A theory is developed for lasing action from ensembles of relativistically or subrelativistically propagating emitters whose motion is bound in the direction(s) transverse to the direction of propagation. These include relativistic electrons and positrons channeled in crystals or other hollow-channel structures, as well as fast ions wherein a bound electron is perturbed by the crystal potential or laser light. Apart from planar-channeled positrons in crystals, the confining potentials for all other emitters in this category are strongly anharmonic. Therefore, their spectral dipolar transitions are nondegenerate, each involving a different pair of nearly discrete levels of the confining potential. This implies that stimulated emission from such systems can exhibit coherence in the Glauber sense. The theoretical framework presented here consists of Heisenberg equations which have the Maxwell-Bloch form with modifications resulting from the high velocity of the emitters. Steady-state semiclassical solutions of these equations are obtained. It is shown that previous approaches, based on the assumption that the cross section for stimulated emission is uniform throughout the system, do not account for the spatial variation of the polarization at high velocities. As a result, these approaches do not yield the correct gain coefficient whenever the characteristic lengths for the dephasing of the dipole oscillation and for emission amplification are comparable. The latter conditions are realizable in structures composed of channels much wider than in crystals. Lasing schemes are investigated and the prospects for achieving gain in these schemes at wavelengths below 100 A ̊ are discussed.
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