  In this paper, we determine the electron beam quality requirements to obtain
exponential radiation amplification in the ion-channel laser, where a
relativistic electron beam wiggles in a focusing ion-channel that can be
created in a wakefield accelerator. The beam energy and wiggler parameter
spreads should be limited. Those spread limits are functions of the Pierce
parameter, which is calculated here without neglecting the radiation
diffraction. Two dimensional and three dimensional simulations of the
self-consistent ion-channel laser confirm our theoretical predictions.
