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Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/fryett_q_series_phase3/DERIVATION_PHASE3.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Phase 3 derivation: outgoing continuation and the first cavity determinant

1. What has changed

Phase 2 constructed the bare rectangular-waveguide Green tensor G_wg(beta, omega) and exposed its guided poles and radiation discontinuity. Phase 3 now performs the beta inverse transform on the outgoing Riemann sheet and projects the result into the physical elliptical holes.

No fitted mirror strength, phenomenological decay constant, or measured Q is used in this step.

2. The outgoing beta contour

Write

\[ u=\frac{\beta}{k_c(\omega)},\qquad k_c=n_c\frac{\omega}{c}. \]

The radiation branch points are fixed at u=+/-1; guided poles lie near u=+/-n_eff/n_c. The outgoing Sommerfeld path passes below every positive-u singularity and above its negative-u partner. The implementation uses

\[ u(t)=t-i h\tanh(t/d)\exp[-(t/L)^8],\qquad -\infty<t<\infty. \]

The dimensional contour is beta(t,omega)=k_c(omega)u(t). Holding u(t) fixed while moving omega from the causal upper half-plane to the resonance lower half-plane is an explicit analytic-continuation homotopy. A pole or branch point crossing this contour is therefore a detectable failure, not an implicit change of sheet.

The spatial waveguide Green tensor is

\[ G_{\rm wg}^+(x-x';\omega)= \int_{\mathcal C_+}\frac{d\beta}{2\pi} e^{i\beta(x-x')}G_{\rm wg}(\beta,\omega). \]

Because the cross-section momentum integrals are evaluated at complex beta, their denominators do not land on the real radiation shell. Guided residues, the radiation cut, and the evanescent principal part are all contained in the single contour integral. The scalar identity

\[ \frac{1}{2\pi}\int_{\mathcal C_+} \frac{e^{i\beta x}}{\beta^2-a^2}\,d\beta =\frac{i e^{iax}}{2a},\qquad x>0, \]

is the first orientation test.

3. Exact projection of the constant elliptical-hole mode

For hole j, let

\[ H_j=\left\{(x,y,z): \frac{(x-x_j)^2}{r_{xj}^2}+\frac{y^2}{r_{yj}^2}\le1, \ |z|\le T/2\right\}, \qquad V_j=\pi r_{xj}r_{yj}T. \]

The first three hole basis functions are

\[ h_{j,s}(\mathbf r)=V_j^{-1/2}\mathbf e_s\,1_{H_j}(\mathbf r), \qquad s\in\{x,y,z\}. \]

At fixed y, the exact x transform is

\[ \int dx\,e^{-i\beta x}h_{j,s} =\frac{e^{-i\beta x_j}}{\sqrt{V_j}} 2a_j(y)\,\operatorname{sinc}[\beta a_j(y)]\mathbf e_s, \]

where

\[ a_j(y)=r_{xj}\sqrt{1-y^2/r_{yj}^2}. \]

The z integral is also exact. Since the etched hole spans the full beam thickness, it couples only to z-Legendre degree zero at constant hole order. The remaining finite y integral is Gauss-Legendre quadrature on [-r_yj,r_yj]; raising its order is a direct convergence axis.

Let C_j(beta) denote this cross-section projection. Analytic continuation requires the bilinear matrix element

\[ [G_H(\omega)]_{is,jt}=\int_{\mathcal C_+}\frac{d\beta}{2\pi} [C_i(-\beta)]_s^T G_{\rm wg}(\beta,\omega)[C_j(\beta)]_t. \]

The left factor is C(-beta)^T, not C(beta)^dagger. They coincide on the real-frequency boundary for real basis functions, but only the former is analytic in complex beta and omega.

4. First determinant and Q

For holes filled by the encapsulating cladding in the SiN reference beam,

\[ \Delta\epsilon=\epsilon_c-\epsilon_b<0. \]

At hole truncation N,

\[ A_N(\omega;g)=I-k_0(\omega)^2\Delta\epsilon\,G_{H,N}(\omega;g), \qquad D_N(\omega;g)=\det A_N(\omega;g). \]

A simple complex zero omega_tilde_N gives

\[ Q_N=-\frac{\operatorname{Re}\omega_tilde_N} {2\operatorname{Im}\omega_tilde_N}. \]

The code searches with the locally projected residual y^H A_N x rather than the raw determinant, whose magnitude is badly scaled at large N.

5. Convergence order and present limitations

The controlled limit is now explicit:

  1. increase the cross-section Legendre degrees and transverse-momentum order;
  2. increase the Sommerfeld contour order while varying its deformation without changing topology;
  3. increase the y quadrature used in ellipse-to-waveguide projection;
  4. replace each constant hole mode by increasing Zernike--Legendre degree;
  5. increase the number of physical mirror holes until the complete Fryett geometry is included;
  6. refine the same simple pole and report Q only after all axes stabilize.

This phase assembles a genuine geometry-dependent determinant, but it does not yet establish that the lowest truncation has a trustworthy pole. In particular, one constant function per vector component cannot resolve the field singularity and boundary layer at an air--silicon interface. The next phase must add the higher ellipse modes and produce the first coupled convergence table.