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 $τ$. Our framework explicitly handles conditionally convergent terms via regularization and addresses the non-holomorphic outlier $σ_4^{(2)}$ 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.

Figures (5)

• Figure 1: Gradient Verification. Relative error of the numerical gradient (blue) versus step size $h$, exhibiting the characteristic "V-shape" trade-off between truncation error ($O(h^2)$) and round-off noise ($O(\epsilon_{\text{mach}}/h)$). In contrast, the analytic QMF gradient (orange) maintains a constant machine-precision error floor ($\sim 10^{-15}$), independent of step size parameters.
• Figure 2: Optimization Efficiency. Convergence of the objective function $J(\tau)$ (Anisotropy) versus number of function evaluations. The analytic gradient engine (orange) drives the optimizer to the solution significantly faster (6.5$\times$ speedup) than the Finite Difference baseline (blue), which suffers from gradient noise near the optimum.
• Figure 3: Modular Optimization Trajectories. Paths taken by the lattice parameter $\tau$ in the complex plane during optimization, overlaid on the modular fundamental domain (shaded). The analytic gradients successfully guide the optimizer from arbitrary initial guesses (circles) to the physically optimal lattice configurations (stars), respecting the modular boundaries.
• Figure 4: Lattice Topology. Top-down view of the dielectric metasurface unit cells. (a) Isotropic square lattice ($\tau_2=1.0$). (b) Anisotropic rectangular lattice ($\tau_2=1.2$), selected to verify the spatial dispersion splitting predicted by the gradient engine.
• Figure 5: Full-Wave Verification of Symmetry Breaking. The Anisotropy Contrast maps, $\Delta S_{ii} = ||S_{ii}^{\phi=0}| - |S_{ii}^{\phi=90}||$, plotted as a function of frequency and incidence angle. Left Column ($\tau_2=1.0$): The dark fields in (a) $\Delta S_{11}$ and (c) $\Delta S_{22}$ indicate negligible polarization difference, confirming the symmetry-protected mode degeneracy of the square lattice. Right Column ($\tau_2=1.2$): The bright bands in (b) $\Delta S_{11}$ and (d) $\Delta S_{22}$ highlight the giant anisotropy engineered by the gradient optimizer. The mode splitting is substantial ($\sim 0.5$ GHz) and consistent across both ports, validating that the QMF-driven geometric perturbation $\tau \to 1.2i$ successfully unlocked the non-local spatial dispersion.