It is shown that the phase of the electromagnetic wave emitted through stimulated emission is intrinsically random. The insensitivity of the phase of the laser field to any disturbance in the laser cavity parameter derives from the fact that stimulated and spontaneous emissions take place concurrently at the same wave vector, the phases of spontaneous emission are mildly bunched, and the central limit theorem can be applied to the phase of the laser field. The two spectral lines observed in the Smith-Purcell free-electron laser experiment show that both classical and quantum-mechanical free-electron lasings, in which the wigglers behave as classical waves and wiggler quanta respectively, take place concurrently at different laser wavelengths in the case of the electric wiggler. It is shown that the coherence of the classical free-electron laser is achieved through modulation of the relativistic electron mass by the electric wiggler. The classical free-electron lasing is calculated using the quantum-augmented classical theory. In this, the probability of stimulated emission is first evaluated by interpreting the classically derived energy exchange between an electron and the laser field from a quantum-mechanical viewpoint. Then the laser gain is obtained from this probability by using a relationship between the two quantities derived by quantum kinetics. The wavelength of the fundamental line of classical free-electron lasing is twice the wavelength of the fundamental line of the free-electron two-quantum Stark emission, which is the quantum free-electron lasing in the electric wiggler. The gain of the classical free-electron lasing appears to scale as λ3w/γ3, where γ is the Lorentz factor of the electron beam and λw is the wavelength of the wiggler.
Structure determination of molecules in an alignment laser field by femtosecond photoelectron diffraction using an X-ray free-electron laser