# 8 × 8 µm y-dipole SiN/SiO2-slab atom-slot campaign

## Physical geometry

The physical design region is an 8 µm × 8 µm patterned Si3N4 core embedded in
a planar SiO2 stack. The SiO2 slab extends across the complete x–y simulation
plane, including longitudinal and transverse PML cells; it is not a finite-width
oxide guide. An explicit immutable air body cuts `-0.5 < x < 0.5 µm` through
the SiO2, the SiN, and the complete y/z simulation extent, including PML. The
atom is centered in that strip and is y polarized.

The source is the corresponding symmetric TE-like waveguide mode. Mirror
parities `(x,y,z)=(+1,-1,+1)` allow a nonzero normal Ey field at the atom and
reduce the full problem to one octant. Both the time monitor and the resonance
fit use Ey; the total-energy monitor supplies the mode-volume numerator and
the atom-local Ey amplitude supplies its denominator. XY and XZ field plots
show Ey at the tracked pole.

Initially `t_SiN=0.100 µm`, with `0.300 µm` of SiO2 above and below. Every
trainable 50 nm topology pixel starts at `rho=0.5`, representing SiN occupancy
over the exact SiO2 background,

`epsilon_core(rho) = epsilon_SiO2 + rho (epsilon_SiN - epsilon_SiO2)`.

The immutable air strip and port pins remain exact materials. Component-aware
Yee subpixel averaging applies a single physical interface consistently at the
staggered Ex, Ey, and Ez locations.

## Stage A — absolute temporal response

A smooth modal pulse is injected and the complete field vector is recorded at
the atom. The fixed target has only Ey and is the causal response of the chosen
Q=10,000, beta=1, small-mode-volume resonator, including its pulse-on transient.
No fitted amplitude is allowed:

`J_time = - sum_t ||E(t) - y_hat Ey,target(t)||^2 / sum_t |Ey,target(t)|^2`.

Raw Adam changes only direct topology pixels. Handoff requires a trustworthy
observable pole with Q at least 100 and a recent gain rate no more than 20% of
the best rate over an optimizer-memory window. This captures approximately 80%
of the readily available stage benefit instead of waiting for a zero gradient.

## Stage B — pixel-only pole Q

Adaptive-trust L-BFGS-B maximizes pure same-pole harmonic Q over topology
pixels only,

`J_Q = log(2 Q_field Q_energy / (Q_field + Q_energy))`.

The field and energy fits must agree and pass residual, observability, and pole
continuity checks. Mode volume, Q/V, and atom-local Purcell remain telemetry.

## Stage C — joint topology and vertical stack

Once the pixel-only Q gain rate has fallen to 20% of its best stage-local rate,
the accepted geometry is preserved while differentiable SiN and per-side SiO2
thickness coordinates are unlocked. The map enforces

`0.050 <= t_SiN <= 0.500 µm`,

`0.050 <= t_SiO2 <= (1.000 - t_SiN)/2 µm`,

and therefore `t_SiN + 2 t_SiO2 <= 1.000 µm` for every proposal.

## Stage D — gradual binarization

After joint-stack Q improvement clearly slows, centered tanh projection starts
at low beta. Every fixed-beta subproblem continues to optimize the same pure-Q
scalar. Beta advances only after recent gain has fallen to 20% of the best rate
at that beta. Existing L-BFGS pairs are retained, but the cross-beta secant is
omitted because the objective map changed.

## Stage E — smooth boundary

Binary-ready bulk is replayed as a topology-free tensor-product cubic-spline
phase field with subpixel cut cells. The chart allows islands to fuse, disappear, or nucleate.
Pure-Q boundary and stack optimization then continues indefinitely.

## Persistence

The accepted state is written atomically, with a separate snapshot every 50 accepted
updates. Optimizer and dashboard services restart automatically. The
dashboard publishes geometry and difference maps, Ey fields, ideal and measured
temporal traces, linear Q, atom-oriented V/(lambda_pole/n_air)^3, Q/V, Purcell,
pole continuity, trust health, binarity, and stack thicknesses.