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A set of mode-coupled excitation equations for the slowly-growing amplitudesof dielectric waveguide eigenmodes is derived as a description of theelectromagnetic signal field of a high-gain free-electron laser, or FEL,including the effects of longitudinal space-charge. This approach of describingthe field basis set has notable advantages for FEL analysis in providing anefficient characterization of eigenmodes, and in allowing a clear connection tofree-space propagation of the input (seeding) and output radiation. Theformulation describes the entire evolution of the radiation wave through thelinear gain regime, prior to the onset of saturation, with arbitrary initialconditions. By virtue of the flexibility in the expansion basis, this techniquecan be used to find the direct coupling and amplification of a particular mode.A simple transformation converts the derived coupled differential excitationequations into a set of coupled algebraic equations and yields a matrixdeterminant equation for the FEL eigenmodes. A quadratic index medium is usedas a model dielectric waveguide to obtain an expression for the predicted spotsize of the dominant system eigenmode, in the approximation that it is a singlegaussian mode.

Virtual dielectric waveguide mode description of a high-gain free-electron laser I: Theory