Propagation of ultrastrong femtosecond laser pulses in PLASMON-X

In this work we present the derivation of the nonlinear equations that describe the propagation of ultrashort laser pulses in a plasma, in the PLASMON-X device, using a fully relativistic hydrodynamic description for electrons. It is shown that for the PLASMONX scheme used for the electron acceleration, it is justified to use a stationary 1-D approximation in the electron hydrodynamic equations, since the pulse width is sufficiently bigger than the pulse length. Furthermore, with the laser power of $W\leq 300$ TW and the initial $130\,\,\mu{\rm m}$ spot size, the nonlinearity is sufficiently weak to allow for the power expansion in the nonlinear Poissons's equation, yielding a version of the nonlocal nonlinear Schr\"{o}dinger eqiation, with a periodic nonlocality. While in a one-dimensional limit the standard wakefield generation is obtained, our two-dimensional numerical studies, including the full nonlinear response, reveal the transverse collapse (or the self-focussing) of the pulse. Under the typical operating conditions, the self-focussing is sufficiently slow to allow the interaction between the laser pulse and the accelerated electrons along an interaction length (in the laboratory frame) that exceeds 1 m.