超穎透鏡具有平面化、厚度薄與高自由度波前調控等優點，可藉由次波長超穎原子控制入射光之相位、振幅與偏振響應，提供傳統折射式光學元件以外的設計方式。然而，多數超穎透鏡於設計與製作完成後，其焦距與光學功能即被固定，限制其於動態成像、焦距掃描及可調式光學系統中的應用。本論文提出一種基於偏振群延遲(Polarization Group Delay, PGD)之反射式可變焦超穎透鏡，並採用Al/SiO2/Al金屬－介電質－金屬結構作為設計平台。透過各向異性超穎原子在不同入射線偏振角下所產生之反射相位差異，以PGD描述反射相位隨偏振角變化的平均趨勢，並結合反射相位、反射率與PGD建立超穎原子資料庫。設計過程中，首先依據起始與終止焦距建立目標相位及目標PGD分布，再利用相位與PGD雙重匹配方法選擇適當的超穎原子，最後透過角頻譜法分析不同偏振角下之焦距、焦斑、聚焦效率與調變傳遞函數。
本研究所設計之超穎透鏡尺寸為500 μm，初始焦距為2000 μm，對應初始數值孔徑約為0.124。模擬結果顯示，藉由調整孔徑內PGD之徑向分布，可分別實現正向與反向變焦，且焦點位置可隨入射偏振角呈現預期之單調變化。其中，正向變焦設計之模擬焦距總變化量約為163.9 μm，相當於初始焦距的8.2%；隨焦距增加，數值孔徑約由0.124降低至0.115。模擬聚焦效率於0°與90°偏振狀態下較高，並於中間偏振角附近下降，顯示中間偏振態之相位匹配與能量集中能力較弱。實際製作之超穎透鏡亦觀察到偏振相依的正向焦點位移趨勢，使用光圈限制入射光束時，量測所得之焦距變化量約為40 μm，相當於約2.0%的焦距調變，但其變焦幅度與中間偏振態表現皆低於模擬結果。此差異可能來自奈米結構製程偏差入射光束品質、背景雜光、光路像差及焦點判定方式。整體而言，本研究建立了一套以PGD為核心之偏振調控可變焦超穎透鏡設計流程，並驗證其於反射式平面光學元件中的應用潛力。

Metalenses offer planar geometry, reduced thickness, and flexible wavefront control through subwavelength meta-atoms. However, most metalenses have fixed focal lengths after fabrication, limiting their use in tunable imaging and focal scanning systems. In this thesis, a reflective varifocal metalens based on polarization group delay (PGD) is proposed using an Al/SiO2/Al metal–insulator–metal structure. The polarization-dependent reflection phase of anisotropic meta-atoms is characterized by PGD, and a database containing reflection phase, reflectance, and PGD is established. Target phase and PGD distributions are generated from the initial and final focal lengths, followed by simultaneous phase–PGD matching for meta-atom selection. Angular spectrum propagation is then used to evaluate the focal position, focusing efficiency, and modulation transfer function at different polarization angles.
The designed metalens has an aperture of 500 μm and an initial focal length of 2000 μm, corresponding to a numerical aperture of approximately 0.124. Numerical results demonstrate both positive and negative varifocal responses by controlling the radial PGD distribution. For the positive varifocal design, the simulated focal shift is approximately 163.9 μm, corresponding to 8.2% of the initial focal length, while the numerical aperture decreases from approximately 0.124 to 0.115. The fabricated device also exhibits a polarization-dependent positive focal shift. Using an iris-limited incident beam, the measured focal variation is approximately 40 μm, corresponding to about 2.0%. The difference between simulation and experiment is mainly attributed to fabrication deviations, material properties, stray light, optical aberrations, and focal-position determination. These results demonstrate the potential of PGD-based reflective metalenses for tunable planar optical systems.

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