全球著名的暢銷小說「哈利波特」中，曾敘述過許多奇幻的魔法，其中最廣為人知的就是隱形斗篷，在過去它似乎只是個的科幻題材；但現今透過最新理論物理與實驗的實證，實現此事已不再是遙不可及。這一系列研究起始於2006年，在國際知名期刊Science之二篇開創性論文Pendry et al. (2006a)及Leonhardt (2006a)。其主要發現利用適當設計的超穎材料可以用來導引電磁波行進的方向，並控制其繞過一封閉區域進而設計出球形隱形斗篷。此舉開啟了光學領域的新里程，稱之為轉換光學，其裝置材料通稱為轉換材料。本論文主要根據此一觀念，證明二維正交曲線座標下轉換材料之通式，另外藉由轉換光學理論提出「假像裝置」，此裝置不僅可隔著空間將某物體隱形，並可同時產生假像，使外部觀察者看起來為一個完全不同的物體，再以推導的曲線正交座標來設計不同的光學裝置。除此之外，文中探討如何透過幾何外形與裝置欲達成之效果來設計轉換材料，依此提出具有平移或旋轉效果的錯覺裝置，同時以數值模擬來展示上述各類裝置的可行性，最後以理論解析解驗證裝置的效應。

Starting with two pioneering papers published in Science by Pendry et al. and Leonhardt in 2006, the new concept has attracted numerous researchers to explore the possibility to control the propagation of electromegnetic waves and optical light. This category of study is now referred to as transformation optics. This thesis is concerned with a number of theoretical aspects that arise in the design of various optical devices. First, following chain-rule operations in partial differential equations, we derive the transformation identity in two-dimensional curvilinear orthogonal coordiantes. Next, we propose several optical illusion devices based on curvilinear coordiantes. A particular example is given to elliptical coordiantes. We demonstrate the effectness of the devices based on numerical finite element calculations. In addition, we propose a unified mathematical formula that allows us to describe several optical devices, such as cloak, rotator, translator, concentrator, in a single framework. This provides new perspectives to interprete the devices in a new way. Lastly, we show that based on curvilinear coordiantes the transformation media will not disturb the exterior fields before the transformation.

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