# Introducing EigenModes from 2D FEM into 3D FEM

This particular FEM question concerns waveguides and FEM 3D simulation. To excite a waveguide with waveport (TE10 and so on), we typically have to solve for eigenvalues ($k$) of helmholtz equation using FEM 2D.$$\bigtriangledown^{2}\phi_{2D}+k^{2}\phi_{2D}=0 ,\quad\phi_{2D}=E_{z}(TM)\quad or \quad H_{z}(TE)$$ Upon solving, $k$ gives the permissible modes for the particular waveport dimensions and $\phi_{2D}$ gives the field patterns. I have got everything working and understood till this point but I am not sure of how to include these results into FEM 3D simulation. This document from researchgate (pdf downloads on clicking) says that the mode patterns $\phi_{2D}$ found using FEM 2D are used as the excitation for FEM 3D. However, I dont quite understand how to include this source into FEM 3D. What I think I must do is incorporate $\phi_{2D}$ (FEM2D) into the in-homogeneous helmholtz equation for FEM 3D as the source $g$ as given below.$$\bigtriangledown^{2}\phi+k^{2}\phi=g$$ But, I am not quite sure if this is correct. I haven't found good sources for waveports either.

• The 2D solution can be used as a Dirichlet boundary condition on the input waveport Jan 1 '18 at 18:43
• @Steve so you mean like assigning nodal values of 3D problem based on eigenvector of 2D problem? But then I would need to just solve the homogeneous case even for 3D right (g=0) since I would already have the nodal values for $\phi$ on the waveport to start with ? I think i'll give it a try. Out of curiosity, if I don't need "g" for waveports, when is it used? Jan 1 '18 at 18:59
• The way I've seen this done is to introduce a sheet current $\mathbf{J}= \mathbf{\hat{z}} \times \mathbf{H}$, (where $\mathbf{H}$ is your calculated mode) inside the waveguide and terminate the waveguide with a PML. It's important that energy reflected from your geometry back to the waveport is absorbed, not re-reflected back inside, which is why the PML (or some other absorbing termination) is necessary. For that reason, I don't think a Dirichlet BC is what you want, but I'm happy to be corrected on this point. Jan 1 '18 at 22:40

I tend to think about this formulation almost like a domain decomposition approach, where in the main computational domain you expand ${\vec E}$ in subsectional FEM basis (Nedelec) and in the adjacent "port" or "infinite waveguide" domain, you expand ${\vec E}$ in a basis of waveguide modes. The modal solution ${\vec E_{2D}}$ plays two roles, it gives you the "shape" of the excitation currents ${\vec J}$/${\vec M}$, but it also provides the change-of-basis coefficients to establish tangential field continuity between the two domains. You don't need to use PML to match, the "absorption" behavior is built into the modal expansion (in fact, I would advise against using PML here, it's less accurate for modes near cutoff and doesn't absorb evanescent waves, but terminating with a modal expansion can handle both of these).