Source
engines/quan_loncar_meep_lab/notes/stack_pole_fails_its_control.md · assembled 2026-09-05 15:22 UTC.
The layered pole failed its first control, and why¶
Resolved 2026-08-02. build_stack_pole had never been asked to return an
answer that was already known. The first time it was, it missed by a factor
of four, and the cause was a boundary condition applied to seven percent of
a symmetry plane.
The control¶
The small silicon fixture is a 700 x 220 nm membrane at n = 3.46 in air with a created level-set defect. The membrane path solves its pole at 1591.675 nm, Q 52.888 in 152,415 unknowns.
A membrane is a layer stack with one etched layer and air underneath, so both builders take the same article, the same settled field, the same aperture faces, and the same settings.
| lambda (nm) | Q | ratio to membrane | |
|---|---|---|---|
| membrane, certified | 1591.675 | 52.888 | -- |
| layered, before | 1546.672 | 12.691 | 0.240 |
| layered, after | 1591.730 | 52.610 | 0.995 |
After the repair the agreement is 0.003% in wavelength and 0.5% in Q, at residual 1.0e-13, and the Ey selector returns it as the first candidate with localization 0.867.
What it was¶
Integrating boundary area over the y = 0 plane, by region name:
| membrane | layered before | layered after | |
|---|---|---|---|
pec_y, constrained |
3.3807 | 6.2738 | 6.7613 |
default, free |
0.0000 | 0.4876 | 0.0001 |
| total | 3.3807 | 6.7614 | 6.7614 |
The plane was fully meshed -- 6.7614 is exactly twice the membrane's -- but 7.2% of it carried no name, and an unnamed boundary in this space is a magnetic wall rather than the intended electric one. The unnamed patch is the beam core's own cross-section at y = 0: 0.865 um^2 of slab, less the holes that straddle the plane.
So the mirror reflected one way over the cladding and the opposite way through the core. That is not a symmetry plane. It scatters the mode it exists to reflect, at exactly the place the field is largest, which is why the high-Q branch was absent from the spectrum entirely rather than merely shifted.
The names were assigned to the outer box before the booleans that split
it, and where the core solid meets y = 0 the boolean produces a fresh face
that inherits nothing. This is the same failure the Bloch controls hit on
their periodic faces, for the same reason, and it is why this was worth
looking for. The repair names exterior faces by position on the glued
shape, so no name depends on propagation surviving a boolean. Constrained
degrees of freedom went from 20,376 to 30,060, twice the membrane's 14,934.
Why it took a measurement to find¶
Everything cheap to check was correct. The two builders mesh the same dielectric to five digits -- core volume, ambient volume, integral of epsilon, PML region volumes. The domain extents agree. The polynomial order is uniformly two in both. The boundary names exist in both meshes. The Dirichlet strings are identical. Reading the code does not reveal it, and the PML sweep actively pointed away from it: the membrane was indifferent to alpha and thickness while the layered path scattered by a factor of two, which reads like an absorber fault.
Only integrating the area carried by each boundary name shows a 7.2% shortfall. The elimination that got there was, in order: geometry, domain, boundary conditions as written, face identification, polynomial order, then the absorber -- with the last step being to stop reading and measure.
What it invalidates¶
Every Device23 pole solved before this repair, which is all of them. Both branches in the two-branch note -- the Q 93-132 family and the converged Q 52.9-55.9 one -- were solved through a broken mirror, as were the pilot and selective fidelity rejections and the 21x disagreement with Tidy3D that motivated this whole line of work.
None of those numbers should be quoted. They must be resolved again on the repaired solver before anything is concluded from them.
Reproducing¶
PYTHONPATH=engines/quan_loncar_meep_lab/package/quan_meep/src \
engines/quan_loncar_meep_lab/tools/control_stack_pole_against_membrane.py \
--fixture small --candidates 24 \
--output benchmarks/small_20260801/stack_pole_membrane_control.json
Exits zero. Eight candidates are not enough for this article; the Arnoldi space converges nothing and the run says so rather than returning a wrong answer.
The mirror itself is pinned by test_the_symmetry_plane_is_entirely_pec so
this cannot regress silently.
The coupling measurement disagrees far worse than the pole did¶
First local two-port measurement of the seed, 2026-08-02, at a coarsened
540,712-unknown driven mesh (curvature_safety 1.0 against the pilot's
2.0), because the acceptance fidelity does not factorize in this machine's
memory:
| quantity | local | FDTD seed |
|---|---|---|
| beta_guided | 0.015129 | 0.878603 |
| beta forward | 0.007015 | -- |
| beta backward | 0.008114 | -- |
A factor of 58. Both campaign gates fail, and feedthrough_qualified is
true, so the port certificate is not what is refusing.
The shape of the error is the clue. The two directions are balanced to within 14% of each other, which is what a symmetric cavity should give, so the numerator's directional split behaves. It is the magnitude that is wrong, and that implicates the denominator rather than the port projection.
beta_guided is guided power over the physical full-y dipole work. The
work done by an ideal point current contains a near-field self-energy term
that does not converge as the mesh is refined -- it grows without bound as
the singular field is resolved. If the denominator carries that term raw,
beta_guided is not a fraction of anything physical and shrinks as the
mesh improves. The engine already knows this: the creation-phase work
records a stack_reference_power Purcell normalization for exactly this
reason.
The sweep refutes that, by its sign¶
| curvature_safety | dof | beta_guided | forward | backward |
|---|---|---|---|---|
| 0.6 | 284,818 | 0.004389 | 0.001907 | 0.002483 |
| 1.0 | 540,712 | 0.015129 | 0.007015 | 0.008114 |
| 1.4 | 836,340 | 0.025930 | 0.016559 | 0.009372 |
beta_guided rises monotonically, roughly doubling for each doubling
of the degree count, with no hint of a plateau. A denominator carrying a
divergent point-source self-energy would make it fall. The hypothesis
above is wrong, and the sign is what kills it.
What grows like that is an under-resolved numerator: the guided power projected onto TE0 recovering as the field becomes representable. None of the three points is converged, so none of them measures the seed's coupling, and extrapolating them does not reach 0.879 at any mesh this machine can hold.
It is not another unnamed boundary¶
The obvious next suspect was the bug already found, in the path not yet
audited -- the driven domain runs use_y_symmetry=False and had never had
its boundary area integrated. It is clean:
| region | half-y | full-y | twice half-y |
|---|---|---|---|
outer |
58.3768 | 116.7537 | 116.7536 |
default |
92.1895 | 184.3849 | 184.3790 |
pec_y |
29.5687 | none, correctly | -- |
Every exterior face is named, and the full-y areas are exactly twice the half-y ones. The repair covers this path too.
So the coupling error is not a boundary condition and not the denominator. It is resolution, in a measurement whose convergence has never been established, and the acceptance fidelity that might settle it does not fit in this machine's memory.
Until that is settled these numbers are diagnostics and nothing more. They are not evidence that the seed couples poorly, and they must not be compared to the 95% gate.
The drive and the driven mesh were never the same article¶
The sweep above hides its own explanation. The drive frequency is the
pole's, and the pole is solved at curvature_safety 2.0. Every driven mesh
in that table is coarser than 2.0, and beta_guided rises as the driven
mesh approaches the mesh the drive came from.
Solving the pole on the driven mesh itself says how far off resonance the
drive was. At curvature_safety 0.6, order one:
| wavelength | Q | localization | |
|---|---|---|---|
drive, from the curvature_safety 2.0 selective pole |
756.734 nm | 1640.617 | 0.702 |
| what the driven mesh actually carries | 763.836 nm | 59.958 | 0.578 |
The drive was 7.10 nm from that mesh's own resonance, against a
linewidth of 12.74 nm. The driven response is a Lorentzian in that
detuning, so this looked like a suppression of about 2.2x applied before
any physics was asked of the measurement. --match-pole-fidelity removes
it by solving the drive pole on the same discretization as the driven
mesh.
Measuring it refutes it¶
Driving that mesh at its own resonance makes the coupling smaller:
| drive | beta_guided | forward | backward | imbalance |
|---|---|---|---|---|
| 756.734 nm, from the selective pole | 0.004389 | 0.001907 | 0.002483 | 30% |
| 763.836 nm, this mesh's own resonance | 0.002984 | 0.001075 | 0.001909 | 78% |
Same driven mesh, same 284,818 unknowns, same port certificate; only the drive frequency changed. The guided fraction is not peaked at the cavity resonance at all, and the third refuted hypothesis of this investigation goes in the record beside the other two.
That result is itself the diagnosis. If the guided power reached the lead through the cavity mode, beta would peak on resonance. It does not, because on this mesh the cavity is too lossy to matter: at Q = 59.958 and V = 6.18 the Purcell factor is 0.74, below one, so the resonance offers the emitter no enhancement and most of what reaches the port never went through the cavity.
It is worth noting what the second row is not. That mesh does carry a cavity branch -- localization 0.578 clears the 0.55 gate -- so this is not the all-order-one pathology where the mode is more present outside the cavity than inside. It is a real cavity, resolved far too badly to mediate the coupling it exists to mediate.
Order is a lever, and a cheaper one than the mesh¶
At near-identical cost, the selective-order fidelity roughly doubles the measured coupling:
| fidelity | cavity order | curvature_safety | dof | beta_guided | direction imbalance |
|---|---|---|---|---|---|
device23_channel_p1_full_y |
1 | 0.6 | 284,818 | 0.004389 | 30% |
device23_p1p5_pml2_cavity2_x1p4 |
2 | 0.4 | 269,652 | 0.009474 | 1.1% |
2.16x more coupling on 5% fewer unknowns. The direction imbalance is the
better evidence: an exactly symmetric cavity must feed both ports equally,
and the selective-order field does so to 1.1% where the order-one fields
disagree by 30% at curvature_safety 0.6 and by 77% at 1.4. The
order-one driven fields are not merely coarse, they are asymmetric in a
device that has no asymmetry.
Nor the denominator: the divergence argument was wrong twice¶
Recorded for the correction rather than deleted: the first explanation offered above blamed a divergent point-source self-energy in the denominator. It is wrong twice over. The sign refutes it, as the sweep shows. And the denominator never carried the divergent term in the first place --
total_power = -0.5 * Im(source . forward) / omega
is the imaginary part of the dipole work, which is the radiated power. The real part of a point source's self-field is what diverges under refinement; the imaginary part is finite and is exactly the quantity an LDOS measurement wants. The code was already right about this.
What actually bounds the number: the driven mesh's own Q¶
Detuning is worth nothing, and order about 2x. Neither is 90x, and the remaining factor is not a defect in the measurement at all. It is the cavity.
For a dipole coupled to one cavity mode and driven on resonance, the guided fraction factorizes into a Purcell competition against everything the dipole radiates that never enters the cavity, and the cavity's own branching into the lead:
beta = [F / (1 + F)] * (Q_total / Q_lead), F = (3 / 4 pi^2)(Q / V)
Both factors rise with Q, so beta falls roughly as Q^2 once F drops below
one. Calibrating Q_lead once against the validated seed --
Q = 1113.003, V = 6.180184, beta = 0.878603 -- gives Q_lead = 1180.5, and
with it a prediction for any mesh whose own Q is known:
| Q the driven mesh carries | predicted beta |
|---|---|
| 60 | 0.022 |
| 120 | 0.061 |
| 300 | 0.200 |
| 600 | 0.448 |
| 1113 | 0.879 (the calibration point) |
At curvature_safety 0.6 the driven mesh carries Q = 59.958, so this
model allows it 0.022. On resonance it measured 0.002984, 7.4x below
even this generous account.
And the ladder refutes this one too¶
Solving each driven mesh's own pole makes the model directly falsifiable, because it supplies the Q the model needs and removes detuning at the same time:
| curvature_safety | driven dof | the mesh's own Q | beta, matched drive | beta, 756.734 nm drive |
|---|---|---|---|---|
| 0.6 | 284,818 | 59.958 | 0.002984 | 0.004389 |
| 1.0 | 540,712 | 65.215 | 0.005636 | 0.015129 |
Between these two meshes Q rises by 8.8% while beta rises by 89% matched and 245% at the fixed drive. A quantity governed by the cavity's loss budget cannot behave like that; beta is not a function of the mesh's Q, and the fourth hypothesis falls with the other three. What beta does track is the degree count -- 1.89x for 1.90x more unknowns, still with no plateau.
The right-hand column carries a second observation worth recording. The off-resonance drive gives more guided power than the resonant one, and the gap widens with refinement, from 1.5x to 2.7x. A cavity-mediated coupling cannot do that either.
What survives is narrower and more useful: the suppression is not in the
cavity's loss budget but in the transport from the cavity to the port. The
cavity is resolved badly and that is real, but it is not what is holding
beta_guided two orders of magnitude down.
The remaining candidate, untested¶
Recorded as a hypothesis, not a finding, because nothing here has measured it. The seed reaches its lead through twenty mirror pairs, and transmission through a Bragg mirror is exponential in the number of periods, so it is exponentially sensitive to how well the mesh carries the evanescent decay -- while Q, dominated on these meshes by other loss, is barely sensitive at all. That asymmetry is exactly the 8.8%-against-245% above. Quantitatively, the 3.4x in beta between those two meshes is 1.85x in amplitude, or 3.1% per period over twenty periods; closing the remaining 58x would need a further 11% per period.
It also accounts for the off-resonance drive winning. The cavity resonance sits inside the mirror's stop band, where transmission is smallest; 756.734 nm is 7 to 12 nm to the blue of it, nearer the band edge, where the mirror passes more. Mirror transmission varies with frequency far faster than the cavity's Lorentzian does, which is why the guided fraction has its maximum away from the resonance rather than on it.
The same error has a second signature already on the record. Under-transmission means the cavity leaks less than it should, which raises Q -- and the certified local pole sits at Q = 1640.617 against the FDTD 1113.003, 47% high, in the direction this predicts.
One root cause would then account for both open discrepancies. Testing it does not need the full article: it needs the mirror unit cell's Bloch transmission per period as a function of mesh, against the band solver, which is a far smaller problem than either measurement it would explain. That is the next thing to do, and it has not been done.
It also states the requirement exactly, and the requirement is uncomfortable: the coupling gate cannot be certified on any mesh that does not first resolve Q. Coupling is downstream of the pole, not independent of it. The coarsest discretization that puts Q in the right range is the certified selective-order pole fidelity, and its full-y driven twin is about 2.0 million unknowns.
None of the local coupling numbers should be quoted as the seed's coupling. They are measurements of their own meshes.
Order does not rescue it: the order-two family climbs the same way¶
The selective-order fidelity gives more coupling per unknown and a far more symmetric field, so the obvious question is whether it also converges where order one did not. It does not.
Both families at the same 756.734 nm drive:
| fidelity | cavity order | dof | beta_guided | forward/backward imbalance |
|---|---|---|---|---|
device23_channel_p1_full_y |
1 | 284,818 | 0.004389 | 30% |
device23_channel_p1_full_y |
1 | 540,712 | 0.015129 | 15% |
device23_channel_p1_full_y |
1 | 836,340 | 0.025930 | 77% |
device23_p1p5_pml2_cavity2_x1p4 |
2 | 269,652 | 0.009474 | 1.1% |
device23_p1p5_pml2_cavity2_x1p4 |
2 | 848,748 | 0.043544 | 3.1% |
Fitted against the degree count, order one gives beta ~ dof^1.65 and the
selective family beta ~ dof^1.33. Order two sits 1.6x to 2.4x above order
one at matched cost, and its ports stay symmetric to a few percent where
order one wanders from 15% to 77% -- both real improvements, and neither of
them a plateau. The order-two curve is the order-one curve shifted up.
Continuing the selective fit to the validated 0.878603 needs 9.6x more unknowns, about 8.1 million driven degrees of freedom. The 848,748-unknown solve already peaked at 38.9 GiB. This machine has 45.
That extrapolation is soft and should be quoted as a floor rather than an estimate, because the exponent is not constant. The order-one family has three points and its local slope falls from 1.930 to 1.235 across them:
| interval | d log beta / d log dof |
|---|---|
| 284,818 -> 540,712 | 1.930 |
| 540,712 -> 836,340 | 1.235 |
A single power law is not a good description of that. The selective family has only two points, so no curvature is visible in it at all and its 1.33 is an average over the same kind of decay. If the selective exponent decays the way order one's does, the requirement grows rather than shrinks -- 9.6 million unknowns at exponent 1.24, 24 million at 0.9, 127 million at 0.6. The alternative reading, that a decaying exponent means the curve is approaching a plateau, is equally consistent with these points and would be much better news; nothing here distinguishes them, and the plateau value is not constrained by two points either. 8.1 million is the most optimistic number the data supports, not the expected one.
So the answer to "does order fix it" is no, and the answer to "can this be reached by refining" is not here.
The next rung, curvature_safety 1.4, meshes to 1,329,929 driven unknowns.
It was stopped by operator decision before its factorization, not
observed to fail: at 38.9 GiB for 848,748 unknowns and three-dimensional
fill growing near N^1.5, it needs about 76 GiB against a 40 GiB cap, and
spending twenty minutes of exclusive machine time to watch a predictable
kill was worth less than the measurement that replaced it. The ceiling here
is an extrapolation from a measured point, and is recorded as one.
Two cheap levers nobody had pulled¶
Both of these follow from noticing where the port plane actually sits. The lead runs from the last hole's outer edge at 4.4283 um to the physical boundary at 4.9801 um, and the port plane is at 4.8301 um.
core_maxh is a lever, and the recorded trap says it is not¶
The campaign's standing note is that maxh does not control this mesh,
that element count tracks contour curvature, and that curvature_safety is
the only lever. That is true of the air ceiling -- 0.48 to 0.36 moved
the degree count by 316 -- and it is false of the core:
core_maxh (um) |
driven dof at curvature_safety 0.6 |
|---|---|
| 0.30 | 284,818 |
| 0.15 | 429,944 |
| 0.08 | 1,351,738 |
A factor of 4.7. The distinction matters because the two levers refine
different places. curvature_safety refines around the forty level-set
contours, all of which lie inside |x| <= 4.43 um. The lead has no contours,
so it keeps whatever core_maxh gives it -- and at 0.30 um, against a
guided wavelength of 770/1.7788 = 433 nm, the port overlap is being
evaluated on a field carried by about 1.4 linear elements per wavelength.
That makes a clean discriminating experiment: refine the lead at fixed hole curvature and see whether the coupling cares.
It does not. The holes matter twice as much as the lead¶
From the same 284,818-unknown baseline at beta 0.004389, spending the
next unknowns in two different places:
| refinement | what it resolves | dof | beta_guided | exponent |
|---|---|---|---|---|
core_maxh 0.30 -> 0.15 |
the lead, where the port plane is | 429,944 | 0.006576 | 0.982 |
curvature_safety 0.6 -> 1.0 |
the forty holes | 540,712 | 0.015129 | 1.930 |
Refining the lead is the less productive of the two, by a factor of 1.97 per decade of unknowns. Carried at the hole exponent, the lead point should have reached 0.009719; it measured 0.006576, 0.68x of that.
So the port plane is not sitting on a starved field, and the seventh explanation goes the way of the other six. The lead is under-resolved in absolute terms -- 1.4 elements per guided wavelength is not defensible -- but it is not what has been holding the coupling down.
The same measurement is positive evidence for what survives. The quantity is most sensitive to resolution in the patterned region, which is where the mirror's evanescent decay per period is determined, and least sensitive in the unpatterned lead the guided mode merely travels along. That is the signature the Bragg-transport account predicts, and it is now the only account still standing.
The port projection can be asked whether it is converged, for free¶
A converged port projection is flat in the plane position: a guided mode
carries the same power through every plane in the lead, so any residual
x-dependence is the numerator's own numerical error. Re-evaluating the same
forward field on extra planes costs no solve at all, which makes this the
cheapest check available on the quantity that has resisted five
explanations. evaluate_channel_gradient now takes
port_plane_scan_offsets_um and reports the powers at each, and
measure_device23_channel.py exposes it as --port-plane-scan-um.
Interference from any backward-propagating content shows up as an
oscillation of period lambda_g/2 = 216 nm; radiative content and
discretization error show up as a drift.
An iterative driven solver was tried, and BDDC does not work here¶
If the acceptance fidelity cannot be factorized, the obvious move is not to
factorize it. quan_meep/iterative_solver.py provides IterativeInverse,
a drop-in for what matrix.Inverse() returns, and
evaluate_channel_gradient gained an opt-in linear_solver argument whose
default is still direct.
The adoption test is that the iterative answer must reproduce the factorized answer on a problem the factorization can still reach. It does not. At 284,818 unknowns:
| solver | time | memory | result |
|---|---|---|---|
| Pardiso, direct | 40.4 s | 10.2 GiB | reference |
| GMRES + BDDC | 3,955.5 s | -- | 1,999 iterations, relative residual 1.86e-01 |
The iterative solution differs from the direct one by 3.27e-01. This is not a solve that needs more iterations; it is a preconditioner that is not controlling the operator. The driven system is complex, indefinite and PML-damped, which is where Krylov methods are least comfortable, and BDDC as configured does not fix it.
The plumbing is kept because it is opt-in, costs the direct path nothing,
and the next preconditioner worth trying -- a complex-shifted Laplacian,
curl curl - (1 + i beta) omega^2 eps applied approximately -- attaches at
exactly the same place.
The pole's Q is not converged either¶
Owed since the repair, because every point of the earlier convergence study
was solved through the broken mirror. Rerun on the repaired solver against
the certified selective-order fidelity, varying curvature_safety, which
the channel work established is the only lever that moves this mesh by
factors:
| curvature_safety | dof | lambda (nm) | Q | localization | residual |
|---|---|---|---|---|---|
| 1.0 | 394,645 | 753.962 | 1461.283 | 0.693 | 2.1e-09 |
| 1.4 | 641,209 | 755.608 | 1885.692 | 0.701 | 2.0e-10 |
| 2.0 | 1,004,130 | 756.734 | 1640.617 | 0.702 | 3.6e-12 |
The two observables behave completely differently.
The wavelength is converging. It is monotone in the mesh, spans only 0.368%, and moves toward the reference as the mesh improves. It is not there yet: all three points sit 1.73% to 2.09% below the validated 770.0297 nm, and the certificate's own gate is 2%, so the coarsest point here would fail it. The remaining disagreement is small, one-sided, and still shrinking.
Q is not converging. It goes 1461 up to 1886 and back down to 1641 -- non-monotone, spanning 1.29x across a 2.5x range of unknowns. This is not solver noise: the relative residual is 1e-9 or better at every point, sixteen candidates converged at every point, and the localization is flat at 0.693, 0.701, 0.702, so the same branch is being tracked throughout. The mesh is genuinely changing the answer by a quarter, in both directions.
So Q = 1640.617 is a number this campaign should not quote. It is one
sample from a family spanning 1461 to 1886, all of which sit 1.31x to 1.69x
above the FDTD 1113.003. The pole certificate passes it because its Q gate
is deliberately 0.5x to 2.0x wide and is documented as a branch and
fidelity sanity window rather than an accuracy claim -- and this ladder is
what that caveat was reserving room for. The certificate stands as written;
what fails is any reading of 1640.617 as the device's Q.
Note which way the disagreement points, because it bears on the section above. Under-transmission through the mirror would make the cavity leak less and read high, and every point here does read high. But refining the mesh does not walk Q down toward 1113; it moves Q around without trend while the wavelength converges smoothly. Whatever sets Q on these meshes is not being fixed by resolution in the range this machine can reach.
Where this leaves the campaign¶
The pole certificate stands. It passed all eleven gates on the repaired solver and nothing measured since has touched it.
The coupling gate does not, and six explanations for it have now been measured rather than argued, and refuted: a divergent denominator, another unnamed boundary, drive detuning, a cavity-Q-limited coupling, the port plane sitting inside the mirror, and low polynomial order. What is left standing is only this much -- the suppression lies in the transport from the cavity to the lead rather than in the cavity's own loss budget, and nothing that fits on this machine measures it. The measurements below are each a correct measurement of their own mesh and none of them is a measurement of the seed:
| fidelity | cav. order | cs | driven dof | drive | beta_guided |
|---|---|---|---|---|---|
device23_channel_p1_full_y |
1 | 0.6 | 284,818 | matched, 763.836 nm | 0.002984 |
device23_channel_p1_full_y |
1 | 1.0 | 540,712 | matched, 768.764 nm | 0.005636 |
device23_channel_p1_full_y |
1 | 0.6 | 284,818 | pole, 756.734 nm | 0.004389 |
device23_p1p5_pml2_cavity2_x1p4 |
2 | 0.4 | 269,652 | pole, 756.734 nm | 0.009474 |
device23_channel_p1_full_y |
1 | 1.0 | 540,712 | pole, 756.734 nm | 0.015129 |
device23_channel_p1_full_y |
1 | 1.4 | 836,340 | pole, 756.734 nm | 0.025930 |
device23_p1p5_pml2_cavity2_x1p4 |
2 | 1.0 | 848,748 | pole, 756.734 nm | 0.043544 |
device23_channel_p1_full_y should not be a channel fidelity for this
campaign at any mesh. It is order one throughout, which is the
discretization whose own pole returns localization 0.367 with more Ey
outside the cavity than inside, and whose driven fields feed the two ports
of a symmetric cavity unequally by 30% to 78%.
What would settle it is a driven solve at the certified pole's own
discretization. Full-y that is about 2.0 million unknowns and does not
factorize in 45 GiB, and the iterative route is blocked above. The one
untried reduction is the y-symmetry the pole already uses: a y-directed
dipole on the y = 0 plane has Ex = Ez = 0 there, so that plane is a
PEC mirror for this problem and the driven domain could be halved to about
1.0 million unknowns, which does factorize. The obstacle is the one
already recorded -- a point current exactly on the mirror has no stable
normal trace in H(curl), which is why the driven path runs full-y today.
Offsetting the source to y = +epsilon puts it back in the interior, and
its PEC image is a same-direction dipole, so the half-domain solve is the
full problem with a source pair that tends to a single doubled dipole as
epsilon shrinks. beta is a power ratio and is indifferent to that
doubling.
That is a plausible route and it is not evidence. It requires the port
functional's y-bookkeeping to be redone for a half domain, and it must be
adopted the way the layered pole was: by reproducing the full-y answer at
curvature_safety 0.6, where both domains still fit, before it is allowed
anywhere near a gate.
The port plane is not inside the mirror¶
Checked because it would have explained everything at once, and it does
not. port_offset = physical_half_x - 0.5 * pml_thickness_x_um puts the
plane at 4.8301 um on this seed:
| feature | x (um) |
|---|---|
| outermost hole centre | 4.3801 |
| last hole outer edge | 4.4283 |
| port plane | 4.8301 |
| PML boundary | 4.9801 |
The plane sits 0.4018 um past the last hole, which is 0.52 free-space wavelengths and 0.93 wavelengths of the n_eff 1.7788 guided mode, inside the declared 0.600 um lead. It is sampling the lead, not the stop band, so the guided fraction is not small because the projection was taken inside the mirror.
Worth noting for later, though not implicated by any measurement yet: the plane is only 0.1500 um from the absorber, 0.35 guided wavelengths, and the functional also samples one grid step either side of it. That is a short run-out for a modal projection to have settled in, but the failure it would produce is a plane-position dependence, and none has been measured.
With this eliminated the surviving candidate is unchanged: the transport from cavity to port, with Bragg transmission through the twenty mirror pairs as the specific mechanism, exponential in period count and so exponentially sensitive to a mesh that barely moves Q.
2026-08-04: the mirror's own bands, and what they cost this account¶
The band solver was pointed at Device23's own mirror cell, and two of the
premises above do not survive the measurement. Detail and controls are in
device23_mirror_bands.md; the consequences here are these.
"Twenty mirror pairs" describes something the seed does not have. Its
pitch converges by cell ten, but ry grows monotonically from 65.03 nm to
136.10 nm at the very last hole. There is no uniform mirror to count
periods of, and the period count that the exponential argument is
exponential in is not twenty.
The exponential premise fails at the resonance. Transmission through a
Bragg mirror is exponential in period count because no propagating mode
exists there. The mirror cell carries a guided, nitride-dominated Bloch
band that crosses the validated 770.030 nm at k/edge 0.912 and the local
pole's 756.734 nm at 0.934, at real wavevector in both cases, stable across
four mesh rungs, two candidate budgets, two confinement floors and a
doubled enclosure. Where a propagating mode exists, transport is not
exponential in the period count and the mesh-sensitivity argument built on
it does not apply.
What is not settled, and what keeps this from being a refutation: that band is not the fundamental. It is narrower across the beam and lower in effective index, and the cavity mode is the fundamental. Whether the coupling actually flows through it is an overlap integral nobody has computed. The mirror's gap is also not yet established -- the small box contaminates the zone edge, which is exactly where a Bragg gap is read.
The subsidiary claim that the off-resonance drive wins because 756.734 nm sits nearer the band edge is likewise unsupported for now. Both wavelengths lie on the same propagating band, 15 nm apart in a band that sweeps 44 nm over the sampled zone, and no stop-band edge has been measured that either of them could be near.