Two-stage topology optimization of log LDOS, following Shaker et al. §3–6.
2D out-of-plane (Ez) Helmholtz in natural units c = ε0 = μ0 = 1, with λ0 = 1 and ω0 = 2π. Finite-difference frequency-domain on a uniform grid, terminated by a reciprocal (complex-symmetric) stretched-coordinate PML so that A = AT. The discrete LDOS is −Im[e†b] from A e = b, reported as a ratio to the same operator in vacuum.
A few hundred to a thousand steps of max log LDOS(ω0) create a resonance near the target. After that we switch to the paper’s frequency-shifted objective: evaluate LDOS at Re[ω*(p)], the eigenfrequency closest to ω0, with the stabilizing bound Re ω* ≥ ω0. Gradients are the Appendix A adjoint formulae (no extra Maxwell solve beyond the LDOS solve and one shift-invert factorization).
Design region 1.5λ × 1.5λ, 0.5λ air pad, 0.25λ PML, resolution 20 pixels/λ. Source 0.05λ above the top edge, centered. ε ∈ [1, 12], conic filter radius 0.1λ, subpixel-smoothed projection. Unshifted β schedule 8 → 16 → 40 → ∞, then shifted at β = ∞.
The conic filter alone does not enforce a minimum feature size. After filtering we project the same field at η = 0.5 − Δη (dilate) and η = 0.5 + Δη (erode) and add λ ⟨(ρd − ρe)²⟩ on the design region to the objective. Physics stays at β = ∞; the penalty uses a finite morphological β so it still has a gradient. See the bake-off.
The dipole sits at the origin. A vertical strip |x| < 0.5 λ0 (0.5 μm on each side if λ0 = 1 μm) is frozen vacuum for all y. Design variables live only outside that strip, in a box 1 λ wide on each side and 2 λ tall, with 0.5 λ air pad and 0.5 λ PML. The density is parameterized in the first quadrant and copied through x = 0 and y = 0, so the permittivity is exactly even. Same filter, SSP, and unshifted→shifted loop. The compact run used 1 λ of design on each side (|x| ≤ 1.5). The wide run extends that to |x| ≤ 5 and |y| ≤ 1 and does not stop on the Fig. 6 bars. Dashboards: wide, compact.
Vector Maxwell on a Yee grid, octant only: PMC at x = 0, PMC at y = 0, PEC at z = 0 for a z-dipole. The discrete curl-curl is C = (curls)T (curls) with SC-PML stretches frozen at ω0, so A = C − ω² M is complex-symmetric and SuperLU’s transpose solve is the adjoint. Design is 2.5D — an in-plane density extruded through a Si3N4 slab of thickness 0.15/0.78 λ0 — with the same A-a35 filter / erosion / dilation and, when the octant has room, the same hard-tied period a = 0.35. See the 3D page and the 2D comparison.
The exact paper structure is not required. The run succeeds when LDOS / vacuum LDOS ≥ 500 and Q ≥ 50,000 on a binarized design.