Hello everone,
I am trying to simulate some nano-photonics device/optical guide. On the input surface (i.e. the boundary on which the light enters the device) the boundary conditions (optical modes) are the solution of a surface eigenvalue problem given by:
\nabla_{\parallel} \times (\nabla_{\parallel} \times E_{\parallel}) - \beta^2 \, \nabla_{\bot} E_{z} + (\beta^2 - k^2) E_{\parallel} = 0
\beta^2 (-\delta_{\parallel} E_{z} + div_{\parallel} E_{\parallel} - k^2 E_{z}) = 0
with f_{\parallel} being only the X and Y component of the respective vector/operator, \beta^2 the eigenvalue and E = (E_{\parallel}, E_z) the eigenvector.
Here all components only depend on x and y but not on z.
I would like to directly compute the solution of the EVP on the respective boundary of the 3-dimensional mesh and then use E_{\parallel} for the boundary values of the main problem.
I looked up tutorial 6.1.2 for the definition of surface PDEs and I'm using the Arnoldi eigensolver accoridng to 1.7.1.
My main question is: Can I acces the 2-dimensional HCurl FE space (only dependend on x and y) for the E_{\parallel} component on a 3-dimensional mesh? Or is there an elegant way to force E_z to be 0?
Currently I'm using a compound FE space consisting of HCurl * H1 (since E_z has more regularity than E_{\parallel}).
This approach was presented
by N. Lebbe page 6.
Thanks for any input
Greetings
Philipp