Modal series expansion of eigensolutions for closed and open periodic waveguides

An efficient approach is presented to compute the eigensolutions of closed and open radiating periodically loaded waveguides by modal series expansion. The eigensolutions of the homogeneous background waveguides serve as basis functions for the fields in the periodic boundaries. The corresponding scattering matrix of an isolated periodic unit cell is determined by commercial software solvers. The scattering matrix is transformed into the transfer matrix, which can be used to formulate an eigenproblem. Open field problems are converted into a closed equivalence while the characteristics of the original open configurations are preserved, even for radiating (leaky) waveguiding structures. Consequently, the original continuous spectrum of modes is represented by a discrete and proper set of modes. An eigenproblem must still be solved but it is considerably reduced in size and complexity for configurations where relatively few expansion modes are sufficient.