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**Unformatted text preview: **le laser and target
technology.
The equation of motion for a moving target (sail) in the
laboratory frame can be obtained with the help of a Lorentz
transformation, similarly to Eq. (28). Neglecting absorption
for simplicity (A ¼ 0) we obtain
d
2I ðtret Þ
1À ð Þ ¼
;
Rð!0 Þ
dt
1þ c2 dX
¼ c;
dt (31) where X is the position of the sail, ¼ V=c is its velocity
in units of c, ¼ ð1 À 2 ÞÀ1=2 , ¼ mi ni ‘ is the mass
density per unit surface, and !0 ¼ !½ð1 À Þ=ð1 þ Þ1=2
is the EM wave (laser) frequency in the rest (sail) frame.
Note that the intensity I is in general a function of the
retarded time tret ¼ t À X=c.
Analytical solutions to Eqs. (31) exist depending on
suitable expressions for Rð!Þ, the simplest case being
Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013 775 that of a perfectly reﬂecting mirror (R ¼ 1) and a pulse of
constant intensity I (Simmons and McInnes, 1993).20 The factor as a function of time is given by ðtÞ ¼ sinhðuÞ þ 1
;
4sinhðuÞ 1
u asinhð3t þ 2Þ; (32)
3 where ðZme a2 =Amp Þ! and has been deﬁned in
0
Eq. (5). Asymptotically, ðtÞ ’ ð3tÞ1=3 [see Fig. 25(a)].
The most signiﬁcant quantities can be obtained for an
arbitrary pulse shape I ðtÞ as a function of the dimensionless
pulse ﬂuence F (the pulse energy per unit surface):
2 Z tret 0 0
I ðt Þdt :
(33)
F ðtret Þ ¼ 2
c 0
The sail velocity , the corresponding energy per nucleon
E ¼ mp c2 ð À 1Þ, and the instantaneous efﬁciency
(i.e., the ratio between the mechanical energy delivered to
the sail and the incident pulse energy)21 are given by ðtret Þ ¼ ½1 þ F ðtret Þ2 À 1
;
½1 þ F ðtret Þ2 þ 1 E ðtret Þ ¼ mp c2
ðtret Þ ¼ F 2 ðtret Þ
;
2½F ðtret Þ þ 1 2ðtret Þ
1
¼1À
:
1 þ ðtret Þ
½F ðtret Þ þ 12 (34) (35) (36) Thus, ! 1 when ðtret Þ ! 1. The ﬁnal energy per nucleon
E max is obtained from the total ﬂuence F 1 ¼ F ðtret ¼ 1Þ.
For a constant intensity F 1 ¼ p , where p is the duration
of the laser pulse. In practical units F 1 ¼ 2:2F1e8 À1 ‘À1 ,
1
10
where F1e8 is the ﬂuence in units of 108 J cmÀ2 , 1 ¼
mi ni =1 g cmÀ3 , and ‘10 ¼ ‘=10 nm. The scalings for E max
are summarized in Fig. 25(b). With present-day or near-term
laser technology, ﬂuence values of 108 J cmÀ3 seem affordable, while target manufacturing can produce ﬁlms of a few
nm thickness, e.g., diamondlike carbon (DLC) foils. These
values yield F 1 > 1 allowing one to approach a regime of
high efﬁciency, relativistic ions, and favorable scaling with
the pulse energy.
The above estimates have been obtained assuming a perfectly reﬂecting sail (R ¼ 1) that, for a given surface density
parameter , limits the laser amplitude to a0 < due to the
onset of relativistic transparency [Eq. (5)] that reduces the
boost on the foil. This effect suggests a0 ¼ as an ‘‘optimal’’
condition for LS acceleration (Macchi, Veghini, and
Pegoraro, 2009; Tr...

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