mode-solver-methods.md

Mode Solver Methods

solve_modes(...) is the main entry point. It validates the Materials grid, resolves frequencies or wavelengths, builds the Yee derivative matrices, chooses the sparse formulation, solves one frequency at a time, and returns a coordinate-aware Result.

Material Grids

• Materials.from_diagonal(...) builds scalar or diagonal-anisotropic grids. These usually use the faster diagonal sparse formulation.
• Materials.from_components(...) accepts full (3\times3) material tensors. Any off-diagonal component routes the solve through the tensorial sparse formulation.
• Materials.from_slice(...) creates a 1D mode-plane slice with an invariant width so integrations and overlaps still have physical weights.
• Materials.from_subpixel_diagonal(...) downsamples a high-resolution diagonal raster with arithmetic, harmonic, geometric, min, or max averaging.

Solver Controls

• num_modes: number of modes returned near the requested target.
• target_neff: center of the shift-invert search. If omitted, MicroMode uses a heuristic based on the largest absolute permittivity component.
• pml: absorbing boundary thickness and stretch profile via PmlSpec.
• boundary: low-edge PEC/PMC symmetry settings via BoundarySpec.
• direction: "+" or "-" propagation; the backward solve flips the appropriate magnetic and longitudinal electric signs.
• components: optional subset of returned field components.
• krylov_dim: dimension of the Arnoldi search space.
• angle_theta, angle_phi, bend_radius, bend_axis: transformation-optics controls that update (\epsilon) and (\mu) before the sparse solve.

Eigenpair Selection

The SciPy solver selects eigenpairs with sparse shift-invert SciPy/ARPACK [1, 2]. For a matrix (A) and shift (\sigma), Arnoldi is applied to

$$ (A-\sigma I)^{-1}, \qquad \lambda = \sigma + 1/\theta, $$

where (\theta) is a Ritz value of the inverse-shifted operator. The diagonal formulation uses (\sigma=-\texttt{target_neff}^2); the tensorial formulation uses (\sigma=\texttt{target_neff}).

Returned modes are sorted by decreasing real effective index, normalized to unit transverse power,

$$ \int (\mathbf{E}\times\mathbf{H}^*)\cdot\hat{\mathbf{n}},dA, $$

and orthogonalized with the unconjugated Lorentz product

$$ L(a,b)=\frac{1}{2}\int \left[(\mathbf{E}_a\times\mathbf{H}_b) +(\mathbf{E}_b\times\mathbf{H}_a)\right]\cdot\hat{\mathbf{n}},dA. $$

Result Helpers

Result exposes the post-processing methods users normally need: n_eff, k_eff, mode_area, pol_fraction, pol_fraction_waveguide, modes_info, to_dataframe(), overlap(), overlap_matrix(), plot_field(), plot_field_components(), to_hdf5(), and Result.from_hdf5().

References

[1] W. E. Arnoldi, "The principle of minimized iterations in the solution of the matrix eigenvalue problem," Quarterly of Applied Mathematics, vol. 9, no. 1, pp. 17-29, 1951. AMS record.

[2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, 1998. doi:10.1137/1.9780898719628.