Mode analysis ridge waveguide


I am trying to compute the modes propagating in a ridge waveguide. In other words I am trying to reproduce this: .

I have also attached a screenshot with the equations I want to solve. There, \alpha is the eigenvalue I want to solve for (together with the eigenvectors, which give me the spatial profile of the modes). Since, I am only interested in the transverse profile of the modes, I am working with a 2D-mesh, where it is bounded on all sides by a PML, since the modes can leak out of the waveguide.

My question is now: How do I implement the “constraint” that \mathbf{E}(x,y,z) = mathbf{\tilde{E}}(x,y)e^{-\alpha z} and E_z=0?

Thank you very much!

Screenshot 2023-08-01 at 09.28.44