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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.03755 |
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| _version_ | 1866908865395687424 |
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| author | Qin, Mingcan Qin, Yifeng |
| author_facet | Qin, Mingcan Qin, Yifeng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03755 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Accelerating Inverse Design of Optical Metasurfaces: Analytic Gradients of Periodic Green's Functions via Quasi-Modular Forms Qin, Mingcan Qin, Yifeng Optics 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. |
| title | Accelerating Inverse Design of Optical Metasurfaces: Analytic Gradients of Periodic Green's Functions via Quasi-Modular Forms |
| topic | Optics |
| url | https://arxiv.org/abs/2603.03755 |