FDFD

Backend lightsmith-fdfd (method fdfd, modules lightsmith.backends.fdfd and lightsmith.backends._fdfd). A 2D finite-difference frequency-domain solver for devices. It gives adjoint gradients with respect to every permittivity pixel, which makes it the engine for inverse design. It runs on the CPU or a CUDA GPU.

Method

• Equation. The scalar 2D Helmholtz equation on a Yee grid: TM solves for E_z, TE for H_z with 1/ε at cell edges. A graded stretched-coordinate PML is added outside the domain (Shin & Fan 2012).
• Geometry. Permittivity comes from supersampled fill fractions. 3D geometries are cut at the domain centre in z.
• Ports. Port modes come from a 1D slab eigensolver on the same discretisation, so they are exact discrete modes. They are launched as unidirectional total-field/scattered-field sources.
• S-parameters. Each port's field is decomposed into forward and backward modal amplitudes. With every port excited, S = C·A⁻¹, which also removes waves that re-enter from imperfect PMLs.
• Linear solvers. SciPy SuperLU on the CPU. On the GPU, a block-tridiagonal Schur-complement LU in PyTorch, suited to devices up to about 500 cells across.
• Gradients. One transposed (adjoint) solve per wavelength, reusing the forward factorisation. FDFD.differentiable_solve(...) exposes them as a PyTorch autograd function.

Options

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

Option	Default	Meaning
polarization	"te"	"te" means E in the xy plane (H_z); "tm" means E_z
dl_nm	None	Cell size. Default λ_min / (cells_per_wavelength · n_max).
cells_per_wavelength	20	Used for the default cell size
pml_cells	16	PML cells added outside the domain, as one value or (x, y). 0 puts a wall on that axis.
pml_reflection	1e-8	Target normal-incidence reflection
solver	"auto"	"scipy", "block", or "auto" (SuperLU on the CPU, block LU on the GPU)
precision	"double"	Factorisation precision of the block solver
subsamples	4	Supersampling per axis (1–16)
max_cells	4000000	Refuse larger grids

Validation

All tests are in tests/test_fdfd.py.

Test	Reference	Tolerance
test_straight_waveguide_transmits (TE and TM)	|S21| = 1; discrete propagation phase	1e-4; |S11| < 1e-3; phase < 1e-3 rad
test_slab_mode_matches_analytic_neff (TE and TM)	analytic slab neff	< 1e-3 at dl = 5 nm; second-order convergence
test_multimode_ports_are_orthogonal	no mode conversion in a straight guide	1e-4
test_power_conservation_closed_guide (TE and TM): lossless scatterers in a parallel-plate guide	power conservation; reciprocity	column sums 1e-6; S = Sᵀ to 1e-8
test_bend_transmission_grows_with_radius	qualitative, plus passivity	T(R = 1.5 µm) > 0.95
test_y_splitter_is_symmetric	symmetry	|P2 − P3| < 1e-6
test_adjoint_gradient_matches_finite_differences (TE and TM)	central differences	1e-3 relative
test_block_solver_matches_scipy_on_cpu	SuperLU	1e-9
test_gds_layout_device	explicit structures	1e-12
test_gpu_matches_cpu (gpu)	CPU	S to 1e-10; gradients 1e-6

Limits

• 2D only (dimension=2); 3D FDFD is not implemented yet. Ports cannot be normal to z.
• Raises an error when:
• the grid exceeds max_cells;
• a port window is too narrow or too close to the PML;
• a port has fewer propagating modes than requested;
• the grid is too coarse for a mode (reduce dl_nm).
• Warns when a port mode is not guided, and when output power exceeds input by more than 1 % in a lossless device (refine dl_nm or thicken the PML).