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Feasibility of ionization injection for Direct Laser Acceleration (DLA) ofelectrons up to hundreds of MeV was studied analytically. Criteria foreffective injection determining range of background and in-channel plasmaparameters, laser intensity, etc. were found using phase portraits of thesystem deduced from the simplified analytical model. The found optimaltrajectory in the phase space corresponds to the electron with low (few eV)initial energy experiencing $\sim$10$^9$ times energy gain. For this to occur,electron density should be a few percent of the critical density, while thein-channel electron density should be $\sim$3 times lower. The analyticallyobtained dependence of the energy gain on the initial electron longitudinal andtransverse momenta corresponds well to the results of exact numericalsimulations of an electron motion in the plasma channel. Developed approach canform the basis for further studies of electron injection in DLA varying plasmaand laser parameters as well as initial electron energies.

Phase space consideration of low energy electron injection for Direct Laser Acceleration