BPM

Backend lightsmith-bpm (method bpm, modules lightsmith.backends.bpm and lightsmith.backends._bpm). A finite-difference beam propagation method for devices whose ports lie on two opposite planes, in 2D and 3D. It suits long, slowly varying devices such as tapers, bends and branches, where light travels mostly along one axis. It runs on the CPU or a CUDA GPU.

BPM is unidirectional: it neglects reflections. Use EME (bidirectional) or FDFD when reflections or abrupt high-contrast junctions matter.

Method

• Propagator. The transverse magnetic field is marched from one port plane to the other with the one-way equation dH_t/dw = i k₀ √N H_t. N = PQ is built from the FDE engine's own Yee-grid operators, so FDE and EME port modes are exact eigenvectors of the propagator in uniform sections, and mode conventions match EME.
• Wide angle. The square root is replaced by a Padé (m, m) approximant (Hadley 1992); m = 0 is the paraxial Fresnel equation. Steps are Crank–Nicolson, factored into m shifted solves. The reference index adapts to the field in every slice (its Rayleigh quotient).
• Slices. The device is cut into piecewise-uniform slices with the EME slicer. Continuously varying segments are staircased so that outlines move about taper_grid_cells_per_slice grid cells per slice. Each slice's operator is factorised once.
• Why H. H is continuous across every interface, so it is carried across slice boundaries. On top of that, each input's power Re⟨E, H*⟩ is rescaled to be continuous at slice boundaries (energy-conserving BPM). Plain H continuity conserves an n-weighted norm instead, so a taper's transmission would be off by about √(n_out / n_in).
• Dimensions.
• dimension=2: slab BPM in the xy plane, te (in-plane E, the default, as in EME) or tm.
• dimension=3: full-vectorial (Hx and Hy coupled, the default) or semi-vectorial (quasi-TE: Hy; quasi-TM: Hx) on the 2D cross-section (Huang & Xu 1993).
• Boundaries. Stretched-coordinate PMLs on the transverse walls.
• Bends. bend_radius_nm propagates along a circular arc through a conformal transformation, as for FDE bend modes. The device is given unrolled.
• S-parameters. One run from the lower port plane fills S(upper ← lower), and a second run from the upper plane fills S(lower ← upper). Entries between ports on the same plane, including reflections, are set to 0.
• Linear algebra. Block-tridiagonal parallel cyclic reduction in the array API: lines run along the longer transverse axis, with dense blocks across the shorter one. The same code runs on NumPy and PyTorch.

Options

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

Option	Default	Meaning
vector	"auto"	3D only: "full" (Hx, Hy) or "semi" (one H component). "auto" means full.
polarization	"auto"	2D: "te" (default) or "tm". 3D semi-vectorial: "te" (dominant Ex, default) or "tm". 3D full-vectorial: filters port modes by TE fraction; "any" by default.
grid_spacing_nm	None	Cross-section grid step, a number or a pair. Default from the two options below.
points_per_wavelength	None	Default 40 in 2D and 15 in 3D
cells_per_feature	6	Used for the default spacing
subpixel	"anisotropic"	Permittivity averaging, as in FDE: "anisotropic", "volume" or "none"
subsamples	32	Supersampling in cut cells
pml_cells	15	PML thickness
pml_reflection	1e-8	Target PML reflection
step_nm	None	Largest propagation step. Default λ / 10.
pade_order	2	0 = paraxial; m = Padé (m, m), costing m solves per step (0–6). (1, 1) suffices for straight guides and gentle tapers; tilted guides, branches and bends need (2, 2).
reference_index	"adaptive"	"adaptive" (per slice, from the field) or a fixed number
taper_grid_cells_per_slice	1.0	How far outlines may move between slices, in grid cells
max_taper_slices	400	Most slices per varying segment
power_continuity	True	Keep each input's power continuous at slice boundaries
directions	"both"	"forward" runs from the lower plane only and fills the reverse direction by reciprocity
bend_radius_nm	None	Propagate along an arc of this radius; positive means the centre of curvature is towards −u
precision	"auto"	"auto" means single on the GPU and double on the CPU
max_grid_cells	400000	Refuse larger cross-section grids
gpu_min_unknowns	2000	Cross-sections with fewer unknowns run on the CPU even when a GPU is selected; provenance["device_note"] says so. 0 = always use the given device.

There are no gradients. Multimode ports are labelled name@j, as in EME. estimate reports the grid, the block size, the number of steps and runs, and the memory of the factorisations.

Validation

All tests are in tests/test_bpm.py. Unless stated, devices are 2D with core n = 2.85 and cladding n = 1.44 at 1550 nm.

Test	Reference	Tolerance
test_operator_blocks_and_solves_match_the_fde_matrices: slab, 3D, PMC mirror, bend, PEC walls	FDE matrices; dense LU	1e-12; solves 1e-11
test_pade_steps_approach_the_exact_one_way_propagator	exact one-way step	|G| = 1 to 1e-12 for lossless media; error falls with the Padé order; lossy media decay
test_straight_slab_is_lossless_with_the_fde_phase: te and tm	|S21| = 1; EME phase	1e-9; phase 1e-9 rad
test_straight_strip_3d_full_vector_keeps_te_and_tm: 500 × 220 nm SOI strip	|S| = 1, no conversion; k₀·neff·L from FDE	2e-3; phase 1e-2 rad (measured 4.6e-3)
test_linear_taper_matches_eme: 500 nm → 3 µm, 3 and 10 µm long	EME, same grid	|S21|² within 0.02 (measured ≤ 0.010); phase 0.01 rad
test_long_taper_is_reciprocal_and_adiabatic (slow): 30 µm, 10 nm grid	EME (0.994); reciprocity	0.01 (measured 0.996 / 0.995)
test_mmi_1x2_self_imaging_length: W = 3 µm, 0.9–1.05 × the paraxial length	EME	same best length (0.95 ×); 50/50 to 1e-9; total within 0.06 of EME; > 0.9
test_slab_bend_loss_matches_the_fde_bend_mode: R = 20 µm, n 1.60 / 1.45	FDE bend mode	2 % (measured 0.2 %)
test_y_branch_splits_evenly: arms diverging at 4.8°	EME (0.936)	50/50 to 1e-9; S21 and S12 within 0.02; total within 0.1 of EME (measured 0.856–0.864)
test_torch_cpu_matches_numpy	NumPy	1e-10
test_gpu_matches_cpu_3d_and_small_2d_falls_back_to_cpu (gpu)	CPU double	1e-4 (single precision)
test_strip_bend_loss_3d_matches_the_fde_bend_mode (slow): SiN 800 × 400 nm, R = 8 µm	FDE bend mode (1244 dB/cm)	2 % (measured 0.4 %)

The test docstrings record where one-way BPM is weakest:

• Short converging tapers (wide → narrow): a 3 µm taper gives 0.68 against EME's 0.77.
• High-contrast branches: the Y-branch total of 0.856–0.864 is converged in grid, slicing and step, but lower than EME's 0.936.
• Bend loss and the PML: with the default pml_reflection of 1e-8, instead of FDE's 1e-6, the 3D bend loss is 6 % lower, because the loss of a leaky mode depends on the truncation.

Limits

• Every port must be normal to the same axis and lie on one of exactly two opposite planes, as for EME. Each port must point into the device.
• Unidirectional: reflections are neglected, and S entries between ports on the same plane are 0. Every result carries a warning that says so.
• At an abrupt junction BPM transmits by mode overlap, an error of the order of the mismatch.
• Raises an error when the cross-section grid exceeds max_grid_cells, or when a port supports fewer modes than its num_modes.
• Warns when Σ|S|² > 1.02 for an input (refine the grid or the step, or raise pade_order), and, for semi-vectorial 3D runs, that polarisation coupling is neglected.
• auto tries EME first, which accepts the same port layouts, so in practice BPM is used when you ask for it with method="bpm".