vector.md

title	Full-vector modes by finite differences
module	mode::vector
summary	The modes of any waveguide cross-section, anisotropic and lossy media included, from the transverse magnetic field on a rectilinear grid.
order	5
papers	cite doi A. B. Fallahkhair, K. S. Li, T. E. Murphy, J. Lightwave Technol. 26, 1423 (2008) 10.1109/JLT.2008.923643 cite doi G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002), part II: dielectric corners (the corner test problems) 10.1109/JLT.2002.800371 cite doi L. Chrostowski, M. Hochberg, Silicon Photonics Design (2015), Fig. 3.14 10.1017/CBO9781316084168 cite doi P. Bienstman et al., Opt. Quantum Electron. 38, 731 (2006), the leaky-wire benchmark 10.1007/s11082-006-9025-9
validation	mode/vector-slab-limit-te mode/vector-slab-limit-tm mode/strip-book mode/hadley-box-low mode/hadley-box-high mode/hadley-corner-low mode/hadley-corner-high mode/leaky-wire-bienstman mode/leaky-wire-bienstman-loss
examples	strip_waveguide hadley_corners group_index leaky_wire_benchmark

The unknowns are the transverse magnetic field, $H_x$ and $H_y$, at the nodes of a rectilinear grid whose spacing need not be uniform. The permittivity is uniform in each cell between nodes: a tensor with $\varepsilon_{xx}, \varepsilon_{xy}, \varepsilon_{yx}, \varepsilon_{yy}$ and $\varepsilon_{zz}$, so anisotropic, lossy and gyrotropic media are all allowed.

The method

Fallahkhair, Li and Murphy discretize the coupled equations for $H_x$ and $H_y$ (their Eqs. 4a–4b) on a nine-point stencil. At each node, the four cells around it may differ, and the interface conditions between them are built into the coefficients (Eqs. 21–36). The result is a sparse eigenproblem (Eq. 8) whose eigenvalues are $\beta^2$:

$$ \begin{pmatrix} A_{xx} & A_{xy} \cr A_{yx} & A_{yy} \end{pmatrix} \begin{pmatrix} H_x \cr H_y \end{pmatrix} = \beta^2 \begin{pmatrix} H_x \cr H_y \end{pmatrix}, \qquad n_\text{eff} = \beta / k_0 . $$

The modes nearest a guess come from shift-and-invert Arnoldi. VectorMode::residual measures a mode against the matrix assembled afresh, $\lVert A h - \beta^2 h\rVert / (|\beta^2|\thinspace\lVert h\rVert)$; a "modes" job records it for every mode, and the studio's Solver panel shows it.

• Convention: the paper uses $e^{+j\omega t}$. Its eigenvalue equations hold in photonoxide's $e^{-i\omega t}$ when written with that convention's permittivity: loss is $+i$, and a gyrotropic term the paper writes as $+j\Delta$ is $-i\Delta$ here.
• A misprint: Eq. (31) divides by $v_{12}$, which the paper never defines. It is $v_{21}$, as the matching term of Eq. (30) shows under the mirror $x \to -x$. A test checks that a structure and its mirror image have the same modes, to 1e-10.
• Edges: beyond each edge the field is zero, or a mirror wall reflects it, or a PML absorbs it.

Accuracy

• Interfaces: second order, in both orientations. A slab turned on its side converges onto the exact slab at orders 2.00 / 2.01 / 2.00; at a 2.5 nm grid the error is 4e-5 (TE) and 5e-6 (TM).
• Corners converge more slowly, as with standard finite differences: their fields' derivatives are singular. On Hadley's four corner problems (series-expansion indices good to 1e-8), about first order at convex corners (a high-index quadrant: boxes, strips), with the error changing sign on the way, and 1.8 falling towards 1.4 at concave ones. Hadley's own equations, on the same grid and unknowns, reach about second order there and sixth at straight interfaces: see high-accuracy modes, for uniform grids of lossless isotropic media.
• The book's strip (500 × 220 nm, 3.473 in 1.444, 1550 nm): TE-like 2.447067, 2.445713, 2.444396, 2.443506 at 20, 10, 5, 2.5 nm, against the book's 2.443 (Lumerical, 20 nm mesh, good to about 1e-3).

Grids and extrapolation

graded_nodes builds a grid fine in a box around the core and growing gently away from it, the interfaces on nodes; keep the growth to a few hundredths per µm, since the scheme loses accuracy where neighbouring spacings differ much. Refining at the corners alone doesn't pay: the core's interior must stay resolved. Because the corners converge at a steady low order, richardson extrapolates from three grids, each half the last, with the order fitted.

On Bienstman et al.'s leaky SOI wire (silicon against air, the hardest corners), the order is 0.67: +4.0e-3, +2.5e-3, +1.5e-3 at 5, 2.5 and 1.25 nm in Re, and 0.97, 0.98, 0.99 of the loss. Extrapolated: 2.412289 + 2.9207e-8 i, against the benchmark's 2.412372 + 2.9135e-8 i (CAMFR and the aperiodic Fourier modal method), 8e-5 and 0.25 % away. The loss also needs the oxide and substrate resolved: grading to 40 nm there misstates it by 3 %, to 10 nm by 0.3 %.

Validation

• A uniform box has its exact discrete eigenvalues, to 1e-9.
• The slab limit, at second order, against the exact slab (both orientations).
• Hadley's four corner problems within 1e-4 at an 80 × 80 grid, and 5e-5 at 160 × 160.
• Chrostowski and Hochberg's strip within 3e-3 at 5 nm.
• Bienstman et al.'s leaky wire, extrapolated: within 8e-5 in Re and 0.25 % in the loss.