Quan–Lončar derivation guide

This is the reading map for the original analytical cavity engine. The linked chapters are not excerpts: the derivation archive preserves each source page in full, including intermediate approximations, rejected branches, numerical convergence checks, and implementation notes.

What the lineage constructs

The derivation starts from the open Maxwell problem for a nanobeam, builds an explicit waveguide Green tensor and a complete hole basis, reduces the system by symmetry, and searches the analytically continued cavity determinant for outgoing poles. Later chapters test which transverse and thickness expansions survive convergence and extract the longitudinal series used by the Quan–Lončar optimizer.

The logical dependency is:

bare waveguide modes
        ↓
waveguide Green tensor
        ↓
complete through-hole basis and interface projection
        ↓
symmetry-reduced multiple-scattering determinant
        ↓
outgoing complex-frequency continuation
        ↓
converged pole and Q series

Core derivation sequence

Chapter	Main result	Why it matters
1. Exact convergent-series target	Defines the full open-cavity series problem and convergence target.	Establishes what an “analytical” Q is supposed to converge to.
2. Exact bare-waveguide Green tensor	Constructs the reference resolvent from guided and radiation sectors.	Supplies propagation between discontinuities without a finite simulation box.
3. Outgoing continuation and first cavity determinant	Continues the scattering problem to complex frequency and defines its poles.	Connects a determinant zero to the QNM frequency and Q.
4. Complete elliptical through-hole modes	Expands the internal hole field in a complete modal basis.	Prevents the hole from being reduced to a scalar reflectivity.
5. Controlled transverse-momentum limit	Identifies a controlled truncation of the transverse spectrum.	Separates a useful reduced model from an uncontrolled shortcut.
6. Exact symmetry reduction	Block-diagonalizes the complete hole basis by physical symmetries.	Reduces cost without changing the selected polarization sector.
7. Contour search in physical sectors	Locates determinant zeros using contour methods.	Avoids mistaking a real-frequency peak or numerical minimum for a pole.
8. Through-thickness convergence	Tests convergence of the vertical representation.	Exposes Q sensitivity to weakly represented radiation channels.
9. Degree-two thickness continuation	Reviews the first nontrivial vertical correction.	Establishes a controlled sequence rather than a one-off fit.
10. Odd thickness degree three	Adds and audits the odd vertical sector.	Detects symmetry leakage and missing channels.
11. Even thickness degree four	Extends the even sector and audits polarization.	Tests whether Q stabilizes as the basis grows.
13. Separated even-x in-plane degree four	Separates the remaining in-plane correction.	Completes the tested high-order reduced basis.
22. Longitudinal extraction and surviving Q series	Extracts the longitudinal structure that remained numerically useful.	This is the closest bridge from the long derivation to the production surrogate.

Every phase README—including phases without a separate derivation manuscript—is available in the full archive. Those pages contain the actual development record and should be consulted when reproducing a numerical choice.

What is and is not established

The lineage establishes a structured open-system pole surrogate with explicit convergence tests. It does not establish that its absolute Q equals FDTD for every finite truncation, nor that its local Q gradient always transfers after a geometry remesh. The productive historical result is weaker and still useful: long analytical BFGS trajectories sometimes produced orders-of-magnitude FDTD improvement even when pointwise Q and gradients disagreed.

For arbitrary boundaries and a solver-independent derivative statement, continue with QNM and arbitrary-boundary shape derivatives.