A virtual dielectric eigenmode expansion of high-gain fel radiation for study of paraxial wave mode coupling

An analytic formulation that uses eigenmodes of a dielectric wave guide to describe the signal field of an FEL is presented. This formulation can provide an efficient characterization of the FEL self-similar eigenmodes and enables a clear descriptive connection to free-space propagation of the input and output radiation. The entire evolution of the radiation wave through the linear gain regime is described with arbitrary initial conditions. By virtue of the flexibility in the expansion basis, this technique can be used to find the direct coupling and amplification of specific modes of interest. A simple transformation converts the derived coupled differential evolution equations into a set of coupled algebraic equations and yields a matrix determinant equation for the FEL eigenmodes. Laguerre-Gaussian modes used as an expansion basis allows investigation of coupling and amplification of optical modes that contain orbital angular momentum, suggesting new regimes of operation for future FELs.