pysie2d 0.2.0

2-D boundary-integral (Müller BIE) solver for electromagnetic scattering from a cylinder in a homogeneous background.

pysie2d

A 2-D surface-integral-equation solver for time-harmonic electromagnetic scattering from a single smooth cylinder — circular or Gielis-superformula cross-section, embedded in a homogeneous background. Validated against analytic Mie theory, with a typed public API and CI.

Scope and non-goals

This package is distilled from a larger private research code; it deliberately covers only the homogeneous-background, single-particle core — the part that can be validated end-to-end against a closed-form reference. Potential extensions in the mid/long-term include slab waveguide backgrounds, multiple-particle simulations, and quasinormal-mode searches based on the surface-integral matrix operator.

Figures

Relative error of the scattering efficiency Q_sca versus the number of boundary points, converging toward analytic Mie theory (both polarisations):

Near field of a Gielis m = 6 star under plane-wave illumination (scattered field outside the boundary, internal field inside):

Relative local density of states (Purcell map) around the same Gielis m = 6 star, at one of its qsca resonances: a line-dipole emitter placed in a red lobe decays faster than in free space (1 + 4·Im S > 1), while blue regions suppress it. The six-fold pattern mirrors the particle's symmetry. The drive and the decay rate of an embedded emitter both come from this map — it is the entry point of quantum-dynamics calculations downstream:

Regenerate them with:

uv run python examples/convergence_study.py
uv run python examples/nearfield_map.py
uv run python examples/purcell_map.py

Formulation (summary)

• The cylinder is invariant along its axis, so Maxwell reduces to a scalar Helmholtz problem for one field component (E_y for TE, H_y for TM).
• The self-consistent field solution is given everywhere in terms of the surface field and its normal derivative; matching across the interface gives a Fredholm integral equation of the second kind.
• Discretising the boundary with nn quadrature points yields a dense 2nn × 2nn complex system M(λ)·ei = rhs, solved directly.
• The logarithmic Green-function singularity is handled analytically in the diagonal terms; complex wavenumbers are supported throughout.
• Lengths are in nm, the time convention is exp(-iωt), and outgoing waves are H_n^{(1)}.

Full details and every sign/layout convention are in docs/conventions.md. The analytic reference is Bohren & Huffman, Absorption and Scattering of Light by Small Particles, ch. 8; the surface-integral formulation follows Valencia et al's formulation.

Validation

The physics test suite compares the solver against analytic Mie theory for a circular cylinder: scattering / extinction / absorption efficiencies, the optical theorem on a lossy particle, energy conservation on a lossless one, the convergence rate, and the 2-D 1/√(kr) far-field decay. At nn = 300 the efficiencies agree with Mie to a few parts in 10³; the error decreases with nn until it reaches the fixed angular-quadrature floor of the far-field integrator. See tests/ for the exact tolerances and the reasoning behind them.

The line-dipole / self-Green machinery (v0.2) is validated the same way: reciprocity of the scattered field (to 10⁻⁶), the free-space limit (LDOS → 1 far from the particle), LDOS positivity, and — the strong anchor — the self-Green function of a circular cylinder against its closed-form Graf-addition-theorem sum on both Re S and Im S. That near-field anchor converges at first order in nn, so it is run at nn = 1000 to reach 1 %; the resolved scattered-field sign convention is recorded in docs/conventions.md.

Install / run / test

Requires Python 3.12 and uv.

uv sync                       # create the environment
uv run pytest                 # run the validation suite
uv run ruff format --check .  # formatting
uv run ruff check .           # lint

Minimal use:

from pysie2d import BIESolver, Geometry, Material

geom = Geometry.gielis(rad=200, n_pts=300, m=0)   # circular cylinder, nm
mat = Material(n_core=1.5, n_clad=1.0, pol=2)      # TE
result = BIESolver(geom, mat).scatter(wavelength=600.0)

print(result.efficiencies())                       # {'qsca', 'qext', 'qabs'}

Line-dipole emitter and Purcell effect:

from pysie2d import BIESolver, Geometry, Material, relative_ldos

geom = Geometry.gielis(rad=200, n_pts=300, m=6, n1=6, n2=12, n3=12)  # Gielis star
solver = BIESolver(geom, Material(n_core=2.0))
print(relative_ldos(solver, wavelength=540.0, x_s=430.0, z_s=0.0))  # LDOS vs free space

Performance

The system is a dense 2nn × 2nn complex matrix; at nn = 300 (a 600 × 600 solve) a single wavelength takes below one second in a modern computer, so wavelength sweeps are cheap serial for loops — no parallelism required.

For a Purcell map, every grid point is a different source position, hence a different right-hand side — but the matrix M(λ) is the same for all of them. relative_ldos_map therefore factorises M once with scipy.linalg.lu_factor and reuses it across all sources (lu_solve), turning what would be an hour-long sweep into a few seconds.

Roadmap

• v0.1.0 — core scattering: plane-wave excitation, near/far fields, cross-section efficiencies, Mie validation, convergence study, CI.
• v0.2.0 — line-dipole (point-source) excitation and the self-Green function → relative LDOS / Purcell maps. (this release)
• v0.3.0 — quasi-normal-mode extraction via Beyn's contour method, validated against analytic Mie resonances.

License

MIT — see LICENSE.