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 core — the part that can be validated end-to-end against a closed-form reference. That core includes quasi-normal-mode extraction from the surface-integral operator (see Quasi-normal modes) and finite clusters of particles (see Clusters of particles), validated against an independent two-cylinder addition-theorem anchor. Potential extensions in the mid/long-term include slab waveguide backgrounds and non-circular external validation against independent solvers (MEEP, dolfinx).

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.

Clusters of particles

A Cluster is a finite arrangement of non-overlapping particles sharing one background. The feature is purely additive: nothing in the single-particle BIESolver / ScatterResult API changes. Each particle keeps its own Geometry, its own n_pts and its own Material; the polarisation and the background index n_clad belong to the problem and must agree across the cluster.

from pysie2d import Cluster, ClusterBIESolver, Geometry, Material

left = Geometry.gielis(rad=200, n_pts=200, m=0, x0=-350.0)    # circle, nm
right = Geometry.gielis(rad=150, n_pts=200, m=4, b=1.0, x0=350.0)
cluster = Cluster([left, right])

mats = [Material(n_core=1.5, n_clad=1.0, pol=2),              # TE, one background
        Material(n_core=2.0, n_clad=1.0, pol=2)]
res = ClusterBIESolver(cluster, mats, pol=2).scatter(wavelength=633.0, angle=30.0)

amp, angles = res.far_field()                # total amplitude of the cluster
print(res.cross_sections())                  # {'c_sca', 'c_ext', 'c_abs'}, nm
print(res.resolution())                      # points per interior wavelength

scatter_dipole(wavelength, x_s, z_s) drives the same cluster with a line dipole, eval_field(x, z) gives the field at arbitrary points inside or outside, and ei_particle(p) hands back one particle's (φ_p, χ_p) sub-vector.

What you need to know before using it:

• Overlapping boundaries raise ClusterOverlapError. The coupled formulation represents a region exterior to every particle and interior to exactly one; where two boundaries cross, that partition does not exist and no resolution repairs it.
• A too-small gap warns, it does not raise. Two nearly touching boundaries put a near-singularity of the cross-block integrand a conformal half-width ~√(2·gap/a) off the real axis, so the quadrature needs about GAP_ENVELOPE_C·√(a/gap) nodes per particle; below that the solution degrades silently, with a well-conditioned matrix and clean single-particle diagnostics. ClusterGapWarning says so. The constant is a measured envelope, not a guess — BIE self-convergence over gap/a ∈ [0.032, 5.06], two polarisations and three size parameters, written up in docs/design/studies/cluster-gap-envelope.md. It was measured on circles; for non-circular facing boundaries the warning is diagnostic only, and says so in its text.
• Ragged resolution warns too. The coupled system is only as accurate as its worst block, and an under-resolved particle is invisible from any other one, so a spread wider than 4× in points per interior wavelength raises ClusterResolutionWarning.
• cross_sections() (absolute, in nm) exists on both the single-particle ScatterResult and the cluster result, which is what makes the two paths comparable. efficiencies() and multipoles() are single-particle only: a cluster has no rad to normalise by and no single centre to expand about.

The validation anchor is the analytic addition-theorem solution for two parallel circular cylinders, pysie2d.reference.two_cylinder — the same role reference.mie plays for the single-particle solver. Like reference.mie it is not exported at top level; import it explicitly to reproduce the validation.

On scale: the dense solve, not assembly, is the cost here — it reaches parity with assembly at Np = 2 and dominates from Np = 3, with Np = 16 still a usable 2.7 s and 625 MiB (performance §6).

A substrate under the particle

background=HalfSpace(eps_sub, z_int) on BIESolver, ClusterBIESolver and QNMSolver puts a half-space below the horizontal plane z = z_int, with the particle (and every source and observation point) above it. The feature is additive: without the argument nothing changes, bit for bit.

from pysie2d import BIESolver, Geometry, HalfSpace, Material, relative_ldos

geom = Geometry.gielis(rad=100, n_pts=256, m=0, z0=115.0)       # 15 nm above z = 0
mat = Material(n_core=2.0, n_clad=1.0, pol=1)
silver = HalfSpace(-18.3 + 0.48j, z_int=0.0)        # absolute complex ε; Re ε < 0 is fine
solver = BIESolver(geom, mat, background=silver)

res = solver.scatter(wavelength=633.0, angle=20.0)  # |angle| < 90°, from the cover
print(res.cross_sections())                         # {'c_sca_up', 'c_abs'}, nm
print(relative_ldos(solver, 633.0, 60.0, 330.0))    # LDOS above the interface

HalfSpace.pec(z_int) is a perfect conductor, evaluated as the exact image term with no integral. Material.from_eps(eps, n_clad, pol) builds a particle from an absolute complex permittivity, which a real n_core cannot express for a metal. eps_sub may be a callable ε(λ_vac) for driven solves; QNMSolver rejects it, because tabulated data is not holomorphic and the contour argument would break silently.

The reflected field is a Sommerfeld integral on a deformed path, so it is valid at complex wavelength — the real-axis integral with the usual Im α ≥ 0 rule returns the incoming wave there (conventions §15). The reflected blocks cost about a tenth of the free-space assembly at nn = 256.

What changes on the results:

• far_field() is the upward amplitude on [−π/2, π/2]; there is no transmitted far field.
• cross_sections() returns {'c_sca_up', 'c_abs'} and efficiencies() {'qsca_up', 'qabs'} — the free-space keys are absent, not raising, because c_ext and the total c_sca are undefined when power also enters the substrate. c_sca_up omits that power, so it is incomplete by construction; c_abs comes from the boundary flux and is exact.
• eval_field returns the scattered field; the incident and reflected-incident waves are the caller's. relative_ldos is still normalised to the unbounded cover, so it equals the substrate-only enhancement with no particle.
• multipoles() and QNMResult.sensitivity raise NotImplementedError; QNMResult.refine works.

Guards: a boundary too coarse for its distance to the interface warns (InterfaceGapWarning: local node spacing over twice the gap above 0.25 for TE, 0.2 for TM, so a flat facet is caught), a search box whose hump cannot clear a singularity raises, and so does a lossless ε_sub = −n_clad² (a surface-plasmon resonance with an infinite pole). These are 2-D line-source fields: absolute LDOS next to an interface is not a 3-D number, while Q, detunings and ratios carry over.

Validation is against the PEC image problem (a particle plus its mirror, solved by ClusterBIESolver, to ~1e-13 in φ, χ and the near field), the PEC closed form, a real-axis QUADPACK reference, and Mie. For a dielectric or lossy substrate the QNM check is consistency only — the poles do not move when the path is deformed differently — until the MEEP comparison of v1.0.

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.
• v0.8.0 — finite clusters of arbitrary particles: coupled assembly, dense solve, and absolute cross-sections, anchored on an independent two-cylinder addition-theorem reference. Purely additive.
• v0.9.0 — a half-space substrate: background=HalfSpace(...) on the single-particle, cluster and QNM solvers, reflected Green function by Sommerfeld integrals on a deformed path (valid at complex wavelength). 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.