provide a ‘‘minimal’’ 1D model, still
in the capacitor approximation, where the electrostatic ﬁeld
is calculated self-consistently and an acceleration of electron bunches similar to that inferred by Brunel is apparent.
We consider the electric ﬁeld as the sum of the electrostatic and driver ﬁelds, e.g., Ex ¼ Ee þ Ed , where Ed ¼
Ed ðtÞ sin!0 t with Ed ðtÞ a suitable temporal envelope,
a step-boundary density proﬁle ni ¼ n0 ÂðxÞ (Z ¼ 1 for
4 In general the main pulse is preceded both by short pulses of
duration similar to the main pulse and by a much longer pedestal
due to ampliﬁed spontaneous emission.
This effect is also commonly referred to as ‘‘vacuum heating.’’
See Gibbon (2005b) for a discussion on the origin of the name. 756 Andrea Macchi, Marco Bor...
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