Support for Floquet/Bloch Ports in Periodic Geometries

[krono-i2]

I'd like to propose an enhancement related to the simulation of periodic structures with oblique plane-wave excitation.

Palace already supports periodic (Bloch) boundary conditions, but currently there is no "Floquet Port" implementation for modal excitation and modal analysis on periodic surfaces.

In commercial FEM/FDTD solvers (CST, HFSS, COMSOL), a Floquet Port provides two key capabilities:

1. It solves the transverse eigenvalue problem on the periodic face (Floquet eigenmodes, with know. eigenvalues).
2. It allows the user to excite a specific Floquet mode (typically the fundamental one) to model oblique plane-wave illumination and to compute meaningful S-parameters for propagating and evanescent diffraction orders.

This feature is essential for:

• periodic arrays,
• metasurfaces and metagratings,
• RCWA-like structures,
• periodic lattices under arbitrary incidence,
• first-Brillouin-zone analysis.

Without a Floquet port, users can impose Bloch phase shifts on the lateral boundaries, but cannot properly define modal excitation nor extract modal transmission/reflection coefficients.

Proposal
I suggest adding a "FloquetPort" or "PeriodicPort" module that would:

• compute Floquet modes (propagating + evanescent) for a periodic surface,
• associate them with user-specified Bloch vectors,
• enable excitation of a selected mode,
• return S-parameters for different diffraction orders.

This feature would significantly improve Palace's capabilities for periodic EM problems.

Thanks for your work and consideration!

[hughcars]

Hi @krono-i2,

I think this sounds like some kind of fusion between a waveport (which is solving a 2D eigenmode problem on the boundary, and then exciting modes for the 3D simulation on the boundary), and a Floquet periodicity (which changes the time harmonic solution ansatz from $E(x,y,z,t) = E(x,y,z) e^{i\omega t}$ to $E(x,y,z,t) = E(x,y,z) e^{i\omega t + i \vec{k} \cdot \vec{x}}$. Floquet periodicity within Palace is not really a boundary condition per se, it's this transformation imposed on the solution space, combined with "perfect periodic" boundary conditions on the remainder field.

If you have a paper which has a simulation involving this type of boundary condition that I could take a look at, I might be able to better appreciate how these pieces all interact with each other.

[krono-i2]

Hi @hughcars, thanks for your kind feedback!

As a reference for the formulation, here is an open-access FEM paper that explicitly uses Bloch/Floquet boundary conditions by introducing the Bloch-modified differential operators:

D. Boffi, M. Conforti, L. Gastaldi,
Modified edge finite elements for photonic crystals, Numerische Mathematik 105, 2006.
PDF (open access):
https://art.torvergata.it/retrieve/dcd6909b-2c6a-c5ab-e053-3a05fe0aa144/boffi-conforti-gastaldi.pdf
In this work the Bloch representation is built into the FEM variational formulation via
$\nabla_\alpha = \nabla + i\alpha$
with $\alpha$ being the Bloch wavevector, and the eigenproblem is solved on a single periodic cell using edge elements.
This is essentially the same mathematical structure used in the implementation I’m proposing.

For convenience, I’m also attaching my notes where I derive the general Bloch/Floquet representation on an arbitrary periodic plane and the resulting Floquet-mode expansion:
Notes on Bloch-Floquet.pdf

[hughcars]

D. Boffi, M. Conforti, L. Gastaldi, Modified edge finite elements for photonic crystals, Numerische Mathematik 105, 2006. PDF (open access): https://art.torvergata.it/retrieve/dcd6909b-2c6a-c5ab-e053-3a05fe0aa144/boffi-conforti-gastaldi.pdf In this work the Bloch representation is built into the FEM variational formulation via ∇ α = ∇ + i α with α being the Bloch wavevector, and the eigenproblem is solved on a single periodic cell using edge elements. This is essentially the same mathematical structure used in the implementation I’m proposing.

That link you provide is broken, but looking at the title and finding it elsewhere, they are using the same notational method used in Palace to derive Floquet conditions, see #125 (comment) and the discussion that follows. That mapping of the grad operator seems to match the notation from Boffi, and your alpha is the floquet vector provided in the configuration interface....? I'm not seeing the linear frequency dependency that you seem to be suggesting, it looks like the standard Floquet condition to me.

[krono-i2]

Let consider an incident plane wave; the incident electric field will be:
$\mathbf{E}(\mathbf{r})=\mathbf{E}_i e^{-j\mathbf{k}_i \cdot \mathbf{r}}$
Where $\mathbf{k}_i=k_0 \hat{\mathbf{k}}_i$ is the incident wave vector, with $k_o = \omega/c_0$ (here is the linear frequency dependency) and $\hat{\mathbf{k}}_i$ indicates the direction of the incident plane wave.

If you assume “almost” periodic behaviour in the plane (x,y) you have that the Bloch-Floquet vector is
$\mathbf{k}_F = \mathbf{k}_i - \mathbf{k}_i \cdot \hat{\mathbf{z}}$

So, also the Bloch-Floquet vector is linearly dependent on frequency.

[krono-i2]

@hughcars, let me know how I can collaborate with the project in order to help building on that topic.

[hughcars]

Hi @krono-i2,

We're discussing this proposal internally presently, (somewhat delayed by the holiday period). I will get back to you with an answer once we have decided what we will do.