Accelerating Inverse Design of Optical Metasurfaces: Analytic Gradients of Periodic Green's Functions via Quasi-Modular Forms

TL;DR

This paper introduces an analytic gradient method for inverse design of optical metasurfaces, leveraging quasi-modular forms to compute exact derivatives of Green's functions, significantly improving speed and accuracy over traditional finite difference methods.

Contribution

The authors develop a novel analytic gradient engine using quasi-modular forms to efficiently and precisely compute derivatives of periodic Green's functions for metasurface design.

Findings

Achieves machine-precision derivatives ($10^{-15}$)

Provides a 6.5× speedup in optimization convergence

Enables robust design in regimes where traditional methods fail

Abstract

The inverse design of nonlocal metasurfaces requires the precise optimization of lattice geometry to engineer spatial dispersion and high-Q resonances. However, gradient-based optimization is frequently bottle-necked by the evaluation of the periodic Dyadic Green's Function (DGF), where traditional Finite Difference (FD) methods suffer from an inherent trade-off between truncation error and numerical instability near spectral singularities. In this work, we present an end-to-end Analytic Gradient Engine for 2D Bravais lattices. By mapping the spectral lattice sums of the Coupled Dipole Approximation (CDA) to the theory of Quasi-Modular Forms (QMF), we derive exact, closed-form expressions for the gradients of the interaction matrix with respect to the modular lattice parameter $\tau$. Our framework explicitly handles conditionally convergent terms via regularization and addresses the non-holomorphic outlier $\sigma_4^{(2)}$ via a hybrid numerical strategy. We further introduce a robust evaluation scheme combining $SL(2, \mathbb{Z})$ domain reduction with automatic error certificates. Experimental validation demonstrates that our engine achieves machine-precision derivatives ($10^{-15}$) and a 6.5$\times$ speedup in optimization convergence compared to finite-difference baselines, enabling the robust design of giant anisotropy in regimes where traditional methods fail.