RCWA

Backend lightsmith-rcwa (method rcwa, modules lightsmith.backends.rcwa and lightsmith.backends._rcwa). Rigorous coupled-wave analysis, also called the Fourier modal method. It solves periodic structures on 1D and 2D lattices, including non-rectangular lattices and conical incidence, and also layer stacks. It is differentiable and runs on a GPU.

Method

• Fourier coefficients of ε and 1/ε.
• 1D: exact segment transforms.
• 2D: closed-form transforms for rectangles, circles, ellipses and polygons when the shapes do not overlap. Otherwise a supersampled raster.
• Factorisation.
• 1D: Li's inverse rule.
• 2D: the normal-vector-field method (Schuster et al. 2007), which metals need. The normal field is a Gaussian-blurred structure-tensor field.
• Optionally, the naive Laurent rule everywhere.
• Layer eigenmodes. Only the matrix functions K = √(PQ) and exp(i d K) are needed. Gradients use the Daleckii–Krein formula, so they stay exact even for degenerate modes.
• Cascade. Layers are combined in a stable Redheffer star-product S-matrix cascade (Rumpf 2011).
• Outputs.
• R, T and A per diffraction order come from the Poynting flux of each harmonic.
• The zeroth-order complex amplitudes follow the TMM conventions.
• Truncation is rectangular: |m| ≤ M and |n| ≤ N.
• Gradients. lightsmith.backends.rcwa.rcwa_tensors(problem, device, thicknesses=, eps_grids=) differentiates with respect to thicknesses and pixelated permittivity maps.

Options

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

Option	Default	Meaning
harmonics	None	Highest diffraction order kept along each lattice direction. An integer applies to both directions; a pair sets them separately. Default 25 for 1D gratings and 7 per direction for 2D lattices.
factorization	"auto"	"auto", "li" and "normal_vector" all mean Li's rule in 1D and the normal-vector method in 2D. "laurent" uses the naive rule everywhere.
fourier	"auto"	2D Fourier coefficients: "analytic" shape transforms, used by "auto" when shapes do not overlap, or "raster"
raster_points	256	Raster points per lattice direction
normal_blur	0.1	Gaussian smoothing of the 2D normal field, as a fraction of the shortest lattice vector
precision	"auto"	"auto" means double
max_batch_bytes	2e9	Wavelengths are solved in batches of about this much memory

Validation

All tests are in tests/test_rcwa.py.

Test	Reference	Tolerance
test_energy_conservation_lossless_1d_conical: θ = 25°, φ = 35°	R + T = 1	1e-12
test_energy_conservation_lossless_2d_hexagonal_lattice	A = 0	1e-11 with the Laurent rule; < 2e-3 at 6 harmonics with the normal-vector rule, shrinking with more harmonics
test_analytic_and_raster_fourier_agree	512-point raster	2e-3
test_rytov_effective_medium_benchmark: ε = 12, fill factor 0.3, P → 0	TMM on the Rytov effective uniaxial slab	2e-5 (TE), 5e-5 (TM)
test_li_factorization_converges_for_metallic_grating: ε = −20 + 1.5i, TM	80-harmonic result	error falls monotonically; < 4e-3 at 40 harmonics; Laurent more than 5× worse
test_crossed_lattice_of_full_width_ridges_reduces_to_1d_li	1D result	1e-10
test_guided_mode_resonance_in_hole_array_converges	convergence in harmonics	peak moves < 1 nm between 3 and 5 harmonics
test_grating_orders_follow_grating_equation	grating equation	exact set of orders
test_gradients_match_finite_differences: degenerate C4-symmetric ε grid	central differences	1e-4 (pixels), 1e-5 (thickness)
test_rcwa_gpu_matches_cpu (gpu)	CPU	1e-9
tests/test_tmm.py::test_tmm_matches_rcwa_for_unpatterned_stack	TMM	1e-12

The FDTD periodic tests also cross-check against RCWA.

Limits

• No incoherent layers; use TMM for those.
• The normal-vector factorisation does not conserve energy exactly at finite truncation. A lossless structure with |A| > 1e-3 triggers a warning: increase harmonics or use precision="double".
• Analytic 2D transforms are valid only for shapes that do not overlap. "auto" falls back to the raster, but forcing "analytic" with overlapping shapes is not checked.
• With an absorbing superstrate, efficiencies are normalised to the incident Poynting flux, and the result carries a warning saying so.