Eigenvalue-accelerated LDOS optimization of high-Q optical resonances

TL;DR

This work tackles the difficulty of accelerating LDOS-based inverse design for high-$Q$ optical resonators, where the Hessian becomes ill-conditioned as resonances sharpen. The authors introduce an eigenfrequency-shifted objective that uses a fast shift-invert eigensolver to locate the nearest resonance $\omega_*(\mathbf{p},\omega_0)$ and optimize $\log\text{LDOS}$ at $\operatorname{Re}[\omega_*]$, subject to a bandwidth constraint. Analytically, the dominant Hessian eigenvalue scales as $O(Q^2)$ in the unshifted case and is eliminated by the shifted formulation; empirically, the method yields orders-of-magnitude faster convergence and can achieve $Q > 10^6$ in 2D and even $Q \approx 4.4\times 10^8$ in 1D with a successive-enlargement strategy. The approach also narrows the gap to theoretical upper bounds and is extensible to other linear-resonant metrics beyond LDOS, enabling faster design of long-lived, strongly localized photonic cavities.

Abstract

We demonstrate a new method that yields orders-of-magnitude acceleration in inverse design (e.g. topology optimization) of high-$Q$ resonant cavities to maximize the local density of states (LDOS), and which is also applicable to other resonant-response metrics. The key idea is that, once conventional LDOS optimization has identified a strong resonance, subsequent optimizations can exploit a fast shift-invert eigensolver to ensure that the LDOS remains centered at the resonance peak. We show that this eliminates ill-conditioning at sharp resonances that otherwise dramatically slows LDOS (and similar) optimization for $Q \gg 100$. Our method is demonstrated by design of $Q > 10^6$ resonant cavities in 1d and 2d dielectric systems.

Figures (6)

• Figure 1: Schematic of new eigenvalue-"shifted" algorithm: from an initial structure (e.g. vacuum) we run $\lesssim 1000$ iterations of the original "unshifted" optimization of LDOS Liang2013 to acquire an unshifted initialization which serves as the starting guess for the shifted optimization of Eqs. (\ref{['eq:shifted_LDOS']},\ref{['eq:shifted_opt']}): maximizing the LDOS at the real part of nearest eigenfrequency $\omega_*$, constrained within some bandwidth BW, to ensure that the objective remains on the resonance peak.
• Figure 2: (a) Schematic 1d optimization problem: maximizing LDOS for an emitter at the center for a design region that spans the whole computational cell except for terminating PML absorbing layers. (b) Unshifted initialization after 200 iterations (left) and shifted (right) and unshifted (middle) optima after $10^5$ iterations: the resonant-mode profile (red, right axis) is superimposed over the permittivity $\varepsilon$ (blue, left axis). (c) LDOS enhancement (left) and resonance $Q$ (right) as a function of the number of sparse-matrix factorizations (dominating the cost of the Maxwell solves) for the shifted (blue) and unshifted (red) algorithms.
• Figure 3: (a) Schematic successive-enlargement algorithm in 1d: LDOS is optimized for a nested sequence of design domains $L_0 < L_1 < \cdots$ (green), which each optimum forming the initial guess for the next optimization. (b) LDOS enhancement as a function of design-region size $L$, exhibiting exponential growth in roughly half-wavelength steps. (c) A hand-designed quarter-wave-stack cavity (left), along with optimized cavities by successive enlargement with the unshifted (middle) and shifted (right) algorithms: the resonant-mode profile (red, right axis) is superimposed over the permittivity $\varepsilon$ (blue, left axis). (d) LDOS enhancement (left) and $Q$ (right) as a function of the number of sparse-matrix factorizations for the successively-enlarged shifted (blue) and unshifted (red) algorithms. Alternating gray and white bands indicate a change in design-region size $L$.
• Figure 4: Eigenvalues of the Hessians for the shifted (blue crosses) and unshifted (red dots) objective functions, computed at the unshifted initialization of our 1d test case (lower-right inset) from Fig. \ref{['fig:1d_cavities']}(left). The largest eigenvalues differ by six orders of magnitude. Upper-left inset verifies that the dominant eigenvalue of the unshifted objective (red dots) scales like $Q^2$ (dashed black line), as predicted in Sec. \ref{['sec:second_derivative']}.
• Figure 5: Schematic example LDOS-optimization problem in 2d: a square $1.5\lambda_0 \times 1.5\lambda_0$ design region (green), surrounded by air ($\varepsilon=1$) and PML absorbers (blue) is optimized to maximize the LDOS of a point source (red dot) $0.05\lambda_0$ outside the design region.