Optical guiding in a sheet-beam free-electron laser

The influence of electron density gradients on optical guiding in free-electron lasers (FEL’s) is considered. We study a model problem of a FEL which employs a sheet electron beam in the absence of a cavity or a waveguide. We calculate the gain and the rate of optical guiding by solving for the eigenvalues and the actual eigenmodes of the system. Similar to what has been found by Moore for a cylindrical beam [Nucl. Instrum. Methods A239, 19 (1985)], we find that two parameters characterize the interaction: a coupling parameter and a detuning parameter. We solve the problem for two density profiles, a uniform density profile and a triangular-shaped density profile. For large and small values of the coupling parameter we obtain the results analytically. For intermediate values of the coupling parameter we obtain numerical results. When the coupling parameter is small, diffraction is large. The gain and the wave profile are then similar for the two density profiles. When the coupling parameter is large, optical guiding is large. The gain for the triangular-shaped beam is then larger by $2^{1/3}$ than the gain for the uniform beam. When the coupling parameter is large, the wave profile for the uniform density beam converges to a certain profile confined to the beam volume. For the triangular-shaped beam the wave profile becomes more and more concentrated near the midplane of the sheet beam. For the solution of the problem we derive an energy integral that determines the domain in the complex plane where nonreal eigenvalues are allowed. We also show the existence of an accumulation point of nonreal eigenvalues in certain cases.