electron mass, and cβ is the transverse electron velocity. It also indicates that the FEL mechanism can be designed to operate over a large range of wavelengths, from centimeters to nanometers. The actual feasibility of operating in a given wavelength range depends on several factors, including the FEL gain and electron-beam quality.

Figure 7 is a schematic of an FEL, with the periodic undulator field indicated by the alternating up and down arrows. The typical peak magnetic field in the undulator is a few kilogauss with a wavelength of λ 0 ≈ several centimeters over N ≈ 100 periods. The length of the undulator is several meters inside the optical resonator mirrors, which are separated by about twice the undulator length, or about 10 meters. The electron-beam path is indicated horizontally. The beam energy can vary from a few MeV to several GeV, although the typical FEL uses an electron-beam energy of around 50 MeV. The transverse dimensions in the FEL schematic are exaggerated with respect to the longitudinal distance shown. The electron-beam radius is typically about 1 mm, and the optical mode waist is about 3 mm. The electron-beam pulse width can be as long as several microseconds for a recirculating electrostatic accelerator. In a radio frequency (RF) accelerator or storage ring, the electron-beam structure is determined by the RF frequency. In an RF accelerator, only those electrons with the correct phase relation in the RF cycle continue to be accelerated. The resulting accelerated electron current is produced in short micropulses of a few picoseconds. The minimum separation between micropulses is one RF cycle, but often there is a much larger separation of many nanoseconds. The stream of micropulses continues for about 10 µs up to about 1 ms because the RF power source is pulsed. This stream of micropulses is called a macropulse. In a storage ring using lower-frequency RF, the micropulse length is typically a nanosecond in length separated by several RF cycles. There is no macropulse in a storage ring. The radiation pulse structure in an FEL reflects that of the electron beam.

FIGURE 7 Schematic of a free electron laser.

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