The general solution of the eigenvalue problem for a high-gain FEL

The exact solution of the eigenvalue equation for a high-gain FEL derived in Xie (Nucl. Instr. and Meth. A 445 (2000) 59) is generalized in order to include the space charge effects. This solution is valid not only for natural undulator focusing, but also for alternating-gradient focusing under some condition that is presented. At such, the obtained solution includes all the important effects in the system of axially homogeneous electron beam and undulator: diffraction, betatron motion, energy spread, space charge and frequency detuning. It is valid for ground TEM00 mode as well as for high-order modes and can be used for calculation of high-gain FEL amplifiers operating in the wavelength regions from far infrared down to X-ray. In addition, a computationally efficient approximate solution for TEM00 mode is derived providing high accuracy (better than 1% in the whole range of parameters). It can be used for quick optimization of FEL amplifiers.