Solving Periodic Eigenproblems by Solving Corresponding Excitation Problems in the Domain of the Eigenvalue

Abstract

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.

Dispersion Analysis of Complex Periodic Structures by Full-Wave Solution of Even-Odd-Mode Excitation Problems for Single Unit Cells

“…Due to symmetry, the field problem in the interval is completely determined if the even and odd fields are known for . Thus, the total fields in this interval are given by the fields obtained by the two simulations according to (14) and (15) The fields given in (14), (15) together with Bloch's theorem and the propagation constant determined in (11) allow to calculate the field distribution of an arbitrarily long waveguiding structure.…”

“…In this case, the structure is excited by a source and the terminal or field data are observed. Frequencies at which the observables become singular represent the solution for the particular phase of the boundary condition [14]. Although this concept is flexible regarding the non-periodic boundary conditions, the singularities must be found in the complex plane by a time consuming iterative process and fully complex boundary conditions are again required to find complex propagation constants.…”

Dispersion Analysis of Double-Sided Open Periodic Media Using Inhomogeneous Plane Wave Excitation

“…Another robust method is trying to find the stationary points of stored energy derived from physical observations. Since eigenvalue decomposition is a numerically expensive procedure, a physics based method for the computation of complex and surface wave modes of planar periodic configurations was addressed in [9,10]. This method was based on the scattering of inhomogeneous plane waves from grounded unit cells and the eigenmodes were explored by exciting the poles of the reflection coefficient.…”

Driven eigenproblem computation in periodic structures accelerated with Brent's method

“…In former researches, full-wave modelling methods based on classical numerical methods like finite-difference time-domain [1], boundary integral-resonant mode expansion [2], finite-difference frequency-domain [3] or method of moments [4] always lead to very slow convergence and often low accuracy. Compared to the purely mathematical approaches of the common eigenproblem solvers, the driven eigenproblem solution treats the eigenproblem from a physical point of view by converting it into an excitation problem and simplifying the eigenvalue determination by analogising the periodic structure to a resonator model [5,6]. According to Floquet's theorem, a periodic structure can be modelled as one appropriately discretised unit cell with periodic boundary conditions with a specific phase shift.…”