Nested 2D finite-element function-spaces formulation for the mode-matching problem of arbitrary cross-section waveguide devices

A large class of microwave, millimeter-wave and terahertz waveguide devices for high-frequency electronic systems are made up of waveguide steps cascaded along the propagation direction, giving rise to diverse modal and numerical analysis techniques to solve Maxwell equations for this problem. In this paper, a novel formulation is proposed to compute the numerical modes in all arbitrary cross-sections and characterize all waveguide steps involved in this kind of structures with a modular and straightforward approach through block-matrix operations. The key idea is expressing the modal fields in terms of 2D nested function spaces (each one for a different cross-section) made up of finite-element basis functions. This leads to finite-element matrices used to compute the modes in all different arbitrary waveguides of the structure from a single inter-cross-section conforming 2D mesh. Moreover, these finite-element matrices and results are used to build directly the mode-matching solution of all steps in the structure. After comparison with analytical results for canonical steps, this flexible and efficient approach is validated with various examples of waveguide devices (two filters, a polarizer, a transformer and a polarization rotator), showing excellent agreement with other numerical methods and measurements.