  Feasibility of ionization injection for Direct Laser Acceleration (DLA) of
electrons up to hundreds of MeV was studied analytically. Criteria for
effective injection determining range of background and in-channel plasma
parameters, laser intensity, etc. were found using phase portraits of the
system deduced from the simplified analytical model. The found optimal
trajectory 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 the
in-channel electron density should be $\sim$3 times lower. The analytically
obtained dependence of the energy gain on the initial electron longitudinal and
transverse momenta corresponds well to the results of exact numerical
simulations of an electron motion in the plasma channel. Developed approach can
form the basis for further studies of electron injection in DLA varying plasma
and laser parameters as well as initial electron energies.
