The requirements of high electron-beam current and quality are crucial in determining the ultimate performance of an FEL. The electron-beam kinetic energy (γ− 1)mc2 can range from a few MeV to several GeV, while the beam current can range from a few amperes to several kiloamperes. The FEL is able to obtain large peak power, on the order of a few gigawatts, because, unlike conventional lasers or microwave tubes, the energy that is not transferred to the radiation field remains in the relativistic electron beam that is transported out of the undulator at nearly the speed of light, or even recovered to improve overall efficiency. The FEL interaction volume contains only light, the undulator magnetic field, and the electron beam, so that unwanted high field effects and thermal distortion of the medium are absent in the FEL. The small duty cycle of most accelerators limits the average power of FELs. The efficiency of FELs has been demonstrated to be greater than 40% at long wavelengths, but most will operate at a lower efficiency of a few percent.

The classical gain in the low-gain FEL oscillator develops from coherent electron bunching on the scale of the radiation wavelength. As the electrons travel through the undulator, they accelerate from side to side and spontaneously radiate in the forward direction. The spontaneous emission process is the same as is used to generate UV and x-rays in the undulators and wigglers of synchrotron radiation sources. On the first few passes through the FEL oscillator, some of this spontaneous radiation is stored in the resonator. As the radiation power grows over many passes, the emission process begins to differ from that of the synchrotron sources and becomes similar to the mechanism used in the early microwave electron tubes.

The electrons entering the operating FEL are forced to move from side to side due to the undulator magnetic field, but now radiate in the presence of the stored transverse radiation field. As the radiation passes over the more slowly moving electrons, the radiation electric field does work on the oscillating electrons. This is the classical analog of quantum stimulated emission. A significant energy exchange can occur only when the undulator and radiation field forces on the relativistic electrons are nearly resonant. The electrons in the beam are initially randomly spread over many radiation wavelengths with many millions of electrons within each section of the beam one wavelength long. The orientation of the radiation's electric field vector and the electron's undulator velocity vector rotates through 2π within each radiation wavelength. These different orientations determine that about half the electrons lose energy to the radiation field, while other electrons gain energy from the co-moving wave. The electrons that gain energy begin to move ahead of the average electron, while the electrons that lose energy begin to fall behind the average. As the faster electrons move ahead of the average and the slower electrons move back, the beam is periodically bunched on the radiation wavelength scale. Since the electron energy loss and gain repeats throughout the electron beam over many radiation wavelengths, the beam becomes bunched on the scale of the radiation wavelength. The emission rate for a perfectly bunched beam of electrons is proportional to the square of the number of electrons, whereas the emission rate for a beam of randomly positioned electrons is only proportional to the number of electrons. The same kind of electron bunching is the goal of the microwave electron tube design. The FEL works in a

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