Hamiltonian analysis of electron self-injection and acceleration into an evolving plasma bubble

Injection and acceleration of the background plasma electrons in laser wakefield accelerators (LWFA) operated in the blowout ('bubble') regime are analysed. Using a model of a slowly expanding spherical plasma bubble propagating with an ultra-relativistic speed, we derive a sufficient condition for the electron injection: the change in the electron's Hamiltonian in the co-moving with the bubble reference frame must exceed its rest mass energy m(e)c(2). We demonstrate the existence of the minimal expansion rate of the bubble needed for electron injection. We demonstrate that if the bubble's expansion is followed by its stabilization or contraction, then a quasi-monoenergetic electron beam can be produced owing to the phase space rotation of the beam inside the bubble. Using particle-in-cell simulations, we verify that the temporal expansion of the bubble is indeed the dominant effect responsible for electron self-injection and trapping in the rarefied plasmas relevant to LWFA with petawatt-class lasers.