to gain and others to lose energy to the optical field via the ponderomotive force. This energy modulation evolves into electron density (current) modulations with a period of one optical wavelength. The electrons are thus longitudinally clumped into microbunches, separated by one optical wavelength along the axis. Whereas an undulator alone would cause the electrons to radiate independently (incoherently), the radiation emitted by the bunched electrons is in phase, and the fields add together coherently. The radiation intensity grows, causing additional microbunching of the electrons, which continue to radiate in phase with each other. This process continues until the electrons are completely microbunched and the radiation reaches a saturated power several orders of magnitude higher than that of the undulator radiation. The wavelength of the radiation emitted can be readily tuned by adjusting the energy of the electron beam or the magnetic-field strength of the undulators. FELs are relativistic machines. The wavelength of the emitted radiation, , is given by or when the wiggler strength parameter , discussed below, is small where is the undulator wavelength (the spatial period of the magnetic field), is the relativistic Lorentz factor and the proportionality constant depends on the undulator geometry and is of the order of 1. This formula can be understood as a combination of two relativistic effects. Imagine you are sitting on an electron passing through the undulator. Due to Lorentz contraction the undulator is shortened by a factor and the electron experiences much shorter undulator wavelength . However, the radiation emitted at this wavelength is observed in the laboratory frame of reference and the relativistic Doppler effect brings the second factor to the above formula. In an X-ray FEL the typical undulator wavelength of 1 cm is transformed to X-ray wavelengths on the order of 1 nm by ≈ 2000, i.e. the electrons have to travel with the speed of 0.9999998c. Wiggler strength parameter K , a dimensionless parameter, defines the wiggler strength as the relationship between the length of a period and the radius of bend, where is the bending radius, is the applied magnetic field, is the electron mass, and is the elementary charge. Expressed in practical units, the dimensionless undulator parameter is . Quantum effects In most cases, the theory of classical electromagnetism adequately accounts for the behavior of free electron lasers. For sufficiently short wavelengths, quantum effects of electron recoil and shot noise may have to