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Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/quan_loncar_series_phase25/NEXT_PHASE_QNM_OBSERVABLES.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Next phase: one normalized pole for Q, mode volume, and feed coupling

1. QNM residue normalization

Let A(omega,p)x=0 and y^H A(omega,p)=0 at a simple pole. Define

N = y^H A_omega x.

For a source-to-field map B and a coefficient-to-field map F®, the pole part of the physical Green response has the form

G_pole(r,r';omega)
  = F(r)x [y^H B(r')] / [N (omega-omega_tilde)].

The first acceptance test is scaling invariance under x -> alpha x and y -> y/alpha*, followed by a direct rank-one residue comparison with (omega-omega_tilde)G at several complex frequency offsets.

2. Complex effective mode volume

For a chosen emitter position r0 and unit polarization e, evaluate the QNM field amplitude E_e=e dot F(r0)x and use the same residue normalization N to define the complex effective volume in a convention consistent with the electric Green tensor. Constants must be fixed by the direct Green-residue test rather than by an arbitrary eigenvector norm.

Required checks:

  • invariance to left/right eigenvector scaling;
  • convergence with hole P1->P2 and thickness z0->z1 bases;
  • convergence of E(r0) under the field-evaluation quadrature;
  • agreement between the residue-derived Purcell factor and the pole response near resonance.

3. Feed-through waveguide coupling

Separate the guided-mode poles of the bare-waveguide beta Green function from the radiation continuum. Construct unit-power right/left port projectors at the two ends of the nanobeam. Their QNM residue amplitudes give the external decay rates kappa_L and kappa_R, while the remaining continuum gives kappa_rad.

Required checks:

-2 Im(omega_tilde) = kappa_L + kappa_R + kappa_rad

within the same frequency convention and within 10% initially, tightened with convergence. The resulting resonant residue is then inserted into the background-plus-pole through coefficient S21(omega) and compared with a direct driven solve.

4. Automatic-differentiation boundary

Once field and port functionals pass the above tests, provide JVP/VJP rules for A_p, F_p, and the port projectors. The pole pullback remains

d omega/dp = -(y^H A_p x)/N,

while mode-volume and coupling derivatives add the explicit derivatives of F, B, and the normalization N. No optimizer should differentiate through the Newton iteration or an eigensolver history.

Immediate implementation order

  1. Add coefficient-to-field and source-to-coefficient maps.
  2. Verify the full Green-function residue and eigenvector-scaling invariance.
  3. Define and converge complex mode volume.
  4. Extract and flux-normalize guided beta-pole port projectors.
  5. Verify the linewidth sum and direct S21 response.
  6. Replace centered geometry partials with analytic/AD JVPs and expose a multi-objective gradient API for Q, Veff, and feed coupling.