Metadata-Version: 2.3
Name: pysie2d
Version: 0.5.0
Summary: 2-D surface-integral solver for electromagnetic scattering from a cylinder in a homogeneous background.
Author: Claudio Silvestre Castro
Author-email: Claudio Silvestre Castro <claudio.silvestre.castro@gmail.com>
License: MIT
Requires-Dist: numpy>=1.26
Requires-Dist: scipy>=1.11
Requires-Python: >=3.12
Description-Content-Type: text/markdown

# pysie2d

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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](#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):

![Convergence to Mie theory](https://raw.githubusercontent.com/claudio-sc/pysie2d/main/figures/convergence_study.png)

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

![Near-field map](https://raw.githubusercontent.com/claudio-sc/pysie2d/main/figures/nearfield_map.png)

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:

![Purcell map](https://raw.githubusercontent.com/claudio-sc/pysie2d/main/figures/purcell_map.png)

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)`:

![QNM spectrum](https://raw.githubusercontent.com/claudio-sc/pysie2d/main/figures/qnm_spectrum.png)

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):

![Wide-window QNM extraction](https://raw.githubusercontent.com/claudio-sc/pysie2d/main/figures/qnm_wide_window.png)

Regenerate them with:

```bash
uv run python examples/convergence_study.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](https://github.com/claudio-sc/pysie2d/blob/main/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.*](https://doi.org/10.1364/JOSAB.20.002150).

## 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](https://github.com/claudio-sc/pysie2d/blob/main/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 converges at
first order in `nn`, in both `Re λ` and `Im λ`.

## Install / run / test

Requires Python 3.12 and [uv](https://docs.astral.sh/uv/).

```bash
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:

```python
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:

```python
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.

```python
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)      # 760.326 + 7.770j, 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](https://github.com/claudio-sc/pysie2d/blob/main/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.

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

def wider(delta):                              # dλ/db, b in its own units
    return Geometry.gielis(rad=200, n_pts=200, theta=theta,
                           m=4, b=1.2 + delta), mat

print(res.sensitivity(wider))
```

The geometry `at` returns must carry `res.geometry.theta` **exactly** — a shape
derivative holds the node set fixed, and differentiating the arc-length
parametrisation along with the physics costs two orders of convergence. Degenerate
poles dispatch to a secular problem rather than raising. `dλ/dp` converges at
first order in `n_pts`, so `richardson_limit` extrapolates two rungs to the limit
for less than the cost of one finer one. Conventions
[§10](https://github.com/claudio-sc/pysie2d/blob/main/docs/conventions.md),
§11 and §12 carry the details and the measured anchors.

## 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](https://github.com/claudio-sc/pysie2d/blob/main/docs/conventions.md) §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. _(latest release)_ **No breaking
  changes**; every v0.4.x call still means what it meant.

### What v0.5 adds to `Geometry`

`Geometry` now records the boundary node angles it was built on, as
`Geometry.theta`, and `Geometry.gielis` accepts `theta=` to build a shape on
angles supplied from elsewhere:

```python
base  = Geometry.gielis(rad=200, n_pts=200, m=4, b=1.2)   # unchanged
wider = Geometry.gielis(rad=200, n_pts=200, m=4, b=1.3,
                        theta=base.theta)                 # new: same node set
```

Both arguments are optional and both are additions — `Geometry.gielis` places
nodes by uniform arc length when you omit `theta`, exactly as before, and
`Geometry(...)` built directly from your own arrays works without one, leaving
`theta` as `None`. Scattering, fields, LDOS and mode extraction never read it.

It exists for shape derivatives. `QNMResult.sensitivity` evaluates `M(p₀−h)` and
`M(p₀+h)` and must do so on the **same** node set; if the nodes are re-placed
between the two, the difference quotient differentiates the arc-length
parametrisation along with the physics. That error term is `O(h)` rather than
`O(h²)`, it is not monotone in `h`, and it **grows** with `n_pts` — the one error
in this package that refinement makes worse. Freezing the nodes takes the
measured convergence rate on `∂M/∂b` from 2.7 to 100.1, against an ideal of 100.

So `sensitivity` refuses a geometry with no node set, and names which one is
missing, rather than falling back to re-inversion — a fallback would return a
wrong answer that looks exactly like a right one. Conventions
[§10](https://github.com/claudio-sc/pysie2d/blob/main/docs/conventions.md).

## License

MIT — see [LICENSE](LICENSE).
