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A formal eigenmode method to solve for electromagnetic fields in a longitudinally translationally invariant multizone dielectric-lined waveguide is presented. The method is specialized to the development of wakefield theory for rectangular dielectric-lined structures which have an arbitrary number of dielectric layers. It is shown that the fields excited by a drive particle moving through the vacuum beam channel in such a structure can simultaneously include both propagating and decaying modes. The decaying modes characterize the short-range self-fields that move together with the particle, while the propagating modes characterize the long-range radiation fields or wakefields that carry energy away from the particle. It is also shown that the formal solution obtained in rectangular structures is applicable to all translationally invariant dielectric-lined waveguides, for example, cylindrical structures. Two important identities which the method relies upon are computationally confirmed for rectangular two-zone dielectric-lined structures. This theory may be employed for calculations of space charge effects, particularly for low or moderate-energy beams where self-field effects cannot be neglected

Theory for wakefields in a multizone dielectric lined waveguide