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One of the challenging issues for analytically modeling electron sources, such as rf photoinjectors, is how to incorporate fully electromagnetic effects which are generated by the electron beam. The main difficulties that arise in finding an analytical solution of the electromagnetic fields are due to the complex shape of the conductor boundary, as well as the complicated structure of the beam density and current. Both of these problems can be handled self-consistently by using an electromagnetic Green’s function method. In this paper, we present a solution to the exact electromagnetic fields, which were derived from the Green’s function, for a simplified electron source conductor geometry, namely a semi-infinite circular pipe with an endcap. We assume that the beam currents are in the axial direction and satisfy the continuity equation in conjunction with the beam charge density, but may have arbitrary spatial and time dependency. We discuss how these analytical methods may be extended to include in the effect of one or multiple irises, which are found in rf photoinjector systems.
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