Model → topology → evidence
How the optimizer works
A radiation-aware slab model supplies fast topology gradients. Every promising density then crosses independent basis and vector-Maxwell gates before it earns a stronger label.
The physical target
Geometry
A patterned dielectric slab of thickness 0.32 λ₀, with n_core = 2, n_clad = 1, and λ₀ = 780 nm. In-plane density is extruded through the slab.
Immutable atom opening
The full strip |x| < 0.5 λ₀ is exactly air in every topology. X/y reflection symmetry and fixed terminal mirrors are hard constraints, not soft penalties.
The 2.5D reduced model
The scalar field is expanded in periodic x/y Fourier harmonics and continuous linear finite elements through the physical slab. Homogeneous upper and lower half-spaces are eliminated analytically with channel-resolved outgoing Dirichlet-to-Neumann maps. That retains vertical radiation loss without a vertical PML or an empirical light-cone penalty.
The open boundary makes the eigenproblem nonlinear in frequency. Continuation stays on one outgoing square-root sheet, and the exact topology derivative includes the frequency derivative of the DtN self-energy. Omitting that term gives the wrong open-system gradient.
Topology and step acceptance
1 · Parameterize
Raw box-bounded variables pass through a compact density filter, smooth Heaviside projection, and reflection projection. Protected air and fixed mirrors are reset exactly after projection.
2 · Differentiate
The nonlinear pole sensitivity is contracted to every physical pixel, then backpropagated through the projection and filter transpose.
3 · Gate
Trust-region proposals are backtracked until score, mode overlap, localization, gap energy, frequency, residual, air, and symmetry gates all pass.
Exact x/y/z reduction for centered-atom discovery
The reflection-symmetric scalar Fourier/FEM operator is now restricted to the centered emitter's even sector in x, y, and z. The orthonormal projection is applied consistently to stiffness, dielectric mass, outgoing DtN self-energy, source, field reconstruction, and both gradients. The current launch basis falls from 1,701 to 275 unknowns; historical checkpoints remain full-basis and cross the boundary by density-only transfer and fresh mode reacquisition.
Parity equivalence: PASS. Full and reduced scalar atom responses agree to roundoff. The full-vector staggered-Yee solver now also has a separately qualified component-correct x/y/z sector: the production system falls from 235,200 physical unknowns to 29,499, while a full/reduced cross-basis check retains 0.96204 complex-field correlation and 1.32% response agreement.
Mode identity is a field question
A nearby frequency or a larger Q does not prove that two discretizations found the same mode. On one basis the optimizer uses an energy-weighted overlap. Across bases it reconstructs complex fields on a common physical grid and requires normalized field correlation. Geometry transfer carries density only; the mode is independently reacquired in the new model.
Evidence pipeline
2.5D candidate
A localized mode and accepted topology step on one reduced basis.
Basis-stable candidate
Analytic and gradient tests pass; independent richer-basis fields correlate above 0.95 and retain the gain.
3D candidate
Density alone is fitted through the 3D topology map, then the full vector mode is reacquired and matched by complex-field overlap.
Device evidence
Requires spectral acceptance plus PML, grid, and domain convergence. The current result has not reached this tier.
Why the full-vector objective changed
The transferred cavity contains a nearby two-mode cluster. Rational pole estimates changed across wide/local windows and disagreed with the real-axis linewidth by 26.1%, so the strict pole objective refused to optimize. The working fallback differentiates the directly resolved Fano linewidth on a fixed sample mask, then uses fresh interleaved frequencies as an independent acceptance set.
The 780 nm atom objective
For the current campaign, the proposal objective is no longer Q. At real frequency f = 1.0 the reduced solver applies a centered emitter, solves the outgoing driven system, and maximizes Im(sᴴA⁻¹s). The Hermitian adjoint reuses the primal sparse LU and backpropagates the log-response gradient through the exact air/symmetry topology map.
A direct-response improvement is not automatically an accepted mode step. Nonlinear pole residual, same-basis identity/localization, continuation frequency, independent richer-basis persistence, and full-vector validation remain separate gates. The current dashboard reports those decisions independently.
The full-vector scale-up gate
Before spending a larger optimization budget, the 780 nm path must pass five checks: outgoing-boundary convergence, a complete atom-response directional derivative, a restartable source-coupled pole cross-validated by the complex real-axis response, density-only grid/domain convergence, and a measured primal/adjoint/backtrack/restart resource envelope.
The pole calculation differentiates the complete effective Maxwell operator on the retarded real axis, including both mirror self-energies. It tracks field identity while moving the real expansion center; a safeguarded secant update is used only after correlated evaluations. The independent AAA fit retains the full complex dipole response, so a nearby weak sharp pole cannot be mistaken for the broad atom-coupled branch.