Self-trapping and acceleration of ions in laser-driven relativistically transparent plasma are investigated with the help of particle-in-cell simulations. A theoretical model based on ion wave breaking is established in describing ion evolution and ion trapping. The threshold for ion trapping is identified. Near the threshold ion trapping is self-regulating and stops when the number of trapped ions is large enough. The model is applied to ion trapping in three-dimensional geometry. Longitudinal distributions of ions and the electric field near the wave breaking point are derived analytically in terms of power-law scalings. The areal density of trapped charge is obtained as a function of the strength of ion wave breaking, which scales with target density for fixed laser intensity. The results of the model are confirmed by the simulations.

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By combining Eqs. (6), (8), and (10)(12), two integrals are obtained, βph(γeβe/μ+γiβi)(γe/μ+γi)=const1. and Ẽ2/2+βph(γeβe/μ+γiβi)=const2. By applying them at the point where βe=βe,max=βph,βi=βi,min, and Ẽ=0, and the point where βe=βi=0,Ẽ=Ẽmax=ẼEWB, one has γi,m=γph2(C1βphC121/γph2),ẼEWB=2(γph1)(1+C2)/μ, where γi,m=1/1βi,min2,C1=1+1/μ1/(μγph), and C2=μ(γi,m1)/(γph1). When μ ≫ 1, one gets C1 = 1, γi,m = 1 and C2 = 0. Then Eq. (13) is obtained.

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See, e.g., Ref. 6e.

30.

In the hole-boring regime,5 all ions in the laser focus are accelerated, therefore, the condition of relativistic transparency discussed in Ref. 18 breaks down, and the plasma becomes opaque.

31.

In a reflection process one has Ji+Jtrap=0, where Ji and Jtrap denote the incoming and reflected current densities in the frame comoving with the laser front, respectively. It is valid in the region where Jtrap0. By using the Lorentz transformation one gets Ji=γf(Jivfqini) and Jtrap=γf(Jtrapvfqintrap), where Ji=qivini and Jtrap=qivtrapntrap are current densities in the laboratory frame. Then the equation ñi(βiβf)+ñtrap(βtrapβf)=0 is obtained.

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The coefficients in Eqs. (43)(46) and (53)(55) may change if this assumption breaks down, however, it is easy to find that the power-law exponents in these equations are still the same.

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