Bounds as blueprints: towards optimal and accelerated photonic inverse design

TL;DR

This work addresses the difficulty of achieving optimal photonic inverse designs due to nonconvexity and ill-conditioning. It introduces verlan, a initialization strategy that harvests information from convex dual limits to seed topology optimization with a dual polarization field $\mathbf P_{\mathcal D}$, yielding a field-based template via $\chi_{\mathrm inf}$ and a practical initialization $\rho_0$. By casting the problem as a quadratically constrained quadratic program (QCQP) in polarization and applying Lagrangian duality, it derives tight dual bounds and a scalable computation strategy including scraping and generalized constraint descent (GCD). Applied to Purcell enhancement in 2D structures, verlan achieves over an order of magnitude improvement over standard TopOpt and nears fundamental limits within a factor of two, while revealing new enhancement mechanisms and robust performance at large domain sizes. This framework opens a practical path to certified, near-optimal photonic designs by integrating global limits with local inverse design across scalable, potentially multi-physics settings.

Abstract

Our ability to structure materials at the nanoscale has, and continues to, enable key advances in optical control. In pursuit of optimal photonic designs, substantial progress has been made on two complementary fronts: bottom-up structural optimizations (inverse design) discover complex high-performing structures but offer no guarantees of optimality; top-down field optimizations (convex relaxations) reveal fundamental performance limits but offer no guarantees that structures meeting the limits exist. We bridge the gap between these two parallel paradigms by introducing a ``verlan'' initialization method that exploits the encoded local and global wave information in duality-based convex relaxations to guide inverse design towards better-performing structures. We illustrate this technique via the challenging problem of Purcell enhancement, maximizing the power extracted from a small emitter in the vicinity of a photonic structure, where ill-conditioning and the presence of competing local maxima lead to sub-optimal designs for adjoint optimization. Structures discovered by our verlan method outperform standard (random) initializations by close to an order of magnitude and approach fundamental performance limits within a factor of two, highlighting the possibility of accessing significant untapped performance improvements.

Figures (5)

• Figure 1: Schematic of verlan method. The optimization landscape of photonic design---here illustrated as a surface, but in practice having thousands of dimensions---is often very complicated, exhibiting ill-conditioning and a large number of local optima which can trap gradient-based (adjoint) algorithms. Limits to device performance can nevertheless be evaluated by transforming the inverse design problem into a quadratically constrained quadratic field optimization program (QCQP), and then applying a convex relaxation (e.g. Lagrange duality) to create a dual limit program. In contrast to structural optimization, the dual program (minimization over Lagrange multipliers) is convex, with a unique global minimum corresponding to the tightest bound. The "polarization" associated with the dual optimum contains global information on the characteristics of optimal fields. Retracing this outer ring of transformations illustrates the major steps involved in our verlan method. First, the original inverse design problem is converted into a QCQP over the polarization field. Next, this QCQP is convexified via Lagrange duality to produce a dual program, which determines a super-optimal dual polarization field $\bm{P}_{\mathcal{D}}$. Using Eq. (\ref{['eq:inference']}), a material profile is then inferred from $\bm{P}_{\mathcal{D}}$ and used to initialize a local optimization (e.g. topology optimization).
• Figure 2: Approaching the limits of Purcell enhancement via verlan design. (a) Scaling of LDOS enhancement as a function of side-length for a point source a distance $d=0.2\lambda$ from a $L\times L$ design region with design material $\chi = 5+10^{-4}i$. Dashed lines show local dual bounds; solid lines show the performance of various inverse designed structures. All ring resonators have optimized width $w$ for a given $L$ (although only a few values of $w$ are noted). Standard inverse design is done using TopOpt with ten random, vacuum, 1/2-slab, and full-slab initializations. The verlan designs originate from global dual bounds (orange), local dual bounds (green), and scraping the local dual to approach strong duality (red) molesky_verlan. Standard TopOpts for $L\geq 4$ achieved its best results using $Q$-ramping (see main text) with the Lorentzian lineshape $Q=\{10, 10^3, \infty\}$ progressively ramped up every $10^3$ iterations; TopOpt after verlan extraction is always performed at the real frequency $\omega$. (b) LDOS enhancement as a function of TopOpt iteration number for the $L=6\lambda$ designs. Results in (a) and (b) show the best verlan designs converging to performance within a factor of 2 of optimal and about 5 times that of standard TopOpt, in significantly fewer iterations. (c) TopOpt initializations, final designs, and dipole fields for all the cases described above at $L=6\lambda$, with performance data for the final designs: LDOS enhancement relative to vacuum $\eta$, mode quality factor $Q$, mode volume $V$, as well as the percent of power extracted attributed to absorption and radiation. Verlan structures have higher $Q$, smaller $V$, and fundamentally different enhancement mechanisms (see Fig. \ref{['fig:mechanism']}) than standard TopOpt.
• Figure 3: Physical mechanisms behind near-optimal Purcell enhancement in verlan designs. The structure and field profiles shown correspond to scraped dual verlan designs for $L=5\lambda$, and exhibit distinct features between the surface and bulk. At the structure surface, there is a thin waveguide-like strip with vertical grating patterns extending into the bulk (green boxes) that concentrates the field using pseudogap confinement: a combination of index guiding in the vertical direction and Bragg reflection in the horizontal direction to center the mode close to the surface. The absorbed power distribution (top green box) shows that most absorption is concentrated in the surface around the target point. The $\log(|\bm{E}|)$ distribution (lower left) further corroborates this, showing the field corralled in vacuum pockets by strips of thin gratings running across the bulk of the structure (blue boxes). These gratings have period around $\lambda/2$ with small grating thickness $h$ and confine the radiation leakage from the surface pseudogap in a delocalized, loss-minimizing fashion; see main text for details.
• Figure 4: Schematic showing the design and background regions surrounded by a perfrectly matched layer (PML) region that enforces Sommerfield radiation conditions, ensuring the Maxwell operator $\mathcal{M}$ is invertible.
• Figure 5: Computation time and limit value as a function of number of projection constraints, for the $L=4\lambda$ case in Fig. \ref{['fig:TOresults']}, for naively calculating a limit leveraging the sparse formulation in Appendix \ref{['asec:sparse']} and the gradient coordinate descent procedure in Appendix \ref{['asec:gcd_final']}. Results show that for a modest number of projection constraints (five), GCD can achieve a bound better than that given by $2^{10}$ projection constraints in approximately one-tenth the time.