“…Taking <β x >~1/2, γ~1, <n(x)>~(α+1)n 0 /2 and <r>~r i /2 during the injection phase, one obtains <Ω>≈-5e<B ϕ >/4m≈-5e 2 (α+1)n 0 r i /64mε 0 , thus Δθ s~< Ω>~ΔT~-ηr i |r i | with η=5e 2 (α+1)n 0 /16mc 2 ε 0 . For simplicity, we assume a homogeneous electron distribution within the cylindrical injection volume of |r i |≤r b (x p ) [27][28][29], where r b (x p ) is the bubble radius at the density peak, satisfying r b 2 (x p )~4σ l σ r (n b0 /αn 0 ) 1/2 for PWFA [42,44] and r b 2 (x p )~4amc 2 ε 0 /αe 2 n 0 for LWFA [51]. Therefore, combining with the fact that ϕ~ϕ p , the polarization at certain ϕ p becomes…”