Inferring Structure via Duality for Photonic Inverse Design

TL;DR

The paper tackles photonic inverse design by formulating Maxwell-constraint problems as QCQPs/SCQPs and leveraging duality and convex relaxations to obtain informative bounds. It introduces a Verlan scheme that gradually transforms a QCQP toward strong duality by updating the objective and scattering operator and coordinating scrape/contract/expand steps to extract dual-informed, globally meaningful initial points for local optimization. Grounded in Sion's minimax theory, it connects dual and primal views and explains how approaching constraint boundaries reduces duality gaps, enabling robust initialization. A partner study reports substantial, order-of-magnitude improvements in power extraction for a dipole near a structured material boundary at relatively large design areas, illustrating the practical impact of leveraging duality-informed structure inference for large-scale photonic devices.

Abstract

Led by a result derived from Sion's minimax theorem concerning constraint violation in quadratically constrained quadratic programs (QCQPs) with at least one constraint bounding the possible solution magnitude, we propose a heuristic scheme for photonic inverse design unifying core ideas from adjoint optimization and convex relaxation bounds. Specifically, through a series of alterations to the underlying constraints and objective, the QCQP associated with a given design problem is gradually transformed so that it becomes strongly dual. Once equivalence between primal and dual programs is achieved, a material geometry is inferred from the solution of the modified QCQP. This inferred structure, due to the complementary relationship between the dual and primal programs, encodes overarching features of the optimization landscape that are otherwise difficult to synthesize, and provides a means of initializing secondary optimization methods informed by the global problem context. An exploratory implementation of the framework, presented in a partner manuscript, is found to achieve dramatic improvements for the exemplary photonic design task of enhancing the amount of power extracted from a dipole source near the boundary of a structured material region -- roughly an order of magnitude compared to randomly initialized adjoint-based topology optimization for areas surpassing $10~λ^{2}$.

Figures (2)

• Figure 1: Pictorial Lagrangian duality. The left part of the figure depicts three sections, $\phi$ multiplier values, of a schematic optimization Lagrangian $\mathcal{L}$ for a two dimensional $\left(x_{1},x_{2}\right)$ QCQP subject to a single feasible equality constraint, the red path intersecting the three sections. The value of the (convex) dual $\mathcal{D}$, determined within each section by the maximum of $\mathcal{L}$ over the associated manifold (represented by the coloured balls), is necessarily at least as large as the maximum of $\mathcal{L}$ over the constraint set, the rippling circle. In the illustration, as indicated by the upper inset, the minimum of $\mathcal{D}$, the maximum of the light blue section, is exactly equal to the maximum of $\mathcal{L}$ along the constraint path. For a general QCQP no such agreement between the maximum of $\mathcal{L}$ within the set of points satisfying all constraints and the minimum of $\mathcal{D}$ need occur, with the two values generally differing by a duality gap. An optimization program exhibits strong duality when the two values agree.
• Figure 2: Schematic protocols. The figure illustrates the scrape and contract protocols of our proposed scheme. (a) A series of scraping steps for the program $\max \left(0,-1\right)^{\dagger}\bm{x},~\bm{x}\in\mathbb{R}^{2}$ subject to the indefinite constraint $4 + 4x_{a} -3x_{b} - 4x^{2}_{b} = 0$ and the compact constraint $1 - x^{2}_{a} - x^{2}_{b} = 0$ (orange disk). The tear-drop shaped purple region is the Sion set, bounding the convex hull depicted as a dashed line. Taking the dual transform of $\mathtt{P}$, the minimum is found to be $\bm{x} = \left(0,-1/3\right)$---the point emphasized by the bullseye. Consequently there is a duality gap of $1/3$ compared to $\mathtt{P}$: $\mathtt{P}$ has two feasible points, the intersection points of the Sion set with the orange circle, with the best objective value being achieved at $\bm{x} = \left(0,0\right)$. Smaller purple circles show the location of the $\bm{x}_{\circledast}$ solution of $\mathtt{D}\left(\mathtt{P}\left(\bm{r}\right)\right)$ for five successive scrapes: $\bm{r}^{\left(n\right)} = \bm{r}^{\left(o\right)} / 2 + \bm{x}_{\circledast}^{\left(o\right)}/ 2$ with $\bm{r}$ initialized as $\bm{s}_{o}$. As the objective vector drifts, $\bm{x}_{\circledast}$ is observed to move towards the boundary of the orange disk. On the fifth modification of $\bm{s}_{o}$ strong duality is achieved. (b) Shape of the Sion set $S_{\mathtt{P}}$ (boundaries demarcated by solid coloured outlines and feasible points) compared to the convex hull $C_{\mathtt{P}}$ (boundaries demarcated by dashed outlines and feasible points) under contraction for an example scattering constraint set in $\mathbb{R}^{2}$. As $\lVert\bm{\mathrm{X}}_{\bullet}^{-1}\rVert$ becomes increasingly positive definite, both $S_{\mathtt{P}}$ and $C_{\mathtt{P}}$ contract towards the trivial feasible point at the origin, the common lowest point of all regions. Simultaneously, $S_{\mathtt{P}}$ increasingly coincides with $C_{\mathtt{P}}$, implying strong duality for any objective with maximum occurring outside of $S_{\mathtt{P}}$.

Theorems & Definitions (6)

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