# Axisymmetric cosine atom-hole optimizer

```{admonition} Live research campaign · spectral pole-acquisition stage
:class: important

This is a true rotational-symmetry reduction of the m=0 TM Maxwell equations,
not a 3-D voxel optimization and not a Cartesian wedge. A z dipole sits at the
center of an air cylinder constrained to remain at least
1.000 µm in diameter. A finite-interval DCT
decodes global cosine coefficients into one radial SiN etch line, which is
filtered, projected, and swept through 2π. Its SiN thickness and the shared
top/bottom SiO₂-cladding thickness are differentiable scalar design variables.
The full physical radius is present from the start; spatial frequencies unlock
progressively rather than introducing new radial shells.
```

<div class="optimizer-metrics">
  <div class="optimizer-metric"><span>campaign iteration</span><strong>846</strong><small>running_unshifted_ldos</small></div>
  <div class="optimizer-metric"><span>cosine controls</span><strong>36 / 36</strong><small>active / available DCT modes · plus three dimensions</small></div>
  <div class="optimizer-metric"><span>active objective</span><strong>target-frequency source-work LDOS discovery</strong><small>automatic fail-closed pole handoff</small></div>
  <div class="optimizer-metric"><span>current stack</span><strong>279.5 / 50.6 nm</strong><small>SiN core / each symmetric SiO₂ layer</small></div>
  <div class="optimizer-metric"><span>air hole</span><strong>1.0001 µm</strong><small>hard lower bound 1.000 µm</small></div>
  <div class="optimizer-metric"><span>latest LDOS</span><strong>1.048</strong><small>vacuum normalized · 780.24 nm</small></div>
  <div class="optimizer-metric"><span>best projected</span><strong>1.05</strong><small>η=0.50</small></div>
  <div class="optimizer-metric"><span>best literal binary</span><strong>1.048</strong><small>100 nm annular cells · 150 nm filter</small></div>
  <div class="optimizer-metric"><span>latest Cartesian audit</span><strong>1.027</strong><small>2.00% difference at iteration 800 · vacuum_normalized_dipole_source_work</small></div>
  <div class="optimizer-metric"><span>pole surveillance</span><strong>Q=36.38 (provisional)</strong><small>shift gate stays closed until fit is trustworthy</small></div>
</div>

## Geometry: from one line to a fabricable 3-D body

![Start, current meridional section, and full revolved mask](../_static/generated/axisymmetric_atom_hole_geometry.png)

The cyan region is the hard air-hole constraint. The current geometry audit
reports `air_hole_clear=True`, `outside_design_air=True`,
and `single_etch_extrusion=True`. The discovery grid
contains only **36 global cosine
coefficients and three dimensions** on a 208 ×
129 r-z field grid. The 36 decoded fabrication samples
are not independent optimization variables. Rotating the line creates the
complete 3-D device; there are no angular sectors or voxels.

## Target-frequency field

![Ez field in the meridional plane](../_static/generated/axisymmetric_atom_hole_field.png)

The displayed field is the literal complex target-frequency FDTD phasor. Its
left half is reconstructed from rotational symmetry only for visualization;
it was not simulated independently.

## Optimization and automatic pole gate

![LDOS, binarization, continuation, and pole history](../_static/generated/axisymmetric_atom_hole_progress.png)

The campaign is a cold zero-coefficient start, decoding to a neutral 50%
density line; no V8 geometry or optimizer moment is inherited. The lowest 12
DCT modes are active through iteration 159, 24 through iteration 319, and all
36 thereafter. β rises from 1 to 64 over 475 updates and reaches the fully
projected stage around iteration 500. Smooth worst-case optimization over
η=0.45, 0.50, and 0.55 begins at β=32.
The cosine block uses one shared RMS normalization so coefficient steps retain
their Maxwell-gradient amplitude ratios; the three physical dimensions retain
independent Adam scaling.

Every 20 updates, a separate 180-period forward ringdown fits both the atomic
field and total electromagnetic energy. Handoff requires all three morphology
projections to be fully projected and to satisfy conservative Q≥40, field
residual ≤10%, energy residual ≤12%, field/energy Q agreement within 1.5×,
bounded decay-window variation, and ≤3% detuning. The shifted stage uses a
memory-safe 100-period differentiable tape to maximize robust fitted Q/V with
stopped-gradient pole recentering. A 180-period fail-closed audit repeats every
20 shifted updates and rejects a candidate if the clean pole is lost.

## Independent 3-D spectral audit

![Cylindrical and Cartesian source-work LDOS spectra](../_static/generated/axisymmetric_atom_hole_spectrum.png)

The solid curves compare the same vacuum-normalized dipole source-work
observable in both solvers. The dashed Cartesian box-flux curve is retained as
an energy-balance diagnostic, but it is not labeled LDOS because residual
stored energy can make finite-pulse flux differ from source work.

## Numerical contract and current limitation

| parameter | value |
| --- | ---: |
| wavelength | 780.24 nm |
| core / fill material | SiN, n=2.05 / SiO₂, n=1.444 |
| initial SiN thickness / allowed range | 250 nm / 100–400 nm |
| initial each-side SiO₂ / allowed range | 125 nm / 50–200 nm |
| air-hole diameter | differentiable, ≥1.000 µm |
| optimization radius | 4.100 µm |
| spectral schedule | 12 → 24 → 36 DCT-II modes |
| grid spacing | 25 nm |
| radial control pitch / filter radius | 100 / 150 nm |
| field cells | 26,832 |
| time steps | 7,119 |
| rotational mode | m=0 TM (Er, Ez, Hφ) |

The cylindrical interior update, axis limit, and differentiable conservative
CPML are unit/analytic tested; the short-pulse late return is suppressed by
more than 500× relative to the PEC boundary. Live LDOS and Q values remain
research metrics until PML-thickness stability and the scheduled Cartesian
3-D octant comparison agree. The 3-D audit records a 41-point local spectrum
and independent pole fit and remains intentionally separate from the hot-loop
objective.

```{admonition} Rejected discovery precursors
:class: warning

V1 used 40 nm fabrication cells and remains rejected for resolution
sensitivity. V2 moved to the 25 nm field grid but linear interpolation created
a literal 25 nm threshold sliver. A later audit correction found that the huge
V1--V4 discrepancies previously shown here were finite-box pulsed flux, not
dipole source-work LDOS. Re-audited target-frequency differences are 11.8%,
7.1%, 5.1%, and 1.5% for V1 through V4. V3 is therefore a valid 100 nm
precursor retired when CPML arrived; V4's 200 nm basis is too coarse for the
SiN quarter-wave scale. Frozen V6 used the 1 µm hole and whole 4 µm-radius
domain. V7 tested a 500 nm hole with Shaker-style successive enlargement. V8
returned to a ≥1 µm hole and added the finite symmetric stack; it was archived
at iteration 543 with best projected LDOS 1.1797. Active V9 starts cold in the
12-mode DCT basis, progressively unlocks the complete 36-mode line, and uses
the longer field-plus-energy gate before differentiable shifted Q/V.
Source-work LDOS, box-flux balance, spectra, and pole fits stay visible.
```

Last atomic update: `2026-08-24T14:06:56.682039+00:00`