[eme] characterize orientation bias under modal truncation
Compare mixed and Galerkin testing against exact polarization/Fresnel interfaces and independent driven FDFD solves. Show complete-basis recovery, omitted-channel power, and the limits of longitudinal refinement, orientation averaging, and Euclidean least squares. Keep the default formulation unchanged and document its approximation limits. Validation: 1689 tests passed, 584 skipped; Ruff and documentation build passed.
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@ -89,3 +89,65 @@ devices. Singular interface and cascade systems also raise contextual errors
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rather than silently substituting pseudoinverses.
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::: meanas.fdfd.eme
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## Orientation bias under modal truncation
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Mixed testing is a projected approximation: H continuity is tested on the left
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and E continuity on the right. Swapping the ports changes the testing spaces
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when the retained modal subspaces are incomplete. This effect survives checks
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of reciprocity and unitarity.
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A reproducible Maxwell counterexample is a uniform isotropic medium with two
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degenerate linear polarizations. Keep one polarization on each port, rotated
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so their overlap is `a=0.8`. There is no material interface: true reflection is
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zero and transmission into the retained receiver is `a`. The other polarization
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carries the remaining transmitted power, `1-a**2=0.36`.
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| Calculation | R_left | R_right | T_left_to_right | T_right_to_left |
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| --- | ---: | ---: | ---: | ---: |
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| Exact full basis, then restrict ports | 0 | 0 | 0.8 | 0.8 |
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| Truncated mixed | 0.219512 | -0.219512 | 0.975610 | 0.975610 |
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| Swap ports, mixed, then reorder | -0.219512 | 0.219512 | 0.975610 | 0.975610 |
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| Truncated left Galerkin | 0 | 0 | 1.25 | 0.8 |
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| Euclidean E/H least squares | 0 | 0 | 0.8 | 0.8 |
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An independent driven 3D FDFD solve with PML agrees with the exact reference
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from either incidence direction: transmission is 0.8 within `1e-12`, with
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reflection below `2e-8` in the test fixture. Keeping both polarizations makes
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mixed testing match the exact complete scattering matrix and restores
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port-relabeling covariance.
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For the single retained polarization, mixed testing yields
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`r=(1-a**2)/(1+a**2)` and `t=2*a/(1+a**2)`. The spurious reflection is quadratic
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in a small angle between adjacent retained polarizations. However, longitudinal
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refinement alone is insufficient: rotating a single retained vector through a
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uniform degenerate medium makes the sampled chain approach artificial following
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of that vector (`|T|` approaches 1), while the physical endpoint overlap remains
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0.8. Keeping the complete degenerate block recovers the physical result at every
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sampling density tested.
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Two tempting corrections are insufficient. Averaging the two port orientations
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cancels reflection in this example but leaves transmission at 0.975610. Plain
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Euclidean E/H least squares gives the correct uniform-medium result, but is not
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a general replacement. A second fixture uses exact degenerate modes of diagonal
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positive epsilon/mu tensors, with electric scaling `(1,2)` on the left and
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`(1,3)` on the right and a 30-degree polarization rotation on both ports. After
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single-mode truncation, least squares has unequal directional transmissions
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(approximately 0.850839 and 1.092163) and a maximum singular value of 1.104947.
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The complete Fresnel solution is reciprocal and lossless.
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The mixed default remains unchanged. Retain complete coupled/degenerate blocks
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when the geometry can mix them, check modal-basis convergence as well as section
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sampling, and compare against independent full-field results. Neither orientation
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averaging nor gain/reciprocity postprocessing recovers omitted-channel physics.
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Orientation discrepancy is a useful diagnostic, not a bound on transmission
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error. These examples use real power-orthonormal propagating modes; for general
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complex modes, the reciprocity constraint involves the modal orthogonality
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matrix, as derived by [Svendsen, Haakestad, and Skaar](https://arxiv.org/abs/1301.2458).
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Reproduce the analytic, full-basis, candidate-formulation, taper-refinement, and
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driven-FDFD checks with:
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```sh
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OPENBLAS_NUM_THREADS=1 python -m pytest meanas/test/test_eme_orientation.py -q
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```
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@ -512,6 +512,12 @@ def get_s(
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use the primal fields. Wavenumbers label the modes and are not used in the
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interface continuity equations. Forward modes imply backward (E, -H).
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Mixed testing assigns different test families to the two ports. Incomplete
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modal subspaces can therefore give different approximations after a port
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exchange, even when each result is reciprocal and unitary. Establish basis
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convergence, especially within coupled or degenerate blocks; do not infer
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physical accuracy from these matrix constraints alone.
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``force_nogain`` caps Euclidean singular values; ``force_reciprocal``
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symmetrizes S. These optional numerical projections are not general power
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or reciprocity tests for open, evanescent, or non-power-normalized bases.
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@ -734,6 +740,8 @@ def get_taper_s(
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Each interval propagates half its length in either adjacent basis. Accuracy
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is controlled by section sampling and basis/PML convergence, not ODE
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tolerances. Standard outgoing/incoming [left, right] ordering is used.
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Longitudinal refinement alone cannot repair omitted coupled/degenerate
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modes or remove the orientation bias of incomplete interface projections.
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"""
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prepared = prepare_taper(sections, dxes, formulation=formulation, measure=measure)
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return evaluate_taper_s(prepared, force_nogain=force_nogain, force_reciprocal=force_reciprocal)
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210
meanas/test/test_eme_orientation.py
Normal file
210
meanas/test/test_eme_orientation.py
Normal file
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@ -0,0 +1,210 @@
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"""Characterize truncation bias against exact reciprocal Maxwell interfaces.
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These tests intentionally document mixed/Galerkin projection limitations; they
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are not passivity-enforcement tests. Both complete modal bases and an independent
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driven FDFD solve establish the reference for the truncated polarization case.
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"""
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import numpy as np
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import pytest
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from numpy.testing import assert_allclose
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from scipy.sparse.linalg import spsolve
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from ..fdmath import vec, unvec
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from ..fdfd import eme, scpml, solvers, waveguide_2d, waveguide_3d
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DXES = [[np.ones(1), np.ones(1)] for _ in range(2)]
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def _rotation(angle):
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return np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])
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def _polarizations(angle, scales=(1., 1.)):
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"""Two power-orthonormal signed-beta=1 modes on a one-cell transverse grid.
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E_xy = sqrt(2) diag(scales) Q, and J^-1 H_xy = sqrt(2) diag(1/scales) Q.
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They solve Maxwell with diagonal epsilon=(1/sx^2,1/sy^2,1) and
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mu=(sy^2,sx^2,1) at omega=1, including nonparallel E/H traces.
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"""
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rotation = _rotation(angle)
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electric = np.diag(scales) @ rotation
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magnetic = np.diag(1 / np.asarray(scales)) @ rotation
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return [
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(np.sqrt(2) * np.array([electric[0, i], electric[1, i], 0], complex),
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np.sqrt(2) * np.array([-magnetic[1, i], magnetic[0, i], 0], complex))
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for i in range(2)
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]
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def _scattering(left, right, formulation='mixed'):
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return eme.get_s(left, np.ones(len(left)), right, np.ones(len(right)), DXES, formulation=formulation)
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def _reorder_ports(matrix):
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n = matrix.shape[0] // 2
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order = [*range(n, 2*n), *range(n)]
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return matrix[np.ix_(order, order)]
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def _fresnel_reference(left_angle, right_angle, left_scales=(1., 1.), right_scales=(1., 1.)):
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"""Independent exact two-polarization interface in the supplied port bases."""
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yl, yr = 1 / np.asarray(left_scales)**2, 1 / np.asarray(right_scales)**2
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reflection = np.diag((yl - yr) / (yl + yr))
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transmission = np.diag(2 * np.sqrt(yl * yr) / (yl + yr))
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ql, qr = _rotation(left_angle), _rotation(right_angle)
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return np.block([[ql.T @ reflection @ ql, ql.T @ transmission @ qr],
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[qr.T @ transmission @ ql, -qr.T @ reflection @ qr]])
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def _trace_least_squares(left, right):
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"""Research comparator: unweighted Euclidean tangential E/H least squares.
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This is deliberately not a public formulation: correcting the isotropic
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polarization example does not establish reciprocity or passivity generally.
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"""
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el = np.column_stack([e[:2] for e, _h in left])
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hl = np.column_stack([h[:2] for _e, h in left])
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er = np.column_stack([e[:2] for e, _h in right])
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hr = np.column_stack([h[:2] for _e, h in right])
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outgoing = np.block([[el, -er], [-hl, -hr]])
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incoming = np.block([[-el, er], [-hl, -hr]])
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return np.linalg.lstsq(outgoing, incoming, rcond=None)[0]
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@pytest.mark.parametrize('overlap', [.5, .8, .95])
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def test_truncated_mixed_polarizations_show_orientation_bias_despite_unitarity(overlap):
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angle = np.arccos(overlap)
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left, right = _polarizations(0)[:1], _polarizations(angle)[:1]
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result = _scattering(left, right)
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reversed_result = _reorder_ports(_scattering(right, left))
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reflection = (1-overlap**2) / (1+overlap**2)
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transmission = 2*overlap / (1+overlap**2)
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assert_allclose(result, [[reflection, transmission], [transmission, -reflection]], atol=1e-14)
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assert_allclose(reversed_result, [[-reflection, transmission], [transmission, reflection]], atol=1e-14)
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assert_allclose(result.T @ result, np.eye(2), atol=1e-14)
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assert_allclose(result, result.T, atol=1e-14)
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assert_allclose(np.linalg.norm(result-reversed_result), 2*np.sqrt(2)*reflection, atol=1e-14)
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@pytest.mark.parametrize(('left_scales', 'right_scales'), [((1., 1.), (1., 1.)), ((1., 2.), (1., 3.))])
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def test_complete_maxwell_bases_recover_fresnel_and_port_relabeling(left_scales, right_scales):
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left_angle, right_angle = np.pi/6, np.arccos(.8)
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left, right = _polarizations(left_angle, left_scales), _polarizations(right_angle, right_scales)
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# Establish that the synthetic fields are actual degenerate Maxwell modes,
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# not arbitrary profiles or mixed vectors with different eigenvalues.
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for modes, scales in ((left, left_scales), (right, right_scales)):
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sx, sy = scales
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epsilon, mu = np.array([1/sx**2, 1/sy**2, 1]), np.array([sy**2, sx**2, 1])
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operator = waveguide_2d.operator_e(1, DXES, epsilon, mu)
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for e, h in modes:
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assert_allclose(operator @ e[:2], e[:2], atol=1e-14)
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assert_allclose(waveguide_2d.exy2h(1, 1, DXES, epsilon, mu) @ e[:2], h, atol=1e-14)
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gram = [[waveguide_2d.lorentz_overlap(a, b, DXES) for b in modes] for a in modes]
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assert_allclose(gram, np.eye(2), atol=1e-14)
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result = _scattering(left, right)
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expected = _fresnel_reference(left_angle, right_angle, left_scales, right_scales)
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assert_allclose(result, expected, atol=1e-14)
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assert_allclose(_reorder_ports(_scattering(right, left)), expected, atol=1e-14)
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assert_allclose(result.T @ result, np.eye(4), atol=1e-14)
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assert_allclose(result, result.T, atol=1e-14)
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def test_missing_polarization_accounts_for_retained_submatrix_power_deficit():
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overlap = .8
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full = _scattering(_polarizations(0), _polarizations(np.arccos(overlap)))
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retained = full[np.ix_([0, 2], [0, 2])]
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assert_allclose(retained, [[0, overlap], [overlap, 0]], atol=1e-14)
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assert_allclose(abs(full[3, 0])**2, 1-overlap**2, atol=1e-14)
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assert_allclose(np.sum(abs(full[:, 0])**2), 1, atol=1e-14)
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assert_allclose(np.sum(abs(retained[:, 0])**2), overlap**2, atol=1e-14)
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def test_posthoc_orientation_averaging_does_not_recover_physical_transmission():
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overlap = .8
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left, right = _polarizations(0)[:1], _polarizations(np.arccos(overlap))[:1]
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averaged = .5 * (_scattering(left, right) + _reorder_ports(_scattering(right, left)))
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assert_allclose(averaged[0, 0], 0, atol=1e-14)
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assert_allclose(averaged[1, 1], 0, atol=1e-14)
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assert abs(averaged[1, 0] - overlap) > .17
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def test_galerkin_testing_does_not_resolve_truncated_polarization_error():
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overlap = .8
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left, right = _polarizations(0)[:1], _polarizations(np.arccos(overlap))[:1]
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result = _scattering(left, right, 'galerkin')
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assert_allclose(result, [[0, overlap], [1/overlap, 0]], atol=1e-14)
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assert not np.allclose(result, result.T)
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def test_trace_least_squares_recovers_uniform_reference_but_is_not_a_general_fix():
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overlap = .8
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left, right = _polarizations(0)[:1], _polarizations(np.arccos(overlap))[:1]
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assert_allclose(_trace_least_squares(left, right), [[0, overlap], [overlap, 0]], atol=1e-14)
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angle = np.pi/6
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left, right = _polarizations(angle, (1., 2.)), _polarizations(angle, (1., 3.))
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reference = _fresnel_reference(angle, angle, (1., 2.), (1., 3.))
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assert_allclose(_trace_least_squares(left, right), reference, atol=1e-14)
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truncated = _trace_least_squares(left[:1], right[:1])
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# All traces come from the exact reciprocal, lossless Maxwell bases above.
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# Euclidean residual minimization nevertheless violates both constraints
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# after truncation. Port-relabeling covariance alone is not sufficient.
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assert np.linalg.norm(truncated-truncated.T) > .3
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assert np.linalg.svd(truncated, compute_uv=False).max() > 1.10
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assert_allclose(_reorder_ports(_trace_least_squares(right[:1], left[:1])), truncated, atol=1e-14)
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@pytest.mark.parametrize('side', [0, 1])
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def test_driven_fdfd_confirms_uniform_polarization_reference(side):
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omega, thickness = np.pi/8, 24
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size = 2*thickness + 64
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physical = [[np.ones(1), np.ones(1), np.ones(size)] for _ in range(2)]
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epsilon = np.ones((3, 1, 1, size))
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solve_widths = physical
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for polarity in (-1, 1):
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solve_widths = scpml.stretch_with_scpml(solve_widths, 2, polarity, omega, thickness=thickness)
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overlap = .8
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profiles = [np.array([1., 0.]), np.array([overlap, np.sqrt(1-overlap**2)])]
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polarity = 1 if side == 0 else -1
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source_index = thickness+4 if side == 0 else thickness+52
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slices = (slice(None), slice(None), slice(source_index, source_index+1))
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mode = waveguide_3d.mode_from_2d(profiles[side], omega, omega=omega, dxes=physical,
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axis=2, polarity=polarity, slices=slices, epsilon=epsilon)
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current = waveguide_3d.compute_source(mode['E'], mode['wavenumber'], omega, physical, 2, polarity, slices, epsilon)
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field = unvec(solvers.generic(omega, solve_widths, vec(current), vec(epsilon), matrix_solver=spsolve), epsilon.shape[1:])
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left_index, right_index = thickness+12, thickness+40
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beta = 2*np.arcsin(omega/2)
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phase = np.exp(-1j*beta)
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def waves(index):
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e0, e1 = field[:2, 0, 0, index], field[:2, 0, 0, index+1]
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forward = (e1 - e0/phase) / (phase - 1/phase)
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return forward, e0-forward
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fl, bl = waves(left_index)
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fr, br = waves(right_index)
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incoming = np.dot(profiles[side], fl if side == 0 else br)
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reflected = np.dot(profiles[side], bl if side == 0 else fr) / incoming
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transmitted = np.dot(profiles[1-side], fr if side == 0 else bl) / incoming
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transmitted *= np.exp(1j*beta*(right_index-left_index))
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assert abs(reflected) < 5e-8
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assert_allclose(transmitted, overlap, atol=1e-12)
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def test_longitudinal_refinement_cannot_repair_an_omitted_degenerate_polarization():
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angle = np.arccos(.8)
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reflection_magnitudes = []
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for count in (3, 9, 33):
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complete = [eme.ModeSection(z, _polarizations(angle*z), [1., 1.]) for z in np.linspace(0, 1, count)]
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truncated = [eme.ModeSection(section.z, section.modes[:1], [1.]) for section in complete]
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full = eme.get_taper_s(complete, DXES)
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expected = _fresnel_reference(0, angle).astype(complex) * np.exp(-1j)
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assert_allclose(full, expected, atol=1e-13)
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small = eme.get_taper_s(truncated, DXES)
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reflection_magnitudes.append(abs(small[0, 0]))
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# There is no material taper: the basis is merely rotated within a
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# degenerate subspace. Restricting the exact result gives |T|=.8, while
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# the truncated chain approaches artificial adiabatic following, |T|=1.
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assert abs(small[1, 0]) > .99
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assert reflection_magnitudes[-1] < reflection_magnitudes[0] / 10
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assert_allclose(abs(expected[2, 0]), .8, atol=1e-14)
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