Device23 derivation guide

Device23 extends the original nanobeam scattering construction to the layered SiO₂/Si₃N₄/organic stack, a finite feedthrough strip, and complex-frequency quasinormal modes. The archive preserves the complete package derivations; this page explains their dependency and present trust boundary.

Dependency map

Fryett phases 1–6: open nanobeam and complete hole basis
                         ↓
stratified reference Green function
                         ↓
layered strip mode + feed-pole subtraction
                         ↓
finite hole-array feed projection
                         ↓
complex-β cavity determinant
                         ↓
complex-frequency continuation and QNM operator

Layered-device sequence

The first six Fryett chapters are shared foundations. Read them in the Quan–Lončar guide, or use the Device23 copies in the full archive when exact package provenance matters.

Chapter	Main result	Implementation role
Layered phase 1: stratified reference Green function	Spectral Green function for the planar material stack.	Replaces the homogeneous cladding reference with SiO₂/Si₃N₄/organic layers.
Layered phase 7: strip and feed-pole subtraction	Isolates the guided feed pole from the layered continuum.	Prevents the desired waveguide channel from being counted as an undifferentiated radiation background.
Layered phase 8: exact finite-hole feed projection	Projects the full finite hole sequence onto the feed representation.	Connects the 40-hole geometry to the layered scattering operator.
Layered phase 9: complex-β cavity determinant	Forms the outgoing determinant with complex longitudinal propagation.	Defines a pole condition for the finite device rather than a real-frequency resonance proxy.
Layered phase 10: complex-frequency feed continuation	Continues the feed pole consistently with complex frequency.	Keeps mode identity meaningful as the resonance moves.
Layered phase 11: complete QNM operator	Assembles the final nonlinear complex-frequency operator.	Supplies the pole and left/right fields needed for Q sensitivity.

Relationship to the free-form optimizer

The package derivation supplies a geometry-to-QNM map. Arbitrary-boundary optimization adds a second layer: a normal velocity is defined on every movable dielectric interface, the complex pole derivative is contracted with that velocity, and the geometry is advected while tracking the same QNM. Holes and cavity-region sidewalls are therefore degrees of freedom of one shape field, not separate scalar radius parameters.

The current Device23 implementation has demonstrated a self-consistent fixed-connectivity Q derivative, but it has also shown why that alone is not a physical certificate: the imaginary pole part is small enough that meshing and geometry-transfer errors can reverse the inferred Q change. The current trust status and the rejected pre-fix evidence are summarized in the Device23 full-boundary Q effort.

The general derivative and its numerical failure modes are derived in QNM and arbitrary-boundary shape derivatives.