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

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.