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. That core now includes quasi-normal-mode extraction from the surface-integral operator (see Quasi-normal modes). Potential extensions in the mid/long-term include slab waveguide backgrounds and multiple-particle simulations.

Figures

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

The same error over size parameter x = 0.1 … 30 and nn, log colour scale. The error falls from 10⁻³ to 10⁻¹² across a narrow front (black contours, 10⁻⁶ solid); past it the solver is at round-off. The white line is 6 points per interior wavelength λ/n_core, which tracks the front's round-off edge from x ≈ 2 up — 10⁻⁶ itself is reached at a median of 4.7:

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:

Quasi-normal modes of a circular cylinder in the complex wavelength plane, extracted from seven search boxes and plotted over the analytic Mie poles of the same cylinder (open marks analytic, filled extracted). The axes are log-log so that each iso-Q contour is a straight line, since Q = Re λ / (2 Im λ):

The same physics, one contour. A single rectangle spanning 600 nm of Re λ returns all eleven TE modes inside it — counting the doubly degenerate pairs — from one call and 128 matrix assemblies. Beyn's method costs 4·n_quad_per_side assemblies whatever is inside the contour, so eleven modes cost no more than one; only the probe count has to exceed the mode count. Grey marks are poles outside the box (three of them leak a rank direction in, hence rank = 14 against 11 modes):

Regenerate them with:

uv run python examples/convergence_study.py
uv run python examples/convergence_map.py
uv run python examples/nearfield_map.py
uv run python examples/purcell_map.py
uv run python examples/qnm_spectrum.py
uv run python examples/qnm_wide_window.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)}.
• Wavelengths are vacuum wavelengths. Material.n_core, Material.n_clad and Material.epsi are absolute; the background index enters through the single conversion Material.wnum_bg(λ_vac) = 2π·n_clad/λ_vac, and the operator sees only background-relative quantities (Material.nc, Material.eps). The Mie size parameter x = 2π·n_clad·rad/λ_vac is exposed as the derived pysie2d.size_parameter (circular geometry only).

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 boundary-integral approach follows Maradudin, Michel, McGurn & Méndez, Enhanced backscattering of light from a random grating, Ann. Phys. 203 (1990) 255–307, developed there for randomly rough surfaces; this implementation uses the closed-surface (particle) form given in Valencia et al..

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. Boundary quadrature is Kress–Martensen product quadrature, so convergence is spectral: on the circle the efficiencies reach double-precision round-off against Mie by nn ≈ 30, and the tests hold them to 10⁻¹². 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, which agree to 10⁻¹³ at nn = 240; the resolved scattered-field sign convention is recorded in docs/conventions.md.

Quasi-normal-mode extraction (v0.4) is anchored the same way, in three independent layers: the analytic Mie poles are located first and their completeness checked against a winding-number count; the contour algorithm is checked on synthetic matrix pencils with known spectra; and only then is the composition tested — that the BIE operator's singularities are the Mie poles, to within its discretisation error and nothing more. That error is spectral too: 10⁻¹³ nm by nn = 40 on the circle, where the contour integral rather than the boundary becomes the accuracy floor. On non-circular shapes, modes are identified by continuation from the circle (conventions §8).

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)  # vacuum nm

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))
# wavelength is vacuum nm; the LDOS is relative to the unbounded background
print(relative_ldos(solver, wavelength=540.0, x_s=430.0, z_s=0.0))

Quasi-normal modes

A quasi-normal mode is a source-free solution: a complex wavelength where the boundary-integral operator M(λ) is singular. QNMSolver finds every mode inside a rectangle of the complex λ-plane by contour integration (Beyn's method) — no initial guess and no scan, at a cost independent of how many modes are inside.

from pysie2d import Geometry, Material, QNMSolver

geom = Geometry.gielis(rad=200, n_pts=200, m=0)
mat = Material(n_core=3.0, n_clad=1.0, pol=2)             # TE
res = QNMSolver(geom, mat).modes(745 + 2j, 775 + 15j)     # box corners, vacuum nm

print(res.wavelengths)      # 760.687 + 7.948j, twice — a degenerate pair
print(res.quality_factors)  # Q = Re λ / (2 Im λ)
print(res.edge_margin)      # contour-quality diagnostic; near zero is a warning

Search boxes must lie in Im λ > 0 (the decaying half-plane under exp(-iωt)) and Re λ > 0 (which keeps M(λ) holomorphic); both are asserted. Poles do not come in conjugate pairs here, and every n ≥ 1 mode of a circle is doubly degenerate.

Read docs/qnm-guide.md before using this — it covers how to place a box, how to read the diagnostics, what refine() does and does not buy you, and the limitations the feature ships with.

Mode sensitivity

QNMResult.sensitivity(at) returns dλ/dp for every mode from the adjoint quotient − uᴴ(∂M/∂p)v / uᴴ(∂M/∂λ)v, re-extracting no eigenvalue. at is a callable δ → (geometry, material), so a shape parameter and a refractive index go through one signature and one code path.

res = QNMSolver(geom, mat).modes(745 + 2j, 775 + 15j).refine()
frozen = res.geometry.parametrisation          # the node map must be frozen

def wider(delta):                              # dλ/db, b in its own units
    return Geometry.gielis(rad=200, n_pts=200, m=4, b=1.0 + delta,
                           parametrisation=frozen), mat

print(res.sensitivity(wider))                  # ≈ λ/2 for both partners

On the circle, stretching one axis is half of a uniform dilation, so each partner moves at λ/2. The geometry at returns must be built on res.geometry.parametrisation — a shape derivative holds the node map fixed, and sensitivity checks the nodes and both map derivatives exactly and raises otherwise. Degenerate poles dispatch to a secular problem rather than raising. dλ/dp converges spectrally in n_pts, like λ itself, so no extrapolation is needed. Conventions §10, §11 and §12 carry the details and the measured anchors.

Multipole decomposition

Outside the circumscribing circle of the particle the scattered field is a sum of outgoing cylindrical harmonics about the particle centre,

ψ_sc(r, θ) = Σ_m c_m · i^m · H_m^(1)(k_bg·r) · e^{imθ}

with θ measured from +z toward +x, as everywhere else in this package. ScatterResult.multipoles() returns those coefficients:

mp = BIESolver(geom, mat).scatter(wavelength=600.0, angle=30.0).multipoles(mmax=8)

mp.c          # signed orders m = −mmax … mmax
mp.c_plus     # symmetric   c_m + c_{−m}  (c_0⁺ = c_0)
mp.c_minus    # antisymmetric c_m − c_{−m}
mp.reconstruct(r, theta)     # the field back, on any larger circle

For pol = 1 (TM) the symmetric coefficients read as magnetic dipole (m = 0), electric dipole (m = 1) and electric quadrupole (m = 2). On a circle the coefficients are the analytic Mie cylinder coefficients exactly — c_m = −(−1)^m·χ_m·e^{imα}, with χ = b_n for TE and a_n for TM — and the tests hold both polarisations to 3·10⁻¹⁵ at normal and oblique incidence and at complex wavelength.

The defaults are the substance: the evaluation circle sits at 3 × the circumscribing radius (which is not Geometry.rad — on a rounded square the circumscribing radius is 2.29 × rad), and the angular grid is sized from both mmax and |k_bg·r0|. Evaluating inside the circumscribing circle is rejected, an angular grid too coarse to separate the retained orders is rejected, and an under-resolved angular spectrum raises RuntimeWarning through the spectrum_tail diagnostic on the result. Each guard stands in front of a measured silent failure — the first is 58 % error at 0.9 × the circumscribing radius.

What it does not give you: the expansion says nothing about the field between the circumscribing circle and the boundary, the coefficients are referred to the particle centre and no translation theorem is implemented, and the only closed-form anchor is Mie on a circle — on a non-circular shape the available check is self-consistency between two circles, which validates the expansion against the solver's own field and not the field itself.

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.

Matrix assembly dominates a single solve, and almost all of that cost is Hankel evaluation. For real arguments H_n^{(1)} = J_n + i·Y_n exactly, and the Cephes J_n/Y_n kernels are an order of magnitude faster than the general complex-argument algorithm — so hank0/hank1/cbesh dispatch on the argument at runtime, making assemble_matrix about 5× faster for a non-absorbing particle and 1.8× for an absorbing one. Complex wavenumbers take the original path and are bit-identical, which is what keeps quasi-normal-mode work possible.

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, and it batches the reuse: one multi-RHS BLAS-3 lu_solve and one vectorised representation-formula evaluation per chunk rather than a per-point loop (7.5× per source point). What would be an hour-long sweep takes 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.
• v0.3.0 — performance: Cephes fast path for real-argument Hankel functions and a batched, factorise-once relative_ldos_map.
• v0.4.0 — vacuum-wavelength and background-index conventions (breaking), and quasi-normal-mode extraction via Beyn's contour method, validated against analytic Mie resonances.
• v0.4.2 — exact scale covariance of the discrete BIE system, recorded as conventions §9.
• v0.5.0 — threaded contour integration in contour_moments, and an adjoint eigenvalue-sensitivity API (dλ/dp per mode) on top of the identity already proved in conventions §9.
• v0.6.0 — spectral convergence: Kress–Martensen product quadrature replaces the first-order diagonal self-patch, and nodes are placed by a smooth Parametrisation map. Breaking — see below.
• v0.7.0 — multipole decomposition of the scattered field into cylindrical harmonics about the particle centre, anchored on analytic Mie in both polarisations. Purely additive.

Migrating from v0.5

v0.5 converged at first order in nn; v0.6 converges spectrally, so every number changes — for the better, typically by many orders of magnitude at the same nn. Three calls change:

# v0.5
geom = Geometry.gielis(rad=200, n_pts=200, m=4, b=1.2)   # uniform arc length
wider = Geometry.gielis(rad=200, n_pts=200, m=4, b=1.3,
                        theta=geom.theta)                # frozen nodes

# v0.6
geom = Geometry.gielis(rad=200, n_pts=200, m=4, b=1.2)   # uniform θ (new default)
wider = Geometry.gielis(rad=200, n_pts=200, m=4, b=1.3,
                        parametrisation=geom.parametrisation)

• theta= is now parametrisation=. The new quadrature needs the node map's first and second derivatives at every node, and those cannot be recovered from an array of angles. Accepting a bare array would return a plausible wrong answer, so theta= is gone, and an angle array passed as parametrisation= raises TypeError with the migration in the message.

• Nodes are equispaced in θ by default, not in arc length: under the new quadrature uniform θ converges faster on every star and superellipse measured within the solver's validity envelope, at low resolution as well as high. For uniform arc length pass parametrisation=Parametrisation.gielis(...) (conventions §13).

• Geometry(...) built from your own arrays takes derivatives with respect to the quadrature parameter t and no longer takes delt; assemble_matrix, assemble_matrix_dwn and assemble_matrix_reference no longer take delt either. Odd m now requires a == b and n2 == n3, since the boundary otherwise does not close.

• n_pts defaults to 100, was 200. The circle anchors reach round-off by nn ≈ 30–40 under the new quadrature, so the old default was sized for a first-order scheme that no longer exists. Pass n_pts explicitly to keep a v0.5 resolution.

richardson_limit still exists but is deprecated: nothing in the package converges at first order any more, and extrapolating a spectrally converged pair makes it worse.

License

MIT — see LICENSE.