Abstract
In a convenient resonator for a two-stage FEL, the cross section is smaller at the second stage than at the first in order to increase the intensity of the pump wave interacting with the electrons. Also, a flared shape is used to decrease mirror loading. By shaping the flares in the form of hyperboloids, the fields configurations can be obtained, checking that mode conversion does not become a problem. Use is made of an oblate speeroidal geometry and a perturbation expansion for the wave equation solutions.
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