--- title: Four-dimensional shifted-ringdown Q/V campaign --- # Four-dimensional shifted-ringdown Q/V campaign ```{admonition} Live optimizer :class: tip This is a new artifact and a new optimizer. It uses the same 8.0 µm Fryett optimization region and Maxwell scene as the prior hybrid-scratch campaign, but starts again from neutral material. The first objective is deliberately pulse-on dominated because the initial structure has no useful ringdown. It then transfers weight to the three source-off field states. A trustworthy $Q\ge100$ pole triggers the eigenfrequency-shifted 4D $Q/V$ objective. No output-mode or beta-factor term is present in this first attempt. ``` | State | Value | |---|---:| | Status | `running_shifted_4d_q_over_v` | | Phase | `shifted_4d_q_over_v` | | Active geometry coordinates | `full_domain_neutral_pixels` | | Completed updates | 20 | | Displayed field window | `shifted_pole_late` | | Objective | 4.9964 | | Pulse-on normalized LDOS | 2.1652 | | Source-off local response | 0.0060772 | | Temporal modal coherence | 1 | | Rank-one fraction | 1 | | Shifted stationarity | 0.5838 | | Energy retention | 0.0078023 | | Effective temporal rank | 1 | | Q | 418.28 | | Physical V | 0.12985 µm³ | | Normalized V | 2.5399 | | Q / normalized V | 164.69 | | Pole wavelength | 0.74229 µm | | Projection beta | 3.379 | | Pulse-on / source-off weights | 0.706 / 0.294 | | DCT x/y rolloff modes | 151 / 18 | | DCT high-frequency floor | 0.952 | | Trust step | 0.08000 | | Boundary handoff | `waiting_for_binary_stable_topology` | | Stable binary-topology states | 0 / 3 | | Boundary step | 0.001 µm | | Boundary topology | 18 material / 1 enclosed voids | | Boundary chart rebases | 0 | | Gray fraction (0.1–0.9) | 93.09% | | Last candidate accepted | True | | Last pole trustworthy | True | | Latest update wall time | 139.4 s | | Last artifact write | 2026-08-28 17:03:09 UTC | The authoritative state is `benchmarks/artifacts/invdes_fryett_4d_qv_qv4d_v2_mild_dct.npz`. “Current” means the accepted geometry; the separately saved candidate is never substituted into the plots. ## Geometry and exact material difference ![Initial, accepted, and difference geometry](../_static/generated/fryett_4d_qv_geometry.png) ## Field used by the current objective ![Current windowed Ey field](../_static/generated/fryett_4d_qv_fields.png) Before pole capture this is explicitly a pulse-on field. After capture it is the final temporal-window state phase-corrected to the fitted pole. ## Temporal response and objective windows ![Pulse, emitter response, energy, and temporal windows](../_static/generated/fryett_4d_qv_temporal.png) ## Full-field temporal correlations ![Temporal Gram matrix and window energies](../_static/generated/fryett_4d_qv_gram.png) ## Objective and pole history ![Discovery, coherence, and shifted Q/V history](../_static/generated/fryett_4d_qv_history.png) ## Exact methodology, equations, and moving components ## Scope and hypothesis This campaign tests one narrow hypothesis: a cavity optimizer that uses the spatial field throughout several time windows can distinguish a collective, single-mode resonance from a large instantaneous field at the emitter. The only terminal electromagnetic performance target in this first experiment is $Q/V$. Waveguide-mode overlap, beta factor, and deliberate mirror opening are not objectives. The terminal feedthrough remains in the geometry only so the simulation scene and optimization region are identical to the preceding eight-micron Fryett scratch campaign. The experiment starts from neutral material density, not from the previous campaign's optimized geometry. All 3,200 cells in the reduced $x$-$y$ design quadrant are present from the first update, except for the same fixed terminal feedthrough. They decode through one material field. A positive-semidefinite multiscale metric correlates proposed pixel motion, but it does not prescribe a beam, hole, period, or material sign. ## Frozen geometry and Maxwell problem | Quantity | Value | |---|---:| | Full optimization region | $8.0\,\mu\mathrm m\times1.0\,\mu\mathrm m\times0.40\,\mu\mathrm m$ | | Reduced design array | $160\times20$ cells | | Maxwell spacing | $25\,\mathrm{nm}$ | | Non-PML interior | $9.10\,\mu\mathrm m\times1.50\,\mu\mathrm m\times1.50\,\mu\mathrm m$ | | Reduced Maxwell grid | $194\times42\times42$ cells | | Symmetry | $(+x,-y,+z)$ mirror parity, factor 8 | | Emitter | centered $y$-polarized point dipole | | Target wavelength | $780.55\,\mathrm{nm}$ | | Materials | the same SiN and embedding cladding as the Fryett reconstruction | Changing any entry in this table requires a new campaign fingerprint and artifact. The previous scratch artifact is never mutated or used as an implicit seed. ## What “using the 4D field” means Writing every field value at every FDTD step would add a huge movie to the adjoint tape without adding information needed by a one-pole test. Instead, the solver accumulates exact differentiable sufficient statistics over four disjoint temporal windows. For window $k$ and demodulation frequency $\omega_c$, $$ u_k(\mathbf r) = \frac{2}{N_k} \sum_{n\in W_k} \begin{bmatrix} \mathbf E(\mathbf r,t_n)\\ \mathbf H(\mathbf r,t_n) \end{bmatrix} e^{+i\omega_c t_n}. $$ All six field components are accumulated throughout the reduced $160\times20\times8$ design volume, which represents the complete $8.0\times1.0\times0.40\,\mu\mathrm m^3$ physical region under the three exact mirror symmetries. The emitter field and total electromagnetic energy are also retained as time traces. Thus the objective consumes every independent sampled spatial cell, field component, and time window relevant to the assumed carrier; it does not retain an unnecessary raw movie. The energy inner product between two window states is $$ \langle u_k,u_l\rangle_W = \frac12\int_{\Omega_d} \left( \mathbf E_k^*\!\cdot\epsilon\mathbf E_l + \mathbf H_k^*\!\cdot\mu\mathbf H_l \right)d^3r, $$ and the temporal Gram matrix is $$ G_{kl}=\langle u_k,u_l\rangle_W. $$ This matrix is small, but every entry is a reduction of the full sampled spatial field. Three complementary diagnostics follow from it. First, the pairwise modal coherence is $$ \eta_{\mathrm{coh}} = \frac{1}{K(K-1)} \sum_{k\ne l} \frac{|G_{kl}|^2}{G_{kk}G_{ll}+\varepsilon}. $$ It equals one when every temporal window contains the same spatial Maxwell state up to a complex scalar. Beating between spatially different modes lowers it. The float32 implementation uses the explicit relative regularizer $$ \varepsilon_E=10^{-10}\max_jG_{jj}, \qquad \widehat u_k=\frac{u_k}{\sqrt{G_{kk}+\varepsilon_E}}, $$ and forms the normalized Gram matrix from $\langle\widehat u_k,\widehat u_l\rangle_W$ before taking its squared magnitude. This is the regularized form of the equation above, arranged to avoid reciprocal-square overflow in the adjoint when a late window is nearly dark. The maximum used in $\varepsilon_E$ is stopped-gradient: it is a numerical scale, not an additional route for one time window to optimize another. Second, the rank-one fraction $$ \eta_1=\frac{\lambda_{\max}(G)}{\operatorname{tr}G} $$ is recorded as an independent diagnostic. The pairwise form is used in the hot-loop objective because it has a better-behaved derivative when the initial late-time field is almost zero. The eigendecomposition itself is stopped-gradient because repeated zero eigenvalues have no unique eigenspace; rank and effective rank are telemetry rather than objective terms. Third, after shifting to the fitted real pole frequency, the stationary-field fraction is $$ \eta_{\mathrm{stat}} = \frac{\left\|\sum_k \widetilde u_k\right\|_W^2} {K\sum_k\|\widetilde u_k\|_W^2+\varepsilon}, \qquad \widetilde u_k =u_k e^{+i(\omega_*-\omega_c)t_k}. $$ This is one only for a constant complex field in the co-rotating frame. It therefore responds to decay and residual beating. Only the real frequency is removed: the factor $e^{+\gamma t}$ is deliberately **not** applied, because doing so would normalize away the lifetime that the optimizer must improve. The window-energy retention diagnostic is $$ R_U=\frac{G_{K-1,K-1}}{G_{00}+\varepsilon}. $$ ## Stage A: pulse-on-assisted discovery The Gaussian emitter pulse is appreciable only during the first roughly $60\,\mathrm{fs}$. The discovery tape therefore contains one driven window and three source-off windows. A fixed-frequency source-work measurement gives $$ L_{\mathrm{on}} = \frac{P_{\mathrm{src}}(\omega_0)} {P_{\mathrm{src,vac}}(\omega_0)}, $$ while the source-off response is measured from the emitter-field coefficients in the three later windows, $$ L_{\mathrm{off}} = \frac{\frac13\sum_{k=1}^{3}|E_y(\mathbf r_0,W_k)|^2} {|E_{y,\mathrm{vac}}(\mathbf r_0,W_0)|^2+\varepsilon}. $$ The initial design has essentially no ringdown, so the continuation begins with 90% weight on $L_{\mathrm{on}}$. Over the first 60 accepted or rejected updates, a smooth schedule transfers weight until the source-off response carries 85%. The discovery objective is $$ J_D = \log\!\left(w_{\mathrm{on}}L_{\mathrm{on}} +w_{\mathrm{off}}L_{\mathrm{off}}+\varepsilon\right) +0.15\log(\eta_{\mathrm{coh}}+\varepsilon) +0.10\log(\eta_{\mathrm{stat}}+\varepsilon) +0.10\log(R_U+\varepsilon). $$ This is explicitly a pre-pole surrogate, not a claim to be exact $Q/V$. $L_{\mathrm{on}}$ prevents a zero-field solution and supplies a useful gradient at initialization. The later terms progressively reward a persistent local response carried by one global spatial mode. Every eight discovery updates, a longer independent pulse-off replay attempts to identify a pole. Discovery ends only when all of the following pass: 1. conservative fitted $Q\ge100$; 2. field- and energy-decay fits pass their residual and window-consistency limits; 3. the two lifetime estimates agree within the stated ratio; 4. the pole remains inside the absolute target-frequency band; 5. full-field temporal coherence and adjacent-window spatial overlap pass; 6. the fitted branch is stable under recentering. The number 100 is not a fundamental threshold. It is the point at which an isolated pole is usually identifiable; the fit-quality tests remain authoritative. ## Stage B: eigenfrequency-shifted 4D Q/V For a tracked pole, $$ E^+(\mathbf r,t) \simeq A\psi(\mathbf r)e^{-i\omega_*t}e^{-\gamma t} +\sum_{m\ne *}A_m\psi_m(\mathbf r)e^{-i\omega_mt}e^{-\gamma_mt}. $$ A fixed-frequency objective develops a narrow detuning direction whose curvature scales as $Q^2$. The campaign uses the time-domain analogue of the Shaker--Johnson shifted-LDOS method: a coarse late-time spectrum selects the frequency branch with stopped gradients, a differentiable translated-window fit obtains $\omega_*$ and $\gamma$, and all temporal states are moved into the $\operatorname{Re}\omega_*$ frame. The final fit remains differentiable through the FDTD fields. Only the discrete mode-selection decision is detached. The physical cavity quantities are $$ Q=\min\!\left( \frac{\operatorname{Re}\omega_*}{2\gamma_E}, \frac{\operatorname{Re}\omega_*}{2\gamma_U} \right), $$ $$ V = \frac{U}{\epsilon(\mathbf r_0)|E_y(\mathbf r_0)|^2}, \qquad \widetilde V=\frac{V}{(\lambda_*/n_{\mathrm{SiN}})^3}, $$ and $$ F_{QV}=\frac{Q}{\widetilde V}. $$ The shifted 4D objective is $$ J_S = \log F_{QV} +0.20\log(\eta_{\mathrm{coh}}+\varepsilon) +0.20\log(\eta_{\mathrm{stat}}+\varepsilon) -P_{\mathrm{fit}}-P_{\mathrm{band}}. $$ The two coherence factors approach one for the desired nondecaying single mode, so the terminal objective becomes proportional to $Q/V$. They act as mechanism selectors and reject multimode beating rather than introducing an output-coupling target. The absolute bandwidth penalty is $$ P_{\mathrm{band}} = \lambda_\omega \left[ \max\left(0, \frac{|\operatorname{Re}\omega_*-\omega_0|}{\Delta\omega}-1 \right) \right]^2. $$ It is not scaled by the shrinking linewidth. This allows the objective to move with the resonance without allowing mode-family drift. The implementation uses the fitted pole to phase-correct the already accumulated equal-duration field windows. Every accepted geometry updates the detector frequency for the next replay. This is the practical FDTD analogue of eigenfrequency shifting. A literal sparse shift-invert eigensolver is reserved for a later handoff if the finite ringdown becomes unable to resolve further lifetime growth. ## Geometry motion and optimizer state The material variables initially are one identity pixel per reduced design cell. A filtered projection produces the physical density. Projection strength grows gradually from a soft discovery field toward a binary design. The same parameter array and material decoder are used in both objective stages until the automatic binary-boundary gate described below passes. The raw adjoint gradient is transformed by a mild, strictly positive DCT multiplier. For DCT indices $(m,n)$, $$ W_{mn} =s+ \frac{1-s}{1+(m/m_{\max})^8+(n/n_{\max})^8}. $$ At update zero, $(m_{\max},n_{\max},s)=(48,12,0.35)$. A smoothstep opens these values to $(159,19,1)$ at update 24, after which the DCT operation is exactly the identity. Thus even the finest initial modes retain at least 35% of their raw gradient, no spatial mode is removed, and there is no selected period. Unlike version 1, there is no emitter-local raw-gradient term: the center has no extra optimizer mobility. The terminal feedthrough cells alone are removed from the direction. Each update performs: 1. replay the current accepted geometry and differentiate its active 4D objective; 2. transform the gradient with the positive-semidefinite multiscale metric; 3. blend chart-local momentum only if the blend retains positive ascent; 4. cap the largest parameter move by the current trust radius; 5. replay exactly one candidate under the same projection and objective; 6. accept only a finite, improving candidate; in the shifted stage also require a trustworthy same-family pole and nondecreasing $Q/V$; 7. grow the radius after a well-predicted accepted step or halve it after rejection; 8. atomically save current, best, candidate, history, traces, geometry, and pole-aligned fields. ## Stage C: automatic subpixel level-set boundary refinement The pixel chart is valuable while the design must create or destroy material components. It is the wrong chart after binarization: a pixel derivative can make gray bulk, whereas the desired degree of freedom is normal motion of a smooth material interface. Version 3 therefore performs an automatic, fail-closed handoff to a level-set boundary after all of these conditions hold on three consecutive trustworthy shifted states: 1. projection strength is $eta=48$; 2. no more than 4% of cells lie in $0.1<\rho<0.9$; 3. the thresholded numbers of material components and enclosed voids are unchanged, including across the $eta=0.47,0.50,0.53$ projection offsets. The remaining fractional cells at this gate form the projected interface layer; they are not interpreted as a third material. A sign-preserving fit constructs a vertex level set $phi_0$ whose zero contour reproduces that layer. Boundary parameters $d_{ij}$ are physical displacements in micrometres, and the local chart is $$ \phi(d)=\phi_0+w(|\phi_0|)d, \qquad w(s)= \begin{cases} \tfrac12[1+\cos(\pi s/b)],&ssetTimeout(() => window.location.reload(), 60000);