負折射、超材料、變換光學與隱形斗篷之探討

Abstract

本論文主要介紹負折射、完美透鏡、變換光學和隱形斗篷的原理與發展史。探討1999-2006年John Pendry教授在這些領域上的相關理論工作；並且介紹Ulf Leonhardt教授2008-2010年變換光學上的工作，包括用微分幾何的語言處理在彎曲時空Maxwell方程的張量形式。 2006年D.Schurig，D.R.Smith和John Pendry等人除了利用變換光學的概念推導出超材料隱形斗篷所要滿足的物質參數之外，也根據幾何光學近似從Maxwell方程推導出光線軌跡的Hamilton正則方程所對應的物質Hamiltonian。這篇著名的論文開啟了人類追求隱形斗篷的夢想，但是當初論文許多地方是模糊且晦澀難懂的，其中物質Hamiltonian和Hamilton正則方程的推導都太過簡略，以致於不好理解。2011年四位數學家Miles M. Crosskeya, Andrew T. Nixonb, Leland M. Schickc, Gregor Kovačičd重新去探討隱形斗篷原理和數學的存在性證明。本論文特別針對球形隱形斗篷和柱形隱形斗篷的張量參數推導Hamilton 正則運動方程，將數學推導和物理原理都解釋清楚。本論文最後對葛墨林教授2009年的相關工作--- 用廣義相對論的語言解釋各異向性非均勻介質的Hamiltonian及證明光線軌跡滿足測地線方程，做解釋和詳細推導。

Abstract This thesis mainly introduces the principle and the history of negative refraction, perfect lens, transformation optics and invisibility cloaking. We explore the theoretical works of Professor John Pendry in these areas during 1999-2006, as well as the works of Professor Ulf Leonhardt during 2008-2010 about transformation optics, which include those using the language of differential geometry for manipulating the tensor form of Maxwell equations in curved spacetime. In 2006, D.Schurig, D.R.Smith and John Pendry et al proposed the concept of transformation optics, which can provide the material parameters for designing invisibility cloaks with metamaterials. Under the assumption of the validity of geometrical optics approximation, they also derived the Hamiltonian from Maxwell equations for obtaining the canonical equations of motion for generating the ray trace. This work triggers the human dreams about making invisibility cloaks. In that work, the derivation of Hamiltonian and Hamilton canonical equations are too brief and compact to understand. In 2011 four mathematicians Miles M. Crosskeya, Andrew T. Nixonb, Leland M. Schickc, Gregor Kovačičd re-investigated the principle of invisibility cloaking and provided the rigorous proof of its existence mathematically. In this thesis, especially for the spherical and cylindrical of invisibility cloaks, we provide the detailed derivations of the tensor parameters, material Hamiltonian and Hamilton E.O.M, and the physical principles are explained thoroughly. Finally, we discuss the works of Professor Molin Ge of applying the language of general relativity to interpret the anisotropic inhomogeneous media, and prove the geodesic equations of light ray thoroughly.