A Waveguide Port Boundary Condition based on approximation space restriction for Finite Element Analysis

TL;DR

The paper addresses open-domain truncation in FEM for waveguide problems by introducing a WPBC based on restricting the port approximation space to waveguide modes. This yields a boundary condition that eliminates double boundary integrals and enables per-mode post-processing, including reflected power, while handling lossy and transverse-anisotropic media. It also provides a practical strategy to determine the number of modes needed to achieve a desired reflection accuracy via a mode-projection error metric. Numerical results in 2D and 3D, including a step-index fibre and a plasmonic nanograting sensor, demonstrate high accuracy and favorable trade-offs between reduced domain size and setup cost, highlighting the WPBC as a viable PML alternative for complex waveguide analyses.

Abstract

A Waveguide Port Boundary Condition (WPBC) based on the restriction of the approximation space is presented in the context of Finite Element Analysis. As well as reducing the computational domain in the same manner as the traditional WPBC, the proposed scheme further reduces the degrees of freedom at the waveguide ports, simplifies the implementation and seamlessly provides post-processing results such as the reflected power in each waveguide mode. The boundary condition is thoroughly derived, and numerical examples are used as a support for the discussion on topics such as the needed number of modes to be employed at a waveguide port. Finally, a nanograting-based plasmonic sensor is analysed to illustrate further possibilities of the scheme.

Figures (13)

• Figure 1: Computational domain representing a slab waveguide discontinuity. Dashed rectangles represent PML regions and the propagation direction is $z$.
• Figure 3: Waveguide port illustration at the $-\hat{\bm{z}}$ boundary
• Figure 4: Illustrating the biorthogonality of the first 200 computed modes of a step-index optical fibre surrounded by a cylindrical PML. \ref{['fig:orth-mat']} shows the magnitude of the integrated non-conjugate cross product of the electric and magnetic fields. In \ref{['fig:orth-mat-conj']} one can see that the conjugate biorthogonality does not hold if the system is non-Hermitian.
• Figure 5: Left: A non-conforming $\textrm{H}^1(\Omega)$ mesh presenting a hanging node. Right: The traces of the basis functions that have non-vanishing traces on hypotenuse of the blue triangular element.
• Figure 6: Detail on the slab waveguide mesh used in the validation of the WPBC. The upper part represents the cladding, the lower part the core. Thick lines represents the unidimensional subdomains $\Gamma_{\mathrm{in}}$, the waveguide port, and $\Gamma_{\mathrm{in}}^{e}$, where the solution is evaluated.