Invisibility cloak with photonic crystals

轉換光學已被理論地證實可用來使物體隱形不被電磁場所偵測，但 是所計算出的隱形斗篷材料的結構參數是非均勻、非等向性甚至在邊 界處會出現極值。為了實現隱形斗篷，簡化的結構參數是必要的，然 而簡化參數意味著隱形效能的降低。針對二維圓柱形的隱形斗篷，在 過去文獻中已提出多種結構參數，這些參數的均勻性或等向性取決於 對空間的關係，這個關係決定了被實現的可行性。比較這些已提出的 結構參數，隱形效能和可被實現的可能性是很難兼顧的。 本篇論文設計出一個操作在可見光波段並同時兼顧隱形效能和 可被實現的可能性之隱形斗篷。藉由考慮在邊界處的阻抗匹配提出 一組簡單的簡化參數，並在隱形斗篷內部加上光子晶體結構來減少 能量穿透進入隱形的區域。在研究中，模擬使用套裝軟體COMSOL MutiphysicsR 的有限元素法解馬克思威爾方程式，計算隱形斗篷在TE 波入射所產生的散射場分佈來評估隱形的效能。與文獻中已提出的結 構參數比較中，在近場及遠場條件下分別針對正向散射、反向散射以 及各個方向的散射做了一系列的比較與討論。模擬比較的結果指出我 們的設計在簡單的結構參數下可以達到很好的隱形效果，證實了在可 見光波段可以同時兼顧隱形效能和可被實現的可能性。Transformation optics has shown taht the ability to cloak an object from incident electromagnetic radiation is theoretically possible. However, the constitutive parameters dictated by the theory are inhomogeneous, anisotropic, and, in some instances, singular at various locations. In order for a cloak to be practically realized, simplified parameter sets are required. However, the simplified parameters result in a degradation in the cloaking function. Constitutive parameters for simplified two-dimensional cylindrical cloaks have been developed and are divided into two categories based on the spatial dependence which represnets the feasibility of practical implementation. Comparing the proposed simplified parameter in the literature, there is a tradeoff between the performance and the possibility for implementation. In this thesis, we design a simplified invisibility cloak operating in the visible light spectrum giving consideration to both the performance and the possibility for implementation. The design is a combinaiton of the simplified parameter by considering impedance match and the photonic crystal structure. During the course of this study, it was noted that all cloak simulations are performed using finite element method (FEM) based numerical methods. In the comparisons with the proposed parameter under the designed PC structure, we compare forward scattering, backward scattering, and scattering in all directions with far field and near field condition respectively. Simulation results illustrate that our design takes into account both the performance and the possibility for implementation in practical in the visible light spectrum.誌謝i 中文摘要ii Abstract iii Contents iv List of Figures vi List of Tables ix 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Invisibility Cloak Theory 8 2.1 Transformation Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Space Invariance Of Maxwell''s Equation . . . . . . . . . . . . . 11 2.1.2 Computation of Material Characteristic . . . . . . . . . . . . . . 11 2.1.3 Control of Electromagnetic Field . . . . . . . . . . . . . . . . . . 13 2.1.4 Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.5 Transformation Between Cartesian and Polar Coordinates . . . . 16 2.2 Infinitely Long Cylindrical Electromagnetic Cloak . . . . . . . . . . . . 17 2.3 Scattering Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Simulation Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Implementation of The Cloak . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Simplified Parameter 27 3.1 Simplified Constraint Equation . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Simplified Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Constraint Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Improved Simplified Parameter . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Designed Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5.1 Impedance Match . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5.2 Transmitted energy in the hidden region . . . . . . . . . . . . . . 44 3.6 Designed Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 Comparisons and Analysis 59 4.1 Forward Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Backward Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Scattering in all directions . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Conclusions 70 References 7