DDA

Backend lightsmith-dda (method dda, module lightsmith.backends.dda). The discrete dipole approximation handles any shape, any number of particles, and orientation averaging. It runs on the CPU or the GPU.

Method

• Lattice. Point dipoles sit on a cubic lattice. Where particles overlap, the first one listed wins.
• Solver. The block-Toeplitz interaction matrix is applied by FFT on a zero-padded doubled grid (Goodman, Draine & Flatau 1991), and the system is solved with BiCGStab. The physics follows Draine & Flatau (1994).
• Polarisability. The lattice dispersion relation (LDR; Draine & Goodman 1993) by default, or Clausius–Mossotti with radiative reaction.
• Cross sections. C_ext and C_abs are computed from the dipole moments, and C_sca = C_ext − C_abs.
• Discretisation. The default center discretisation rescales the lattice so that the dipole volume equals the true volume.
• Extrapolation (optional). Extrapolates to zero dipole size (Yurkin et al. 2006).
• Orientation average. Uses Fibonacci-sphere incidence directions, each with two polarisations.
• Fields. Near and far fields come from the finest lattice. Near-field points inside particles are NaN.

The same code runs on NumPy, PyTorch (CPU or CUDA) and CuPy through the array API.

Options

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

Option	Default	Meaning
dipoles_per_diameter	None	Lattice sites across the smallest particle's largest dimension. Default: max(16, enough for |m| k d ≤ 0.5).
polarizability	"ldr"	"ldr", or "cm_rr" (Clausius–Mossotti + radiative reaction)
discretization	"center"	"center" (staircase with volume rescaling), or "weighted" (experimental: boundary sites mixed by volume fraction)
precision	"auto"	"auto" means double on the CPU and single on the GPU
tol	None	BiCGStab relative residual. Default 1e-8 (double) or 1e-5 (single).
max_iter	10000	BiCGStab iteration cap
max_dipoles	5000000	Refuse problems larger than this
orientations	32	Incidence directions (× 2 polarisations) for orientation_average
extrapolate	None	List of dipoles_per_diameter values to extrapolate to zero dipole size, for example [12, 16, 24]. Experimental: staircase noise makes it unreliable for metal spheres.

There are no gradients.

Validation

Test	Reference	Tolerance
tests/test_core.py::test_dda_matches_mie_for_dielectric_sphere: r = 40 nm, n = 1.5 + 0.02i in water; 12 dipoles per diameter	Mie	C_ext, C_abs < 1 %
test_core.py::test_dda_lossless_has_no_absorption: n = 1.6	C_abs = 0	< 1e-6 C_ext
test_core.py::test_dda_gpu_matches_cpu (gpu)	CPU	1e-3 relative
tests/test_dda.py::test_far_field_integrates_to_c_sca	∫ dC/dΩ = C_sca	1e-3
test_dda.py::test_shell_of_core_material_equals_sphere	plain sphere	1e-10
test_dda.py::test_orientation_average_of_sphere: 4 orientations	fixed incidence	1e-3
test_dda.py::test_near_field_tends_to_incident_and_masks_inside	|E|² → 1 far away; NaN inside	0.02 absolute
test_dda.py::test_spheroid_polarization_anisotropy: Drude rod	qualitative: the longitudinal peak lies red of the transverse peak	—

The test suite has no DDA accuracy test against Mie for metal spheres. DDA converges slowly for metals, so check convergence by increasing dipoles_per_diameter.

Limits

• No substrate.
• Field outputs cannot be combined with orientation_average.
• Raises an error above max_dipoles (lower dipoles_per_diameter or raise the cap). Also raises an error when no dipole falls inside a particle (increase dipoles_per_diameter).
• Warns when BiCGStab breaks down or does not converge (try precision="double"), and when results are extrapolated.