  A set of mode-coupled excitation equations for the slowly-growing amplitudes
of dielectric waveguide eigenmodes is derived as a description of the
electromagnetic signal field of a high-gain free-electron laser, or FEL,
including the effects of longitudinal space-charge. This approach of describing
the field basis set has notable advantages for FEL analysis in providing an
efficient characterization of eigenmodes, and in allowing a clear connection to
free-space propagation of the input (seeding) and output radiation. The
formulation describes the entire evolution of the radiation wave through the
linear gain regime, prior to the onset of saturation, 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 a particular mode.
A simple transformation converts the derived coupled differential excitation
equations into a set of coupled algebraic equations and yields a matrix
determinant equation for the FEL eigenmodes. A quadratic index medium is used
as a model dielectric waveguide to obtain an expression for the predicted spot
size of the dominant system eigenmode, in the approximation that it is a single
gaussian mode.
