本論文旨在解決傳統微奈米光學正向設計的侷限性，該方法高度仰賴使用全波電磁模擬進行耗時的試錯過程。為克服純數據驅動之深度學習模型中常見的運算瓶頸與缺乏物理一致性之問題，本研究提出了一套基於物理資訊神經網路（PINNs）的逆向設計框架。透過將亥姆霍茲方程式（Helmholtz equation）的殘差直接嵌入損失函數中，神經網路能夠在無需龐大真實標籤資料集的情況下，自主學習並預測出嚴格遵守物理定律的電磁場分佈。我們建立了一套自動化流程，利用DeepSDF與MaxwellNet架構，針對目標焦距為8 µm的非球面平凸微透鏡進行材料分佈的最佳化。此流程整合了連續結構生成、Python空間二值化處理，以及MATLAB幾何對稱重構技術，藉此建構出精確的點雲模型，以供有限差分時域法（FDTD）進行模擬驗證。模擬結果顯示，經PINN最佳化的透鏡能精準地將 8 µm 的中紅外入射平面波聚焦於目標焦點，並達到極高的尖峰電場強度。為嚴格驗證模型的物理一致性，我們使用了 600 nm 至 400 nm 的可見光波段進行跨頻譜驗證。該框架精確預測了波動光學現象，包括較短波長下的縱向色差，同時在波長逼近 400 nm 時，成功界定出 FDTD 演算法的數值空間取樣極限。此外，透鏡曲率修改與複雜的離軸斜入射測試證明了該模型已全面掌握波干涉與空間相位傳播的底層原理。它有效地呈現了焦距偏移、焦深拉長以及離軸像差，且未對特定的初始設計參數產生過度擬合。總結來說，本研究建立了一個高精確度、受物理約束且運算穩定的深度學習求解器，可用於自由曲面微透鏡的自動化逆向設計。在此框架中，MaxwellNet即扮演了如同傳統 FDTD 軟體（如 VSim 或 Meep）的核心電磁數值運算工具角色，藉由其物理驅動的快速推論能力，大幅加速了結構優化的迭代過程。此框架為未來發展受實際製程限制的奈米光學元件，提供了穩固的理論與技術基礎。

This thesis addresses the limitations of traditional forward design in micro/nano-optics, which relies heavily on time-consuming trial-and-error using full-wave electromagnetic simulations. To overcome the computational bottlenecks and the lack of physical consistency often found in purely data-driven deep learning models, this study proposes an inverse design framework based on Physics-Informed Neural Networks (PINNs). By embedding the residuals of the Helmholtz equation directly into the loss function, the neural network autonomously learns to predict electromagnetic field distributions that strictly obey physical laws without requiring massive ground-truth datasets. We developed an automated pipeline utilizing DeepSDF and MaxwellNet architectures to optimize the material distribution for a non-spherical plano-convex microlens with a target focal length of 8 µm. The process integrates continuous structure generation, spatial binarization in Python, and geometric symmetry reconstruction in MATLAB to create an accurate point cloud model for Finite-Difference Time-Domain (FDTD) simulation validation.
Simulation results demonstrate that the PINN-optimized lens successfully and precisely focuses an 8 µm mid-infrared incident plane wave at the target focal point, achieving a high peak electric field intensity. To rigorously verify the model's physical consistency and generalization capabilities, we conducted cross-spectrum validations using visible light wavelengths ranging from 600 nm to 400 nm. The framework accurately predicted wave-optical phenomena, including longitudinal chromatic aberration at shorter wavelengths, while successfully identifying the numerical spatial sampling limits of the FDTD algorithm when approaching 400 nm. Furthermore, structural curvature modifications and complex off-axis incident tests (e.g., ±10° tilt angles) proved that the model comprehensively captured the underlying principles of wave interference and spatial phase propagation. It effectively represented focal shifts, focal depth elongation, and off-axis aberrations without overfitting to specific initial design parameters.
In conclusion, this study establishes a highly accurate, physically constrained, and computationally stable deep learning solver for the automated inverse design of free-form microlenses. This framework provides a robust theoretical and technical foundation for the future development of three-dimensional full-vector modulation, broadband achromatic designs, and fabrication-constrained nanophotonic components.

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