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High-power microwave generation via a relativistic electron beam propagating through a periodic focusing quadrupole transport (PFQT) system is studied. Transverse confinement of e-beam in a PFQT system is provided by segmented cylindrical electrodes with applied oscillatory voltages Vosc(t) over 90° segments. With regard to transverse confinement, there is no applied axial guide magnetic field. An analytic theory of laser gain in the low-gain-per-pass limit is developed. Numerical computations of electron trajectories and gain are also presented. It is shown that the presence of transverse oscillation frequency due to PFQT leads to significant gain enhancement under appropriate conditions.