Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/fryett_q_series_phase1/DERIVATION.md
Snapshot: design23-v1.0.0. The body below is unabridged.

An exact convergent-series target for the Fryett nanobeam cavity

1. Benchmark and scope

Fryett et al. report an encapsulated silicon-nitride nanobeam cavity with a simulated resonance near 740 nm, simulated quality factor

\[ Q_{\rm paper}\simeq 10^5, \]

and mode volume approximately

\[ V\simeq 2.5(\lambda/n)^3. \]

The stated material and mirror parameters are

\[ n_b=2.0,\quad n_c=1.47,\quad W=450\ {\rm nm},\quad T=330\ {\rm nm}, \]

\[ a_m=233\ {\rm nm},\quad d_x=100\ {\rm nm},\quad d_{y,m}=300\ {\rm nm}. \]

Here b means SiN beam and c means encapsulating cladding. The beam axis is x, its in-plane transverse direction is y, and the vertical direction is z.

The goal is not a heuristic expression such as Q = exp(alpha N). The goal is an exact implicit function

\[ Q(\mathbf p)=-\frac{\Re\widetilde\omega(\mathbf p)} {2\Im\widetilde\omega(\mathbf p)}, \]

where the complex pole widetilde omega(p) is defined by an outgoing Maxwell operator and is approached by an explicitly refinable sequence.

2. A geometry ambiguity that must remain visible

The article says that the major hole diameter and period are tapered linearly, that the mirror major/minor diameters are 300/100 nm, and that the innermost major diameter is 100 nm with the innermost holes “separated by 140 nm.” It also describes a Bragg region of 40 holes.

The previous project bundle used a different, more specific reconstruction:

• five taper holes per side with transverse semiaxes [70, 90, 110, 130, 150] nm;
• controlled center intervals [215, 222, 224.75, 227.5, 230.25] nm;
• 40 repeated mirror holes per side;
• a 233 nm mirror period.

Those choices cannot all be inferred uniquely from the quoted article text. In particular, 140 nm was treated as an optimized transverse diameter in the old reconstruction, not as the separation of the central holes. The meaning of “40 holes” is also potentially total-versus-per-side ambiguous.

Therefore this phase treats the geometry as a parameter vector and preserves the old reconstruction only as a named candidate. A numerical claim of reproducing Q=10^5 must state the complete list of hole centers and radii.

3. Exact dielectric function

Use the unperforated rectangular waveguide as the background:

\[ \epsilon_0(y,z) =\epsilon_c+(\epsilon_b-\epsilon_c) \mathbf 1_{|y|<W/2}\mathbf 1_{|z|<T/2}, \]

with epsilon_b=n_b^2 and epsilon_c=n_c^2.

Hole j is the elliptical cylinder

\[ H_j(\mathbf p)= \left\{(x,y,z): \frac{(x-x_j)^2}{r_{x,j}^2}+\frac{y^2}{r_{y,j}^2}\leq 1, \quad |z|\leq T/2 \right\}. \]

For the Fryett geometry r_x=50 nm is nominally fixed and r_y is tapered. The complete dielectric is exactly

\[ \epsilon_{\mathbf p}(\mathbf r) =\epsilon_0(y,z)+\Delta\epsilon_{\mathbf p}(\mathbf r), \]

\[ \Delta\epsilon_{\mathbf p}(\mathbf r) =(\epsilon_c-\epsilon_b) \sum_{j=1}^{N_h}\mathbf 1_{H_j(\mathbf p)}(\mathbf r). \tag{1} \]

For a mirror-symmetric cavity one can use positive-side coordinates

\[ x_1=s_0/2,\qquad x_j=s_0/2+\sum_{\ell=1}^{j-1}s_\ell, \]

and add holes at -x_j. Thus a natural optimization vector is

\[ \mathbf p=(r_{y,1},\ldots,r_{y,N_t}, s_0,\ldots,s_{N_t-1},a_m,W,T,n_b,n_c,\ldots). \]

4. Exact analytical geometry transform

Define the Fourier transform by

\[ \widehat f(\mathbf k)=\int f(\mathbf r)e^{-i\mathbf k\cdot\mathbf r}\,d^3r. \]

For one hole define

\[ \rho_j=\sqrt{(k_xr_{x,j})^2+(k_yr_{y,j})^2}. \]

Direct integration over the elliptical disk and vertical interval gives

\[ \widehat{\mathbf 1}_{H_j}(\mathbf k) =e^{-ik_xx_j} \left[2\pi r_{x,j}r_{y,j}\frac{J_1(\rho_j)}{\rho_j}\right] \left[\frac{2\sin(k_zT/2)}{k_z}\right]. \tag{2} \]

The continuous limits are J1(rho)/rho -> 1/2 and 2 sin(k_z T/2)/k_z -> T. At zero wavevector, Eq. (2) is exactly the hole volume pi r_x r_y T.

The full geometry dependence is consequently

\[ \widehat{\Delta\epsilon}_{\mathbf p}(\mathbf k) =(\epsilon_c-\epsilon_b) \sum_j e^{-ik_xx_j} \left[2\pi r_{x,j}r_{y,j}\frac{J_1(\rho_j)}{\rho_j}\right] \left[\frac{2\sin(k_zT/2)}{k_z}\right]. \tag{3} \]

This is already a useful exact analytical representation of the complete finite set of holes as a function of every center and radius.

For example,

\[ \frac{\partial \widehat{\mathbf 1}_{H_j}}{\partial x_j} =-ik_x\widehat{\mathbf 1}_{H_j}. \tag{4} \]

Writing f(rho)=J1(rho)/rho,

\[ f'(\rho)=\frac{J_0(\rho)}{\rho} -\frac{2J_1(\rho)}{\rho^2}, \]

and

\[ \frac{\partial\rho_j}{\partial r_{y,j}} =\frac{k_y^2r_{y,j}}{\rho_j}, \]

gives an exact radius derivative of Eq. (2). The code implements the removable singularities with small-argument Bessel series.

For completeness, the rectangular bare-waveguide contrast also has the exact two-dimensional transform

\[ \widehat{\Delta\epsilon}_{\rm beam}(k_y,k_z) =(\epsilon_b-\epsilon_c)WT \operatorname{sinc}(k_yW/2)\operatorname{sinc}(k_zT/2), \tag{5} \]

where the sinc functions here are unnormalized sin(x)/x.

5. Outgoing Maxwell pole equation

Use time dependence exp(-i omega t) and nonmagnetic media. Define the bare waveguide Maxwell operator

\[ \mathcal L_0(\omega) =\nabla\times\nabla\times -\frac{\omega^2}{c^2}\epsilon_0(y,z). \]

Let

\[ \mathbf G_0^+(\omega)=\mathcal L_0(\omega)^{-1}_{\rm outgoing} \]

be its outgoing dyadic Green operator on the correct analytic sheet. The full source-free Maxwell equation is equivalent to the homogeneous Lippmann--Schwinger equation

\[ \mathbf E =\frac{\omega^2}{c^2}\mathbf G_0^+(\omega) \Delta\epsilon_{\mathbf p}\mathbf E. \tag{6} \]

Restricting the unknown to the finite union of holes Omega(p) defines

\[ K(\omega,\mathbf p) =\frac{\omega^2}{c^2} \mathbf G_0^+(\omega)\Delta\epsilon_{\mathbf p}. \]

The exact pole condition is

\[ D(\omega,\mathbf p) =\det_{\rm reg}[I-K(\omega,\mathbf p)]=0. \tag{7} \]

det_reg is written deliberately. The electric dyadic Green tensor has a singular local term. A rigorous implementation must extract the Maxwell self term and use either an ordinary Fredholm determinant of the resulting trace-class operator or an appropriate regularized determinant such as det_2. Calling an unregularized collocation determinant “exact” would be wrong.

The analytic Fredholm framework makes the desired geometry-to-Q function

\[ \boxed{ Q(\mathbf p) =-\frac{\Re\widetilde\omega(\mathbf p)} {2\Im\widetilde\omega(\mathbf p)}, \quad D(\widetilde\omega(\mathbf p),\mathbf p)=0, \quad \Im\widetilde\omega<0. } \tag{8} \]

Equation (8) is implicit and ugly, but it is an exact analytical definition.

6. Where the radiation continuum appears

The bare-waveguide Green tensor is not a sum over only a few guided modes. Its spectral representation has the schematic form

\[ \mathbf G_0^+(\mathbf r,\mathbf r';\omega) =\sum_{g\in{\rm guided}} \frac{\mathbf e_g(\mathbf r_\perp) \otimes\mathbf e_g^\ddagger(\mathbf r'_\perp)} {2i\beta_gN_g} e^{i\beta_g|x-x'|} \]

\[ \quad+ \sum_{\sigma}\int_{\mathcal C_{\rm rad}} \frac{\mathbf e_{\alpha\sigma}(\mathbf r_\perp) \otimes\mathbf e_{\alpha\sigma}^\ddagger(\mathbf r'_\perp)} {2i\beta_{\alpha\sigma}N_{\alpha\sigma}} e^{i\beta_{\alpha\sigma}|x-x'|}\,d\alpha. \tag{9} \]

The contour and square-root branches enforce outgoing propagation and evanescent decay. The dual symbol allows the normalization to be biorthogonal after analytic continuation to complex frequency.

Equation (9) explains the earlier failure. A handful of guided or PML modes can approximate the near field while getting the tiny continuum coupling that sets Q completely wrong. No arbitrary imaginary part is assigned to a free-space mode; the imaginary part of the cavity pole is generated by the outgoing continuum in Eq. (9).

7. The actual convergent series

Map every hole to a fixed unit cylinder using

\[ u=(x-x_j)/r_{x,j},\qquad v=y/r_{y,j},\qquad \zeta=2z/T. \]

On the unit disk use a complete set of Zernike polynomials Z_mn(u,v). On [-1,1] use normalized Legendre polynomials P_l(zeta), and include three vector components. After the correct Maxwell self-term treatment, this gives a complete local basis

\[ \boldsymbol\phi_{jmn\ell s}(\mathbf r;\mathbf p) =C_{jmn\ell}(\mathbf p) Z_{mn}(u,v)P_\ell(\zeta)\mathbf e_s. \tag{10} \]

Projection of Eq. (6) gives

\[ A_{ab}^{(N)}(\omega,\mathbf p) =\delta_{ab} -\frac{\omega^2}{c^2}(\epsilon_c-\epsilon_b) \int_{\Omega}\!d^3r\int_{\Omega}\!d^3r' \boldsymbol\phi_a^*(\mathbf r) \cdot\mathbf G_0^+(\mathbf r,\mathbf r';\omega) \cdot\boldsymbol\phi_b(\mathbf r'). \tag{11} \]

The finite pole and Q are

\[ D_N(\omega,\mathbf p)=\det A^{(N)}(\omega,\mathbf p), \]

\[ D_N(\widetilde\omega_N,\mathbf p)=0, \qquad Q_N(\mathbf p) =-\frac{\Re\widetilde\omega_N}{2\Im\widetilde\omega_N}. \tag{12} \]

The intended exact result is the joint limit

\[ Q(\mathbf p) =\lim_{N_{xy},N_z,N_{\rm rad},N_{\rm ev}\to\infty} Q_{N_{xy},N_z,N_{\rm rad},N_{\rm ev}}(\mathbf p). \tag{13} \]

Here N_rad is a radiation-continuum quadrature order, not a fictional finite number of physical radiation modes.

There is also a formal trace series. Where its local convergence conditions hold,

\[ \det(I-K) =\exp\left[-\sum_{m=1}^{\infty}\frac{\operatorname{Tr}K^m}{m}\right], \tag{14} \]

or, for the two-regularized determinant,

\[ \det_2(I-K) =\exp\left[-\sum_{m=2}^{\infty}\frac{\operatorname{Tr}K^m}{m}\right]. \tag{15} \]

At a resonance the naive trace-log series is not generally the best numerical evaluation because an eigenvalue reaches one. Equations (14)--(15) define the local analytical series and its continuation; Eq. (12) is the stable computational realization.

8. Exact pole and Q gradients

Let x and y be right and left null vectors at a simple pole:

\[ A(\widetilde\omega,\mathbf p)\mathbf x=0, \qquad \mathbf y^\dagger A(\widetilde\omega,\mathbf p)=0. \]

Implicit differentiation gives

\[ \frac{d\widetilde\omega}{dp_j} =-\frac{\mathbf y^\dagger A_{p_j}\mathbf x} {\mathbf y^\dagger A_{\omega}\mathbf x}. \tag{16} \]

If widetilde omega=omega_r+i omega_i, with omega_i<0, then

\[ \frac{dQ}{dp_j} =-\frac{1}{2\omega_i}\frac{d\omega_r}{dp_j} +\frac{\omega_r}{2\omega_i^2}\frac{d\omega_i}{dp_j}. \tag{17} \]

Equations (2)--(5) provide analytical geometry derivatives inside A_p. Equation (16) means that the final optimizer does not need to differentiate a long FDTD ringdown or finite-difference every geometry variable.

9. Relation to cell scattering matrices

Partitioning the basis indices in Eq. (11) by hole or cell produces a block matrix. Eliminating internal blocks using Schur complements produces the same cell scattering maps that can be cascaded using Redheffer star products.

Thus cell cascading is retained, but its cell matrices should be derived from the converged global operator. The scattering model is then a stable block factorization of Eq. (11), rather than a separate approximation whose missing radiation channels are unknown.

10. Concrete state at the end of phase 1

Completed:

1. An exact parameterized dielectric function for the finite hole array.
2. Exact Bessel/sinc Fourier factors for every hole and their geometry derivatives.
3. The outgoing bare-waveguide volume-integral pole equation.
4. A specific complete local basis and finite determinant sequence Q_N(p).
5. Exact implicit derivatives of the pole and Q.
6. Executable tests for the analytical geometry factors and pole identities.

Not completed:

1. Rigorous extraction of the singular Maxwell self term.
2. Construction and normalization of the full guided-plus-continuum bare-waveguide Green tensor.
3. Numerical evaluation of Eq. (11).
4. Resolution of the paper's finite-geometry ambiguity.
5. Recovery of the reported Q ~ 100,000.

The next phase should solve items 1--2 first for a single elliptical hole, and verify its guided reflection, transmission, and total radiated power against a full-wave solver before assembling the full Fryett cavity.

11. Primary benchmark source

T. K. Fryett, Y. Chen, J. Whitehead, Z. M. Peycke, X. Xu, and A. Majumdar, “Encapsulated Silicon Nitride Nanobeam Cavity for Hybrid Nanophotonics,” ACS Photonics 5, 2176–2181 (2018), DOI: 10.1021/acsphotonics.8b00036; arXiv:1709.02032.