And mcinnes 1993 are useful to illustrate the most

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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 ފ1=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 ފ2 À 1 ; ½1 þ F ðtret ފ2 þ 1 E ðtret Þ ¼ mp c2 ðtret Þ ¼ F 2 ðtret Þ ; 2½F ðtret Þ þ 1Š 2 ðtret Þ 1 ¼1À : 1 þ ðtret Þ ½F ðtret Þ þ 1Š2 (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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