refactor(notebooks): use internal adam and optimize helper - batch 4

[marcorudolphflex]

follow-up from #463

Replaces the optax Adam with our Adam and makes use of the optimize helper where easily possible. This is the fourth batch, covering the 12 notebooks currently in this PR where develop still had real optax usage.

Results are roughly unchanged overall. Notable differences are sorted by risk first; images can be inserted/updated inline as needed.

Percent differences below use (branch - develop) / develop. For max objectives, positive is better; for min objectives/costs, negative is better.

Status	Notebook	Direction	develop	branch	diff	diff %	Note
⚠️	Autograd27Smatrix	min	1.0909e-02	3.6146e-02	+2.5237e-02	+231.34%	objective, worse
❓	TidyFab0GC	max	3.0199e-01	5.1519e-01	+2.1320e-01	+70.60%	objective, unexpectedly better
❓	Metalens	max	6.9960e+01	1.2850e+02	+5.854e+01	+83.68%	saved output shifted; old develop output did not match the current two-source code
	Autograd19ApodizedCoupler	max	4.799e-01	4.801e-01	+2.0e-04	+0.04%	objective; coupling efficiency develop -1.98 dB vs. branch -1.99 dB
	Autograd7Metalens	max	2.2378e+01	2.3344e+01	+9.660e-01	+4.32%	objective, roughly unchanged
	Autograd25WaveguideCrossing	max	9.6852e-01	9.4947e-01	-1.905e-02	-1.97%	objective; changed fwidth from f0 / 20 to f0 / 5
	Autograd26DiffractiveBeamSplitter	max	8.792e-01	8.790e-01	-2.0e-04	-0.02%	loss value; cost min changes by +0.02%; final efficiency 0.91 to 0.89
	Autograd28FiberLens	max	4.0488e-02	4.0460e-02	-2.8e-05	-0.07%	final flux, plot/result nearly unchanged
	BilayerSiNSiGC	max	8.134e-01	8.139e-01	+5.0e-04	+0.06%	objective, effectively unchanged
	RFAutograd1RectangularPatchAntenna	max	20.812315	20.717626	-9.4689e-02	-0.45%	final nonzero FOM, roughly unchanged
	Autograd13Metasurface	min	1.194e-01	1.185e-01	-9.0e-04	-0.75%	final loss, roughly unchanged
	Autograd23FabricationAwareInvdes	max	0.09930831775707324	0.09930831775707324	0	0.00%	initial objective check; no credits for full rerun

⚠️ Autograd27Smatrix.ipynb

Objective (min.): develop 1.0909e-02 vs. branch 3.6146e-02 (+231.34%)

develop

branch

❓ TidyFab0GC.ipynb

Objective (max.): develop 3.0199e-01 vs. branch 5.1519e-01 (+70.60%)

Unexpectedly better. This may be from client/solver changes rather than the optimizer refactor.

❓ Metalens.ipynb

Objective (max., saved notebook output): develop 6.9960e+01 vs. branch 1.2850e+02 (+83.68%)

Branch output is internally consistent with the current source and the rerun (120.32 to 128.50, 30 optimization steps). The large shift is not expected from the optimizer refactor; the old develop output cells did not match the current code path with both source_x and source_y. I checked with Filipe that this is okay to keep, and the changed results are explainable from that stale-output/code mismatch.

Autograd19ApodizedCoupler.ipynb

Objective (max.): develop 4.799e-01 vs. branch 4.801e-01 (+0.04%)

Maximum coupling efficiency: develop -1.98 dB vs. branch -1.99 dB

Rerun is back in line with develop.

Autograd7Metalens.ipynb

Objective (max.): develop 2.2378e+01 vs. branch 2.3344e+01 (+4.32%)

Plot/result roughly unchanged.

Autograd25WaveguideCrossing.ipynb: had to change fwidth from f0 / 20 to f0 / 5; otherwise the transmission output got glitchy values above 1.

Objective (max.): develop 9.6852e-01 vs. branch 9.4947e-01 (-1.97%)

develop:

branch:

Autograd26DiffractiveBeamSplitter.ipynb

develop:

Loss value (maximize): 0.8792
  Cost (minimize): 9.2696e-01
  Fab penalty: 9.9816e-01
  Efficiency: 0.9999

branch:

Loss value (maximize): 0.8790
  Cost (minimize): 9.2712e-01
  Fab penalty: 9.9816e-01
  Efficiency: 0.9999

Later final-efficiency output: develop 0.91 vs. branch 0.89.

Autograd28FiberLens.ipynb

Objective / final flux (max.): develop 4.0488e-02 vs. branch 4.0460e-02 (-0.07%)

Plot/result nearly unchanged.

BilayerSiNSiGC.ipynb

Objective (max.): develop 8.134e-01 vs. branch 8.139e-01 (+0.06%)

Effectively unchanged.

RFAutograd1RectangularPatchAntenna.ipynb

Final nonzero FOM: develop 20.812315 vs. branch 20.717626 (-0.45%).

Optimization a little less smooth now, result differs a bit:

Autograd13Metasurface.ipynb

Final loss (min.): develop 1.194e-01 vs. branch 1.185e-01 (-0.75%).

Autograd23FabricationAwareInvdes.ipynb

Did not rerun this notebook because I do not have credits for the prefab/fabrication-aware run path. No result shift is expected from this diff.

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[marcorudolphflex]

will rerun Autograd27 with other seed to check if we can still see that objective setback.

[marcorudolphflex]

Superseded by #513, which keeps only the no-brainer subset and leaves Metalens, Autograd27Smatrix, and Autograd25WaveguideCrossing out for separate discussion.