Proton acceleration in a slow wakefield

We propose and analyze a mechanism to accelerate protons in a low-phase-velocity wakefield. The wakefield is shock-excited by the interaction of two counter-propagating laser pulses in a plasma density gradient. The laser pulses consist of a forward-propagating short pulse (less than a plasma period) and a counter-propagating long pulse. The beating of these pulses generates a slow forward-propagating wakefield that can trap and accelerate protons. The trapping and acceleration is accomplished by appropriately tapering both the plasma density and the amplitude of the backward-propagating pulse. An example is presented in which the trapping and accelerating wakefield has a phase velocity varying from Vph≈0  to ≈0.15 c   (∼10 MeV   proton  energy) over a distance of ∼1 cm. The required laser intensities, pulse durations, pulse energies, and plasma densities are relatively modest. Instabilities such as the Raman instability are mitigated because of the large plasma density gradients. Numerical solutions of the wakefield equation and simulations using turboWAVE are carried out to support our model.