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Eigenvalues of periodically constituted waveguides with complicated material distribution are efficiently determined, where eigensolvers in software packages show difficulties in finding the eigensolutions. Moreover, the treated waveguide structures are open, causing even more trouble with available numerical solvers or rather demanding procedures within analytical approaches. Since open configurations are essential for the evaluation of radiating structures, we suggest to convert the open problem setting into a closed equivalent configuration but without disturbing the actual field behaviour. Thus, the original continuous mode spectrum is turned into a discrete and proper set of eigenmodes. The field solution of the periodically modified waveguide is obtained by series expansion with the eigensolutions of the homogeneous background waveguide as basis functions. An eigenproblem is still to solve, but it is considerably reduced in size and easy to solve by the use of a-priori information.

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Abstract-Periodic eigenproblems describing the dispersion behavior of periodically loaded waveguiding structures are considered as resonating systems. In analogy to resonators, their eigenvalues and eigensolutions are determined by solving corresponding excitation problems directly in the domain of the eigenvalue. Arbitrary excitations can be chosen in order to excite the desired modal solutions, where in particular lumped ports and volumetric current distributions are considered. The method is employed together with a doubly periodic hybrid finite element boundary integral technique, which is able to consider complex propagation constants in the periodic boundary conditions and the Green's functions. Other numerical solvers such as commercial simulation packages can also be employed with the proposed procedure, where complex propagation constants are typically not directly supported. However, for propagating waves with relatively small attenuation, it is shown that the attenuation constant can be determined by perturbation methods. Numerical results for composite right/left-handed waveguides and for the leaky modes of a grounded dielectric slab are presented.