Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma

We analyze the propagation of a short intense laser pulse in underdense cold plasma. When no electron cavitation is present, a global invariant H is obtained, and its relation with self-focusing is studied. For relativistic self-focusing, H<0 is a sufficient and necessary condition. For relativistic and ponderomotive self-focusing, H<0 is sufficient but not necessary. Numerical simulations are performed to confirm the above points.