隱形一直是科幻小說或電影中的熱門題材。直到2006年這個概念不再只是個空想，而是真實的在微波頻段的電磁波中實現了。開啟這扇大門的是Leonhardt 以及 Pendry 這兩組研究團隊，在2006年發表兩篇研究在Science 期刊雜誌上。其中所提到光學隱形斗篷的概念更成為近來非常引人關注與感興趣的研究課題。主要探討電磁波Maxwell方程式與幾何座標轉換之間的結合與應用，透過此方法能設計出適當的材料參數來達到控制電磁波的目的。這個概念逐漸發展成為一個新興的領域－轉換光學(transformation optics)，而所對應設計出材料則被稱為轉換材料(transformation media)。轉換材料的理論可以延伸至不同物理現象，本論文將詳細介紹傳導、電磁波與其他彈性問題的推導與應用，並探討其在應力波傳問題中的可行性。本論文亦著重於討論轉換函數的性質與應用，理論上轉換光學的行為是由座標轉換來控制，因此所對應的幾何形狀、材料參數等皆可透過調整轉換函數來進行調整與最佳化設計。在幾何問題上，本文提出了簡易的任意多邊形與特殊幾何曲線的設計方式，並將此設計方式推廣至異向性的背景材料之中。在材料參數設計上，本文推導出在參數簡化後仍符合阻抗匹配條件的高階轉換函數，並將此函數整理成一通用公式，使其能方便延伸到各種形狀轉換材料進行參數調整。最後，本文提出一個理論分析架構，解析多層轉換材料的場量分布，並進一步去探討層狀轉換材料的等效行為，結果證明只要轉換函數連續，層狀轉換材料的等效的行為是決定在邊界條件而非函數本身。

To make objects invisible to human eyes has been long a subject of science fiction. But just in 2006, this imagination has been materialized in the range of microwave radiation. This was attributed two pioneering papers in Science by Leonhardt and Pendry et al. in 2006, in which they proposed an ingenious idea to control electromagnetic waves by a specially designed materials. The general concept that leads to the design of proper materials is now referred to as transformation optics. In this thesis, we present theoretical frameworks for different physical phenomena, including conduction, electromagnetic waves and elastodynamics. Specifically, we are concerned with the study of transformation functions. From the theoretical point of view, the behavior of transformation optics is governed by the function we select, so the geometric configurations and material properties could be refined by adjusting transformation functions. We develop a simple theory form that enable us to design cloaks with arbitrarily complex geometry. In addition, we propose a methodology of cloaking with curvilinearly anisotropic background materials. Also, we propose a unified derivation of the quadratic function for arbitrary geometry. This quadratic function can be applied to design optimal reduced parameters that impedance matching condition is fulfilled at outer boundary. The restriction of transformation medium theory in stress wave propagation is also discussed. Lastly, we present a framework based on a Fourier-Bessel analysis to prove the equivalency of different geometric configurations in transformation optics.

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