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1 29 Paraxial wave equation The laser fields of the pump (i = 0) and injection (i = 1) laser pulses are described by the normalized vector potentials ai = eAi /me c2 . c denotes the complex conjugate. In general this integral cannot be solved analytically. However for laser pulses with a slowly varying envelope compared to the carrier wavelength λ0 , there exists a set of gaussian modes that are exact solutions of Maxwell’s equation. 7) 1) group velocity of a laser pulse in a plasma, taken along the propagation axis.

Normalized vector potential ai , wakefield Ez /E0 , transverse momentum uy and axial momentum uz is shown before collision. 33) As an example, analytical solutions for φb can be found in the linear limit (for the case of square pulse profiles, without diffraction) and equal frequencies ∆ω = 0. 2. 2: Normalized wakefield Ez /E0 and density variation δn/n0 is shown during head-on collision and for the parameters of Fig. 1. Note that the resolution within one period of the density inside the beatwave is sampled by 20 grid points and 100 macroparticles per cell.

3 a ˆ0 a ˆ1 . 4. 25, βb = 0, and L0 = λp . Chirikov island overlap criterion) [59]. 48) ub− ≤ uz (H = 1) . 49) If this occurs, then there exists a phase space path that can take an electron from the cold fluid orbit, through the beat wave separatrix, and finally on a trapped orbit within the 2D separatrix of the wakefield. The trapping threshold can be solved analytically. In the limit βb 1, Eqs. 50) where β∗ = √ [u2z (Hf ) − u2z (H = 1)] / 2 uz (Hf ) 2 + 2ˆ a20 + u2z (Hf ) − uz (H = 1) 2 + 2ˆ a20 + u2z (H = 1) .