t-bar electrode

[vvahidd]

simulate a unit cell with eigen solver, periodic BC, to get impedance and index

https://opg.optica.org/oe/fulltext.cfm?uri=oe-23-11-14263

@mdmaas @dot-cross

[cdaunt]

Other useful T-bar modulator . Multiple ways to extract n_eff

https://api-depositonce.tu-berlin.de/server/api/core/bitstreams/7ad0bec6-fa23-4385-b7b3-f29f32d84714/content

[mdmaas]

I've been checking this out and running some tests on a simple CPW line, which we can use for validation.

We need to use the Bloch-Floquet solver with a certain value of $\phi$ and solve for the eigenmodes. The eigensolver will return the frequency $f_r$ of each mode, at which we can compute the effective index $n_\text{eff}$ and characteristic impedance $Z_0$ in different ways.

Note that if we want $n_\text{eff}$ and $Z_0$ for a given frequency $f$, we need to iterate and find a suitable value of $\phi$. We could either try to guess it, if the exact frequency doesn't matter, or we can run a parametric sweep for various values of $\phi$ and interpolate the results to obtain the exact value.

Now, for obtaining $n_\text{eff}$ and $Z_0$ from Palace in the most direct way possible (without implementing ad-hoc post-processing integration routines along voltage lines or the like) we can use the ratio of electric and magnetic energies that Palace already reports in domain-E.csv:

Effective index:

$$n_\text{eff} = \frac{c \phi}{\omega d L_0}$$

where $\phi$ is the user-prescribed Bloch phase, $\omega = 2\pi f_r$ is obtained from the eigenvalue, $d$ is the unit-cell length in mesh units, and $L_0$ is the mesh-to-SI scale factor. This is exact -- no field integration needed beyond what the eigensolver already computes.

Characteristic impedance:

$$Z_0 = \frac{\omega d L_0}{\phi} \sqrt{\frac{\mathcal{E}_H}{\mathcal{E}_E}}$$

where $\mathcal{E}_E$ = E_elec and $\mathcal{E}_H$ = E_mag from Palace's domain-E.csv. These are the peak electric and magnetic energies integrated over the unit cell volume. The $\sqrt{\mathcal{E}_H / \mathcal{E}_E}$ ratio is equivalent to $\sqrt{L/C}$ (the standard impedance definition), and the prefactor $\omega d L_0 / \phi = \omega / \beta = v_p$ converts per-unit-length quantities to the correct units. The factor-of-2 convention in Palace's energy definition (peak, not time-averaged) cancels in the ratio.

Implementation.

I'm running some tests in a simple CPW configuration for quick validation, as we can compute the impedance in different ways and test convergence, independence of $d$, etc.

I have a small utility function that sets two faces as periodic using gmsh. The key is to use:

gmsh.model.mesh.setPeriodic(2, [ss], [ms], affine)

where affine is the 4×4 affine transformation matrix that maps the master surface to the slave surface.

On the Palace side, the JSON needs a Periodic block with the FloquetWaveVector:

"Periodic": {
    "FloquetWaveVector": [kx, 0.0, 0.0],
    "BoundaryPairs": [
        {
            "DonorAttributes": [<master_pg>],
            "ReceiverAttributes": [<slave_pg>]
        }
    ]
}

where FloquetWaveVector is in radians per mesh unit. So if $\phi$ is the desired Bloch phase advance and $d$ is the cell length in mesh units, then kx = phi / d

Next steps

• Make sure EigenmodeSim is working.
• Add periodic BCs on the Gmsh side.
• Create either a new Bloch-FloquetSim, or have EigenmodeSim accept optional FloquetWaveVector and BoundaryPairs.
• Post-processing should be super simple: just use the values we get from domain-E.csv.
• Run some tests for CPW
• Add this feature to the T-bar notebook, which should have the option to model a single unit cell.

[dot-cross]

I changed the tbar parameterization. Right now I am fighting a Palace Error related to the mesh, which seems like a regression since the simulation worked well before.

[mdmaas]

@dot-cross you might want to check out if #79 is of any help, as I think it could be related to the problem you showed me (it looked like an unresolved boolean).