PWE

Backend lightsmith-pwe (method pwe, modules lightsmith.backends.pwe and lightsmith.backends._pwe). Plane-wave expansion for photonic band structures. It covers 2D TE and TM, and 3D, for lossless, non-dispersive dielectric crystals.

Method

PWE solves Maxwell's eigenproblem in a transverse plane-wave basis, so div H = 0 holds exactly and no spurious modes appear. It has two discretisations:

• dense: the Ho–Chan–Soukoulis inverse rule followed by batched dense eigensolves. The Fourier coefficients of ε are analytic for inclusions that do not overlap: circles, ellipses, rectangles, polygons, balls, boxes and vertical prisms. Other inclusions use a supersampled raster.
• lobpcg: matrix-free FFT operators with an MPB-style subpixel-smoothed inverse-ε tensor. The block LOBPCG eigensolver uses a preconditioner and warm starts along the k-path.

auto picks between them:

• num_plane_waves alone selects dense; resolution alone selects lobpcg.
• Otherwise, TM and empty cells use dense when the matrix is small (size ≤ 2500), and TE and 3D use lobpcg.
• The reason: for TE at curved interfaces, dense converges slowly, with about 1 % error at 1200 plane waves, against about 0.3 % for lobpcg at 32 points per a.

Helpers in lightsmith.backends.pwe:

• band_gaps(result);
• density_of_states(problem, ...);
• supercell(problem, repeats, remove, add), for line and point defects;
• bands_from_permittivity(eps, ...), which is differentiable with PyTorch for non-degenerate bands.

Options

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

Option	Default	Meaning
solver	"auto"	"dense", "lobpcg" or "auto"
num_plane_waves	None	Dense basis size, completed to a full shell. Default 400 (2D) or 600 (3D) per unit cell.
resolution	None	LOBPCG grid points per a. Default 32 (2D) or 16 (3D).
eps_fourier	"auto"	"analytic" or "raster". "auto" uses analytic when it is valid and inclusions do not overlap.
subsamples	None	Raster supersampling. Default 4 (2D) or 2 (3D).
reference_wavelength_nm	None	Wavelength at which materials are evaluated. Default 2a, with a warning if a material is dispersive.
precision	"auto"	Double, except single for dense on a GPU (about 1e-5 in a/λ). LOBPCG always uses double.
tol	None	LOBPCG relative residual. Default 1e-6 (double) or 1e-4 (single).
max_iter	400	LOBPCG iterations per k-point
extra_bands	None	LOBPCG block padding. Default max(2, num_bands/4).

Validation

All tests are in tests/test_pwe.py.

Test	Reference	Tolerance
test_empty_lattice_2d_free_photon_bands (both solvers)	analytic free-photon bands	1e-6
test_empty_lattice_3d_fcc_free_photon_bands (both solvers)	analytic free-photon bands	1e-6
test_square_rods_tm_gap_joannopoulos_and_convergence: alumina rods, ε = 8.9, r = 0.2a	Joannopoulos et al., Photonic Crystals, ch. 5: gap 0.3224–0.4425, 31.4 %	gap edges 5e-3 relative; ratio 0.003
test_square_rods_eps12_matches_mpb_tutorial: ε = 12	MPB tutorial: 0.282623–0.419335	1 % (converged: 0.28066–0.41717)
test_triangular_air_holes_complete_gap: ε = 13, r = 0.48a	book: complete gap 0.4297–0.5199, 18.6 %	1 %; ratio 0.01
test_lobpcg_matches_dense_for_tm	dense solver	3e-3
test_te_subpixel_smoothing_converges	resolution 64	< 3e-3 at resolution 32
test_supercell_line_defect_mode_in_gap: W1 waveguide	one guided band in the gap	qualitative
test_density_of_states	integral = number of bands; zero in the gap	—
test_bands_from_permittivity_is_differentiable	central differences	1e-5
test_gpu_matches_cpu (gpu)	CPU	2e-4 (dense, single precision); 1e-6 (LOBPCG)
test_diamond_lattice_complete_gap_3d (slow): ε = 11.56 spheres	gap 0.2815–0.3144	5e-3

Limits

• Positive permittivity only. Metallic crystals need a dispersive solver.
• Loss is ignored and dispersive materials are evaluated at reference_wavelength_nm; both trigger warnings.
• In 2D, out-of-plane k is refused.
• Analytic Fourier coefficients do not support sloped sidewalls.
• band_gaps reports true gaps only if the k-path traces the edge of the irreducible Brillouin zone.