In a free‐electron laser with a long, linear wiggler, the external focusing required to keep the electron beam from dispersing can seriously degrade the performance of the laser. The transverse focusing modulates the longitudinal velocity of each electron, periodically and nonadiabatically changing the phase of the electron with respect to the electromagnetic wave. Phase changes of order unity over a betatron period can strongly detrap or debunch electrons and greatly reduce the gain of a linear wiggler amplifier. The modulation of the electron’s longitudinal velocity can be prevented if focusing in the plane of the electron’s wiggle motion is provided by parabolically curved magnet pole faces. The focusing and resonance effects of curved pole faces are analytically calculated and numerically confirmed. Numerical simulations of linear wiggler amplifiers are presented to illustrate the effect of the curved pole faces on amplifier performance.