feat: variational effective index approximation for 2D FDTD

[thanojo]

Motivation

gsim's 2D EIM mode (solver.is_3d=False) collapses the z-dimension by assigning a uniform effective index to each in-plane material region. This just assumes the user knows the effective indices to place here and ignores how the vertical (slab) mode profile weights local permittivity.

The variational EIM (Hammer & Ivanova, Opt. Quant. Electron. 41, 267, 2009) is a rigorous improvement. It computes a spatially-varying effective permittivity map directly from the 3D stack:

$$\varepsilon_\text{eff}(x,y) = n_\text{eff}^2(\boldsymbol{r}) + \frac{\int dz,\left[\varepsilon(x,y,z) - \varepsilon(\boldsymbol{r},z)\right],|\Phi_r(z)|^2}{\int dz,|\Phi_r(z)|^2}$$

where $\boldsymbol{r} = (x_r, y_r)$ is a fixed reference point in the waveguide, and $n_\text{eff}(\boldsymbol{r})$ and $\Phi_r(z)$ are the effective index and fundamental slab mode profile solved at that point.

Scope

• Compute the effective permittivity map from a component's 3D layer stack and a chosen reference point via a slab mode solver (in z), per wavelength
• Fit the resulting frequency-dependent $\varepsilon_\text{eff}(x,y,\lambda)$ to a dispersive material model compatible with MEEP
• Expose this as an option in the existing 2D simulation workflow

Notebook

The existing 2D FDTD notebook meep_2d.ipynb doesn't actually contain an effective index approximation, it just uses the bulk indices.
As a first step, we could add the correct index approximation directly to this notebook, or we could add a new one.

Then, we could follow the three examples from the paper directly:

1. 2D Bragg grating (paper Fig. 3): 2D structure reduced to 1D, validates reflection/transmission spectra
2. Ring resonator (paper Fig. 6): 3D to 2D reduction, validates FSR matching
3. Strip-to-slot converter: counter-example showing where single-reference-mode approximation breaks down

Limitations

• Above only works for TE polarization; TM requires a different weight kernel (but could be added)
• Not suitable for mode-converting transitions where a single reference mode cannot represent both endpoints

Reference

Hammer & Ivanova, Opt. Quant. Electron. 41, 267 (2009) doi:10.1007/s11082-009-9349-3