3D free electron laser gain for an electron beam with finite emittance and energy spread

A generalization of the analytical 3-D gain derivation for an arbitrary free electron laser (FEL) with a Hermite-Gaussian TEM wave is presented for the low gain, small signal regime. The finite emittance and energy spread of the electron beam as well as the orbit distortions and misalignments are rigorously taken into account. The paraxial approximation is used both for the electron beam and the optical wave. The expression for FEL gain is analytically reduced from an integral over eight dimensions to a two dimensional integral along the FEL. Some peculiar FEL modes are used for illustration of this method.