# Fryett encapsulated nanobeam · staged inverse design
```{admonition} Live campaign
:class: important
This page is rebuilt from the persistent optimizer checkpoints and published every minute.
Stage 1 is preserved as **completed_handoff_to_boundary_stage** at iteration **23**;
its last checkpoint is `2026-08-24T18:35:05.063319+00:00`. The later sections contain the independent
boundary-normal branches and the verified low-radiation mode-volume campaign.
```
Q/V1.5312e+06 µm⁻³direct differentiated objective
best Q/V1.5312e+06 µm⁻³best accepted checkpoint or qualified seed
conservative Q1.1306e+06minimum of two decay fits
pole769.260 nmshifted tracking center
mode volume12.98 (λ/n)³center-field definition
beam width400.4 nm400–700 nm bounded control
largest hole shift10.87 nmrelative to paper centers
minimum bridge116.01 nmhard bound ≥15 nm
physical variables15150 × (x, major, minor) + width
symmetry saving8×x/y/z mirror planes
## Download the fabrication layouts
[**Download every GDS file as one ZIP bundle**](../_static/downloads/fryett/fryett_gds_bundle.zip)
· [combined all-stages GDS library](../_static/downloads/fryett/fryett_all_stages.gds)
· [machine-readable manifest and checksums](../_static/downloads/fryett/manifest.json)
| Preserved geometry | State | Direct GDS |
|---|---|---|
| Published arXiv v1 seed | unoptimized | [download](../_static/downloads/fryett/fryett_arxiv_v1_seed.gds) |
| Published ACS-text seed | unoptimized alternative | [download](../_static/downloads/fryett/fryett_acs_final_seed.gds) |
| 151-parameter ellipse result | optimized Stage 1 handoff | [download](../_static/downloads/fryett/fryett_ellipse_optimized.gds) |
| Level-set zero-motion handoff | Stage 2A start | [download](../_static/downloads/fryett/fryett_level_set_start.gds) |
| Level-set terminal cavity | optimized Stage 2A | [download](../_static/downloads/fryett/fryett_level_set_optimized.gds) |
| Cubic-spline zero-motion handoff | Stage 2B start | [download](../_static/downloads/fryett/fryett_spline_start.gds) |
| Cubic-spline terminal cavity | optimized Stage 2B | [download](../_static/downloads/fryett/fryett_spline_optimized.gds) |
| Low-radiation spline handoff | immutable Stage 3 start | [download](../_static/downloads/fryett/fryett_low_radiation_initial.gds) |
| Low-radiation current cavity | live accepted Stage 3 state | [download](../_static/downloads/fryett/fryett_low_radiation_current.gds) |
All layouts use micrometre coordinates and a 1 nm database unit. Layer 1/0 is
the authoritative positive-SiN device; layer 2/0 contains the 100 hole
apertures separately for inspection or a holes-only process. The masks are
fully unfolded and include 2 µm of fixed 450 nm-wide feedthrough at each end.
GDS is only the in-plane single-etch mask: the 330 nm SiN thickness and the
oxide/organic cladding stack shown above are process metadata, not GDS layers.
The bundle contains a README with this layer map and tape-out caveats. It was
generated from **9** preserved geometries.
## What was reproduced
The seed is the encapsulated Si₃N₄ nanobeam from
[Fryett *et al.*, arXiv:1709.02032](https://arxiv.org/abs/1709.02032): a
330 nm × 450 nm beam in index-1.47 cladding, 233 nm Bragg pitch, 100 × 300 nm
elliptical mirror holes, a roughly 2.04 µm taper, and 40 saturated mirror
periods per side. The full simulation retains all 50 holes per side and the
beam continues through the x PML.
The preprint and final journal prose disagree on the innermost ellipse. Both
literal variants were screened. The selected **arxiv_v1** seed has a
dominant FDTDX pole at **780.550 nm**.
On the 2 ps qualification tape its conservative differentiable fit gives
**Q=1.5598e+05**, **V=0.8198 µm³**,
and **Q/V=1.9027e+05 µm⁻³**. The paper's
nominal 740 nm target is quoted rather than forced: the 25 nm Yee-grid
reconstruction is visibly red-shifted.
## Why this branch restarted
The preceding continuous-pixel branch improved Q/V from **1.9027e+05** to **2.7691e+05 µm⁻³**, then rejected **3** consecutive trial steps. It was stopped as plateaued. This campaign does not inherit that density, momentum, or pole history.
The replacement begins at exactly the paper ellipse parameters. Its optimizer
moments, learning-rate history, accepted-step history, and shifted tracking
state all begin at iteration zero.
## Layer stack used by the simulation
The physical device places the **330 nm Si₃N₄ beam on thermal SiO₂** and
encapsulates it from above with **PMMA**. Both surrounding materials have
approximately the same refractive index, 1.47, so they form an optically
continuous cladding. FDTDX therefore represents them—and the material filling
every through-etched opening—as one homogeneous index-1.47 background. This
avoids inserting a fictitious interface at the beam midplane. The trainable
x-y mask is extruded only through the 330 nm nitride thickness; neither
cladding is a design variable.
## Geometry: published seed, current ellipses, and exact change
**GDS downloads:** [arXiv seed](../_static/downloads/fryett/fryett_arxiv_v1_seed.gds) ·
[ACS seed alternative](../_static/downloads/fryett/fryett_acs_final_seed.gds) ·
[optimized ellipse handoff](../_static/downloads/fryett/fryett_ellipse_optimized.gds)
This branch is deliberately **not** a free-form pixel search. Each of the 50
positive-side holes has an independent center position, longitudinal diameter,
and transverse diameter. Mirror symmetry generates the negative side, and one
more variable changes the beam width: **151 physical controls** in total. The
current longitudinal diameters span **80.8–120.5 nm**
and the transverse diameters span **145.5–359.8 nm**.
Position bounds of ±30 nm and longitudinal diameters of 60–140 nm guarantee
that ellipses cannot cross; even the worst allowed central pair retains a
15 nm Si₃N₄ bridge. Transverse diameters are bounded to 80–360 nm. The beam is
400–700 nm wide through the cavity and returns smoothly to the fixed 450 nm
feedthrough over the final 750 nm, so it still crosses the x PML without a
dielectric termination. The displayed mask is the physically unfolded design.
## Center field, including the complete vertical section
The centered dipole and objective use **Ey**, the cavity polarization reported
for this design. The x/y mode parity is `(magnetic, electric)` and identical
upper/lower cladding supplies z magnetic parity, giving FDTDX symmetry
`(1, -1, 1)`. These archived Stage-1 panels were demodulated at the original
780.550 nm pulse carrier and are retained only as historical driven-response
plots; they must not be interpreted as the later fitted pole. The corrected
pole-aligned Stage-2 fields are displayed above with both detector and fitted
wavelengths printed on the figure.
## Q/V objective and shifted pole tracking
```{admonition} Exploratory stage now frozen
:class: warning
This objective followed the moving cavity pole from 780.550 nm toward 769 nm.
Its Q/V history is useful shape-search evidence, but it is **not** accepted as
fixed-emitter LDOS improvement. The matched shifted-LDOS branches below correct
that objective error and compare two boundary parameterizations.
```
Each accepted step maximizes `log(Q/V)` at the cavity center. Q is the lower of
an Ey complex-pole decay fit and an independent total-energy decay fit. V uses
the late modal energy divided by the local center-field intensity and local
Si₃N₄ permittivity. The fit follows the measured pole between steps with a
stopped-gradient frequency shift, avoiding the ill-conditioned pole-motion
term while retaining differentiation through the short FDTD ringdown.
The forward geometry is binary Si₃N₄/cladding and is rasterized with a 6×6
subcell area average. A narrow smooth surrogate exists only in the backward
pass to provide derivatives of the ellipse boundaries and beam edge; it never
turns a hole into a gray material parameter. Bounded Adam steps use monotonic
backtracking, and the learning rate halves after a rejected attempt rather than
repeating the same failed step indefinitely. Trial steps are accepted only
when Q/V does not decrease and both decay fits pass residual, consistency,
Q-ratio, and tracking-window gates.
## Matched-start method comparison
Both curves include their independently replayed zero-displacement handoff as
step zero. Comparing by adjoint step measures optimizer efficiency; the final
panel separately shows accumulated GPU wall time, including line-search
forward trials. Neither branch inherits geometry or L-BFGS curvature from the
other.
## Stage 2A — level-set boundary optimization with L-BFGS
```{admonition} Active continuation
:class: important
This is a clean optimizer stage below the preserved parametric record. It
inherits the complete iteration-23
geometry but no ellipse controls, Adam moments, or superseded boundary steps.
The stage is currently
**stopped_plateau_user_requested** at boundary iteration
**79**. Last checkpoint: `2026-08-25T11:12:52.341919+00:00`.
```
```{admonition} Superseded frequency-correction attempt
:class: note
The first boundary attempt fixed the objective at 780.550 nm instead of the
boundary handoff's own pole. It spent **6**
steps broadening Q from 893,193 to
**965.69** while trying to cover that
11 nm detuning. It is archived intact, but its history is excluded from the
fresh curve below because frequency correction is not the present goal.
```
```{admonition} Superseded hard-raster attempt
:class: note
The next fixed-line attempt was stopped after **7**
nominally accepted steps because its hard 6×6 forward occupancy never changed:
the density and fields remained bitwise identical. Its checkpoint is preserved,
but none of its displacement, momentum, or history enters this restart.
```
shifted peak LDOS27265differentiated objective at the tracked pole
handoff peak LDOS9921.5continuous-subpixel zero-displacement audit
initial-line LDOS9.0102e-05diagnostic at 769.622 nm; not optimized
conservative Q2.1931e+06minimum of field and energy fits
handoff Q8.5692e+05pole 769.622 nm before boundary motion
tracked pole766.570 nmconstrained to the starting mode family, not one linewidth
mode volume12.22 (λ/n)³diagnostic at the tracked pole
largest normal move1.886 nmsince last signed-distance rebase
optimizerL-BFGS0 curvature pairs · 38 accepted steps
move trust region0.0025 nmArmijo line search selects the actual gradient step
step controlleradaptive_expansion_rho_v2changed after 27 completed updates · next α=1
topology1 solid / 50 holesone-to-one lineage enforced
### Boundary-stage geometry
**GDS downloads:** [stage start](../_static/downloads/fryett/fryett_level_set_start.gds) ·
[final optimized boundary](../_static/downloads/fryett/fryett_level_set_optimized.gds)
The middle panel is the one-time signed-distance conversion of the parametric
handoff. Fractional edge cells are continuously differentiable geometric 6×6
subcell coverage, not grey design variables. Uniform cells remain exactly
binary. Its mean absolute fill difference from the exact parametric
raster is **0.633%**. An independent zero-displacement replay
retains **Q=8.5692e+05** at **769.622 nm**, so this is the same
high-Q pole family rather than the rejected thresholded handoff. From that
point onward, each parameter is a signed normal
displacement of an existing interface. Positive motion grows Si₃N₄ and
negative motion retreats it. The outer 750 nm feedthrough is fixed so the beam
still enters the x PML without a termination.
The initial topology has one connected Si₃N₄ body and 50 enclosed holes in the
stored x/y quadrant. Symmetry-plane half-holes are counted after unfolding.
Every proposed step must preserve one-to-one component and hole lineage, so a
hole cannot vanish while another appears elsewhere.
### Eigenfrequency-shifted objective
The handoff pole at **769.622 nm** identifies the mode family. At every
iteration, differentiable ringdown fitting returns the current pole
`ω*(geometry)`, and the optimized quantity is
`log LDOS_peak(Re[ω*(geometry)], geometry)`.
This is the time-domain analogue of the Shaker eigenfrequency-shifted
formulation: the evaluation follows the peak, eliminating the narrow
fixed-frequency ridge. A broad ±2.0%
frequency constraint prevents mode switching without forcing the pole to stay
within a physical linewidth. The initial-frequency Lorentzian is displayed
only as a diagnostic.
The boundary controls pass through a 50 nm spatial filter, then through the
continuous subpixel fill calculation used by the actual FDTD permittivity.
L-BFGS constructs its direction from exact autodiff gradients and up to
8 accepted secant pairs. An Armijo line search chooses
the accepted distance; the geometric move limit is only a topology/fabrication
trust region. Equal-score or unchanged-material proposals are not accepted.
The dashed orange marker records the controller change after
**27 completed boundary updates**. From that point,
the first line-search scale may exceed one, only two forward trials are
allowed, and the trust radius follows actual/predicted objective agreement.
It is no longer collapsed merely because a smaller Armijo trial succeeded.
The level-set radius begins this continuation at 2.5 nm and may grow to
7.5 nm.
Per the present campaign protocol, no periodic exact-binary validation solve
is performed during this stage.
### Boundary-stage fields
These are pole-aligned field audits, not the original fixed-carrier phasors.
The detector wavelength and independently fitted wavelength are printed above
the panels. The audit stores the geometry it actually measured and can trail
the newest accepted boundary by at most one shared-GPU update. Panels are
normalized independently for spatial inspection; scalar LDOS and Q values
come from the unnormalized ringdown tape.
### Zoomed boundary field with the etched geometry
[](../_static/generated/fryett_boundary_field_xy_zoom.png)
[Open the full-resolution xy PNG](../_static/generated/fryett_boundary_field_xy_zoom.png).
The x and y axes use the same physical scale, the view is restricted to
**-2.5 to +2.5 µm**, and the cyan silhouette is the exact
`pole_field_density` checkpoint measured by the field audit. It therefore
shows directly whether each antinode lies in SiN, a through-etched hole, or
the surrounding cladding.
[](../_static/generated/fryett_boundary_field_xz_zoom.png)
[Open the full-resolution xz PNG](../_static/generated/fryett_boundary_field_xz_zoom.png).
This panel also has equal physical scaling. Its cyan silhouette is the 330 nm
SiN cross-section evaluated on the y=0 symmetry plane, including the gaps made
by every hole crossing that plane.
## Stage 2B — explicit cubic-spline boundary optimization
```{admonition} Independent matched-start campaign
:class: important
This branch starts from the same terminal stage-1 ellipse geometry and uses
the same FDTD scene, symmetry, shifted peak-LDOS objective, 2% pole guard,
L-BFGS memory, and Armijo acceptance rule as Stage 2A. It is currently
**stopped_plateau_user_requested** at update **74**. Last checkpoint:
`2026-08-25T11:18:26.709277+00:00`.
```
shifted peak LDOS30249tracked-pole objective
handoff peak LDOS9921.5independent spline replay
initial-line LDOS4.5007e-05diagnostic only
conservative Q2.3767e+06minimum field/energy fit
tracked pole765.425 nmhandoff 769.622 nm
mode volume11.94 (λ/n)³at the tracked pole
physical controls54850 centers + 50×9 radii + 48 beam-edge knots
L-BFGS memory0 pairslatest accepted step 0 nm
next trust limit0.0025 nmagreement-ratio controller · maximum 10.0 nm
step controlleradaptive_expansion_rho_v2changed after 22 completed updates · next α=1
topology1 solid / 50 holessame lineage gate
### Spline geometry
**GDS downloads:** [stage start](../_static/downloads/fryett/fryett_spline_start.gds) ·
[final optimized spline](../_static/downloads/fryett/fryett_spline_optimized.gds)
Each positive-side hole has one movable x center and nine cubic radial
displacement knots. Reflecting the radial spline at angles 0 and π makes its
slope vanish on the y symmetry plane. Forty-eight cubic controls move the
positive-quadrant beam edge, with an analytic taper forcing displacement and
slope back to zero at the fixed PML feedthrough. The initial spline raster
differs from the stage-1 parametric raster by **0.633%**
mean fill.
Unlike the 8,092-control level set, this **548-control**
model cannot create fine independent corrugations between spline knots. That
is the intended tradeoff: a smaller search space and smooth boundaries in
exchange for less free-form geometric freedom.
### Spline optimization history
This is the spline worker's complete independent history, not the shared
comparison plot. Step zero is its own zero-displacement FDTD handoff replay.
The upper row shows the differentiated shifted peak-LDOS objective, all three
ringdown Q estimates, normalized and physical mode volume, and tracked pole.
The lower row separates the fixed-handoff-frequency LDOS diagnostic from the
objective and exposes the accepted Armijo step, gradient scale, accumulated
L-BFGS memory, and GPU solve time. A falling fixed-frequency LDOS is therefore
not mistaken for failure when the optimized pole has shifted.
The orange **next-step trust limit** is an optimizer control, not a measured
device property. A full Armijo step expands it by 20%; a backtracked accepted
step resets it to 1.5 times the motion that actually worked; and a completely
rejected proposal halves it. The resulting sawtooth prevents repeated large
geometry proposals while allowing the cap to recover after successful full
steps.
The dashed orange marker records the replacement of that legacy rule after
**22 completed spline updates**. The active controller
may expand the L-BFGS scale above one, chooses an aggressive trial before
spending a forward solve, and permits at most two FDTD line-search trials. Its
trust radius stays unchanged after an adequately predicted accepted step,
grows 1.5× only when a good step uses most of the available radius, and shrinks
only after poor actual/predicted agreement or complete rejection. The spline
radius restarted at 4 nm and may grow to
10.0 nm.
### Spline fields
These are pole-aligned field audits, not the original 780.55 nm fixed-carrier
phasors. The detector wavelength and independently fitted wavelength are
printed above the panels. The audit can trail the latest accepted geometry by
at most one shared-GPU update.
### Zoomed spline field with the etched geometry
[](../_static/generated/fryett_spline_field_xy_zoom.png)
[Open the full-resolution xy PNG](../_static/generated/fryett_spline_field_xy_zoom.png).
The plot uses a true 1:1 spatial scale and overlays the geometry stored with
this pole field, not the other optimizer's mask.
[](../_static/generated/fryett_spline_field_xz_zoom.png)
[Open the full-resolution xz PNG](../_static/generated/fryett_spline_field_xz_zoom.png).
The two optimizers take turns under one process-shared GPU lock. Each turn is
one complete adjoint update plus any required forward Armijo trials; neither
worker can overlap a Maxwell solve with the other.
## Stage 3 — low-radiation subspace mode-volume optimization
```{admonition} Last independent replay passed
:class: important
The independent best-design replay **passed**
the Q, pole-tracking, fit-quality, and topology gates. The audited checkpoint is
`2026-08-26T10:49:12.588135+00:00`.
```
current Q104,063hard floor 100,000 · audited 104,063
current peak LDOS1,651.0tracked-pole Purcell factor · audited 1,650.7
current normalized V9.5794V / (λ/n)³ · audited 9.581
audited physical V0.514470 µm³23.143% below 0.669387 µm³
starting Q2,376,739same tracked mode family
audited pole754.534 nminside the tracking band
learned low-radiation rank1radiation-gradient and accepted-secant basis
optimizer attempts10156 accepted · 45 rejected
GPU solve time15.84 hcampaign total
projectionenabledopt-in and independently disableable
active boundarycubic splinelast changed at attempt 95
### Spline handoff geometry
**GDS download:** [initial Stage 3 cavity](../_static/downloads/fryett/fryett_low_radiation_initial.gds)
### Current checkpoint geometry
**GDS download:** current accepted cavity — update 101
These are native boundary exports, not contours traced from the dashboard
pixels. The initial file is the immutable spline start; the current file is
rebuilt from the latest accepted cubic spline whenever this
dashboard refreshes.
### Added and removed dielectric
Red is dielectric added relative to the spline handoff; blue is dielectric
removed. White cells are unchanged.
### Audited xy field
### Audited xz field
Each geometry and field occupies its own full-width row. Both field panels use
equal physical x/transverse scaling, one shared absolute color normalization,
an automatic crop around the localized mode, and the matching audited geometry
in cyan. While the optimizer is running, the current geometry can lead the
independent field audit; the field rows update to the new best when the batch
finishes its exact replay.
This branch starts from the terminal cubic-spline cavity and minimizes mode
volume while enforcing **Q ≥ 100,000**.
The last independent replay has a **23.143%** smaller physical
mode volume, **Q=104,063**, peak LDOS **1,650.7**, and
normalized **V=9.581**. The field rows use that
independently replayed geometry rather than an in-loop estimate.
This uses the same 2×4 history layout as the cubic-spline campaign above:
tracked-pole peak LDOS, conservative Q, normalized and physical mode volume,
pole wavelength, fixed-emitter LDOS, accepted motion and trust limit,
optimizer conditioning, and GPU solve cost. Peak LDOS is the cavity Purcell
factor at the tracked pole; the lower-left fixed-emitter panel is diagnostic
only. Normalized volume is **V / (λ/n)³** at each fitted pole.
After **31 completed attempts**, Q
reached the hard floor and the proposal rule changed at the dashed green line.
The current full `∇log(Q)` now defines an explicit feasible half-space. The
L-BFGS mode-volume proposal is projected into it with a
2.0% first-order Q
margin, and a candidate is accepted only when the forward solve measures
**Q_new ≥ Q_current**. This is stronger than merely remaining above the
absolute floor.
After **63 completed attempts**, the accepted spline boundary was
converted exactly into a zero-displacement free level set, marked by the
dotted purple line. The material-map change at conversion was at most
**5.96e-07**. Its independent
Maxwell handoff replay **passed**. Coordinate-dependent L-BFGS
curvature, radiation secants, and the learned dark basis were cleared; the full
Q gradient restarted the dark constraint in the new
7,565-variable space.
After **95 completed attempts**, the accepted free boundary was
projected back into the 548
cubic-spline coordinates, marked by the dotted blue line. The closest spline
fit changed the fill map by RMS
**0.00015** and maximum
**0.00353**. Its independent
Maxwell handoff replay **passed**. Coordinate-dependent curvature and
the learned dark basis were restarted, while the objective, nondecreasing-Q
rule, history, and every forward acceptance gate remained unchanged.
The adjoint solve still supplies the ordinary first-order mode-volume and
radiation-loss gradients. The optional projector learns directions that are
radiation-sensitive from the current gradient and accepted secants, then
removes those components from the proposed mode-volume step. This makes a
coordinated many-boundary change available to the optimizer without forming a
dense electromagnetic Hessian. Every trial still passes the full forward
Maxwell audit, the hard-Q constraint, an Armijo decrease test, pole and fit
checks, and the original topology gate.