Transverse mode frequency pulling in free electron lasers

We consider the spectral evolution of transverse modes in free electron lasers. While the gain maxima of the various transverse modes occur at different frequencies, coupling between the modes via the electron beam causes the modes to shift frequency such that the final laser frequency differs from what one would expect if only the TEM ₀₀ mode were present. This can strongly effect the turn-on behavior in lasers with short electron pulses.