ldos-opt · Shaker et al. reproduction

Validation

Solver checks that must pass before a long 2D run.

Vacuum Green function: Im G(x₀,x₀) → 1/4.
PML: outgoing cylindrical wave; 1D Bragg cavity Q not PML-limited.
1D quarter-wave stack eigen-Q.
Unshifted ∂LDOS/∂ε vs finite difference.
∂ω_*/∂ε and the full shifted gradient vs FD.
SSP at β = ∞ is binary except a 1-pixel interface; VJP matches FD.
‖A − A^T‖ at roundoff.

Solver checks (2026-08-12)

FDFD / adjoint
- 2D operator ||A − Aᵀ|| = 4.8e-13 on a 60×60 grid.
- Im G(x0,x0) ≈ 0.253 versus analytic 1/4 at 20 px/λ.
- 1D five-pair λ/4 stack: Q = 3.55e5.
- Unshifted ∂LDOS/∂ε finite-difference relative error ≤ 6e-7.
- SSP VJPs match finite differences at finite β and at β = ∞.

Optimization
- 1D shifted log-LDOS on a 5λ domain: LDOS/vac ~ 3e7, Q ~ 1e8.
- 2D Fig. 6 geometry, 1.5λ design, 0.5λ PML (rebuilt from saved p):
  LDOS/vac = 544, Q = 3.42e4. Not PML-limited.
- 2D successive enlargement to 2λ (fig6-grow), independently rebuilt
  on 0.5λ and 0.75λ PML:
  LDOS/vac = 500.2, Q_eig = 7.40e4, Q_FWHM ≈ 8.9e4.
  Lorentzian peak sits at Re ω*. PML energy fraction < 0.1%.
- Slot-gap (gap-sym): dipole at the origin, vacuum strip |x| < 0.5 λ0
  for all y, design only outside, 4-fold mirror symmetry. Independently
  rebuilt from saved p:
  LDOS/vac = 1154.7, Q_eig = 5.106e4, Q_FWHM = 5.103e4.
  Peak sits at Re ω*/ω0 = 1.0014. Gap |ε−1| = 0, |ε−ε(−x)| ~ 3e-13,
  PML energy fraction 0.065%.

- Wide slot |x|≤5, |y|≤1 (gap-wide), unregularized: LDOS/vac ~ 2e6
  and Q_eig ~ 5e7 on a speckled design (82 solid pieces, 12 islands
  smaller than 4 px). Real field, tiny PML energy, but not fabricable.
- SiN (ε=4) slot, λ/10 filter: existing designs sit at ~0.08λ min
  width (1.6 px at 20/λ) — the grid floor, not a comfortable λ/10.
- Slow successive enlargement from sin-enl-best-s4, outer-period
  tiling, ΔL=0.25λ: Q 7.3e4 → 1.2e5 → 4.9e5 → 9.9e5 at |x|≤5,
  LDOS/vac = 1.29e5. Rebuild: source-weighted, PML energy 0.02%.
  Fast 0.5λ jumps had been copying the cavity defect into the new
  ring and Q collapsed.
- Erosion/dilation bake-off: λ ∈ {1,3,10} at R=0.1 left p frozen
  (filtered density sat at ~0.5, penalty gradient too small vs the
  high-Q basin). R=0.15, λ=3 (fab-r15) cut islands 82→26 and brought
  Q down to 6.2e4 with LDOS/vac = 1895. Opened-init from speckle
  killed the cavity. Continuation gap-wide-fab uses R=0.15, λ=3,
  plus a grayness term ⟨4ρ_f(1−ρ_f)⟩.

Eigensolver note: “nearest to ω0” can latch onto dark/PML modes with
huge fake Q. We only count a design after rebuilding ε from p and
re-picking the source-coupled mode.