FDE

Backend lightsmith-fde (method fde, modules lightsmith.backends.fde and lightsmith.backends._fde). A full-vector finite-difference eigenmode solver for waveguide cross-sections, straight or bent. The EME and FDFD solvers reuse it for port modes.

Method

• Formulation. The transverse E field on a 2D Yee grid (Zhu & Brown 2002). It solves (QP)E = n²E.
• Interfaces. Subpixel permittivity averaging: cells cut by an interface are supersampled. The default anisotropic scheme takes the harmonic mean along each E component and the arithmetic mean across it.
• Boundaries. Walls are PEC, or a PML implemented as complex coordinate stretching. Mirror planes can be PEC or PMC.
• Bends (about y). The exact conformal-map transformation: ε' = ε·diag(h, h, 1/h) and μ' = diag(h, h, 1/h), with h = 1 + (x − x_c)/R.
• Eigensolver. Shift-invert Arnoldi (SciPy splu plus ARPACK) around target_neff².
• Group index. A perturbation formula that includes material dispersion, or a central finite difference in wavelength.
• Slabs. window_size_nm[1] == 0 gives a slab (1D) problem.
• Normalisation. Fields are normalised to unit power, ½ Re∫E×H*·z dA = 1.

Options

options["fde"] or options["lightsmith-fde"].

Option	Default	Meaning
grid_spacing_nm	None	Grid step, as one value or (x, y). Default: the smaller of λ_min/(n_max · points_per_wavelength) and the smallest feature / cells_per_feature.
points_per_wavelength	20	Used for the default spacing
cells_per_feature	8	Used for the default spacing
subpixel	"anisotropic"	"anisotropic", "volume" or "none"
subsamples	32	Samples per axis in cut cells
boundary	"auto"	"auto" means PML for bends and PEC otherwise; "pec" or "pml" force one
pml_cells	12	PML thickness
pml_reflection	1e-6	Target normal-incidence reflection of the PML
symmetry	("none", "none")	Mirror planes through the window centre, normal to x and to y: "none", "pec" or "pmc"
polarization	"any"	"te" keeps modes with te_fraction > 0.5 (dominant Ex); "tm" keeps those below
group_index	"auto"	"perturbation", "finite_difference" or "none". "auto" uses the perturbation formula without a PML and the finite difference with one.
track_modes	False	Order modes at each wavelength by overlap with the previous wavelength
bend_reference_x_nm	None	The x position at which the bend radius is measured. Default: the window centre.
max_cells	1000000	Refuse larger grids

The solver runs on the CPU. There are no gradients.

Slab polarisation naming

For a slab normal to x, te_fraction is the share of Ex. The physical slab TE mode (Ey) is therefore selected with polarization="tm".

Validation

All tests are in tests/test_fde.py.

Test	Reference	Tolerance
test_slab_matches_dispersion_relation (TE and TM): n = 2.0/1.45, d = 600 nm	analytic slab dispersion relation	< 1e-4 at dx = 2.5 nm; second-order convergence
test_slab_group_index_matches_analytic_derivative	analytic dispersion derivative	2e-4
test_step_index_fibre_he11: n = 1.50/1.45, a = 1.5 µm	exact vector HE11 eigenvalue equation (Snyder & Love)	2e-6
test_si_strip_te0_and_grid_convergence: 500 × 220 nm Si strip in SiO2, 1550 nm	published neff 2.44523	< 1e-3 at dx = 10 nm; 3.9 < ng < 4.2
test_si_strip_fine_grid (slow): dx = 5 nm	neff 2.44523	3e-4
test_symmetry_matches_full_window_and_group_index_methods_agree	full window; the other group-index method	1e-9; 2e-3
test_bend_loss_decreases_with_radius: slab bend, R = 2–5 µm	exponential trend in R	strictly decreasing; log-linear fit residual < 0.3
test_port_modes_match_mode_problem	ModeProblem	1e-6

Limits

• Bends only about the y axis, and no x-symmetry plane in a bend.
• Material dispersion enters only through the group index; materials are evaluated at each wavelength.
• Raises an error above max_cells: increase grid_spacing_nm, use symmetry, or raise the cap.
• Warnings:
• fewer modes found than requested;
• a mode is not guided (neff ≤ the background index);
• more than 20 % of |E|² is in the PML.