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engines/quan_loncar_v1/package/quan_loncar_optimizer/engine/fryett_q_series_phase4/DERIVATION_PHASE4.md · assembled 2026-07-29 15:57 UTC.

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Phase 4 derivation: complete modes inside an elliptical through-hole

1. Normalized hole coordinates

For hole j, introduce

\[ x-x_j=r_{xj}\rho\cos\theta,\qquad y=r_{yj}\rho\sin\theta,\qquad 0\le\rho\le1, \]

and z in [-T/2,T/2]. The Jacobian is r_xj r_yj rho.

The real disk modes are

\[ Z_{nm}^{c}(\rho,\theta)=R_n^m(\rho)\cos(m\theta), \qquad Z_{nm}^{s}(\rho,\theta)=R_n^m(\rho)\sin(m\theta), \]

where 0 <= m <= n and n-m is even. The sine mode is omitted for m=0. Their squared norms on the unit disk are

\[ N_{n0}=\frac{\pi}{n+1},\qquad N_{nm}=\frac{\pi}{2(n+1)}\quad(m>0). \]

The radial polynomial is evaluated stably as

\[ R_n^m(\rho)=(-1)^{(n-m)/2}\rho^m P_{(n-m)/2}^{(m,0)}(1-2\rho^2). \]

The normalized scalar basis in the physical hole is

\[ h_{j,nm\sigma\ell}= \frac{Z_{nm}^{\sigma}(\rho,\theta)} {\sqrt{r_{xj}r_{yj}N_{nm}}} \sqrt{\frac{2\ell+1}{T}}P_\ell(2z/T). \]

Tensoring it with the three Cartesian unit vectors gives a complete vector basis of L2(H_j)^3 as the retained degrees tend to infinity.

2. Projection into the waveguide basis

At fixed longitudinal momentum beta, the projection is

\[ C_{a,j\mu}(\beta)= \int_{H_j}d^3r\,\psi_a(y,z) e^{-i\beta x}h_{j\mu}(x,y,z). \]

Because the hole spans the complete beam thickness and uses the same normalized Legendre functions as the waveguide,

\[ \int_{-T/2}^{T/2}dz\,\phi_{i_z}(z)\phi_\ell(z) =\delta_{i_z\ell}. \]

Thus the thickness projection is exact. The remaining disk integral is

\[ \frac{\sqrt{r_xr_y}}{\sqrt{N_{nm}}}e^{-i\beta x_j} \int_0^1\rho d\rho\int_0^{2\pi}d\theta\, \phi_{i_y}(r_y\rho\sin\theta) Z_{nm}^{\sigma}(\rho,\theta) e^{-i\beta r_x\rho\cos\theta}. \]

It is entire in beta. Gauss--Legendre radial quadrature and periodic angular trapezoidal quadrature therefore form a spectrally convergent evaluation.

At (n,m,l)=(0,0,0), this basis is exactly 1/sqrt(pi r_x r_y T). Consequently Phase 4 reduces algebraically and numerically to the Phase-3 constant-hole projection at degree zero.

3. Cavity determinant

With all holes concatenated in C_N, the outgoing projected Green matrix is

\[ G_{H,N}^{+}(\omega)= \int_{\mathcal C_+}\frac{d\beta}{2\pi} C_N(-\beta)^T G_{\rm wg}(\beta,\omega)C_N(\beta). \]

The finite pole equation remains

\[ D_N(\omega;g)=\det\left[ I-k_0^2(\epsilon_c-\epsilon_b)G_{H,N}^{+}(\omega) \right]=0. \]

The transpose rather than a Hermitian conjugate is required for analytic continuation. Reciprocity predicts a complex-symmetric operator, furnishing a stringent numerical check.

4. First branch-tracking sequence

The Phase-4 runner begins with a real-frequency singular-value scan. It then uses the smallest left/right singular vectors to solve the local scalar equation

\[ y^\dagger A_N(\omega)x=0 \]

and refreshes the vectors after each outer iteration. Each higher truncation starts from the preceding complex root. This is a first test of whether one simple branch can be followed; it is not yet proof of convergence.

When several zeros are nearby, choosing the smallest singular value can change branches between two truncations. The refined tracker labels the branch by maximal overlap of its right null vector with the preceding right null vector. It accepts a row only when the tracked eigenvalue, the smallest singular value, and the lower-half-plane sign all pass simultaneously.

The required final study must vary axes independently around a substantially higher common baseline and include all physical holes. A stable Q may be compared with the paper only after the same pole survives those variations.

5. Resolved longitudinal contour rule

The first attempted table exposed a failure of a single global tangent-map quadrature: changing its order can place a new beta node extremely close to a guided pole, producing a huge operator norm instead of convergence. Phase 4 therefore also supplies a panelized Sommerfeld rule on [-U,U]. Its cutoff, number of panels, and order per panel are separate and explicit. Increasing U controls the omitted high-|beta| tail; refining the panels resolves the radiation interval and guided-pole neighborhoods without moving every node nonlocally.

An equal-width panel sweep still moved all panel boundaries relative to the singularities. The final Phase-4 rule therefore fixes breakpoints clustered around u=1 and the first guided-mode interval. Its convergence sequence raises Gauss order inside the unchanged segments. This is the appropriate longitudinal quadrature axis for the first trustworthy table.