Variational Principles of Electromagnetic Field Excitation in Waveguides

Interaction of radiation and plasma waves in many electron devices takes place inside an open or closed cylinder (waveguide) of some arbitrary cross-section. A well-known example is the free-electron laser, in which the electromagnetic field interacts with an electron beam in the presence of a periodic magneto-static wiggler field, generating high power coherent radiation. In order to achieve lasing, the radiation is being excited inside a resonator, dictating boundary conditions for both forward and backward waves. Solution of the electromagnetic radiation field inside the resonator, requires simultaneous integration of the coupled excitation equations of forward and backward waves [1]. However, it becomes difficult to accommodate the different boundary conditions for both forward and backward modes in the same numerical integration scheme. Although the radiation power is built gradually in the direction of the electron beam propagation, the natural boundary conditions for the backward waves are given at the end of the interaction region. Thus it is desirable to develop a numerical procedure that allows non-local boundary conditions. We suggest employing variational methods for calculating the total electromagnetic field, including excitation of forward and backward waves. Our developed variational principle is based on formulation of the electromagnetic field action in the space-frequency domain. Further more, the expression for the action is developed utilizing a modal representation of the total electromagnetic field in terms of the eigenmodes of the geometry in which the radiation is excited