High-power optical beam equations for a free-electron-laser oscillator cavity

The three-dimensional optical pattern generated in a single pulse by a high-power free-electron-laser system when the e^--beam radius is small compared to the laser-beam radius is shown to be determined by the solution of a simple one-dimensional integral equation. This strong-diffraction-limit equation is of the convolution form and, while it is mostly easily solved numerically, it is shown to have nontrivial analytic solutions of interest.