Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/fryett_q_series_phase2/DERIVATION_PHASE2.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Phase 2: the exact bare-waveguide Green tensor

1. Purpose

Phase 1 reduced the finite Fryett cavity to

\[ D(\omega,\mathbf p) =\det_{\rm reg}\left[I-\frac{\omega^2}{c^2} G_{\rm wg}^+(\omega)\Delta\epsilon_{\rm holes}(\mathbf p)\right]=0. \tag{1} \]

This phase constructs G_wg^+, the Green tensor of the unperforated 450-by-330 nm SiN beam embedded in index 1.47 material. The construction must contain discrete guided poles and the continuous radiation spectrum without assigning artificial linewidths to either.

2. Fourier reduction along the beam

The unperforated beam is invariant in x. Write

\[ G_{\rm wg}(x,\mathbf r_\perp;x',\mathbf r'_\perp;\omega) =\int_{\mathcal C_\beta}\frac{d\beta}{2\pi} e^{i\beta(x-x')} \widetilde G_{\rm wg}(\beta; \mathbf r_\perp,\mathbf r'_\perp;\omega). \tag{2} \]

The contour C_beta, not a finite computational box, enforces the outgoing condition. The transverse problem at each beta has compact dielectric support: only the rectangle |y|<W/2, |z|<T/2 differs from the homogeneous cladding.

3. Exact homogeneous-cladding symbol

Let

\[ k_c=n_c\omega/c,\qquad \mathbf k=(\beta,q_y,q_z),\qquad k^2=\mathbf k\cdot\mathbf k. \]

For the operator

\[ L_c=\nabla\times\nabla\times-k_c^2I, \]

define longitudinal and transverse projectors

\[ P_L=\frac{\mathbf k\mathbf k^T}{k^2},\qquad P_T=I-P_L. \]

Its exact Fourier-space inverse is

\[ \boxed{ \widetilde G_c(\beta,q_y,q_z;\omega) =\frac{P_T}{k^2-k_c^2}-\frac{P_L}{k_c^2}. } \tag{3} \]

Indeed,

\[ \left[(k^2-k_c^2)I-\mathbf k\mathbf k^T\right] \widetilde G_c=I. \]

The second term in Eq. (3) is the longitudinal distributional/self part of the electric dyadic Green tensor. Keeping it in Fourier space avoids evaluating a point-sampled 1/R^3 singularity and then guessing a depolarization correction.

The radiation shell is

\[ q_y^2+q_z^2+\beta^2=k_c^2. \tag{4} \]

Equation (4) becomes a branch continuum after integrating over transverse momentum. For real guided beta with |beta|>k_c, the denominator never vanishes on the real transverse plane.

4. Compact cross-section volume-integral equation

The core susceptibility relative to cladding is

\[ \chi_b=n_b^2-n_c^2. \]

Restricting fields to the rectangular core gives

\[ \widetilde G_{\rm wg} =\widetilde G_c +\widetilde G_c k_0^2\chi_b\widetilde G_{\rm wg}, \]

and therefore

\[ \boxed{ \widetilde G_{\rm wg}(\beta,\omega) =\left[I-k_0^2\chi_b\widetilde G_c(\beta,\omega)\right]^{-1} \widetilde G_c(\beta,\omega). } \tag{5} \]

This is a Dyson resummation of all scattering from the uniform beam cross-section. The guided propagation constants satisfy

\[ \det A_{\rm wg}(\beta,\omega)=0, \qquad A_{\rm wg}=I-k_0^2\chi_b\widetilde G_c. \tag{6} \]

Thus guided modes are not inserted by hand. They are poles of the same resolvent that contains the radiation continuum.

5. Complete transverse Galerkin sequence

On the interval |y|<W/2, use

\[ \phi_m^{(W)}(y) =\sqrt{\frac{2m+1}{W}}P_m(2y/W). \]

Its exact Fourier transform is

\[ \Phi_m^{(W)}(q_y) =\sqrt{W(2m+1)}(-i)^m j_m(q_yW/2), \tag{7} \]

where j_m is a spherical Bessel function. Use the analogous basis in z and three Cartesian vector components:

\[ \boldsymbol\phi_{mns}(y,z) =\phi_m^{(W)}(y)\phi_n^{(T)}(z)\mathbf e_s. \tag{8} \]

This is complete in the vector L2 space of the rectangular cross-section. The homogeneous Green matrix is

\[ [G_c^{(N)}]_{mns,m'n's'} =\int_{\mathbb R^2}\frac{dq_y dq_z}{(2\pi)^2} \Phi_m^*(q_y)\Phi_n^*(q_z) [\widetilde G_c]_{ss'} \Phi_{m'}(q_y)\Phi_{n'}(q_z). \tag{9} \]

Equations (3), (7), and (9) make every matrix element an explicit convergent two-dimensional integral of Bessel and rational functions.

The finite sequence is

\[ G_{\rm wg}^{(N)} =\left[I-k_0^2\chi_bG_c^{(N)}\right]^{-1}G_c^{(N)}, \qquad N=(N_y,N_z,N_q). \tag{10} \]

Here N_q controls the infinite transverse-momentum quadrature. No PML size or PML strength enters Eq. (10).

The prototype supplies two quadrature families. A tangent map covers the whole real momentum axis compactly. A panelized finite-cutoff rule is slower but makes the omitted high-momentum tail explicit. The latter is the appropriate route for a defensible convergence table: increase polynomial order, panel resolution, and cutoff independently. The slowly convergent longitudinal projector means that agreement from a single tangent-map order is not an error estimate.

6. Guided-pole residues

At a simple guided pole beta_g, let

\[ A(\beta_g)x_g=0,\qquad y_g^\dagger A(\beta_g)=0. \]

Then

\[ \operatorname*{Res}_{\beta=\beta_g} \widetilde G_{\rm wg} =\frac{x_g y_g^\dagger G_c(\beta_g)} {y_g^\dagger A_\beta(\beta_g)x_g}. \tag{11} \]

This supplies the discrete term in the real-space Green tensor with its normalization fixed by the pole derivative. It avoids separately normalizing open waveguide modes by an inconsistent finite-box norm.

7. Radiation continuum and outgoing sheet

For causal frequency Im(omega)>0, Eq. (2) is initially defined with the real beta axis and the corresponding analytic square-root choices. As the frequency is continued toward the real axis, guided poles approach it and the transverse radiation shell creates branch points at

\[ \beta=\pm k_c. \]

For x>x', the outgoing contour is deformed into the upper beta half-plane; for x<x', symmetry gives the corresponding lower-half-plane continuation. The deformation separates Eq. (2) into

\[ G_{\rm wg}^+ =i\sum_{g,+}e^{i\beta_g|x-x'|} \operatorname*{Res}_{\beta_g}\widetilde G_{\rm wg} +G_{\rm rad}^++G_{\rm ev}. \tag{12} \]

For a cavity pole with Im(omega)<0, Eq. (12) must be analytically continued without letting the contour jump across its guided poles or radiation branch cuts. Directly substituting a lower-half-plane frequency into a real-axis quadrature generally lands on the wrong sheet. This is the central remaining implementation issue.

For a real beta strictly inside the light cone, the prototype now evaluates the transverse radiation shell without adding a small numerical loss. Writing

\[ \kappa^2=k_c^2-\beta^2, \]

and denoting the basis-projected numerator of Eq. (3) by H(q), the radial identity used is

\[ \int_0^Q\frac{qH(q)\,dq}{q^2-\kappa^2-i0} =\int_0^Q \frac{q[H(q)-H(\kappa)]}{q^2-\kappa^2}\,dq +\frac{H(\kappa)}{2} \log\!\frac{Q^2-\kappa^2}{\kappa^2} +\frac{i\pi}{2}H(\kappa). \tag{13} \]

The first integrand is regular at q=kappa; the last term is the exact on-shell radiation contribution. The incoming boundary value changes the sign of that term. Consequently the radiation spectral density is

\[ \rho_{\rm rad}(\beta) =\frac{\widetilde G_{\rm wg}^+(\beta) -\widetilde G_{\rm wg}^-(\beta)}{2\pi i}. \tag{14} \]

For a passive real-frequency waveguide, this matrix must be Hermitian positive semidefinite. That condition is now an internal test.

8. Connection to the finite holes

The phase-1 hole basis is Fourier transformed in x. If Phi_a(beta,y,z;p) is the transformed basis function for hole degree of freedom a, then the exact cavity Green matrix is

\[ [\mathcal G_{\rm holes}]_{ab} =\int_{\mathcal C_\beta}\frac{d\beta}{2\pi} \int d^2r_\perp d^2r'_\perp \Phi_a^*(\beta,\mathbf r_\perp;\mathbf p) \widetilde G_{\rm wg}(\beta;mathbf r_\perp,\mathbf r'_\perp) \Phi_b(\beta,\mathbf r'_\perp;\mathbf p). \tag{15} \]

Substitution into phase 1 produces

\[ D_N(\omega,\mathbf p) =\det\left[I-\frac{\omega^2}{c^2} (\epsilon_c-\epsilon_b)\mathcal G_{\rm holes}^{(N)} \right]=0. \tag{16} \]

Equations (3), (7), (9), (11), (13), (15), and (16) are the requested analytical mess: nested Bessel-function continuum integrals, matrix inverses, pole residues, and a final nonlinear determinant whose root gives Q.

9. Implemented and remaining

Implemented in this phase:

1. Exact homogeneous Maxwell Green symbol, including the longitudinal term.
2. Exact Fourier transforms of normalized rectangular Legendre bases.
3. Infinite transverse-momentum quadrature.
4. Vector Galerkin matrices for the homogeneous cladding Green tensor.
5. Dyson-resummed rectangular-waveguide Green matrices.
6. Guided-pole searches above the cladding light line.
7. Simple-pole residue and explicit beta-contour quadrature algebra.
8. Principal-value plus exact on-shell evaluation inside the radiation cone.
9. Outgoing-minus-incoming radiation spectral-density matrices.

Still required:

1. Symmetry- and overlap-based tracking of guided branches.
2. A robust deformed beta contour around +/- k_c.
3. Analytic continuation of that contour to lower-half-plane cavity poles.
4. Projection of the elliptical hole basis into the cross-section basis.
5. Substitution into the full Fryett determinant and recovery of Q~10^5.

No FDTD or single-hole comparison is performed here, following the decision to put that verification gate temporarily on hold.

10. References

• D. P. Nyquist, “Orthogonality and amplitude spectrum of radiation modes along open-boundary dielectric waveguides,” JOSA 71, 49–55 (1981).
• W. C. Chew, Lectures on Theory of Microwave and Optical Waveguides, arXiv:2107.09672.
• P. Ylä-Oijala et al., “On the Computation of Power in Volume Integral Equation Formulations,” arXiv:1406.7260.
• P. T. Kristensen et al., “Modeling electromagnetic resonators using quasinormal modes,” arXiv:1910.05412.