Optical gain, phase shift, and profile in free-electron lasers

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Abstract
The gain, phase shift, wave-front curvature, and radius of the radiation envelope in a free-electron-laser amplifier are obtained in the small-signal regime. The electron beam is assumed to have a Gaussian density distribution in the transverse direction. Numerical calculations indicate that the radius and curvature of the radiation beam entering a wiggler asymptote to unique spatially constant values after a finite transition region. However, in the asymptotic region the wave fronts are divergent. Analytical expressions for the gain, phase shift, curvature, and spot size are derived. It is shown analytically that small perturbations of the optical waist and curvature about the matched value are spatially damped out, indicating the stability of the matched envelope. When the electron-beam envelope is modulated in space, the optical spot size oscillates with an almost identical wavelength but is delayed in phase. In the case of small-amplitude long-wavelength betatron modulation of the electron-beam envelope, generation of optical sidebands in wave-number space is examined and the effect on the dispersion characteristics of the primary wave is found to be negligible for typical experimental parameters.

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