Single-mode efficiency in free-electron lasers

For free-electron laser (FEL) devices with dilute and cold electron beams, an expression for the single-mode efficiency, ∊, is defined. The connection with the full (quantum mechanical) gain, G, is such that ∊/G is equal to the ratio of the total photon energy to the kinetic electron-beam energy “trapped” in the interaction volume V. Both ∊ and G are discussed quantitatively, using the example of a quantum model of the Smith-Purcell FEL: it is shown that unlike G, ∊ depends on the total photon energy trapped in V even in the classical regime.