Counting 1D free electron laser growing modes in the presence of space charge

Knowing the number and structure of the free electron laser's (FEL) growing modes is of fundamental importance to its theoretical description and its applications. The one-dimensional FEL dispersion relation, including the space-charge effects, is well established in the FEL theory, but questions remain about the number of the growing solutions (modes). In this paper, we provide the definite answers to the latter question and identify the amplification bandwidth of these modes.