bump version to v0.12

- Properly account for rmin vs r0
- Change return values to match waveguide_2d
- Change operator definition to look more like waveguide_2d

remaining TODO:
- Fix docs
- Further consolidate operators vs waveguide_2d
- Figure out E/H field conversions

.gitignore

@@ -54,6 +54,10 @@ coverage.xml
 
 # documentation
 doc/
+site/
+_doc_mathimg/
+doc.md
+doc.htex
 
 # PyBuilder
 target/

README.md

@@ -56,6 +56,21 @@ linear systems, ideally with double precision.
 
 Install from PyPI with pip:
 ```bash
+pip3 install meanas
+```
+
+Optional extras:
+
+- `meanas[test]`: pytest and coverage
+- `meanas[docs]`: MkDocs-based documentation toolchain
+- `meanas[examples]`: optional runtime dependencies used by the tracked examples
+- `meanas[dev]`: the union of `test`, `docs`, and `examples`, plus local lint/docs-publish helpers
+
+Examples:
+```bash
+pip3 install 'meanas[test]'
+pip3 install 'meanas[docs]'
+pip3 install 'meanas[examples]'
 pip3 install 'meanas[dev]'
 ```
 
@@ -80,9 +95,13 @@ source my_venv/bin/activate
 # Install in-place (-e, editable) from ./meanas, including development dependencies ([dev])
 pip3 install --user -e './meanas[dev]'
 
-# Run tests
+# Fast local iteration: excludes slower 3D/integration/example-smoke checks
 cd meanas
-python3 -m pytest -rsxX | tee test_results.txt
+python3 -m pytest -q -m "not complete"
+
+# Complete pre-commit confidence run: includes the slower integration tests and
+# tracked example smoke tests
+python3 -m pytest -q | tee test_results.txt
 ```
 
 #### See also:
@@ -94,6 +113,125 @@ python3 -m pytest -rsxX | tee test_results.txt
 
 ## Use
 
-See `examples/` for some simple examples; you may need additional
-packages such as [gridlock](https://mpxd.net/code/jan/gridlock)
-to run the examples.
+`meanas` is a collection of finite-difference electromagnetics tools:
+
+- `meanas.fdfd`: frequency-domain wave equations, sparse operators, SCPML, and
+  iterative solves for driven problems.
+- `meanas.fdfd.waveguide_2d` / `meanas.fdfd.waveguide_3d`: waveguide mode
+  solvers, mode-source construction, and overlap windows for port-based
+  excitation and analysis.
+- `meanas.fdtd`: Yee-step updates, CPML boundaries, flux/energy accounting, and
+  on-the-fly phasor extraction for comparing time-domain runs against FDFD.
+- `meanas.fdmath`: low-level finite-difference operators, vectorization helpers,
+  and derivations shared by the FDTD and FDFD layers.
+
+For most users, the tracked examples under `examples/` are the right entry
+point. The library API is primarily a toolbox; the module docstrings and API
+pages are there to document the mathematical conventions and derivations behind
+those tools.
+
+## Documentation
+
+API and workflow docs are generated from the package docstrings with
+[MkDocs](https://www.mkdocs.org/), [Material for MkDocs](https://squidfunk.github.io/mkdocs-material/),
+and [mkdocstrings](https://mkdocstrings.github.io/).
+
+Install the docs toolchain with:
+
+```bash
+pip3 install -e './meanas[docs]'
+```
+
+Then build the docs site with:
+
+```bash
+./make_docs.sh
+```
+
+This produces:
+
+- a normal multi-page site under `site/`
+- a combined printable single-page HTML site under `site/print_page/`
+- an optional fully inlined `site/standalone.html` when `htmlark` is available
+
+The docs build uses a local MathJax bundle vendored under `docs/assets/`, so
+the rendered HTML does not rely on external services for equation rendering.
+
+The tracked examples under `examples/` are the intended entry points for users:
+
+- `examples/fdtd.py`: broadband FDTD pulse excitation, phasor extraction, and a
+  residual check against the matching FDFD operator.
+- `examples/waveguide.py`: waveguide mode solving, unidirectional mode-source
+  construction, overlap readout, and FDTD/FDFD comparison on a guided structure.
+- `examples/waveguide_real.py`: real-valued continuous-wave FDTD on a straight
+  guide, with late-time monitor slices, guided-core windows, and mode-weighted
+  errors compared directly against real fields reconstructed from the matching
+  FDFD solution, plus a guided-mode / orthogonal-residual split.
+- `examples/eme.py`: straight-interface mode matching / EME, including port
+  mode solving, interface scattering, and modal field visualization.
+- `examples/eme_bend.py`: straight-to-bent waveguide mode matching with
+  cylindrical bend modes, interface scattering, and a cascaded bend-network
+  example built with `scikit-rf`.
+- `examples/fdfd.py`: direct frequency-domain waveguide excitation and overlap /
+  Poynting analysis without a time-domain run.
+
+Several examples rely on optional packages such as
+[gridlock](https://mpxd.net/code/jan/gridlock).
+
+### Frequency-domain waveguide workflow
+
+For a structure with a constant cross-section in one direction:
+
+1. Build `dxes` and the diagonal `epsilon` / `mu` distributions on the Yee grid.
+2. Solve the port mode with `meanas.fdfd.waveguide_3d.solve_mode(...)`.
+3. Build a unidirectional source with `compute_source(...)`.
+4. Build a matching overlap window with `compute_overlap_e(...)`.
+5. Solve the full FDFD problem and project the result onto the overlap window or
+   evaluate plane flux with `meanas.fdfd.functional.poynting_e_cross_h(...)`.
+
+### Time-domain phasor workflow
+
+For a broadband or continuous-wave FDTD run:
+
+1. Advance the fields with `meanas.fdtd.maxwell_e/maxwell_h` or
+   `updates_with_cpml(...)`.
+2. Inject electric current using the same sign convention used throughout the
+   examples and library: `E -= dt * J / epsilon`.
+3. Accumulate the desired phasor with `accumulate_phasor(...)` or the Yee-aware
+   wrappers `accumulate_phasor_e/h/j(...)`.
+4. Build the matching FDFD operator on the stretched `dxes` if CPML/SCPML is
+   part of the simulation, and compare the extracted phasor to the FDFD field or
+   residual.
+
+This is the primary FDTD/FDFD equivalence workflow. The phasor extraction step
+filters the time-domain run down to the guided `+\omega` content that FDFD
+solves for directly, so it is the cleanest apples-to-apples comparison.
+
+### Real-field reconstruction workflow
+
+For a continuous-wave real-valued FDTD run:
+
+1. Build the analytic source phasor for the structure, for example with
+   `waveguide_3d.compute_source(...)`.
+2. Run the real-valued FDTD simulation using the real part of that source.
+3. Solve the matching FDFD problem from the analytic source phasor on the
+   stretched `dxes`.
+4. Reconstruct late real `E/H/J` snapshots with
+   `reconstruct_real_e/h/j(...)` and compare those directly against the
+   real-valued FDTD fields, ideally on a monitor window or mode-weighted norm
+   centered on the guided field rather than on the full transverse plane. When
+   needed, split the monitor field into guided-mode and orthogonal residual
+   pieces to see whether the remaining mismatch is actually in the mode or in
+   weak nonguided tails.
+
+This is a stricter diagnostic, not the primary equivalence benchmark. A raw
+monitor slice contains both the guided field and the remaining orthogonal
+content on that plane,
+
+$$ E_{\text{monitor}} = E_{\text{guided}} + E_{\text{residual}} , $$
+
+so its full-plane instantaneous error is naturally noisier than the extracted
+phasor comparison even when the underlying guided `+\omega` content matches
+well.
+
+`examples/waveguide_real.py` is the reference implementation of this workflow.

docs/api/eigensolvers.md

@@ -0,0 +1,3 @@
+# eigensolvers
+
+::: meanas.eigensolvers

docs/api/fdfd.md

@@ -0,0 +1,15 @@
+# fdfd
+
+::: meanas.fdfd
+
+## Core operator layers
+
+::: meanas.fdfd.functional
+
+::: meanas.fdfd.operators
+
+::: meanas.fdfd.solvers
+
+::: meanas.fdfd.scpml
+
+::: meanas.fdfd.farfield

docs/api/fdmath.md

@@ -0,0 +1,13 @@
+# fdmath
+
+::: meanas.fdmath
+
+## Functional and sparse operators
+
+::: meanas.fdmath.functional
+
+::: meanas.fdmath.operators
+
+::: meanas.fdmath.vectorization
+
+::: meanas.fdmath.types

docs/api/fdtd.md

@@ -0,0 +1,15 @@
+# fdtd
+
+::: meanas.fdtd
+
+## Core update and analysis helpers
+
+::: meanas.fdtd.base
+
+::: meanas.fdtd.pml
+
+::: meanas.fdtd.boundaries
+
+::: meanas.fdtd.energy
+
+::: meanas.fdtd.phasor

docs/api/index.md

@@ -0,0 +1,14 @@
+# API Overview
+
+The package is documented directly from its docstrings. The most useful entry
+points are:
+
+- [meanas](meanas.md): top-level package overview
+- [eigensolvers](eigensolvers.md): generic eigenvalue utilities used by the mode solvers
+- [fdfd](fdfd.md): frequency-domain operators, sources, PML, solvers, and far-field transforms
+- [waveguides](waveguides.md): straight, cylindrical, and 3D waveguide mode helpers
+- [fdtd](fdtd.md): timestepping, CPML, energy/flux helpers, and phasor extraction
+- [fdmath](fdmath.md): shared discrete operators, vectorization helpers, and derivation background
+
+The waveguide and FDTD pages are the best places to start if you want the
+mathematical derivations rather than just the callable reference.

docs/api/meanas.md

@@ -0,0 +1,3 @@
+# meanas
+
+::: meanas

docs/api/waveguides.md

@@ -0,0 +1,7 @@
+# waveguides
+
+::: meanas.fdfd.waveguide_2d
+
+::: meanas.fdfd.waveguide_3d
+
+::: meanas.fdfd.waveguide_cyl

docs/assets/vendor/mathjax/manifest.json

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docs/index.md

@@ -0,0 +1,61 @@
+# meanas
+
+`meanas` is a Python package for finite-difference electromagnetic simulation.
+It combines:
+
+- `meanas.fdfd` for frequency-domain operators, sources, waveguide modes, and SCPML
+- `meanas.fdtd` for Yee-grid timestepping, CPML, energy/flux accounting, and phasor extraction
+- `meanas.fdmath` for the shared discrete operators and derivations underneath both solvers
+
+This documentation is built directly from the package docstrings. The API pages
+are the source of truth for the mathematical derivations and calling
+conventions.
+
+## Examples and API Map
+
+For most users, the tracked examples under `examples/` are the right entry
+point. They show the intended combinations of tools for solving complete
+problems.
+
+Relevant starting examples:
+
+- `examples/fdtd.py` for broadband pulse excitation and phasor extraction
+- `examples/waveguide.py` for guided phasor-domain FDTD/FDFD comparison
+- `examples/waveguide_real.py` for real-valued continuous-wave FDTD compared
+  against real fields reconstructed from an FDFD solution, including guided-core,
+  mode-weighted, and guided-mode / residual comparisons
+- `examples/eme.py` for straight-interface mode matching / EME and modal
+  scattering between two nearby waveguide cross-sections
+- `examples/eme_bend.py` for straight-to-bent mode matching with cylindrical
+  bend modes and a cascaded bend-network example
+- `examples/fdfd.py` for direct frequency-domain waveguide excitation
+
+For solver equivalence, prefer the phasor-based examples first. They compare
+the extracted `+\omega` content of the FDTD run directly against the FDFD
+solution and are the main accuracy benchmarks in the test suite.
+
+`examples/waveguide_real.py` answers a different, stricter question: how well a
+late raw real snapshot matches `Re(E_\omega e^{i\omega t})` on a monitor plane.
+That diagnostic is useful, but it also includes orthogonal residual structure
+that the phasor comparison intentionally filters out.
+
+The API pages are better read as a toolbox map and derivation reference:
+
+- Use the [FDTD API](api/fdtd.md) for time-domain stepping, CPML, phasor
+  extraction, and real-field reconstruction from FDFD phasors.
+- Use the [FDFD API](api/fdfd.md) for driven frequency-domain solves and sparse
+  operator algebra.
+- Use the [Waveguide API](api/waveguides.md) for mode solving, port sources,
+  and overlap windows.
+- Use the [fdmath API](api/fdmath.md) for the lower-level finite-difference
+  operators and the shared discrete derivations underneath both solvers.
+
+## Build outputs
+
+The docs build generates two HTML views from the same source:
+
+- a normal multi-page site
+- a print-oriented combined page under `site/print_page/`
+
+If `htmlark` is installed, `./make_docs.sh` also writes a fully inlined
+`site/standalone.html`.

docs/javascripts/mathjax.js

@@ -0,0 +1,19 @@
+window.MathJax = {
+  loader: {
+    load: ["input/tex-full", "output/chtml"]
+  },
+  tex: {
+    processEscapes: true,
+    processEnvironments: true
+  },
+  options: {
+    ignoreHtmlClass: ".*|",
+    processHtmlClass: "arithmatex"
+  }
+};
+
+document$.subscribe(() => {
+  MathJax.typesetPromise?.(
+    document.querySelectorAll(".arithmatex")
+  ).catch((error) => console.error(error));
+});

docs/stylesheets/extra.css

@@ -0,0 +1,64 @@
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+  overflow-x: auto;
+}
+
+.md-typeset .doc-contents {
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+}
+
+.md-typeset h1 code,
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+}
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+  --md-code-bg-color: #050505;
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+
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examples/eme.py

@@ -0,0 +1,221 @@
+"""
+Mode-matching / EME example for a straight rib-waveguide interface.
+
+This example shows the intended user-facing workflow for `meanas.fdfd.eme` on a
+simple straight interface:
+
+1. build two nearby waveguide cross-sections on a Yee grid,
+2. solve a small set of guided modes on each side,
+3. normalize those modes into E/H port fields,
+4. assemble the interface scattering matrix with `meanas.fdfd.eme.get_s(...)`,
+5. inspect the dominant modal coupling numerically and visually.
+"""
+
+from __future__ import annotations
+
+import importlib
+from typing import TYPE_CHECKING
+
+import numpy
+from numpy import pi
+
+import gridlock
+from gridlock import Extent
+
+from meanas.fdfd import eme, waveguide_2d
+from meanas.fdmath import unvec
+
+if TYPE_CHECKING:
+    from types import ModuleType
+
+
+WL = 1310.0
+DX = 40.0
+WIDTH = 400.0
+THF = 161.0
+THP = 77.0
+EPS_SI = 3.51 ** 2
+EPS_OX = 1.453 ** 2
+MODE_NUMBERS = numpy.array([0])
+
+
+def require_optional(name: str, package_name: str | None = None) -> ModuleType:
+    package_name = package_name or name
+    try:
+        return importlib.import_module(name)
+    except ImportError as exc:  # pragma: no cover - user environment guard
+        raise SystemExit(
+            f"This example requires the optional package '{package_name}'. "
+            "Install example dependencies with `pip install -e './meanas[examples]'`.",
+        ) from exc
+
+
+def build_geometry(
+        *,
+        dx: float = DX,
+        width: float = WIDTH,
+        thf: float = THF,
+        thp: float = THP,
+        eps_si: float = EPS_SI,
+        eps_ox: float = EPS_OX,
+        ) -> tuple[gridlock.Grid, numpy.ndarray, list[list[numpy.ndarray]], float]:
+    x0 = (width / 2) % dx
+    omega = 2 * pi / WL
+
+    grid = gridlock.Grid(
+        [
+            numpy.arange(-800, 800 + dx, dx),
+            numpy.arange(-400, 400 + dx, dx),
+            numpy.arange(-2 * dx, 2 * dx + dx, dx),
+        ],
+        periodic=True,
+    )
+    epsilon = grid.allocate(eps_ox)
+
+    grid.draw_cuboid(
+        epsilon,
+        foreground=eps_si,
+        x=Extent(center=x0, span=width + 1200),
+        y=Extent(min=0, max=thf),
+        z=Extent(min=-1e6, max=0),
+    )
+    grid.draw_cuboid(
+        epsilon,
+        foreground=eps_ox,
+        x=Extent(max=-width / 2, span=300),
+        y=Extent(min=thp, max=1e6),
+        z=Extent(min=-1e6, max=0),
+    )
+    grid.draw_cuboid(
+        epsilon,
+        foreground=eps_ox,
+        x=Extent(min=width / 2, span=300),
+        y=Extent(min=thp, max=1e6),
+        z=Extent(min=-1e6, max=0),
+    )
+
+    grid.draw_cuboid(
+        epsilon,
+        foreground=eps_si,
+        x=Extent(max=-(width / 2 + 600), span=240),
+        y=Extent(min=0, max=thf),
+        z=Extent(min=0, max=1e6),
+    )
+    grid.draw_cuboid(
+        epsilon,
+        foreground=eps_si,
+        x=Extent(max=width / 2 + 600, span=240),
+        y=Extent(min=0, max=thf),
+        z=Extent(min=0, max=1e6),
+    )
+
+    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes_2d = [[d[0], d[1]] for d in dxes]
+    return grid, epsilon, dxes_2d, omega
+
+
+def solve_cross_section_modes(
+        epsilon_slice: numpy.ndarray,
+        *,
+        omega: float,
+        dxes_2d: list[list[numpy.ndarray]],
+        mode_numbers: numpy.ndarray = MODE_NUMBERS,
+        ) -> tuple[list[tuple[numpy.ndarray, numpy.ndarray]], numpy.ndarray]:
+    e_xys, wavenumbers = waveguide_2d.solve_modes(
+        epsilon=epsilon_slice.ravel(),
+        omega=omega,
+        dxes=dxes_2d,
+        mode_numbers=mode_numbers,
+    )
+    eh_fields = [
+        waveguide_2d.normalized_fields_e(
+            e_xy,
+            wavenumber=wavenumber,
+            dxes=dxes_2d,
+            omega=omega,
+            epsilon=epsilon_slice.ravel(),
+        )
+        for e_xy, wavenumber in zip(e_xys, wavenumbers, strict=True)
+    ]
+    return eh_fields, wavenumbers
+
+
+def print_summary(ss: numpy.ndarray, wavenumbers_left: numpy.ndarray, wavenumbers_right: numpy.ndarray, omega: float) -> None:
+    n_left = len(wavenumbers_left)
+    left_neff = numpy.real(wavenumbers_left / omega)
+    right_neff = numpy.real(wavenumbers_right / omega)
+
+    print('left effective indices:', ', '.join(f'{value:.5f}' for value in left_neff[:4]))
+    print('right effective indices:', ', '.join(f'{value:.5f}' for value in right_neff[:4]))
+
+    reflection = abs(ss[0, 0]) ** 2
+    transmission = abs(ss[n_left, 0]) ** 2
+    total_output = numpy.sum(abs(ss[:, 0]) ** 2)
+    print(f'fundamental left-incident reflection |S_00|^2      = {reflection:.6f}')
+    print(f'fundamental left-to-right transmission |S_{n_left},0|^2 = {transmission:.6f}')
+    print(f'fundamental left-incident total output power      = {total_output:.6f}')
+
+    strongest = numpy.argsort(abs(ss[n_left:, 0]) ** 2)[::-1][:3]
+    print('dominant transmitted right-side modes for left mode 0:')
+    for mode_index in strongest:
+        print(f'  mode {mode_index}: |S|^2 = {abs(ss[n_left + mode_index, 0]) ** 2:.6f}')
+
+
+def plot_results(
+        *,
+        pyplot: ModuleType,
+        ss: numpy.ndarray,
+        left_mode: tuple[numpy.ndarray, numpy.ndarray],
+        right_mode: tuple[numpy.ndarray, numpy.ndarray],
+        shape_2d: tuple[int, int],
+        ) -> None:
+    fig_s, ax_s = pyplot.subplots()
+    image = ax_s.imshow(abs(ss) ** 2, origin='lower', cmap='magma')
+    fig_s.colorbar(image, ax=ax_s)
+    ax_s.set_title('Interface scattering magnitude |S|^2')
+    ax_s.set_xlabel('Incident mode index')
+    ax_s.set_ylabel('Outgoing mode index')
+
+    e_left = unvec(left_mode[0], shape_2d)
+    e_right = unvec(right_mode[0], shape_2d)
+    left_intensity = numpy.sum(abs(e_left) ** 2, axis=0).T
+    right_intensity = numpy.sum(abs(e_right) ** 2, axis=0).T
+
+    fig_modes, axes = pyplot.subplots(1, 2, figsize=(10, 4))
+    left_plot = axes[0].imshow(left_intensity, origin='lower', cmap='viridis')
+    fig_modes.colorbar(left_plot, ax=axes[0])
+    axes[0].set_title('Left fundamental mode |E|^2')
+    right_plot = axes[1].imshow(right_intensity, origin='lower', cmap='viridis')
+    fig_modes.colorbar(right_plot, ax=axes[1])
+    axes[1].set_title('Right fundamental mode |E|^2')
+    if pyplot.get_backend().lower().endswith('agg'):
+        pyplot.close(fig_s)
+        pyplot.close(fig_modes)
+    else:
+        pyplot.show()
+
+
+def main() -> None:
+    pyplot = require_optional('matplotlib.pyplot', package_name='matplotlib')
+
+    grid, epsilon, dxes_2d, omega = build_geometry()
+    left_slice = epsilon[:, :, :, 1]
+    right_slice = epsilon[:, :, :, -2]
+
+    left_modes, wavenumbers_left = solve_cross_section_modes(left_slice, omega=omega, dxes_2d=dxes_2d)
+    right_modes, wavenumbers_right = solve_cross_section_modes(right_slice, omega=omega, dxes_2d=dxes_2d)
+
+    ss = eme.get_s(left_modes, wavenumbers_left, right_modes, wavenumbers_right, dxes=dxes_2d)
+
+    print_summary(ss, wavenumbers_left, wavenumbers_right, omega)
+    plot_results(
+        pyplot=pyplot,
+        ss=ss,
+        left_mode=left_modes[0],
+        right_mode=right_modes[0],
+        shape_2d=grid.shape[:2],
+    )
+
+
+if __name__ == '__main__':
+    main()

examples/eme_bend.py

@@ -0,0 +1,314 @@
+"""
+Mode-matching / EME example for a straight-to-bent waveguide interface.
+
+This example demonstrates a cylindrical-waveguide EME workflow:
+
+1. build a rib-waveguide cross-section,
+2. solve straight port modes with `waveguide_2d`,
+3. solve bend modes with `waveguide_cyl`,
+4. assemble the straight-to-bend interface scattering matrix with
+   `meanas.fdfd.eme.get_s(...)`,
+5. optionally cascade a straight section, bend section, and interface pair into
+   a compact multiport network using `scikit-rf`.
+"""
+
+from __future__ import annotations
+
+import importlib
+from typing import TYPE_CHECKING
+
+import numpy
+from numpy import pi
+from scipy import sparse
+
+import gridlock
+from gridlock import Extent
+
+from meanas.fdfd import eme, waveguide_2d, waveguide_cyl
+from meanas.fdmath import unvec
+
+if TYPE_CHECKING:
+    from types import ModuleType
+
+
+WL = 1310.0
+DX = 40.0
+RADIUS = 6e3
+WIDTH = 400.0
+THF = 161.0
+THP = 77.0
+EPS_SI = 3.51 ** 2
+EPS_OX = 1.453 ** 2
+MODE_NUMBERS = numpy.array([0])
+STRAIGHT_SECTION_LENGTH = 12e3
+BEND_ANGLE = pi / 2
+
+
+def require_optional(name: str, package_name: str | None = None) -> ModuleType:
+    package_name = package_name or name
+    try:
+        return importlib.import_module(name)
+    except ImportError as exc:  # pragma: no cover - user environment guard
+        raise SystemExit(
+            f"This example requires the optional package '{package_name}'. "
+            "Install example dependencies with `pip install -e './meanas[examples]'`.",
+        ) from exc
+
+
+def build_geometry(
+        *,
+        dx: float = DX,
+        width: float = WIDTH,
+        thf: float = THF,
+        thp: float = THP,
+        eps_si: float = EPS_SI,
+        eps_ox: float = EPS_OX,
+        ) -> tuple[gridlock.Grid, numpy.ndarray, list[list[numpy.ndarray]], float]:
+    x0 = (width / 2) % dx
+    omega = 2 * pi / WL
+
+    grid = gridlock.Grid(
+        [
+            numpy.arange(-800, 800 + dx, dx),
+            numpy.arange(-400, 400 + dx, dx),
+            numpy.arange(-2 * dx, 2 * dx + dx, dx),
+        ],
+        periodic=True,
+    )
+    epsilon = grid.allocate(eps_ox)
+
+    grid.draw_cuboid(
+        epsilon,
+        foreground=eps_si,
+        x=Extent(center=x0, span=width + 1200),
+        y=Extent(min=0, max=thf),
+        z=Extent(min=-1e6, max=0),
+    )
+    grid.draw_cuboid(
+        epsilon,
+        foreground=eps_ox,
+        x=Extent(max=-width / 2, span=300),
+        y=Extent(min=thp, max=1e6),
+        z=Extent(min=-1e6, center=0),
+    )
+    grid.draw_cuboid(
+        epsilon,
+        foreground=eps_ox,
+        x=Extent(min=width / 2, span=300),
+        y=Extent(min=thp, max=1e6),
+        z=Extent(min=-1e6, center=0),
+    )
+
+    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    dxes_2d = [[d[0], d[1]] for d in dxes]
+    return grid, epsilon, dxes_2d, omega
+
+
+def solve_straight_modes(
+        epsilon_slice: numpy.ndarray,
+        *,
+        omega: float,
+        dxes_2d: list[list[numpy.ndarray]],
+        mode_numbers: numpy.ndarray = MODE_NUMBERS,
+        ) -> tuple[list[tuple[numpy.ndarray, numpy.ndarray]], numpy.ndarray]:
+    e_xys, wavenumbers = waveguide_2d.solve_modes(
+        epsilon=epsilon_slice.ravel(),
+        omega=omega,
+        dxes=dxes_2d,
+        mode_numbers=mode_numbers,
+    )
+    eh_fields = [
+        waveguide_2d.normalized_fields_e(
+            e_xy,
+            wavenumber=wavenumber,
+            dxes=dxes_2d,
+            omega=omega,
+            epsilon=epsilon_slice.ravel(),
+        )
+        for e_xy, wavenumber in zip(e_xys, wavenumbers, strict=True)
+    ]
+    return eh_fields, wavenumbers
+
+
+def solve_bend_modes(
+        epsilon_slice: numpy.ndarray,
+        *,
+        omega: float,
+        dxes_2d: list[list[numpy.ndarray]],
+        rmin: float,
+        mode_numbers: numpy.ndarray = MODE_NUMBERS,
+        ) -> tuple[list[tuple[numpy.ndarray, numpy.ndarray]], numpy.ndarray, numpy.ndarray]:
+    e_xys, angular_wavenumbers = waveguide_cyl.solve_modes(
+        epsilon=epsilon_slice.ravel(),
+        omega=omega,
+        dxes=dxes_2d,
+        mode_numbers=mode_numbers,
+        rmin=rmin,
+    )
+    linear_wavenumbers = waveguide_cyl.linear_wavenumbers(
+        e_xys=e_xys,
+        angular_wavenumbers=angular_wavenumbers,
+        rmin=rmin,
+        epsilon=epsilon_slice.ravel(),
+        dxes=dxes_2d,
+    )
+    eh_fields = [
+        waveguide_cyl.normalized_fields_e(
+            e_xy,
+            angular_wavenumber=angular_wavenumber,
+            dxes=dxes_2d,
+            omega=omega,
+            epsilon=epsilon_slice.ravel(),
+            rmin=rmin,
+        )
+        for e_xy, angular_wavenumber in zip(e_xys, angular_wavenumbers, strict=True)
+    ]
+    return eh_fields, linear_wavenumbers, angular_wavenumbers
+
+
+def build_cascaded_network(
+        skrf: ModuleType,
+        *,
+        interface_s: numpy.ndarray,
+        straight_wavenumbers: numpy.ndarray,
+        bend_angular_wavenumbers: numpy.ndarray,
+        n_modes: int,
+        ) -> object:
+    net_sb = skrf.Network(f=[1 / WL], s=interface_s[numpy.newaxis, ...])
+    net_bs = net_sb.copy()
+    net_bs.renumber(numpy.arange(2 * n_modes), numpy.roll(numpy.arange(2 * n_modes), n_modes))
+
+    straight_phase = sparse.diags_array(numpy.exp(-1j * straight_wavenumbers[:n_modes] * STRAIGHT_SECTION_LENGTH))
+    bend_phase = sparse.diags_array(numpy.exp(-1j * bend_angular_wavenumbers[:n_modes] * BEND_ANGLE))
+    zero = numpy.zeros((n_modes, n_modes), dtype=complex)
+
+    straight_s = numpy.block([[zero, straight_phase.toarray()], [straight_phase.toarray(), zero]])
+    bend_s = numpy.block([[zero, bend_phase.toarray()], [bend_phase.toarray(), zero]])
+    net_straight = skrf.Network(f=[1 / WL], s=straight_s[numpy.newaxis, ...])
+    net_bend = skrf.Network(f=[1 / WL], s=bend_s[numpy.newaxis, ...])
+
+    return skrf.network.cascade_list([net_straight, net_sb, net_bend, net_bs, net_straight])
+
+
+def print_summary(
+        interface_s: numpy.ndarray,
+        cascaded_s: numpy.ndarray,
+        straight_wavenumbers: numpy.ndarray,
+        bend_linear_wavenumbers: numpy.ndarray,
+        bend_angular_wavenumbers: numpy.ndarray,
+        omega: float,
+        n_modes: int,
+        ) -> None:
+    straight_neff = numpy.real(straight_wavenumbers / omega)
+    bend_neff = numpy.real(bend_linear_wavenumbers / omega)
+    print('straight effective indices:', ', '.join(f'{value:.5f}' for value in straight_neff[:4]))
+    print('bend effective indices   :', ', '.join(f'{value:.5f}' for value in bend_neff[:4]))
+    print('bend angular wavenumbers :', ', '.join(f'{value:.5e}' for value in bend_angular_wavenumbers[:4]))
+
+    interface_transmission = abs(interface_s[n_modes, 0]) ** 2
+    interface_reflection = abs(interface_s[0, 0]) ** 2
+    print(f'interface fundamental transmission |S_{n_modes},0|^2 = {interface_transmission:.6f}')
+    print(f'interface fundamental reflection   |S_00|^2          = {interface_reflection:.6f}')
+
+    total_cascaded_output = numpy.sum(abs(cascaded_s[:, 0]) ** 2)
+    bend_through = abs(cascaded_s[n_modes, 0]) ** 2
+    bend_reflection = abs(cascaded_s[0, 0]) ** 2
+    print(f'cascaded bend through power        |S_{n_modes},0|^2 = {bend_through:.6f}')
+    print(f'cascaded bend reflection           |S_00|^2          = {bend_reflection:.6f}')
+    print(f'cascaded left-incident total output power            = {total_cascaded_output:.6f}')
+
+
+def plot_results(
+        *,
+        pyplot: ModuleType,
+        interface_s: numpy.ndarray,
+        cascaded_s: numpy.ndarray,
+        straight_mode: tuple[numpy.ndarray, numpy.ndarray],
+        bend_mode: tuple[numpy.ndarray, numpy.ndarray],
+        shape_2d: tuple[int, int],
+        ) -> None:
+    fig_s, axes = pyplot.subplots(1, 2, figsize=(12, 4))
+    interface_plot = axes[0].imshow(abs(interface_s) ** 2, origin='lower', cmap='magma')
+    fig_s.colorbar(interface_plot, ax=axes[0])
+    axes[0].set_title('Straight-to-bend interface |S|^2')
+    axes[0].set_xlabel('Incident mode index')
+    axes[0].set_ylabel('Outgoing mode index')
+
+    cascaded_plot = axes[1].imshow(abs(cascaded_s) ** 2, origin='lower', cmap='magma')
+    fig_s.colorbar(cascaded_plot, ax=axes[1])
+    axes[1].set_title('Cascaded bend network |S|^2')
+    axes[1].set_xlabel('Incident mode index')
+    axes[1].set_ylabel('Outgoing mode index')
+
+    straight_e = unvec(straight_mode[0], shape_2d)
+    bend_e = unvec(bend_mode[0], shape_2d)
+    straight_intensity = numpy.sum(abs(straight_e) ** 2, axis=0).T
+    bend_intensity = numpy.sum(abs(bend_e) ** 2, axis=0).T
+
+    fig_modes, axes_modes = pyplot.subplots(1, 2, figsize=(10, 4))
+    straight_plot = axes_modes[0].imshow(straight_intensity, origin='lower', cmap='viridis')
+    fig_modes.colorbar(straight_plot, ax=axes_modes[0])
+    axes_modes[0].set_title('Straight fundamental mode |E|^2')
+    bend_plot = axes_modes[1].imshow(bend_intensity, origin='lower', cmap='viridis')
+    fig_modes.colorbar(bend_plot, ax=axes_modes[1])
+    axes_modes[1].set_title('Bent fundamental mode |E|^2')
+    if pyplot.get_backend().lower().endswith('agg'):
+        pyplot.close(fig_s)
+        pyplot.close(fig_modes)
+    else:
+        pyplot.show()
+
+
+def main() -> None:
+    pyplot = require_optional('matplotlib.pyplot', package_name='matplotlib')
+    skrf = require_optional('skrf', package_name='scikit-rf')
+
+    grid, epsilon, dxes_2d, omega = build_geometry()
+    epsilon_slice = epsilon[:, :, :, 2]
+    rmin = RADIUS + grid.xyz[0].min()
+
+    straight_modes, straight_wavenumbers = solve_straight_modes(epsilon_slice, omega=omega, dxes_2d=dxes_2d)
+    bend_modes, bend_linear_wavenumbers, bend_angular_wavenumbers = solve_bend_modes(
+        epsilon_slice,
+        omega=omega,
+        dxes_2d=dxes_2d,
+        rmin=rmin,
+    )
+
+    interface_s = eme.get_s(
+        straight_modes,
+        straight_wavenumbers,
+        bend_modes,
+        bend_linear_wavenumbers,
+        dxes=dxes_2d,
+    )
+    cascaded = build_cascaded_network(
+        skrf,
+        interface_s=interface_s,
+        straight_wavenumbers=straight_wavenumbers,
+        bend_angular_wavenumbers=bend_angular_wavenumbers,
+        n_modes=len(MODE_NUMBERS),
+    )
+    cascaded_s = cascaded.s[0]
+
+    print_summary(
+        interface_s,
+        cascaded_s,
+        straight_wavenumbers,
+        bend_linear_wavenumbers,
+        bend_angular_wavenumbers,
+        omega,
+        len(MODE_NUMBERS),
+    )
+    plot_results(
+        pyplot=pyplot,
+        interface_s=interface_s,
+        cascaded_s=cascaded_s,
+        straight_mode=straight_modes[0],
+        bend_mode=bend_modes[0],
+        shape_2d=grid.shape[:2],
+    )
+
+
+if __name__ == '__main__':
+    main()

examples/fdfd0.py

@@ -0,0 +1,103 @@
+import numpy
+from numpy.linalg import norm
+from matplotlib import pyplot, colors
+import logging
+
+import meanas
+from meanas import fdtd
+from meanas.fdmath import vec, unvec
+from meanas.fdfd import waveguide_3d, functional, scpml, operators
+from meanas.fdfd.solvers import generic as generic_solver
+
+import gridlock
+
+
+logging.basicConfig(level=logging.DEBUG)
+logging.getLogger('matplotlib').setLevel(logging.WARNING)
+
+__author__ = 'Jan Petykiewicz'
+
+
+def pcolor(ax, v) -> None:
+    mappable = ax.pcolor(v, cmap='seismic', norm=colors.CenteredNorm())
+    ax.axis('equal')
+    ax.get_figure().colorbar(mappable)
+
+
+def test0(solver=generic_solver):
+    dx = 50                # discretization (nm/cell)
+    pml_thickness = 10     # (number of cells)
+
+    wl = 1550               # Excitation wavelength
+    omega = 2 * numpy.pi / wl
+
+    # Device design parameters
+    radii = (1, 0.6)
+    th = 220
+    center = [0, 0, 0]
+
+    # refractive indices
+    n_ring = numpy.sqrt(12.6)  # ~Si
+    n_air = 4.0   # air
+
+    # Half-dimensions of the simulation grid
+    xyz_max = numpy.array([1.2, 1.2, 0.3]) * 1000 + pml_thickness * dx
+
+    # Coordinates of the edges of the cells.
+    half_edge_coords = [numpy.arange(dx/2, m + dx, step=dx) for m in xyz_max]
+    edge_coords = [numpy.hstack((-h[::-1], h)) for h in half_edge_coords]
+
+    # #### Create the grid, mask, and draw the device ####
+    grid = gridlock.Grid(edge_coords)
+    epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
+    grid.draw_cylinder(
+        epsilon,
+        h = dict(axis='z', center=center[2], span=th),
+        radius = max(radii),
+        center2d = center[:2],
+        foreground = n_ring ** 2,
+        num_points = 24,
+        )
+    grid.draw_cylinder(
+        epsilon,
+        h = dict(axis='z', center=center[2], span=th * 1.1),
+        radius = min(radii),
+        center2d = center[:2],
+        foreground = n_air ** 2,
+        num_points = 24,
+        )
+
+    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    for a in (0, 1, 2):
+        for p in (-1, 1):
+            dxes = meanas.fdfd.scpml.stretch_with_scpml(dxes, axis=a, polarity=p, omega=omega,
+                                                        thickness=pml_thickness)
+
+    J = [numpy.zeros_like(epsilon[0], dtype=complex) for _ in range(3)]
+    J[1][15, grid.shape[1]//2, grid.shape[2]//2] = 1
+
+
+    #
+    # Solve!
+    #
+    sim_args = dict(
+        omega = omega,
+        dxes = dxes,
+        epsilon = vec(epsilon),
+        )
+    x = solver(J=vec(J), **sim_args)
+
+    A = operators.e_full(omega, dxes, vec(epsilon)).tocsr()
+    b = -1j * omega * vec(J)
+    print('Norm of the residual is ', norm(A @ x - b) / norm(b))
+
+    E = unvec(x, grid.shape)
+
+    #
+    # Plot results
+    #
+    grid.visualize_slice(E.real, plane=dict(z=0), which_shifts=1, pcolormesh_args=dict(norm=colors.CenteredNorm(), cmap='bwr'))
+
+
+if __name__ == '__main__':
+    test0()

examples/fdfd1.py

@@ -1,10 +1,13 @@
 import importlib
+import logging
 import numpy
 from numpy.linalg import norm
+from matplotlib import pyplot, colors
+import logging
 
 import meanas
 from meanas import fdtd
-from meanas.fdmath import vec, unvec
+from meanas.fdmath import vec, unvec, fdfield_t
 from meanas.fdfd import waveguide_3d, functional, scpml, operators
 from meanas.fdfd.solvers import generic as generic_solver
 
@@ -12,7 +15,6 @@ import gridlock
 
 from matplotlib import pyplot
 
-import logging
 
 logging.basicConfig(level=logging.DEBUG)
 logging.getLogger('matplotlib').setLevel(logging.WARNING)
@@ -20,82 +22,6 @@ logging.getLogger('matplotlib').setLevel(logging.WARNING)
 __author__ = 'Jan Petykiewicz'
 
 
-def test0(solver=generic_solver):
-    dx = 50                # discretization (nm/cell)
-    pml_thickness = 10     # (number of cells)
-
-    wl = 1550               # Excitation wavelength
-    omega = 2 * numpy.pi / wl
-
-    # Device design parameters
-    radii = (1, 0.6)
-    th = 220
-    center = [0, 0, 0]
-
-    # refractive indices
-    n_ring = numpy.sqrt(12.6)  # ~Si
-    n_air = 4.0   # air
-
-    # Half-dimensions of the simulation grid
-    xyz_max = numpy.array([1.2, 1.2, 0.3]) * 1000 + pml_thickness * dx
-
-    # Coordinates of the edges of the cells.
-    half_edge_coords = [numpy.arange(dx/2, m + dx, step=dx) for m in xyz_max]
-    edge_coords = [numpy.hstack((-h[::-1], h)) for h in half_edge_coords]
-
-    # #### Create the grid, mask, and draw the device ####
-    grid = gridlock.Grid(edge_coords)
-    epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
-    grid.draw_cylinder(epsilon,
-                       surface_normal=2,
-                       center=center,
-                       radius=max(radii),
-                       thickness=th,
-                       eps=n_ring**2,
-                       num_points=24)
-    grid.draw_cylinder(epsilon,
-                       surface_normal=2,
-                       center=center,
-                       radius=min(radii),
-                       thickness=th*1.1,
-                       eps=n_air ** 2,
-                       num_points=24)
-
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
-    for a in (0, 1, 2):
-        for p in (-1, 1):
-            dxes = meanas.fdfd.scpml.stretch_with_scpml(dxes, axis=a, polarity=p, omega=omega,
-                                                        thickness=pml_thickness)
-
-    J = [numpy.zeros_like(epsilon[0], dtype=complex) for _ in range(3)]
-    J[1][15, grid.shape[1]//2, grid.shape[2]//2] = 1
-
-
-    '''
-    Solve!
-    '''
-    sim_args = {
-        'omega': omega,
-        'dxes': dxes,
-        'epsilon': vec(epsilon),
-    }
-    x = solver(J=vec(J), **sim_args)
-
-    A = operators.e_full(omega, dxes, vec(epsilon)).tocsr()
-    b = -1j * omega * vec(J)
-    print('Norm of the residual is ', norm(A @ x - b))
-
-    E = unvec(x, grid.shape)
-
-    '''
-    Plot results
-    '''
-    pyplot.figure()
-    pyplot.pcolor(numpy.real(E[1][:, :, grid.shape[2]//2]), cmap='seismic')
-    pyplot.axis('equal')
-    pyplot.show()
-
-
 def test1(solver=generic_solver):
     dx = 40                 # discretization (nm/cell)
     pml_thickness = 10      # (number of cells)
@@ -122,7 +48,7 @@ def test1(solver=generic_solver):
     # #### Create the grid and draw the device ####
     grid = gridlock.Grid(edge_coords)
     epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
-    grid.draw_cuboid(epsilon, center=center, dimensions=[8e3, w, th], eps=n_wg**2)
+    grid.draw_cuboid(epsilon, x=dict(center=0, span=8e3), y=dict(center=0, span=w), z=dict(center=0, span=th), foreground=n_wg**2)
 
     dxes = [grid.dxyz, grid.autoshifted_dxyz()]
     for a in (0, 1, 2):
@@ -156,22 +82,14 @@ def test1(solver=generic_solver):
     # grid.draw_cuboid(pmcg, center=[700, 0, 0], dimensions=[80, 1e8, 1e8], eps=1)
     # grid.visualize_isosurface(pmcg)
 
-    def pcolor(v) -> None:
-        vmax = numpy.max(numpy.abs(v))
-        pyplot.pcolor(v, cmap='seismic', vmin=-vmax, vmax=vmax)
-        pyplot.axis('equal')
-        pyplot.colorbar()
-
-    ss = (1, slice(None), J.shape[2]//2+6, slice(None))
-#    pyplot.figure()
-#    pcolor(J3[ss].T.imag)
-#    pyplot.figure()
-#    pcolor((numpy.abs(J3).sum(axis=2).sum(axis=0) > 0).astype(float).T)
+    grid.visualize_slice(J.imag, plane=dict(y=6*dx), which_shifts=1, pcolormesh_args=dict(norm=colors.CenteredNorm(), cmap='bwr'))
+    fig, ax = pyplot.subplots()
+    ax.pcolormesh((numpy.abs(J).sum(axis=2).sum(axis=0) > 0).astype(float).T, cmap='hot')
     pyplot.show(block=True)
 
-    '''
-    Solve!
-    '''
+    #
+    # Solve!
+    #
     sim_args = {
         'omega': omega,
         'dxes': dxes,
@@ -188,20 +106,18 @@ def test1(solver=generic_solver):
 
     E = unvec(x, grid.shape)
 
-    '''
-    Plot results
-    '''
+    #
+    # Plot results
+    #
     center = grid.pos2ind([0, 0, 0], None).astype(int)
-    pyplot.figure()
-    pyplot.subplot(2, 2, 1)
-    pcolor(numpy.real(E[1][center[0], :, :]).T)
-    pyplot.subplot(2, 2, 2)
-    pyplot.plot(numpy.log10(numpy.abs(E[1][:, center[1], center[2]]) + 1e-10))
-    pyplot.grid(alpha=0.6)
-    pyplot.ylabel('log10 of field')
-    pyplot.subplot(2, 2, 3)
-    pcolor(numpy.real(E[1][:, :, center[2]]).T)
-    pyplot.subplot(2, 2, 4)
+    fig, axes = pyplot.subplots(2, 2)
+    grid.visualize_slice(E.real, plane=dict(x=0), which_shifts=1, ax=axes[0, 0], finalize=False, pcolormesh_args=dict(norm=colors.CenteredNorm(), cmap='bwr'))
+    grid.visualize_slice(E.real, plane=dict(z=0), which_shifts=1, ax=axes[0, 1], finalize=False, pcolormesh_args=dict(norm=colors.CenteredNorm(), cmap='bwr'))
+#    pcolor(axes[0, 0], numpy.real(E[1][center[0], :, :]).T)
+#    pcolor(axes[0, 1], numpy.real(E[1][:, :, center[2]]).T)
+    axes[1, 0].plot(numpy.log10(numpy.abs(E[1][:, center[1], center[2]]) + 1e-10))
+    axes[1, 0].grid(alpha=0.6)
+    axes[1, 0].set_ylabel('log10 of field')
 
     def poyntings(E):
         H = functional.e2h(omega, dxes)(E)
@@ -215,24 +131,28 @@ def test1(solver=generic_solver):
         return s0, s1, s2
 
     s0x, s1x, s2x = poyntings(E)
-    pyplot.plot(s0x[0].sum(axis=2).sum(axis=1), label='s0', marker='.')
-    pyplot.plot(s1x[0].sum(axis=2).sum(axis=1), label='s1', marker='.')
-    pyplot.plot(s2x[0].sum(axis=2).sum(axis=1), label='s2', marker='.')
-    pyplot.plot(E[1][:, center[1], center[2]].real.T, label='Ey', marker='x')
-    pyplot.grid(alpha=0.6)
-    pyplot.legend()
-    pyplot.show()
+    ax = axes[1, 1]
+    ax.plot(s0x[0].sum(axis=2).sum(axis=1), label='s0', marker='.')
+    ax.plot(s1x[0].sum(axis=2).sum(axis=1), label='s1', marker='.')
+    ax.plot(s2x[0].sum(axis=2).sum(axis=1), label='s2', marker='.')
+    ax.plot(E[1][:, center[1], center[2]].real.T, label='Ey', marker='x')
+    ax.grid(alpha=0.6)
+    ax.legend()
+
+    p_in = (-E * J.conj()).sum() / 2 * (dx * dx * dx)
+    print(f'{p_in=}')
 
     q = []
     for i in range(-5, 30):
         e_ovl_rolled = numpy.roll(e_overlap, i, axis=1)
-        q += [numpy.abs(vec(E) @ vec(e_ovl_rolled).conj())]
-    pyplot.figure()
-    pyplot.plot(q, marker='.')
-    pyplot.grid(alpha=0.6)
-    pyplot.title('Overlap with mode')
-    pyplot.show()
-    print('Average overlap with mode:', sum(q)/len(q))
+        q += [numpy.abs(vec(E).conj() @ vec(e_ovl_rolled))]
+    fig, ax = pyplot.subplots()
+    ax.plot(q, marker='.')
+    ax.grid(alpha=0.6)
+    ax.set_title('Overlap with mode')
+    print('Average overlap with mode:', sum(q[8:32])/len(q[8:32]))
+
+    pyplot.show(block=True)
 
 
 def module_available(name):
@@ -240,9 +160,6 @@ def module_available(name):
 
 
 if __name__ == '__main__':
-    #test0()
-#    test1()
-
     if module_available('opencl_fdfd'):
         from opencl_fdfd import cg_solver as opencl_solver
         test1(opencl_solver)
@@ -253,3 +170,4 @@ if __name__ == '__main__':
     #     test1(magma_solver)
     else:
         test1()
+

examples/fdtd.py

@@ -1,18 +1,30 @@
 """
-Example code for running an OpenCL FDTD simulation
+Example code for a broadband FDTD run with phasor extraction.
 
-See main() for simulation setup.
+This script shows the intended low-level workflow for:
+
+1. building a Yee-grid simulation with CPML on all faces,
+2. driving it with an electric-current pulse,
+3. extracting a single-frequency phasor on the fly, and
+4. checking that phasor against the matching stretched-grid FDFD operator.
 """
 
 import sys
 import time
+import copy
 
 import numpy
 import h5py
+from numpy.linalg import norm
 
 from meanas import fdtd
 from meanas.fdtd import cpml_params, updates_with_cpml
-from masque import Pattern, shapes
+from meanas.fdtd.misc import gaussian_packet
+
+from meanas.fdfd.operators import e_full
+from meanas.fdfd.scpml import stretch_with_scpml
+from meanas.fdmath import vec
+from masque import Pattern, Circle, Polygon
 import gridlock
 import pcgen
 
@@ -41,8 +53,7 @@ def perturbed_l3(a: float, radius: float, **kwargs) -> Pattern:
         `masque.Pattern` object containing the L3 design
     """
 
-    default_args = {'hole_dose':    1,
-                    'trench_dose':  1,
+    default_args = {
         'hole_layer':   0,
         'trench_layer': 1,
         'shifts_a':     (0.15, 0, 0.075),
@@ -53,38 +64,39 @@ def perturbed_l3(a: float, radius: float, **kwargs) -> Pattern:
         }
     kwargs = {**default_args, **kwargs}
 
-    xyr = pcgen.l3_shift_perturbed_defect(mirror_dims=kwargs['xy_size'],
+    xyr = pcgen.l3_shift_perturbed_defect(
+        mirror_dims=kwargs['xy_size'],
         perturbed_radius=kwargs['perturbed_radius'],
         shifts_a=kwargs['shifts_a'],
-                                          shifts_r=kwargs['shifts_r'])
+        shifts_r=kwargs['shifts_r'],
+        )
     xyr *= a
     xyr[:, 2] *= radius
 
     pat = Pattern()
-    pat.name = f'L3p-a{a:g}r{radius:g}rp{kwargs["perturbed_radius"]:g}'
-    pat.shapes += [shapes.Circle(radius=r, offset=(x, y),
-                                 dose=kwargs['hole_dose'],
-                                 layer=kwargs['hole_layer'])
+    #pat.name = f'L3p-a{a:g}r{radius:g}rp{kwargs["perturbed_radius"]:g}'
+    pat.shapes[(kwargs['hole_layer'], 0)] += [
+        Circle(radius=r, offset=(x, y))
         for x, y, r in xyr]
 
     maxes = numpy.max(numpy.fabs(xyr), axis=0)
-    pat.shapes += [shapes.Polygon.rectangle(
+    pat.shapes[(kwargs['trench_layer'], 0)] += [
+        Polygon.rectangle(
             lx=(2 * maxes[0]), ly=kwargs['trench_width'],
-        offset=(0, s * (maxes[1] + a + kwargs['trench_width'] / 2)),
-        dose=kwargs['trench_dose'], layer=kwargs['trench_layer'])
+            offset=(0, s * (maxes[1] + a + kwargs['trench_width'] / 2))
+            )
         for s in (-1, 1)]
     return pat
 
 
-def main():
+def main() -> None:
     dtype = numpy.float32
-    max_t = 8000            # number of timesteps
+    max_t = 3600            # number of timesteps
 
     dx = 40                 # discretization (nm/cell)
     pml_thickness = 8       # (number of cells)
 
     wl = 1550               # Excitation wavelength and fwhm
-    dwl = 200
 
     # Device design parameters
     xy_size = numpy.array([10, 10])
@@ -107,68 +119,97 @@ def main():
 
     # #### Create the grid, mask, and draw the device ####
     grid = gridlock.Grid(edge_coords)
-    epsilon = grid.allocate(n_air**2, dtype=dtype)
-    grid.draw_slab(epsilon,
-                   surface_normal=2,
-                   center=[0, 0, 0],
-                   thickness=th,
-                   eps=n_slab**2)
+    epsilon = grid.allocate(n_air ** 2, dtype=dtype)
+    grid.draw_slab(
+        epsilon,
+        slab = dict(axis='z', center=0, span=th),
+        foreground = n_slab ** 2,
+        )
+
     mask = perturbed_l3(a, r)
+    grid.draw_polygons(
+        epsilon,
+        slab = dict(axis='z', center=0, span=2 * th),
+        foreground = n_air ** 2,
+        offset2d = (0, 0),
+        polygons = mask.as_polygons(library=None),
+        )
 
-    grid.draw_polygons(epsilon,
-                       surface_normal=2,
-                       center=[0, 0, 0],
-                       thickness=2 * th,
-                       eps=n_air**2,
-                       polygons=mask.as_polygons())
+    print(f'{grid.shape=}')
 
-    print(grid.shape)
-
-    dt = .99/numpy.sqrt(3)
-    e = [numpy.zeros_like(epsilon[0], dtype=dtype) for _ in range(3)]
-    h = [numpy.zeros_like(epsilon[0], dtype=dtype) for _ in range(3)]
+    dt = dx * 0.99 / numpy.sqrt(3)
+    ee = numpy.zeros_like(epsilon, dtype=complex)
+    hh = numpy.zeros_like(epsilon, dtype=complex)
 
     dxes = [grid.dxyz, grid.autoshifted_dxyz()]
 
     # PMLs in every direction
-    pml_params = [[cpml_params(axis=dd, polarity=pp, dt=dt,
-                               thickness=pml_thickness, epsilon_eff=1.0**2)
+    pml_params = [
+        [cpml_params(axis=dd, polarity=pp, dt=dt, thickness=pml_thickness, epsilon_eff=n_air ** 2)
          for pp in (-1, +1)]
         for dd in range(3)]
-    update_E, update_H = updates_with_cpml(cpml_params=pml_params, dt=dt,
-                                           dxes=dxes, epsilon=epsilon)
+    update_E, update_H = updates_with_cpml(cpml_params=pml_params, dt=dt, dxes=dxes, epsilon=epsilon, dtype=complex)
 
-    # Source parameters and function
-    w = 2 * numpy.pi * dx / wl
-    fwhm = dwl * w * w / (2 * numpy.pi * dx)
-    alpha = (fwhm ** 2) / 8 * numpy.log(2)
-    delay = 7/numpy.sqrt(2 * alpha)
+    # sample_interval = numpy.floor(1 / (2 * 1 / wl * dt)).astype(int)
+    # print(f'Save time interval would be {sample_interval} * dt = {sample_interval * dt:3g}')
 
-    def field_source(i):
-        t0 = i * dt - delay
-        return numpy.sin(w * t0) * numpy.exp(-alpha * t0**2)
+
+    # Build the pulse directly at the current half-steps and normalize that
+    # scalar waveform so its extracted temporal phasor is exactly 1 at omega.
+    source_phasor, _delay = gaussian_packet(wl=wl, dwl=100, dt=dt, turn_on=1e-5)
+    aa, cc, ss = source_phasor(numpy.arange(max_t) + 0.5)
+    source_waveform = aa * (cc + 1j * ss)
+    omega = 2 * numpy.pi / wl
+    pulse_scale = fdtd.temporal_phasor_scale(source_waveform, omega, dt, offset_steps=0.5)[0]
+
+    j_source = numpy.zeros_like(epsilon, dtype=complex)
+    j_source[1, *(grid.shape // 2)] = epsilon[1, *(grid.shape // 2)]
+    jph = numpy.zeros((1, *epsilon.shape), dtype=complex)
+    eph = numpy.zeros((1, *epsilon.shape), dtype=complex)
+    hph = numpy.zeros((1, *epsilon.shape), dtype=complex)
 
     # #### Run a bunch of iterations ####
     output_file = h5py.File('simulation_output.h5', 'w')
     start = time.perf_counter()
-    for t in range(max_t):
-        update_E(e, h, epsilon)
+    for tt in range(max_t):
+        update_E(ee, hh, epsilon)
 
-        e[1][tuple(grid.shape//2)] += field_source(t)
-        update_H(e, h)
+        # Electric-current injection uses E -= dt * J / epsilon, which is the
+        # same sign convention used by the matching FDFD right-hand side.
+        j_step = pulse_scale * source_waveform[tt] * j_source
+        ee -= dt * j_step / epsilon
+        update_H(ee, hh)
 
-        print('iteration {}: average {} iterations per sec'.format(t, (t+1)/(time.perf_counter()-start)))
+        avg_rate = (tt + 1) / (time.perf_counter() - start)
         sys.stdout.flush()
 
-        if t % 20 == 0:
-            r = sum([(f * f * e).sum() for f, e in zip(e, epsilon)])
-            print('E sum', r)
+        if tt % 200 == 0:
+            print(f'iteration {tt}: average {avg_rate} iterations per sec')
+            E_energy_sum = (ee.conj() * ee * epsilon).sum().real
+            print(f'{E_energy_sum=}')
 
         # Save field slices
-        if (t % 20 == 0 and (max_t - t <= 1000 or t <= 2000)) or t == max_t-1:
-            print('saving E-field')
-            for j, f in enumerate(e):
-                output_file['/E{}_t{}'.format('xyz'[j], t)] = f[:, :, round(f.shape[2]/2)]
+        if (tt % 20 == 0 and (max_t - tt <= 1000 or tt <= 2000)) or tt == max_t - 1:
+            print(f'saving E-field at iteration {tt}')
+            output_file[f'/E_t{tt}'] = ee[:, :, :, ee.shape[3] // 2]
+
+        fdtd.accumulate_phasor_j(jph, omega, dt, j_step, tt)
+        fdtd.accumulate_phasor_e(eph, omega, dt, ee, tt + 1)
+        fdtd.accumulate_phasor_h(hph, omega, dt, hh, tt + 1)
+
+    Eph = eph[0]
+    Jph = jph[0]
+    b = -1j * omega * Jph
+    dxes_fdfd = copy.deepcopy(dxes)
+    for pp in (-1, +1):
+        for dd in range(3):
+            stretch_with_scpml(dxes_fdfd, axis=dd, polarity=pp, omega=omega, epsilon_effective=n_air ** 2, thickness=pml_thickness)
+    # Compare the extracted phasor to the FDFD operator on the stretched grid,
+    # not the unstretched Yee spacings used by the raw time-domain update.
+    A = e_full(omega=omega, dxes=dxes_fdfd, epsilon=epsilon)
+    residual = norm(A @ vec(Eph) - vec(b)) / norm(vec(b))
+    print(f'FDFD residual is {residual}')
+
 
 if __name__ == '__main__':
     main()

examples/tcyl.py

@@ -3,7 +3,7 @@ import numpy
 from numpy.linalg import norm
 
 from meanas.fdmath import vec, unvec
-from meanas.fdfd import waveguide_mode, functional, scpml
+from meanas.fdfd import waveguide_cyl, functional, scpml
 from meanas.fdfd.solvers import generic as generic_solver
 
 import gridlock
@@ -37,29 +37,34 @@ def test1(solver=generic_solver):
     xyz_max = numpy.array([800, y_max, z_max]) + (pml_thickness + 2) * dx
 
     # Coordinates of the edges of the cells.
-    half_edge_coords = [numpy.arange(dx/2, m + dx/2, step=dx) for m in xyz_max]
+    half_edge_coords = [numpy.arange(dx / 2, m + dx / 2, step=dx) for m in xyz_max]
     edge_coords = [numpy.hstack((-h[::-1], h)) for h in half_edge_coords]
     edge_coords[0] = numpy.array([-dx, dx])
 
     # #### Create the grid and draw the device ####
     grid = gridlock.Grid(edge_coords)
     epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
-    grid.draw_cuboid(epsilon, center=center, dimensions=[8e3, w, th], eps=n_wg**2)
+    grid.draw_cuboid(epsilon, center=center, dimensions=[8e3, w, th], foreground=n_wg**2)
 
     dxes = [grid.dxyz, grid.autoshifted_dxyz()]
     for a in (1, 2):
         for p in (-1, 1):
-            dxes = scmpl.stretch_with_scpml(dxes, omega=omega, axis=a, polarity=p,
-                                            thickness=pml_thickness)
+            dxes = scpml.stretch_with_scpml(
+                dxes,
+                omega=omega,
+                axis=a,
+                polarity=p,
+                thickness=pml_thickness,
+                )
 
     wg_args = {
         'omega': omega,
         'dxes': [(d[1], d[2]) for d in dxes],
-        'epsilon': vec(g.transpose([1, 2, 0]) for g in epsilon),
+        'epsilon': vec(epsilon.transpose([0, 2, 3, 1])),
         'r0': r0,
     }
 
-    wg_results = waveguide_mode.solve_waveguide_mode_cylindrical(mode_number=0, **wg_args)
+    wg_results = waveguide_cyl.solve_mode(mode_number=0, **wg_args)
 
     E = wg_results['E']
 
@@ -70,20 +75,17 @@ def test1(solver=generic_solver):
     '''
     Plot results
     '''
-    def pcolor(v):
+    def pcolor(fig, ax, v, title):
         vmax = numpy.max(numpy.abs(v))
-        pyplot.pcolor(v.T, cmap='seismic', vmin=-vmax, vmax=vmax)
-        pyplot.axis('equal')
-        pyplot.colorbar()
+        mappable = ax.pcolormesh(v.T, cmap='seismic', vmin=-vmax, vmax=vmax)
+        ax.set_aspect('equal', adjustable='box')
+        ax.set_title(title)
+        ax.figure.colorbar(mappable)
 
-    pyplot.figure()
-    pyplot.subplot(2, 2, 1)
-    pcolor(numpy.real(E[0][:, :]))
-    pyplot.subplot(2, 2, 2)
-    pcolor(numpy.real(E[1][:, :]))
-    pyplot.subplot(2, 2, 3)
-    pcolor(numpy.real(E[2][:, :]))
-    pyplot.subplot(2, 2, 4)
+    fig, axes = pyplot.subplots(2, 2)
+    pcolor(fig, axes[0][0], numpy.real(E[0]), 'Ex')
+    pcolor(fig, axes[0][1], numpy.real(E[1]), 'Ey')
+    pcolor(fig, axes[1][0], numpy.real(E[2]), 'Ez')
     pyplot.show()
 
 

examples/waveguide.py

@@ -0,0 +1,342 @@
+"""
+Example code for guided-wave FDFD and FDTD comparison.
+
+This example is the reference workflow for:
+
+1. solving a waveguide port mode,
+2. turning that mode into a one-sided source and overlap window,
+3. comparing a direct FDFD solve against a time-domain phasor extracted from FDTD.
+"""
+from typing import Callable
+import logging
+import time
+import copy
+
+import numpy
+import h5py
+from numpy.linalg import norm
+
+import gridlock
+import meanas
+from meanas import fdtd, fdfd
+from meanas.fdtd import cpml_params, updates_with_cpml
+from meanas.fdtd.misc import gaussian_packet
+
+from meanas.fdmath import vec, unvec, vcfdfield_t, cfdfield_t, fdfield_t, dx_lists_t
+from meanas.fdfd import waveguide_3d, functional, scpml, operators
+from meanas.fdfd.solvers import generic as generic_solver
+from meanas.fdfd.operators import e_full
+from meanas.fdfd.scpml import stretch_with_scpml
+
+
+logging.basicConfig(level=logging.DEBUG)
+for pp in ('matplotlib', 'PIL'):
+    logging.getLogger(pp).setLevel(logging.WARNING)
+
+logger = logging.getLogger(__name__)
+
+
+def pcolor(vv, fig=None, ax=None) -> None:
+    if fig is None:
+        assert ax is None
+        fig, ax = pyplot.subplots()
+    mb = ax.pcolor(vv, cmap='seismic', norm=colors.CenteredNorm())
+    fig.colorbar(mb)
+    ax.set_aspect('equal')
+
+
+def draw_grid(
+        *,
+        dx: float,
+        pml_thickness: int,
+        n_wg: float = 3.476,        # Si index @ 1550
+        n_cladding: float = 1.00,    # Air index
+        wg_w: float = 400,
+        wg_th: float = 200,
+        ) -> tuple[gridlock.Grid, fdfield_t]:
+    """ Create the grid and draw the device  """
+    # Half-dimensions of the simulation grid
+    xyz_max = numpy.array([800, 900, 600]) + (pml_thickness + 2) * dx
+
+    # Coordinates of the edges of the cells.
+    half_edge_coords = [numpy.arange(dx / 2, m + dx / 2, step=dx) for m in xyz_max]
+    edge_coords = [numpy.hstack((-h[::-1], h)) for h in half_edge_coords]
+
+    grid = gridlock.Grid(edge_coords)
+    epsilon = grid.allocate(n_cladding**2, dtype=numpy.float32)
+    grid.draw_cuboid(
+        epsilon,
+        x = dict(center=0, span=8e3),
+        y = dict(center=0, span=wg_w),
+        z = dict(center=0, span=wg_th),
+        foreground = n_wg ** 2,
+        )
+
+    return grid, epsilon
+
+
+def get_waveguide_mode(
+        *,
+        grid: gridlock.Grid,
+        dxes: dx_lists_t,
+        omega: float,
+        epsilon: fdfield_t,
+        ) -> tuple[vcfdfield_t, vcfdfield_t]:
+    """Create a mode source and overlap window for one forward-going port."""
+    dims = numpy.array([[-10, -20, -15],
+                        [-10,  20,  15]]) * [[numpy.median(numpy.real(dx)) for dx in dxes[0]]]
+    ind_dims = (grid.pos2ind(dims[0], which_shifts=None).astype(int),
+                grid.pos2ind(dims[1], which_shifts=None).astype(int))
+    wg_args = dict(
+        slices = [slice(i, f+1) for i, f in zip(*ind_dims)],
+        dxes = dxes,
+        axis = 0,
+        polarity = +1,
+        )
+
+    wg_results = waveguide_3d.solve_mode(mode_number=0, omega=omega, epsilon=epsilon, **wg_args)
+    J = waveguide_3d.compute_source(E=wg_results['E'], wavenumber=wg_results['wavenumber'],
+                                    omega=omega, epsilon=epsilon, **wg_args)
+
+    # compute_overlap_e() returns the normalized upstream overlap window used to
+    # project another field onto this same guided mode.
+    e_overlap = waveguide_3d.compute_overlap_e(E=wg_results['E'], wavenumber=wg_results['wavenumber'], **wg_args)
+    return J, e_overlap
+
+
+def main(
+        *,
+        solver: Callable = generic_solver,
+        dx: float = 40,                  # discretization (nm / cell)
+        pml_thickness: int = 10,         # (number of cells)
+        wl: float = 1550,                # Excitation wavelength
+        wg_w: float = 600,               # Waveguide width
+        wg_th: float = 220,              # Waveguide thickness
+        ):
+    omega = 2 * numpy.pi / wl
+
+    grid, epsilon = draw_grid(dx=dx, pml_thickness=pml_thickness)
+
+    # Add SCPML stretching to the FDFD grid; this matches the CPML-backed FDTD
+    # run below so the two solvers see the same absorbing boundary profile.
+    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    for a in (0, 1, 2):
+        for p in (-1, 1):
+            dxes = scpml.stretch_with_scpml(dxes, omega=omega, axis=a, polarity=p, thickness=pml_thickness)
+
+
+    J, e_overlap = get_waveguide_mode(grid=grid, dxes=dxes, omega=omega, epsilon=epsilon)
+
+
+    pecg = numpy.zeros_like(epsilon)
+    # pecg.draw_cuboid(pecg, center=[700, 0, 0], dimensions=[80, 1e8, 1e8], eps=1)
+    # pecg.visualize_isosurface(pecg)
+
+    pmcg = numpy.zeros_like(epsilon)
+    # grid.draw_cuboid(pmcg, center=[700, 0, 0], dimensions=[80, 1e8, 1e8], eps=1)
+    # grid.visualize_isosurface(pmcg)
+
+
+#    ss = (1, slice(None), J.shape[2]//2+6, slice(None))
+#    pcolor(J3[ss].T.imag)
+#    pcolor((numpy.abs(J3).sum(axis=(0, 2)) > 0).astype(float).T)
+#    pyplot.show(block=True)
+
+    E_fdfd = fdfd_solve(
+        omega = omega,
+        dxes = dxes,
+        epsilon = epsilon,
+        J = J,
+        pec = pecg,
+        pmc = pmcg,
+        )
+
+
+    #
+    # Plot results
+    #
+    center = grid.pos2ind([0, 0, 0], None).astype(int)
+    fig, axes = pyplot.subplots(2, 2)
+    pcolor(numpy.real(E[1][center[0], :, :]).T, fig=fig, ax=axes[0, 0])
+    axes[0, 1].plot(numpy.log10(numpy.abs(E[1][:, center[1], center[2]]) + 1e-10))
+    axes[0, 1].grid(alpha=0.6)
+    axes[0, 1].set_ylabel('log10 of field')
+    pcolor(numpy.real(E[1][:, :, center[2]]).T, fig=fig, ax=axes[1, 0])
+
+    def poyntings(E):
+        H = functional.e2h(omega, dxes)(E)
+        poynting = fdtd.poynting(e=E, h=H.conj(), dxes=dxes)
+        cross1 = operators.poynting_e_cross(vec(E), dxes) @ vec(H).conj()
+        cross2 = operators.poynting_h_cross(vec(H), dxes) @ vec(E).conj() * -1
+        s1 = 0.5 * unvec(numpy.real(cross1), grid.shape)
+        s2 = 0.5 * unvec(numpy.real(cross2), grid.shape)
+        s0 = 0.5 * poynting.real
+#        s2 = poynting.imag
+        return s0, s1, s2
+
+    s0x, s1x, s2x = poyntings(E)
+    axes[1, 1].plot(s0x[0].sum(axis=2).sum(axis=1), label='s0', marker='.')
+    axes[1, 1].plot(s1x[0].sum(axis=2).sum(axis=1), label='s1', marker='.')
+    axes[1, 1].plot(s2x[0].sum(axis=2).sum(axis=1), label='s2', marker='.')
+    axes[1, 1].plot(E[1][:, center[1], center[2]].real.T, label='Ey', marker='x')
+    axes[1, 1].grid(alpha=0.6)
+    axes[1, 1].legend()
+
+    q = []
+    for i in range(-5, 30):
+        e_ovl_rolled = numpy.roll(e_overlap, i, axis=1)
+        q += [numpy.abs(vec(E) @ vec(e_ovl_rolled).conj())]
+    fig, ax = pyplot.subplots()
+    ax.plot(q, marker='.')
+    ax.grid(alpha=0.6)
+    ax.set_title('Overlap with mode')
+
+    logger.info('Average overlap with mode:', sum(q[8:32]) / len(q[8:32]))
+
+    pyplot.show(block=True)
+
+
+def fdfd_solve(
+        *,
+        omega: float,
+        dxes = dx_lists_t,
+        epsilon: fdfield_t,
+        J: cfdfield_t,
+        pec: fdfield_t,
+        pmc: fdfield_t,
+        ) -> cfdfield_t:
+    """ Construct and run the solve """
+    sim_args = dict(
+        omega = omega,
+        dxes = dxes,
+        epsilon = vec(epsilon),
+        pec = vec(pecg),
+        pmc = vec(pmcg),
+        )
+
+    x = solver(J=vec(J), **sim_args)
+
+    b = -1j * omega * vec(J)
+    A = operators.e_full(**sim_args).tocsr()
+    logger.info('Norm of the residual is ', norm(A @ x - b) / norm(b))
+
+    E = unvec(x, epsilon.shape[1:])
+    return E
+
+
+def main2():
+    dtype = numpy.float32
+    max_t = 3600            # number of timesteps
+
+    dx = 40                 # discretization (nm/cell)
+    pml_thickness = 8       # (number of cells)
+
+    wl = 1550               # Excitation wavelength and fwhm
+    dwl = 100
+
+    # Device design parameters
+    xy_size = numpy.array([10, 10])
+    a = 430
+    r = 0.285
+    th = 170
+
+    # refractive indices
+    n_slab = 3.408  # InGaAsP(80, 50) @ 1550nm
+    n_cladding = 1.0        # air
+
+    # Half-dimensions of the simulation grid
+    xy_max = (xy_size + 1) * a * [1, numpy.sqrt(3)/2]
+    z_max = 1.6 * a
+    xyz_max = numpy.hstack((xy_max, z_max)) + pml_thickness * dx
+
+    # Coordinates of the edges of the cells. The fdtd package can only do square grids at the moment.
+    half_edge_coords = [numpy.arange(dx/2, m + dx, step=dx) for m in xyz_max]
+    edge_coords = [numpy.hstack((-h[::-1], h)) for h in half_edge_coords]
+
+    # #### Create the grid, mask, and draw the device ####
+    grid = gridlock.Grid(edge_coords)
+    epsilon = grid.allocate(n_cladding ** 2, dtype=dtype)
+    grid.draw_slab(
+        epsilon,
+        slab = dict(axis='z', center=0, span=th),
+        foreground = n_slab ** 2,
+        )
+
+
+    print(f'{grid.shape=}')
+
+    dt = dx * 0.99 / numpy.sqrt(3)
+    ee = numpy.zeros_like(epsilon, dtype=complex)
+    hh = numpy.zeros_like(epsilon, dtype=complex)
+
+    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+
+    # PMLs in every direction
+    pml_params = [
+        [cpml_params(axis=dd, polarity=pp, dt=dt, thickness=pml_thickness, epsilon_eff=n_cladding ** 2)
+         for pp in (-1, +1)]
+        for dd in range(3)]
+    update_E, update_H = updates_with_cpml(cpml_params=pml_params, dt=dt, dxes=dxes, epsilon=epsilon, dtype=complex)
+
+    # sample_interval = numpy.floor(1 / (2 * 1 / wl * dt)).astype(int)
+    # print(f'Save time interval would be {sample_interval} * dt = {sample_interval * dt:3g}')
+
+
+    # Sample the pulse at the current half-steps and normalize that scalar
+    # waveform so the extracted temporal phasor is exactly 1 at omega.
+    source_phasor, _delay = gaussian_packet(wl=wl, dwl=100, dt=dt, turn_on=1e-5)
+    aa, cc, ss = source_phasor(numpy.arange(max_t) + 0.5)
+    source_waveform = aa * (cc + 1j * ss)
+    omega = 2 * numpy.pi / wl
+    pulse_scale = fdtd.temporal_phasor_scale(source_waveform, omega, dt, offset_steps=0.5)[0]
+
+    j_source = numpy.zeros_like(epsilon, dtype=complex)
+    j_source[1, *(grid.shape // 2)] = epsilon[1, *(grid.shape // 2)]
+    jph = numpy.zeros((1, *epsilon.shape), dtype=complex)
+    eph = numpy.zeros((1, *epsilon.shape), dtype=complex)
+    hph = numpy.zeros((1, *epsilon.shape), dtype=complex)
+
+    # #### Run a bunch of iterations ####
+    output_file = h5py.File('simulation_output.h5', 'w')
+    start = time.perf_counter()
+    for tt in range(max_t):
+        update_E(ee, hh, epsilon)
+
+        # Electric-current injection uses E -= dt * J / epsilon, which is the
+        # sign convention matched by the FDFD source term -1j * omega * J.
+        j_step = pulse_scale * source_waveform[tt] * j_source
+        ee -= dt * j_step / epsilon
+        update_H(ee, hh)
+
+        avg_rate = (tt + 1) / (time.perf_counter() - start)
+
+        if tt % 200 == 0:
+            print(f'iteration {tt}: average {avg_rate} iterations per sec')
+            E_energy_sum = (ee.conj() * ee * epsilon).sum().real
+            print(f'{E_energy_sum=}')
+
+        # Save field slices
+        if (tt % 20 == 0 and (max_t - tt <= 1000 or tt <= 2000)) or tt == max_t - 1:
+            print(f'saving E-field at iteration {tt}')
+            output_file[f'/E_t{tt}'] = ee[:, :, :, ee.shape[3] // 2]
+
+        fdtd.accumulate_phasor_j(jph, omega, dt, j_step, tt)
+        fdtd.accumulate_phasor_e(eph, omega, dt, ee, tt + 1)
+        fdtd.accumulate_phasor_h(hph, omega, dt, hh, tt + 1)
+
+    Eph = eph[0]
+    Jph = jph[0]
+    b = -1j * omega * Jph
+    dxes_fdfd = copy.deepcopy(dxes)
+    for pp in (-1, +1):
+        for dd in range(3):
+            stretch_with_scpml(dxes_fdfd, axis=dd, polarity=pp, omega=omega, epsilon_effective=n_cladding ** 2, thickness=pml_thickness)
+    # Residuals must be checked on the stretched FDFD grid, because the FDTD run
+    # already includes those same absorbing layers through CPML.
+    A = e_full(omega=omega, dxes=dxes_fdfd, epsilon=epsilon)
+    residual = norm(A @ vec(Eph) - vec(b)) / norm(vec(b))
+    print(f'FDFD residual is {residual}')
+
+
+if __name__ == '__main__':
+    main()

examples/waveguide_real.py

@@ -0,0 +1,219 @@
+"""
+Real-valued straight-waveguide FDTD/FDFD comparison.
+
+This example shows the user-facing "compare real FDTD against reconstructed real
+FDFD" workflow:
+
+1. build a straight waveguide on a uniform Yee grid,
+2. drive it with a real-valued continuous-wave mode source,
+3. solve the matching FDFD problem from the analytic source phasor, and
+4. compare late real monitor slices against `fdtd.reconstruct_real_e/h(...)`.
+
+Unlike the phasor-based examples, this script does not use extracted phasors as
+the main output. It is a stricter diagnostic: the comparison target is the raw
+real field itself, with full-plane, mode-weighted, guided-mode, and orthogonal-
+residual errors reported. Strong phasor agreement can coexist with visibly
+larger raw-snapshot error because the latter includes weak nonguided tails on
+the monitor plane.
+"""
+
+import numpy
+
+from meanas import fdfd, fdtd
+from meanas.fdfd import functional, scpml, waveguide_3d
+from meanas.fdmath import vec, unvec
+
+
+DT = 0.25
+PERIOD_STEPS = 64
+OMEGA = 2 * numpy.pi / (PERIOD_STEPS * DT)
+CPML_THICKNESS = 3
+SHAPE = (3, 37, 13, 13)
+SOURCE_SLICES = (slice(5, 6), slice(None), slice(None))
+MONITOR_SLICES = (slice(30, 31), slice(None), slice(None))
+WARMUP_PERIODS = 16
+SOURCE_PHASE = 0.4
+CORE_SLICES = (slice(None), slice(None), slice(4, 9), slice(4, 9))
+
+
+def build_uniform_dxes(shape: tuple[int, int, int, int]) -> list[list[numpy.ndarray]]:
+    return [[numpy.ones(shape[axis + 1]) for axis in range(3)] for _ in range(2)]
+
+
+def build_epsilon(shape: tuple[int, int, int, int]) -> numpy.ndarray:
+    epsilon = numpy.ones(shape, dtype=float)
+    y0 = (shape[2] - 3) // 2
+    z0 = (shape[3] - 3) // 2
+    epsilon[:, :, y0:y0 + 3, z0:z0 + 3] = 12.0
+    return epsilon
+
+
+def build_stretched_dxes(base_dxes: list[list[numpy.ndarray]]) -> list[list[numpy.ndarray]]:
+    stretched_dxes = [[dx.copy() for dx in group] for group in base_dxes]
+    for axis in (0, 1, 2):
+        for polarity in (-1, 1):
+            stretched_dxes = scpml.stretch_with_scpml(
+                stretched_dxes,
+                axis=axis,
+                polarity=polarity,
+                omega=OMEGA,
+                epsilon_effective=1.0,
+                thickness=CPML_THICKNESS,
+            )
+    return stretched_dxes
+
+
+def build_cpml_params() -> list[list[dict[str, numpy.ndarray | float]]]:
+    return [
+        [fdtd.cpml_params(axis=axis, polarity=polarity, dt=DT, thickness=CPML_THICKNESS, epsilon_eff=1.0)
+         for polarity in (-1, 1)]
+        for axis in range(3)
+    ]
+
+
+def weighted_rel_err(observed: numpy.ndarray, reference: numpy.ndarray, weight: numpy.ndarray) -> float:
+    return numpy.linalg.norm((observed - reference) * weight) / numpy.linalg.norm(reference * weight)
+
+
+def project_onto_mode(observed: numpy.ndarray, mode: numpy.ndarray) -> tuple[complex, numpy.ndarray, numpy.ndarray]:
+    coefficient = numpy.vdot(mode, observed) / numpy.vdot(mode, mode)
+    guided = coefficient * mode
+    residual = observed - guided
+    return coefficient, guided, residual
+
+
+def main() -> None:
+    epsilon = build_epsilon(SHAPE)
+    base_dxes = build_uniform_dxes(SHAPE)
+    stretched_dxes = build_stretched_dxes(base_dxes)
+
+    source_mode = waveguide_3d.solve_mode(
+        0,
+        omega=OMEGA,
+        dxes=base_dxes,
+        axis=0,
+        polarity=1,
+        slices=SOURCE_SLICES,
+        epsilon=epsilon,
+    )
+    j_mode = waveguide_3d.compute_source(
+        E=source_mode['E'],
+        wavenumber=source_mode['wavenumber'],
+        omega=OMEGA,
+        dxes=base_dxes,
+        axis=0,
+        polarity=1,
+        slices=SOURCE_SLICES,
+        epsilon=epsilon,
+    )
+    # A small global phase aligns the real-valued source with the late-cycle
+    # raw-snapshot diagnostic. The underlying phasor problem is unchanged.
+    j_mode *= numpy.exp(1j * SOURCE_PHASE)
+    monitor_mode = waveguide_3d.solve_mode(
+        0,
+        omega=OMEGA,
+        dxes=base_dxes,
+        axis=0,
+        polarity=1,
+        slices=MONITOR_SLICES,
+        epsilon=epsilon,
+    )
+    e_weight = numpy.abs(monitor_mode['E'][:, MONITOR_SLICES[0], :, :])
+    h_weight = numpy.abs(monitor_mode['H'][:, MONITOR_SLICES[0], :, :])
+    e_mode = monitor_mode['E'][:, MONITOR_SLICES[0], :, :]
+    h_mode = monitor_mode['H'][:, MONITOR_SLICES[0], :, :]
+
+    e_fdfd = unvec(
+        fdfd.solvers.generic(
+            J=vec(j_mode),
+            omega=OMEGA,
+            dxes=stretched_dxes,
+            epsilon=vec(epsilon),
+            matrix_solver_opts={'atol': 1e-10, 'rtol': 1e-7},
+        ),
+        SHAPE[1:],
+    )
+    h_fdfd = functional.e2h(OMEGA, stretched_dxes)(e_fdfd)
+
+    update_e, update_h = fdtd.updates_with_cpml(
+        cpml_params=build_cpml_params(),
+        dt=DT,
+        dxes=base_dxes,
+        epsilon=epsilon,
+    )
+
+    e_field = numpy.zeros_like(epsilon)
+    h_field = numpy.zeros_like(epsilon)
+    total_steps = (WARMUP_PERIODS + 1) * PERIOD_STEPS
+    e_errors: list[float] = []
+    h_errors: list[float] = []
+    e_core_errors: list[float] = []
+    h_core_errors: list[float] = []
+    e_weighted_errors: list[float] = []
+    h_weighted_errors: list[float] = []
+    e_guided_errors: list[float] = []
+    h_guided_errors: list[float] = []
+    e_residual_errors: list[float] = []
+    h_residual_errors: list[float] = []
+
+    for step in range(total_steps):
+        update_e(e_field, h_field, epsilon)
+
+        # Real-valued FDTD uses the real part of the analytic mode source.
+        t_half = (step + 0.5) * DT
+        j_real = (j_mode.real * numpy.cos(OMEGA * t_half) - j_mode.imag * numpy.sin(OMEGA * t_half)).real
+        e_field -= DT * j_real / epsilon
+
+        update_h(e_field, h_field)
+
+        if step >= total_steps - PERIOD_STEPS // 4:
+            reconstructed_e = fdtd.reconstruct_real_e(
+                e_fdfd[:, MONITOR_SLICES[0], :, :],
+                OMEGA,
+                DT,
+                step + 1,
+            )
+            reconstructed_h = fdtd.reconstruct_real_h(
+                h_fdfd[:, MONITOR_SLICES[0], :, :],
+                OMEGA,
+                DT,
+                step + 1,
+            )
+
+            e_monitor = e_field[:, MONITOR_SLICES[0], :, :]
+            h_monitor = h_field[:, MONITOR_SLICES[0], :, :]
+            e_errors.append(numpy.linalg.norm(e_monitor - reconstructed_e) / numpy.linalg.norm(reconstructed_e))
+            h_errors.append(numpy.linalg.norm(h_monitor - reconstructed_h) / numpy.linalg.norm(reconstructed_h))
+            e_core_errors.append(
+                numpy.linalg.norm(e_monitor[CORE_SLICES] - reconstructed_e[CORE_SLICES])
+                / numpy.linalg.norm(reconstructed_e[CORE_SLICES]),
+            )
+            h_core_errors.append(
+                numpy.linalg.norm(h_monitor[CORE_SLICES] - reconstructed_h[CORE_SLICES])
+                / numpy.linalg.norm(reconstructed_h[CORE_SLICES]),
+            )
+            e_weighted_errors.append(weighted_rel_err(e_monitor, reconstructed_e, e_weight))
+            h_weighted_errors.append(weighted_rel_err(h_monitor, reconstructed_h, h_weight))
+            e_guided_coeff, _, e_residual = project_onto_mode(e_monitor, e_mode)
+            e_guided_coeff_ref, _, e_residual_ref = project_onto_mode(reconstructed_e, e_mode)
+            h_guided_coeff, _, h_residual = project_onto_mode(h_monitor, h_mode)
+            h_guided_coeff_ref, _, h_residual_ref = project_onto_mode(reconstructed_h, h_mode)
+            e_guided_errors.append(abs(e_guided_coeff - e_guided_coeff_ref) / abs(e_guided_coeff_ref))
+            h_guided_errors.append(abs(h_guided_coeff - h_guided_coeff_ref) / abs(h_guided_coeff_ref))
+            e_residual_errors.append(numpy.linalg.norm(e_residual - e_residual_ref) / numpy.linalg.norm(e_residual_ref))
+            h_residual_errors.append(numpy.linalg.norm(h_residual - h_residual_ref) / numpy.linalg.norm(h_residual_ref))
+
+    print(f'late-cycle monitor E errors: min={min(e_errors):.4f} max={max(e_errors):.4f}')
+    print(f'late-cycle monitor H errors: min={min(h_errors):.4f} max={max(h_errors):.4f}')
+    print(f'late-cycle core-window E errors: min={min(e_core_errors):.4f} max={max(e_core_errors):.4f}')
+    print(f'late-cycle core-window H errors: min={min(h_core_errors):.4f} max={max(h_core_errors):.4f}')
+    print(f'late-cycle mode-weighted E errors: min={min(e_weighted_errors):.4f} max={max(e_weighted_errors):.4f}')
+    print(f'late-cycle mode-weighted H errors: min={min(h_weighted_errors):.4f} max={max(h_weighted_errors):.4f}')
+    print(f'late-cycle guided-mode E coefficient errors: min={min(e_guided_errors):.4f} max={max(e_guided_errors):.4f}')
+    print(f'late-cycle guided-mode H coefficient errors: min={min(h_guided_errors):.4f} max={max(h_guided_errors):.4f}')
+    print(f'late-cycle orthogonal-residual E errors: min={min(e_residual_errors):.4f} max={max(e_residual_errors):.4f}')
+    print(f'late-cycle orthogonal-residual H errors: min={min(h_residual_errors):.4f} max={max(h_residual_errors):.4f}')
+
+
+if __name__ == '__main__':
+    main()

make_docs.sh

@@ -2,18 +2,20 @@
 
 set -Eeuo pipefail
 
-cd ~/projects/meanas
+ROOT="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
+cd "$ROOT"
 
-# Approach 1: pdf to html?
-#pdoc3 --pdf --force --template-dir pdoc_templates -o doc . | \
-#    pandoc --metadata=title:"meanas" --toc --toc-depth=4 --from=markdown+abbreviations --to=html --output=doc.html --gladtex -s -
+DOCS_TMP="$(mktemp -d)"
+cleanup() {
+    rm -rf "$DOCS_TMP"
+}
+trap cleanup EXIT
 
-# Approach 2: pdf to html with gladtex
-rm -rf _doc_mathimg
-pdoc --pdf --force --template-dir pdoc_templates -o doc . > doc.md
-pandoc --metadata=title:"meanas" --from=markdown+abbreviations --to=html --output=doc.htex --gladtex -s --css pdoc_templates/pdoc.css doc.md
-gladtex -a -n -d _doc_mathimg -c white doc.htex
+python3 "$ROOT/scripts/prepare_docs_sources.py" "$ROOT/meanas" "$DOCS_TMP"
 
-# Approach 3: html with gladtex
-#pdoc3 --html --force --template-dir pdoc_templates -o doc .
-#find doc -iname '*.html' -exec gladtex -a -n -d _mathimg -c white {} \;
+MKDOCSTRINGS_PYTHON_PATH="$DOCS_TMP" mkdocs build --clean
+
+PRINT_PAGE='site/print_page/index.html'
+if [[ -f "$PRINT_PAGE" ]] && command -v htmlark >/dev/null 2>&1; then
+    htmlark "$PRINT_PAGE" -o site/standalone.html
+fi

meanas/__init__.py

@@ -1,18 +1,19 @@
 """
 Electromagnetic simulation tools
 
-See the readme or `import meanas; help(meanas)` for more info.
+See the tracked examples for end-to-end workflows, and `help(meanas)` for the
+toolbox overview and API derivations.
 """
 
 import pathlib
 
-__version__ = '0.8'
+__version__ = '0.12'
 __author__ = 'Jan Petykiewicz'
 
 
 try:
-    with open(pathlib.Path(__file__).parent / 'README.md', 'r') as f:
+    readme_path = pathlib.Path(__file__).parent / 'README.md'
+    with readme_path.open('r') as f:
         __doc__ = f.read()
 except Exception:
     pass
-

meanas/eigensolvers.py

@@ -1,12 +1,12 @@
 """
 Solvers for eigenvalue / eigenvector problems
 """
-from typing import Callable
+from collections.abc import Callable
 import numpy
 from numpy.typing import NDArray, ArrayLike
 from numpy.linalg import norm
-from scipy import sparse              # type: ignore
-import scipy.sparse.linalg as spalg   # type: ignore
+from scipy import sparse
+import scipy.sparse.linalg as spalg
 
 
 def power_iteration(
@@ -25,8 +25,9 @@ def power_iteration(
     Returns:
         (Largest-magnitude eigenvalue, Corresponding eigenvector estimate)
     """
+    rng = numpy.random.default_rng()
     if guess_vector is None:
-        v = numpy.random.rand(operator.shape[0]) + 1j * numpy.random.rand(operator.shape[0])
+        v = rng.random(operator.shape[0]) + 1j * rng.random(operator.shape[0])
     else:
         v = guess_vector
 
@@ -63,10 +64,10 @@ def rayleigh_quotient_iteration(
         (eigenvalues, eigenvectors)
     """
     try:
-        (operator - sparse.eye(operator.shape[0]))
+        (operator - sparse.eye_array(operator.shape[0]))
 
-        def shift(eigval: float) -> sparse:
-            return eigval * sparse.eye(operator.shape[0])
+        def shift(eigval: float) -> sparse.sparray:
+            return eigval * sparse.eye_array(operator.shape[0])
 
         if solver is None:
             solver = spalg.spsolve
@@ -129,12 +130,12 @@ def signed_eigensolve(
 
     # Try to combine, use general LinearOperator if we fail
     try:
-        shifted_operator = operator + shift * sparse.eye(operator.shape[0])
+        shifted_operator = operator + shift * sparse.eye_array(operator.shape[0])
     except TypeError:
         shifted_operator = operator + spalg.LinearOperator(shape=operator.shape,
                                                            matvec=lambda v: shift * v)
 
-    shifted_eigenvalues, eigenvectors = spalg.eigs(shifted_operator, which='LM', k=how_many, ncv=50)
+    shifted_eigenvalues, eigenvectors = spalg.eigs(shifted_operator, which='LM', k=how_many, ncv=2 * how_many + 50)
     eigenvalues = shifted_eigenvalues - shift
 
     k = eigenvalues.argsort()

meanas/fdfd/__init__.py

@@ -1,4 +1,4 @@
-"""
+r"""
 Tools for finite difference frequency-domain (FDFD) simulations and calculations.
 
 These mostly involve picking a single frequency, then setting up and solving a
@@ -9,9 +9,12 @@ Submodules:
 
 - `operators`, `functional`: General FDFD problem setup.
 - `solvers`: Solver interface and reference implementation.
-- `scpml`: Stretched-coordinate perfectly matched layer (scpml) boundary conditions
+- `scpml`: Stretched-coordinate perfectly matched layer (SCPML) boundary conditions.
 - `waveguide_2d`: Operators and mode-solver for waveguides with constant cross-section.
-- `waveguide_3d`: Functions for transforming `waveguide_2d` results into 3D.
+- `waveguide_3d`: Functions for transforming `waveguide_2d` results into 3D,
+  including mode-source and overlap-window construction.
+- `farfield`, `bloch`, `eme`: specialized helper modules for near/far transforms,
+  Bloch-periodic problems, and eigenmode expansion.
 
 
 ================================================================
@@ -19,77 +22,80 @@ Submodules:
 From the "Frequency domain" section of `meanas.fdmath`, we have
 
 $$
- \\begin{aligned}
- \\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
- \\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{H}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
- \\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
- \\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{M}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega l \\Delta_t} \\\\
- \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-    -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &= -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\
- \\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
- \\end{aligned}
+ \begin{aligned}
+ \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
+ \tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
+ \tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
+ \tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\
+ \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
+    -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
+ \Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
+ \end{aligned}
 $$
 
 resulting in
 
 $$
- \\begin{aligned}
- \\tilde{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
-   \\hat{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
- \\end{aligned}
+ \begin{aligned}
+ \tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
+   \hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
+ \end{aligned}
 $$
 
 Maxwell's equations are then
 
 $$
-  \\begin{aligned}
-  \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=
-         \\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2} \\hat{B}_{\\vec{r} + \\frac{1}{2}}
-                                                          - \\hat{M}_{\\vec{r} + \\frac{1}{2}}  \\\\
-  \\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=
-        -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2} \\tilde{D}_{\\vec{r}}
-                                                          + \\tilde{J}_{\\vec{r}} \\\\
-  \\tilde{\\nabla} \\cdot \\hat{B}_{\\vec{r} + \\frac{1}{2}} &= 0 \\\\
-  \\hat{\\nabla} \\cdot \\tilde{D}_{\\vec{r}} &= \\rho_{\\vec{r}}
- \\end{aligned}
+  \begin{aligned}
+  \tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
+         \imath \Omega e^{-\imath \omega \Delta_t / 2} \hat{B}_{\vec{r} + \frac{1}{2}}
+                                                     - \hat{M}_{\vec{r} + \frac{1}{2}}  \\
+  \hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
+        -\imath \Omega e^{ \imath \omega \Delta_t / 2} \tilde{D}_{\vec{r}}
+                                                     + \tilde{J}_{\vec{r}} \\
+  \tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\
+  \hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}}
+ \end{aligned}
 $$
 
-With $\\Delta_t \\to 0$, this simplifies to
+With $\Delta_t \to 0$, this simplifies to
 
 $$
- \\begin{aligned}
- \\tilde{E}_{l, \\vec{r}} &\\to \\tilde{E}_{\\vec{r}} \\\\
- \\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{H}_{\\vec{r} + \\frac{1}{2}} \\\\
- \\tilde{J}_{l, \\vec{r}} &\\to \\tilde{J}_{\\vec{r}} \\\\
- \\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{M}_{\\vec{r} + \\frac{1}{2}} \\\\
- \\Omega &\\to \\omega \\\\
- \\tilde{\\partial}_t &\\to -\\imath \\omega \\\\
-   \\hat{\\partial}_t &\\to -\\imath \\omega \\\\
- \\end{aligned}
+ \begin{aligned}
+ \tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\
+ \tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\
+ \tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\
+ \tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\
+ \Omega &\to \omega \\
+ \tilde{\partial}_t &\to -\imath \omega \\
+   \hat{\partial}_t &\to -\imath \omega \\
+ \end{aligned}
 $$
 
 and then
 
 $$
-  \\begin{aligned}
-  \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=
-         \\imath \\omega \\hat{B}_{\\vec{r} + \\frac{1}{2}}
-                       - \\hat{M}_{\\vec{r} + \\frac{1}{2}}  \\\\
-  \\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=
-        -\\imath \\omega \\tilde{D}_{\\vec{r}}
-                       + \\tilde{J}_{\\vec{r}} \\\\
- \\end{aligned}
+  \begin{aligned}
+  \tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
+         \imath \omega \hat{B}_{\vec{r} + \frac{1}{2}}
+                     - \hat{M}_{\vec{r} + \frac{1}{2}}  \\
+  \hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
+        -\imath \omega \tilde{D}_{\vec{r}}
+                     + \tilde{J}_{\vec{r}} \\
+ \end{aligned}
 $$
 
 $$
- \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-    -\\omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\omega \\tilde{J}_{\\vec{r}} \\\\
+ \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
+    -\omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \omega \tilde{J}_{\vec{r}} \\
 $$
 
-# TODO FDFD?
-# TODO PML
-
-
 """
-from . import solvers, operators, functional, scpml, waveguide_2d, waveguide_3d
+from . import (
+    solvers as solvers,
+    operators as operators,
+    functional as functional,
+    scpml as scpml,
+    waveguide_2d as waveguide_2d,
+    waveguide_3d as waveguide_3d,
+    )
 # from . import farfield, bloch TODO

meanas/fdfd/bloch.py

@@ -94,18 +94,19 @@ This module contains functions for generating and solving the
 
 """
 
-from typing import Callable, Any, cast, Sequence
+from typing import Any, cast
+from collections.abc import Callable, Sequence
 import logging
 import numpy
 from numpy import pi, real, trace
 from numpy.fft import fftfreq
 from numpy.typing import NDArray, ArrayLike
-import scipy                            # type: ignore
-import scipy.optimize                   # type: ignore
-from scipy.linalg import norm           # type: ignore
-import scipy.sparse.linalg as spalg     # type: ignore
+import scipy
+import scipy.optimize
+from scipy.linalg import norm
+import scipy.sparse.linalg as spalg
 
-from ..fdmath import fdfield_t, cfdfield_t
+from ..fdmath import fdfield, cfdfield, cfdfield_t
 
 
 logger = logging.getLogger(__name__)
@@ -114,7 +115,6 @@ logger = logging.getLogger(__name__)
 try:
     import pyfftw.interfaces.numpy_fft  # type: ignore
     import pyfftw.interfaces            # type: ignore
-    import multiprocessing
     logger.info('Using pyfftw')
 
     pyfftw.interfaces.cache.enable()
@@ -136,6 +136,14 @@ except ImportError:
     logger.info('Using numpy fft')
 
 
+def _assemble_hmn_vector(
+        h_m: NDArray[numpy.complex128],
+        h_n: NDArray[numpy.complex128],
+        ) -> NDArray[numpy.complex128]:
+    stacked = numpy.concatenate((numpy.ravel(h_m), numpy.ravel(h_n)))
+    return stacked[:, None]
+
+
 def generate_kmn(
         k0: ArrayLike,
         G_matrix: ArrayLike,
@@ -155,7 +163,7 @@ def generate_kmn(
             All are given in the xyz basis (e.g. `|k|[0,0,0] = norm(G_matrix @ k0)`).
     """
     k0 = numpy.array(k0)
-    G_matrix = numpy.array(G_matrix, copy=False)
+    G_matrix = numpy.asarray(G_matrix)
 
     Gi_grids = numpy.array(numpy.meshgrid(*(fftfreq(n, 1 / n) for n in shape[:3]), indexing='ij'))
     Gi = numpy.moveaxis(Gi_grids, 0, -1)
@@ -183,8 +191,8 @@ def generate_kmn(
 def maxwell_operator(
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None
+        epsilon: fdfield,
+        mu: fdfield | None = None
         ) -> Callable[[NDArray[numpy.complex128]], NDArray[numpy.complex128]]:
     """
     Generate the Maxwell operator
@@ -232,13 +240,13 @@ def maxwell_operator(
             Raveled conv(1/mu_k, ik x conv(1/eps_k, ik x h_mn)), returned
             and overwritten in-place of `h`.
         """
-        hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
+        hin_m, hin_n = (hi.reshape(shape) for hi in numpy.split(h, 2))
 
         #{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
 
         # cross product and transform into xyz basis
         d_xyz = (n * hin_m
-               - m * hin_n) * k_mag         # noqa: E128
+               - m * hin_n) * k_mag
 
         # divide by epsilon
         temp = ifftn(d_xyz, axes=range(3))   # reuses d_xyz if using pyfftw
@@ -253,8 +261,8 @@ def maxwell_operator(
             h_m, h_n = b_m, b_n
         else:
             # transform from mn to xyz
-            b_xyz = (m * b_m[:, :, :, None]
-                   + n * b_n[:, :, :, None])    # noqa: E128
+            b_xyz = (m * b_m
+                   + n * b_n)    # noqa
 
             # divide by mu
             temp = ifftn(b_xyz, axes=range(3))
@@ -265,10 +273,7 @@ def maxwell_operator(
             h_m = numpy.sum(h_xyz * m, axis=3)
             h_n = numpy.sum(h_xyz * n, axis=3)
 
-        h.shape = (h.size,)
-        h = numpy.concatenate((h_m.ravel(), h_n.ravel()), axis=None, out=h)     # ravel and merge
-        h.shape = (h.size, 1)
-        return h
+        return _assemble_hmn_vector(h_m, h_n)
 
     return operator
 
@@ -276,7 +281,7 @@ def maxwell_operator(
 def hmn_2_exyz(
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
         ) -> Callable[[NDArray[numpy.complex128]], cfdfield_t]:
     """
     Generate an operator which converts a vectorized spatial-frequency-space
@@ -303,12 +308,13 @@ def hmn_2_exyz(
     k_mag, m, n = generate_kmn(k0, G_matrix, shape)
 
     def operator(h: NDArray[numpy.complex128]) -> cfdfield_t:
-        hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
+        hin_m, hin_n = (hi.reshape(shape) for hi in numpy.split(h, 2))
         d_xyz = (n * hin_m
-               - m * hin_n) * k_mag         # noqa: E128
+               - m * hin_n) * k_mag
 
         # divide by epsilon
-        return numpy.array([ei for ei in numpy.moveaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3, 0)])         # TODO avoid copy
+        exyz = numpy.moveaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3, 0)
+        return cfdfield_t(exyz)
 
     return operator
 
@@ -316,7 +322,7 @@ def hmn_2_exyz(
 def hmn_2_hxyz(
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t
+        epsilon: fdfield,
         ) -> Callable[[NDArray[numpy.complex128]], cfdfield_t]:
     """
     Generate an operator which converts a vectorized spatial-frequency-space
@@ -341,10 +347,10 @@ def hmn_2_hxyz(
     _k_mag, m, n = generate_kmn(k0, G_matrix, shape)
 
     def operator(h: NDArray[numpy.complex128]) -> cfdfield_t:
-        hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
+        hin_m, hin_n = (hi.reshape(shape) for hi in numpy.split(h, 2))
         h_xyz = (m * hin_m
-               + n * hin_n)     # noqa: E128
-        return numpy.array([ifftn(hi) for hi in numpy.moveaxis(h_xyz, 3, 0)])
+               + n * hin_n)
+        return cfdfield_t(numpy.array([ifftn(hi) for hi in numpy.moveaxis(h_xyz, 3, 0)]))
 
     return operator
 
@@ -352,8 +358,8 @@ def hmn_2_hxyz(
 def inverse_maxwell_operator_approx(
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None,
+        epsilon: fdfield,
+        mu: fdfield | None = None,
         ) -> Callable[[NDArray[numpy.complex128]], NDArray[numpy.complex128]]:
     """
     Generate an approximate inverse of the Maxwell operator,
@@ -394,7 +400,7 @@ def inverse_maxwell_operator_approx(
         Returns:
             Raveled ik x conv(eps_k, ik x conv(mu_k, h_mn))
         """
-        hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
+        hin_m, hin_n = (hi.reshape(shape) for hi in numpy.split(h, 2))
 
         #{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
 
@@ -402,8 +408,8 @@ def inverse_maxwell_operator_approx(
             b_m, b_n = hin_m, hin_n
         else:
             # transform from mn to xyz
-            h_xyz = (m * hin_m[:, :, :, None]
-                   + n * hin_n[:, :, :, None])  # noqa: E128
+            h_xyz = (m * hin_m
+                   + n * hin_n)  # noqa
 
             # multiply by mu
             temp = ifftn(h_xyz, axes=range(3))
@@ -411,12 +417,12 @@ def inverse_maxwell_operator_approx(
             b_xyz = fftn(temp, axes=range(3))
 
             # transform back to mn
-            b_m = numpy.sum(b_xyz * m, axis=3)
-            b_n = numpy.sum(b_xyz * n, axis=3)
+            b_m = numpy.sum(b_xyz * m, axis=3, keepdims=True)
+            b_n = numpy.sum(b_xyz * n, axis=3, keepdims=True)
 
         # cross product and transform into xyz basis
         e_xyz = (n * b_m
-               - m * b_n) / k_mag  # noqa: E128
+               - m * b_n) / k_mag
 
         # multiply by epsilon
         temp = ifftn(e_xyz, axes=range(3))
@@ -427,10 +433,7 @@ def inverse_maxwell_operator_approx(
         h_m = numpy.sum(d_xyz * n, axis=3, keepdims=True) / +k_mag
         h_n = numpy.sum(d_xyz * m, axis=3, keepdims=True) / -k_mag
 
-        h.shape = (h.size,)
-        h = numpy.concatenate((h_m, h_n), axis=None, out=h)
-        h.shape = (h.size, 1)
-        return h
+        return _assemble_hmn_vector(h_m, h_n)
 
     return operator
 
@@ -440,15 +443,15 @@ def find_k(
         tolerance: float,
         direction: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None,
+        epsilon: fdfield,
+        mu: fdfield | None = None,
         band: int = 0,
         k_bounds: tuple[float, float] = (0, 0.5),
         k_guess: float | None = None,
         solve_callback: Callable[..., None] | None = None,
         iter_callback: Callable[..., None] | None = None,
         v0: NDArray[numpy.complex128] | None = None,
-        ) -> tuple[float, float, NDArray[numpy.complex128], NDArray[numpy.complex128]]:
+        ) -> tuple[NDArray[numpy.float64], float, NDArray[numpy.complex128], NDArray[numpy.complex128]]:
     """
     Search for a bloch vector that has a given frequency.
 
@@ -471,7 +474,7 @@ def find_k(
         `(k, actual_frequency, eigenvalues, eigenvectors)`
         The found k-vector and its frequency, along with all eigenvalues and eigenvectors.
     """
-    direction = numpy.array(direction) / norm(direction)
+    direction = numpy.array(direction) / norm(direction)  # type: ignore[operator]
 
     k_bounds = tuple(sorted(k_bounds))    # type: ignore    # we know the length already...
     assert len(k_bounds) == 2
@@ -493,23 +496,23 @@ def find_k(
 
     res = scipy.optimize.minimize_scalar(
         lambda x: abs(get_f(x, band) - frequency),
-        k_guess,
-        method='Bounded',
+        method='bounded',
         bounds=k_bounds,
         options={'xatol': abs(tolerance)},
         )
 
     assert n is not None
     assert v is not None
-    return float(res.x * direction), float(res.fun + frequency), n, v
+    actual_frequency = get_f(float(res.x), band)
+    return direction * float(res.x), float(actual_frequency), n, v  # type: ignore[operator,return-value]
 
 
 def eigsolve(
         num_modes: int,
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None,
+        epsilon: fdfield,
+        mu: fdfield | None = None,
         tolerance: float = 1e-7,
         max_iters: int = 10000,
         reset_iters: int = 100,
@@ -538,7 +541,7 @@ def eigsolve(
         `(eigenvalues, eigenvectors)` where `eigenvalues[i]` corresponds to the
         vector `eigenvectors[i, :]`
     """
-    k0 = numpy.array(k0, copy=False)
+    k0 = numpy.asarray(k0)
 
     h_size = 2 * epsilon[0].size
 
@@ -561,11 +564,12 @@ def eigsolve(
     prev_theta = 0.5
     D = numpy.zeros(shape=y_shape, dtype=complex)
 
+    rng = numpy.random.default_rng()
     Z: NDArray[numpy.complex128]
     if y0 is None:
-        Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
+        Z = rng.random(y_shape) + 1j * rng.random(y_shape)
     else:
-        Z = numpy.array(y0, copy=False).T
+        Z = numpy.asarray(y0).T
 
     while True:
         Z *= num_modes / norm(Z)
@@ -573,7 +577,7 @@ def eigsolve(
         try:
             U = numpy.linalg.inv(ZtZ)
         except numpy.linalg.LinAlgError:
-            Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
+            Z = rng.random(y_shape) + 1j * rng.random(y_shape)
             continue
 
         trace_U = real(trace(U))
@@ -646,17 +650,16 @@ def eigsolve(
 
         Qi_memo: list[float | None] = [None, None]
 
-        def Qi_func(theta: float) -> float:
-            nonlocal Qi_memo
+        def Qi_func(theta: float, Qi_memo=Qi_memo, ZtZ=ZtZ, DtD=DtD, symZtD=symZtD) -> float:   # noqa: ANN001
             if Qi_memo[0] == theta:
-                return cast(float, Qi_memo[1])
+                return cast('float', Qi_memo[1])
 
             c = numpy.cos(theta)
             s = numpy.sin(theta)
             Q = c * c * ZtZ + s * s * DtD + 2 * s * c * symZtD
             try:
                 Qi = numpy.linalg.inv(Q)
-            except numpy.linalg.LinAlgError:
+            except numpy.linalg.LinAlgError as err:
                 logger.info('taylor Qi')
                 # if c or s small, taylor expand
                 if c < 1e-4 * s and c != 0:
@@ -666,12 +669,12 @@ def eigsolve(
                     ZtZi = numpy.linalg.inv(ZtZ)
                     Qi = ZtZi / (c * c) - 2 * s / (c * c * c) * (ZtZi @ (ZtZi @ symZtD).conj().T)
                 else:
-                    raise Exception('Inexplicable singularity in trace_func')
+                    raise Exception('Inexplicable singularity in trace_func') from err
             Qi_memo[0] = theta
-            Qi_memo[1] = cast(float, Qi)
-            return cast(float, Qi)
+            Qi_memo[1] = cast('float', Qi)
+            return cast('float', Qi)
 
-        def trace_func(theta: float) -> float:
+        def trace_func(theta: float, ZtAZ=ZtAZ, DtAD=DtAD, symZtAD=symZtAD) -> float:           # noqa: ANN001
             c = numpy.cos(theta)
             s = numpy.sin(theta)
             Qi = Qi_func(theta)
@@ -680,15 +683,24 @@ def eigsolve(
             return numpy.abs(trace)
 
         if False:
-            def trace_deriv(theta):
+            def trace_deriv(
+                    theta: float,
+                    sgn: int = sgn,
+                    ZtAZ=ZtAZ,           # noqa: ANN001
+                    DtAD=DtAD,           # noqa: ANN001
+                    symZtD=symZtD,       # noqa: ANN001
+                    symZtAD=symZtAD,     # noqa: ANN001
+                    ZtZ=ZtZ,             # noqa: ANN001
+                    DtD=DtD,             # noqa: ANN001
+                    ) -> float:
                 Qi = Qi_func(theta)
                 c2 = numpy.cos(2 * theta)
                 s2 = numpy.sin(2 * theta)
-                F = -0.5*s2 *  (ZtAZ - DtAD) + c2 * symZtAD
+                F = -0.5 * s2 * (ZtAZ - DtAD) + c2 * symZtAD
                 trace_deriv = _rtrace_AtB(Qi, F)
 
                 G = Qi @ F.conj().T @ Qi.conj().T
-                H = -0.5*s2 * (ZtZ - DtD) + c2 * symZtD
+                H = -0.5 * s2 * (ZtZ - DtD) + c2 * symZtD
                 trace_deriv -= _rtrace_AtB(G, H)
 
                 trace_deriv *= 2
@@ -696,12 +708,12 @@ def eigsolve(
 
             U_sZtD = U @ symZtD
 
-            dE = 2.0 * (_rtrace_AtB(U, symZtAD) -
-                        _rtrace_AtB(ZtAZU, U_sZtD))
+            dE = 2.0 * (_rtrace_AtB(U, symZtAD)
+                        - _rtrace_AtB(ZtAZU, U_sZtD))
 
-            d2E = 2 * (_rtrace_AtB(U, DtAD) -
-                       _rtrace_AtB(ZtAZU, U @ (DtD - 4 * symZtD @ U_sZtD)) -
-                   4 * _rtrace_AtB(U, symZtAD @ U_sZtD))
+            d2E = 2 * (_rtrace_AtB(U, DtAD)
+                       - _rtrace_AtB(ZtAZU, U @ (DtD - 4 * symZtD @ U_sZtD))
+                       - 4 * _rtrace_AtB(U, symZtAD @ U_sZtD))
 
             # Newton-Raphson to find a root of the first derivative:
             theta = -dE / d2E
@@ -722,7 +734,12 @@ def eigsolve(
                 amax=pi,
                 )
 
-        result = scipy.optimize.minimize_scalar(trace_func, bounds=(0, pi), tol=tolerance)
+        result = scipy.optimize.minimize_scalar(
+            trace_func,
+            method='bounded',
+            bounds=(0, pi),
+            options={'xatol': tolerance},
+            )
         new_E = result.fun
         theta = result.x
 
@@ -751,7 +768,7 @@ def eigsolve(
         v = eigvecs[:, i]
         n = eigvals[i]
         v /= norm(v)
-        Av = (scipy_op @ v.copy())[:, 0]
+        Av = numpy.asarray(scipy_op @ v.copy()).reshape(-1)
         eigness = norm(Av - (v.conj() @ Av) * v)
         f = numpy.sqrt(-numpy.real(n))
         df = numpy.sqrt(-numpy.real(n) + eigness)
@@ -781,7 +798,7 @@ def linmin(x_guess, f0, df0, x_max, f_tol=0.1, df_tol=min(tolerance, 1e-6), x_to
                                           x_min, x_max, isave, dsave)
         for i in range(int(1e6)):
             if task != 'F':
-                logging.info('search converged in {} iterations'.format(i))
+                logging.info(f'search converged in {i} iterations')
                 break
             fx = f(x, dfx)
             x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task,
@@ -799,3 +816,62 @@ def _rtrace_AtB(
 def _symmetrize(A: NDArray[numpy.complex128]) -> NDArray[numpy.complex128]:
     return (A + A.conj().T) * 0.5
 
+
+
+def inner_product(
+        eL: cfdfield,
+        hL: cfdfield,
+        eR: cfdfield,
+        hR: cfdfield,
+        ) -> complex:
+    # assumes x-axis propagation
+
+    assert numpy.array_equal(eR.shape, hR.shape)
+    assert numpy.array_equal(eL.shape, hL.shape)
+    assert numpy.array_equal(eR.shape, eL.shape)
+
+    # Cross product, times 2 since it's <p | n>, then divide by 4. # TODO might want to abs() this?
+    norm2R = (eR[1] * hR[2] - eR[2] * hR[1]).sum() / 2
+    norm2L = (eL[1] * hL[2] - eL[2] * hL[1]).sum() / 2
+
+    # eRxhR_x = numpy.cross(eR.reshape(3, -1), hR.reshape(3, -1), axis=0).reshape(eR.shape)[0] / normR
+    # logger.info(f'power {eRxhR_x.sum() / 2})
+
+    eR_norm = eR / numpy.sqrt(abs(norm2R))
+    hR_norm = hR / numpy.sqrt(abs(norm2R))
+    eL_norm = eL / numpy.sqrt(abs(norm2L))
+    hL_norm = hL / numpy.sqrt(abs(norm2L))
+
+    # (eR x hL)[0] and (eL x hR)[0]
+    eRxhL_x = eR_norm[1] * hL_norm[2] - eR_norm[2] * hL_norm[1]
+    eLxhR_x = eL_norm[1] * hR_norm[2] - eL_norm[2] * hR_norm[1]
+
+    #return 1j *  (eRxhL_x - eLxhR_x).sum() / numpy.sqrt(norm2R * norm2L)
+    #return (eRxhL_x.sum() - eLxhR_x.sum()) / numpy.sqrt(norm2R * norm2L)
+    return eLxhR_x.sum() - eRxhL_x.sum()
+
+
+def trq(
+        eI: cfdfield,
+        hI: cfdfield,
+        eO: cfdfield,
+        hO: cfdfield,
+        ) -> tuple[complex, complex]:
+    pp = inner_product(eO,  hO, eI,  hI)
+    pn = inner_product(eO,  hO, eI, -hI)
+    np = inner_product(eO, -hO, eI,  hI)
+    nn = inner_product(eO, -hO, eI, -hI)
+
+    assert numpy.allclose(pp, -nn, atol=1e-12, rtol=1e-12)
+    assert numpy.allclose(pn, -np, atol=1e-12, rtol=1e-12)
+
+    logger.info(f'''
+        {pp=:4g} {pn=:4g}
+        {nn=:4g} {np=:4g}
+            {nn * pp / pn=:4g}    {-np=:4g}
+        ''')
+
+    r = -pp / pn    # -<Pp|Bp>/<Pn/Bp> = -(-pp) / (-pn)
+    t = (np - nn * pp / pn) / 4
+
+    return t, r

meanas/fdfd/eme.py

@@ -0,0 +1,190 @@
+"""
+Low-level mode-matching helpers for waveguide / EME workflows.
+
+These helpers operate on already-solved and already-normalized port fields.
+They do not build geometries or solve modes themselves; downstream users are
+expected to supply compatible `(E, H)` modal field pairs from
+`waveguide_2d`, `waveguide_3d`, or `waveguide_cyl`.
+
+The returned matrices follow the usual port ordering:
+
+- `get_tr(...)` returns `(T, R)` for left-incident modes.
+- `get_abcd(...)` returns the 2-port block transfer matrix built from the two
+  directional `T/R` solves.
+- `get_s(...)` returns the full block scattering matrix
+  `[[R12, T12], [T21, R21]]`.
+
+This module is intentionally a thin library layer rather than an integrated
+simulation suite. It provides the overlap algebra that downstream users can
+compose into larger workflows.
+"""
+
+from collections.abc import Sequence
+import numpy
+from numpy.typing import NDArray
+from scipy import sparse
+
+from ..fdmath import dx_lists2_t, vcfdfield2
+from .waveguide_2d import inner_product
+
+type wavenumber_seq = Sequence[complex] | NDArray[numpy.complexfloating] | NDArray[numpy.floating]
+
+
+def _validate_port_modes(
+        name: str,
+        ehs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers: wavenumber_seq,
+        ) -> tuple[tuple[int, ...], tuple[int, ...]]:
+    if len(ehs) != len(wavenumbers):
+        raise ValueError(f'{name} mode list and wavenumber list must have the same length')
+    if not ehs:
+        raise ValueError(f'{name} must contain at least one mode')
+
+    e_shape: tuple[int, ...] | None = None
+    h_shape: tuple[int, ...] | None = None
+    for index, mode in enumerate(ehs):
+        if len(mode) != 2:
+            raise ValueError(f'{name}[{index}] must be a 2-tuple of (E, H) modal fields')
+        e_field, h_field = mode
+        mode_e_shape = numpy.shape(e_field)
+        mode_h_shape = numpy.shape(h_field)
+        if mode_e_shape != mode_h_shape:
+            raise ValueError(f'{name}[{index}] has mismatched E/H field shapes')
+        if e_shape is None:
+            e_shape = mode_e_shape
+            h_shape = mode_h_shape
+        elif mode_e_shape != e_shape or mode_h_shape != h_shape:
+            raise ValueError(f'{name} modal fields must all share the same shape')
+
+    assert e_shape is not None
+    assert h_shape is not None
+    return e_shape, h_shape
+
+
+def get_tr(
+        ehLs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_L: wavenumber_seq,
+        ehRs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_R: wavenumber_seq,
+        dxes: dx_lists2_t,
+        ) -> tuple[NDArray[numpy.complex128], NDArray[numpy.complex128]]:
+    """
+    Compute left-incident transmission and reflection matrices.
+
+    Args:
+        ehLs: Left-port modes as `(E, H)` field pairs.
+        wavenumbers_L: Propagation constants for `ehLs`.
+        ehRs: Right-port modes as `(E, H)` field pairs.
+        wavenumbers_R: Propagation constants for `ehRs`.
+        dxes: Two-dimensional Yee-cell edge lengths for the shared port plane.
+
+    Returns:
+        `(T12, R12)` where columns index left-incident modes and rows index
+        outgoing right-going / left-going modes respectively.
+
+    Raises:
+        ValueError: If the port mode lists are empty, malformed, or defined on
+            incompatible field shapes.
+    """
+    left_e_shape, left_h_shape = _validate_port_modes('ehLs', ehLs, wavenumbers_L)
+    right_e_shape, right_h_shape = _validate_port_modes('ehRs', ehRs, wavenumbers_R)
+    if left_e_shape != right_e_shape or left_h_shape != right_h_shape:
+        raise ValueError('left and right modal fields must share the same E/H shapes')
+
+    nL = len(wavenumbers_L)
+    nR = len(wavenumbers_R)
+    A12 = numpy.zeros((nL, nR), dtype=complex)
+    A21 = numpy.zeros((nL, nR), dtype=complex)
+    B11 = numpy.zeros((nL,), dtype=complex)
+    for ll in range(nL):
+        eL, hL = ehLs[ll]
+        B11[ll] = inner_product(eL, hL, dxes=dxes, conj_h=False)
+        for rr in range(nR):
+            eR, hR = ehRs[rr]
+            A12[ll, rr] = inner_product(eL, hR, dxes=dxes, conj_h=False)    # TODO optimize loop?
+            A21[ll, rr] = inner_product(eR, hL, dxes=dxes, conj_h=False)
+
+    # tt0 = 2 * numpy.linalg.pinv(A21 + numpy.conj(A12))
+    tt0, _resid, _rank, _sing = numpy.linalg.lstsq(A21 + A12, numpy.diag(2 * B11), rcond=None)
+
+    U, st, V = numpy.linalg.svd(tt0)
+    gain = st > 1
+    st[gain] = 1 / st[gain]
+    tt = U @ numpy.diag(st) @ V
+
+    # rr = 0.5 * (A21 - numpy.conj(A12)) @ tt
+    rr = numpy.diag(0.5 / B11) @ (A21 - A12) @ tt
+
+    return tt, rr
+
+
+def get_abcd(
+        ehLs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_L: wavenumber_seq,
+        ehRs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_R: wavenumber_seq,
+        **kwargs,
+        ) -> sparse.sparray:
+    """
+    Build the 2-port block transfer matrix for an interface.
+
+    The blocks are assembled from the forward and reverse `get_tr(...)`
+    solutions using the standard
+
+    `[[A, B], [C, D]] = [[T12 - R21 T21^-1 R12, R21 T21^-1], [-T21^-1 R12, T21^-1]]`
+
+    convention.
+    """
+    t12, r12 = get_tr(ehLs, wavenumbers_L, ehRs, wavenumbers_R, **kwargs)
+    t21, r21 = get_tr(ehRs, wavenumbers_R, ehLs, wavenumbers_L, **kwargs)
+    t21i = numpy.linalg.pinv(t21)
+    A = t12 - r21 @ t21i @ r12
+    B = r21 @ t21i
+    C = -t21i @ r12
+    D = t21i
+    return sparse.block_array(
+        [
+            [sparse.csr_array(A), sparse.csr_array(B)],
+            [sparse.csr_array(C), sparse.csr_array(D)],
+            ],
+        format='csr',
+        )
+
+
+def get_s(
+        ehLs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_L: wavenumber_seq,
+        ehRs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_R: wavenumber_seq,
+        force_nogain: bool = False,
+        force_reciprocal: bool = False,
+        **kwargs,
+        ) -> NDArray[numpy.complex128]:
+    """
+    Build the full block scattering matrix for a two-sided interface.
+
+    The returned matrix is ordered as `[[R12, T12], [T21, R21]]`, where the
+    first block-row/column corresponds to the left port and the second to the
+    right port.
+
+    Args:
+        force_nogain: If `True`, clamp singular values of the assembled
+            scattering matrix to at most one.
+        force_reciprocal: If `True`, symmetrize the assembled matrix as
+            `0.5 * (S + S.T)`.
+    """
+    t12, r12 = get_tr(ehLs, wavenumbers_L, ehRs, wavenumbers_R, **kwargs)
+    t21, r21 = get_tr(ehRs, wavenumbers_R, ehLs, wavenumbers_L, **kwargs)
+
+    ss = numpy.block([[r12, t12],
+                      [t21, r21]])
+
+    if force_nogain:
+        # force S @ S.H diagonal
+        U, sing, Vh = numpy.linalg.svd(ss)
+        ss = U @ numpy.diag(numpy.minimum(sing, 1.0)) @ Vh
+
+    if force_reciprocal:
+        ss = 0.5 * (ss + ss.T)
+
+    return ss

meanas/fdfd/farfield.py

@@ -1,20 +1,24 @@
 """
 Functions for performing near-to-farfield transformation (and the reverse).
 """
-from typing import Any, Sequence, cast
+from typing import Any, cast
+from collections.abc import Sequence
 import numpy
 from numpy.fft import fft2, fftshift, fftfreq, ifft2, ifftshift
 from numpy import pi
+from numpy.typing import NDArray
+from numpy import complexfloating
 
-from ..fdmath import cfdfield_t
+type farfield_slice = NDArray[complexfloating]
+type transverse_slice_pair = Sequence[farfield_slice]
 
 
 def near_to_farfield(
-        E_near: cfdfield_t,
-        H_near: cfdfield_t,
+        E_near: transverse_slice_pair,
+        H_near: transverse_slice_pair,
         dx: float,
         dy: float,
-        padded_size: list[int] | int | None = None
+        padded_size: Sequence[int] | int | None = None
         ) -> dict[str, Any]:
     """
     Compute the farfield, i.e. the distribution of the fields after propagation
@@ -55,14 +59,14 @@ def near_to_farfield(
         raise Exception('H_near must be a length-2 list of ndarrays')
 
     s = E_near[0].shape
-    if not all(s == f.shape for f in E_near + H_near):
+    if not all(s == f.shape for f in [*E_near, *H_near]):
         raise Exception('All fields must be the same shape!')
 
     if padded_size is None:
         padded_size = (2**numpy.ceil(numpy.log2(s))).astype(int)
     if not hasattr(padded_size, '__len__'):
         padded_size = (padded_size, padded_size)            # type: ignore  # checked if sequence
-    padded_shape = cast(Sequence[int], padded_size)
+    padded_shape = cast('Sequence[int]', padded_size)
 
     En_fft = [fftshift(fft2(fftshift(Eni), s=padded_shape)) for Eni in E_near]
     Hn_fft = [fftshift(fft2(fftshift(Hni), s=padded_shape)) for Hni in H_near]
@@ -75,25 +79,22 @@ def near_to_farfield(
     kx, ky = numpy.meshgrid(kxx, kyy, indexing='ij')
     kxy2 = kx * kx + ky * ky
     kxy = numpy.sqrt(kxy2)
-    kz = numpy.sqrt(k * k - kxy2)
+    kz = numpy.sqrt(numpy.maximum(0, k * k - kxy2))
 
-    sin_th = ky / kxy
-    cos_th = kx / kxy
+    sin_th = numpy.divide(ky, kxy, out=numpy.zeros_like(ky), where=kxy != 0)
+    cos_th = numpy.divide(kx, kxy, out=numpy.ones_like(kx), where=kxy != 0)
     cos_phi = kz / k
 
-    sin_th[numpy.logical_and(kx == 0, ky == 0)] = 0
-    cos_th[numpy.logical_and(kx == 0, ky == 0)] = 1
-
     # Normalized vector potentials N, L
     N = [-Hn_fft[1] * cos_phi * cos_th + Hn_fft[0] * cos_phi * sin_th,
-          Hn_fft[1] * sin_th + Hn_fft[0] * cos_th]                      # noqa: E127
+          Hn_fft[1] * sin_th + Hn_fft[0] * cos_th]                      # noqa
     L = [ En_fft[1] * cos_phi * cos_th - En_fft[0] * cos_phi * sin_th,
-         -En_fft[1] * sin_th - En_fft[0] * cos_th]                      # noqa: E128
+         -En_fft[1] * sin_th - En_fft[0] * cos_th]                      # noqa
 
     E_far = [-L[1] - N[0],
-              L[0] - N[1]]      # noqa: E127
+              L[0] - N[1]]      # noqa
     H_far = [-E_far[1],
-              E_far[0]]         # noqa: E127
+              E_far[0]]         # noqa
 
     theta = numpy.arctan2(ky, kx)
     phi = numpy.arccos(cos_phi)
@@ -111,8 +112,8 @@ def near_to_farfield(
     outputs = {
         'E': E_far,
         'H': H_far,
-        'dkx': kx[1] - kx[0],
-        'dky': ky[1] - ky[0],
+        'dkx': float(kxx[1] - kxx[0]),
+        'dky': float(kyy[1] - kyy[0]),
         'kx': kx,
         'ky': ky,
         'theta': theta,
@@ -123,11 +124,11 @@ def near_to_farfield(
 
 
 def far_to_nearfield(
-        E_far: cfdfield_t,
-        H_far: cfdfield_t,
+        E_far: transverse_slice_pair,
+        H_far: transverse_slice_pair,
         dkx: float,
         dky: float,
-        padded_size: list[int] | int | None = None
+        padded_size: Sequence[int] | int | None = None
         ) -> dict[str, Any]:
     """
     Compute the farfield, i.e. the distribution of the fields after propagation
@@ -164,32 +165,29 @@ def far_to_nearfield(
         raise Exception('H_far must be a length-2 list of ndarrays')
 
     s = E_far[0].shape
-    if not all(s == f.shape for f in E_far + H_far):
+    if not all(s == f.shape for f in [*E_far, *H_far]):
         raise Exception('All fields must be the same shape!')
 
     if padded_size is None:
         padded_size = (2 ** numpy.ceil(numpy.log2(s))).astype(int)
     if not hasattr(padded_size, '__len__'):
         padded_size = (padded_size, padded_size)            # type: ignore  # checked if sequence
-    padded_shape = cast(Sequence[int], padded_size)
+    padded_shape = cast('Sequence[int]', padded_size)
 
     k = 2 * pi
-    kxs = fftshift(fftfreq(s[0], 1 / (s[0] * dkx)))
-    kys = fftshift(fftfreq(s[0], 1 / (s[1] * dky)))
+    kxs = dkx * fftshift(fftfreq(s[0], d=1 / s[0]))
+    kys = dky * fftshift(fftfreq(s[1], d=1 / s[1]))
 
     kx, ky = numpy.meshgrid(kxs, kys, indexing='ij')
     kxy2 = kx * kx + ky * ky
     kxy = numpy.sqrt(kxy2)
 
-    kz = numpy.sqrt(k * k - kxy2)
+    kz = numpy.sqrt(numpy.maximum(0, k * k - kxy2))
 
-    sin_th = ky / kxy
-    cos_th = kx / kxy
+    sin_th = numpy.divide(ky, kxy, out=numpy.zeros_like(ky), where=kxy != 0)
+    cos_th = numpy.divide(kx, kxy, out=numpy.ones_like(kx), where=kxy != 0)
     cos_phi = kz / k
 
-    sin_th[numpy.logical_and(kx == 0, ky == 0)] = 0
-    cos_th[numpy.logical_and(kx == 0, ky == 0)] = 1
-
     theta = numpy.arctan2(ky, kx)
     phi = numpy.arccos(cos_phi)
     theta[numpy.logical_and(kx == 0, ky == 0)] = 0
@@ -205,25 +203,45 @@ def far_to_nearfield(
 
     # Normalized vector potentials N, L
     L = [0.5 * E_far[1],
-        -0.5 * E_far[0]]    # noqa: E128
+        -0.5 * E_far[0]]    # noqa
     N = [L[1],
-        -L[0]]  # noqa: E128
+        -L[0]]  # noqa
 
-    En_fft = [-( L[0] * sin_th + L[1] * cos_phi * cos_th) / cos_phi,
-              -(-L[0] * cos_th + L[1] * cos_phi * sin_th) / cos_phi]
+    En_fft = [
+        numpy.divide(
+            -(L[0] * sin_th + L[1] * cos_phi * cos_th),
+            cos_phi,
+            out=numpy.zeros_like(L[0]),
+            where=cos_phi != 0,
+            ),
+        numpy.divide(
+            -(-L[0] * cos_th + L[1] * cos_phi * sin_th),
+            cos_phi,
+            out=numpy.zeros_like(L[0]),
+            where=cos_phi != 0,
+            ),
+        ]
 
-    Hn_fft = [( N[0] * sin_th + N[1] * cos_phi * cos_th) / cos_phi,
-              (-N[0] * cos_th + N[1] * cos_phi * sin_th) / cos_phi]
-
-    for i in range(2):
-        En_fft[i][cos_phi == 0] = 0
-        Hn_fft[i][cos_phi == 0] = 0
+    Hn_fft = [
+        numpy.divide(
+            N[0] * sin_th + N[1] * cos_phi * cos_th,
+            cos_phi,
+            out=numpy.zeros_like(N[0]),
+            where=cos_phi != 0,
+            ),
+        numpy.divide(
+            -N[0] * cos_th + N[1] * cos_phi * sin_th,
+            cos_phi,
+            out=numpy.zeros_like(N[0]),
+            where=cos_phi != 0,
+            ),
+        ]
 
     E_near = [ifftshift(ifft2(ifftshift(Ei), s=padded_shape)) for Ei in En_fft]
     H_near = [ifftshift(ifft2(ifftshift(Hi), s=padded_shape)) for Hi in Hn_fft]
 
     dx = 2 * pi / (s[0] * dkx)
-    dy = 2 * pi / (s[0] * dky)
+    dy = 2 * pi / (s[1] * dky)
 
     outputs = {
         'E': E_near,
@@ -233,4 +251,3 @@ def far_to_nearfield(
     }
 
     return outputs
-

meanas/fdfd/functional.py

@@ -5,10 +5,10 @@ Functional versions of many FDFD operators. These can be useful for performing
 The functions generated here expect `cfdfield_t` inputs with shape (3, X, Y, Z),
 e.g. E = [E_x, E_y, E_z] where each (complex) component has shape (X, Y, Z)
 """
-from typing import Callable
+from collections.abc import Callable
 import numpy
 
-from ..fdmath import dx_lists_t, fdfield_t, cfdfield_t, cfdfield_updater_t
+from ..fdmath import dx_lists_t, cfdfield_t, fdfield, cfdfield, cfdfield_updater_t
 from ..fdmath.functional import curl_forward, curl_back
 
 
@@ -18,8 +18,8 @@ __author__ = 'Jan Petykiewicz'
 def e_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None,
+        epsilon: fdfield,
+        mu: fdfield | None = None,
         ) -> cfdfield_updater_t:
     """
     Wave operator for use with E-field. See `operators.e_full` for details.
@@ -37,26 +37,25 @@ def e_full(
     ch = curl_back(dxes[1])
     ce = curl_forward(dxes[0])
 
-    def op_1(e: cfdfield_t) -> cfdfield_t:
+    def op_1(e: cfdfield) -> cfdfield_t:
         curls = ch(ce(e))
-        return curls - omega ** 2 * epsilon * e
+        return cfdfield_t(curls - omega ** 2 * epsilon * e)
 
     def op_mu(e: cfdfield_t) -> cfdfield_t:
-        curls = ch(mu * ce(e))          # type: ignore   # mu = None ok because we don't return the function
-        return curls - omega ** 2 * epsilon * e
+        curls = ch(ce(e) / mu)          # type: ignore   # mu = None ok because we don't return the function
+        return cfdfield_t(curls - omega ** 2 * epsilon * e)
 
     if mu is None:
         return op_1
-    else:
     return op_mu
 
 
 def eh_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None,
-        ) -> Callable[[cfdfield_t, cfdfield_t], tuple[cfdfield_t, cfdfield_t]]:
+        epsilon: fdfield,
+        mu: fdfield | None = None,
+        ) -> Callable[[cfdfield, cfdfield], tuple[cfdfield_t, cfdfield_t]]:
     """
     Wave operator for full (both E and H) field representation.
     See `operators.eh_full`.
@@ -74,24 +73,23 @@ def eh_full(
     ch = curl_back(dxes[1])
     ce = curl_forward(dxes[0])
 
-    def op_1(e: cfdfield_t, h: cfdfield_t) -> tuple[cfdfield_t, cfdfield_t]:
-        return (ch(h) - 1j * omega * epsilon * e,
-                ce(e) + 1j * omega * h)
+    def op_1(e: cfdfield, h: cfdfield) -> tuple[cfdfield_t, cfdfield_t]:
+        return (cfdfield_t(ch(h) - 1j * omega * epsilon * e),
+                cfdfield_t(ce(e) + 1j * omega * h))
 
-    def op_mu(e: cfdfield_t, h: cfdfield_t) -> tuple[cfdfield_t, cfdfield_t]:
-        return (ch(h) - 1j * omega * epsilon * e,
-                ce(e) + 1j * omega * mu * h)            # type: ignore   # mu=None ok
+    def op_mu(e: cfdfield, h: cfdfield) -> tuple[cfdfield_t, cfdfield_t]:
+        return (cfdfield_t(ch(h) - 1j * omega * epsilon * e),
+                cfdfield_t(ce(e) + 1j * omega * mu * h))            # type: ignore   # mu=None ok
 
     if mu is None:
         return op_1
-    else:
     return op_mu
 
 
 def e2h(
         omega: complex,
         dxes: dx_lists_t,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         ) -> cfdfield_updater_t:
     """
     Utility operator for converting the `E` field into the `H` field.
@@ -108,22 +106,21 @@ def e2h(
     """
     ce = curl_forward(dxes[0])
 
-    def e2h_1_1(e: cfdfield_t) -> cfdfield_t:
-        return ce(e) / (-1j * omega)
+    def e2h_1_1(e: cfdfield) -> cfdfield_t:
+        return cfdfield_t(ce(e) / (-1j * omega))
 
-    def e2h_mu(e: cfdfield_t) -> cfdfield_t:
-        return ce(e) / (-1j * omega * mu)       # type: ignore   # mu=None ok
+    def e2h_mu(e: cfdfield) -> cfdfield_t:
+        return cfdfield_t(ce(e) / (-1j * omega * mu))       # type: ignore   # mu=None ok
 
     if mu is None:
         return e2h_1_1
-    else:
     return e2h_mu
 
 
 def m2j(
         omega: complex,
         dxes: dx_lists_t,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         ) -> cfdfield_updater_t:
     """
     Utility operator for converting magnetic current `M` distribution
@@ -142,30 +139,42 @@ def m2j(
     ch = curl_back(dxes[1])
 
     def m2j_mu(m: cfdfield_t) -> cfdfield_t:
-        J = ch(m / mu) / (-1j * omega)          # type: ignore  # mu=None ok
-        return J
+        J = ch(m / mu) / (1j * omega)           # type: ignore  # mu=None ok
+        return cfdfield_t(J)
 
     def m2j_1(m: cfdfield_t) -> cfdfield_t:
-        J = ch(m) / (-1j * omega)
-        return J
+        J = ch(m) / (1j * omega)
+        return cfdfield_t(J)
 
     if mu is None:
         return m2j_1
-    else:
     return m2j_mu
 
 
 def e_tfsf_source(
-        TF_region: fdfield_t,
+        TF_region: fdfield,
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None,
+        epsilon: fdfield,
+        mu: fdfield | None = None,
         ) -> cfdfield_updater_t:
-    """
+    r"""
     Operator that turns an E-field distribution into a total-field/scattered-field
     (TFSF) source.
 
+    If `A` is the full wave operator from `e_full(...)` and `Q` is the diagonal
+    mask selecting the total-field region, then the TFSF source is the commutator
+
+    $$
+    \frac{A Q - Q A}{-i \omega} E.
+    $$
+
+    This vanishes in the interior of the total-field and scattered-field regions
+    and is supported only at their shared boundary, where the mask discontinuity
+    makes `A` and `Q` fail to commute. The returned current is therefore the
+    distributed source needed to inject the desired total field without also
+    forcing the scattered-field region.
+
     Args:
         TF_region: mask which is set to 1 in the total-field region, and 0 elsewhere
                    (i.e. in the scattered-field region).
@@ -179,20 +188,25 @@ def e_tfsf_source(
         Function `f` which takes an E field and returns a current distribution,
         `f(E)` -> `J`
     """
-    # TODO documentation
     A = e_full(omega, dxes, epsilon, mu)
 
-    def op(e: cfdfield_t) -> cfdfield_t:
+    def op(e: cfdfield) -> cfdfield_t:
         neg_iwj = A(TF_region * e) - TF_region * A(e)
-        return neg_iwj / (-1j * omega)
+        return cfdfield_t(neg_iwj / (-1j * omega))
     return op
 
 
-def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], cfdfield_t]:
-    """
+def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield, cfdfield], cfdfield_t]:
+    r"""
     Generates a function that takes the single-frequency `E` and `H` fields
-    and calculates the cross product `E` x `H` = $E \\times H$ as required
-    for the Poynting vector, $S = E \\times H$
+    and calculates the cross product `E` x `H` = $E \times H$ as required
+    for the Poynting vector, $S = E \times H$.
+
+    On the Yee grid, the electric and magnetic components are not stored at the
+    same locations. This helper therefore applies the same one-cell electric-field
+    shifts used by the sparse `operators.poynting_e_cross(...)` construction so
+    that the discrete cross product matches the face-centered energy flux used in
+    `meanas.fdtd.energy.poynting(...)`.
 
     Note:
         This function also shifts the input `E` field by one cell as required
@@ -200,7 +214,7 @@ def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], c
 
     Note:
         If `E` and `H` are peak amplitudes as assumed elsewhere in this code,
-        the time-average of the poynting vector is `<S> = Re(S)/2 = Re(E x H) / 2`.
+        the time-average of the poynting vector is `<S> = Re(S)/2 = Re(E x H*) / 2`.
         The factor of `1/2` can be omitted if root-mean-square quantities are used
         instead.
 
@@ -208,9 +222,10 @@ def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], c
         dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
 
     Returns:
-        Function `f` that returns E x H as required for the poynting vector.
+        Function `f` that returns the staggered-grid cross product `E \times H`.
+        For time-average power, call it as `f(E, H.conj())` and take `Re(...) / 2`.
     """
-    def exh(e: cfdfield_t, h: cfdfield_t) -> cfdfield_t:
+    def exh(e: cfdfield, h: cfdfield) -> cfdfield_t:
         s = numpy.empty_like(e)
         ex = e[0] * dxes[0][0][:, None, None]
         ey = e[1] * dxes[0][1][None, :, None]
@@ -221,5 +236,5 @@ def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], c
         s[0] = numpy.roll(ey, -1, axis=0) * hz - numpy.roll(ez, -1, axis=0) * hy
         s[1] = numpy.roll(ez, -1, axis=1) * hx - numpy.roll(ex, -1, axis=1) * hz
         s[2] = numpy.roll(ex, -1, axis=2) * hy - numpy.roll(ey, -1, axis=2) * hx
-        return s
+        return cfdfield_t(s)
     return exh

meanas/fdfd/operators.py

@@ -1,7 +1,7 @@
 """
 Sparse matrix operators for use with electromagnetic wave equations.
 
-These functions return sparse-matrix (`scipy.sparse.spmatrix`) representations of
+These functions return sparse-matrix (`scipy.sparse.sparray`) representations of
  a variety of operators, intended for use with E and H fields vectorized using the
  `meanas.fdmath.vectorization.vec()` and `meanas.fdmath.vectorization.unvec()` functions.
 
@@ -28,9 +28,9 @@ The following operators are included:
 """
 
 import numpy
-import scipy.sparse as sparse       # type: ignore
+from scipy import sparse
 
-from ..fdmath import vec, dx_lists_t, vfdfield_t, vcfdfield_t
+from ..fdmath import vec, dx_lists_t, vfdfield, vcfdfield
 from ..fdmath.operators import shift_with_mirror, shift_circ, curl_forward, curl_back
 
 
@@ -40,19 +40,19 @@ __author__ = 'Jan Petykiewicz'
 def e_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
-        mu: vfdfield_t | None = None,
-        pec: vfdfield_t | None = None,
-        pmc: vfdfield_t | None = None,
-        ) -> sparse.spmatrix:
-    """
+        epsilon: vfdfield | vcfdfield,
+        mu: vfdfield | None = None,
+        pec: vfdfield | None = None,
+        pmc: vfdfield | None = None,
+        ) -> sparse.sparray:
+    r"""
     Wave operator
-     $$ \\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon $$
+     $$ \nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon $$
 
         del x (1/mu * del x) - omega**2 * epsilon
 
      for use with the E-field, with wave equation
-     $$ (\\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon) E = -\\imath \\omega J $$
+     $$ (\nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon) E = -\imath \omega J $$
 
         (del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J
 
@@ -77,20 +77,20 @@ def e_full(
     ce = curl_forward(dxes[0])
 
     if pec is None:
-        pe = sparse.eye(epsilon.size)
+        pe = sparse.eye_array(epsilon.size)
     else:
-        pe = sparse.diags(numpy.where(pec, 0, 1))     # Set pe to (not PEC)
+        pe = sparse.diags_array(numpy.where(pec, 0, 1))     # Set pe to (not PEC)
 
     if pmc is None:
-        pm = sparse.eye(epsilon.size)
+        pm = sparse.eye_array(epsilon.size)
     else:
-        pm = sparse.diags(numpy.where(pmc, 0, 1))     # set pm to (not PMC)
+        pm = sparse.diags_array(numpy.where(pmc, 0, 1))     # set pm to (not PMC)
 
-    e = sparse.diags(epsilon)
+    e = sparse.diags_array(epsilon)
     if mu is None:
-        m_div = sparse.eye(epsilon.size)
+        m_div = sparse.eye_array(epsilon.size)
     else:
-        m_div = sparse.diags(1 / mu)
+        m_div = sparse.diags_array(1 / mu)
 
     op = pe @ (ch @ pm @ m_div @ ce - omega**2 * e) @ pe
     return op
@@ -98,7 +98,7 @@ def e_full(
 
 def e_full_preconditioners(
         dxes: dx_lists_t,
-        ) -> tuple[sparse.spmatrix, sparse.spmatrix]:
+        ) -> tuple[sparse.sparray, sparse.sparray]:
     """
     Left and right preconditioners `(Pl, Pr)` for symmetrizing the `e_full` wave operator.
 
@@ -118,27 +118,27 @@ def e_full_preconditioners(
                  dxes[1][0][:, None, None] * dxes[1][1][None, :, None] * dxes[0][2][None, None, :]]
 
     p_vector = numpy.sqrt(vec(p_squared))
-    P_left = sparse.diags(p_vector)
-    P_right = sparse.diags(1 / p_vector)
+    P_left = sparse.diags_array(p_vector)
+    P_right = sparse.diags_array(1 / p_vector)
     return P_left, P_right
 
 
 def h_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
-        mu: vfdfield_t | None = None,
-        pec: vfdfield_t | None = None,
-        pmc: vfdfield_t | None = None,
-        ) -> sparse.spmatrix:
-    """
+        epsilon: vfdfield,
+        mu: vfdfield | None = None,
+        pec: vfdfield | None = None,
+        pmc: vfdfield | None = None,
+        ) -> sparse.sparray:
+    r"""
     Wave operator
-     $$ \\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu $$
+     $$ \nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu $$
 
         del x (1/epsilon * del x) - omega**2 * mu
 
      for use with the H-field, with wave equation
-     $$ (\\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu) E = \\imath \\omega M $$
+     $$ (\nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu) E = \imath \omega M $$
 
         (del x (1/epsilon * del x) - omega**2 * mu) E = i * omega * M
 
@@ -161,20 +161,20 @@ def h_full(
     ce = curl_forward(dxes[0])
 
     if pec is None:
-        pe = sparse.eye(epsilon.size)
+        pe = sparse.eye_array(epsilon.size)
     else:
-        pe = sparse.diags(numpy.where(pec, 0, 1))    # set pe to (not PEC)
+        pe = sparse.diags_array(numpy.where(pec, 0, 1))    # set pe to (not PEC)
 
     if pmc is None:
-        pm = sparse.eye(epsilon.size)
+        pm = sparse.eye_array(epsilon.size)
     else:
-        pm = sparse.diags(numpy.where(pmc, 0, 1))    # Set pe to (not PMC)
+        pm = sparse.diags_array(numpy.where(pmc, 0, 1))    # Set pe to (not PMC)
 
-    e_div = sparse.diags(1 / epsilon)
+    e_div = sparse.diags_array(1 / epsilon)
     if mu is None:
-        m = sparse.eye(epsilon.size)
+        m = sparse.eye_array(epsilon.size)
     else:
-        m = sparse.diags(mu)
+        m = sparse.diags_array(mu)
 
     A = pm @ (ce @ pe @ e_div @ ch - omega**2 * m) @ pm
     return A
@@ -183,33 +183,33 @@ def h_full(
 def eh_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
-        mu: vfdfield_t | None = None,
-        pec: vfdfield_t | None = None,
-        pmc: vfdfield_t | None = None,
-        ) -> sparse.spmatrix:
-    """
+        epsilon: vfdfield,
+        mu: vfdfield | None = None,
+        pec: vfdfield | None = None,
+        pmc: vfdfield | None = None,
+        ) -> sparse.sparray:
+    r"""
     Wave operator for `[E, H]` field representation. This operator implements Maxwell's
      equations without cancelling out either E or H. The operator is
-    $$  \\begin{bmatrix}
-        -\\imath \\omega \\epsilon  &  \\nabla \\times      \\\\
-        \\nabla \\times             &  \\imath \\omega \\mu
-        \\end{bmatrix} $$
+    $$  \begin{bmatrix}
+        -\imath \omega \epsilon  &  \nabla \times      \\
+        \nabla \times            &  \imath \omega \mu
+        \end{bmatrix} $$
 
         [[-i * omega * epsilon,  del x         ],
          [del x,                 i * omega * mu]]
 
     for use with a field vector of the form `cat(vec(E), vec(H))`:
-    $$  \\begin{bmatrix}
-        -\\imath \\omega \\epsilon  &  \\nabla \\times      \\\\
-        \\nabla \\times             &  \\imath \\omega \\mu
-        \\end{bmatrix}
-        \\begin{bmatrix} E \\\\
+    $$  \begin{bmatrix}
+        -\imath \omega \epsilon  &  \nabla \times      \\
+        \nabla \times            &  \imath \omega \mu
+        \end{bmatrix}
+        \begin{bmatrix} E \\
                         H
-        \\end{bmatrix}
-        = \\begin{bmatrix} J \\\\
+        \end{bmatrix}
+        = \begin{bmatrix} J \\
                          -M
-          \\end{bmatrix} $$
+          \end{bmatrix} $$
 
     Args:
         omega: Angular frequency of the simulation
@@ -227,25 +227,27 @@ def eh_full(
         Sparse matrix containing the wave operator.
     """
     if pec is None:
-        pe = sparse.eye(epsilon.size)
+        pe = sparse.eye_array(epsilon.size)
     else:
-        pe = sparse.diags(numpy.where(pec, 0, 1))    # set pe to (not PEC)
+        pe = sparse.diags_array(numpy.where(pec, 0, 1))    # set pe to (not PEC)
 
     if pmc is None:
-        pm = sparse.eye(epsilon.size)
+        pm = sparse.eye_array(epsilon.size)
     else:
-        pm = sparse.diags(numpy.where(pmc, 0, 1))    # set pm to (not PMC)
+        pm = sparse.diags_array(numpy.where(pmc, 0, 1))    # set pm to (not PMC)
 
     iwe = pe @ (1j * omega * sparse.diags(epsilon)) @ pe
-    iwm = 1j * omega
-    if mu is not None:
-        iwm *= sparse.diags(mu)
+    if mu is None:
+        iwm = 1j * omega * sparse.eye(epsilon.size)
+    else:
+        iwm = 1j * omega * sparse.diags(mu)
+
     iwm = pm @ iwm @ pm
 
     A1 = pe @ curl_back(dxes[1]) @ pm
     A2 = pm @ curl_forward(dxes[0]) @ pe
 
-    A = sparse.bmat([[-iwe, A1],
+    A = sparse.block_array([[-iwe, A1],
                             [A2,  iwm]])
     return A
 
@@ -253,9 +255,9 @@ def eh_full(
 def e2h(
         omega: complex,
         dxes: dx_lists_t,
-        mu: vfdfield_t | None = None,
-        pmc: vfdfield_t | None = None,
-        ) -> sparse.spmatrix:
+        mu: vfdfield | None = None,
+        pmc: vfdfield | None = None,
+        ) -> sparse.sparray:
     """
     Utility operator for converting the E field into the H field.
     For use with `e_full()` -- assumes that there is no magnetic current M.
@@ -274,10 +276,10 @@ def e2h(
     op = curl_forward(dxes[0]) / (-1j * omega)
 
     if mu is not None:
-        op = sparse.diags(1 / mu) @ op
+        op = sparse.diags_array(1 / mu) @ op
 
     if pmc is not None:
-        op = sparse.diags(numpy.where(pmc, 0, 1)) @ op
+        op = sparse.diags_array(numpy.where(pmc, 0, 1)) @ op
 
     return op
 
@@ -285,8 +287,8 @@ def e2h(
 def m2j(
         omega: complex,
         dxes: dx_lists_t,
-        mu: vfdfield_t | None = None,
-        ) -> sparse.spmatrix:
+        mu: vfdfield | None = None,
+        ) -> sparse.sparray:
     """
     Operator for converting a magnetic current M into an electric current J.
     For use with eg. `e_full()`.
@@ -302,78 +304,108 @@ def m2j(
     op = curl_back(dxes[1]) / (1j * omega)
 
     if mu is not None:
-        op = op @ sparse.diags(1 / mu)
+        op = op @ sparse.diags_array(1 / mu)
 
     return op
 
 
-def poynting_e_cross(e: vcfdfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
-    """
-    Operator for computing the Poynting vector, containing the
-    (E x) portion of the Poynting vector.
+def poynting_e_cross(e: vcfdfield, dxes: dx_lists_t) -> sparse.sparray:
+    r"""
+    Operator for computing the staggered-grid `(E \times)` part of the Poynting vector.
+
+    On the Yee grid the E and H components live on different edges, so the
+    electric field must be shifted by one cell in the appropriate direction
+    before forming the discrete cross product. This sparse operator encodes that
+    shifted cross product directly and is the matrix equivalent of
+    `functional.poynting_e_cross_h(...)`.
 
     Args:
         e: Vectorized E-field for the ExH cross product
         dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
 
     Returns:
-        Sparse matrix containing (E x) portion of Poynting cross product.
+        Sparse matrix containing the `(E \times)` part of the staggered Poynting
+        cross product.
     """
     shape = [len(dx) for dx in dxes[0]]
 
-    fx, fy, fz = [shift_circ(i, shape, 1) for i in range(3)]
+    fx, fy, fz = (shift_circ(i, shape, 1) for i in range(3))
 
     dxag = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
-    Ex, Ey, Ez = [ei * da for ei, da in zip(numpy.split(e, 3), dxag)]
+    dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
+    Ex, Ey, Ez = (ei * da for ei, da in zip(numpy.split(e, 3), dxag, strict=True))
 
     block_diags = [[ None,     fx @ -Ez, fx @  Ey],
                    [ fy @  Ez, None,     fy @ -Ex],
                    [ fz @ -Ey, fz @  Ex, None]]
-    block_matrix = sparse.bmat([[sparse.diags(x) if x is not None else None for x in row]
+    block_matrix = sparse.block_array([[sparse.diags_array(x) if x is not None else None for x in row]
                                         for row in block_diags])
-    P = block_matrix @ sparse.diags(numpy.concatenate(dxag))
+    P = block_matrix @ sparse.diags_array(numpy.concatenate(dxbg))
     return P
 
 
-def poynting_h_cross(h: vcfdfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
-    """
-    Operator for computing the Poynting vector, containing the (H x) portion of the Poynting vector.
+def poynting_h_cross(h: vcfdfield, dxes: dx_lists_t) -> sparse.sparray:
+    r"""
+    Operator for computing the staggered-grid `(H \times)` part of the Poynting vector.
+
+    Together with `poynting_e_cross(...)`, this provides the matrix form of the
+    Yee-grid cross product used in the flux helpers. The two are related by the
+    usual antisymmetry of the cross product,
+
+    $$
+    H \times E = -(E \times H),
+    $$
+
+    once the same staggered field placement is used on both sides.
 
     Args:
         h: Vectorized H-field for the HxE cross product
         dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
 
     Returns:
-        Sparse matrix containing (H x) portion of Poynting cross product.
+        Sparse matrix containing the `(H \times)` part of the staggered Poynting
+        cross product.
     """
     shape = [len(dx) for dx in dxes[0]]
 
-    fx, fy, fz = [shift_circ(i, shape, 1) for i in range(3)]
+    fx, fy, fz = (shift_circ(i, shape, 1) for i in range(3))
 
     dxag = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
     dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
-    Hx, Hy, Hz = [sparse.diags(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg)]
+    Hx, Hy, Hz = (sparse.diags_array(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg, strict=True))
 
-    P = (sparse.bmat(
+    P = (sparse.block_array(
         [[ None,    -Hz @ fx,   Hy @ fx],
          [ Hz @ fy,  None,     -Hx @ fy],
          [-Hy @ fz,  Hx @ fz,   None]])
-         @ sparse.diags(numpy.concatenate(dxag)))
+         @ sparse.diags_array(numpy.concatenate(dxag)))
     return P
 
 
 def e_tfsf_source(
-        TF_region: vfdfield_t,
+        TF_region: vfdfield,
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
-        mu: vfdfield_t | None = None,
-        ) -> sparse.spmatrix:
-    """
+        epsilon: vfdfield,
+        mu: vfdfield | None = None,
+        ) -> sparse.sparray:
+    r"""
     Operator that turns a desired E-field distribution into a
      total-field/scattered-field (TFSF) source.
 
-    TODO: Reference Rumpf paper
+    Let `A` be the full wave operator from `e_full(...)`, and let
+    `Q = \mathrm{diag}(TF_region)` be the projector onto the total-field region.
+    Then the TFSF current operator is the commutator
+
+    $$
+    \frac{A Q - Q A}{-i \omega}.
+    $$
+
+    Inside regions where `Q` is locally constant, `A` and `Q` commute and the
+    source vanishes. Only cells at the TF/SF boundary contribute nonzero current,
+    which is exactly the desired distributed source for injecting the chosen
+    field into the total-field region without directly forcing the
+    scattered-field region.
 
     Args:
         TF_region: Mask, which is set to 1 inside the total-field region and 0 in the
@@ -385,27 +417,31 @@ def e_tfsf_source(
 
     Returns:
         Sparse matrix that turns an E-field into a current (J) distribution.
-
     """
-    # TODO documentation
     A = e_full(omega, dxes, epsilon, mu)
-    Q = sparse.diags(TF_region)
+    Q = sparse.diags_array(TF_region)
     return (A @ Q - Q @ A) / (-1j * omega)
 
 
 def e_boundary_source(
-        mask: vfdfield_t,
+        mask: vfdfield,
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
-        mu: vfdfield_t | None = None,
+        epsilon: vfdfield,
+        mu: vfdfield | None = None,
         periodic_mask_edges: bool = False,
-        ) -> sparse.spmatrix:
-    """
+        ) -> sparse.sparray:
+    r"""
     Operator that turns an E-field distrubtion into a current (J) distribution
       along the edges (external and internal) of the provided mask. This is just an
       `e_tfsf_source()` with an additional masking step.
 
+    Equivalently, this helper first constructs the TFSF commutator source for the
+    full mask and then zeroes out all cells except the mask boundary. The
+    boundary is defined as the set of cells whose mask value changes under a
+    one-cell shift in any Cartesian direction. With `periodic_mask_edges=False`
+    the shifts mirror at the domain boundary; with `True` they wrap periodically.
+
     Args:
         mask: The current distribution is generated at the edges of the mask,
               i.e. any points where shifting the mask by one cell in any direction
@@ -423,10 +459,10 @@ def e_boundary_source(
     shape = [len(dxe) for dxe in dxes[0]]
     jmask = numpy.zeros_like(mask, dtype=bool)
 
-    def shift_rot(axis: int, polarity: int) -> sparse.spmatrix:
+    def shift_rot(axis: int, polarity: int) -> sparse.sparray:
         return shift_circ(axis=axis, shape=shape, shift_distance=polarity)
 
-    def shift_mir(axis: int, polarity: int) -> sparse.spmatrix:
+    def shift_mir(axis: int, polarity: int) -> sparse.sparray:
         return shift_with_mirror(axis=axis, shape=shape, shift_distance=polarity)
 
     shift = shift_rot if periodic_mask_edges else shift_mir
@@ -435,7 +471,7 @@ def e_boundary_source(
         if shape[axis] == 1:
             continue
         for polarity in (-1, +1):
-            r = shift(axis, polarity) - sparse.eye(numpy.prod(shape))  # shifted minus original
+            r = shift(axis, polarity) - sparse.eye_array(numpy.prod(shape))  # shifted minus original
             r3 = sparse.block_diag((r, r, r))
             jmask = numpy.logical_or(jmask, numpy.abs(r3 @ mask))
 
@@ -446,5 +482,4 @@ def e_boundary_source(
 #             (numpy.roll(mask, -1, axis=2) != mask) |
 #             (numpy.roll(mask, +1, axis=2) != mask))
 
-    return sparse.diags(jmask.astype(int)) @ full
-
+    return sparse.diags_array(jmask.astype(int)) @ full

meanas/fdfd/scpml.py

@@ -2,7 +2,7 @@
 Functions for creating stretched coordinate perfectly matched layer (PML) absorbers.
 """
 
-from typing import Sequence, Callable
+from collections.abc import Sequence, Callable
 
 import numpy
 from numpy.typing import NDArray
@@ -128,6 +128,11 @@ def stretch_with_scpml(
     dx_ai = dxes[0][axis].astype(complex)
     dx_bi = dxes[1][axis].astype(complex)
 
+    if thickness == 0:
+        dxes[0][axis] = dx_ai
+        dxes[1][axis] = dx_bi
+        return dxes
+
     pos = numpy.hstack((0, dx_ai.cumsum()))
     pos_a = (pos[:-1] + pos[1:]) / 2
     pos_b = pos[:-1]
@@ -153,9 +158,6 @@ def stretch_with_scpml(
         def l_d(x: NDArray[numpy.float64]) -> NDArray[numpy.float64]:
             return (x - bound) / (pos[-1] - bound)
 
-        if thickness == 0:
-            slc = slice(None)
-        else:
         slc = slice(-thickness, None)
 
     dx_ai[slc] *= 1 + 1j * s_function(l_d(pos_a[slc])) / d / s_correction

meanas/fdfd/solvers.py

@@ -2,15 +2,16 @@
 Solvers and solver interface for FDFD problems.
 """
 
-from typing import Callable, Dict, Any, Optional
+from typing import Any
+from collections.abc import Callable
 import logging
 
 import numpy
 from numpy.typing import ArrayLike, NDArray
 from numpy.linalg import norm
-import scipy.sparse.linalg          # type: ignore
+import scipy.sparse.linalg
 
-from ..fdmath import dx_lists_t, vfdfield_t, vcfdfield_t
+from ..fdmath import dx_lists_t, vfdfield, vcfdfield, vcfdfield_t
 from . import operators
 
 
@@ -18,7 +19,7 @@ logger = logging.getLogger(__name__)
 
 
 def _scipy_qmr(
-        A: scipy.sparse.csr_matrix,
+        A: scipy.sparse.csr_array,
         b: ArrayLike,
         **kwargs: Any,
         ) -> NDArray[numpy.float64]:
@@ -34,30 +35,32 @@ def _scipy_qmr(
         Guess for solution (returned even if didn't converge)
     """
 
-    '''
-    Report on our progress
-    '''
+    #
+    #Report on our progress
+    #
     ii = 0
 
     def log_residual(xk: ArrayLike) -> None:
         nonlocal ii
         ii += 1
         if ii % 100 == 0:
-            logger.info('Solver residual at iteration {} : {}'.format(ii, norm(A @ xk - b)))
+            cur_norm = norm(A @ xk - b) / norm(b)
+            logger.info(f'Solver residual at iteration {ii} : {cur_norm}')
 
     if 'callback' in kwargs:
+        callback = kwargs['callback']
+
         def augmented_callback(xk: ArrayLike) -> None:
             log_residual(xk)
-            kwargs['callback'](xk)
+            callback(xk)
 
         kwargs['callback'] = augmented_callback
     else:
         kwargs['callback'] = log_residual
 
-    '''
-    Run the actual solve
-    '''
-
+    #
+    # Run the actual solve
+    #
     x, _ = scipy.sparse.linalg.qmr(A, b, **kwargs)
     return x
 
@@ -65,14 +68,16 @@ def _scipy_qmr(
 def generic(
         omega: complex,
         dxes: dx_lists_t,
-        J: vcfdfield_t,
-        epsilon: vfdfield_t,
-        mu: Optional[vfdfield_t] = None,
-        pec: Optional[vfdfield_t] = None,
-        pmc: Optional[vfdfield_t] = None,
+        J: vcfdfield,
+        epsilon: vfdfield,
+        mu: vfdfield | None = None,
+        *,
+        pec: vfdfield | None = None,
+        pmc: vfdfield | None = None,
         adjoint: bool = False,
         matrix_solver: Callable[..., ArrayLike] = _scipy_qmr,
-        matrix_solver_opts: Optional[Dict[str, Any]] = None,
+        matrix_solver_opts: dict[str, Any] | None = None,
+        E_guess: vcfdfield | None = None,
         ) -> vcfdfield_t:
     """
     Conjugate gradient FDFD solver using CSR sparse matrices.
@@ -92,13 +97,15 @@ def generic(
              (at H-field locations; non-zero value indicates PMC is present)
         adjoint: If true, solves the adjoint problem.
         matrix_solver: Called as `matrix_solver(A, b, **matrix_solver_opts) -> x`,
-                where `A`: `scipy.sparse.csr_matrix`;
+                where `A`: `scipy.sparse.csr_array`;
                       `b`: `ArrayLike`;
                       `x`: `ArrayLike`;
                 Default is a wrapped version of `scipy.sparse.linalg.qmr()`
                  which doesn't return convergence info and logs the residual
                  every 100 iterations.
         matrix_solver_opts: Passed as kwargs to `matrix_solver(...)`
+        E_guess: Guess at the solution E-field. `matrix_solver` must accept an
+                `x0` argument with the same purpose.
 
     Returns:
         E-field which solves the system.
@@ -113,17 +120,24 @@ def generic(
     Pl, Pr = operators.e_full_preconditioners(dxes)
 
     if adjoint:
-        A = (Pl @ A0 @ Pr).H
-        b = Pr.H @ b0
+        A = (Pl @ A0 @ Pr).T.conjugate()
+        b = Pr.T.conjugate() @ b0
     else:
         A = Pl @ A0 @ Pr
         b = Pl @ b0
 
+    if E_guess is not None:
+        if adjoint:
+            x0 = Pr.T.conjugate() @ E_guess
+        else:
+            x0 = Pl @ E_guess
+        matrix_solver_opts['x0'] = x0
+
     x = matrix_solver(A.tocsr(), b, **matrix_solver_opts)
 
     if adjoint:
-        x0 = Pl.H @ x
+        x0 = Pl.T.conjugate() @ x
     else:
         x0 = Pr @ x
 
-    return x0
+    return vcfdfield_t(x0)

meanas/fdfd/waveguide_3d.py

@@ -3,15 +3,40 @@ Tools for working with waveguide modes in 3D domains.
 
 This module relies heavily on `waveguide_2d` and mostly just transforms
 its parameters into 2D equivalents and expands the results back into 3D.
+
+The intended workflow is:
+
+1. Select a single-cell slice normal to the propagation axis.
+2. Solve the corresponding 2D mode problem with `solve_mode(...)`.
+3. Turn that mode into a one-sided source with `compute_source(...)`.
+4. Build an overlap window with `compute_overlap_e(...)` for port readout.
+
+`polarity` is part of the public convention throughout this module:
+
+- `+1` means forward propagation toward increasing index along `axis`
+- `-1` means backward propagation toward decreasing index along `axis`
+
+That same convention controls which side of the selected slice is used for the
+overlap window and how the expanded field is phased.
 """
-from typing import Sequence, Any
+from typing import Any, TypedDict, cast
+import warnings
+from collections.abc import Sequence
 import numpy
 from numpy.typing import NDArray
+from numpy import complexfloating
 
-from ..fdmath import vec, unvec, dx_lists_t, fdfield_t, cfdfield_t
+from ..fdmath import vec, unvec, dx_lists_t, cfdfield_t, fdfield, cfdfield
 from . import operators, waveguide_2d
 
 
+class Waveguide3DMode(TypedDict):
+    wavenumber: complex
+    wavenumber_2d: complex
+    H: NDArray[complexfloating]
+    E: NDArray[complexfloating]
+
+
 def solve_mode(
         mode_number: int,
         omega: complex,
@@ -19,10 +44,10 @@ def solve_mode(
         axis: int,
         polarity: int,
         slices: Sequence[slice],
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None,
-        ) -> dict[str, complex | NDArray[numpy.float_]]:
-    """
+        epsilon: fdfield,
+        mu: fdfield | None = None,
+        ) -> Waveguide3DMode:
+    r"""
     Given a 3D grid, selects a slice from the grid and attempts to
     solve for an eigenmode propagating through that slice.
 
@@ -33,27 +58,31 @@ def solve_mode(
         axis: Propagation axis (0=x, 1=y, 2=z)
         polarity: Propagation direction (+1 for +ve, -1 for -ve)
         slices: `epsilon[tuple(slices)]` is used to select the portion of the grid to use
-                as the waveguide cross-section. `slices[axis]` should select only one item.
+            as the waveguide cross-section. `slices[axis]` must select exactly one item.
         epsilon: Dielectric constant
         mu: Magnetic permeability (default 1 everywhere)
 
     Returns:
-        ```
-        {
-            'E': list[NDArray[numpy.float_]],
-            'H': list[NDArray[numpy.float_]],
-            'wavenumber': complex,
-        }
-        ```
+        Dictionary containing:
+
+        - `E`: full-grid electric field for the solved mode
+        - `H`: full-grid magnetic field for the solved mode
+        - `wavenumber`: propagation constant corrected for the discretized
+            propagation axis
+        - `wavenumber_2d`: propagation constant of the reduced 2D eigenproblem
+
+    Notes:
+        The returned fields are normalized through the `waveguide_2d`
+        normalization convention before being expanded back to 3D.
     """
     if mu is None:
         mu = numpy.ones_like(epsilon)
 
     slices = tuple(slices)
 
-    '''
-    Solve the 2D problem in the specified plane
-    '''
+    #
+    # Solve the 2D problem in the specified plane
+    #
     # Define rotation to set z as propagation direction
     order = numpy.roll(range(3), 2 - axis)
     reverse_order = numpy.roll(range(3), axis - 2)
@@ -71,9 +100,10 @@ def solve_mode(
     }
     e_xy, wavenumber_2d = waveguide_2d.solve_mode(mode_number, **args_2d)
 
-    '''
-    Apply corrections and expand to 3D
-    '''
+    #
+    # Apply corrections and expand to 3D
+    #
+
     # Correct wavenumber to account for numerical dispersion.
     wavenumber = 2 / dx_prop * numpy.arcsin(wavenumber_2d * dx_prop / 2)
 
@@ -92,11 +122,12 @@ def solve_mode(
     # Expand E, H to full epsilon space we were given
     E = numpy.zeros_like(epsilon, dtype=complex)
     H = numpy.zeros_like(epsilon, dtype=complex)
-    for a, o in enumerate(reverse_order):
-        E[(a, *slices)] = e[o][:, :, None].transpose(reverse_order)
-        H[(a, *slices)] = h[o][:, :, None].transpose(reverse_order)
+    for aa, oo in enumerate(reverse_order):
+        iii = cast('tuple[slice | int]', (aa, *slices))
+        E[iii] = e[oo][:, :, None].transpose(reverse_order)
+        H[iii] = h[oo][:, :, None].transpose(reverse_order)
 
-    results = {
+    results: Waveguide3DMode = {
         'wavenumber': wavenumber,
         'wavenumber_2d': wavenumber_2d,
         'H': H,
@@ -106,15 +137,15 @@ def solve_mode(
 
 
 def compute_source(
-        E: cfdfield_t,
+        E: cfdfield,
         wavenumber: complex,
         omega: complex,
         dxes: dx_lists_t,
         axis: int,
         polarity: int,
         slices: Sequence[slice],
-        epsilon: fdfield_t,
-        mu: fdfield_t | None = None,
+        epsilon: fdfield,
+        mu: fdfield | None = None,
         ) -> cfdfield_t:
     """
     Given an eigenmode obtained by `solve_mode`, returns the current source distribution
@@ -132,7 +163,14 @@ def compute_source(
         mu: Magnetic permeability (default 1 everywhere)
 
     Returns:
-        J distribution for the unidirectional source
+        `J` distribution for a one-sided electric-current source.
+
+    Notes:
+        The source is built from the expanded mode field and a boundary-source
+        operator. The resulting current is intended to be injected with the
+        same sign convention used elsewhere in the package:
+
+        `E -= dt * J / epsilon`
     """
     E_expanded = expand_e(E=E, dxes=dxes, wavenumber=wavenumber, axis=axis,
                           polarity=polarity, slices=slices)
@@ -148,66 +186,113 @@ def compute_source(
 
     masked_e2j = operators.e_boundary_source(mask=vec(mask), omega=omega, dxes=dxes, epsilon=vec(epsilon), mu=vec(mu))
     J = unvec(masked_e2j @ vec(E_expanded), E.shape[1:])
-    return J
+    return cfdfield_t(J)
 
 
 def compute_overlap_e(
-        E: cfdfield_t,
-        wavenumber: complex,
-        dxes: dx_lists_t,
-        axis: int,
-        polarity: int,
-        slices: Sequence[slice],
-        ) -> cfdfield_t:                 # TODO DOCS
-    """
-    Given an eigenmode obtained by `solve_mode`, calculates an overlap_e for the
-    mode orthogonality relation Integrate(((E x H_mode) + (E_mode x H)) dot dn)
-    [assumes reflection symmetry].
-
-    TODO: add reference
-
-    Args:
-        E: E-field of the mode
-        H: H-field of the mode (advanced by half of a Yee cell from E)
-        wavenumber: Wavenumber of the mode
-        omega: Angular frequency of the simulation
-        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
-        axis: Propagation axis (0=x, 1=y, 2=z)
-        polarity: Propagation direction (+1 for +ve, -1 for -ve)
-        slices: `epsilon[tuple(slices)]` is used to select the portion of the grid to use
-                as the waveguide cross-section. slices[axis] should select only one item.
-        mu: Magnetic permeability (default 1 everywhere)
-
-    Returns:
-        overlap_e such that `numpy.sum(overlap_e * other_e.conj())` computes the overlap integral
-    """
-    slices = tuple(slices)
-
-    Ee = expand_e(E=E, wavenumber=wavenumber, dxes=dxes,
-                  axis=axis, polarity=polarity, slices=slices)
-
-    start, stop = sorted((slices[axis].start, slices[axis].start - 2 * polarity))
-
-    slices2_l = list(slices)
-    slices2_l[axis] = slice(start, stop)
-    slices2 = (slice(None), *slices2_l)
-
-    Etgt = numpy.zeros_like(Ee)
-    Etgt[slices2] = Ee[slices2]
-
-    Etgt /= (Etgt.conj() * Etgt).sum()
-    return Etgt
-
-
-def expand_e(
-        E: cfdfield_t,
+        E: cfdfield,
         wavenumber: complex,
         dxes: dx_lists_t,
         axis: int,
         polarity: int,
         slices: Sequence[slice],
         ) -> cfdfield_t:
+    r"""
+    Build an overlap field for projecting another 3D electric field onto a mode.
+
+    The returned field is intended for the discrete overlap expression
+
+    $$
+    \sum \mathrm{overlap\_e} \; E_\mathrm{other}^*
+    $$
+
+    where the sum is over the full Yee-grid field storage.
+
+    The construction uses a two-cell window immediately upstream of the selected
+    slice:
+
+    - for `polarity=+1`, the two cells just before `slices[axis].start`
+    - for `polarity=-1`, the two cells just after `slices[axis].stop`
+
+    The window is clipped to the simulation domain if necessary. A clipped but
+    non-empty window raises `RuntimeWarning`; an empty window raises
+    `ValueError`.
+
+    The derivation below assumes reflection symmetry and the standard waveguide
+    overlap relation involving
+
+    $$
+    \int ((E \times H_\mathrm{mode}) + (E_\mathrm{mode} \times H)) \cdot dn.
+    $$
+
+    E x H_mode + E_mode x H
+    -> Ex Hmy - EyHmx + Emx Hy - Emy Hx  (Z-prop)
+    Ex we/B Emx + Ex i/B dy Hmz - Ey (-we/B Emy) - Ey i/B dx Hmz
+    we/B (Ex Emx + Ey Emy) + i/B (Ex dy Hmz - Ey dx Hmz)
+    we/B (Ex Emx + Ey Emy) + i/B (Ex dy (dx Emy - dy Emx) - Ey dx (dx Emy - dy Emx))
+    we/B (Ex Emx + Ey Emy) + i/B (Ex dy dx Emy - Ex dy dy Emx - Ey dx dx Emy -  Ey dx dy Emx)
+
+    Ex j/wu (-jB Emx - dx Emz) - Ey j/wu (dy Emz + jB Emy)
+    B/wu (Ex Emx + Ey Emy) - j/wu (Ex dx Emz + Ey dy Emz)
+
+
+    Args:
+        E: E-field of the mode
+        wavenumber: Wavenumber of the mode
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
+        axis: Propagation axis (0=x, 1=y, 2=z)
+        polarity: Propagation direction (+1 for +ve, -1 for -ve)
+        slices: `epsilon[tuple(slices)]` is used to select the portion of the grid to use
+                as the waveguide cross-section. slices[axis] should select only one item.
+
+    Returns:
+        `overlap_e` normalized so that `numpy.sum(overlap_e * E.conj()) == 1`
+        over the retained overlap window.
     """
+    slices = tuple(slices)
+
+    Ee = expand_e(E=E, wavenumber=wavenumber, dxes=dxes,
+                  axis=axis, polarity=polarity, slices=slices)
+
+    axis_size = E.shape[axis + 1]
+    if polarity > 0:
+        start = slices[axis].start - 2
+        stop = slices[axis].start
+    else:
+        start = slices[axis].stop
+        stop = slices[axis].stop + 2
+
+    clipped_start = max(0, start)
+    clipped_stop = min(axis_size, stop)
+    if clipped_start >= clipped_stop:
+        raise ValueError('Requested overlap window lies outside the domain')
+    if clipped_start != start or clipped_stop != stop:
+        warnings.warn('Requested overlap window was clipped to fit within the domain', RuntimeWarning, stacklevel=2)
+
+    slices2_l = list(slices)
+    slices2_l[axis] = slice(clipped_start, clipped_stop)
+    slices2 = (slice(None), *slices2_l)
+
+    Etgt = numpy.zeros_like(Ee)
+    Etgt[slices2] = Ee[slices2]
+
+    # Note: We normalize so that (Etgt @ E.conj()) == 1, so (Etgt @ Etgt.conj) != 1
+    norm = (Etgt.conj() * Etgt).sum()
+    if norm == 0:
+        raise ValueError('Requested overlap window contains no overlap field support')
+    Etgt = Etgt / norm
+    return cfdfield_t(Etgt)
+
+
+def expand_e(
+        E: cfdfield,
+        wavenumber: complex,
+        dxes: dx_lists_t,
+        axis: int,
+        polarity: int,
+        slices: Sequence[slice],
+        ) -> cfdfield_t:
+    r"""
     Given an eigenmode obtained by `solve_mode`, expands the E-field from the 2D
     slice where the mode was calculated to the entire domain (along the propagation
     axis). This assumes the epsilon cross-section remains constant throughout the
@@ -225,6 +310,16 @@ def expand_e(
 
     Returns:
         `E`, with the original field expanded along the specified `axis`.
+
+    Notes:
+        This helper assumes that the waveguide cross-section remains constant
+        along the propagation axis and applies the phase factor
+
+        $$
+        e^{-i \, \mathrm{polarity} \, wavenumber \, \Delta z}
+        $$
+
+        to each copied slice.
     """
     slices = tuple(slices)
 
@@ -245,4 +340,4 @@ def expand_e(
     slices_in = (slice(None), *slices)
 
     E_expanded[slices_exp] = phase_E * numpy.array(E)[slices_in]
-    return E_expanded
+    return cfdfield_t(E_expanded)

meanas/fdfd/waveguide_cyl.py

@@ -1,46 +1,181 @@
-"""
+r"""
 Operators and helper functions for cylindrical waveguides with unchanging cross-section.
 
-WORK IN PROGRESS, CURRENTLY BROKEN
+Waveguide operator is derived according to 10.1364/OL.33.001848.
 
-As the z-dependence is known, all the functions in this file assume a 2D grid
- (i.e. `dxes = [[[dr_e_0, dx_e_1, ...], [dy_e_0, ...]], [[dr_h_0, ...], [dy_h_0, ...]]]`).
+As in `waveguide_2d`, the propagation dependence is separated from the
+transverse solve. Here the propagation coordinate is the bend angle `\theta`,
+and the fields are assumed to have the form
+
+$$
+\vec{E}(r, y, \theta), \vec{H}(r, y, \theta) \propto e^{-\imath m \theta},
+$$
+
+where `m` is the angular wavenumber returned by `solve_mode(s)`. It is often
+convenient to introduce the corresponding linear wavenumber
+
+$$
+\beta = \frac{m}{r_{\min}},
+$$
+
+so that the cylindrical problem resembles the straight-waveguide problem with
+additional metric factors.
+
+Those metric factors live on the staggered radial Yee grids. If the left edge of
+the computational window is at `r = r_{\min}`, define the electric-grid and
+magnetic-grid radial sample locations by
+
+$$
+\begin{aligned}
+r_a(n) &= r_{\min} + \sum_{j \le n} \Delta r_{e, j}, \\
+r_b\!\left(n + \tfrac{1}{2}\right) &= r_{\min} + \tfrac{1}{2}\Delta r_{e, n}
+  + \sum_{j < n} \Delta r_{h, j},
+\end{aligned}
+$$
+
+and from them the diagonal metric matrices
+
+$$
+\begin{aligned}
+T_a &= \operatorname{diag}(r_a / r_{\min}), \\
+T_b &= \operatorname{diag}(r_b / r_{\min}).
+\end{aligned}
+$$
+
+With the same forward/backward derivative notation used in `waveguide_2d`, the
+coordinate-transformed discrete curl equations used here are
+
+$$
+\begin{aligned}
+-\imath \omega \mu_{rr} H_r &= \tilde{\partial}_y E_z + \imath \beta T_a^{-1} E_y, \\
+-\imath \omega \mu_{yy} H_y &= -\imath \beta T_b^{-1} E_r
+  - T_b^{-1} \tilde{\partial}_r (T_a E_z), \\
+-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_r E_y - \tilde{\partial}_y E_r, \\
+\imath \beta H_y &= -\imath \omega T_b \epsilon_{rr} E_r - T_b \hat{\partial}_y H_z, \\
+\imath \beta H_r &= \imath \omega T_a \epsilon_{yy} E_y
+  - T_b T_a^{-1} \hat{\partial}_r (T_b H_z), \\
+\imath \omega E_z &= T_a \epsilon_{zz}^{-1}
+  \left(\hat{\partial}_r H_y - \hat{\partial}_y H_r\right).
+\end{aligned}
+$$
+
+The first three equations are the cylindrical analogue of the straight-guide
+relations for `H_r`, `H_y`, and `H_z`. The next two are the metric-weighted
+versions of the straight-guide identities for `\imath \beta H_y` and
+`\imath \beta H_r`, and the last equation plays the same role as the
+longitudinal `E_z` reconstruction in `waveguide_2d`.
+
+Following the same elimination steps as in `waveguide_2d`, apply
+`\imath \beta \tilde{\partial}_r` and `\imath \beta \tilde{\partial}_y` to the
+equation for `E_z`, substitute for `\imath \beta H_r` and `\imath \beta H_y`,
+and then eliminate `H_z` with
+
+$$
+H_z = \frac{1}{-\imath \omega \mu_{zz}}
+\left(\tilde{\partial}_r E_y - \tilde{\partial}_y E_r\right).
+$$
+
+This yields the transverse electric eigenproblem implemented by
+`cylindrical_operator(...)`:
+
+$$
+\beta^2
+\begin{bmatrix} E_r \\ E_y \end{bmatrix}
+=
+\left(
+\omega^2
+\begin{bmatrix}
+T_b^2 \mu_{yy} \epsilon_{xx} & 0 \\
+0 & T_a^2 \mu_{xx} \epsilon_{yy}
+\end{bmatrix}
++
+\begin{bmatrix}
+-T_b \mu_{yy} \hat{\partial}_y \\
+ T_a \mu_{xx} \hat{\partial}_x
+\end{bmatrix}
+T_b \mu_{zz}^{-1}
+\begin{bmatrix}
+-\tilde{\partial}_y & \tilde{\partial}_x
+\end{bmatrix}
++
+\begin{bmatrix}
+\tilde{\partial}_x \\
+\tilde{\partial}_y
+\end{bmatrix}
+T_a \epsilon_{zz}^{-1}
+\begin{bmatrix}
+\hat{\partial}_x T_b \epsilon_{xx} &
+\hat{\partial}_y T_a \epsilon_{yy}
+\end{bmatrix}
+\right)
+\begin{bmatrix} E_r \\ E_y \end{bmatrix}.
+$$
+
+Since `\beta = m / r_{\min}`, the solver implemented in this file returns the
+angular wavenumber `m`, while the operator itself is most naturally written in
+terms of the linear quantity `\beta`. The helpers below reconstruct the full
+field components from the solved transverse eigenvector and then normalize the
+mode to unit forward power with the same discrete longitudinal Poynting inner
+product used by `waveguide_2d`.
+
+As in the straight-waveguide case, all functions here assume a 2D grid:
+
+`dxes = [[[dr_e_0, dr_e_1, ...], [dy_e_0, ...]], [[dr_h_0, ...], [dy_h_0, ...]]]`.
 """
-# TODO update module docs
+from typing import Any, cast
+from collections.abc import Sequence
+import logging
 
 import numpy
-import scipy.sparse as sparse       # type: ignore
+from numpy.typing import NDArray, ArrayLike
+from scipy import sparse
 
-from ..fdmath import vec, unvec, dx_lists_t, fdfield_t, vfdfield_t, cfdfield_t
+from ..fdmath import vec, unvec, dx_lists2_t, vcfdslice_t, vfdslice, vcfdslice, vcfdfield2
 from ..fdmath.operators import deriv_forward, deriv_back
 from ..eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
+from . import waveguide_2d
+
+logger = logging.getLogger(__name__)
 
 
 def cylindrical_operator(
-        omega: complex,
-        dxes: dx_lists_t,
-        epsilon: vfdfield_t,
-        r0: float,
-        ) -> sparse.spmatrix:
-    """
+        omega: float,
+        dxes: dx_lists2_t,
+        epsilon: vfdslice,
+        rmin: float,
+        ) -> sparse.sparray:
+    r"""
     Cylindrical coordinate waveguide operator of the form
 
-    TODO
+    $$
+        (\omega^2 \begin{bmatrix} T_b T_b \mu_{yy} \epsilon_{xx} & 0 \\
+                                                                0 & T_a T_a \mu_{xx} \epsilon_{yy} \end{bmatrix} +
+                  \begin{bmatrix} -T_b \mu_{yy} \hat{\partial}_y \\
+                                   T_a \mu_{xx} \hat{\partial}_x \end{bmatrix} T_b \mu_{zz}^{-1}
+                  \begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
+          \begin{bmatrix} \tilde{\partial}_x \\
+                          \tilde{\partial}_y \end{bmatrix} T_a \epsilon_{zz}^{-1}
+                     \begin{bmatrix} \hat{\partial}_x T_b \epsilon_{xx} & \hat{\partial}_y T_a \epsilon_{yy} \end{bmatrix})
+        \begin{bmatrix} E_r \\
+                        E_y \end{bmatrix}
+    $$
 
     for use with a field vector of the form `[E_r, E_y]`.
 
     This operator can be used to form an eigenvalue problem of the form
-        A @ [E_r, E_y] = wavenumber**2 * [E_r, E_y]
+        A @ [E_r, E_y] = beta**2 * [E_r, E_y]
 
     which can then be solved for the eigenmodes of the system
-    (an `exp(-i * wavenumber * theta)` theta-dependence is assumed for the fields).
+    (an `exp(-i * angular_wavenumber * theta)` theta-dependence is assumed for
+    the fields, with `beta = angular_wavenumber / rmin`).
+
+    (NOTE: See module docs and 10.1364/OL.33.001848)
 
     Args:
         omega: The angular frequency of the system
         dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
         epsilon: Vectorized dielectric constant grid
-        r0: Radius of curvature for the simulation. This should be the minimum value of
-            r within the simulation domain.
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
 
     Returns:
         Sparse matrix representation of the operator
@@ -49,95 +184,409 @@ def cylindrical_operator(
     Dfx, Dfy = deriv_forward(dxes[0])
     Dbx, Dby = deriv_back(dxes[1])
 
-    rx = r0 + numpy.cumsum(dxes[0][0])
-    ry = r0 + dxes[0][0] / 2.0 + numpy.cumsum(dxes[1][0])
-    tx = rx / r0
-    ty = ry / r0
-
-    Tx = sparse.diags(vec(tx[:, None].repeat(dxes[0][1].size, axis=1)))
-    Ty = sparse.diags(vec(ty[:, None].repeat(dxes[1][1].size, axis=1)))
+    Ta, Tb = dxes2T(dxes=dxes, rmin=rmin)
 
     eps_parts = numpy.split(epsilon, 3)
-    eps_x = sparse.diags(eps_parts[0])
-    eps_y = sparse.diags(eps_parts[1])
-    eps_z_inv = sparse.diags(1 / eps_parts[2])
-
-    pa = sparse.vstack((Dfx, Dfy)) @ Tx @ eps_z_inv @ sparse.hstack((Dbx, Dby))
-    pb = sparse.vstack((Dfx, Dfy)) @ Tx @ eps_z_inv @ sparse.hstack((Dby, Dbx))
-    a0 = Ty @ eps_x + omega**-2 * Dby @ Ty @ Dfy
-    a1 = Tx @ eps_y + omega**-2 * Dbx @ Ty @ Dfx
-    b0 = Dbx @ Ty @ Dfy
-    b1 = Dby @ Ty @ Dfx
-
-    diag = sparse.block_diag
+    eps_x = sparse.diags_array(eps_parts[0])
+    eps_y = sparse.diags_array(eps_parts[1])
+    eps_z_inv = sparse.diags_array(1 / eps_parts[2])
 
     omega2 = omega * omega
+    diag = sparse.block_diag
 
-    op = (omega2 * diag((Tx, Ty)) + pa) @ diag((a0, a1)) + \
-        - (sparse.bmat(((None, Ty), (Tx, None))) + pb / omega2) @ diag((b0, b1))
+    sq0 = omega2 * diag((Tb @ Tb @ eps_x,
+                         Ta @ Ta @ eps_y))
+    lin0 = sparse.vstack((-Tb @ Dby, Ta @ Dbx)) @ Tb @ sparse.hstack((-Dfy, Dfx))
+    lin1 = sparse.vstack((Dfx, Dfy)) @ Ta @ eps_z_inv @ sparse.hstack((Dbx @ Tb @ eps_x,
+                                                                       Dby @ Ta @ eps_y))
+    op = sq0 + lin0 + lin1
     return op
 
 
-def solve_mode(
-        mode_number: int,
-        omega: complex,
-        dxes: dx_lists_t,
-        epsilon: vfdfield_t,
-        r0: float,
-        ) -> dict[str, complex | cfdfield_t]:
+def solve_modes(
+        mode_numbers: Sequence[int],
+        omega: float,
+        dxes: dx_lists2_t,
+        epsilon: vfdslice,
+        rmin: float,
+        mode_margin: int = 2,
+        ) -> tuple[NDArray[numpy.complex128], NDArray[numpy.complex128]]:
     """
-    TODO: fixup
     Given a 2d (r, y) slice of epsilon, attempts to solve for the eigenmode
      of the bent waveguide with the specified mode number.
 
     Args:
-        mode_number: Number of the mode, 0-indexed
+        mode_numbers: Mode numbers to solve, 0-indexed.
         omega: Angular frequency of the simulation
         dxes: Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types.
             The first coordinate is assumed to be r, the second is y.
         epsilon: Dielectric constant
-        r0: Radius of curvature for the simulation. This should be the minimum value of
+        rmin: Radius of curvature for the simulation. This should be the minimum value of
             r within the simulation domain.
 
     Returns:
-        ```
-        {
-            'E': list[NDArray[numpy.complex_]],
-            'H': list[NDArray[numpy.complex_]],
-            'wavenumber': complex,
-        }
-        ```
+        e_xys: NDArray of vfdfield_t specifying fields. First dimension is mode number.
+        angular_wavenumbers: list of wavenumbers in 1/rad units.
     """
 
-    '''
-    Solve for the largest-magnitude eigenvalue of the real operator
-    '''
+    #
+    # Solve for the largest-magnitude eigenvalue of the real operator
+    #
     dxes_real = [[numpy.real(dx) for dx in dxi] for dxi in dxes]
 
-    A_r = cylindrical_operator(numpy.real(omega), dxes_real, numpy.real(epsilon), r0)
-    eigvals, eigvecs = signed_eigensolve(A_r, mode_number + 3)
-    e_xy = eigvecs[:, -(mode_number + 1)]
+    A_r = cylindrical_operator(numpy.real(omega), dxes_real, numpy.real(epsilon), rmin=rmin)
+    eigvals, eigvecs = signed_eigensolve(A_r, max(mode_numbers) + mode_margin)
+    keep_inds = -(numpy.array(mode_numbers) + 1)
+    e_xys = eigvecs[:, keep_inds].T
+    eigvals = eigvals[keep_inds]
 
-    '''
-    Now solve for the eigenvector of the full operator, using the real operator's
-     eigenvector as an initial guess for Rayleigh quotient iteration.
-    '''
-    A = cylindrical_operator(omega, dxes, epsilon, r0)
-    eigval, e_xy = rayleigh_quotient_iteration(A, e_xy)
+    #
+    # Now solve for the eigenvector of the full operator, using the real operator's
+    #  eigenvector as an initial guess for Rayleigh quotient iteration.
+    #
+    A = cylindrical_operator(omega, dxes, epsilon, rmin=rmin)
+    for nn in range(len(mode_numbers)):
+        eigvals[nn], e_xys[nn, :] = rayleigh_quotient_iteration(A, e_xys[nn, :])
 
     # Calculate the wave-vector (force the real part to be positive)
-    wavenumber = numpy.sqrt(eigval)
-    wavenumber *= numpy.sign(numpy.real(wavenumber))
+    wavenumbers = numpy.sqrt(eigvals)
+    wavenumbers *= numpy.sign(numpy.real(wavenumbers))
 
-    # TODO: Perform correction on wavenumber to account for numerical dispersion.
+    # Wavenumbers assume the mode is at rmin, which is unlikely
+    # Instead, return the wavenumber in inverse radians
+    angular_wavenumbers = wavenumbers * cast('complex', rmin)
 
-    shape = [d.size for d in dxes[0]]
-    e_xy = numpy.hstack((e_xy, numpy.zeros(shape[0] * shape[1])))
-    fields = {
-        'wavenumber': wavenumber,
-        'E': unvec(e_xy, shape),
-        # 'E': unvec(e, shape),
-        # 'H': unvec(h, shape),
-    }
+    order = angular_wavenumbers.argsort()[::-1]
+    e_xys = e_xys[order]
+    angular_wavenumbers = angular_wavenumbers[order]
 
-    return fields
+    return e_xys, angular_wavenumbers
+
+
+def solve_mode(
+        mode_number: int,
+        *args: Any,
+        **kwargs: Any,
+        ) -> tuple[vcfdfield2, complex]:
+    """
+    Wrapper around `solve_modes()` that solves for a single mode.
+
+    Args:
+       mode_number: 0-indexed mode number to solve for
+       *args: passed to `solve_modes()`
+       **kwargs: passed to `solve_modes()`
+
+    Returns:
+        (e_xy, angular_wavenumber)
+    """
+    kwargs['mode_numbers'] = [mode_number]
+    e_xys, angular_wavenumbers = solve_modes(*args, **kwargs)
+    return e_xys[0], angular_wavenumbers[0]
+
+
+def linear_wavenumbers(
+        e_xys: Sequence[vcfdfield2] | NDArray[numpy.complex128],
+        angular_wavenumbers: ArrayLike,
+        epsilon: vfdslice,
+        dxes: dx_lists2_t,
+        rmin: float,
+        ) -> NDArray[numpy.complex128]:
+    """
+    Calculate linear wavenumbers (1/distance) based on angular wavenumbers (1/rad)
+      and the mode's energy distribution.
+
+    Args:
+        e_xys: Vectorized mode fields with shape (num_modes, 2 * x *y)
+        angular_wavenumbers: Wavenumbers assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. They should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        epsilon: Vectorized dielectric constant grid with shape (3, x, y)
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+
+    Returns:
+        NDArray containing the calculated linear (1/distance) wavenumbers
+    """
+    angular_wavenumbers = numpy.asarray(angular_wavenumbers)
+    mode_radii = numpy.empty_like(angular_wavenumbers, dtype=float)
+
+    shape2d = (len(dxes[0][0]), len(dxes[0][1]))
+    epsilon2d = unvec(epsilon, shape2d)[:2]
+    grid_radii = rmin + numpy.cumsum(dxes[0][0])
+    for ii in range(angular_wavenumbers.size):
+        efield = unvec(e_xys[ii], shape2d, 2)
+        energy = numpy.real((efield * efield.conj()) * epsilon2d)
+        energy_vs_x = energy.sum(axis=(0, 2))
+        mode_radii[ii] = (grid_radii * energy_vs_x).sum() / energy_vs_x.sum()
+
+    logger.info(f'{mode_radii=}')
+    lin_wavenumbers = angular_wavenumbers / mode_radii
+    return lin_wavenumbers
+
+
+def exy2h(
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists2_t,
+        rmin: float,
+        epsilon: vfdslice,
+        mu: vfdslice | None = None
+        ) -> sparse.sparray:
+    """
+    Operator which transforms the vector `e_xy` containing the vectorized E_r and E_y fields,
+     into a vectorized H containing all three H components
+
+    Args:
+        angular_wavenumber: Wavenumber assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. It should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        omega: The angular frequency of the system
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+        epsilon: Vectorized dielectric constant grid
+        mu: Vectorized magnetic permeability grid (default 1 everywhere)
+
+    Returns:
+        Sparse matrix representing the operator.
+    """
+    e2hop = e2h(angular_wavenumber=angular_wavenumber, omega=omega, dxes=dxes, rmin=rmin, mu=mu)
+    return e2hop @ exy2e(angular_wavenumber=angular_wavenumber, omega=omega, dxes=dxes, rmin=rmin, epsilon=epsilon)
+
+
+def exy2e(
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists2_t,
+        rmin: float,
+        epsilon: vfdslice,
+        ) -> sparse.sparray:
+    """
+    Operator which transforms the vector `e_xy` containing the vectorized E_r and E_y fields,
+     into a vectorized E containing all three E components
+
+    Unlike the straight waveguide case, the H_z components do not cancel and must be calculated
+    from E_r and E_y in order to then calculate E_z.
+
+    Args:
+        angular_wavenumber: Wavenumber assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. It should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        omega: The angular frequency of the system
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+        epsilon: Vectorized dielectric constant grid
+
+    Returns:
+        Sparse matrix representing the operator.
+    """
+    Dfx, Dfy = deriv_forward(dxes[0])
+    Dbx, Dby = deriv_back(dxes[1])
+    wavenumber = angular_wavenumber / rmin
+
+    Ta, Tb = dxes2T(dxes=dxes, rmin=rmin)
+    Tai = sparse.diags_array(1 / Ta.diagonal())
+    #Tbi = sparse.diags_array(1 / Tb.diagonal())
+
+    epsilon_parts = numpy.split(epsilon, 3)
+    epsilon_x, epsilon_y = (sparse.diags_array(epsi) for epsi in epsilon_parts[:2])
+    epsilon_z_inv = sparse.diags_array(1 / epsilon_parts[2])
+
+    n_pts = dxes[0][0].size * dxes[0][1].size
+    zeros = sparse.coo_array((n_pts, n_pts))
+
+    mu_z = numpy.ones(n_pts)
+    mu_z_inv = sparse.diags_array(1 / mu_z)
+    exy2hz = 1 / (-1j * omega) * mu_z_inv @ sparse.hstack((Dfy, -Dfx))
+    hxy2ez = 1 / (1j * omega) * epsilon_z_inv @ sparse.hstack((Dby, -Dbx))
+
+    exy2hy = Tb / (1j * wavenumber) @ (-1j * omega * sparse.hstack((epsilon_x, zeros)) - Dby @ exy2hz)
+    exy2hx = Tb / (1j * wavenumber) @ ( 1j * omega * sparse.hstack((zeros, epsilon_y)) - Tai @ Dbx @ Tb @ exy2hz)
+
+    exy2ez = hxy2ez @ sparse.vstack((exy2hx, exy2hy))
+
+    op = sparse.vstack((sparse.eye_array(2 * n_pts),
+                        exy2ez))
+    return op
+
+
+def e2h(
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists2_t,
+        rmin: float,
+        mu: vfdslice | None = None
+        ) -> sparse.sparray:
+    r"""
+    Returns an operator which, when applied to a vectorized E eigenfield, produces
+     the vectorized H eigenfield.
+
+    This operator is created directly from the initial coordinate-transformed equations:
+    $$
+    \begin{aligned}
+    -\imath \omega \mu_{rr} H_r &= \tilde{\partial}_y E_z + \imath \beta T_a^{-1} E_y, \\
+    -\imath \omega \mu_{yy} H_y &= -\imath \beta T_b^{-1} E_r
+      - T_b^{-1} \tilde{\partial}_r (T_a E_z), \\
+    -\imath \omega \mu_{zz} H_z &= \tilde{\partial}_r E_y - \tilde{\partial}_y E_r,
+    \end{aligned}
+    $$
+
+    Args:
+        angular_wavenumber: Wavenumber assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. It should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        omega: The angular frequency of the system
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+        mu: Vectorized magnetic permeability grid (default 1 everywhere)
+
+    Returns:
+        Sparse matrix representation of the operator.
+    """
+    Dfx, Dfy = deriv_forward(dxes[0])
+    Ta, Tb = dxes2T(dxes=dxes, rmin=rmin)
+    Tai = sparse.diags_array(1 / Ta.diagonal())
+    Tbi = sparse.diags_array(1 / Tb.diagonal())
+
+    jB = 1j * angular_wavenumber / rmin
+    op = sparse.block_array([[    None, -jB * Tai,           -Dfy],
+                             [jB * Tbi,      None, Tbi @ Dfx @ Ta],
+                             [     Dfy,      -Dfx,           None]]) / (-1j * omega)
+    if mu is not None:
+        op = sparse.diags_array(1 / mu) @ op
+    return op
+
+
+def dxes2T(
+        dxes: dx_lists2_t,
+        rmin: float,
+        ) -> tuple[NDArray[numpy.float64], NDArray[numpy.float64]]:
+    r"""
+    Construct the cylindrical metric matrices $T_a$ and $T_b$.
+
+    `T_a` is sampled on the E-grid radial locations, while `T_b` is sampled on
+    the staggered H-grid radial locations. These are the diagonal matrices that
+    convert the straight-waveguide algebra into its cylindrical counterpart.
+
+    Args:
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+
+    Returns:
+        Sparse diagonal matrices `(T_a, T_b)`.
+    """
+    ra = rmin + numpy.cumsum(dxes[0][0])                      # Radius at Ey points
+    rb = rmin + dxes[0][0] / 2.0 + numpy.cumsum(dxes[1][0])   # Radius at Ex points
+    ta = ra / rmin
+    tb = rb / rmin
+
+    Ta = sparse.diags_array(vec(ta[:, None].repeat(dxes[0][1].size, axis=1)))
+    Tb = sparse.diags_array(vec(tb[:, None].repeat(dxes[1][1].size, axis=1)))
+    return Ta, Tb
+
+
+def normalized_fields_e(
+        e_xy: vcfdfield2,
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists2_t,
+        rmin: float,
+        epsilon: vfdslice,
+        mu: vfdslice | None = None,
+        prop_phase: float = 0,
+        ) -> tuple[vcfdslice_t, vcfdslice_t]:
+    r"""
+    Given a vector `e_xy` containing the vectorized E_r and E_y fields,
+    returns normalized, vectorized E and H fields for the system.
+
+    Args:
+        e_xy: Vector containing E_r and E_y fields
+        angular_wavenumber: Wavenumber assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. It should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        omega: The angular frequency of the system
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+        epsilon: Vectorized dielectric constant grid
+        mu: Vectorized magnetic permeability grid (default 1 everywhere)
+        prop_phase: Phase shift `(dz * corrected_wavenumber)` over 1 cell in propagation direction.
+                    Default 0 (continuous propagation direction, i.e. dz->0).
+
+    Returns:
+        `(e, h)`, where each field is vectorized, normalized,
+        and contains all three vector components.
+
+    Notes:
+        The normalization step is delegated to `_normalized_fields(...)`, which
+        enforces unit forward power under the discrete inner product
+
+        $$
+        \frac{1}{2}\int (E_r H_y^* - E_y H_r^*) \, dr \, dy.
+        $$
+
+        The angular wavenumber `m` is first converted into the full three-component
+        fields, then the overall complex phase and sign are fixed so the result is
+        reproducible for symmetric modes.
+    """
+    e = exy2e(angular_wavenumber=angular_wavenumber, omega=omega, dxes=dxes, rmin=rmin, epsilon=epsilon) @ e_xy
+    h = exy2h(angular_wavenumber=angular_wavenumber, omega=omega, dxes=dxes, rmin=rmin, epsilon=epsilon, mu=mu) @ e_xy
+    e_norm, h_norm = _normalized_fields(
+        e=e, h=h, dxes=dxes, epsilon=epsilon, prop_phase=prop_phase,
+    )
+    return e_norm, h_norm
+
+
+def _normalized_fields(
+        e: vcfdslice,
+        h: vcfdslice,
+        dxes: dx_lists2_t,
+        epsilon: vfdslice,
+        prop_phase: float = 0,
+        ) -> tuple[vcfdslice_t, vcfdslice_t]:
+    r"""
+    Normalize a cylindrical waveguide mode to unit forward power.
+
+    The cylindrical helpers reuse the straight-waveguide inner product after the
+    field reconstruction step. The extra metric factors have already been folded
+    into the reconstructed `e`/`h` fields through `dxes2T(...)` and the
+    cylindrical `exy2e(...)` / `exy2h(...)` operators, so the same discrete
+    longitudinal Poynting integral can be used here.
+
+    The normalization procedure is:
+
+    1. Flip the magnetic field sign so the reconstructed `(e, h)` pair follows
+       the same forward-power convention as `waveguide_2d`.
+    2. Compute the time-averaged forward power with
+       `waveguide_2d.inner_product(..., conj_h=True)`.
+    3. Scale by `1 / sqrt(S_z)` so the resulting mode has unit forward power.
+    4. Remove the arbitrary complex phase and apply a quadrant-sum sign heuristic
+       so symmetric modes choose a stable representative.
+
+    `prop_phase` has the same meaning as in `waveguide_2d`: it compensates for
+    the half-cell longitudinal staggering between the E and H fields when the
+    propagation direction is itself discretized.
+    """
+    h *= -1
+    shape = [s.size for s in dxes[0]]
+
+    # Find time-averaged Sz and normalize to it
+    # H phase is adjusted by a half-cell forward shift for Yee cell, and 1-cell reverse shift for Poynting
+    Sz_tavg = waveguide_2d.inner_product(e, h, dxes=dxes, prop_phase=prop_phase, conj_h=True).real   # Note, using linear poynting vector
+    assert Sz_tavg > 0, f'Found a mode propagating in the wrong direction! {Sz_tavg=}'
+
+    energy = numpy.real(epsilon * e.conj() * e)
+
+    norm_amplitude = 1 / numpy.sqrt(Sz_tavg)
+    norm_angle = -numpy.angle(e[energy.argmax()])       # Will randomly add a negative sign when mode is symmetric
+
+    # Try to break symmetry to assign a consistent sign [experimental]
+    E_weighted = unvec(e * energy * numpy.exp(1j * norm_angle), shape)
+    sign = numpy.sign(E_weighted[:,
+                                 :max(shape[0] // 2, 1),
+                                 :max(shape[1] // 2, 1)].real.sum())
+    assert sign != 0
+
+    norm_factor = sign * norm_amplitude * numpy.exp(1j * norm_angle)
+
+    e *= norm_factor
+    h *= norm_factor
+    return vcfdslice_t(e), vcfdslice_t(h)

meanas/fdmath/__init__.py

@@ -1,4 +1,4 @@
-"""
+r"""
 
 Basic discrete calculus for finite difference (fd) simulations.
 
@@ -43,11 +43,11 @@ Scalar derivatives and cell shifts
 ----------------------------------
 
 Define the discrete forward derivative as
- $$ [\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
+ $$ [\tilde{\partial}_x f]_{m + \frac{1}{2}} = \frac{1}{\Delta_{x, m}} (f_{m + 1} - f_m) $$
  where $f$ is a function defined at discrete locations on the x-axis (labeled using $m$).
- The value at $m$ occupies a length $\\Delta_{x, m}$ along the x-axis. Note that $m$
+ The value at $m$ occupies a length $\Delta_{x, m}$ along the x-axis. Note that $m$
  is an index along the x-axis, _not_ necessarily an x-coordinate, since each length
- $\\Delta_{x, m}, \\Delta_{x, m+1}, ...$ is independently chosen.
+ $\Delta_{x, m}, \Delta_{x, m+1}, ...$ is independently chosen.
 
 If we treat `f` as a 1D array of values, with the `i`-th value `f[i]` taking up a length `dx[i]`
 along the x-axis, the forward derivative is
@@ -56,13 +56,13 @@ along the x-axis, the forward derivative is
 
 
 Likewise, discrete reverse derivative is
- $$ [\\hat{\\partial}_x f ]_{m - \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m} - f_{m - 1}) $$
+ $$ [\hat{\partial}_x f ]_{m - \frac{1}{2}} = \frac{1}{\Delta_{x, m}} (f_{m} - f_{m - 1}) $$
  or
 
     deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
 
 The derivatives' values are shifted by a half-cell relative to the original function, and
-will have different cell widths if all the `dx[i]` ( $\\Delta_{x, m}$ ) are not
+will have different cell widths if all the `dx[i]` ( $\Delta_{x, m}$ ) are not
 identical:
 
     [figure: derivatives and cell sizes]
@@ -88,19 +88,19 @@ identical:
 
 In the above figure,
  `f0 =` $f_0$, `f1 =` $f_1$
- `Df0 =` $[\\tilde{\\partial}f]_{0 + \\frac{1}{2}}$
- `Df1 =` $[\\tilde{\\partial}f]_{1 + \\frac{1}{2}}$
- `df0 =` $[\\hat{\\partial}f]_{0 - \\frac{1}{2}}$
+ `Df0 =` $[\tilde{\partial}f]_{0 + \frac{1}{2}}$
+ `Df1 =` $[\tilde{\partial}f]_{1 + \frac{1}{2}}$
+ `df0 =` $[\hat{\partial}f]_{0 - \frac{1}{2}}$
  etc.
 
-The fractional subscript $m + \\frac{1}{2}$ is used to indicate values defined
+The fractional subscript $m + \frac{1}{2}$ is used to indicate values defined
  at shifted locations relative to the original $m$, with corresponding lengths
- $$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
+ $$ \Delta_{x, m + \frac{1}{2}} = \frac{1}{2} * (\Delta_{x, m} + \Delta_{x, m + 1}) $$
 
-Just as $m$ is not itself an x-coordinate, neither is $m + \\frac{1}{2}$;
+Just as $m$ is not itself an x-coordinate, neither is $m + \frac{1}{2}$;
 carefully note the positions of the various cells in the above figure vs their labels.
 If the positions labeled with $m$ are considered the "base" or "original" grid,
-the positions labeled with $m + \\frac{1}{2}$ are said to lie on a "dual" or
+the positions labeled with $m + \frac{1}{2}$ are said to lie on a "dual" or
 "derived" grid.
 
 For the remainder of the `Discrete calculus` section, all figures will show
@@ -113,12 +113,12 @@ Gradients and fore-vectors
 --------------------------
 
 Expanding to three dimensions, we can define two gradients
-  $$ [\\tilde{\\nabla} f]_{m,n,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
-                                    \\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
-                                    \\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}}  $$
-  $$ [\\hat{\\nabla} f]_{m,n,p} = \\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
-                                  \\vec{y} [\\hat{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
-                                  \\vec{z} [\\hat{\\partial}_z f]_{m,n,p + \\frac{1}{2}}  $$
+  $$ [\tilde{\nabla} f]_{m,n,p} = \vec{x} [\tilde{\partial}_x f]_{m + \frac{1}{2},n,p} +
+                                  \vec{y} [\tilde{\partial}_y f]_{m,n + \frac{1}{2},p} +
+                                  \vec{z} [\tilde{\partial}_z f]_{m,n,p + \frac{1}{2}}  $$
+  $$ [\hat{\nabla} f]_{m,n,p} = \vec{x} [\hat{\partial}_x f]_{m + \frac{1}{2},n,p} +
+                                \vec{y} [\hat{\partial}_y f]_{m,n + \frac{1}{2},p} +
+                                \vec{z} [\hat{\partial}_z f]_{m,n,p + \frac{1}{2}}  $$
 
  or
 
@@ -144,12 +144,12 @@ y in y, and z in z.
 
 We call the resulting object a "fore-vector" or "back-vector", depending
 on the direction of the shift. We write it as
-  $$ \\tilde{g}_{m,n,p} = \\vec{x} g^x_{m + \\frac{1}{2},n,p} +
-                          \\vec{y} g^y_{m,n + \\frac{1}{2},p} +
-                          \\vec{z} g^z_{m,n,p + \\frac{1}{2}} $$
-  $$ \\hat{g}_{m,n,p} = \\vec{x} g^x_{m - \\frac{1}{2},n,p} +
-                        \\vec{y} g^y_{m,n - \\frac{1}{2},p} +
-                        \\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
+  $$ \tilde{g}_{m,n,p} = \vec{x} g^x_{m + \frac{1}{2},n,p} +
+                         \vec{y} g^y_{m,n + \frac{1}{2},p} +
+                         \vec{z} g^z_{m,n,p + \frac{1}{2}} $$
+  $$ \hat{g}_{m,n,p} = \vec{x} g^x_{m - \frac{1}{2},n,p} +
+                       \vec{y} g^y_{m,n - \frac{1}{2},p} +
+                       \vec{z} g^z_{m,n,p - \frac{1}{2}} $$
 
 
     [figure: gradient / fore-vector]
@@ -172,15 +172,15 @@ Divergences
 
 There are also two divergences,
 
-  $$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p}
-               = [\\tilde{\\partial}_x g^x]_{m,n,p} +
-                 [\\tilde{\\partial}_y g^y]_{m,n,p} +
-                 [\\tilde{\\partial}_z g^z]_{m,n,p}   $$
+  $$ d_{n,m,p} = [\tilde{\nabla} \cdot \hat{g}]_{n,m,p}
+               = [\tilde{\partial}_x g^x]_{m,n,p} +
+                 [\tilde{\partial}_y g^y]_{m,n,p} +
+                 [\tilde{\partial}_z g^z]_{m,n,p}   $$
 
-  $$ d_{n,m,p} = [\\hat{\\nabla} \\cdot \\tilde{g}]_{n,m,p}
-               = [\\hat{\\partial}_x g^x]_{m,n,p} +
-                 [\\hat{\\partial}_y g^y]_{m,n,p} +
-                 [\\hat{\\partial}_z g^z]_{m,n,p}  $$
+  $$ d_{n,m,p} = [\hat{\nabla} \cdot \tilde{g}]_{n,m,p}
+               = [\hat{\partial}_x g^x]_{m,n,p} +
+                 [\hat{\partial}_y g^y]_{m,n,p} +
+                 [\hat{\partial}_z g^z]_{m,n,p}  $$
 
  or
 
@@ -203,7 +203,7 @@ where `g = [gx, gy, gz]` is a fore- or back-vector field.
 
 Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
 is defined at the back-vector's (fore-vector's) location $(m,n,p)$ and not at the locations of its components
-$(m \\pm \\frac{1}{2},n,p)$ etc.
+$(m \pm \frac{1}{2},n,p)$ etc.
 
     [figure: divergence]
                                     ^^
@@ -227,23 +227,23 @@ Curls
 
 The two curls are then
 
-  $$ \\begin{aligned}
-     \\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
-     [\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
-        \\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
-     &+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
-     &+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
-     \\end{aligned} $$
+  $$ \begin{aligned}
+     \hat{h}_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &= \\
+     [\tilde{\nabla} \times \tilde{g}]_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &=
+        \vec{x} (\tilde{\partial}_y g^z_{m,n,p + \frac{1}{2}} - \tilde{\partial}_z g^y_{m,n + \frac{1}{2},p}) \\
+     &+ \vec{y} (\tilde{\partial}_z g^x_{m + \frac{1}{2},n,p} - \tilde{\partial}_x g^z_{m,n,p + \frac{1}{2}}) \\
+     &+ \vec{z} (\tilde{\partial}_x g^y_{m,n + \frac{1}{2},p} - \tilde{\partial}_y g^z_{m + \frac{1}{2},n,p})
+     \end{aligned} $$
 
  and
 
-  $$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
-     [\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
+  $$ \tilde{h}_{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}} =
+     [\hat{\nabla} \times \hat{g}]_{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}} $$
 
-  where $\\hat{g}$ and $\\tilde{g}$ are located at $(m,n,p)$
-  with components at $(m \\pm \\frac{1}{2},n,p)$ etc.,
-  while $\\hat{h}$ and $\\tilde{h}$ are located at $(m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$
-  with components at $(m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$ etc.
+  where $\hat{g}$ and $\tilde{g}$ are located at $(m,n,p)$
+  with components at $(m \pm \frac{1}{2},n,p)$ etc.,
+  while $\hat{h}$ and $\tilde{h}$ are located at $(m \pm \frac{1}{2}, n \pm \frac{1}{2}, p \pm \frac{1}{2})$
+  with components at $(m, n \pm \frac{1}{2}, p \pm \frac{1}{2})$ etc.
 
 
     [code: curls]
@@ -287,27 +287,27 @@ Maxwell's Equations
 
 If we discretize both space (m,n,p) and time (l), Maxwell's equations become
 
- $$ \\begin{aligned}
-  \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
-                                                                         - \\hat{M}_{l, \\vec{r} + \\frac{1}{2}}  \\\\
-  \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
-                                                                             + \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
-  \\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
-  \\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
- \\end{aligned} $$
+ $$ \begin{aligned}
+  \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &= -\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}
+                                                                   - \hat{M}_{l, \vec{r} + \frac{1}{2}}  \\
+  \hat{\nabla} \times \hat{H}_{l-\frac{1}{2},\vec{r} + \frac{1}{2}} &= \hat{\partial}_t \tilde{D}_{l, \vec{r}}
+                                                                   + \tilde{J}_{l-\frac{1}{2},\vec{r}} \\
+  \tilde{\nabla} \cdot \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} &= 0 \\
+  \hat{\nabla} \cdot \tilde{D}_{l,\vec{r}} &= \rho_{l,\vec{r}}
+ \end{aligned} $$
 
  with
 
- $$ \\begin{aligned}
-  \\hat{B}_{\\vec{r}} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
-  \\tilde{D}_{\\vec{r}} &= \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}}
- \\end{aligned} $$
+ $$ \begin{aligned}
+  \hat{B}_{\vec{r}} &= \mu_{\vec{r} + \frac{1}{2}} \cdot \hat{H}_{\vec{r} + \frac{1}{2}} \\
+  \tilde{D}_{\vec{r}} &= \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}}
+ \end{aligned} $$
 
-where the spatial subscripts are abbreviated as $\\vec{r} = (m, n, p)$ and
-$\\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2})$,
-$\\tilde{E}$ and $\\hat{H}$ are the electric and magnetic fields,
-$\\tilde{J}$ and $\\hat{M}$ are the electric and magnetic current distributions,
-and $\\epsilon$ and $\\mu$ are the dielectric permittivity and magnetic permeability.
+where the spatial subscripts are abbreviated as $\vec{r} = (m, n, p)$ and
+$\vec{r} + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2})$,
+$\tilde{E}$ and $\hat{H}$ are the electric and magnetic fields,
+$\tilde{J}$ and $\hat{M}$ are the electric and magnetic current distributions,
+and $\epsilon$ and $\mu$ are the dielectric permittivity and magnetic permeability.
 
 The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
 The time derivatives can be expanded to form the update equations:
@@ -369,34 +369,34 @@ Each component forms its own grid, offset from the others:
 
 The divergence equations can be derived by taking the divergence of the curl equations
 and combining them with charge continuity,
- $$ \\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho = 0 $$
+ $$ \hat{\nabla} \cdot \tilde{J} + \hat{\partial}_t \rho = 0 $$
  implying that the discrete Maxwell's equations do not produce spurious charges.
 
 
 Wave equation
 -------------
 
-Taking the backward curl of the $\\tilde{\\nabla} \\times \\tilde{E}$ equation and
-replacing the resulting $\\hat{\\nabla} \\times \\hat{H}$ term using its respective equation,
-and setting $\\hat{M}$ to zero, we can form the discrete wave equation:
+Taking the backward curl of the $\tilde{\nabla} \times \tilde{E}$ equation and
+replacing the resulting $\hat{\nabla} \times \hat{H}$ term using its respective equation,
+and setting $\hat{M}$ to zero, we can form the discrete wave equation:
 
 $$
-  \\begin{aligned}
-  \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-      -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
-                          - \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}  \\\\
-  \\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-    -\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}  \\\\
-  \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-    \\hat{\\nabla} \\times (-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}})  \\\\
-  \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-    -\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}  \\\\
-  \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-    -\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
-  \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
-            + \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}}
-            &= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
-  \\end{aligned}
+  \begin{aligned}
+  \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
+     -\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}
+                       - \hat{M}_{l-1, \vec{r} + \frac{1}{2}}  \\
+  \mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
+   -\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}  \\
+  \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
+   \hat{\nabla} \times (-\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}})  \\
+  \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
+   -\tilde{\partial}_t \hat{\nabla} \times \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}  \\
+  \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
+   -\tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \tilde{E}_{l, \vec{r}} + \hat{\partial}_t \tilde{J}_{l-\frac{1}{2},\vec{r}} \\
+  \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}})
+           + \tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \cdot \tilde{E}_{l, \vec{r}}
+           &= \tilde{\partial}_t \tilde{J}_{l - \frac{1}{2}, \vec{r}}
+  \end{aligned}
 $$
 
 
@@ -406,27 +406,27 @@ Frequency domain
 We can substitute in a time-harmonic fields
 
 $$
- \\begin{aligned}
- \\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
- \\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
- \\end{aligned}
+ \begin{aligned}
+ \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
+ \tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t}
+ \end{aligned}
 $$
 
 resulting in
 
 $$
- \\begin{aligned}
- \\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
-   \\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
- \\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
- \\end{aligned}
+ \begin{aligned}
+ \tilde{\partial}_t &\Rightarrow (e^{ \imath \omega \Delta_t} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{-\imath \omega \Delta_t / 2} = -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
+   \hat{\partial}_t &\Rightarrow (1 - e^{-\imath \omega \Delta_t}) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{ \imath \omega \Delta_t / 2} = -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
+ \Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
+ \end{aligned}
 $$
 
 This gives the frequency-domain wave equation,
 
 $$
- \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-    -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\
+ \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
+    -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
 $$
 
 
@@ -436,48 +436,48 @@ Plane waves and Dispersion relation
 With uniform material distribution and no sources
 
 $$
- \\begin{aligned}
- \\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
- \\epsilon_{\\vec{r}} &= \\epsilon \\\\
- \\tilde{J}_{\\vec{r}} &= 0 \\\\
- \\end{aligned}
+ \begin{aligned}
+ \mu_{\vec{r} + \frac{1}{2}} &= \mu \\
+ \epsilon_{\vec{r}} &= \epsilon \\
+ \tilde{J}_{\vec{r}} &= 0 \\
+ \end{aligned}
 $$
 
 the frequency domain wave equation simplifies to
 
-$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} - \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
+$$ \hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}} - \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 $$
 
-Since $\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0$, we can simplify
+Since $\hat{\nabla} \cdot \tilde{E}_{\vec{r}} = 0$, we can simplify
 
 $$
- \\begin{aligned}
- \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}
-   &= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
-   &= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
-   &= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}
- \\end{aligned}
+ \begin{aligned}
+ \hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}}
+  &= \tilde{\nabla}(\hat{\nabla} \cdot \tilde{E}_{\vec{r}}) - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\
+  &= - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\
+  &= - \tilde{\nabla}^2 \tilde{E}_{\vec{r}}
+ \end{aligned}
 $$
 
 and we get
 
-$$  \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} + \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
+$$  \tilde{\nabla}^2 \tilde{E}_{\vec{r}} + \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 $$
 
 We can convert this to three scalar-wave equations of the form
 
-$$ (\\tilde{\\nabla}^2 + K^2) \\phi_{\\vec{r}} = 0 $$
+$$ (\tilde{\nabla}^2 + K^2) \phi_{\vec{r}} = 0 $$
 
-with $K^2 = \\Omega^2 \\mu \\epsilon$. Now we let
+with $K^2 = \Omega^2 \mu \epsilon$. Now we let
 
-$$  \\phi_{\\vec{r}} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)}  $$
+$$  \phi_{\vec{r}} = A e^{\imath (k_x m \Delta_x + k_y n \Delta_y + k_z p \Delta_z)}  $$
 
 resulting in
 
 $$
- \\begin{aligned}
- \\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\
-   \\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\
- K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
- \\end{aligned}
+ \begin{aligned}
+ \tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\
+   \hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\
+ K_x &= 2 \sin(k_x \Delta_x / 2) / \Delta_x \\
+ \end{aligned}
 $$
 
 with similar expressions for the y and z dimnsions (and $K_y, K_z$).
@@ -485,20 +485,20 @@ with similar expressions for the y and z dimnsions (and $K_y, K_z$).
 This implies
 
 $$
-  \\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\
-  K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
+  \tilde{\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\
+  K_x^2 + K_y^2 + K_z^2 = \Omega^2 \mu \epsilon = \Omega^2 / c^2
 $$
 
-where $c = \\sqrt{\\mu \\epsilon}$.
+where $c = \sqrt{\mu \epsilon}$.
 
-Assuming real $(k_x, k_y, k_z), \\omega$ will be real only if
+Assuming real $(k_x, k_y, k_z), \omega$ will be real only if
 
-$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
+$$ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) $$
 
-If $\\Delta_x = \\Delta_y = \\Delta_z$, this simplifies to $c \\Delta_t < \\Delta_x / \\sqrt{3}$.
+If $\Delta_x = \Delta_y = \Delta_z$, this simplifies to $c \Delta_t < \Delta_x / \sqrt{3}$.
 This last form can be interpreted as enforcing causality; the distance that light
-travels in one timestep (i.e., $c \\Delta_t$) must be less than the diagonal
-of the smallest cell ( $\\Delta_x / \\sqrt{3}$ when on a uniform cubic grid).
+travels in one timestep (i.e., $c \Delta_t$) must be less than the diagonal
+of the smallest cell ( $\Delta_x / \sqrt{3}$ when on a uniform cubic grid).
 
 
 Grid description
@@ -515,8 +515,8 @@ to make the illustration simpler; we need at least two cells in the x dimension
 demonstrate how nonuniform `dx` affects the various components.
 
 Place the E fore-vectors at integer indices $r = (m, n, p)$ and the H back-vectors
-at fractional indices $r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2},
-p + \\frac{1}{2})$. Remember that these are indices and not coordinates; they can
+at fractional indices $r + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2},
+p + \frac{1}{2})$. Remember that these are indices and not coordinates; they can
 correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths.
 
 Draw lines to denote the planes on which the H components and back-vectors are defined.
@@ -718,14 +718,14 @@ composed of the three diagonal tensor components:
 or
 
 $$
- \\epsilon = \\begin{bmatrix} \\epsilon_{xx} & 0 & 0 \\\\
-                              0 & \\epsilon_{yy} & 0 \\\\
-                              0 & 0 & \\epsilon_{zz} \\end{bmatrix}
+ \epsilon = \begin{bmatrix} \epsilon_{xx} & 0 & 0 \\
+                            0 & \epsilon_{yy} & 0 \\
+                            0 & 0 & \epsilon_{zz} \end{bmatrix}
 $$
 $$
- \\mu = \\begin{bmatrix} \\mu_{xx} & 0 & 0 \\\\
-                         0 & \\mu_{yy} & 0 \\\\
-                         0 & 0 & \\mu_{zz} \\end{bmatrix}
+ \mu = \begin{bmatrix} \mu_{xx} & 0 & 0 \\
+                         0 & \mu_{yy} & 0 \\
+                         0 & 0 & \mu_{zz} \end{bmatrix}
 $$
 
 where the off-diagonal terms (e.g. `epsilon_xy`) are assumed to be zero.
@@ -741,8 +741,46 @@ the true values can be multiplied back in after the simulation is complete if no
 normalized results are needed.
 """
 
-from .types import fdfield_t, vfdfield_t, cfdfield_t, vcfdfield_t, dx_lists_t, dx_lists_mut
-from .types import fdfield_updater_t, cfdfield_updater_t
-from .vectorization import vec, unvec
-from . import operators, functional, types, vectorization
+from .types import (
+      fdfield_t as    fdfield_t,
+     vfdfield_t as   vfdfield_t,
+     cfdfield_t as   cfdfield_t,
+    vcfdfield_t as  vcfdfield_t,
+      fdfield2_t as   fdfield2_t,
+     vfdfield2_t as  vfdfield2_t,
+     cfdfield2_t as  cfdfield2_t,
+    vcfdfield2_t as vcfdfield2_t,
+      fdfield   as    fdfield,
+     vfdfield   as   vfdfield,
+     cfdfield   as   cfdfield,
+    vcfdfield   as  vcfdfield,
+      fdfield2   as   fdfield2,
+     vfdfield2   as  vfdfield2,
+     cfdfield2   as  cfdfield2,
+    vcfdfield2   as vcfdfield2,
+      fdslice_t as   fdslice_t,
+     vfdslice_t as  vfdslice_t,
+     cfdslice_t as  cfdslice_t,
+    vcfdslice_t as vcfdslice_t,
+      fdslice   as   fdslice,
+     vfdslice   as  vfdslice,
+     cfdslice   as  cfdslice,
+    vcfdslice   as vcfdslice,
+    dx_lists_t  as dx_lists_t,
+    dx_lists2_t as dx_lists2_t,
+    dx_lists_mut  as dx_lists_mut,
+    dx_lists2_mut as dx_lists2_mut,
+    fdfield_updater_t as fdfield_updater_t,
+    cfdfield_updater_t as cfdfield_updater_t,
+    )
+from .vectorization import (
+    vec as vec,
+    unvec as unvec,
+    )
+from . import (
+    operators as operators,
+    functional as functional,
+    types as types,
+    vectorization as vectorization,
+    )
 

meanas/fdmath/functional.py

@@ -3,16 +3,18 @@ Math functions for finite difference simulations
 
 Basic discrete calculus etc.
 """
-from typing import Sequence, Callable
+from typing import TypeVar
+from collections.abc import Sequence, Callable
 
 import numpy
 from numpy.typing import NDArray
+from numpy import floating, complexfloating
 
-from .types import fdfield_t, fdfield_updater_t
+from .types import fdfield, fdfield_updater_t
 
 
 def deriv_forward(
-        dx_e: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> tuple[fdfield_updater_t, fdfield_updater_t, fdfield_updater_t]:
     """
     Utility operators for taking discretized derivatives (backward variant).
@@ -36,7 +38,7 @@ def deriv_forward(
 
 
 def deriv_back(
-        dx_h: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> tuple[fdfield_updater_t, fdfield_updater_t, fdfield_updater_t]:
     """
     Utility operators for taking discretized derivatives (forward variant).
@@ -59,10 +61,13 @@ def deriv_back(
     return derivs
 
 
+TT = TypeVar('TT', bound='NDArray[floating | complexfloating]')
+
+
 def curl_forward(
-        dx_e: Sequence[NDArray[numpy.float_]] | None = None,
-        ) -> fdfield_updater_t:
-    """
+        dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
+        ) -> Callable[[TT], TT]:
+    r"""
     Curl operator for use with the E field.
 
     Args:
@@ -71,11 +76,11 @@ def curl_forward(
 
     Returns:
         Function `f` for taking the discrete forward curl of a field,
-        `f(E)` -> curlE $= \\nabla_f \\times E$
+        `f(E)` -> curlE $= \nabla_f \times E$
     """
     Dx, Dy, Dz = deriv_forward(dx_e)
 
-    def ce_fun(e: fdfield_t) -> fdfield_t:
+    def ce_fun(e: TT) -> TT:
         output = numpy.empty_like(e)
         output[0] = Dy(e[2])
         output[1] = Dz(e[0])
@@ -89,9 +94,9 @@ def curl_forward(
 
 
 def curl_back(
-        dx_h: Sequence[NDArray[numpy.float_]] | None = None,
-        ) -> fdfield_updater_t:
-    """
+        dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
+        ) -> Callable[[TT], TT]:
+    r"""
     Create a function which takes the backward curl of a field.
 
     Args:
@@ -100,11 +105,11 @@ def curl_back(
 
     Returns:
         Function `f` for taking the discrete backward curl of a field,
-        `f(H)` -> curlH $= \\nabla_b \\times H$
+        `f(H)` -> curlH $= \nabla_b \times H$
     """
     Dx, Dy, Dz = deriv_back(dx_h)
 
-    def ch_fun(h: fdfield_t) -> fdfield_t:
+    def ch_fun(h: TT) -> TT:
         output = numpy.empty_like(h)
         output[0] = Dy(h[2])
         output[1] = Dz(h[0])
@@ -118,11 +123,11 @@ def curl_back(
 
 
 def curl_forward_parts(
-        dx_e: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> Callable:
     Dx, Dy, Dz = deriv_forward(dx_e)
 
-    def mkparts_fwd(e: fdfield_t) -> tuple[tuple[fdfield_t, fdfield_t], ...]:
+    def mkparts_fwd(e: fdfield) -> tuple[tuple[fdfield, fdfield], ...]:
         return ((-Dz(e[1]),  Dy(e[2])),
                 ( Dz(e[0]), -Dx(e[2])),
                 (-Dy(e[0]),  Dx(e[1])))
@@ -131,11 +136,11 @@ def curl_forward_parts(
 
 
 def curl_back_parts(
-        dx_h: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> Callable:
     Dx, Dy, Dz = deriv_back(dx_h)
 
-    def mkparts_back(h: fdfield_t) -> tuple[tuple[fdfield_t, fdfield_t], ...]:
+    def mkparts_back(h: fdfield) -> tuple[tuple[fdfield, fdfield], ...]:
         return ((-Dz(h[1]),  Dy(h[2])),
                 ( Dz(h[0]), -Dx(h[2])),
                 (-Dy(h[0]),  Dx(h[1])))

meanas/fdmath/operators.py

@@ -3,19 +3,20 @@ Matrix operators for finite difference simulations
 
 Basic discrete calculus etc.
 """
-from typing import Sequence
+from collections.abc import Sequence
 import numpy
 from numpy.typing import NDArray
-import scipy.sparse as sparse   # type: ignore
+from numpy import floating, complexfloating
+from scipy import sparse
 
-from .types import vfdfield_t
+from .types import vfdfield
 
 
 def shift_circ(
         axis: int,
         shape: Sequence[int],
         shift_distance: int = 1,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Utility operator for performing a circular shift along a specified axis by a
      specified number of elements.
@@ -29,12 +30,12 @@ def shift_circ(
         Sparse matrix for performing the circular shift.
     """
     if len(shape) not in (2, 3):
-        raise Exception('Invalid shape: {}'.format(shape))
+        raise Exception(f'Invalid shape: {shape}')
     if axis not in range(len(shape)):
-        raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
+        raise Exception(f'Invalid direction: {axis}, shape is {shape}')
 
-    shifts = [abs(shift_distance) if a == axis else 0 for a in range(3)]
-    shifted_diags = [(numpy.arange(n) + s) % n for n, s in zip(shape, shifts)]
+    shifts = [abs(shift_distance) if a == axis else 0 for a in range(len(shape))]
+    shifted_diags = [(numpy.arange(n) + s) % n for n, s in zip(shape, shifts, strict=True)]
     ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
 
     n = numpy.prod(shape)
@@ -43,7 +44,7 @@ def shift_circ(
 
     vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
 
-    d = sparse.csr_matrix(vij, shape=(n, n))
+    d = sparse.csr_array(vij, shape=(n, n))
 
     if shift_distance < 0:
         d = d.T
@@ -55,7 +56,7 @@ def shift_with_mirror(
         axis: int,
         shape: Sequence[int],
         shift_distance: int = 1,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Utility operator for performing an n-element shift along a specified axis, with mirror
     boundary conditions applied to the cells beyond the receding edge.
@@ -69,12 +70,11 @@ def shift_with_mirror(
         Sparse matrix for performing the shift-with-mirror.
     """
     if len(shape) not in (2, 3):
-        raise Exception('Invalid shape: {}'.format(shape))
+        raise Exception(f'Invalid shape: {shape}')
     if axis not in range(len(shape)):
-        raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
+        raise Exception(f'Invalid direction: {axis}, shape is {shape}')
     if shift_distance >= shape[axis]:
-        raise Exception('Shift ({}) is too large for axis {} of size {}'.format(
-                        shift_distance, axis, shape[axis]))
+        raise Exception(f'Shift ({shift_distance}) is too large for axis {axis} of size {shape[axis]}')
 
     def mirrored_range(n: int, s: int) -> NDArray[numpy.int_]:
         v = numpy.arange(n) + s
@@ -82,8 +82,8 @@ def shift_with_mirror(
         v = numpy.where(v < 0, - 1 - v, v)
         return v
 
-    shifts = [shift_distance if a == axis else 0 for a in range(3)]
-    shifted_diags = [mirrored_range(n, s) for n, s in zip(shape, shifts)]
+    shifts = [shift_distance if a == axis else 0 for a in range(len(shape))]
+    shifted_diags = [mirrored_range(n, s) for n, s in zip(shape, shifts, strict=True)]
     ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
 
     n = numpy.prod(shape)
@@ -92,13 +92,13 @@ def shift_with_mirror(
 
     vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
 
-    d = sparse.csr_matrix(vij, shape=(n, n))
+    d = sparse.csr_array(vij, shape=(n, n))
     return d
 
 
 def deriv_forward(
-        dx_e: Sequence[NDArray[numpy.float_]],
-        ) -> list[sparse.spmatrix]:
+        dx_e: Sequence[NDArray[floating | complexfloating]],
+        ) -> list[sparse.sparray]:
     """
     Utility operators for taking discretized derivatives (forward variant).
 
@@ -114,18 +114,18 @@ def deriv_forward(
 
     dx_e_expanded = numpy.meshgrid(*dx_e, indexing='ij')
 
-    def deriv(axis: int) -> sparse.spmatrix:
-        return shift_circ(axis, shape, 1) - sparse.eye(n)
+    def deriv(axis: int) -> sparse.sparray:
+        return shift_circ(axis, shape, 1) - sparse.eye_array(n)
 
-    Ds = [sparse.diags(+1 / dx.ravel(order='C')) @ deriv(a)
+    Ds = [sparse.diags_array(+1 / dx.ravel(order='C')) @ deriv(a)
           for a, dx in enumerate(dx_e_expanded)]
 
     return Ds
 
 
 def deriv_back(
-        dx_h: Sequence[NDArray[numpy.float_]],
-        ) -> list[sparse.spmatrix]:
+        dx_h: Sequence[NDArray[floating | complexfloating]],
+        ) -> list[sparse.sparray]:
     """
     Utility operators for taking discretized derivatives (backward variant).
 
@@ -141,18 +141,18 @@ def deriv_back(
 
     dx_h_expanded = numpy.meshgrid(*dx_h, indexing='ij')
 
-    def deriv(axis: int) -> sparse.spmatrix:
-        return shift_circ(axis, shape, -1) - sparse.eye(n)
+    def deriv(axis: int) -> sparse.sparray:
+        return shift_circ(axis, shape, -1) - sparse.eye_array(n)
 
-    Ds = [sparse.diags(-1 / dx.ravel(order='C')) @ deriv(a)
+    Ds = [sparse.diags_array(-1 / dx.ravel(order='C')) @ deriv(a)
           for a, dx in enumerate(dx_h_expanded)]
 
     return Ds
 
 
 def cross(
-        B: Sequence[sparse.spmatrix],
-        ) -> sparse.spmatrix:
+        B: Sequence[sparse.sparray],
+        ) -> sparse.sparray:
     """
     Cross product operator
 
@@ -164,13 +164,14 @@ def cross(
         Sparse matrix corresponding to (B x), where x is the cross product.
     """
     n = B[0].shape[0]
-    zero = sparse.csr_matrix((n, n))
-    return sparse.bmat([[zero, -B[2], B[1]],
+    zero = sparse.csr_array((n, n))
+    return sparse.block_array([
+            [zero, -B[2], B[1]],
             [B[2], zero, -B[0]],
             [-B[1], B[0], zero]])
 
 
-def vec_cross(b: vfdfield_t) -> sparse.spmatrix:
+def vec_cross(b: vfdfield) -> sparse.sparray:
     """
     Vector cross product operator
 
@@ -182,11 +183,11 @@ def vec_cross(b: vfdfield_t) -> sparse.spmatrix:
         Sparse matrix corresponding to (b x), where x is the cross product.
 
     """
-    B = [sparse.diags(c) for c in numpy.split(b, 3)]
+    B = [sparse.diags_array(c) for c in numpy.split(b, 3)]
     return cross(B)
 
 
-def avg_forward(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
+def avg_forward(axis: int, shape: Sequence[int]) -> sparse.sparray:
     """
     Forward average operator `(x4 = (x4 + x5) / 2)`
 
@@ -198,13 +199,13 @@ def avg_forward(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
         Sparse matrix for forward average operation.
     """
     if len(shape) not in (2, 3):
-        raise Exception('Invalid shape: {}'.format(shape))
+        raise Exception(f'Invalid shape: {shape}')
 
     n = numpy.prod(shape)
-    return 0.5 * (sparse.eye(n) + shift_circ(axis, shape))
+    return 0.5 * (sparse.eye_array(n) + shift_circ(axis, shape))
 
 
-def avg_back(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
+def avg_back(axis: int, shape: Sequence[int]) -> sparse.sparray:
     """
     Backward average operator `(x4 = (x4 + x3) / 2)`
 
@@ -219,8 +220,8 @@ def avg_back(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
 
 
 def curl_forward(
-        dx_e: Sequence[NDArray[numpy.float_]],
-        ) -> sparse.spmatrix:
+        dx_e: Sequence[NDArray[floating | complexfloating]],
+        ) -> sparse.sparray:
     """
     Curl operator for use with the E field.
 
@@ -235,8 +236,8 @@ def curl_forward(
 
 
 def curl_back(
-        dx_h: Sequence[NDArray[numpy.float_]],
-        ) -> sparse.spmatrix:
+        dx_h: Sequence[NDArray[floating | complexfloating]],
+        ) -> sparse.sparray:
     """
     Curl operator for use with the H field.
 

meanas/fdmath/types.py

@@ -1,26 +1,65 @@
 """
 Types shared across multiple submodules
 """
-from typing import Sequence, Callable, MutableSequence
-import numpy
+from typing import NewType
+from collections.abc import Sequence, Callable, MutableSequence
 from numpy.typing import NDArray
+from numpy import floating, complexfloating
 
 
 # Field types
-fdfield_t = NDArray[numpy.float_]
+fdfield_t = NewType('fdfield_t', NDArray[floating])
+type fdfield = fdfield_t | NDArray[floating]
 """Vector field with shape (3, X, Y, Z) (e.g. `[E_x, E_y, E_z]`)"""
 
-vfdfield_t = NDArray[numpy.float_]
+vfdfield_t = NewType('vfdfield_t', NDArray[floating])
+type vfdfield = vfdfield_t | NDArray[floating]
 """Linearized vector field (single vector of length 3*X*Y*Z)"""
 
-cfdfield_t = NDArray[numpy.complex_]
+cfdfield_t = NewType('cfdfield_t', NDArray[complexfloating])
+type cfdfield = cfdfield_t | NDArray[complexfloating]
 """Complex vector field with shape (3, X, Y, Z) (e.g. `[E_x, E_y, E_z]`)"""
 
-vcfdfield_t = NDArray[numpy.complex_]
+vcfdfield_t = NewType('vcfdfield_t', NDArray[complexfloating])
+type vcfdfield = vcfdfield_t | NDArray[complexfloating]
 """Linearized complex vector field (single vector of length 3*X*Y*Z)"""
 
 
-dx_lists_t = Sequence[Sequence[NDArray[numpy.float_]]]
+fdslice_t = NewType('fdslice_t', NDArray[floating])
+type fdslice = fdslice_t | NDArray[floating]
+"""Vector field slice with shape (3, X, Y) (e.g. `[E_x, E_y, E_z]` at a single Z position)"""
+
+vfdslice_t = NewType('vfdslice_t', NDArray[floating])
+type vfdslice = vfdslice_t | NDArray[floating]
+"""Linearized vector field slice (single vector of length 3*X*Y)"""
+
+cfdslice_t = NewType('cfdslice_t', NDArray[complexfloating])
+type cfdslice = cfdslice_t | NDArray[complexfloating]
+"""Complex vector field slice with shape (3, X, Y) (e.g. `[E_x, E_y, E_z]` at a single Z position)"""
+
+vcfdslice_t = NewType('vcfdslice_t', NDArray[complexfloating])
+type vcfdslice = vcfdslice_t | NDArray[complexfloating]
+"""Linearized complex vector field slice (single vector of length 3*X*Y)"""
+
+
+fdfield2_t = NewType('fdfield2_t', NDArray[floating])
+type fdfield2 = fdfield2_t | NDArray[floating]
+"""2D Vector field with shape (2, X, Y) (e.g. `[E_x, E_y]`)"""
+
+vfdfield2_t = NewType('vfdfield2_t', NDArray[floating])
+type vfdfield2 = vfdfield2_t | NDArray[floating]
+"""2D Linearized vector field (single vector of length 2*X*Y)"""
+
+cfdfield2_t = NewType('cfdfield2_t', NDArray[complexfloating])
+type cfdfield2 = cfdfield2_t | NDArray[complexfloating]
+"""2D Complex vector field with shape (2, X, Y) (e.g. `[E_x, E_y]`)"""
+
+vcfdfield2_t = NewType('vcfdfield2_t', NDArray[complexfloating])
+type vcfdfield2 = vcfdfield2_t | NDArray[complexfloating]
+"""2D Linearized complex vector field (single vector of length 2*X*Y)"""
+
+
+dx_lists_t = Sequence[Sequence[NDArray[floating | complexfloating]]]
 """
  'dxes' datastructure which contains grid cell width information in the following format:
 
@@ -31,12 +70,26 @@ dx_lists_t = Sequence[Sequence[NDArray[numpy.float_]]]
    and `dy_h[0]` is the y-width of the `y=0` cells, as used when calculating dH/dy, etc.
 """
 
-dx_lists_mut = MutableSequence[MutableSequence[NDArray[numpy.float_]]]
+dx_lists2_t = Sequence[Sequence[NDArray[floating | complexfloating]]]
+"""
+ 2D 'dxes' datastructure which contains grid cell width information in the following format:
+
+     [[[dx_e[0], dx_e[1], ...], [dy_e[0], ...]],
+      [[dx_h[0], dx_h[1], ...], [dy_h[0], ...]]]
+
+   where `dx_e[0]` is the x-width of the `x=0` cells, as used when calculating dE/dx,
+   and `dy_h[0]` is the y-width of the `y=0` cells, as used when calculating dH/dy, etc.
+"""
+
+dx_lists_mut = MutableSequence[MutableSequence[NDArray[floating | complexfloating]]]
 """Mutable version of `dx_lists_t`"""
 
+dx_lists2_mut = MutableSequence[MutableSequence[NDArray[floating | complexfloating]]]
+"""Mutable version of `dx_lists2_t`"""
 
-fdfield_updater_t = Callable[..., fdfield_t]
-"""Convenience type for functions which take and return an fdfield_t"""
 
-cfdfield_updater_t = Callable[..., cfdfield_t]
-"""Convenience type for functions which take and return an cfdfield_t"""
+fdfield_updater_t = Callable[..., fdfield]
+"""Convenience type for functions which take and return a real `fdfield`"""
+
+cfdfield_updater_t = Callable[..., cfdfield]
+"""Convenience type for functions which take and return a complex `cfdfield`"""

meanas/fdmath/vectorization.py

@@ -4,37 +4,60 @@ and a 1D array representation of that field `[f_x0, f_x1, f_x2,... f_y0,... f_z0
 Vectorized versions of the field use row-major (ie., C-style) ordering.
 """
 
-from typing import overload, Sequence
+from typing import overload
+from collections.abc import Sequence
 import numpy
-from numpy.typing import ArrayLike
+from numpy.typing import ArrayLike, NDArray
 
-from .types import fdfield_t, vfdfield_t, cfdfield_t, vcfdfield_t
+from .types import (
+    fdfield_t, vfdfield_t, cfdfield_t, vcfdfield_t,
+    fdslice_t, vfdslice_t, cfdslice_t, vcfdslice_t,
+    fdfield2_t, vfdfield2_t, cfdfield2_t, vcfdfield2_t,
+    )
 
 
 @overload
 def vec(f: None) -> None:
-    pass
+    pass  # pragma: no cover
 
 @overload
 def vec(f: fdfield_t) -> vfdfield_t:
-    pass
+    pass  # pragma: no cover
 
 @overload
 def vec(f: cfdfield_t) -> vcfdfield_t:
-    pass
+    pass  # pragma: no cover
 
 @overload
-def vec(f: ArrayLike) -> vfdfield_t | vcfdfield_t:
-    pass
+def vec(f: fdfield2_t) -> vfdfield2_t:
+    pass  # pragma: no cover
 
-def vec(f: fdfield_t | cfdfield_t | ArrayLike | None) -> vfdfield_t | vcfdfield_t | None:
+@overload
+def vec(f: cfdfield2_t) -> vcfdfield2_t:
+    pass  # pragma: no cover
+
+@overload
+def vec(f: fdslice_t) -> vfdslice_t:
+    pass  # pragma: no cover
+
+@overload
+def vec(f: cfdslice_t) -> vcfdslice_t:
+    pass  # pragma: no cover
+
+@overload
+def vec(f: ArrayLike) -> NDArray:
+    pass  # pragma: no cover
+
+def vec(
+        f: fdfield_t | cfdfield_t | fdfield2_t | cfdfield2_t | fdslice_t | cfdslice_t | ArrayLike | None,
+        ) -> vfdfield_t | vcfdfield_t | vfdfield2_t | vcfdfield2_t | vfdslice_t | vcfdslice_t | NDArray | None:
     """
-    Create a 1D ndarray from a 3D vector field which spans a 1-3D region.
+    Create a 1D ndarray from a vector field which spans a 1-3D region.
 
     Returns `None` if called with `f=None`.
 
     Args:
-        f: A vector field, `[f_x, f_y, f_z]` where each `f_` component is a 1- to
+        f: A vector field, e.g. `[f_x, f_y, f_z]` where each `f_` component is a 1- to
             3-D ndarray (`f_*` should all be the same size). Doesn't fail with `f=None`.
 
     Returns:
@@ -42,37 +65,61 @@ def vec(f: fdfield_t | cfdfield_t | ArrayLike | None) -> vfdfield_t | vcfdfield_
     """
     if f is None:
         return None
-    return numpy.ravel(f, order='C')
+    return numpy.ravel(f, order='C')        # type: ignore
 
 
 @overload
-def unvec(v: None, shape: Sequence[int]) -> None:
+def unvec(v: None, shape: Sequence[int], nvdim: int = 3) -> None:
+    pass  # pragma: no cover
+
+@overload
+def unvec(v: vfdfield_t, shape: Sequence[int], nvdim: int = 3) -> fdfield_t:
+    pass  # pragma: no cover
+
+@overload
+def unvec(v: vcfdfield_t, shape: Sequence[int], nvdim: int = 3) -> cfdfield_t:
+    pass  # pragma: no cover
+
+@overload
+def unvec(v: vfdfield2_t, shape: Sequence[int], nvdim: int = 3) -> fdfield2_t:
     pass
 
 @overload
-def unvec(v: vfdfield_t, shape: Sequence[int]) -> fdfield_t:
+def unvec(v: vcfdfield2_t, shape: Sequence[int], nvdim: int = 3) -> cfdfield2_t:
     pass
 
 @overload
-def unvec(v: vcfdfield_t, shape: Sequence[int]) -> cfdfield_t:
+def unvec(v: vfdslice_t, shape: Sequence[int], nvdim: int = 3) -> fdslice_t:
     pass
 
-def unvec(v: vfdfield_t | vcfdfield_t | None, shape: Sequence[int]) -> fdfield_t | cfdfield_t | None:
+@overload
+def unvec(v: vcfdslice_t, shape: Sequence[int], nvdim: int = 3) -> cfdslice_t:
+    pass
+
+@overload
+def unvec(v: ArrayLike, shape: Sequence[int], nvdim: int = 3) -> NDArray:
+    pass
+
+def unvec(
+        v: vfdfield_t | vcfdfield_t | vfdfield2_t | vcfdfield2_t | vfdslice_t | vcfdslice_t | ArrayLike | None,
+        shape: Sequence[int],
+        nvdim: int = 3,
+        ) -> fdfield_t | cfdfield_t | fdfield2_t | cfdfield2_t | fdslice_t | cfdslice_t | NDArray | None:
     """
-    Perform the inverse of vec(): take a 1D ndarray and output a 3D field
-     of form `[f_x, f_y, f_z]` where each of `f_*` is a len(shape)-dimensional
+    Perform the inverse of vec(): take a 1D ndarray and output an `nvdim`-component field
+     of form e.g. `[f_x, f_y, f_z]` (`nvdim=3`) where each of `f_*` is a len(shape)-dimensional
      ndarray.
 
     Returns `None` if called with `v=None`.
 
     Args:
-        v: 1D ndarray representing a 3D vector field of shape shape (or None)
+        v: 1D ndarray representing a vector field of shape shape (or None)
         shape: shape of the vector field
+        nvdim: Number of components in each vector
 
     Returns:
         `[f_x, f_y, f_z]` where each `f_` is a `len(shape)` dimensional ndarray (or `None`)
     """
     if v is None:
         return None
-    return v.reshape((3, *shape), order='C')
-
+    return v.reshape((nvdim, *shape), order='C')  # type: ignore

meanas/fdtd/__init__.py

@@ -1,4 +1,4 @@
-"""
+r"""
 Utilities for running finite-difference time-domain (FDTD) simulations
 
 See the discussion of `Maxwell's Equations` in `meanas.fdmath` for basic
@@ -11,9 +11,9 @@ Timestep
 From the discussion of "Plane waves and the Dispersion relation" in `meanas.fdmath`,
 we have
 
-$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
+$$ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) $$
 
-or, if $\\Delta_x = \\Delta_y = \\Delta_z$, then $c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}}$.
+or, if $\Delta_x = \Delta_y = \Delta_z$, then $c \Delta_t < \frac{\Delta_x}{\sqrt{3}}$.
 
 Based on this, we can set
 
@@ -27,81 +27,81 @@ Poynting Vector and Energy Conservation
 
 Let
 
-$$ \\begin{aligned}
-  \\tilde{S}_{l, l', \\vec{r}} &=& &\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}}  \\\\
-  &=&  &\\vec{x} (\\tilde{E}^y_{l,m+1,n,p} \\hat{H}^z_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^z_{l,m+1,n,p} \\hat{H}^y_{l', \\vec{r} + \\frac{1}{2}}) \\\\
-  & &+ &\\vec{y} (\\tilde{E}^z_{l,m,n+1,p} \\hat{H}^x_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^x_{l,m,n+1,p} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) \\\\
-  & &+ &\\vec{z} (\\tilde{E}^x_{l,m,n,p+1} \\hat{H}^y_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^y_{l,m,n,p+1} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}})
-   \\end{aligned}
+$$ \begin{aligned}
+  \tilde{S}_{l, l', \vec{r}} &=& &\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}}  \\
+  &=&  &\vec{x} (\tilde{E}^y_{l,m+1,n,p} \hat{H}^z_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^z_{l,m+1,n,p} \hat{H}^y_{l', \vec{r} + \frac{1}{2}}) \\
+  & &+ &\vec{y} (\tilde{E}^z_{l,m,n+1,p} \hat{H}^x_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^x_{l,m,n+1,p} \hat{H}^z_{l', \vec{r} + \frac{1}{2}}) \\
+  & &+ &\vec{z} (\tilde{E}^x_{l,m,n,p+1} \hat{H}^y_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^y_{l,m,n,p+1} \hat{H}^z_{l', \vec{r} + \frac{1}{2}})
+   \end{aligned}
 $$
 
-where $\\vec{r} = (m, n, p)$ and $\\otimes$ is a modified cross product
-in which the $\\tilde{E}$ terms are shifted as indicated.
+where $\vec{r} = (m, n, p)$ and $\otimes$ is a modified cross product
+in which the $\tilde{E}$ terms are shifted as indicated.
 
 By taking the divergence and rearranging terms, we can show that
 
 $$
-  \\begin{aligned}
-  \\hat{\\nabla} \\cdot \\tilde{S}_{l, l', \\vec{r}}
-   &= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}})  \\\\
-   &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}} -
-      \\tilde{E}_{l, \\vec{r}} \\cdot \\hat{\\nabla} \\times \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
-   &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot
-            (-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -
-                \\hat{M}_{l, \\vec{r} + \\frac{1}{2}}) -
-      \\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_{\\vec{r}} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
-                \\tilde{J}_{l', \\vec{r}}) \\\\
-   &= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
-      \\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})
-      - \\hat{H}_{l'} \\cdot \\hat{M}_{l} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\
-  \\end{aligned}
+  \begin{aligned}
+  \hat{\nabla} \cdot \tilde{S}_{l, l', \vec{r}}
+   &= \hat{\nabla} \cdot (\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}})  \\
+   &= \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l, \vec{r}} -
+      \tilde{E}_{l, \vec{r}} \cdot \hat{\nabla} \times \hat{H}_{l', \vec{r} + \frac{1}{2}} \\
+   &= \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot
+          (-\tilde{\partial}_t \mu_{\vec{r} + \frac{1}{2}} \hat{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} -
+              \hat{M}_{l, \vec{r} + \frac{1}{2}}) -
+      \tilde{E}_{l, \vec{r}} \cdot (\hat{\partial}_t \tilde{\epsilon}_{\vec{r}} \tilde{E}_{l'+\frac{1}{2}, \vec{r}} +
+              \tilde{J}_{l', \vec{r}}) \\
+   &= \hat{H}_{l'} \cdot (-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) -
+      \tilde{E}_l \cdot (\epsilon / \Delta_t )(\tilde{E}_{l'+\frac{1}{2}} - \tilde{E}_{l'-\frac{1}{2}})
+      - \hat{H}_{l'} \cdot \hat{M}_{l} - \tilde{E}_l \cdot \tilde{J}_{l'} \\
+  \end{aligned}
 $$
 
 where in the last line the spatial subscripts have been dropped to emphasize
 the time subscripts $l, l'$, i.e.
 
 $$
-  \\begin{aligned}
-  \\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\
-  \\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}}  \\\\
-  \\tilde{\\epsilon} &= \\tilde{\\epsilon}_{\\vec{r}}  \\\\
-  \\end{aligned}
+  \begin{aligned}
+  \tilde{E}_l &= \tilde{E}_{l, \vec{r}} \\
+  \hat{H}_l &= \tilde{H}_{l, \vec{r} + \frac{1}{2}}  \\
+  \tilde{\epsilon} &= \tilde{\epsilon}_{\vec{r}}  \\
+  \end{aligned}
 $$
 
 etc.
-For $l' = l + \\frac{1}{2}$ we get
+For $l' = l + \frac{1}{2}$ we get
 
 $$
-  \\begin{aligned}
-  \\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}
-   &= \\hat{H}_{l + \\frac{1}{2}} \\cdot
-            (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
-      \\tilde{E}_l \\cdot (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} - \\tilde{E}_l)
-      - \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
-   &= (-\\mu / \\Delta_t)(\\hat{H}^2_{l + \\frac{1}{2}} - \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}) -
-      (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} \\cdot \\tilde{E}_l - \\tilde{E}^2_l)
-      - \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
-   &= -(\\mu \\hat{H}^2_{l + \\frac{1}{2}}
-       +\\epsilon \\tilde{E}_{l+1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
-      +(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
-       +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
-      - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
-      - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
-  \\end{aligned}
+  \begin{aligned}
+  \hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}}
+   &= \hat{H}_{l + \frac{1}{2}} \cdot
+            (-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) -
+      \tilde{E}_l \cdot (\epsilon / \Delta_t)(\tilde{E}_{l+1} - \tilde{E}_l)
+      - \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\
+   &= (-\mu / \Delta_t)(\hat{H}^2_{l + \frac{1}{2}} - \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}) -
+      (\epsilon / \Delta_t)(\tilde{E}_{l+1} \cdot \tilde{E}_l - \tilde{E}^2_l)
+      - \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\
+   &= -(\mu \hat{H}^2_{l + \frac{1}{2}}
+       +\epsilon \tilde{E}_{l+1} \cdot \tilde{E}_l) / \Delta_t \\
+      +(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}
+       +\epsilon \tilde{E}^2_l) / \Delta_t \\
+      - \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
+      - \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
+  \end{aligned}
 $$
 
-and for $l' = l - \\frac{1}{2}$,
+and for $l' = l - \frac{1}{2}$,
 
 $$
-  \\begin{aligned}
-  \\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}}
-   &=  (\\mu \\hat{H}^2_{l - \\frac{1}{2}}
-       +\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
-      -(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
-       +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
-      - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
-      - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
-  \\end{aligned}
+  \begin{aligned}
+  \hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}}
+   &=  (\mu \hat{H}^2_{l - \frac{1}{2}}
+       +\epsilon \tilde{E}_{l-1} \cdot \tilde{E}_l) / \Delta_t \\
+      -(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}
+       +\epsilon \tilde{E}^2_l) / \Delta_t \\
+      - \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
+      - \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
+  \end{aligned}
 $$
 
 These two results form the discrete time-domain analogue to Poynting's theorem.
@@ -109,25 +109,25 @@ They hint at the expressions for the energy, which can be calculated at the same
 time-index as either the E or H field:
 
 $$
- \\begin{aligned}
- U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\
- U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\
- \\end{aligned}
+ \begin{aligned}
+ U_l &= \epsilon \tilde{E}^2_l + \mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} \\
+ U_{l + \frac{1}{2}} &= \epsilon \tilde{E}_l \cdot \tilde{E}_{l + 1} + \mu \hat{H}^2_{l + \frac{1}{2}} \\
+ \end{aligned}
 $$
 
 Rewriting the Poynting theorem in terms of the energy expressions,
 
 $$
-  \\begin{aligned}
-  (U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
-   &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
-      - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
-      - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
-  (U_l - U_{l-\\frac{1}{2}}) / \\Delta_t
-   &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
-      - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
-      - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
- \\end{aligned}
+  \begin{aligned}
+  (U_{l+\frac{1}{2}} - U_l) / \Delta_t
+   &= -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\
+      - \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
+      - \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
+  (U_l - U_{l-\frac{1}{2}}) / \Delta_t
+   &= -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\
+      - \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
+      - \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
+ \end{aligned}
 $$
 
 This result is exact and should practically hold to within numerical precision. No time-
@@ -144,23 +144,131 @@ It is often useful to excite the simulation with an arbitrary broadband pulse an
 extract the frequency-domain response by performing an on-the-fly Fourier transform
 of the time-domain fields.
 
+`accumulate_phasor` in `meanas.fdtd.phasor` performs the phase accumulation for one
+or more target frequencies, while leaving source normalization and simulation-loop
+policy to the caller. `temporal_phasor(...)` and `temporal_phasor_scale(...)`
+apply the same Fourier sum to a scalar waveform, which is useful when a pulsed
+source envelope must be normalized before being applied to a point source or
+mode source. `real_injection_scale(...)` is the corresponding helper for the
+common real-valued injection pattern `numpy.real(scale * waveform)`. Convenience
+wrappers `accumulate_phasor_e`, `accumulate_phasor_h`, and `accumulate_phasor_j`
+apply the usual Yee time offsets. `reconstruct_real(...)` and the corresponding
+`reconstruct_real_e/h/j(...)` wrappers perform the inverse operation, converting
+phasors back into real-valued field snapshots at explicit Yee-aligned times.
+For scalar `omega`, the reconstruction helpers accept either a plain field
+phasor or the batched `(1, *sample_shape)` form used by the accumulator helpers.
+The helpers accumulate
+
+$$ \Delta_t \sum_l w_l e^{-i \omega t_l} f_l $$
+
+with caller-provided sample times and weights. In this codebase the matching
+electric-current convention is typically `E -= dt * J / epsilon` in FDTD and
+`-i \omega J` on the right-hand side of the FDFD wave equation.
+
+For FDTD/FDFD equivalence, this phasor path is the primary comparison workflow.
+It isolates the guided `+\omega` response that the frequency-domain solver
+targets directly, regardless of whether the underlying FDTD run used real- or
+complex-valued fields.
+
+For exact pulsed FDTD/FDFD equivalence it is often simplest to keep the
+injected source, fields, and CPML auxiliary state complex-valued. The
+`real_injection_scale(...)` helper is instead for the more ordinary one-run
+real-valued source path, where the intended positive-frequency waveform is
+injected through `numpy.real(scale * waveform)` and any remaining negative-
+frequency leakage is controlled by the pulse bandwidth and run length.
+
+`reconstruct_real(...)` is for a different question: given a phasor, what late
+real-valued field snapshot should it produce? That raw-snapshot comparison is
+stricter and noisier because a monitor plane generally contains both the guided
+field and the remaining orthogonal content,
+
+$$ E_{\text{monitor}} = E_{\text{guided}} + E_{\text{residual}} . $$
+
+Phasor/modal comparisons mostly validate the guided `+\omega` term. Raw
+real-field comparisons expose both terms at once, so they should be treated as
+secondary diagnostics rather than the main solver-equivalence benchmark.
+
 The Ricker wavelet (normalized second derivative of a Gaussian) is commonly used for the pulse
 shape. It can be written
 
-$$ f_r(t) = (1 - \\frac{1}{2} (\\omega (t - \\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2} $$
+$$ f_r(t) = (1 - \frac{1}{2} (\omega (t - \tau))^2) e^{-(\frac{\omega (t - \tau)}{2})^2} $$
 
-with $\\tau > \\frac{2 * \\pi}{\\omega}$ as a minimum delay to avoid a discontinuity at
-t=0 (assuming the source is off for t<0 this gives $\\sim 10^{-3}$ error at t=0).
+with $\tau > \frac{2 * \pi}{\omega}$ as a minimum delay to avoid a discontinuity at
+t=0 (assuming the source is off for t<0 this gives $\sim 10^{-3}$ error at t=0).
 
 
 
 Boundary conditions
 ===================
-# TODO notes about boundaries / PMLs
+
+`meanas.fdtd` exposes two boundary-related building blocks:
+
+- `conducting_boundary(...)` for simple perfect-electric-conductor style field
+  clamping at one face of the domain.
+- `cpml_params(...)` and `updates_with_cpml(...)` for convolutional perfectly
+  matched layers (CPMLs) attached to one or more faces of the Yee grid.
+
+`updates_with_cpml(...)` accepts a three-by-two table of CPML parameter blocks:
+
+```
+cpml_params[axis][polarity_index]
+```
+
+where `axis` is `0`, `1`, or `2` and `polarity_index` corresponds to `(-1, +1)`.
+Passing `None` for one entry disables CPML on that face while leaving the other
+directions unchanged. This is how mixed boundary setups such as "absorbing in x,
+periodic in y/z" are expressed.
+
+When comparing an FDTD run against an FDFD solve, use the same stretched
+coordinate system in both places:
+
+1. Build the FDTD update with the desired CPML parameters.
+2. Stretch the FDFD `dxes` with the matching SCPML transform.
+3. Compare the extracted phasor against the FDFD field or residual on those
+   stretched `dxes`.
+
+The electric-current sign convention used throughout the examples and tests is
+
+$$
+E \leftarrow E - \Delta_t J / \epsilon
+$$
+
+which matches the FDFD right-hand side
+
+$$
+-i \omega J.
+$$
 """
 
-from .base import maxwell_e, maxwell_h
-from .pml import cpml_params, updates_with_cpml
-from .energy import (poynting, poynting_divergence, energy_hstep, energy_estep,
-                     delta_energy_h2e, delta_energy_j)
-from .boundaries import conducting_boundary
+from .base import (
+    maxwell_e as maxwell_e,
+    maxwell_h as maxwell_h,
+    )
+from .pml import (
+    cpml_params as cpml_params,
+    updates_with_cpml as updates_with_cpml,
+    )
+from .energy import (
+    poynting as poynting,
+    poynting_divergence as poynting_divergence,
+    energy_hstep as energy_hstep,
+    energy_estep as energy_estep,
+    delta_energy_h2e as delta_energy_h2e,
+    delta_energy_j as delta_energy_j,
+    )
+from .boundaries import (
+    conducting_boundary as conducting_boundary,
+    )
+from .phasor import (
+    accumulate_phasor as accumulate_phasor,
+    accumulate_phasor_e as accumulate_phasor_e,
+    accumulate_phasor_h as accumulate_phasor_h,
+    accumulate_phasor_j as accumulate_phasor_j,
+    real_injection_scale as real_injection_scale,
+    reconstruct_real as reconstruct_real,
+    reconstruct_real_e as reconstruct_real_e,
+    reconstruct_real_h as reconstruct_real_h,
+    reconstruct_real_j as reconstruct_real_j,
+    temporal_phasor as temporal_phasor,
+    temporal_phasor_scale as temporal_phasor_scale,
+    )

meanas/fdtd/base.py

@@ -3,7 +3,7 @@ Basic FDTD field updates
 
 
 """
-from ..fdmath import dx_lists_t, fdfield_t, fdfield_updater_t
+from ..fdmath import dx_lists_t, fdfield, fdfield_updater_t
 from ..fdmath.functional import curl_forward, curl_back
 
 
@@ -47,7 +47,7 @@ def maxwell_e(
     else:
         curl_h_fun = curl_back()
 
-    def me_fun(e: fdfield_t, h: fdfield_t, epsilon: fdfield_t | float) -> fdfield_t:
+    def me_fun(e: fdfield, h: fdfield, epsilon: fdfield | float) -> fdfield:
         """
         Update the E-field.
 
@@ -103,7 +103,7 @@ def maxwell_h(
     else:
         curl_e_fun = curl_forward()
 
-    def mh_fun(e: fdfield_t, h: fdfield_t, mu: fdfield_t | float | None = None) -> fdfield_t:
+    def mh_fun(e: fdfield, h: fdfield, mu: fdfield | float | None = None) -> fdfield:
         """
         Update the H-field.
 

meanas/fdtd/boundaries.py

@@ -6,7 +6,7 @@ Boundary conditions
 
 from typing import Any
 
-from ..fdmath import fdfield_t, fdfield_updater_t
+from ..fdmath import fdfield, fdfield_updater_t
 
 
 def conducting_boundary(
@@ -15,26 +15,32 @@ def conducting_boundary(
         ) -> tuple[fdfield_updater_t, fdfield_updater_t]:
     dirs = [0, 1, 2]
     if direction not in dirs:
-        raise Exception('Invalid direction: {}'.format(direction))
+        raise ValueError(f'Invalid direction: {direction}')
     dirs.remove(direction)
     u, v = dirs
 
+    boundary_slice: list[Any]
+    shifted1_slice: list[Any]
+    shifted2_slice: list[Any]
+
     if polarity < 0:
-        boundary_slice = [slice(None)] * 3      # type: list[Any]
-        shifted1_slice = [slice(None)] * 3      # type: list[Any]
+        boundary_slice = [slice(None)] * 3
+        shifted1_slice = [slice(None)] * 3
         boundary_slice[direction] = 0
         shifted1_slice[direction] = 1
+        boundary = tuple(boundary_slice)
+        shifted1 = tuple(shifted1_slice)
 
-        def en(e: fdfield_t) -> fdfield_t:
-            e[direction][boundary_slice] = 0
-            e[u][boundary_slice] = e[u][shifted1_slice]
-            e[v][boundary_slice] = e[v][shifted1_slice]
+        def en(e: fdfield) -> fdfield:
+            e[direction][boundary] = 0
+            e[u][boundary] = e[u][shifted1]
+            e[v][boundary] = e[v][shifted1]
             return e
 
-        def hn(h: fdfield_t) -> fdfield_t:
-            h[direction][boundary_slice] = h[direction][shifted1_slice]
-            h[u][boundary_slice] = 0
-            h[v][boundary_slice] = 0
+        def hn(h: fdfield) -> fdfield:
+            h[direction][boundary] = h[direction][shifted1]
+            h[u][boundary] = 0
+            h[v][boundary] = 0
             return h
 
         return en, hn
@@ -42,26 +48,29 @@ def conducting_boundary(
     if polarity > 0:
         boundary_slice = [slice(None)] * 3
         shifted1_slice = [slice(None)] * 3
-        shifted2_slice = [slice(None)] * 3      # type: list[Any]
+        shifted2_slice = [slice(None)] * 3
         boundary_slice[direction] = -1
         shifted1_slice[direction] = -2
         shifted2_slice[direction] = -3
+        boundary = tuple(boundary_slice)
+        shifted1 = tuple(shifted1_slice)
+        shifted2 = tuple(shifted2_slice)
 
-        def ep(e: fdfield_t) -> fdfield_t:
-            e[direction][boundary_slice] = -e[direction][shifted2_slice]
-            e[direction][shifted1_slice] = 0
-            e[u][boundary_slice] = e[u][shifted1_slice]
-            e[v][boundary_slice] = e[v][shifted1_slice]
+        def ep(e: fdfield) -> fdfield:
+            e[direction][boundary] = -e[direction][shifted2]
+            e[direction][shifted1] = 0
+            e[u][boundary] = e[u][shifted1]
+            e[v][boundary] = e[v][shifted1]
             return e
 
-        def hp(h: fdfield_t) -> fdfield_t:
-            h[direction][boundary_slice] = h[direction][shifted1_slice]
-            h[u][boundary_slice] = -h[u][shifted2_slice]
-            h[u][shifted1_slice] = 0
-            h[v][boundary_slice] = -h[v][shifted2_slice]
-            h[v][shifted1_slice] = 0
+        def hp(h: fdfield) -> fdfield:
+            h[direction][boundary] = h[direction][shifted1]
+            h[u][boundary] = -h[u][shifted2]
+            h[u][shifted1] = 0
+            h[v][boundary] = -h[v][shifted2]
+            h[v][shifted1] = 0
             return h
 
         return ep, hp
 
-    raise Exception('Bad polarity: {}'.format(polarity))
+    raise ValueError(f'Bad polarity: {polarity}')

meanas/fdtd/energy.py

@@ -1,18 +1,32 @@
 import numpy
 
-from ..fdmath import dx_lists_t, fdfield_t
+from ..fdmath import dx_lists_t, fdfield_t, fdfield
 from ..fdmath.functional import deriv_back
 
 
-# TODO documentation
+"""
+Energy- and flux-accounting helpers for Yee-grid FDTD fields.
+
+These functions complement the derivation in `meanas.fdtd`:
+
+- `poynting(...)` and `poynting_divergence(...)` evaluate the discrete flux terms
+  from the exact time-domain Poynting identity.
+- `energy_hstep(...)` / `energy_estep(...)` evaluate the two staggered energy
+  expressions.
+- `delta_energy_*` helpers evaluate the corresponding energy changes between
+  adjacent half-steps.
+
+The helpers are intended for diagnostics, regression tests, and consistency
+checks between source work, field energy, and flux through cell faces.
+"""
 
 
 def poynting(
-        e: fdfield_t,
-        h: fdfield_t,
+        e: fdfield,
+        h: fdfield,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
-    """
+    r"""
     Calculate the poynting vector `S` ($S$).
 
     This is the energy transfer rate (amount of energy `U` per `dt` transferred
@@ -43,17 +57,18 @@ def poynting(
     (see `meanas.tests.test_fdtd.test_poynting_planes`)
 
     The full relationship is
+
     $$
-      \\begin{aligned}
-      (U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
-       &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
-          - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
-          - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
-      (U_l - U_{l-\\frac{1}{2}}) / \\Delta_t
-       &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
-          - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
-          - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
-     \\end{aligned}
+      \begin{aligned}
+      (U_{l+\frac{1}{2}} - U_l) / \Delta_t
+       &= -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\
+          - \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
+          - \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
+      (U_l - U_{l-\frac{1}{2}}) / \Delta_t
+       &= -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\
+          - \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
+          - \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
+     \end{aligned}
     $$
 
     These equalities are exact and should practically hold to within numerical precision.
@@ -84,14 +99,14 @@ def poynting(
     s[0] = numpy.roll(ey, -1, axis=0) * hz - numpy.roll(ez, -1, axis=0) * hy
     s[1] = numpy.roll(ez, -1, axis=1) * hx - numpy.roll(ex, -1, axis=1) * hz
     s[2] = numpy.roll(ex, -1, axis=2) * hy - numpy.roll(ey, -1, axis=2) * hx
-    return s
+    return fdfield_t(s)
 
 
 def poynting_divergence(
-        s: fdfield_t | None = None,
+        s: fdfield | None = None,
         *,
-        e: fdfield_t | None = None,
-        h: fdfield_t | None = None,
+        e: fdfield | None = None,
+        h: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -122,15 +137,15 @@ def poynting_divergence(
 
     Dx, Dy, Dz = deriv_back()
     ds = Dx(s[0]) + Dy(s[1]) + Dz(s[2])
-    return ds
+    return fdfield_t(ds)
 
 
 def energy_hstep(
-        e0: fdfield_t,
-        h1: fdfield_t,
-        e2: fdfield_t,
-        epsilon: fdfield_t | None = None,
-        mu: fdfield_t | None = None,
+        e0: fdfield,
+        h1: fdfield,
+        e2: fdfield,
+        epsilon: fdfield | None = None,
+        mu: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -150,15 +165,15 @@ def energy_hstep(
         Energy, at the time of the H-field `h1`.
     """
     u = dxmul(e0 * e2, h1 * h1, epsilon, mu, dxes)
-    return u
+    return fdfield_t(u)
 
 
 def energy_estep(
-        h0: fdfield_t,
-        e1: fdfield_t,
-        h2: fdfield_t,
-        epsilon: fdfield_t | None = None,
-        mu: fdfield_t | None = None,
+        h0: fdfield,
+        e1: fdfield,
+        h2: fdfield,
+        epsilon: fdfield | None = None,
+        mu: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -178,17 +193,17 @@ def energy_estep(
         Energy, at the time of the E-field `e1`.
     """
     u = dxmul(e1 * e1, h0 * h2, epsilon, mu, dxes)
-    return u
+    return fdfield_t(u)
 
 
 def delta_energy_h2e(
         dt: float,
-        e0: fdfield_t,
-        h1: fdfield_t,
-        e2: fdfield_t,
-        h3: fdfield_t,
-        epsilon: fdfield_t | None = None,
-        mu: fdfield_t | None = None,
+        e0: fdfield,
+        h1: fdfield,
+        e2: fdfield,
+        h3: fdfield,
+        epsilon: fdfield | None = None,
+        mu: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -211,17 +226,17 @@ def delta_energy_h2e(
     de = e2 * (e2 - e0) / dt
     dh = h1 * (h3 - h1) / dt
     du = dxmul(de, dh, epsilon, mu, dxes)
-    return du
+    return fdfield_t(du)
 
 
 def delta_energy_e2h(
         dt: float,
-        h0: fdfield_t,
-        e1: fdfield_t,
-        h2: fdfield_t,
-        e3: fdfield_t,
-        epsilon: fdfield_t | None = None,
-        mu: fdfield_t | None = None,
+        h0: fdfield,
+        e1: fdfield,
+        h2: fdfield,
+        e3: fdfield,
+        epsilon: fdfield | None = None,
+        mu: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -244,21 +259,31 @@ def delta_energy_e2h(
     de = e1 * (e3 - e1) / dt
     dh = h2 * (h2 - h0) / dt
     du = dxmul(de, dh, epsilon, mu, dxes)
-    return du
+    return fdfield_t(du)
 
 
 def delta_energy_j(
-        j0: fdfield_t,
-        e1: fdfield_t,
+        j0: fdfield,
+        e1: fdfield,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
-    """
-    Calculate
+    r"""
+    Calculate the electric-current work term $J \cdot E$ on the Yee grid.
 
-    Note that each value of $J$ contributes to the energy twice (i.e. once per field update)
-    despite only causing the value of $E$ to change once (same for $M$ and $H$).
+    This is the source contribution that appears beside the flux divergence in
+    the discrete Poynting identities documented in `meanas.fdtd`.
 
+    Note that each value of `J` contributes twice in a full Yee cycle (once per
+    half-step energy balance) even though it directly changes `E` only once.
 
+    Args:
+        j0: Electric-current density sampled at the same half-step as the
+            current work term.
+        e1: Electric field sampled at the matching integer timestep.
+        dxes: Grid description; defaults to unit spacing.
+
+    Returns:
+        Per-cell source-work contribution with the scalar field shape.
     """
     if dxes is None:
         dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
@@ -267,16 +292,30 @@ def delta_energy_j(
           * dxes[0][0][:, None, None]
           * dxes[0][1][None, :, None]
           * dxes[0][2][None, None, :])
-    return du
+    return fdfield_t(du)
 
 
 def dxmul(
-        ee: fdfield_t,
-        hh: fdfield_t,
-        epsilon: fdfield_t | float | None = None,
-        mu: fdfield_t | float | None = None,
+        ee: fdfield,
+        hh: fdfield,
+        epsilon: fdfield | float | None = None,
+        mu: fdfield | float | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
+    """
+    Multiply E- and H-like field products by material weights and cell volumes.
+
+    Args:
+        ee: Three-component electric-field product, such as `e0 * e2`.
+        hh: Three-component magnetic-field product, such as `h1 * h1`.
+        epsilon: Electric material weight; defaults to `1`.
+        mu: Magnetic material weight; defaults to `1`.
+        dxes: Grid description; defaults to unit spacing.
+
+    Returns:
+        Scalar field containing the weighted electric plus magnetic contribution
+        for each Yee cell.
+    """
     if epsilon is None:
         epsilon = 1
     if mu is None:
@@ -292,4 +331,4 @@ def dxmul(
               * dxes[1][0][:, None, None]
               * dxes[1][1][None, :, None]
               * dxes[1][2][None, None, :])
-    return result
+    return fdfield_t(result)

meanas/fdtd/misc.py

@@ -0,0 +1,180 @@
+from collections.abc import Callable
+import logging
+from typing import cast
+
+import numpy
+from numpy.typing import NDArray, ArrayLike
+from numpy import pi
+
+
+logger = logging.getLogger(__name__)
+
+
+type pulse_scalar_t = float | NDArray[numpy.floating]
+pulse_fn_t = Callable[[ArrayLike], tuple[pulse_scalar_t, pulse_scalar_t, pulse_scalar_t]]
+
+
+def _scalar_or_array(values: NDArray[numpy.floating]) -> pulse_scalar_t:
+    if values.ndim == 0:
+        return float(values)
+    return cast('NDArray[numpy.floating]', values)
+
+
+def gaussian_packet(
+        wl: float,
+        dwl: float,
+        dt: float,
+        turn_on: float = 1e-10,
+        one_sided: bool = False,
+        ) -> tuple[pulse_fn_t, float]:
+    """
+    Gaussian pulse (or gaussian ramp) for FDTD excitation
+
+    exp(-a*t*t) ==> exp(-omega * omega / (4 * a))   [fourier, ignoring leading const.]
+
+    FWHM_time is 2 * sqrt(2 * log(2)) * sqrt(2 / a)
+    FWHM_omega is 2 * sqrt(2 * log(2)) * sqrt(2 * a)  = 4 * sqrt(log(2) * a)
+
+    Args:
+        wl: wavelength
+        dwl: Gaussian's FWHM in wavelength space
+        dt: Timestep
+        turn_on: Max allowable amplitude at t=0
+        one_sided: If `True`, source amplitude never decreases after reaching max
+
+    Returns:
+        Source function: src(timestep) -> (envelope[tt], cos[... * tt], sin[... * tt])
+        Delay: number of initial timesteps for which envelope[tt] will be 0
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    # dt * dw = 4 * ln(2)
+
+    omega = 2 * pi / wl
+    freq = 1 / wl
+    fwhm_omega = dwl * omega * omega / (2 * pi)         # dwl -> d_omega (approx)
+    alpha = (fwhm_omega * fwhm_omega) * numpy.log(2) / 8
+    delay = numpy.sqrt(-numpy.log(turn_on) / alpha)
+    delay = numpy.ceil(delay * freq) / freq     # force delay to integer number of periods to maintain phase
+    logger.info(f'src_time {2 * delay / dt}')
+
+    def source_phasor(ii: ArrayLike) -> tuple[pulse_scalar_t, pulse_scalar_t, pulse_scalar_t]:
+        ii_array = numpy.asarray(ii, dtype=float)
+        t0 = ii_array * dt - delay
+        envelope = numpy.sqrt(numpy.sqrt(2 * alpha / pi)) * numpy.exp(-alpha * t0 * t0)
+
+        if one_sided:
+            envelope = numpy.where(t0 > 0, 1.0, envelope)
+
+        cc = numpy.cos(omega * t0)
+        ss = numpy.sin(omega * t0)
+
+        return _scalar_or_array(envelope), _scalar_or_array(cc), _scalar_or_array(ss)
+
+    # nrm = numpy.exp(-omega * omega / alpha) / 2
+
+    return source_phasor, delay
+
+
+def ricker_pulse(
+        wl: float,
+        dt: float,
+        turn_on: float = 1e-10,
+        ) -> tuple[pulse_fn_t, float]:
+    """
+    Ricker wavelet (second derivative of a gaussian pulse)
+
+    t0 = ii * dt - delay
+    R = w_peak * t0 / 2
+    f(t) = (1 - 2 * (pi * f_peak * t0) ** 2) * exp(-(pi * f_peak * t0)**2
+         = (1 - (w_peak * t0)**2 / 2 exp(-(w_peak * t0 / 2) **2)
+         = (1 - 2 * R * R) * exp(-R * R)
+
+    # NOTE: don't use cosine/sine for J, just for phasor readout
+
+    Args:
+        wl: wavelength
+        dt: Timestep
+        turn_on: Max allowable amplitude at t=0
+
+    Returns:
+        Source function: src(timestep) -> (envelope[tt], cos[... * tt], sin[... * tt])
+        Delay: number of initial timesteps for which envelope[tt] will be 0
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    omega = 2 * pi / wl
+    freq = 1 / wl
+
+    from scipy.optimize import root_scalar
+    delay_results = root_scalar(
+        lambda tt: (1 - omega * omega * tt * tt / 2) * numpy.exp(-omega * omega * tt * tt / 4) - turn_on,
+        x0=0,
+        x1=-2 / omega,
+        )
+
+    delay = delay_results.root
+    delay = numpy.ceil(delay * freq) / freq     # force delay to integer number of periods to maintain phase
+
+    def source_phasor(ii: ArrayLike) -> tuple[pulse_scalar_t, pulse_scalar_t, pulse_scalar_t]:
+        ii_array = numpy.asarray(ii, dtype=float)
+        t0 = ii_array * dt - delay
+        rr = omega * t0 / 2
+        ff = (1 - 2 * rr * rr) * numpy.exp(-rr * rr)
+
+        cc = numpy.cos(omega * t0)
+        ss = numpy.sin(omega * t0)
+
+        return _scalar_or_array(ff), _scalar_or_array(cc), _scalar_or_array(ss)
+
+    return source_phasor, delay
+
+
+def gaussian_beam(
+        xyz: list[NDArray],
+        center: ArrayLike,
+        waist_radius: float,
+        wl: float,
+        tilt: float = 0,
+        ) -> NDArray[numpy.complex128]:
+    """
+    Gaussian beam
+    (solution to paraxial Helmholtz equation)
+
+    Default (no tilt) corresponds to a beam propagating in the -z direction.
+
+    Args:
+        xyz: List of [[x0, x1, ...], [y0, ...], [z0, ...]] positions specifying grid
+            locations at which the field will be sampled.
+        center: [x, y, z] location of beam waist
+        waist_radius: Beam radius at the waist
+        wl: Wavelength
+        tilt: Rotation around y axis. Default (0) has beam propagating in -z direction.
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    w0 = waist_radius
+    grids = numpy.asarray(numpy.meshgrid(*xyz, indexing='ij'))
+    grids -= numpy.asarray(center)[:, None, None, None]
+
+    rot = numpy.array([
+        [ numpy.cos(tilt), 0, numpy.sin(tilt)],
+        [               0, 1,               0],
+        [-numpy.sin(tilt), 0, numpy.cos(tilt)],
+        ])
+
+    xx, yy, zz = numpy.einsum('ij,jxyz->ixyz', rot, grids)
+    r2 = xx * xx + yy * yy
+    z2 = zz * zz
+
+    zr = pi * w0 * w0 / wl
+    zr2 = zr * zr
+    wz2 = w0 * w0 * (1 + z2 / zr2)
+    wz = numpy.sqrt(wz2)        # == fwhm(z) / sqrt(2 * ln(2))
+
+    kk = 2 * pi / wl
+    inv_Rz = numpy.divide(zz, z2 + zr2, out=numpy.zeros_like(zz), where=(z2 + zr2) != 0)
+    gouy = numpy.arctan(zz / zr)
+
+    gaussian = w0 / wz * numpy.exp(-r2 / wz2) * numpy.exp(1j * (kk * zz + kk * r2 * inv_Rz / 2 - gouy))
+
+    row = gaussian[:, :, gaussian.shape[2] // 2]
+    norm = numpy.linalg.norm(row)
+    return gaussian / norm

meanas/fdtd/phasor.py

@@ -0,0 +1,319 @@
+"""
+Helpers for extracting single- or multi-frequency phasors from FDTD samples.
+
+These helpers are intentionally low-level: callers own the accumulator arrays and
+the sampling policy. The accumulated quantity is
+
+    dt * sum(weight * exp(-1j * omega * t_step) * sample_step)
+
+where `t_step = (step + offset_steps) * dt`.
+
+The usual Yee offsets are:
+
+- `accumulate_phasor_e(..., step=l)` for `E_l`
+- `accumulate_phasor_h(..., step=l)` for `H_{l + 1/2}`
+- `accumulate_phasor_j(..., step=l)` for `J_{l + 1/2}`
+
+`temporal_phasor(...)` and `temporal_phasor_scale(...)` apply the same Fourier
+sum to a 1D scalar waveform. They are useful for normalizing a pulsed source
+before that scalar waveform is applied to a point source or spatial mode source.
+`real_injection_scale(...)` is a companion helper for the common real-valued
+injection pattern `numpy.real(scale * waveform)`, where `waveform` is the
+analytic positive-frequency signal and the injected real current should still
+produce a desired phasor response.
+`reconstruct_real(...)` and its `E/H/J` wrappers perform the inverse operation:
+they turn one or more phasors back into real-valued field snapshots at explicit
+Yee-aligned sample times. For a scalar target frequency they accept either a
+plain field phasor or the batched `(1, *sample_shape)` form used elsewhere in
+this module.
+
+These helpers do not choose warmup/accumulation windows or pulse-envelope
+normalization. They also do not impose a current sign convention. In this
+codebase, electric-current injection normally appears as `E -= dt * J / epsilon`,
+which matches the FDFD right-hand side `-1j * omega * J`.
+"""
+from collections.abc import Sequence
+
+import numpy
+from numpy.typing import ArrayLike, NDArray
+
+
+def _normalize_omegas(
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        ) -> NDArray[numpy.complexfloating]:
+    omega_array = numpy.atleast_1d(numpy.asarray(omegas, dtype=complex))
+    if omega_array.ndim != 1 or omega_array.size == 0:
+        raise ValueError('omegas must be a scalar or non-empty 1D sequence')
+    return omega_array
+
+
+def _normalize_weight(
+        weight: ArrayLike,
+        omega_shape: tuple[int, ...],
+        ) -> NDArray[numpy.complexfloating]:
+    weight_array = numpy.asarray(weight, dtype=complex)
+    if weight_array.ndim == 0:
+        return numpy.full(omega_shape, weight_array, dtype=complex)
+    if weight_array.shape == omega_shape:
+        return weight_array
+    raise ValueError(f'weight must be scalar or have shape {omega_shape}, got {weight_array.shape}')
+
+
+def _normalize_temporal_samples(
+        samples: ArrayLike,
+        ) -> NDArray[numpy.complexfloating]:
+    sample_array = numpy.asarray(samples, dtype=complex)
+    if sample_array.ndim != 1 or sample_array.size == 0:
+        raise ValueError('samples must be a non-empty 1D sequence')
+    return sample_array
+
+
+def _validate_reconstruction_inputs(
+        phasors: ArrayLike,
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        ) -> tuple[NDArray[numpy.complexfloating], NDArray[numpy.complexfloating], bool]:
+    if dt <= 0:
+        raise ValueError('dt must be positive')
+
+    omega_array = _normalize_omegas(omegas)
+    phasor_array = numpy.asarray(phasors, dtype=complex)
+    expected_leading = omega_array.size
+    if phasor_array.ndim == 0:
+        raise ValueError(
+            f'phasors must have shape ({expected_leading}, *sample_shape) or sample_shape for scalar omega, got {phasor_array.shape}',
+        )
+    added_axis = False
+    if expected_leading == 1 and (phasor_array.ndim == 1 or phasor_array.shape[0] != expected_leading):
+        phasor_array = phasor_array[numpy.newaxis, ...]
+        added_axis = True
+    elif phasor_array.shape[0] != expected_leading:
+        raise ValueError(
+            f'phasors must have shape ({expected_leading}, *sample_shape) or sample_shape for scalar omega, got {phasor_array.shape}',
+        )
+    return omega_array, phasor_array, added_axis
+
+
+def accumulate_phasor(
+        accumulator: NDArray[numpy.complexfloating],
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        sample: ArrayLike,
+        step: int,
+        *,
+        offset_steps: float = 0.0,
+        weight: ArrayLike = 1.0,
+        ) -> NDArray[numpy.complexfloating]:
+    """
+    Add one time-domain sample into a phasor accumulator.
+
+    The added quantity is
+
+        dt * weight * exp(-1j * omega * t_step) * sample
+
+    where `t_step = (step + offset_steps) * dt`.
+
+    Note:
+        This helper already multiplies by `dt`. If the caller's normalization
+        factor was derived from a discrete sum that already includes `dt`, pass
+        `weight / dt` here.
+    """
+    if dt <= 0:
+        raise ValueError('dt must be positive')
+
+    omega_array = _normalize_omegas(omegas)
+    sample_array = numpy.asarray(sample)
+    expected_shape = (omega_array.size, *sample_array.shape)
+    if accumulator.shape != expected_shape:
+        raise ValueError(f'accumulator must have shape {expected_shape}, got {accumulator.shape}')
+
+    weight_array = _normalize_weight(weight, omega_array.shape)
+    time = (step + offset_steps) * dt
+    phase = numpy.exp(-1j * omega_array * time)
+    scaled = dt * (weight_array * phase).reshape((-1,) + (1,) * sample_array.ndim)
+    accumulator += scaled * sample_array
+    return accumulator
+
+
+def temporal_phasor(
+        samples: ArrayLike,
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        *,
+        start_step: int = 0,
+        offset_steps: float = 0.0,
+        ) -> NDArray[numpy.complexfloating]:
+    """
+    Fourier-project a 1D temporal waveform onto one or more angular frequencies.
+
+    The returned quantity is
+
+        dt * sum(exp(-1j * omega * t_step) * samples[step_index])
+
+    where `t_step = (start_step + step_index + offset_steps) * dt`.
+    """
+    if dt <= 0:
+        raise ValueError('dt must be positive')
+
+    omega_array = _normalize_omegas(omegas)
+    sample_array = _normalize_temporal_samples(samples)
+    steps = start_step + numpy.arange(sample_array.size, dtype=float) + offset_steps
+    phase = numpy.exp(-1j * omega_array[:, None] * (steps[None, :] * dt))
+    return dt * (phase @ sample_array)
+
+
+def temporal_phasor_scale(
+        samples: ArrayLike,
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        *,
+        start_step: int = 0,
+        offset_steps: float = 0.0,
+        target: ArrayLike = 1.0,
+        ) -> NDArray[numpy.complexfloating]:
+    """
+    Return the scalar multiplier that gives a desired temporal phasor response.
+
+    The returned scale satisfies
+
+        temporal_phasor(scale * samples, omegas, dt, ...) == target
+
+    for each target frequency. The result keeps a leading frequency axis even
+    when `omegas` is scalar.
+    """
+    response = temporal_phasor(samples, omegas, dt, start_step=start_step, offset_steps=offset_steps)
+    target_array = _normalize_weight(target, response.shape)
+    if numpy.any(numpy.abs(response) <= numpy.finfo(float).eps):
+        raise ValueError('cannot normalize a waveform with zero temporal phasor response')
+    return target_array / response
+
+
+def real_injection_scale(
+        samples: ArrayLike,
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        *,
+        start_step: int = 0,
+        offset_steps: float = 0.0,
+        target: ArrayLike = 1.0,
+        ) -> NDArray[numpy.complexfloating]:
+    """
+    Return the scale for a real-valued injection built from an analytic waveform.
+
+    If the time-domain source is applied as
+
+        numpy.real(scale * samples[step])
+
+    then the desired positive-frequency phasor is obtained by compensating for
+    the 1/2 factor between the real-valued source and its analytic component:
+
+        scale = 2 * target / temporal_phasor(samples, ...)
+
+    This helper normalizes only the intended positive-frequency component. Any
+    residual negative-frequency leakage is controlled by the waveform design and
+    the accumulation window.
+    """
+    response = temporal_phasor(samples, omegas, dt, start_step=start_step, offset_steps=offset_steps)
+    target_array = _normalize_weight(target, response.shape)
+    if numpy.any(numpy.abs(response) <= numpy.finfo(float).eps):
+        raise ValueError('cannot normalize a waveform with zero temporal phasor response')
+    return 2 * target_array / response
+
+
+def reconstruct_real(
+        phasors: ArrayLike,
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        step: int,
+        *,
+        offset_steps: float = 0.0,
+        ) -> NDArray[numpy.floating]:
+    """
+    Reconstruct a real-valued field snapshot from one or more phasors.
+
+    The returned quantity is
+
+        real(phasor * exp(1j * omega * t_step))
+
+    where `t_step = (step + offset_steps) * dt`.
+
+    For multi-frequency inputs, the leading frequency axis is preserved. For a
+    scalar `omega`, callers may pass either `(1, *sample_shape)` or
+    `sample_shape`; the return shape matches that choice.
+    """
+    omega_array, phasor_array, added_axis = _validate_reconstruction_inputs(phasors, omegas, dt)
+    time = (step + offset_steps) * dt
+    phase = numpy.exp(1j * omega_array * time).reshape((-1,) + (1,) * (phasor_array.ndim - 1))
+    reconstructed = numpy.real(phasor_array * phase)
+    if added_axis:
+        return reconstructed[0]
+    return reconstructed
+
+
+def accumulate_phasor_e(
+        accumulator: NDArray[numpy.complexfloating],
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        sample: ArrayLike,
+        step: int,
+        *,
+        weight: ArrayLike = 1.0,
+        ) -> NDArray[numpy.complexfloating]:
+    """Accumulate an E-field sample taken at integer timestep `step`."""
+    return accumulate_phasor(accumulator, omegas, dt, sample, step, offset_steps=0.0, weight=weight)
+
+
+def accumulate_phasor_h(
+        accumulator: NDArray[numpy.complexfloating],
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        sample: ArrayLike,
+        step: int,
+        *,
+        weight: ArrayLike = 1.0,
+        ) -> NDArray[numpy.complexfloating]:
+    """Accumulate an H-field sample corresponding to `H_{step + 1/2}`."""
+    return accumulate_phasor(accumulator, omegas, dt, sample, step, offset_steps=0.5, weight=weight)
+
+
+def accumulate_phasor_j(
+        accumulator: NDArray[numpy.complexfloating],
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        sample: ArrayLike,
+        step: int,
+        *,
+        weight: ArrayLike = 1.0,
+        ) -> NDArray[numpy.complexfloating]:
+    """Accumulate a current sample corresponding to `J_{step + 1/2}`."""
+    return accumulate_phasor(accumulator, omegas, dt, sample, step, offset_steps=0.5, weight=weight)
+
+
+def reconstruct_real_e(
+        phasors: ArrayLike,
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        step: int,
+        ) -> NDArray[numpy.floating]:
+    """Reconstruct a real E-field snapshot taken at integer timestep `step`."""
+    return reconstruct_real(phasors, omegas, dt, step, offset_steps=0.0)
+
+
+def reconstruct_real_h(
+        phasors: ArrayLike,
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        step: int,
+        ) -> NDArray[numpy.floating]:
+    """Reconstruct a real H-field snapshot corresponding to `H_{step + 1/2}`."""
+    return reconstruct_real(phasors, omegas, dt, step, offset_steps=0.5)
+
+
+def reconstruct_real_j(
+        phasors: ArrayLike,
+        omegas: float | complex | Sequence[float | complex] | NDArray,
+        dt: float,
+        step: int,
+        ) -> NDArray[numpy.floating]:
+    """Reconstruct a real current snapshot corresponding to `J_{step + 1/2}`."""
+    return reconstruct_real(phasors, omegas, dt, step, offset_steps=0.5)

meanas/fdtd/pml.py

@@ -1,18 +1,29 @@
 """
-PML implementations
+Convolutional perfectly matched layer (CPML) support for FDTD updates.
 
-#TODO discussion of PMLs
-#TODO cpml documentation
+The helpers in this module construct per-face CPML parameters and then wrap the
+standard Yee updates with the additional auxiliary `psi` fields needed by the
+CPML recurrence.
 
+The intended call pattern is:
+
+1. Build a `cpml_params[axis][polarity_index]` table with `cpml_params(...)`.
+2. Pass that table into `updates_with_cpml(...)` together with `dt`, `dxes`, and
+   `epsilon`.
+3. Advance the returned `update_E` / `update_H` closures in the simulation loop.
+
+Each face can be enabled or disabled independently by replacing one table entry
+with `None`.
 """
 # TODO retest pmls!
 
-from typing import Callable, Sequence, Any
+from typing import Any
+from collections.abc import Callable, Sequence
 from copy import deepcopy
 import numpy
 from numpy.typing import NDArray, DTypeLike
 
-from ..fdmath import fdfield_t, dx_lists_t
+from ..fdmath import fdfield, dx_lists_t
 from ..fdmath.functional import deriv_forward, deriv_back
 
 
@@ -31,18 +42,41 @@ def cpml_params(
         ma: float = 1,
         cfs_alpha: float = 0,
         ) -> dict[str, Any]:
+    """
+    Construct the parameter block for one CPML face.
+
+    Args:
+        axis: Which Cartesian axis the CPML is normal to (`0`, `1`, or `2`).
+        polarity: Which face along that axis (`-1` for the low-index face,
+            `+1` for the high-index face).
+        dt: Timestep used by the Yee update.
+        thickness: Number of Yee cells occupied by the CPML region.
+        ln_R_per_layer: Logarithmic attenuation target per layer.
+        epsilon_eff: Effective permittivity used when choosing the CPML scaling.
+        mu_eff: Effective permeability used when choosing the CPML scaling.
+        m: Polynomial grading exponent for `sigma` and `kappa`.
+        ma: Polynomial grading exponent for the complex-frequency shift `alpha`.
+        cfs_alpha: Maximum complex-frequency shift parameter.
+
+    Returns:
+        Dictionary with:
+
+        - `param_e`: `(p0, p1, p2)` arrays for the E update
+        - `param_h`: `(p0, p1, p2)` arrays for the H update
+        - `region`: slice tuple selecting the CPML cells on that face
+    """
 
     if axis not in range(3):
-        raise Exception('Invalid axis: {}'.format(axis))
+        raise ValueError(f'Invalid axis: {axis}')
 
     if polarity not in (-1, 1):
-        raise Exception('Invalid polarity: {}'.format(polarity))
+        raise ValueError(f'Invalid polarity: {polarity}')
 
     if thickness <= 2:
-        raise Exception('It would be wise to have a pml with 4+ cells of thickness')
+        raise ValueError('It would be wise to have a pml with 4+ cells of thickness')
 
     if epsilon_eff <= 0:
-        raise Exception('epsilon_eff must be positive')
+        raise ValueError('epsilon_eff must be positive')
 
     sigma_max = -ln_R_per_layer / 2 * (m + 1)
     kappa_max = numpy.sqrt(epsilon_eff * mu_eff)
@@ -56,8 +90,6 @@ def cpml_params(
         xh -= 0.5
         xe = xe[::-1]
         xh = xh[::-1]
-    else:
-        raise Exception('Bad polarity!')
 
     expand_slice_l: list[Any] = [None, None, None]
     expand_slice_l[axis] = slice(None)
@@ -81,8 +113,6 @@ def cpml_params(
         region_list[axis] = slice(None, thickness)
     elif polarity > 0:
         region_list[axis] = slice(-thickness, None)
-    else:
-        raise Exception('Bad polarity!')
     region = tuple(region_list)
 
     return {
@@ -96,11 +126,31 @@ def updates_with_cpml(
         cpml_params: Sequence[Sequence[dict[str, Any] | None]],
         dt: float,
         dxes: dx_lists_t,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
         *,
         dtype: DTypeLike = numpy.float32,
-        ) -> tuple[Callable[[fdfield_t, fdfield_t, fdfield_t], None],
-                   Callable[[fdfield_t, fdfield_t, fdfield_t], None]]:
+        ) -> tuple[Callable[..., None], Callable[..., None]]:
+    """
+    Build Yee-step update closures augmented with CPML terms.
+
+    Args:
+        cpml_params: Three-by-two sequence indexed as `[axis][polarity_index]`.
+            Entries are the dictionaries returned by `cpml_params(...)`; use
+            `None` to disable CPML on one face.
+        dt: Timestep.
+        dxes: Yee-grid spacing lists `[dx_e, dx_h]`.
+        epsilon: Electric material distribution used by the E update.
+        dtype: Storage dtype for the auxiliary CPML state arrays.
+
+    Returns:
+        `(update_E, update_H)` closures with the same call shape as the basic
+        Yee updates:
+
+        - `update_E(e, h, epsilon)`
+        - `update_H(e, h, mu)`
+
+        The closures retain the CPML auxiliary state internally.
+    """
 
     Dfx, Dfy, Dfz = deriv_forward(dxes[1])
     Dbx, Dby, Dbz = deriv_back(dxes[1])
@@ -111,7 +161,7 @@ def updates_with_cpml(
     params_H: list[list[tuple[Any, Any, Any, Any]]] = deepcopy(params_E)
 
     for axis in range(3):
-        for pp, polarity in enumerate((-1, 1)):
+        for pp, _polarity in enumerate((-1, 1)):
             cpml_param = cpml_params[axis][pp]
             if cpml_param is None:
                 psi_E[axis][pp] = (None, None)
@@ -136,9 +186,9 @@ def updates_with_cpml(
     pH = numpy.empty_like(epsilon, dtype=dtype)
 
     def update_E(
-            e: fdfield_t,
-            h: fdfield_t,
-            epsilon: fdfield_t,
+            e: fdfield,
+            h: fdfield,
+            epsilon: fdfield,
             ) -> None:
         dyHx = Dby(h[0])
         dzHx = Dbz(h[0])
@@ -182,9 +232,9 @@ def updates_with_cpml(
         e[2] += dt / epsilon[2] * (dxHy - dyHx + pE[2])
 
     def update_H(
-            e: fdfield_t,
-            h: fdfield_t,
-            mu: fdfield_t = numpy.ones(3),
+            e: fdfield,
+            h: fdfield,
+            mu: fdfield | tuple[int, int, int] = (1, 1, 1),
             ) -> None:
         dyEx = Dfy(e[0])
         dzEx = Dfz(e[0])

meanas/test/_bloch_case.py

@@ -0,0 +1,37 @@
+import numpy
+
+
+SHAPE = (2, 2, 2)
+G_MATRIX = numpy.eye(3)
+K0_GENERAL = numpy.array([0.1, 0.2, 0.3], dtype=float)
+K0_AXIAL = numpy.array([0.0, 0.0, 0.25], dtype=float)
+K0_X = numpy.array([0.1, 0.0, 0.0], dtype=float)
+EPSILON = numpy.ones((3, *SHAPE), dtype=float)
+MU = numpy.stack([
+    numpy.linspace(2.0, 2.7, numpy.prod(SHAPE)).reshape(SHAPE),
+    numpy.linspace(2.1, 2.8, numpy.prod(SHAPE)).reshape(SHAPE),
+    numpy.linspace(2.2, 2.9, numpy.prod(SHAPE)).reshape(SHAPE),
+])
+H_SIZE = 2 * numpy.prod(SHAPE)
+H_MN = (numpy.arange(H_SIZE) + 0.25j).astype(complex)
+ZERO_H_MN = numpy.zeros_like(H_MN)
+Y0 = (numpy.arange(H_SIZE, dtype=float) + 1j * numpy.linspace(0.1, 0.9, H_SIZE))[None, :]
+Y0_TWO_MODE = numpy.vstack(
+    [
+        numpy.arange(H_SIZE, dtype=float) + 1j * numpy.linspace(0.1, 0.9, H_SIZE),
+        numpy.linspace(2.0, 3.5, H_SIZE) - 0.5j * numpy.arange(H_SIZE, dtype=float),
+    ],
+)
+
+
+def build_overlap_fixture() -> tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray, numpy.ndarray]:
+    e_in = numpy.zeros((3, *SHAPE), dtype=complex)
+    h_in = numpy.zeros_like(e_in)
+    e_out = numpy.zeros_like(e_in)
+    h_out = numpy.zeros_like(e_in)
+
+    e_in[1] = 1.0
+    h_in[2] = 2.0
+    e_out[1] = 3.0
+    h_out[2] = 4.0
+    return e_in, h_in, e_out, h_out

meanas/test/_fdfd_case.py

@@ -0,0 +1,49 @@
+import numpy
+
+from ..fdmath import vec, unvec
+
+
+OMEGA = 1 / 1500
+SHAPE = (2, 3, 2)
+DXES = [
+    [numpy.array([1.0, 1.5]), numpy.array([0.75, 1.25, 1.5]), numpy.array([1.2, 0.8])],
+    [numpy.array([0.9, 1.4]), numpy.array([0.8, 1.1, 1.4]), numpy.array([1.0, 0.7])],
+]
+
+EPSILON = numpy.stack([
+    numpy.linspace(1.0, 2.2, numpy.prod(SHAPE)).reshape(SHAPE),
+    numpy.linspace(1.1, 2.3, numpy.prod(SHAPE)).reshape(SHAPE),
+    numpy.linspace(1.2, 2.4, numpy.prod(SHAPE)).reshape(SHAPE),
+])
+MU = numpy.stack([
+    numpy.linspace(2.0, 3.2, numpy.prod(SHAPE)).reshape(SHAPE),
+    numpy.linspace(2.1, 3.3, numpy.prod(SHAPE)).reshape(SHAPE),
+    numpy.linspace(2.2, 3.4, numpy.prod(SHAPE)).reshape(SHAPE),
+])
+
+E_FIELD = (numpy.arange(3 * numpy.prod(SHAPE)).reshape((3, *SHAPE)) + 0.5j).astype(complex)
+H_FIELD = (numpy.arange(3 * numpy.prod(SHAPE)).reshape((3, *SHAPE)) * 0.25 - 0.75j).astype(complex)
+
+PEC = numpy.zeros((3, *SHAPE), dtype=float)
+PEC[1, 0, 1, 0] = 1.0
+PMC = numpy.zeros((3, *SHAPE), dtype=float)
+PMC[2, 1, 2, 1] = 1.0
+
+TF_REGION = numpy.zeros((3, *SHAPE), dtype=float)
+TF_REGION[:, 0, 1, 0] = 1.0
+
+BOUNDARY_SHAPE = (3, 4, 3)
+BOUNDARY_DXES = [
+    [numpy.array([1.0, 1.5, 0.8]), numpy.array([0.75, 1.25, 1.5, 0.9]), numpy.array([1.2, 0.8, 1.1])],
+    [numpy.array([0.9, 1.4, 1.0]), numpy.array([0.8, 1.1, 1.4, 1.0]), numpy.array([1.0, 0.7, 1.3])],
+]
+BOUNDARY_EPSILON = numpy.stack([
+    numpy.linspace(1.0, 2.2, numpy.prod(BOUNDARY_SHAPE)).reshape(BOUNDARY_SHAPE),
+    numpy.linspace(1.1, 2.3, numpy.prod(BOUNDARY_SHAPE)).reshape(BOUNDARY_SHAPE),
+    numpy.linspace(1.2, 2.4, numpy.prod(BOUNDARY_SHAPE)).reshape(BOUNDARY_SHAPE),
+])
+BOUNDARY_FIELD = (numpy.arange(3 * numpy.prod(BOUNDARY_SHAPE)).reshape((3, *BOUNDARY_SHAPE)) + 0.5j).astype(complex)
+
+
+def apply_fdfd_matrix(op, field: numpy.ndarray, shape: tuple[int, ...] = SHAPE) -> numpy.ndarray:
+    return unvec(op @ vec(field), shape)

meanas/test/_solver_cases.py

@@ -0,0 +1,56 @@
+import dataclasses
+import numpy
+from scipy import sparse
+import scipy.sparse.linalg as spalg
+
+from ._test_builders import unit_dxes
+
+
+@dataclasses.dataclass(frozen=True)
+class DiagonalEigenCase:
+    operator: sparse.csr_matrix
+    linear_operator: spalg.LinearOperator
+    guess_default: numpy.ndarray
+    guess_sparse: numpy.ndarray
+
+
+@dataclasses.dataclass(frozen=True)
+class SolverPlumbingCase:
+    omega: float
+    a0: sparse.csr_matrix
+    pl: sparse.csr_matrix
+    pr: sparse.csr_matrix
+    j: numpy.ndarray
+    guess: numpy.ndarray
+    solver_result: numpy.ndarray
+    dxes: tuple[tuple[numpy.ndarray, ...], tuple[numpy.ndarray, ...]]
+    epsilon: numpy.ndarray
+
+
+def diagonal_eigen_case() -> DiagonalEigenCase:
+    operator = sparse.diags([5.0, 3.0, 1.0, -2.0]).tocsr()
+    linear_operator = spalg.LinearOperator(
+        shape=operator.shape,
+        dtype=complex,
+        matvec=lambda vv: operator @ vv,
+    )
+    return DiagonalEigenCase(
+        operator=operator,
+        linear_operator=linear_operator,
+        guess_default=numpy.array([0.0, 1.0, 1e-6, 0.0], dtype=complex),
+        guess_sparse=numpy.array([1.0, 0.1, 0.0, 0.0], dtype=complex),
+    )
+
+
+def solver_plumbing_case() -> SolverPlumbingCase:
+    return SolverPlumbingCase(
+        omega=2.0,
+        a0=sparse.csr_matrix(numpy.array([[1.0 + 2.0j, 2.0], [3.0 - 1.0j, 4.0]])),
+        pl=sparse.csr_matrix(numpy.array([[2.0, 0.0], [0.0, 3.0j]])),
+        pr=sparse.csr_matrix(numpy.array([[0.5, 0.0], [0.0, -2.0j]])),
+        j=numpy.array([1.0 + 0.5j, -2.0]),
+        guess=numpy.array([0.25 - 0.75j, 1.5 + 0.5j]),
+        solver_result=numpy.array([3.0 - 1.0j, -4.0 + 2.0j]),
+        dxes=unit_dxes((1, 1, 1)),
+        epsilon=numpy.ones(2),
+    )

meanas/test/_test_builders.py

@@ -0,0 +1,22 @@
+import numpy
+
+
+def real_ramp(shape: tuple[int, ...], *, scale: float = 1.0, offset: float = 0.0) -> numpy.ndarray:
+    return numpy.arange(numpy.prod(shape), dtype=float).reshape(shape, order='C') * scale + offset
+
+
+def complex_ramp(
+        shape: tuple[int, ...],
+        *,
+        scale: float = 1.0,
+        offset: float = 0.0,
+        imag_scale: float = 0.0,
+        imag_offset: float = 0.0,
+        ) -> numpy.ndarray:
+    real = real_ramp(shape, scale=scale, offset=offset)
+    imag = real_ramp(shape, scale=imag_scale, offset=imag_offset)
+    return (real + 1j * imag).astype(complex)
+
+
+def unit_dxes(shape: tuple[int, ...]) -> tuple[tuple[numpy.ndarray, ...], tuple[numpy.ndarray, ...]]:
+    return tuple(tuple(numpy.ones(length) for length in shape) for _ in range(2))  # type: ignore[return-value]

meanas/test/conftest.py

@@ -3,12 +3,13 @@
 Test fixtures
 
 """
-from typing import Iterable, Any
+# ruff: noqa: ARG001
+from typing import Any
 import numpy
 from numpy.typing import NDArray
 import pytest       # type: ignore
 
-from .utils import PRNG
+from .utils import make_prng
 
 
 FixtureRequest = Any
@@ -20,18 +21,18 @@ FixtureRequest = Any
                         (5, 5, 5),
                         # (7, 7, 7),
                        ])
-def shape(request: FixtureRequest) -> Iterable[tuple[int, ...]]:
-    yield (3, *request.param)
+def shape(request: FixtureRequest) -> tuple[int, ...]:
+    return (3, *request.param)
 
 
 @pytest.fixture(scope='module', params=[1.0, 1.5])
-def epsilon_bg(request: FixtureRequest) -> Iterable[float]:
-    yield request.param
+def epsilon_bg(request: FixtureRequest) -> float:
+    return request.param
 
 
 @pytest.fixture(scope='module', params=[1.0, 2.5])
-def epsilon_fg(request: FixtureRequest) -> Iterable[float]:
-    yield request.param
+def epsilon_fg(request: FixtureRequest) -> float:
+    return request.param
 
 
 @pytest.fixture(scope='module', params=['center', '000', 'random'])
@@ -40,7 +41,8 @@ def epsilon(
         shape: tuple[int, ...],
         epsilon_bg: float,
         epsilon_fg: float,
-        ) -> Iterable[NDArray[numpy.float64]]:
+        ) -> NDArray[numpy.float64]:
+    prng = make_prng()
     is3d = (numpy.array(shape) == 1).sum() == 0
     if is3d:
         if request.param == '000':
@@ -56,21 +58,23 @@ def epsilon(
     elif request.param == '000':
         epsilon[:, 0, 0, 0] = epsilon_fg
     elif request.param == 'random':
-        epsilon[:] = PRNG.uniform(low=min(epsilon_bg, epsilon_fg),
+        epsilon[:] = prng.uniform(
+            low=min(epsilon_bg, epsilon_fg),
             high=max(epsilon_bg, epsilon_fg),
-                                  size=shape)
+            size=shape,
+        )
 
-    yield epsilon
+    return epsilon
 
 
 @pytest.fixture(scope='module', params=[1.0])  # 1.5
-def j_mag(request: FixtureRequest) -> Iterable[float]:
-    yield request.param
+def j_mag(request: FixtureRequest) -> float:
+    return request.param
 
 
 @pytest.fixture(scope='module', params=[1.0, 1.5])
-def dx(request: FixtureRequest) -> Iterable[float]:
-    yield request.param
+def dx(request: FixtureRequest) -> float:
+    return request.param
 
 
 @pytest.fixture(scope='module', params=['uniform', 'centerbig'])
@@ -78,7 +82,8 @@ def dxes(
         request: FixtureRequest,
         shape: tuple[int, ...],
         dx: float,
-        ) -> Iterable[list[list[NDArray[numpy.float64]]]]:
+        ) -> list[list[NDArray[numpy.float64]]]:
+    prng = make_prng()
     if request.param == 'uniform':
         dxes = [[numpy.full(s, dx) for s in shape[1:]] for _ in range(2)]
     elif request.param == 'centerbig':
@@ -87,8 +92,7 @@ def dxes(
             for ax in (0, 1, 2):
                 dxes[eh][ax][dxes[eh][ax].size // 2] *= 1.1
     elif request.param == 'random':
-        dxe = [PRNG.uniform(low=1.0 * dx, high=1.1 * dx, size=s) for s in shape[1:]]
+        dxe = [prng.uniform(low=1.0 * dx, high=1.1 * dx, size=s) for s in shape[1:]]
         dxh = [(d + numpy.roll(d, -1)) / 2 for d in dxe]
         dxes = [dxe, dxh]
-    yield dxes
-
+    return dxes

meanas/test/test_bloch_foundations.py

@@ -0,0 +1,73 @@
+import numpy
+from numpy.linalg import norm
+
+from ..fdfd import bloch
+from ._bloch_case import EPSILON, G_MATRIX, H_MN, K0_AXIAL, K0_GENERAL, MU, SHAPE, ZERO_H_MN
+from .utils import assert_close
+
+def test_generate_kmn_general_case_returns_orthonormal_basis() -> None:
+    k_mag, m_vecs, n_vecs = bloch.generate_kmn(K0_GENERAL, G_MATRIX, SHAPE)
+
+    assert k_mag.shape == SHAPE + (1,)
+    assert m_vecs.shape == SHAPE + (3,)
+    assert n_vecs.shape == SHAPE + (3,)
+    assert numpy.isfinite(k_mag).all()
+    assert numpy.isfinite(m_vecs).all()
+    assert numpy.isfinite(n_vecs).all()
+
+    assert_close(norm(m_vecs.reshape(-1, 3), axis=1), 1.0)
+    assert_close(norm(n_vecs.reshape(-1, 3), axis=1), 1.0)
+    assert_close(numpy.sum(m_vecs * n_vecs, axis=3), 0.0, atol=1e-12)
+
+
+def test_generate_kmn_z_aligned_uses_default_transverse_basis() -> None:
+    k_mag, m_vecs, n_vecs = bloch.generate_kmn(K0_AXIAL, G_MATRIX, (1, 1, 1))
+
+    assert numpy.isfinite(k_mag).all()
+    assert_close(m_vecs[0, 0, 0], [0.0, 1.0, 0.0])
+    assert_close(numpy.sum(m_vecs * n_vecs, axis=3), 0.0, atol=1e-12)
+    assert_close(norm(n_vecs.reshape(-1, 3), axis=1), 1.0)
+
+
+def test_maxwell_operator_returns_finite_column_vector_without_mu() -> None:
+    operator = bloch.maxwell_operator(K0_GENERAL, G_MATRIX, EPSILON)
+
+    result = operator(H_MN.copy())
+    zero_result = operator(ZERO_H_MN.copy())
+
+    assert result.shape == (2 * numpy.prod(SHAPE), 1)
+    assert numpy.isfinite(result).all()
+    assert_close(zero_result, 0.0)
+
+
+def test_maxwell_operator_returns_finite_column_vector_with_mu() -> None:
+    operator = bloch.maxwell_operator(K0_GENERAL, G_MATRIX, EPSILON, MU)
+
+    result = operator(H_MN.copy())
+    zero_result = operator(ZERO_H_MN.copy())
+
+    assert result.shape == (2 * numpy.prod(SHAPE), 1)
+    assert numpy.isfinite(result).all()
+    assert_close(zero_result, 0.0)
+
+
+def test_inverse_maxwell_operator_returns_finite_column_vector_for_both_mu_branches() -> None:
+    for mu in (None, MU):
+        operator = bloch.inverse_maxwell_operator_approx(K0_GENERAL, G_MATRIX, EPSILON, mu)
+
+        result = operator(H_MN.copy())
+        zero_result = operator(ZERO_H_MN.copy())
+
+        assert result.shape == (2 * numpy.prod(SHAPE), 1)
+        assert numpy.isfinite(result).all()
+        assert_close(zero_result, 0.0)
+
+
+def test_bloch_field_converters_return_finite_fields() -> None:
+    e_field = bloch.hmn_2_exyz(K0_GENERAL, G_MATRIX, EPSILON)(H_MN.copy())
+    h_field = bloch.hmn_2_hxyz(K0_GENERAL, G_MATRIX, EPSILON)(H_MN.copy())
+
+    assert e_field.shape == (3, *SHAPE)
+    assert h_field.shape == (3, *SHAPE)
+    assert numpy.isfinite(e_field).all()
+    assert numpy.isfinite(h_field).all()

meanas/test/test_bloch_interactions.py

@@ -0,0 +1,315 @@
+import numpy
+import pytest
+from numpy.testing import assert_allclose
+from types import SimpleNamespace
+
+from ..fdfd import bloch
+from ._bloch_case import EPSILON, G_MATRIX, H_SIZE, K0_X, Y0, Y0_TWO_MODE, build_overlap_fixture
+from .utils import assert_close
+
+
+def test_rtrace_atb_matches_real_frobenius_inner_product() -> None:
+    a_mat = numpy.array([[1.0 + 2.0j, 3.0 - 1.0j], [2.0j, 4.0]], dtype=complex)
+    b_mat = numpy.array([[5.0 - 1.0j, 1.0 + 1.0j], [2.0, 3.0j]], dtype=complex)
+    expected = numpy.real(numpy.sum(a_mat.conj() * b_mat))
+
+    assert bloch._rtrace_AtB(a_mat, b_mat) == expected
+
+
+def test_symmetrize_returns_hermitian_average() -> None:
+    matrix = numpy.array([[1.0 + 2.0j, 3.0 - 1.0j], [2.0j, 4.0]], dtype=complex)
+    result = bloch._symmetrize(matrix)
+
+    assert_close(result, 0.5 * (matrix + matrix.conj().T))
+    assert_close(result, result.conj().T)
+
+
+def test_inner_product_is_nonmutating_and_obeys_sign_symmetry() -> None:
+    e_in, h_in, e_out, h_out = build_overlap_fixture()
+    originals = (e_in.copy(), h_in.copy(), e_out.copy(), h_out.copy())
+
+    pp = bloch.inner_product(e_out, h_out, e_in, h_in)
+    pn = bloch.inner_product(e_out, h_out, e_in, -h_in)
+    np_term = bloch.inner_product(e_out, -h_out, e_in, h_in)
+    nn = bloch.inner_product(e_out, -h_out, e_in, -h_in)
+
+    assert_close(pp, 0.8164965809277263 + 0.0j)
+    assert_close(pp, -nn, atol=1e-12, rtol=1e-12)
+    assert_close(pn, -np_term, atol=1e-12, rtol=1e-12)
+    assert numpy.array_equal(e_in, originals[0])
+    assert numpy.array_equal(h_in, originals[1])
+    assert numpy.array_equal(e_out, originals[2])
+    assert numpy.array_equal(h_out, originals[3])
+
+
+def test_trq_returns_expected_transmission_and_reflection() -> None:
+    e_in, h_in, e_out, h_out = build_overlap_fixture()
+
+    transmission, reflection = bloch.trq(e_in, h_in, e_out, h_out)
+
+    assert_close(transmission, 0.9797958971132713 + 0.0j, atol=1e-12, rtol=1e-12)
+    assert_close(reflection, 0.2 + 0.0j, atol=1e-12, rtol=1e-12)
+
+
+def test_eigsolve_returns_finite_modes_with_small_residual() -> None:
+    callback_count = 0
+
+    def callback() -> None:
+        nonlocal callback_count
+        callback_count += 1
+
+    eigvals, eigvecs = bloch.eigsolve(
+        1,
+        K0_X,
+        G_MATRIX,
+        EPSILON,
+        tolerance=1e-6,
+        max_iters=50,
+        y0=Y0,
+        callback=callback,
+    )
+
+    operator = bloch.maxwell_operator(K0_X, G_MATRIX, EPSILON)
+    eigvec = eigvecs[0] / numpy.linalg.norm(eigvecs[0])
+    residual = numpy.linalg.norm(operator(eigvec).reshape(-1) - eigvals[0] * eigvec) / numpy.linalg.norm(eigvec)
+
+    assert eigvals.shape == (1,)
+    assert eigvecs.shape == (1, H_SIZE)
+    assert numpy.isfinite(eigvals).all()
+    assert numpy.isfinite(eigvecs).all()
+    assert residual < 1e-5
+    assert callback_count > 0
+
+
+def test_eigsolve_without_initial_guess_returns_finite_modes() -> None:
+    eigvals, eigvecs = bloch.eigsolve(
+        1,
+        K0_X,
+        G_MATRIX,
+        EPSILON,
+        tolerance=1e-6,
+        max_iters=20,
+        y0=None,
+    )
+
+    operator = bloch.maxwell_operator(K0_X, G_MATRIX, EPSILON)
+    eigvec = eigvecs[0] / numpy.linalg.norm(eigvecs[0])
+    residual = numpy.linalg.norm(operator(eigvec).reshape(-1) - eigvals[0] * eigvec) / numpy.linalg.norm(eigvec)
+
+    assert eigvals.shape == (1,)
+    assert eigvecs.shape == (1, H_SIZE)
+    assert numpy.isfinite(eigvals).all()
+    assert numpy.isfinite(eigvecs).all()
+    assert residual < 1e-5
+
+
+def test_eigsolve_recovers_from_singular_initial_subspace(monkeypatch: pytest.MonkeyPatch) -> None:
+    class FakeRng:
+        def __init__(self) -> None:
+            self.calls = 0
+
+        def random(self, shape: tuple[int, ...]) -> numpy.ndarray:
+            self.calls += 1
+            return numpy.arange(numpy.prod(shape), dtype=float).reshape(shape) + self.calls
+
+    fake_rng = FakeRng()
+    singular_y0 = numpy.vstack([Y0_TWO_MODE[0], Y0_TWO_MODE[0]])
+    monkeypatch.setattr(bloch.numpy.random, 'default_rng', lambda: fake_rng)
+
+    eigvals, eigvecs = bloch.eigsolve(
+        2,
+        K0_X,
+        G_MATRIX,
+        EPSILON,
+        tolerance=1e-6,
+        max_iters=20,
+        y0=singular_y0,
+    )
+
+    assert fake_rng.calls == 2
+    assert eigvals.shape == (2,)
+    assert eigvecs.shape == (2, H_SIZE)
+    assert numpy.isfinite(eigvals).all()
+    assert numpy.isfinite(eigvecs).all()
+
+
+def test_eigsolve_reconditions_large_trace_initial_subspace(monkeypatch: pytest.MonkeyPatch) -> None:
+    original_inv = bloch.numpy.linalg.inv
+    original_sqrtm = bloch.scipy.linalg.sqrtm
+    sqrtm_calls = 0
+    inv_calls = 0
+
+    def inv_with_large_first_trace(matrix: numpy.ndarray) -> numpy.ndarray:
+        nonlocal inv_calls
+        inv_calls += 1
+        if inv_calls == 1:
+            return numpy.eye(matrix.shape[0], dtype=complex) * 1e9
+        return original_inv(matrix)
+
+    def sqrtm_wrapper(matrix: numpy.ndarray) -> numpy.ndarray:
+        nonlocal sqrtm_calls
+        sqrtm_calls += 1
+        return original_sqrtm(matrix)
+
+    monkeypatch.setattr(bloch.numpy.linalg, 'inv', inv_with_large_first_trace)
+    monkeypatch.setattr(bloch.scipy.linalg, 'sqrtm', sqrtm_wrapper)
+
+    eigvals, eigvecs = bloch.eigsolve(
+        2,
+        K0_X,
+        G_MATRIX,
+        EPSILON,
+        tolerance=1e-6,
+        max_iters=20,
+        y0=Y0_TWO_MODE,
+    )
+
+    assert sqrtm_calls >= 2
+    assert eigvals.shape == (2,)
+    assert eigvecs.shape == (2, H_SIZE)
+    assert numpy.isfinite(eigvals).all()
+    assert numpy.isfinite(eigvecs).all()
+
+
+def test_eigsolve_qi_memoization_reuses_cached_theta(monkeypatch: pytest.MonkeyPatch) -> None:
+    def fake_minimize_scalar(func, method: str, bounds: tuple[float, float], options: dict[str, float]) -> SimpleNamespace:
+        theta = 0.3
+        first = func(theta)
+        second = func(theta)
+        assert_allclose(second, first)
+        return SimpleNamespace(fun=second, x=theta)
+
+    monkeypatch.setattr(bloch.scipy.optimize, 'minimize_scalar', fake_minimize_scalar)
+
+    eigvals, eigvecs = bloch.eigsolve(
+        1,
+        K0_X,
+        G_MATRIX,
+        EPSILON,
+        tolerance=1e-6,
+        max_iters=1,
+        y0=Y0,
+    )
+
+    assert eigvals.shape == (1,)
+    assert eigvecs.shape == (1, H_SIZE)
+    assert numpy.isfinite(eigvals).all()
+    assert numpy.isfinite(eigvecs).all()
+
+
+@pytest.mark.parametrize('theta', [numpy.pi / 2 - 1e-8, 1e-8])
+def test_eigsolve_qi_taylor_expansions_return_finite_modes(monkeypatch: pytest.MonkeyPatch, theta: float) -> None:
+    original_inv = bloch.numpy.linalg.inv
+    inv_calls = 0
+
+    def inv_raise_once_for_q(matrix: numpy.ndarray) -> numpy.ndarray:
+        nonlocal inv_calls
+        inv_calls += 1
+        if inv_calls == 3:
+            raise numpy.linalg.LinAlgError('forced singular Q')
+        return original_inv(matrix)
+
+    def fake_minimize_scalar(func, method: str, bounds: tuple[float, float], options: dict[str, float]) -> SimpleNamespace:
+        value = func(theta)
+        return SimpleNamespace(fun=value, x=theta)
+
+    monkeypatch.setattr(bloch.numpy.linalg, 'inv', inv_raise_once_for_q)
+    monkeypatch.setattr(bloch.scipy.optimize, 'minimize_scalar', fake_minimize_scalar)
+
+    eigvals, eigvecs = bloch.eigsolve(
+        1,
+        K0_X,
+        G_MATRIX,
+        EPSILON,
+        tolerance=1e-6,
+        max_iters=1,
+        y0=Y0,
+    )
+
+    assert eigvals.shape == (1,)
+    assert eigvecs.shape == (1, H_SIZE)
+    assert numpy.isfinite(eigvals).all()
+    assert numpy.isfinite(eigvecs).all()
+
+
+def test_eigsolve_qi_inexplicable_singularity_raises(monkeypatch: pytest.MonkeyPatch) -> None:
+    original_inv = bloch.numpy.linalg.inv
+    inv_calls = 0
+
+    def inv_raise_once_for_q(matrix: numpy.ndarray) -> numpy.ndarray:
+        nonlocal inv_calls
+        inv_calls += 1
+        if inv_calls == 3:
+            raise numpy.linalg.LinAlgError('forced singular Q')
+        return original_inv(matrix)
+
+    def fake_minimize_scalar(func, method: str, bounds: tuple[float, float], options: dict[str, float]) -> SimpleNamespace:
+        func(numpy.pi / 4)
+        raise AssertionError('unreachable after trace_func exception')
+
+    monkeypatch.setattr(bloch.numpy.linalg, 'inv', inv_raise_once_for_q)
+    monkeypatch.setattr(bloch.scipy.optimize, 'minimize_scalar', fake_minimize_scalar)
+
+    with pytest.raises(Exception, match='Inexplicable singularity in trace_func'):
+        bloch.eigsolve(
+            1,
+            K0_X,
+            G_MATRIX,
+            EPSILON,
+            tolerance=1e-6,
+            max_iters=1,
+            y0=Y0,
+        )
+
+
+def test_find_k_returns_vector_frequency_and_callbacks() -> None:
+    target_eigvals, _target_eigvecs = bloch.eigsolve(
+        1,
+        K0_X,
+        G_MATRIX,
+        EPSILON,
+        tolerance=1e-6,
+        max_iters=50,
+        y0=Y0,
+    )
+    target_frequency = float(numpy.sqrt(abs(numpy.real(target_eigvals[0]))))
+
+    solve_calls = 0
+    iter_calls = 0
+
+    def solve_callback(k_mag: float, eigvals: numpy.ndarray, eigvecs: numpy.ndarray, frequency: float) -> None:
+        nonlocal solve_calls
+        solve_calls += 1
+        assert eigvals.shape == (1,)
+        assert eigvecs.shape == (1, H_SIZE)
+        assert isinstance(k_mag, float)
+        assert isinstance(frequency, float)
+
+    def iter_callback() -> None:
+        nonlocal iter_calls
+        iter_calls += 1
+
+    found_k, found_frequency, eigvals, eigvecs = bloch.find_k(
+        target_frequency,
+        1e-4,
+        [1, 0, 0],
+        G_MATRIX,
+        EPSILON,
+        band=0,
+        k_bounds=(0.05, 0.15),
+        v0=Y0,
+        solve_callback=solve_callback,
+        iter_callback=iter_callback,
+    )
+
+    assert found_k.shape == (3,)
+    assert numpy.isfinite(found_k).all()
+    assert_close(numpy.cross(found_k, [1.0, 0.0, 0.0]), 0.0, atol=1e-12, rtol=1e-12)
+    assert_close(found_k, K0_X, atol=1e-4, rtol=1e-4)
+    assert abs(found_frequency - target_frequency) <= 1e-4
+    assert eigvals.shape == (1,)
+    assert eigvecs.shape == (1, H_SIZE)
+    assert numpy.isfinite(eigvals).all()
+    assert numpy.isfinite(eigvecs).all()
+    assert solve_calls > 0
+    assert iter_calls > 0

meanas/test/test_eigensolvers_numerics.py

@@ -0,0 +1,101 @@
+import numpy
+from numpy.linalg import norm
+import pytest
+
+from ._solver_cases import diagonal_eigen_case
+from .utils import assert_close
+from ..eigensolvers import power_iteration, rayleigh_quotient_iteration, signed_eigensolve
+
+
+def test_rayleigh_quotient_iteration_with_linear_operator() -> None:
+    case = diagonal_eigen_case()
+
+    def dense_solver(
+            shifted_operator,
+            rhs: numpy.ndarray,
+            ) -> numpy.ndarray:
+        basis = numpy.eye(case.operator.shape[0], dtype=complex)
+        columns = [shifted_operator.matvec(basis[:, ii]) for ii in range(case.operator.shape[0])]
+        dense_matrix = numpy.column_stack(columns)
+        return numpy.linalg.lstsq(dense_matrix, rhs, rcond=None)[0]
+
+    eigval, eigvec = rayleigh_quotient_iteration(
+        case.linear_operator,
+        case.guess_default,
+        iterations=8,
+        solver=dense_solver,
+        )
+
+    residual = norm(case.operator @ eigvec - eigval * eigvec)
+    assert abs(eigval - 3.0) < 1e-12
+    assert residual < 1e-12
+
+
+def test_signed_eigensolve_negative_returns_largest_negative_mode() -> None:
+    case = diagonal_eigen_case()
+
+    eigvals, eigvecs = signed_eigensolve(case.operator, how_many=1, negative=True)
+
+    assert eigvals.shape == (1,)
+    assert eigvecs.shape == (4, 1)
+    assert abs(eigvals[0] + 2.0) < 1e-12
+    assert abs(eigvecs[3, 0]) > 0.99
+
+
+def test_rayleigh_quotient_iteration_uses_default_linear_operator_solver() -> None:
+    case = diagonal_eigen_case()
+
+    eigval, eigvec = rayleigh_quotient_iteration(
+        case.linear_operator,
+        case.guess_default,
+        iterations=8,
+    )
+
+    residual = norm(case.operator @ eigvec - eigval * eigvec)
+    assert abs(eigval - 3.0) < 1e-12
+    assert residual < 1e-12
+
+
+def test_signed_eigensolve_linear_operator_fallback_returns_dominant_positive_mode() -> None:
+    case = diagonal_eigen_case()
+
+    eigvals, eigvecs = signed_eigensolve(case.linear_operator, how_many=1)
+
+    assert eigvals.shape == (1,)
+    assert eigvecs.shape == (4, 1)
+    assert_close(eigvals[0], 5.0, atol=1e-12, rtol=1e-12)
+    assert abs(eigvecs[0, 0]) > 0.99
+
+
+def test_power_iteration_finds_dominant_mode() -> None:
+    case = diagonal_eigen_case()
+    eigval, eigvec = power_iteration(case.operator, guess_vector=numpy.ones(4, dtype=complex), iterations=20)
+
+    assert eigval == pytest.approx(5.0, rel=1e-6)
+    assert abs(eigvec[0]) > abs(eigvec[1])
+
+
+def test_rayleigh_quotient_iteration_refines_known_sparse_mode() -> None:
+    case = diagonal_eigen_case()
+
+    def solver(matrix, rhs: numpy.ndarray) -> numpy.ndarray:
+        return numpy.linalg.lstsq(matrix.toarray(), rhs, rcond=None)[0]
+
+    eigval, eigvec = rayleigh_quotient_iteration(
+        case.operator,
+        case.guess_sparse,
+        iterations=8,
+        solver=solver,
+    )
+
+    residual = numpy.linalg.norm(case.operator @ eigvec - eigval * eigvec)
+    assert eigval == pytest.approx(3.0, rel=1e-6)
+    assert residual < 1e-8
+
+
+def test_signed_eigensolve_returns_largest_positive_modes() -> None:
+    case = diagonal_eigen_case()
+    eigvals, eigvecs = signed_eigensolve(case.operator, how_many=2)
+
+    assert_close(eigvals, [3.0, 5.0], atol=1e-6)
+    assert eigvecs.shape == (4, 2)

meanas/test/test_eme_numerics.py

@@ -0,0 +1,491 @@
+from typing import cast
+
+import numpy
+import pytest
+from scipy import sparse
+
+from ..fdmath import vec
+from ..fdfd import eme, waveguide_2d, waveguide_cyl
+from ._test_builders import complex_ramp, unit_dxes
+from .utils import assert_close
+
+
+SHAPE = (3, 2, 2)
+DXES = unit_dxes((2, 2))
+WAVENUMBERS_L = numpy.array([1.0, 0.8])
+WAVENUMBERS_R = numpy.array([0.9, 0.7])
+OMEGA = 1 / 1500
+REAL_DXES = unit_dxes((5, 5))
+
+
+def _mode(scale: float) -> tuple[numpy.ndarray, numpy.ndarray]:
+    e_field = complex_ramp(SHAPE, offset=1.0 + scale)
+    h_field = complex_ramp(SHAPE, scale=0.2, offset=2.0, imag_offset=0.05 * scale)
+    return vec(e_field), vec(h_field)
+
+
+def _mode_sets() -> tuple[list[tuple[numpy.ndarray, numpy.ndarray]], list[tuple[numpy.ndarray, numpy.ndarray]]]:
+    left_modes = [_mode(0.0), _mode(0.7)]
+    right_modes = [_mode(1.4), _mode(2.1)]
+    return left_modes, right_modes
+
+
+def _gain_only_tr(*args, **kwargs) -> tuple[numpy.ndarray, numpy.ndarray]:
+    return numpy.array([[2.0, 0.0], [0.0, 0.5]]), numpy.zeros((2, 2))
+
+
+def _gain_and_reflection_tr(*args, **kwargs) -> tuple[numpy.ndarray, numpy.ndarray]:
+    return numpy.array([[2.0, 0.0], [0.0, 0.5]]), numpy.array([[0.0, 1.0], [2.0, 0.0]])
+
+
+def _nonsymmetric_tr(left_marker: object):
+    def fake_get_tr(_eh_left, wavenumbers_left, _eh_right, _wavenumbers_right, **kwargs):
+        if wavenumbers_left is left_marker:
+            return (
+                numpy.array([[1.0, 2.0], [0.5, 1.0]]),
+                numpy.array([[0.0, 1.0], [2.0, 0.0]]),
+            )
+        return (
+            numpy.array([[1.0, -1.0], [0.0, 1.0]]),
+            numpy.array([[0.0, 0.5], [1.5, 0.0]]),
+        )
+
+    return fake_get_tr
+
+
+def _dummy_modes() -> tuple[list[tuple[numpy.ndarray, numpy.ndarray]], numpy.ndarray]:
+    return [_mode(0.0), _mode(0.7)], numpy.array([1.0, 0.5])
+
+
+def test_get_tr_returns_finite_bounded_transfer_matrices() -> None:
+    left_modes, right_modes = _mode_sets()
+
+    transmission, reflection = eme.get_tr(
+        left_modes,
+        WAVENUMBERS_L,
+        right_modes,
+        WAVENUMBERS_R,
+        dxes=DXES,
+    )
+
+    singular_values = numpy.linalg.svd(transmission, compute_uv=False)
+
+    assert transmission.shape == (2, 2)
+    assert reflection.shape == (2, 2)
+    assert numpy.isfinite(transmission).all()
+    assert numpy.isfinite(reflection).all()
+    assert (singular_values <= 1.0 + 1e-12).all()
+
+
+def test_get_abcd_matches_explicit_block_formula() -> None:
+    left_modes, right_modes = _mode_sets()
+    t12, r12 = eme.get_tr(left_modes, WAVENUMBERS_L, right_modes, WAVENUMBERS_R, dxes=DXES)
+    t21, r21 = eme.get_tr(right_modes, WAVENUMBERS_R, left_modes, WAVENUMBERS_L, dxes=DXES)
+    t21_inv = numpy.linalg.pinv(t21)
+
+    expected = numpy.block([
+        [t12 - r21 @ t21_inv @ r12, r21 @ t21_inv],
+        [-t21_inv @ r12, t21_inv],
+    ])
+    abcd = eme.get_abcd(left_modes, WAVENUMBERS_L, right_modes, WAVENUMBERS_R, dxes=DXES)
+
+    assert sparse.issparse(abcd)
+    assert abcd.shape == (4, 4)
+    assert_close(abcd.toarray(), expected)
+
+
+def test_get_s_plain_matches_block_assembly_from_get_tr() -> None:
+    left_modes, right_modes = _mode_sets()
+    t12, r12 = eme.get_tr(left_modes, WAVENUMBERS_L, right_modes, WAVENUMBERS_R, dxes=DXES)
+    t21, r21 = eme.get_tr(right_modes, WAVENUMBERS_R, left_modes, WAVENUMBERS_L, dxes=DXES)
+    expected = numpy.block([[r12, t12], [t21, r21]])
+
+    ss = eme.get_s(left_modes, WAVENUMBERS_L, right_modes, WAVENUMBERS_R, dxes=DXES)
+
+    assert ss.shape == (4, 4)
+    assert numpy.isfinite(ss).all()
+    assert_close(ss, expected)
+
+
+def test_get_s_force_nogain_caps_singular_values(monkeypatch) -> None:
+    monkeypatch.setattr(eme, 'get_tr', _gain_only_tr)
+    modes, wavenumbers = _dummy_modes()
+
+    plain_s = eme.get_s(modes, wavenumbers, modes, wavenumbers)
+    clipped_s = eme.get_s(modes, wavenumbers, modes, wavenumbers, force_nogain=True)
+
+    plain_singular_values = numpy.linalg.svd(plain_s, compute_uv=False)
+    clipped_singular_values = numpy.linalg.svd(clipped_s, compute_uv=False)
+
+    assert plain_singular_values.max() > 1.0
+    assert (clipped_singular_values <= 1.0 + 1e-12).all()
+    assert numpy.isfinite(clipped_s).all()
+
+
+def test_get_s_force_reciprocal_symmetrizes_output(monkeypatch) -> None:
+    left = numpy.array([1.0, 0.5])
+    right = numpy.array([0.9, 0.4])
+    modes, _wavenumbers = _dummy_modes()
+
+    monkeypatch.setattr(eme, 'get_tr', _nonsymmetric_tr(left))
+    ss = eme.get_s(modes, left, modes, right, force_reciprocal=True)
+
+    assert_close(ss, ss.T)
+
+
+def test_get_s_force_nogain_and_reciprocal_returns_finite_output(monkeypatch) -> None:
+    monkeypatch.setattr(eme, 'get_tr', _gain_and_reflection_tr)
+    modes, wavenumbers = _dummy_modes()
+    ss = eme.get_s(modes, wavenumbers, modes, wavenumbers, force_nogain=True, force_reciprocal=True)
+
+    assert ss.shape == (4, 4)
+    assert numpy.isfinite(ss).all()
+    assert_close(ss, ss.T)
+    assert (numpy.linalg.svd(ss, compute_uv=False) <= 1.0 + 1e-12).all()
+
+
+def test_get_tr_rejects_length_mismatches() -> None:
+    left_modes, right_modes = _mode_sets()
+
+    with pytest.raises(ValueError, match='same length'):
+        eme.get_tr(left_modes[:1], WAVENUMBERS_L, right_modes, WAVENUMBERS_R, dxes=DXES)
+
+
+def test_get_tr_rejects_malformed_mode_tuples() -> None:
+    bad_modes = cast(list[tuple[numpy.ndarray, numpy.ndarray]], [(numpy.ones(4, dtype=complex),)])
+
+    with pytest.raises(ValueError, match='2-tuple'):
+        eme.get_tr(bad_modes, [1.0], bad_modes, [1.0], dxes=DXES)
+
+
+def test_get_tr_rejects_incompatible_field_shapes() -> None:
+    left_modes = [(numpy.ones(4, dtype=complex), numpy.ones(4, dtype=complex))]
+    right_modes = [(numpy.ones(6, dtype=complex), numpy.ones(6, dtype=complex))]
+
+    with pytest.raises(ValueError, match='same E/H shapes'):
+        eme.get_tr(left_modes, [1.0], right_modes, [1.0], dxes=DXES)
+
+
+def _build_real_epsilon() -> numpy.ndarray:
+    epsilon = numpy.ones((3, 5, 5), dtype=float)
+    epsilon[:, 2, 1] = 2.0
+    return vec(epsilon)
+
+
+def _build_straight_mode() -> tuple[tuple[numpy.ndarray, numpy.ndarray], complex, numpy.ndarray]:
+    epsilon = _build_real_epsilon()
+    e_xy, wavenumber = waveguide_2d.solve_mode(
+        0,
+        omega=OMEGA,
+        dxes=REAL_DXES,
+        epsilon=epsilon,
+    )
+    e_field, h_field = waveguide_2d.normalized_fields_e(
+        e_xy,
+        wavenumber=wavenumber,
+        omega=OMEGA,
+        dxes=REAL_DXES,
+        epsilon=epsilon,
+    )
+    return (e_field, h_field), wavenumber, epsilon
+
+
+def _build_bend_mode() -> tuple[tuple[numpy.ndarray, numpy.ndarray], complex]:
+    epsilon = vec(numpy.ones((3, 5, 5), dtype=float))
+    rmin = 10.0
+    e_xy, angular_wavenumber = waveguide_cyl.solve_mode(
+        0,
+        omega=OMEGA,
+        dxes=REAL_DXES,
+        epsilon=epsilon,
+        rmin=rmin,
+    )
+    linear_wavenumber = waveguide_cyl.linear_wavenumbers(
+        [e_xy],
+        [angular_wavenumber],
+        epsilon=epsilon,
+        dxes=REAL_DXES,
+        rmin=rmin,
+    )[0]
+    e_field, h_field = waveguide_cyl.normalized_fields_e(
+        e_xy,
+        angular_wavenumber=angular_wavenumber,
+        omega=OMEGA,
+        dxes=REAL_DXES,
+        epsilon=epsilon,
+        rmin=rmin,
+    )
+    return (e_field, h_field), linear_wavenumber
+
+
+def _build_uniform_mode(index: float) -> tuple[tuple[numpy.ndarray, numpy.ndarray], complex]:
+    area = 25.0
+    e_field = numpy.zeros((3, 5, 5), dtype=complex)
+    h_field = numpy.zeros((3, 5, 5), dtype=complex)
+    e_field[0] = numpy.sqrt(2.0 / (index * area))
+    h_field[1] = numpy.sqrt(2.0 * index / area)
+    return (vec(e_field), vec(h_field)), complex(index * OMEGA)
+
+
+def _interface_s(n_left: float, n_right: float) -> numpy.ndarray:
+    left_mode, left_beta = _build_uniform_mode(n_left)
+    right_mode, right_beta = _build_uniform_mode(n_right)
+    return eme.get_s([left_mode], [left_beta], [right_mode], [right_beta], dxes=REAL_DXES)
+
+
+def _interface_abcd(n_left: float, n_right: float) -> numpy.ndarray:
+    left_mode, left_beta = _build_uniform_mode(n_left)
+    right_mode, right_beta = _build_uniform_mode(n_right)
+    return eme.get_abcd([left_mode], [left_beta], [right_mode], [right_beta], dxes=REAL_DXES).toarray()
+
+
+def _expected_interface_s(n_left: float, n_right: float) -> numpy.ndarray:
+    reflection = (n_left - n_right) / (n_left + n_right)
+    transmission = 2 * numpy.sqrt(n_left * n_right) / (n_left + n_right)
+    return numpy.array(
+        [
+            [reflection, transmission],
+            [transmission, -reflection],
+        ],
+        dtype=complex,
+    )
+
+
+def _propagation_abcd(beta: complex, length: float) -> numpy.ndarray:
+    phase = numpy.exp(-1j * beta * length)
+    return numpy.array(
+        [
+            [phase, 0.0],
+            [0.0, phase ** -1],
+        ],
+        dtype=complex,
+    )
+
+
+def _abcd_to_s(abcd: numpy.ndarray) -> numpy.ndarray:
+    aa = abcd[0, 0]
+    bb = abcd[0, 1]
+    cc = abcd[1, 0]
+    dd = abcd[1, 1]
+    t21 = 1.0 / dd
+    r21 = bb / dd
+    r12 = -cc / dd
+    t12 = aa - bb * cc / dd
+    return numpy.array(
+        [
+            [r12, t12],
+            [t21, r21],
+        ],
+        dtype=complex,
+    )
+
+
+def _expected_bragg_reflector_s(n_low: float, n_high: float, periods: int) -> numpy.ndarray:
+    ratio = n_high / n_low
+    reflection = (1 - ratio ** (2 * periods)) / (1 + ratio ** (2 * periods))
+    transmission = ((-1) ** periods) * 2 * ratio ** periods / (1 + ratio ** (2 * periods))
+    return numpy.array(
+        [
+            [reflection, transmission],
+            [transmission, -reflection],
+        ],
+        dtype=complex,
+    )
+
+
+def test_get_s_is_near_identity_for_identical_solved_straight_modes() -> None:
+    mode, wavenumber, _epsilon = _build_straight_mode()
+
+    ss = eme.get_s([mode], [wavenumber], [mode], [wavenumber], dxes=REAL_DXES)
+
+    assert ss.shape == (2, 2)
+    assert numpy.isfinite(ss).all()
+    assert abs(ss[0, 0]) < 1e-6
+    assert abs(ss[1, 1]) < 1e-6
+    assert abs(abs(ss[0, 1]) - 1.0) < 1e-6
+    assert abs(abs(ss[1, 0]) - 1.0) < 1e-6
+    assert numpy.linalg.svd(ss, compute_uv=False).max() <= 1.0 + 1e-10
+
+
+def test_get_s_returns_finite_passive_output_for_small_straight_to_bend_fixture() -> None:
+    straight_mode, straight_wavenumber, _epsilon = _build_straight_mode()
+    bend_mode, bend_wavenumber = _build_bend_mode()
+
+    ss = eme.get_s([straight_mode], [straight_wavenumber], [bend_mode], [bend_wavenumber], dxes=REAL_DXES)
+
+    assert ss.shape == (2, 2)
+    assert numpy.isfinite(ss).all()
+    assert numpy.linalg.svd(ss, compute_uv=False).max() <= 1.0 + 1e-10
+
+
+def test_get_s_matches_analytic_fresnel_interface_for_uniform_one_mode_ports() -> None:
+    """
+    For power-normalized one-mode ports at normal incidence, the interface matrix is
+
+        r12 = (n_left - n_right) / (n_left + n_right)
+        r21 = -r12
+        t12 = t21 = 2 * sqrt(n_left * n_right) / (n_left + n_right)
+
+    so
+
+        S = [[r12, t12], [t21, r21]].
+    """
+    ss = _interface_s(1.0, 2.0)
+    expected = _expected_interface_s(1.0, 2.0)
+
+    assert ss.shape == (2, 2)
+    assert numpy.isfinite(ss).all()
+    assert_close(ss, expected, atol=1e-6, rtol=1e-6)
+    assert numpy.linalg.svd(ss, compute_uv=False).max() <= 1.0 + 1e-10
+
+
+def test_quarter_wave_matching_layer_is_nearly_reflectionless_at_design_frequency() -> None:
+    """
+    A single quarter-wave matching layer with
+
+        n1 = sqrt(n0 * n2),  beta1 * L = pi / 2
+
+    cancels the two interface reflections at the design wavelength, so the
+    normal-incidence stack should satisfy `r = 0` and `|t| = 1`.
+    """
+    n0 = 1.0
+    n1 = numpy.sqrt(2.0)
+    n2 = 2.0
+    interface_01 = _interface_abcd(n0, n1)
+    interface_12 = _interface_abcd(n1, n2)
+    _mode_1, beta_1 = _build_uniform_mode(float(n1))
+    quarter_wave_length = numpy.pi / (2 * numpy.real(beta_1))
+
+    stack_abcd = interface_01 @ _propagation_abcd(beta_1, quarter_wave_length) @ interface_12
+    ss = _abcd_to_s(stack_abcd)
+
+    assert ss.shape == (2, 2)
+    assert numpy.isfinite(ss).all()
+    assert abs(ss[0, 0]) < 1e-12
+    assert abs(ss[1, 1]) < 1e-12
+    assert abs(abs(ss[0, 1]) - 1.0) < 1e-12
+    assert abs(abs(ss[1, 0]) - 1.0) < 1e-12
+    assert numpy.linalg.svd(ss, compute_uv=False).max() <= 1.0 + 1e-10
+
+
+def test_quarter_wave_ar_layer_reduces_reflection_relative_to_abrupt_interface() -> None:
+    """
+    Compare the abrupt interface `n0 -> n2` against the quarter-wave matching-layer
+    stack `n0 -> sqrt(n0 n2) -> n2` at the same design wavelength.
+
+    For the canonical `n0 = 1`, `n2 = 2` case, the abrupt interface has
+
+        |r_abrupt| = |(n0 - n2) / (n0 + n2)| = 1 / 3,
+
+    while the quarter-wave matching layer should cancel the interface reflections
+    so that `|r_ar|` is essentially zero and `|t_ar|` is correspondingly larger.
+    """
+    n0 = 1.0
+    n2 = 2.0
+    abrupt = _interface_s(n0, n2)
+
+    n1 = numpy.sqrt(n0 * n2)
+    interface_01 = _interface_abcd(n0, n1)
+    interface_12 = _interface_abcd(n1, n2)
+    _mode_1, beta_1 = _build_uniform_mode(float(n1))
+    quarter_wave_length = numpy.pi / (2 * numpy.real(beta_1))
+    ar_stack = _abcd_to_s(interface_01 @ _propagation_abcd(beta_1, quarter_wave_length) @ interface_12)
+
+    abrupt_reflection = abs(abrupt[0, 0])
+    abrupt_transmission = abs(abrupt[1, 0])
+    ar_reflection = abs(ar_stack[0, 0])
+    ar_transmission = abs(ar_stack[1, 0])
+
+    assert numpy.linalg.svd(abrupt, compute_uv=False).max() <= 1.0 + 1e-10
+    assert numpy.linalg.svd(ar_stack, compute_uv=False).max() <= 1.0 + 1e-10
+    assert ar_reflection < abrupt_reflection
+    assert ar_transmission > abrupt_transmission
+    assert ar_reflection < 1e-12
+    assert abs(abrupt_reflection - (1.0 / 3.0)) < 1e-12
+
+
+def test_half_wave_uniform_slab_restores_unit_transmission_between_matched_media() -> None:
+    """
+    For matched exterior media `n0 = n2`, a half-wave slab with
+
+        beta1 * L = pi
+
+    contributes only a global phase, so the stack returns to `r = 0` and
+    `|t| = 1` at the design wavelength.
+    """
+    n0 = 1.0
+    n1 = 2.0
+    interface_01 = _interface_abcd(n0, n1)
+    interface_10 = _interface_abcd(n1, n0)
+    _mode_1, beta_1 = _build_uniform_mode(n1)
+    half_wave_length = numpy.pi / numpy.real(beta_1)
+
+    stack_abcd = interface_01 @ _propagation_abcd(beta_1, half_wave_length) @ interface_10
+    ss = _abcd_to_s(stack_abcd)
+
+    assert ss.shape == (2, 2)
+    assert numpy.isfinite(ss).all()
+    assert abs(ss[0, 0]) < 1e-12
+    assert abs(ss[1, 1]) < 1e-12
+    assert abs(abs(ss[0, 1]) - 1.0) < 1e-12
+    assert abs(abs(ss[1, 0]) - 1.0) < 1e-12
+    assert numpy.linalg.svd(ss, compute_uv=False).max() <= 1.0 + 1e-10
+
+
+def test_strong_uniform_index_mismatch_behaves_like_near_termination() -> None:
+    """
+    In the large-index-ratio limit, the same Fresnel formulas approach a hard-wall
+    reflector:
+
+        |r| -> 1,  |t| -> 0  as  n_right / n_left -> infinity.
+    """
+    ss = _interface_s(1.0, 100.0)
+    expected = _expected_interface_s(1.0, 100.0)
+
+    assert ss.shape == (2, 2)
+    assert numpy.isfinite(ss).all()
+    assert_close(ss, expected, atol=1e-6, rtol=1e-6)
+    assert abs(ss[0, 0]) > 0.9
+    assert abs(ss[1, 0]) < 0.25
+    assert numpy.linalg.svd(ss, compute_uv=False).max() <= 1.0 + 1e-10
+
+
+def test_quarter_wave_bragg_reflector_matches_closed_form_stopband_response() -> None:
+    """
+    For `N` quarter-wave high/low periods at the Bragg wavelength with identical
+    low-index incident and exit media (`n0 = ns = n_low`),
+
+        M_pair = diag(-(n_low / n_high), -(n_high / n_low))
+        M_stack = M_pair ** N
+
+    which yields the closed-form scattering amplitudes
+
+        r = (1 - (n_high / n_low) ** (2N)) / (1 + (n_high / n_low) ** (2N))
+        t = 2 * (n_high / n_low) ** N / (1 + (n_high / n_low) ** (2N)).
+
+    The reflector should therefore sit deep in the stopband with `|r|` near 1 and
+    `|t|` correspondingly small.
+    """
+    n_low = 1.0
+    n_high = 2.0
+    periods = 5
+    interface_lh = _interface_abcd(n_low, n_high)
+    interface_hl = _interface_abcd(n_high, n_low)
+    _mode_h, beta_h = _build_uniform_mode(n_high)
+    _mode_l, beta_l = _build_uniform_mode(n_low)
+    quarter_wave_high = numpy.pi / (2 * numpy.real(beta_h))
+    quarter_wave_low = numpy.pi / (2 * numpy.real(beta_l))
+
+    stack_abcd = numpy.eye(2, dtype=complex)
+    for _ in range(periods):
+        stack_abcd = stack_abcd @ interface_lh @ _propagation_abcd(beta_h, quarter_wave_high)
+        stack_abcd = stack_abcd @ interface_hl @ _propagation_abcd(beta_l, quarter_wave_low)
+    ss = _abcd_to_s(stack_abcd)
+    expected = _expected_bragg_reflector_s(n_low, n_high, periods)
+
+    assert ss.shape == (2, 2)
+    assert numpy.isfinite(ss).all()
+    assert_close(ss, expected, atol=1e-12, rtol=1e-12)
+    assert abs(ss[0, 0]) > 0.99
+    assert abs(ss[1, 0]) < 0.1
+    assert numpy.linalg.svd(ss, compute_uv=False).max() <= 1.0 + 1e-10

meanas/test/test_examples_smoke.py

@@ -0,0 +1,47 @@
+from pathlib import Path
+import os
+import subprocess
+import sys
+
+import pytest
+
+
+pytestmark = pytest.mark.complete
+
+REPO_ROOT = Path(__file__).resolve().parents[2]
+
+
+def _run_example(example_name: str, tmp_path: Path) -> subprocess.CompletedProcess[str]:
+    env = os.environ.copy()
+    env['MPLBACKEND'] = 'Agg'
+    env['MPLCONFIGDIR'] = str(tmp_path / f'mpl-{example_name}')
+    return subprocess.run(
+        [sys.executable, str(REPO_ROOT / 'examples' / example_name)],
+        cwd=REPO_ROOT,
+        env=env,
+        text=True,
+        capture_output=True,
+        check=False,
+        timeout=60,
+    )
+
+
+def test_eme_example_smoke_runs(tmp_path: Path) -> None:
+    pytest.importorskip('matplotlib')
+
+    result = _run_example('eme.py', tmp_path)
+
+    assert result.returncode == 0, result.stdout + result.stderr
+    assert 'left effective indices:' in result.stdout
+    assert 'fundamental left-to-right transmission' in result.stdout
+
+
+def test_eme_bend_example_smoke_runs(tmp_path: Path) -> None:
+    pytest.importorskip('matplotlib')
+    pytest.importorskip('skrf')
+
+    result = _run_example('eme_bend.py', tmp_path)
+
+    assert result.returncode == 0, result.stdout + result.stderr
+    assert 'straight effective indices:' in result.stdout
+    assert 'cascaded bend through power' in result.stdout

meanas/test/test_fdfd.py

@@ -1,4 +1,4 @@
-from typing import Iterable
+# ruff: noqa: ARG001
 import dataclasses
 import pytest       # type: ignore
 import numpy
@@ -6,7 +6,7 @@ from numpy.typing import NDArray
 #from numpy.testing import assert_allclose, assert_array_equal
 
 from .. import fdfd
-from ..fdmath import vec, unvec
+from ..fdmath import vec, unvec, vcfdfield, vfdfield, dx_lists_t
 from .utils import assert_close  # , assert_fields_close
 from .conftest import FixtureRequest
 
@@ -61,24 +61,24 @@ def test_poynting_planes(sim: 'FDResult') -> None:
 # Also see conftest.py
 
 @pytest.fixture(params=[1 / 1500])
-def omega(request: FixtureRequest) -> Iterable[float]:
-    yield request.param
+def omega(request: FixtureRequest) -> float:
+    return request.param
 
 
 @pytest.fixture(params=[None])
-def pec(request: FixtureRequest) -> Iterable[NDArray[numpy.float64] | None]:
-    yield request.param
+def pec(request: FixtureRequest) -> NDArray[numpy.float64] | None:
+    return request.param
 
 
 @pytest.fixture(params=[None])
-def pmc(request: FixtureRequest) -> Iterable[NDArray[numpy.float64] | None]:
-    yield request.param
+def pmc(request: FixtureRequest) -> NDArray[numpy.float64] | None:
+    return request.param
 
 
 #@pytest.fixture(scope='module',
 #                params=[(25, 5, 5)])
-#def shape(request):
-#    yield (3, *request.param)
+#def shape(request: FixtureRequest):
+#    return (3, *request.param)
 
 
 @pytest.fixture(params=['diag'])        # 'center'
@@ -86,7 +86,7 @@ def j_distribution(
         request: FixtureRequest,
         shape: tuple[int, ...],
         j_mag: float,
-        ) -> Iterable[NDArray[numpy.float64]]:
+        ) -> NDArray[numpy.float64]:
     j = numpy.zeros(shape, dtype=complex)
     center_mask = numpy.zeros(shape, dtype=bool)
     center_mask[:, shape[1] // 2, shape[2] // 2, shape[3] // 2] = True
@@ -96,22 +96,22 @@ def j_distribution(
     elif request.param == 'diag':
         j[numpy.roll(center_mask, [1, 1, 1], axis=(1, 2, 3))] = (1 + 1j) * j_mag
         j[numpy.roll(center_mask, [-1, -1, -1], axis=(1, 2, 3))] = (1 - 1j) * j_mag
-    yield j
+    return j
 
 
 @dataclasses.dataclass()
 class FDResult:
     shape: tuple[int, ...]
-    dxes: list[list[NDArray[numpy.float64]]]
-    epsilon: NDArray[numpy.float64]
+    dxes: dx_lists_t
+    epsilon: vfdfield
     omega: complex
-    j: NDArray[numpy.complex128]
-    e: NDArray[numpy.complex128]
-    pmc: NDArray[numpy.float64] | None
-    pec: NDArray[numpy.float64] | None
+    j: vcfdfield
+    e: vcfdfield
+    pmc: vfdfield | None
+    pec: vfdfield | None
 
 
-@pytest.fixture()
+@pytest.fixture
 def sim(
         request: FixtureRequest,
         shape: tuple[int, ...],
@@ -141,11 +141,11 @@ def sim(
     j_vec = vec(j_distribution)
     eps_vec = vec(epsilon)
     e_vec = fdfd.solvers.generic(
-        J=j_vec,
-        omega=omega,
-        dxes=dxes,
-        epsilon=eps_vec,
-        matrix_solver_opts={'atol': 1e-15, 'tol': 1e-11},
+        J = j_vec,
+        omega = omega,
+        dxes = dxes,
+        epsilon = eps_vec,
+        matrix_solver_opts = dict(atol=1e-15, rtol=1e-11),
         )
     e = unvec(e_vec, shape[1:])
 

meanas/test/test_fdfd_algebra_helpers.py

@@ -0,0 +1,213 @@
+import numpy
+
+from ..fdmath import vec, unvec
+from ..fdmath import functional as fd_functional
+from ..fdfd import operators, scpml
+from ._fdfd_case import (
+    BOUNDARY_DXES,
+    BOUNDARY_EPSILON,
+    BOUNDARY_FIELD,
+    BOUNDARY_SHAPE,
+    DXES,
+    EPSILON,
+    E_FIELD,
+    MU,
+    H_FIELD,
+    OMEGA,
+    PEC,
+    PMC,
+    SHAPE,
+    apply_fdfd_matrix,
+)
+from .utils import assert_close, assert_fields_close
+
+
+def _dense_e_full(mu: numpy.ndarray | None) -> numpy.ndarray:
+    ce = fd_functional.curl_forward(DXES[0])
+    ch = fd_functional.curl_back(DXES[1])
+    pe = numpy.where(PEC, 0.0, 1.0)
+    pm = numpy.where(PMC, 0.0, 1.0)
+
+    masked_e = pe * E_FIELD
+    curl_term = ce(masked_e)
+    if mu is not None:
+        curl_term = curl_term / mu
+    curl_term = pm * curl_term
+    curl_term = ch(curl_term)
+    return pe * (curl_term - OMEGA**2 * EPSILON * masked_e)
+
+
+def _dense_h_full(mu: numpy.ndarray | None) -> numpy.ndarray:
+    ce = fd_functional.curl_forward(DXES[0])
+    ch = fd_functional.curl_back(DXES[1])
+    pe = numpy.where(PEC, 0.0, 1.0)
+    pm = numpy.where(PMC, 0.0, 1.0)
+    magnetic = numpy.ones_like(EPSILON) if mu is None else mu
+
+    masked_h = pm * H_FIELD
+    curl_term = ch(masked_h)
+    curl_term = pe * (curl_term / EPSILON)
+    curl_term = ce(curl_term)
+    return pm * (curl_term - OMEGA**2 * magnetic * masked_h)
+
+
+def _normalized_distance(u: numpy.ndarray, size: int, thickness: int) -> numpy.ndarray:
+    return ((thickness - u).clip(0) + (u - (size - thickness)).clip(0)) / thickness
+
+
+def test_h_full_matches_dense_reference_with_and_without_mu() -> None:
+    for mu in (None, MU):
+        matrix_result = apply_fdfd_matrix(
+            operators.h_full(OMEGA, DXES, vec(EPSILON), None if mu is None else vec(mu), vec(PEC), vec(PMC)),
+            H_FIELD,
+        )
+        dense_result = _dense_h_full(mu)
+        assert_fields_close(matrix_result, dense_result, atol=1e-10, rtol=1e-10)
+
+
+def test_e_full_matches_dense_reference_with_masks() -> None:
+    for mu in (None, MU):
+        matrix_result = apply_fdfd_matrix(
+            operators.e_full(OMEGA, DXES, vec(EPSILON), None if mu is None else vec(mu), vec(PEC), vec(PMC)),
+            E_FIELD,
+        )
+        dense_result = _dense_e_full(mu)
+        assert_fields_close(matrix_result, dense_result, atol=1e-10, rtol=1e-10)
+
+
+def test_h_full_without_masks_matches_dense_reference() -> None:
+    ce = fd_functional.curl_forward(DXES[0])
+    ch = fd_functional.curl_back(DXES[1])
+    dense_result = ce(ch(H_FIELD) / EPSILON) - OMEGA**2 * MU * H_FIELD
+    matrix_result = apply_fdfd_matrix(
+        operators.h_full(OMEGA, DXES, vec(EPSILON), vec(MU)),
+        H_FIELD,
+    )
+    assert_fields_close(matrix_result, dense_result, atol=1e-10, rtol=1e-10)
+
+
+def test_eh_full_matches_manual_block_operator_with_masks() -> None:
+    pe = numpy.where(PEC, 0.0, 1.0)
+    pm = numpy.where(PMC, 0.0, 1.0)
+    ce = fd_functional.curl_forward(DXES[0])
+    ch = fd_functional.curl_back(DXES[1])
+
+    matrix_result = operators.eh_full(OMEGA, DXES, vec(EPSILON), vec(MU), vec(PEC), vec(PMC)) @ numpy.concatenate(
+        [vec(E_FIELD), vec(H_FIELD)],
+    )
+    matrix_e, matrix_h = (unvec(part, SHAPE) for part in numpy.split(matrix_result, 2))
+
+    dense_e = pe * ch(pm * H_FIELD) - pe * (1j * OMEGA * EPSILON * (pe * E_FIELD))
+    dense_h = pm * ce(pe * E_FIELD) + pm * (1j * OMEGA * MU * (pm * H_FIELD))
+
+    assert_fields_close(matrix_e, dense_e, atol=1e-10, rtol=1e-10)
+    assert_fields_close(matrix_h, dense_h, atol=1e-10, rtol=1e-10)
+
+
+def test_e2h_pmc_mask_matches_masked_unmasked_result() -> None:
+    pmc_complement = numpy.where(PMC, 0.0, 1.0)
+    unmasked = apply_fdfd_matrix(operators.e2h(OMEGA, DXES, vec(MU)), E_FIELD)
+    masked = apply_fdfd_matrix(operators.e2h(OMEGA, DXES, vec(MU), vec(PMC)), E_FIELD)
+
+    assert_fields_close(masked, pmc_complement * unmasked, atol=1e-10, rtol=1e-10)
+
+
+def test_poynting_h_cross_matches_negative_e_cross_relation() -> None:
+    h_cross_e = apply_fdfd_matrix(operators.poynting_h_cross(vec(H_FIELD), DXES), E_FIELD)
+    e_cross_h = apply_fdfd_matrix(operators.poynting_e_cross(vec(E_FIELD), DXES), H_FIELD)
+
+    assert_fields_close(h_cross_e, -e_cross_h, atol=1e-10, rtol=1e-10)
+
+
+def test_e_boundary_source_interior_mask_is_independent_of_periodic_edges() -> None:
+    mask = numpy.zeros((3, *BOUNDARY_SHAPE), dtype=float)
+    mask[:, 1, 1, 1] = 1.0
+
+    periodic = operators.e_boundary_source(vec(mask), OMEGA, BOUNDARY_DXES, vec(BOUNDARY_EPSILON), periodic_mask_edges=True)
+    mirrored = operators.e_boundary_source(vec(mask), OMEGA, BOUNDARY_DXES, vec(BOUNDARY_EPSILON), periodic_mask_edges=False)
+
+    assert_close(periodic.toarray(), mirrored.toarray())
+
+
+def test_e_boundary_source_periodic_edges_add_opposite_face_response() -> None:
+    mask = numpy.zeros((3, *BOUNDARY_SHAPE), dtype=float)
+    mask[:, 0, 1, 1] = 1.0
+
+    periodic = operators.e_boundary_source(vec(mask), OMEGA, BOUNDARY_DXES, vec(BOUNDARY_EPSILON), periodic_mask_edges=True)
+    mirrored = operators.e_boundary_source(vec(mask), OMEGA, BOUNDARY_DXES, vec(BOUNDARY_EPSILON), periodic_mask_edges=False)
+    diff = unvec((periodic - mirrored) @ vec(BOUNDARY_FIELD), BOUNDARY_SHAPE)
+
+    assert numpy.isfinite(diff).all()
+    assert_close(diff[:, 1:-1, :, :], 0.0)
+    assert numpy.linalg.norm(diff[:, -1, :, :]) > 0
+
+
+def test_prepare_s_function_matches_closed_form_polynomial() -> None:
+    ln_r = -12.0
+    order = 3.0
+    distances = numpy.array([0.0, 0.25, 0.5, 1.0])
+    s_function = scpml.prepare_s_function(ln_R=ln_r, m=order)
+    expected = (order + 1) * ln_r / 2 * distances**order
+
+    assert_close(s_function(distances), expected)
+
+
+def test_uniform_grid_scpml_matches_expected_stretch_profile() -> None:
+    s_function = scpml.prepare_s_function(ln_R=-12.0, m=3.0)
+    dxes = scpml.uniform_grid_scpml((6, 4, 3), (2, 0, 1), omega=2.0, epsilon_effective=4.0, s_function=s_function)
+    correction = numpy.sqrt(4.0) * 2.0
+
+    for axis, size, thickness in ((0, 6, 2), (2, 3, 1)):
+        grid = numpy.arange(size, dtype=float)
+        expected_a = 1 + 1j * s_function(_normalized_distance(grid, size, thickness)) / correction
+        expected_b = 1 + 1j * s_function(_normalized_distance(grid + 0.5, size, thickness)) / correction
+        assert_close(dxes[0][axis], expected_a)
+        assert_close(dxes[1][axis], expected_b)
+
+    assert_close(dxes[0][1], 1.0)
+    assert_close(dxes[1][1], 1.0)
+    assert numpy.isfinite(dxes[0][0]).all()
+    assert numpy.isfinite(dxes[1][0]).all()
+
+
+def test_uniform_grid_scpml_default_s_function_matches_explicit_default() -> None:
+    implicit = scpml.uniform_grid_scpml((6, 4, 3), (2, 0, 1), omega=2.0)
+    explicit = scpml.uniform_grid_scpml((6, 4, 3), (2, 0, 1), omega=2.0, s_function=scpml.prepare_s_function())
+
+    for implicit_group, explicit_group in zip(implicit, explicit, strict=True):
+        for implicit_axis, explicit_axis in zip(implicit_group, explicit_group, strict=True):
+            assert_close(implicit_axis, explicit_axis)
+
+
+def test_stretch_with_scpml_only_modifies_requested_front_edge() -> None:
+    s_function = scpml.prepare_s_function(ln_R=-12.0, m=3.0)
+    base = [[numpy.ones(6), numpy.ones(4), numpy.ones(3)] for _ in range(2)]
+    stretched = scpml.stretch_with_scpml(base, axis=0, polarity=1, omega=2.0, epsilon_effective=4.0, thickness=2, s_function=s_function)
+
+    assert_close(stretched[0][0][2:], 1.0)
+    assert_close(stretched[1][0][2:], 1.0)
+    assert_close(stretched[0][0][-2:], 1.0)
+    assert_close(stretched[1][0][-2:], 1.0)
+    assert numpy.linalg.norm(stretched[0][0][:2] - 1.0) > 0
+    assert numpy.linalg.norm(stretched[1][0][:2] - 1.0) > 0
+
+
+def test_stretch_with_scpml_only_modifies_requested_back_edge() -> None:
+    s_function = scpml.prepare_s_function(ln_R=-12.0, m=3.0)
+    base = [[numpy.ones(6), numpy.ones(4), numpy.ones(3)] for _ in range(2)]
+    stretched = scpml.stretch_with_scpml(base, axis=0, polarity=-1, omega=2.0, epsilon_effective=4.0, thickness=2, s_function=s_function)
+
+    assert_close(stretched[0][0][:4], 1.0)
+    assert_close(stretched[1][0][:4], 1.0)
+    assert numpy.linalg.norm(stretched[0][0][-2:] - 1.0) > 0
+    assert numpy.linalg.norm(stretched[1][0][-2:] - 1.0) > 0
+
+
+def test_stretch_with_scpml_thickness_zero_is_noop() -> None:
+    s_function = scpml.prepare_s_function(ln_R=-12.0, m=3.0)
+    base = [[numpy.ones(6), numpy.ones(4), numpy.ones(3)] for _ in range(2)]
+    stretched = scpml.stretch_with_scpml(base, axis=0, polarity=-1, omega=2.0, epsilon_effective=4.0, thickness=0, s_function=s_function)
+
+    for grid_group in stretched:
+        for axis_grid in grid_group:
+            assert_close(axis_grid, 1.0)

meanas/test/test_fdfd_farfield.py

@@ -0,0 +1,100 @@
+import numpy
+import pytest
+
+from ..fdfd import farfield
+
+
+NEAR_SHAPE = (2, 3)
+E_NEAR = [numpy.zeros(NEAR_SHAPE, dtype=complex), numpy.zeros(NEAR_SHAPE, dtype=complex)]
+H_NEAR = [numpy.zeros(NEAR_SHAPE, dtype=complex), numpy.zeros(NEAR_SHAPE, dtype=complex)]
+
+
+def test_near_to_farfield_rejects_wrong_length_inputs() -> None:
+    with pytest.raises(Exception, match='E_near must be a length-2 list'):
+        farfield.near_to_farfield(E_NEAR[:1], H_NEAR, dx=0.2, dy=0.3)
+
+    with pytest.raises(Exception, match='H_near must be a length-2 list'):
+        farfield.near_to_farfield(E_NEAR, H_NEAR[:1], dx=0.2, dy=0.3)
+
+
+def test_near_to_farfield_rejects_mismatched_shapes() -> None:
+    bad_h_near = [H_NEAR[0], numpy.zeros((2, 4), dtype=complex)]
+
+    with pytest.raises(Exception, match='All fields must be the same shape'):
+        farfield.near_to_farfield(E_NEAR, bad_h_near, dx=0.2, dy=0.3)
+
+
+def test_near_to_farfield_uses_default_and_scalar_padding_shapes() -> None:
+    default_result = farfield.near_to_farfield(E_NEAR, H_NEAR, dx=0.2, dy=0.3)
+    scalar_result = farfield.near_to_farfield(E_NEAR, H_NEAR, dx=0.2, dy=0.3, padded_size=8)
+
+    assert default_result['E'][0].shape == (2, 4)
+    assert default_result['H'][0].shape == (2, 4)
+    assert scalar_result['E'][0].shape == (8, 8)
+    assert scalar_result['H'][0].shape == (8, 8)
+
+
+def test_far_to_nearfield_rejects_wrong_length_inputs() -> None:
+    ff = farfield.near_to_farfield(E_NEAR, H_NEAR, dx=0.2, dy=0.3, padded_size=8)
+
+    with pytest.raises(Exception, match='E_far must be a length-2 list'):
+        farfield.far_to_nearfield(ff['E'][:1], ff['H'], ff['dkx'], ff['dky'])
+
+    with pytest.raises(Exception, match='H_far must be a length-2 list'):
+        farfield.far_to_nearfield(ff['E'], ff['H'][:1], ff['dkx'], ff['dky'])
+
+
+def test_far_to_nearfield_rejects_mismatched_shapes() -> None:
+    ff = farfield.near_to_farfield(E_NEAR, H_NEAR, dx=0.2, dy=0.3, padded_size=8)
+    bad_h_far = [ff['H'][0], numpy.zeros((8, 4), dtype=complex)]
+
+    with pytest.raises(Exception, match='All fields must be the same shape'):
+        farfield.far_to_nearfield(ff['E'], bad_h_far, ff['dkx'], ff['dky'])
+
+
+def test_far_to_nearfield_uses_default_and_scalar_padding_shapes() -> None:
+    ff = farfield.near_to_farfield(E_NEAR, H_NEAR, dx=0.2, dy=0.3, padded_size=8)
+    default_result = farfield.far_to_nearfield(
+        [field.copy() for field in ff['E']],
+        [field.copy() for field in ff['H']],
+        ff['dkx'],
+        ff['dky'],
+    )
+    scalar_result = farfield.far_to_nearfield(
+        [field.copy() for field in ff['E']],
+        [field.copy() for field in ff['H']],
+        ff['dkx'],
+        ff['dky'],
+        padded_size=4,
+    )
+
+    assert default_result['E'][0].shape == (8, 8)
+    assert default_result['H'][0].shape == (8, 8)
+    assert scalar_result['E'][0].shape == (4, 4)
+    assert scalar_result['H'][0].shape == (4, 4)
+
+
+def test_farfield_roundtrip_supports_rectangular_arrays() -> None:
+    e_near = [numpy.zeros((4, 8), dtype=complex), numpy.zeros((4, 8), dtype=complex)]
+    h_near = [numpy.zeros((4, 8), dtype=complex), numpy.zeros((4, 8), dtype=complex)]
+    e_near[0][1, 3] = 1.0 + 0.25j
+    h_near[1][2, 5] = -0.5j
+
+    ff = farfield.near_to_farfield(e_near, h_near, dx=0.2, dy=0.3, padded_size=(4, 8))
+    restored = farfield.far_to_nearfield(
+        [field.copy() for field in ff['E']],
+        [field.copy() for field in ff['H']],
+        ff['dkx'],
+        ff['dky'],
+        padded_size=(4, 8),
+    )
+
+    assert isinstance(ff['dkx'], float)
+    assert isinstance(ff['dky'], float)
+    assert ff['E'][0].shape == (4, 8)
+    assert restored['E'][0].shape == (4, 8)
+    assert restored['H'][0].shape == (4, 8)
+    assert restored['dx'] == pytest.approx(0.2)
+    assert restored['dy'] == pytest.approx(0.3)
+    assert numpy.isfinite(restored['E'][0]).all()
+    assert numpy.isfinite(restored['H'][0]).all()

meanas/test/test_fdfd_functional.py

@@ -0,0 +1,122 @@
+import numpy
+
+from ..fdmath import unvec, vec
+from ..fdfd import functional, operators
+from ._fdfd_case import DXES, EPSILON, E_FIELD, H_FIELD, MU, OMEGA, SHAPE, TF_REGION, apply_fdfd_matrix
+from .utils import assert_fields_close
+
+
+ATOL = 1e-9
+RTOL = 1e-9
+
+
+def assert_fields_match(actual: numpy.ndarray, expected: numpy.ndarray) -> None:
+    assert_fields_close(actual, expected, atol=ATOL, rtol=RTOL)
+
+
+def test_e_full_matches_sparse_operator_without_mu() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.e_full(OMEGA, DXES, vec(EPSILON)),
+        E_FIELD,
+    )
+    functional_result = functional.e_full(OMEGA, DXES, EPSILON)(E_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)
+
+
+def test_e_full_matches_sparse_operator_with_mu() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.e_full(OMEGA, DXES, vec(EPSILON), vec(MU)),
+        E_FIELD,
+    )
+    functional_result = functional.e_full(OMEGA, DXES, EPSILON, MU)(E_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)
+
+
+def test_eh_full_matches_sparse_operator_with_mu() -> None:
+    matrix_result = operators.eh_full(OMEGA, DXES, vec(EPSILON), vec(MU)) @ numpy.concatenate([vec(E_FIELD), vec(H_FIELD)])
+    matrix_e, matrix_h = (unvec(part, SHAPE) for part in numpy.split(matrix_result, 2))
+    functional_e, functional_h = functional.eh_full(OMEGA, DXES, EPSILON, MU)(E_FIELD, H_FIELD)
+
+    assert_fields_match(functional_e, matrix_e)
+    assert_fields_match(functional_h, matrix_h)
+
+
+def test_eh_full_matches_sparse_operator_without_mu() -> None:
+    matrix_result = operators.eh_full(OMEGA, DXES, vec(EPSILON)) @ numpy.concatenate([vec(E_FIELD), vec(H_FIELD)])
+    matrix_e, matrix_h = (unvec(part, SHAPE) for part in numpy.split(matrix_result, 2))
+    functional_e, functional_h = functional.eh_full(OMEGA, DXES, EPSILON)(E_FIELD, H_FIELD)
+
+    assert_fields_match(functional_e, matrix_e)
+    assert_fields_match(functional_h, matrix_h)
+
+
+def test_e2h_matches_sparse_operator_with_mu() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.e2h(OMEGA, DXES, vec(MU)),
+        E_FIELD,
+    )
+    functional_result = functional.e2h(OMEGA, DXES, MU)(E_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)
+
+
+def test_e2h_matches_sparse_operator_without_mu() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.e2h(OMEGA, DXES),
+        E_FIELD,
+    )
+    functional_result = functional.e2h(OMEGA, DXES)(E_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)
+
+
+def test_m2j_matches_sparse_operator_without_mu() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.m2j(OMEGA, DXES),
+        H_FIELD,
+    )
+    functional_result = functional.m2j(OMEGA, DXES)(H_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)
+
+
+def test_m2j_matches_sparse_operator_with_mu() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.m2j(OMEGA, DXES, vec(MU)),
+        H_FIELD,
+    )
+    functional_result = functional.m2j(OMEGA, DXES, MU)(H_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)
+
+
+def test_e_tfsf_source_matches_sparse_operator_without_mu() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.e_tfsf_source(vec(TF_REGION), OMEGA, DXES, vec(EPSILON)),
+        E_FIELD,
+    )
+    functional_result = functional.e_tfsf_source(TF_REGION, OMEGA, DXES, EPSILON)(E_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)
+
+
+def test_e_tfsf_source_matches_sparse_operator_with_mu() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.e_tfsf_source(vec(TF_REGION), OMEGA, DXES, vec(EPSILON), vec(MU)),
+        E_FIELD,
+    )
+    functional_result = functional.e_tfsf_source(TF_REGION, OMEGA, DXES, EPSILON, MU)(E_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)
+
+
+def test_poynting_e_cross_h_matches_sparse_operator() -> None:
+    matrix_result = apply_fdfd_matrix(
+        operators.poynting_e_cross(vec(E_FIELD), DXES),
+        H_FIELD,
+    )
+    functional_result = functional.poynting_e_cross_h(DXES)(E_FIELD, H_FIELD)
+
+    assert_fields_match(functional_result, matrix_result)

meanas/test/test_fdfd_pml.py

@@ -1,11 +1,11 @@
-from typing import Iterable
+# ruff: noqa: ARG001
 import pytest       # type: ignore
 import numpy
 from numpy.typing import NDArray
 from numpy.testing import assert_allclose
 
 from .. import fdfd
-from ..fdmath import vec, unvec, dx_lists_mut
+from ..fdmath import vec, unvec, dx_lists_mut, vfdfield, cfdfield_t
 #from .utils import assert_close, assert_fields_close
 from .test_fdfd import FDResult
 from .conftest import FixtureRequest
@@ -44,49 +44,51 @@ def test_pml(sim: FDResult, src_polarity: int) -> None:
 # Also see conftest.py
 
 @pytest.fixture(params=[1 / 1500])
-def omega(request: FixtureRequest) -> Iterable[float]:
-    yield request.param
+def omega(request: FixtureRequest) -> float:
+    return request.param
 
 
 @pytest.fixture(params=[None])
-def pec(request: FixtureRequest) -> Iterable[NDArray[numpy.float64] | None]:
-    yield request.param
+def pec(request: FixtureRequest) -> NDArray[numpy.float64] | None:
+    return request.param
 
 
 @pytest.fixture(params=[None])
-def pmc(request: FixtureRequest) -> Iterable[NDArray[numpy.float64] | None]:
-    yield request.param
+def pmc(request: FixtureRequest) -> NDArray[numpy.float64] | None:
+    return request.param
 
 
 @pytest.fixture(params=[(30, 1, 1),
                         (1, 30, 1),
                         (1, 1, 30)])
-def shape(request: FixtureRequest) -> Iterable[tuple[int, ...]]:
-    yield (3, *request.param)
+def shape(request: FixtureRequest) -> tuple[int, int, int]:
+    return (3, *request.param)
 
 
 @pytest.fixture(params=[+1, -1])
-def src_polarity(request: FixtureRequest) -> Iterable[int]:
-    yield request.param
+def src_polarity(request: FixtureRequest) -> int:
+    return request.param
 
 
-@pytest.fixture()
+@pytest.fixture
 def j_distribution(
         request: FixtureRequest,
         shape: tuple[int, ...],
-        epsilon: NDArray[numpy.float64],
+        epsilon: vfdfield,
         dxes: dx_lists_mut,
         omega: float,
         src_polarity: int,
-        ) -> Iterable[NDArray[numpy.complex128]]:
+        ) -> cfdfield_t:
     j = numpy.zeros(shape, dtype=complex)
 
     dim = numpy.where(numpy.array(shape[1:]) > 1)[0][0]    # Propagation axis
     other_dims = [0, 1, 2]
     other_dims.remove(dim)
 
-    dx_prop = (dxes[0][dim][shape[dim + 1] // 2]
-             + dxes[1][dim][shape[dim + 1] // 2]) / 2   # noqa: E128   # TODO is this right for nonuniform dxes?
+    dx_prop = (
+        dxes[0][dim][shape[dim + 1] // 2]
+        + dxes[1][dim][shape[dim + 1] // 2]
+    ) / 2   # TODO is this right for nonuniform dxes?
 
     # Mask only contains components orthogonal to propagation direction
     center_mask = numpy.zeros(shape, dtype=bool)
@@ -106,18 +108,18 @@ def j_distribution(
 
     j = fdfd.waveguide_3d.compute_source(E=e, wavenumber=wavenumber_corrected, omega=omega, dxes=dxes,
                                          axis=dim, polarity=src_polarity, slices=slices, epsilon=epsilon)
-    yield j
+    return j
 
 
-@pytest.fixture()
+@pytest.fixture
 def epsilon(
         request: FixtureRequest,
         shape: tuple[int, ...],
         epsilon_bg: float,
         epsilon_fg: float,
-        ) -> Iterable[NDArray[numpy.float64]]:
+        ) -> NDArray[numpy.float64]:
     epsilon = numpy.full(shape, epsilon_fg, dtype=float)
-    yield epsilon
+    return epsilon
 
 
 @pytest.fixture(params=['uniform'])
@@ -127,7 +129,7 @@ def dxes(
         dx: float,
         omega: float,
         epsilon_fg: float,
-        ) -> Iterable[list[list[NDArray[numpy.float64]]]]:
+        ) -> list[list[NDArray[numpy.float64]]]:
     if request.param == 'uniform':
         dxes = [[numpy.full(s, dx) for s in shape[1:]] for _ in range(2)]
     dim = numpy.where(numpy.array(shape[1:]) > 1)[0][0]    # Propagation axis
@@ -141,10 +143,10 @@ def dxes(
                 epsilon_effective=epsilon_fg,
                 thickness=10,
                 )
-    yield dxes
+    return dxes
 
 
-@pytest.fixture()
+@pytest.fixture
 def sim(
         request: FixtureRequest,
         shape: tuple[int, ...],
@@ -162,7 +164,7 @@ def sim(
         omega=omega,
         dxes=dxes,
         epsilon=eps_vec,
-        matrix_solver_opts={'atol': 1e-15, 'tol': 1e-11},
+        matrix_solver_opts={'atol': 1e-15, 'rtol': 1e-11},
         )
     e = unvec(e_vec, shape[1:])
 

meanas/test/test_fdfd_solvers.py

@@ -0,0 +1,128 @@
+from typing import cast
+
+import numpy
+
+from ..fdfd import solvers
+from ._solver_cases import solver_plumbing_case
+from .utils import assert_close
+
+
+def test_scipy_qmr_wraps_user_callback_without_recursion(monkeypatch) -> None:
+    seen: list[tuple[float, ...]] = []
+
+    def fake_qmr(a, b: numpy.ndarray, **kwargs):
+        kwargs['callback'](numpy.array([1.0, 2.0]))
+        return numpy.array([3.0, 4.0]), 0
+
+    monkeypatch.setattr(solvers.scipy.sparse.linalg, 'qmr', fake_qmr)
+    result = solvers._scipy_qmr(
+        solver_plumbing_case().a0,
+        numpy.array([1.0, 0.0]),
+        callback=lambda xk: seen.append(tuple(xk)),
+    )
+
+    assert_close(result, [3.0, 4.0])
+    assert seen == [(1.0, 2.0)]
+
+
+def test_scipy_qmr_installs_logging_callback_when_missing(monkeypatch) -> None:
+    callback_seen: list[numpy.ndarray] = []
+
+    def fake_qmr(a, b: numpy.ndarray, **kwargs):
+        callback = kwargs['callback']
+        callback(numpy.array([5.0, 6.0]))
+        callback_seen.append(b.copy())
+        return numpy.array([7.0, 8.0]), 0
+
+    monkeypatch.setattr(solvers.scipy.sparse.linalg, 'qmr', fake_qmr)
+    result = solvers._scipy_qmr(solver_plumbing_case().a0, numpy.array([1.0, 0.0]))
+
+    assert_close(result, [7.0, 8.0])
+    assert len(callback_seen) == 1
+
+
+def test_generic_forward_preconditions_system_and_guess(monkeypatch) -> None:
+    case = solver_plumbing_case()
+    captured: dict[str, numpy.ndarray | float | object] = {}
+
+    monkeypatch.setattr(solvers.operators, 'e_full', lambda *args, **kwargs: case.a0)
+    monkeypatch.setattr(solvers.operators, 'e_full_preconditioners', lambda dxes: (case.pl, case.pr))
+
+    def fake_solver(a, b: numpy.ndarray, **kwargs):
+        captured['a'] = a
+        captured['b'] = b
+        captured['x0'] = kwargs['x0']
+        captured['atol'] = kwargs['atol']
+        return case.solver_result
+
+    result = solvers.generic(
+        omega=case.omega,
+        dxes=case.dxes,
+        J=case.j,
+        epsilon=case.epsilon,
+        matrix_solver=fake_solver,
+        matrix_solver_opts={'atol': 1e-12},
+        E_guess=case.guess,
+    )
+
+    assert_close(cast(object, captured['a']).toarray(), (case.pl @ case.a0 @ case.pr).toarray())  # type: ignore[attr-defined]
+    assert_close(cast(numpy.ndarray, captured['b']), case.pl @ (-1j * case.omega * case.j))
+    assert_close(cast(numpy.ndarray, captured['x0']), case.pl @ case.guess)
+    assert captured['atol'] == 1e-12
+    assert_close(result, case.pr @ case.solver_result)
+
+
+def test_generic_adjoint_preconditions_system_and_guess(monkeypatch) -> None:
+    case = solver_plumbing_case()
+    captured: dict[str, numpy.ndarray | float | object] = {}
+
+    monkeypatch.setattr(solvers.operators, 'e_full', lambda *args, **kwargs: case.a0)
+    monkeypatch.setattr(solvers.operators, 'e_full_preconditioners', lambda dxes: (case.pl, case.pr))
+
+    def fake_solver(a, b: numpy.ndarray, **kwargs):
+        captured['a'] = a
+        captured['b'] = b
+        captured['x0'] = kwargs['x0']
+        captured['rtol'] = kwargs['rtol']
+        return case.solver_result
+
+    result = solvers.generic(
+        omega=case.omega,
+        dxes=case.dxes,
+        J=case.j,
+        epsilon=case.epsilon,
+        matrix_solver=fake_solver,
+        matrix_solver_opts={'rtol': 1e-9},
+        E_guess=case.guess,
+        adjoint=True,
+    )
+
+    expected_matrix = (case.pl @ case.a0 @ case.pr).T.conjugate()
+    assert_close(cast(object, captured['a']).toarray(), expected_matrix.toarray())  # type: ignore[attr-defined]
+    assert_close(cast(numpy.ndarray, captured['b']), case.pr.T.conjugate() @ (-1j * case.omega * case.j))
+    assert_close(cast(numpy.ndarray, captured['x0']), case.pr.T.conjugate() @ case.guess)
+    assert captured['rtol'] == 1e-9
+    assert_close(result, case.pl.T.conjugate() @ case.solver_result)
+
+
+def test_generic_without_guess_does_not_inject_x0(monkeypatch) -> None:
+    case = solver_plumbing_case()
+    captured: dict[str, object] = {}
+
+    monkeypatch.setattr(solvers.operators, 'e_full', lambda *args, **kwargs: case.a0)
+    monkeypatch.setattr(solvers.operators, 'e_full_preconditioners', lambda dxes: (case.pl, case.pr))
+
+    def fake_solver(a, b: numpy.ndarray, **kwargs):
+        captured['kwargs'] = kwargs
+        return numpy.array([1.0, -1.0])
+
+    result = solvers.generic(
+        omega=1.0,
+        dxes=case.dxes,
+        J=numpy.array([2.0, 3.0]),
+        epsilon=case.epsilon,
+        matrix_solver=fake_solver,
+    )
+
+    assert 'x0' not in cast(dict[str, object], captured['kwargs'])
+    assert_close(result, case.pr @ numpy.array([1.0, -1.0]))

meanas/test/test_fdmath_functional.py

@@ -0,0 +1,60 @@
+import numpy
+
+from ..fdmath import functional as fd_functional
+from ..fdmath import operators as fd_operators
+from ..fdmath import vec, unvec
+from .utils import assert_close, assert_fields_close
+
+
+SHAPE = (2, 3, 2)
+DX_E = [numpy.array([1.0, 1.5]), numpy.array([0.75, 1.25, 1.5]), numpy.array([1.2, 0.8])]
+DX_H = [numpy.array([0.9, 1.4]), numpy.array([0.8, 1.1, 1.4]), numpy.array([1.0, 0.7])]
+
+SCALAR_FIELD = (
+    numpy.arange(numpy.prod(SHAPE)).reshape(SHAPE)
+    + 0.1j * numpy.arange(numpy.prod(SHAPE)).reshape(SHAPE)
+).astype(complex)
+VECTOR_FIELD = (numpy.arange(3 * numpy.prod(SHAPE)).reshape((3, *SHAPE)) + 0.25j).astype(complex)
+
+
+def test_deriv_forward_without_dx_matches_numpy_roll() -> None:
+    for axis, deriv in enumerate(fd_functional.deriv_forward()):
+        expected = numpy.roll(SCALAR_FIELD, -1, axis=axis) - SCALAR_FIELD
+        assert_close(deriv(SCALAR_FIELD), expected)
+
+
+def test_deriv_back_without_dx_matches_numpy_roll() -> None:
+    for axis, deriv in enumerate(fd_functional.deriv_back()):
+        expected = SCALAR_FIELD - numpy.roll(SCALAR_FIELD, 1, axis=axis)
+        assert_close(deriv(SCALAR_FIELD), expected)
+
+
+def test_curl_parts_sum_to_full_curl() -> None:
+    curl_forward = fd_functional.curl_forward(DX_E)(VECTOR_FIELD)
+    curl_back = fd_functional.curl_back(DX_H)(VECTOR_FIELD)
+    forward_parts = fd_functional.curl_forward_parts(DX_E)(VECTOR_FIELD)
+    back_parts = fd_functional.curl_back_parts(DX_H)(VECTOR_FIELD)
+
+    for axis in range(3):
+        assert_close(forward_parts[axis][0] + forward_parts[axis][1], curl_forward[axis])
+        assert_close(back_parts[axis][0] + back_parts[axis][1], curl_back[axis])
+
+
+def test_derivatives_match_sparse_operators_on_nonuniform_grid() -> None:
+    for axis, deriv in enumerate(fd_functional.deriv_forward(DX_E)):
+        matrix_result = (fd_operators.deriv_forward(DX_E)[axis] @ SCALAR_FIELD.ravel(order='C')).reshape(SHAPE, order='C')
+        assert_close(deriv(SCALAR_FIELD), matrix_result, atol=1e-12, rtol=1e-12)
+
+    for axis, deriv in enumerate(fd_functional.deriv_back(DX_H)):
+        matrix_result = (fd_operators.deriv_back(DX_H)[axis] @ SCALAR_FIELD.ravel(order='C')).reshape(SHAPE, order='C')
+        assert_close(deriv(SCALAR_FIELD), matrix_result, atol=1e-12, rtol=1e-12)
+
+
+def test_curls_match_sparse_operators_on_nonuniform_grid() -> None:
+    curl_forward = fd_functional.curl_forward(DX_E)(VECTOR_FIELD)
+    curl_back = fd_functional.curl_back(DX_H)(VECTOR_FIELD)
+    matrix_forward = unvec(fd_operators.curl_forward(DX_E) @ vec(VECTOR_FIELD), SHAPE)
+    matrix_back = unvec(fd_operators.curl_back(DX_H) @ vec(VECTOR_FIELD), SHAPE)
+
+    assert_fields_close(curl_forward, matrix_forward, atol=1e-12, rtol=1e-12)
+    assert_fields_close(curl_back, matrix_back, atol=1e-12, rtol=1e-12)

meanas/test/test_fdmath_operators.py

@@ -0,0 +1,90 @@
+import numpy
+import pytest
+
+from ..fdmath import operators, unvec, vec
+from ._test_builders import real_ramp
+from .utils import assert_close
+
+
+SHAPE = (2, 3, 2)
+SCALAR_FIELD = real_ramp(SHAPE)
+VECTOR_LEFT = real_ramp((3, *SHAPE), offset=0.5)
+VECTOR_RIGHT = real_ramp((3, *SHAPE), scale=1 / 3, offset=2.0)
+
+
+def _apply_scalar_matrix(op: operators.sparse.spmatrix) -> numpy.ndarray:
+    return (op @ SCALAR_FIELD.ravel(order='C')).reshape(SHAPE, order='C')
+
+
+def _mirrored_indices(size: int, shift_distance: int) -> numpy.ndarray:
+    indices = numpy.arange(size) + shift_distance
+    indices = numpy.where(indices >= size, 2 * size - indices - 1, indices)
+    indices = numpy.where(indices < 0, -1 - indices, indices)
+    return indices
+
+
+@pytest.mark.parametrize(('axis', 'shift_distance'), [(0, 1), (1, -1), (2, 1)])
+def test_shift_circ_matches_numpy_roll(axis: int, shift_distance: int) -> None:
+    matrix_result = _apply_scalar_matrix(operators.shift_circ(axis, SHAPE, shift_distance))
+    expected = numpy.roll(SCALAR_FIELD, -shift_distance, axis=axis)
+    assert_close(matrix_result, expected)
+
+
+@pytest.mark.parametrize(('axis', 'shift_distance'), [(0, 1), (1, -1), (2, 1)])
+def test_shift_with_mirror_matches_explicit_mirrored_indices(axis: int, shift_distance: int) -> None:
+    matrix_result = _apply_scalar_matrix(operators.shift_with_mirror(axis, SHAPE, shift_distance))
+    indices = [numpy.arange(length) for length in SHAPE]
+    indices[axis] = _mirrored_indices(SHAPE[axis], shift_distance)
+    expected = SCALAR_FIELD[numpy.ix_(*indices)]
+    assert_close(matrix_result, expected)
+
+
+@pytest.mark.parametrize(
+    ('args', 'message'),
+    [
+        ((0, (2,), 1), 'Invalid shape'),
+        ((3, SHAPE, 1), 'Invalid direction'),
+    ],
+    )
+def test_shift_circ_rejects_invalid_arguments(args: tuple[int, tuple[int, ...], int], message: str) -> None:
+    with pytest.raises(Exception, match=message):
+        operators.shift_circ(*args)
+
+
+@pytest.mark.parametrize(
+    ('args', 'message'),
+    [
+        ((0, (2,), 1), 'Invalid shape'),
+        ((3, SHAPE, 1), 'Invalid direction'),
+        ((0, SHAPE, SHAPE[0]), 'too large'),
+    ],
+    )
+def test_shift_with_mirror_rejects_invalid_arguments(args: tuple[int, tuple[int, ...], int], message: str) -> None:
+    with pytest.raises(Exception, match=message):
+        operators.shift_with_mirror(*args)
+
+
+def test_vec_cross_matches_pointwise_cross_product() -> None:
+    matrix_result = unvec(operators.vec_cross(vec(VECTOR_LEFT)) @ vec(VECTOR_RIGHT), SHAPE)
+    expected = numpy.empty_like(VECTOR_LEFT)
+    expected[0] = VECTOR_LEFT[1] * VECTOR_RIGHT[2] - VECTOR_LEFT[2] * VECTOR_RIGHT[1]
+    expected[1] = VECTOR_LEFT[2] * VECTOR_RIGHT[0] - VECTOR_LEFT[0] * VECTOR_RIGHT[2]
+    expected[2] = VECTOR_LEFT[0] * VECTOR_RIGHT[1] - VECTOR_LEFT[1] * VECTOR_RIGHT[0]
+    assert_close(matrix_result, expected)
+
+
+def test_avg_forward_matches_half_sum_with_forward_neighbor() -> None:
+    matrix_result = _apply_scalar_matrix(operators.avg_forward(1, SHAPE))
+    expected = 0.5 * (SCALAR_FIELD + numpy.roll(SCALAR_FIELD, -1, axis=1))
+    assert_close(matrix_result, expected)
+
+
+def test_avg_back_matches_half_sum_with_backward_neighbor() -> None:
+    matrix_result = _apply_scalar_matrix(operators.avg_back(1, SHAPE))
+    expected = 0.5 * (SCALAR_FIELD + numpy.roll(SCALAR_FIELD, 1, axis=1))
+    assert_close(matrix_result, expected)
+
+
+def test_avg_forward_rejects_invalid_shape() -> None:
+    with pytest.raises(Exception, match='Invalid shape'):
+        operators.avg_forward(0, (2,))

meanas/test/test_fdmath_vectorization.py

@@ -0,0 +1,46 @@
+import numpy
+
+from ..fdmath import unvec, vec
+from ._test_builders import complex_ramp, real_ramp
+from .utils import assert_close
+
+
+SHAPE = (2, 3, 2)
+FIELD = real_ramp((3, *SHAPE))
+COMPLEX_FIELD = complex_ramp((3, *SHAPE), imag_scale=0.5)
+
+
+def test_vec_and_unvec_return_none_for_none_input() -> None:
+    assert vec(None) is None
+    assert unvec(None, SHAPE) is None
+
+
+def test_real_field_round_trip_preserves_shape_and_values() -> None:
+    vector = vec(FIELD)
+    assert vector is not None
+    restored = unvec(vector, SHAPE)
+    assert restored is not None
+    assert restored.shape == (3, *SHAPE)
+    assert_close(restored, FIELD)
+
+
+def test_complex_field_round_trip_preserves_shape_and_values() -> None:
+    vector = vec(COMPLEX_FIELD)
+    assert vector is not None
+    restored = unvec(vector, SHAPE)
+    assert restored is not None
+    assert restored.shape == (3, *SHAPE)
+    assert_close(restored, COMPLEX_FIELD)
+
+
+def test_unvec_with_two_components_round_trips_vector() -> None:
+    vector = numpy.arange(2 * numpy.prod(SHAPE), dtype=float)
+    field = unvec(vector, SHAPE, nvdim=2)
+    assert field is not None
+    assert field.shape == (2, *SHAPE)
+    assert_close(vec(field), vector)
+
+
+def test_vec_accepts_arraylike_input() -> None:
+    arraylike = [[[1, 2], [3, 4]], [[5, 6], [7, 8]]]
+    assert_close(vec(arraylike), numpy.ravel(arraylike, order='C'))

meanas/test/test_fdtd.py

@@ -1,4 +1,5 @@
-from typing import Iterable, Any
+# ruff: noqa: ARG001
+from typing import Any
 import dataclasses
 import pytest       # type: ignore
 import numpy
@@ -6,7 +7,7 @@ from numpy.typing import NDArray
 #from numpy.testing import assert_allclose, assert_array_equal
 
 from .. import fdtd
-from .utils import assert_close, assert_fields_close, PRNG
+from .utils import assert_close, assert_fields_close, make_prng
 from .conftest import FixtureRequest
 
 
@@ -101,7 +102,7 @@ def test_poynting_divergence(sim: 'TDResult') -> None:
 def test_poynting_planes(sim: 'TDResult') -> None:
     mask = (sim.js[0] != 0).any(axis=0)
     if mask.sum() > 1:
-        pytest.skip('test_poynting_planes can only test single point sources, got {}'.format(mask.sum()))
+        pytest.skip(f'test_poynting_planes can only test single point sources, got {mask.sum()}')
 
     args: dict[str, Any] = {
         'dxes': sim.dxes,
@@ -150,8 +151,8 @@ def test_poynting_planes(sim: 'TDResult') -> None:
 
 
 @pytest.fixture(params=[0.3])
-def dt(request: FixtureRequest) -> Iterable[float]:
-    yield request.param
+def dt(request: FixtureRequest) -> float:
+    return request.param
 
 
 @dataclasses.dataclass()
@@ -168,8 +169,8 @@ class TDResult:
 
 
 @pytest.fixture(params=[(0, 4, 8)])  # (0,)
-def j_steps(request: FixtureRequest) -> Iterable[tuple[int, ...]]:
-    yield request.param
+def j_steps(request: FixtureRequest) -> tuple[int, ...]:
+    return request.param
 
 
 @pytest.fixture(params=['center', 'random'])
@@ -177,18 +178,19 @@ def j_distribution(
         request: FixtureRequest,
         shape: tuple[int, ...],
         j_mag: float,
-        ) -> Iterable[NDArray[numpy.float64]]:
+        ) -> NDArray[numpy.float64]:
+    prng = make_prng()
     j = numpy.zeros(shape)
     if request.param == 'center':
         j[:, shape[1] // 2, shape[2] // 2, shape[3] // 2] = j_mag
     elif request.param == '000':
         j[:, 0, 0, 0] = j_mag
     elif request.param == 'random':
-        j[:] = PRNG.uniform(low=-j_mag, high=j_mag, size=shape)
-    yield j
+        j[:] = prng.uniform(low=-j_mag, high=j_mag, size=shape)
+    return j
 
 
-@pytest.fixture()
+@pytest.fixture
 def sim(
         request: FixtureRequest,
         shape: tuple[int, ...],
@@ -199,8 +201,7 @@ def sim(
         j_steps: tuple[int, ...],
         ) -> TDResult:
     is3d = (numpy.array(shape) == 1).sum() == 0
-    if is3d:
-        if dt != 0.3:
+    if is3d and dt != 0.3:
         pytest.skip('Skipping dt != 0.3 because test is 3D (for speed)')
 
     sim = TDResult(

meanas/test/test_fdtd_base.py

@@ -0,0 +1,43 @@
+from ..fdmath import functional as fd_functional
+from ..fdtd import base
+from ._test_builders import real_ramp
+from .utils import assert_close
+
+
+DT = 0.25
+SHAPE = (3, 2, 2, 2)
+E_FIELD = real_ramp(SHAPE, scale=1 / 5)
+H_FIELD = real_ramp(SHAPE, scale=1 / 7, offset=1 / 7)
+EPSILON = 1.5 + E_FIELD / 10.0
+MU_FIELD = 2.0 + H_FIELD / 8.0
+MU_SCALAR = 3.0
+
+
+def test_maxwell_e_without_dxes_matches_unit_spacing_update() -> None:
+    updater = base.maxwell_e(dt=DT)
+    expected = E_FIELD + DT * fd_functional.curl_back()(H_FIELD) / EPSILON
+
+    updated = updater(E_FIELD.copy(), H_FIELD.copy(), EPSILON)
+
+    assert_close(updated, expected)
+
+
+def test_maxwell_h_without_dxes_and_without_mu_matches_unit_spacing_update() -> None:
+    updater = base.maxwell_h(dt=DT)
+    expected = H_FIELD - DT * fd_functional.curl_forward()(E_FIELD)
+
+    updated = updater(E_FIELD.copy(), H_FIELD.copy())
+
+    assert_close(updated, expected)
+
+
+def test_maxwell_h_without_dxes_accepts_scalar_and_field_mu() -> None:
+    updater = base.maxwell_h(dt=DT)
+
+    updated_scalar = updater(E_FIELD.copy(), H_FIELD.copy(), MU_SCALAR)
+    expected_scalar = H_FIELD - DT * fd_functional.curl_forward()(E_FIELD) / MU_SCALAR
+    assert_close(updated_scalar, expected_scalar)
+
+    updated_field = updater(E_FIELD.copy(), H_FIELD.copy(), MU_FIELD)
+    expected_field = H_FIELD - DT * fd_functional.curl_forward()(E_FIELD) / MU_FIELD
+    assert_close(updated_field, expected_field)