It is shown using the photon concept that free-electron lasing (or net stimulated bremsstrahlung) is unrelated to the electron phase with respect to the laser wave, while the net acceleration (or net two-photon absorption) in an RF acceleration cavity depends on the electron phase with respect to the RF wave. The gain formula for the free-electron laser using a magnetic wiggler (MFEL) derived using the recently developed quantum-augmented classical theory in which the electron phase is ignored is in excellent agreement with that obtained quantum-mechanically. It is found by means of this theory that if an electric wiggler is added to a MFEL, the synchronization between the transverse velocity and the laser wave, which is required for coherence of the laser light, is not affected, while the laser gain is enhanced owing to the increase in the amplitude of the energy modulation by the electric wiggler. As a configuration of this turbo-MFEL, a two-beam elliptical wake-field cavity is proposed. An electron beam injected in the antiparallel direction along the lasing-beam path in this cavity lases through transverse wiggling by the transverse wake field and energy modulation by the longitudinal wake produced by relativistic drivingbeam bunches. This laser (WFEL) becomes of greater advantage compared with the MFEL as the laser wavelength is made shorter. It is also shown that the amplification of the WFEL is much greater than that of the present MFEL if we can produce a wake field whose longitudinal component has field strength greater than 1 MV m–1.