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**Unformatted text preview: **The nonlinear, relativistic index nNL ¼ ð1 À ne =
nc Þ1=2
becomes imaginary when ne > nc , showing an increase
of the cutoff density for a plane monochromatic wave: this
effect is known as relativistic self-induced transparency or,
brieﬂy, relativistic transparency. However, the problem of
laser penetration inside a plasma is not trivial (Cattani
et al., 2000; Goloviznin and Schep, 2000; Shen and Xu,
2001) because of both the nonlinearity in the wave equation
and the self-consistent modiﬁcation of the plasma density
proﬁle due to radiation pressure effects. These latter may be
described via the ponderomotive force (PF).2 In an oscillating, quasimonochromatic electromagnetic ﬁeld described by
a dimensionless vector potential aðr; tÞ whose envelope is
sufﬁciently smooth in space and time, the relativistic PF is
[see, e.g., Bauer, Mulser, and Steeb (1995) and Mulser and
Bauer (2010)]
fp ¼ Àme c2 rð1 þ hai2 Þ1=2 : (4) For a plane wave impinging on an overdense plasma, the
resulting PF, more effective on the lightest particles, is in the
inward direction and tends to push and pile up electrons
inside the plasma. Ponderomotive effects will be further
discussed below (see Secs. II.B and II.C).
In a multidimensional geometry, a laser pulse of ﬁnite
width may produce a density depression around the propagation axis also because of the ponderomotive force pushing the electrons in the radial direction. Jointly with the
relativistic effect and target expansion driven by electron
heating, this mechanism may lead to a transition to transparency as soon as the electron density drops below the cutoff
value (Fuchs et al., 1999). Investigations of ion acceleration
in the transparency regime are described in Sec. IV.C.
1 Consistently with our deﬁnitions, given the value for I , the peak
value of the dimensionless vector potential of the plane wave will be
pﬃﬃﬃ
given by a0 for linear polarization and by a0 = 2 for circular
polarization.
2
Throughout this review we refer to the ponderomotive force as
the slowly varying, effective force describing the cycle-averaged
motion of the ‘‘oscillation center’’ of a charged particle in an
oscillating nonuniform ﬁeld, over a time scale longer than the
oscillation period. ‘‘Fast’’ oscillating components are not included
in the deﬁnition of ponderomotive force here adopted. Andrea Macchi, Marco Borghesi, and Matteo Passoni: Ion acceleration by superintense laser-plasma . . . The penetration of the laser pulse depends not only on the
electron density but also on the target size when the latter
becomes close to or smaller than one wavelength. As a simple
but useful example, the nonlinear transmission and reﬂection
coefﬁcients can be calculated analytically for a subwavelength foil modeled as a Dirac deltalike density proﬁle
(Vshivkov et al., 1998; Macchi, Veghini, and Pegoraro,
2009), obtaining a transparency threshold
a0 > ne ‘
;
nc (5) where ‘ is the thick...

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