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[waveguide] correct modal reconstruction and calibrate full port bases

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Forgejo Actions 2026-10-28 19:03:37 -07:00
commit 4452dad609
7 changed files with 257 additions and 94 deletions
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93
meanas/fdfd/waveguide_2d.py
View file

@ -801,77 +801,40 @@ def sensitivity(
wavenumber: complex,
omega: complex,
dxes: dx_lists2_t,
epsilon: vfdslice,
mu: vfdslice | None = None,
epsilon: vfdslice | vcfdslice,
mu: vfdslice | vcfdslice | None = None,
) -> vcfdslice_t:
r"""
Given a waveguide structure (`dxes`, `epsilon`, `mu`) and mode fields
(`e_norm`, `h_norm`, `wavenumber`, `omega`), calculates the sensitivity of the wavenumber
$\beta$ to changes in the dielectric structure $\epsilon$.
Calculate the analytic sensitivity of an isolated mode's beta to epsilon.
The output is a vector of the same size as `vec(epsilon)`, with each element specifying the
sensitivity of `wavenumber` to changes in the corresponding element in `vec(epsilon)`, i.e.
For the reciprocal diagonal waveguide problem ``A_E E_xy = beta**2 E_xy``,
the Yee-weighted Lorentz left eigenvector is the unconjugated row
$$sens_{i} = \frac{\partial\beta}{\partial\epsilon_i}$$
$$ L^T = [H_y \Delta x_e \Delta y_h,\; -H_x \Delta x_h \Delta y_e]. $$
An adjoint approach is used to calculate the sensitivity; the derivation is provided here:
Differentiating the eigenproblem and projecting with this left vector gives
Starting with the eigenvalue equation
$$\beta^2 E_{xy} = A_E E_{xy}$$
where $A_E$ is the waveguide operator from `operator_e()`, and $E_{xy} = \begin{bmatrix} E_x \\
E_y \end{bmatrix}$,
we can differentiate with respect to one of the $\epsilon$ elements (i.e. at one Yee grid point), $\epsilon_i$:
$$
(2 \beta) \partial_{\epsilon_i}(\beta) E_{xy} + \beta^2 \partial_{\epsilon_i} E_{xy}
= \partial_{\epsilon_i}(A_E) E_{xy} + A_E \partial_{\epsilon_i} E_{xy}
$$
We then multiply by $H_{yx}^\star = \begin{bmatrix}H_y^\star \\ -H_x^\star \end{bmatrix}$ from the left:
$$
(2 \beta) \partial_{\epsilon_i}(\beta) H_{yx}^\star E_{xy} + \beta^2 H_{yx}^\star \partial_{\epsilon_i} E_{xy}
= H_{yx}^\star \partial_{\epsilon_i}(A_E) E_{xy} + H_{yx}^\star A_E \partial_{\epsilon_i} E_{xy}
$$
On a graded Yee grid, the left projection includes the quadrature weights:
$H_y^\star \Delta x_e \Delta y_h$ and $-H_x^\star \Delta x_h \Delta y_e$.
These weights multiply the left vector, before it acts on any derivative operators.
With these weights understood, $H_{yx}^\star$ is a left-eigenvector of $A_E$. This can be verified by inspecting
the form of `operator_h` ($A_H$) and comparing its conjugate transpose to `operator_e` ($A_E$). Also, note
$H_{yx}^\star \cdot E_{xy} = H^\star \times E$ recalls the mode orthogonality relation. See doi:10.5194/ars-9-85-201
for a similar approach. Therefore,
$$
H_{yx}^\star A_E \partial_{\epsilon_i} E_{xy} = \beta^2 H_{yx}^\star \partial_{\epsilon_i} E_{xy}
$$
and we can simplify to
$$
\partial_{\epsilon_i}(\beta)
= \frac{1}{2 \beta} \frac{H_{yx}^\star \partial_{\epsilon_i}(A_E) E_{xy} }{H_{yx}^\star E_{xy}}
$$
This expression can be quickly calculated for all $i$ by writing out the various terms of
$\partial_{\epsilon_i} A_E$ and recognizing that the vector-matrix-vector products (i.e. scalars)
$sens_i = \vec{v}_{left} \partial_{\epsilon_i} (\epsilon_{xyz}) \vec{v}_{right}$, indexed by $i$, can be expressed as
elementwise multiplications $\vec{sens} = \vec{v}_{left} \star \vec{v}_{right}$
$$ \frac{\partial\beta}{\partial\epsilon_i}
= \frac{L^T (\partial_{\epsilon_i} A_E) E_{xy}}
{2\beta L^T E_{xy}}. $$
This pairing is bilinear: neither H nor the coordinate weights are
conjugated. It therefore applies to complex PML stretches and reciprocal
lossy materials as well as lossless grids. Quadrature weights multiply the
left projection before derivative operators act. Overall field scaling
cancels between numerator and denominator.
Args:
e_norm: Normalized, vectorized E_xyz field for the mode. E.g. as returned by `normalized_fields_e`.
h_norm: Normalized, vectorized H_xyz field for the mode. E.g. as returned by `normalized_fields_e`.
wavenumber: Propagation constant for the mode. The z-axis is assumed to be continuous (i.e. without numerical dispersion).
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
mu: Vectorized magnetic permeability grid (default 1 everywhere)
e_norm: Vectorized E_xyz eigenfield for the mode.
h_norm: Matching vectorized H_xyz eigenfield at the same phase reference.
wavenumber: Nonzero beta of the continuous-z eigenproblem.
omega: Angular frequency of the system.
dxes: Two-dimensional Yee widths, including any complex PML stretches.
epsilon: Vectorized diagonal permittivity.
mu: Optional vectorized diagonal permeability; defaults to one.
Returns:
Sparse matrix representation of the operator.
Vector ``d beta / d epsilon_i`` in the same component order as epsilon.
"""
if mu is None:
mu = numpy.ones_like(epsilon)
@ -891,12 +854,12 @@ def sensitivity(
ev_xy = numpy.concatenate(numpy.split(e_norm, 3)[:2])
hx, hy, _hz = numpy.split(h_norm, 3)
# The Yee-weighted H projection is the left eigenvector on a graded grid.
hv_yx_conj = numpy.conj(numpy.concatenate([hy, -hx])) * numpy.concatenate([da_exxhyy, da_eyyhxx])
hv_yx = numpy.concatenate([hy, -hx]) * numpy.concatenate([da_exxhyy, da_eyyhxx])
sens_xy1 = (hv_yx_conj @ (omega * omega * mu_yx)) * ev_xy
sens_xy2 = (hv_yx_conj @ sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby))) * ev_xy
sens_z = (hv_yx_conj @ sparse.vstack((Dfx, Dfy)) @ (-eps_z_inv * eps_z_inv)) * (sparse.hstack((Dbx, Dby)) @ eps_xy @ ev_xy)
norm = hv_yx_conj @ ev_xy
sens_xy1 = (hv_yx @ (omega * omega * mu_yx)) * ev_xy
sens_xy2 = (hv_yx @ sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby))) * ev_xy
sens_z = (hv_yx @ sparse.vstack((Dfx, Dfy)) @ (-eps_z_inv * eps_z_inv)) * (sparse.hstack((Dbx, Dby)) @ eps_xy @ ev_xy)
norm = hv_yx @ ev_xy
sens_tot = numpy.concatenate([sens_xy1 + sens_xy2, sens_z]) / (2 * wavenumber * norm)
return vcfdslice_t(sens_tot)

97
meanas/fdfd/waveguide_3d.py
View file

@ -143,6 +143,9 @@ def mode_from_2d(
``polarity`` is +1 toward increasing indices and -1 toward decreasing
indices. ``normalization`` follows waveguide_2d.normalized_fields_e.
Lossless modes at or beyond the longitudinal grid band edge cannot be
power-normalized. Lorentz normalization can represent these modes
beyond that edge and selects the decaying branch for real beta.
"""
if polarity not in (-1, 1):
raise ValueError('polarity must be -1 or +1')
@ -154,7 +157,12 @@ def mode_from_2d(
raise ValueError('the 2D eigenvector and wavenumber must be finite and match the cross-section')
wavenumber_2d = complex(wavenumber_2d)
# Correct wavenumber to account for numerical dispersion.
wavenumber = 2 / dx_prop * numpy.arcsin(wavenumber_2d * dx_prop / 2)
argument = wavenumber_2d * dx_prop / 2
if numpy.isreal(argument) and abs(argument.real) >= 1:
if normalization == 'power':
raise ValueError('mode is at or beyond the longitudinal grid band edge; refine the propagation grid')
argument = complex(argument.real, -0.0)
wavenumber = 2 / dx_prop * numpy.arcsin(argument)
shape = [d.size for d in args_2d['dxes'][0]]
ve, vh = waveguide_2d.normalized_fields_e(e_xy, wavenumber=wavenumber_2d, prop_phase=dx_prop * wavenumber, normalization=normalization, **args_2d)
@ -372,12 +380,18 @@ def expand_e(
$$
to each copied slice.
Transverse electric samples use forward intervals dx_e[i] between
indices i and i+1. The longitudinal electric component uses backward
intervals dx_h[i] between indices i-1 and i on its staggered grid.
"""
slices = tuple(slices)
# Determine phase factors for parallel slices
a_shape = numpy.roll([1, -1, 1, 1], axis)
a_E = numpy.real(dxes[0][axis]).cumsum()
# dx_e[i] spans transverse electric samples i -> i+1. The longitudinal
# component lies on the complementary grid, with backward intervals dx_h.
a_E = numpy.concatenate(([0.0], numpy.real(dxes[0][axis][:-1]).cumsum()))
r_E = a_E - a_E[slices[axis]]
iphi = polarity * -1j * wavenumber
phase_E = numpy.exp(iphi * r_E).reshape(a_shape)
@ -392,6 +406,9 @@ def expand_e(
slices_in = (slice(None), *slices)
E_expanded[slices_exp] = phase_E * numpy.array(E)[slices_in]
a_long = numpy.real(dxes[1][axis]).cumsum()
phase_long = numpy.exp(iphi * (a_long - a_long[slices[axis]])).reshape(a_shape[1:])
E_expanded[(axis, *slices_exp_l)] = phase_long * E[(axis, *slices)]
return cfdfield_t(E_expanded)
@ -550,10 +567,41 @@ def _decompose_port_field(
matrix, overlaps, mode_count = decomposition
field_vec = vec(field)
measured = numpy.asarray([field_vec @ vec(overlap).conj() for overlap in overlaps])
coeffs, _resid, _rank, _sing = numpy.linalg.lstsq(matrix, measured, rcond=None)
coeffs, _resid, rank, _sing = numpy.linalg.lstsq(matrix, measured, rcond=None)
if rank != matrix.shape[1]:
raise numpy.linalg.LinAlgError('monitor modal basis is rank deficient')
return coeffs[:mode_count], coeffs[mode_count:]
def _port_power_gram(
modes: Sequence[Waveguide3DMode],
dxes: dx_lists_t,
axis: int,
polarity: int,
slices: Sequence[slice],
) -> NDArray[numpy.complex128]:
"""Hermitian flux metric, including interference between nonorthogonal modes."""
u, v = (axis + 1) % 3, (axis + 2) % 3
def width(family: int, component: int) -> NDArray:
values = numpy.real(dxes[family][component][slices[component]])
shape = [1, 1, 1]
shape[component] = values.size
return values.reshape(shape)
area_uv = width(0, u) * width(1, v)
area_vu = width(1, u) * width(0, v)
dx = float(numpy.real(dxes[0][axis][slices[axis]]).item())
gram = numpy.empty((len(modes), len(modes)), dtype=complex)
for row, test in enumerate(modes):
h = test['H'][(slice(None), *slices)].conj()
for column, mode in enumerate(modes):
e = mode['E'][(slice(None), *slices)]
phase = numpy.exp(-1j * polarity * mode['wavenumber'] * dx)
gram[row, column] = 0.5 * polarity * phase * numpy.sum(e[u] * h[v] * area_uv - e[v] * h[u] * area_vu)
return 0.5 * (gram + gram.conj().T)
def _copy_cross_section(
target: NDArray,
cross_section: NDArray,
@ -637,8 +685,11 @@ def get_s_fdfd(
Extract a two-port guided-mode scattering matrix from local 3D FDFD solves.
The returned S-matrix is ordered as `[[R_L, T_R_to_L], [T_L_to_R, R_R]]`, with
reference planes at the supplied monitor slices. Each column is normalized
by the incident modal amplitude measured on the corresponding input monitor.
reference planes at the supplied monitor slices. Calibration solves
``S @ incoming_amplitudes = raw_outgoing_amplitudes`` using every measured
incoming coefficient on both ports, including mixed or degenerate modes.
Diagnostic guided powers use the full input/output flux Gram matrices,
retaining interference between nonorthogonal modal basis vectors.
"""
if incident_amplitude_atol <= 0:
raise ValueError('incident_amplitude_atol must be positive')
@ -710,8 +761,8 @@ def get_s_fdfd(
slices=right_monitor,
)
s = numpy.zeros((n_total, n_total), dtype=complex)
raw_outgoing = numpy.zeros_like(s)
raw_outgoing = numpy.zeros((n_total, n_total), dtype=complex)
incoming_amplitudes = numpy.zeros_like(raw_outgoing)
incident_amplitudes = numpy.zeros((n_total,), dtype=complex)
electric_fields: list[cfdfield_t] = []
magnetic_fields: list[cfdfield_t] = []
@ -747,26 +798,39 @@ def get_s_fdfd(
e_field = unvec(solvers.generic(**solve_kwargs), shape)
left_in, left_out = _decompose_port_field(left_decomposition, e_field)
right_in, right_out = _decompose_port_field(right_decomposition, e_field)
if column < n_left:
incident = left_in[column]
else:
incident = right_in[column - n_left]
if abs(incident) <= incident_amplitude_atol:
incoming_column = numpy.concatenate((left_in, right_in))
if numpy.linalg.norm(incoming_column) <= incident_amplitude_atol:
raise ValueError(f'incident modal amplitude for column {column} is zero')
raw_column = numpy.concatenate((left_out, right_out))
incident_amplitudes[column] = incident
incident_amplitudes[column] = incoming_column[column]
incoming_amplitudes[:, column] = incoming_column
raw_outgoing[:, column] = raw_column
s[:, column] = raw_column / incident
if return_diagnostics:
electric_fields.append(cfdfield_t(e_field))
magnetic_fields.append(cfdfield_t(functional.e2h(omega, solve_dxes, mu=mu)(e_field)))
if not numpy.isfinite(incoming_amplitudes).all() or not numpy.isfinite(raw_outgoing).all():
raise ValueError('modal calibration requires finite incoming and outgoing amplitudes')
try:
s = numpy.linalg.solve(incoming_amplitudes.T, raw_outgoing.T).T
except numpy.linalg.LinAlgError as error:
raise numpy.linalg.LinAlgError('incoming excitation matrix is singular') from error
if not return_diagnostics:
return s
guided_powers = numpy.sum(numpy.abs(s) ** 2, axis=0).real
zero = numpy.zeros((n_left, n_right), dtype=complex)
input_power_gram = numpy.block([
[_port_power_gram(left_in_modes, dxes, axis, +1, left_monitor), zero],
[zero.T, _port_power_gram(right_in_modes, dxes, axis, -1, right_monitor)],
])
output_power_gram = numpy.block([
[_port_power_gram(left_out_modes, dxes, axis, -1, left_monitor), zero],
[zero.T, _port_power_gram(right_out_modes, dxes, axis, +1, right_monitor)],
])
guided_powers = numpy.diag(s.conj().T @ output_power_gram @ s).real / numpy.diag(input_power_gram).real
diagnostics: dict[str, Any] = {
'axis': axis,
'left_mode_numbers': left_modes_i,
@ -781,6 +845,9 @@ def get_s_fdfd(
'electric_fields': tuple(electric_fields),
'magnetic_fields': tuple(magnetic_fields),
'incident_amplitudes': incident_amplitudes,
'incoming_amplitudes': incoming_amplitudes,
'incoming_power_gram': input_power_gram,
'outgoing_power_gram': output_power_gram,
'raw_outgoing_amplitudes': raw_outgoing,
'guided_powers': guided_powers,
'unresolved_powers': 1.0 - guided_powers,

24
meanas/fdfd/waveguide_cyl.py
View file

@ -86,7 +86,7 @@ large-radius-compatible identities
$$
E_z =
\frac{1}{\imath \beta} T_a \epsilon_{zz}^{-1}
\frac{1}{\imath \beta} \epsilon_{zz}^{-1}
\begin{bmatrix}
\hat{\partial}_r T_b \epsilon_{rr} &
\hat{\partial}_y T_a \epsilon_{yy}
@ -304,6 +304,11 @@ def linear_wavenumbers(
Calculate linear wavenumbers (1/distance) based on angular wavenumbers (1/rad)
and the mode's energy distribution.
The effective radius is weighted by transverse electric energy, using the
physical Yee cell areas for each component. Radial E uses dx_e * dy_h;
transverse E uses dx_h * dy_e. Real coordinate widths define these areas
and radii when complex PML stretching is present.
Args:
e_xys: Vectorized mode fields with shape (num_modes, 2 * x *y)
angular_wavenumbers: Wavenumbers assuming fields have theta-dependence of
@ -321,14 +326,19 @@ def linear_wavenumbers(
shape2d = (len(dxes[0][0]), len(dxes[0][1]))
epsilon2d = unvec(epsilon, shape2d)[:2]
ra = rmin + numpy.cumsum(dxes[0][0])
rb = rmin + dxes[0][0] / 2.0 + numpy.concatenate((
numpy.zeros(1, dtype=dxes[1][0].dtype),
numpy.cumsum(dxes[1][0][:-1]),
physical_widths = [[numpy.real(dx) for dx in family] for family in dxes]
ra = rmin + numpy.cumsum(physical_widths[0][0])
rb = rmin + physical_widths[0][0] / 2.0 + numpy.concatenate((
numpy.zeros(1),
numpy.cumsum(physical_widths[1][0][:-1]),
))
areas = numpy.stack((
numpy.outer(physical_widths[0][0], physical_widths[1][1]),
numpy.outer(physical_widths[1][0], physical_widths[0][1]),
))
for ii in range(angular_wavenumbers.size):
efield = unvec(e_xys[ii], shape2d, 2)
energy = numpy.real((efield * efield.conj()) * epsilon2d)
energy = numpy.real((efield * efield.conj()) * epsilon2d) * areas
er_energy_vs_r = energy[0].sum(axis=1)
ey_energy_vs_r = energy[1].sum(axis=1)
energy_vs_r = er_energy_vs_r + ey_energy_vs_r
@ -405,7 +415,7 @@ def exy2e(
n_pts = dxes[0][0].size * dxes[0][1].size
exy2ez = (
Ta @ epsilon_z_inv
epsilon_z_inv
@ sparse.hstack((Dbx @ Tb @ epsilon_x,
Dby @ Ta @ epsilon_y))
/ (1j * wavenumber)

45
meanas/test/test_eme_numerics.py
View file

@ -2,6 +2,7 @@ from typing import cast
import numpy
import pytest
from scipy.sparse.linalg import spsolve
from ..fdmath import vec
from ..fdfd import eme, waveguide_2d, waveguide_3d, waveguide_cyl
@ -547,6 +548,50 @@ def test_get_s_fdfd_from_2d_ports_identical_guides_are_nearly_transmissive() ->
assert abs(abs(ss[0, 1]) - 1.0) < 0.03
def test_multimode_fdfd_calibrates_the_full_incoming_matrix(monkeypatch) -> None:
# Exactly degenerate polarizations have different source/monitor bases.
# A uniform guide must still have zero interface reflection and conserve flux.
original_solve = waveguide_2d.solve_mode
seed = 0
def seeded_solve(*args, **kwargs):
nonlocal seed
seed += 1
return original_solve(*args, **kwargs, rng=seed)
decompositions = []
original_decomposition = waveguide_3d._port_decomposition
def capture_decomposition(*args, **kwargs):
result = original_decomposition(*args, **kwargs)
decompositions.append(result)
return result
monkeypatch.setattr(waveguide_2d, 'solve_mode', seeded_solve)
monkeypatch.setattr(waveguide_3d, '_port_decomposition', capture_decomposition)
epsilon = numpy.ones((3, 2, 2))
widths = [[numpy.full(2, 0.5), numpy.full(2, 0.5)] for _ in range(2)]
ss, diagnostics = waveguide_3d.get_s_fdfd_from_2d_ports(
epsilon, epsilon, omega=0.5, dxes_2d=widths, mode_numbers=[0, 1],
pml_cells=4, pml_buffer_cells=1, source_monitor_cells=2, monitor_interface_cells=2,
source_scales=[2, 0.3, 1j, 0.7j], matrix_solver=spsolve, return_diagnostics=True,
)
incoming, outgoing = [], []
for field in diagnostics['electric_fields']:
li, lo = waveguide_3d._decompose_port_field(decompositions[0], field)
ri, ro = waveguide_3d._decompose_port_field(decompositions[1], field)
incoming.append(numpy.concatenate((li, ri)))
outgoing.append(numpy.concatenate((lo, ro)))
incoming, outgoing = numpy.column_stack(incoming), numpy.column_stack(outgoing)
assert numpy.linalg.norm(incoming - numpy.diag(numpy.diag(incoming))) > 0.1
assert_close(ss @ incoming, outgoing, atol=1e-11, rtol=1e-11)
assert_close(diagnostics['incoming_amplitudes'], incoming, atol=1e-12)
assert_close(ss[:2, :2], 0, atol=1e-11)
assert_close(ss[2:, 2:], 0, atol=1e-11)
assert_close(ss.conj().T @ diagnostics['outgoing_power_gram'] @ ss, diagnostics['incoming_power_gram'], atol=1e-11)
assert_close(diagnostics['guided_powers'], 1, atol=1e-11)
def test_get_s_fdfd_from_2d_ports_mismatch_returns_finite_diagnostics() -> None:
left = _fdfd_cross_section(2.25)
right = _fdfd_cross_section(1.5)

1
meanas/test/test_waveguide_2d_modes.py
View file

@ -1261,6 +1261,7 @@ def test_richer_complex_basis_reduces_driven_fdfd_monitor_residual() -> None:
dxes_2d=dxes_2d,
mode_numbers=[0],
axis=2,
prop_dx=0.15, # Resolve both port modes below the longitudinal band edge.
pml_cells=4,
pml_buffer_cells=2,
source_monitor_cells=4,

10
meanas/test/test_waveguide_2d_numerics.py
View file

@ -1,10 +1,11 @@
import numpy
import pytest
from numpy.linalg import norm
from numpy.testing import assert_allclose
from scipy import linalg
from ..fdmath import vec
from ..fdfd import waveguide_2d
from ..fdfd import waveguide_2d, scpml
OMEGA = 1 / 1500
@ -121,7 +122,8 @@ def test_waveguide_2d_sensitivity_matches_finite_difference() -> None:
assert 0.4 < ratio < 1.8
def test_sensitivity_matches_central_differences_on_graded_yee_mesh() -> None:
@pytest.mark.parametrize('absorption', ['none', 'pml', 'material'])
def test_sensitivity_matches_central_differences_on_graded_yee_mesh(absorption) -> None:
rng = numpy.random.default_rng(4)
shape = (4, 3)
omega = 0.7
@ -129,6 +131,10 @@ def test_sensitivity_matches_central_differences_on_graded_yee_mesh() -> None:
magnetic_widths = [rng.uniform(0.2, 1.2, n) for n in shape]
electric_widths = [0.5 * (dx + numpy.roll(dx, -1)) for dx in magnetic_widths]
dxes = [electric_widths, magnetic_widths]
if absorption == 'pml':
dxes = scpml.stretch_with_scpml(dxes, axis=0, polarity=-1, omega=omega, thickness=1, max_imag_ratio=0.3)
elif absorption == 'material':
epsilon = epsilon * (1 - 0.15j * rng.uniform(0.5, 1.5, epsilon.size))
values, vectors = linalg.eig(waveguide_2d.operator_e(omega, dxes, epsilon).toarray())
selected = numpy.argmax(values.real)
beta = numpy.sqrt(values[selected])

81
meanas/test/test_waveguide_mode_helpers.py
View file

@ -153,8 +153,8 @@ def test_waveguide_3d_solve_mode_and_expand_e_are_phase_consistent(
line_selection[axis] = slice(None)
values = expanded[component][tuple(line_selection)]
ratios = values[1:] / values[:-1]
phase_positions = numpy.real(dxes[0][axis]).cumsum()
expected_ratios = numpy.exp(-1j * result['wavenumber'] * numpy.diff(phase_positions))
intervals = numpy.real(dxes[1][axis][1:] if component == axis else dxes[0][axis][:-1])
expected_ratios = numpy.exp(-1j * result['wavenumber'] * intervals)
numpy.testing.assert_allclose(ratios, expected_ratios, rtol=1e-6, atol=1e-9)
@ -422,7 +422,7 @@ def test_waveguide_cyl_reconstruction_is_finite_radius_consistent(use_mu: bool)
eps_r, eps_y, eps_z = numpy.split(epsilon, 3)
mu_r, mu_y, mu_z = numpy.split(mu_full, 3)
ez_rhs = ta * (
ez_rhs = (
(Dbx @ (tb * eps_r * er))
+ (Dby @ (ta * eps_y * ey))
) / eps_z
@ -469,8 +469,79 @@ def test_waveguide_cyl_linear_wavenumbers_use_component_radii() -> None:
rmin=rmin,
)
numpy.testing.assert_allclose(er_linear[0], angular_wavenumber[0] / rb.mean())
numpy.testing.assert_allclose(ey_linear[0], angular_wavenumber[0] / ra.mean())
numpy.testing.assert_allclose(er_linear[0], angular_wavenumber[0] / numpy.average(rb, weights=dxes[0][0]))
numpy.testing.assert_allclose(ey_linear[0], angular_wavenumber[0] / numpy.average(ra, weights=dxes[1][0]))
@pytest.mark.parametrize('use_mu', [False, True])
def test_cylindrical_modes_obey_the_full_ampere_curl(use_mu) -> None:
shape, omega, rmin = (4, 3), 0.7, 2.0
dxes = [[numpy.full(n, 0.4) for n in shape] for _ in range(2)]
rng = numpy.random.default_rng(4)
epsilon = vec(1 + rng.uniform(0, 3, (3, *shape)))
mu = vec(1 + rng.uniform(0, 2, (3, *shape))) if use_mu else numpy.ones_like(epsilon)
matrix = waveguide_cyl.cylindrical_operator(omega, dxes, epsilon, rmin, mu).toarray()
values, vectors = linalg.eig(matrix)
selected = numpy.argmax(values.real)
beta = numpy.sqrt(values[selected])
electric = waveguide_cyl.exy2e(beta * rmin, dxes, rmin, epsilon) @ vectors[:, selected]
magnetic = waveguide_cyl.e2h(beta * rmin, omega, dxes, rmin, mu) @ electric
ta, tb = (metric.diagonal() for metric in waveguide_cyl.dxes2T(dxes, rmin))
dbx, dby = deriv_back(dxes[1])
hr, hy, hz = numpy.split(magnetic, 3)
# Physical cylindrical Ampere curl, independently of electric reconstruction.
curl_h = numpy.concatenate((
1j * beta * hy / tb + dby @ hz,
-1j * beta * hr / ta - (dbx @ (tb * hz)) / ta,
-dby @ hr + dbx @ hy,
))
numpy.testing.assert_allclose(curl_h, 1j * omega * epsilon * electric, atol=1e-11, rtol=1e-10)
def test_cylindrical_mean_radius_is_invariant_under_constant_density_refinement() -> None:
for widths in ([1.0, 3.0], [1.0, 1.0, 1.0, 1.0]):
n = len(widths)
dxes = [[numpy.array(widths), numpy.ones(1)] for _ in range(2)]
electric_xy = numpy.concatenate((numpy.ones(n), numpy.zeros(n)))[None, :].astype(complex)
beta = waveguide_cyl.linear_wavenumbers(electric_xy, [12.0], numpy.ones(3 * n), dxes, rmin=10.0)
numpy.testing.assert_allclose(beta, 1.0)
@pytest.mark.parametrize('axis', [0, 1, 2])
@pytest.mark.parametrize('polarity', [-1, 1])
def test_graded_expansion_uses_each_components_actual_yee_intervals(axis, polarity) -> None:
shape = [1, 1, 1]
shape[axis] = 4
dxes = [[numpy.ones(n) for n in shape] for _ in range(2)]
dxes[0][axis] = numpy.array([1.0, 2.0, 3.0, 4.0])
dxes[1][axis] = numpy.array([0.6, 0.8, 1.0, 1.2])
source = [slice(None)] * 3
source[axis] = slice(1, 2)
field = numpy.zeros((3, *shape), dtype=complex)
field[(slice(None), *source)] = 1 + 0.5j
expanded = waveguide_3d.expand_e(field, 0.2, dxes, axis, polarity, source)
for component in range(3):
line = expanded[component].ravel()
intervals = dxes[1][axis][1:] if component == axis else dxes[0][axis][:-1]
numpy.testing.assert_allclose(line[1:] / line[:-1], numpy.exp(-1j * polarity * 0.2 * intervals))
numpy.testing.assert_array_equal(expanded[(slice(None), *source)], field[(slice(None), *source)])
@pytest.mark.parametrize('beta', [2.0, 3.0])
def test_grid_band_edge_rejects_power_normalization_and_selects_decay(beta) -> None:
shape = (1, 1, 5)
dxes = [[numpy.ones(n) for n in shape] for _ in range(2)]
kwargs = dict(omega=1.0, dxes=dxes, axis=2, polarity=1,
slices=(slice(None), slice(None), slice(0, 1)), epsilon=numpy.full((3, *shape), beta ** 2))
with pytest.raises(ValueError, match='grid band edge'):
waveguide_3d.mode_from_2d(numpy.array([1, 0], dtype=complex), beta, **kwargs)
mode = waveguide_3d.mode_from_2d(numpy.array([1, 0], dtype=complex), beta, normalization='lorentz', **kwargs)
assert mode['wavenumber'].imag <= 0
expanded = waveguide_3d.expand_e(mode['E'], mode['wavenumber'], dxes, 2, 1, kwargs['slices'])
ratio = abs(expanded[0, 0, 0, 1] / expanded[0, 0, 0, 0])
assert ratio <= 1 + 1e-14
if beta > 2:
assert ratio < 1
def test_waveguide_cyl_exy2e_and_exy2h_return_finite_full_fields() -> None: