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研究生:賴珍寧
研究生(外文):Jane-Nine Lai
論文名稱:電磁波隱形斗蓬之數學模型
論文名稱(外文):The Mathematical Model of Electromagnetic Cloaking
指導教授:羅廷剛
指導教授(外文):Ting-Kung Luo
口試委員:陸林天曹景懿
口試委員(外文):Lin-Tian LuhG.Tsaur
口試日期:2017-05-24
學位類別:碩士
校院名稱:靜宜大學
系所名稱:財務與計算數學系
學門:數學及統計學門
學類:其他數學及統計學類
論文種類:學術論文
論文出版年:2017
畢業學年度:105
語文別:英文
論文頁數:108
中文關鍵詞:隱形斗篷電磁波變換光學
外文關鍵詞:Invisibility cloakElectromagnetic waveTransformation optics
相關次數:
  • 被引用被引用:0
  • 點閱點閱:366
  • 評分評分:
  • 下載下載:3
  • 收藏至我的研究室書目清單書目收藏:0
自從2006 年首次實現了單頻和極化微波的電磁波隱形斗蓬,超材料隱形斗篷的研究一直在增長。
本論文的目的是為了對以變換光學為基礎的電磁波隱形斗蓬之基礎理論的理論數學模型作出貢獻,並詳細介紹了麥克斯韋方程的相關理論。
首先介紹從最小作用量原理得出的幾何光學的背景和理論。徹底推導了斯奈爾定律的公式和Matlab 程式,並提供了詳細的解釋,從而獲得了折射定律的工作知識。
為了詳細說明電磁波的傳播,提出傅立葉光學。研究了光速的推導和折射率,從而可以徹底地描述由超材料製成的斗蓬中折射率的複雜分佈。
透過引入坐標變換,提出了由超材料具有的廣範本構參數的可能性所提供的變換光學理論。
麥克斯韋方程之不變量的證明以三種方法提出。基本理論的介紹之後是我們對物質參數的計算和數值模擬的工作。然後通過創建單頻電磁波隱形斗蓬來掲示設計程序。並給定模擬實例來說明結論。
Research on metamaterial invisibility cloak has been growing ever since electromagnetic cloaking was first realised for microwaves of one frequency and polarization in 2006.
The aim of this thesis is to contribute to the theoretical mathematical model of the basic theory of transformation optics-based electromagnetic cloaking, and the relevant theory of Maxwell equation is described in detail.
Background and theory of geometric optics derived from Principle of Least Action are introduced first. Formulas and Matlab code of Snell’s Law are thoroughly derived and detailed explanations are provided, so that the working knowledge of law of refraction can be acquired.
In order to give a detailed description of the propagation of electromagnetic wave, the Fourier’s optics is proposed. The derivation of speed of light and index of refraction are studied, so that the complicated distribution of refractive index in the cloaking made of matamaterials can be described thoroughly.
The theory of transformation optics offered by the vast possibilities of the constitutive parameters of metamaterial is celebrated by the introduction of coordinate transformation.
A proof of the form invariance of Maxwell’s equations with sources is presented in three kinds of method. The introduction to the basic theory is followed by works of our computation of material parameters and numerical simulation. Design procedure is then demonstrated by creating electromagnetic cloaking with single frequency. Simulated examples are given to illustrate the conclusion.
Abstract Acknowledgment I
Abstract III
Introduction 1
Thesis Outline 6
Part 1 Theory of Fermat’s Principle
Principle of Least Action 8
Lagrangian Mechanics 8
Newtonian Mechanics 12
Hamiltonian Mechanics 12
Geometrical Optics 14
Fermat’s principle 15
Law of Reflection 15
Snell’s Law 17
Snell’s Law for single frequency 20
Optimal Optical Path in Matlab 21
Part 2 Theory of Transformation Optics
Transformation optics 25
Matamaterial 26
Subwavelength 27
Matamaterials and Coordinate Transformation 27
Negative Refraction 29
Coordinate Transformation 32
Scale factor 32
Gradient in Curvilinear Coordinates 34
Divergence in Curvilinear Coordinates 36
Curl in Curvilinear Coordinates 38
Proof of Maxwell Invariance in differential form 41
Proof of Maxwell Invariance in Einstein Convention 44
Proof of Maxwell Invariance in phase form 51
Part 3 Electromagnetic Wave
Wave 55
Fourier Transform 59
Fourier Optics 62
Maxwell Equation 63
Refractive Index and the speed of light 66
The solution to uniform plane wave 68
Plane Wave in Matlab 69
Electric field, Magnetic field and Poynting vector are mutually orthogonal 70
Derivation of Electromagnetic Waves 71
Poynting Vector 73
Parameterization of Rays 75
Electromagnetic Wave in Matlab 76
Part 4 Calculation and Simulation
Calculation of Invisibility Cloak 77
Hamiltonian and ray equation 81
Snell’s Law at the Air-Cloak Interface 83
Define Spatial Transform 85
Calculate Effective Material Properties 87
Map Properties to Engineered Materials 90
Simulation
Spherical Cloak in Matlab 91
Cylindrical Cloak in Matlab 93
Calculation of Material properties
Spherical Cloak 96
Cylindrical Cloak 98
Experiment 99
Simplification of Material’s Parameter 100
Simulation of Research Team 101
Conclusion 103
Bakhtiyar Orazbayev, Advanced metamaterials for high resolution focusing and invisibility cloaks
N. S. Manton ,The Principle of Least Action in Dynamics,DAMTP, Centre for Mathematical Sciences, University of Cambridge,
ADRIAN CHO, High-Tech Materials Could Render Objects Invisible, Science 312,
Ulf Leonhardt,Optical Conformal Mapping, Science 312, 1777 (2006);
Branimir IVSIC, Zvonimir SIPUS, Juraj BARTOLIC Bandwidth of Invisible Cloak Realized with Split Ring Resonators
Matjaz ̆ Boz ̆ic ̆,Invisibility cloak, UNIVERSITY OF LJUBLJANA,FACULTY OF MATHEMATICS AND PHYSICS DEPARTMENT OF PHYSICS, Seminar 2009/2010
J. B. Pendry,1* D. Schurig, D. R. Smith, Controlling Electromagnetic Fields,Science, 312, 2006
Darryl D Holm ,Geometric Mechanics Part I: Dynamics and Symmetry Mathematics
Department Imperial College London, 2011
Ulf Leonhardt and Tomáš Tyc, Broadband Invisibility by Non-Euclidean Cloaking Science 323, 2009
Romain Fleury and Andrea Al`u*, Cloaking and Invisibility: A Review Progress In Electromagnetics Research, Vol. 147, 171–202, 201
The Principle of Least Action
https://www.jfoadi.me.uk/documents/lecture_mathphys2_04.pdf
The Lagrangian Formalism,
http://www.damtp.cam.ac.uk/user/tong/dynamics/two.pdf
D.G. Simpson,Lagrangian and Hamiltonian Mechanics, Ph.D Department of Physical Sciencesand Engineering Prince George’s Community College
Geometrical Optics / Mirror and Lenses http://www.vorc.fcu.edu.tw/
wSite/publicfile/Attachment/f1427708912322pdf
Fermat's Principle and Refraction
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/Fermat.html#c2
Leno S. Pedrotti ,Basic Geometrical Optics, CORD,Waco, Texas
George B. Thomas, Maurice D. Weir,Joel R. Hass Thomas’ Calculus Twelfth Edition
Justin Peatross, Michael Ware,Physics of Light and Optics, Brigham Young University
https://en.wikipedia.org/wiki/Snell%27s_law
Ting-Chung Poon • Taegeun Kim ENGINEERING OPTICS WITH MAT LAB*
A V Kildishev, V M Shalaev ,Transformation optics and metamaterials
Janos Perczel, Invisibility cloaking without superluminal propagation
University of St Andrews, 2011
John Pendry, Metamaterials, transformation optics, & cloaks of invisibility,
Can We make Harry Porter Invisible?
https://www.google.com.tw/#q=invisibility+cloak+ppt
ALEXEY KHLOPOTIN, SENAD RAZANICA,
Designing Materials for Mechanical Invisibility Cloaks
Sarah Harvey, NEGATIVE-INDEX METAMATERIALS
Imperial College London Faculty of Natural Sciences Department of Physics,
Transformation Optics Photonics
http://www.cmth.ph.ic.ac.uk/photonics/Newphotonics/TransOptics.htmlhttp:
//www.cmth.ph.ic.ac.uk/photonics/Newphotonics/pdf/newlens_rev.pdf
Matamaterials, transformation optics & cloaks of invisibility
John Pendry, Imperial College, London
D. Schurig, J. B. Pendry2, D. R. Smith
Calculation of material properties and ray tracing in transformation media
https://en.wikipedia.org/wiki/Negative-index_metamaterial
D. R. Smith, J. B. Pendry, M. C. K. Wiltshire Metamaterials and Negative Refractive Index
Lin Xu and Huanyang Chen* Conformal transformation optics
Coordinate Transformation Based Electromagnetic Design and Applications by Wenxuan Tang, Queen Mary, University of London United Kingdom October 2012
Vector Calculus & General Coordinate Systems
http://www.colorado.edu/ASEN/asen5227_offline/slides/126-174.pdf
Gradient: definition and properties MIT OpenCourseWare
https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall
-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-deriva
tives/session-35-gradient-definition-perpendicular-to-level-curves/MIT18_02S
C_notes_18.pdf
Vector Calculus Examples Using MATLAB
Gradient: definition and properties, MIT OpenCourseWare
https://en.wikipedia.org/wiki/Divergence
Curl (mathematics) https://en.wikipedia.org/wiki/Curl_(mathematics)
NEGATIVE-INDEX METAMATERIALS Sarah Harvey
Metamaterials, transformation optics & Cloaks of invisibility,John Pendry, Imperial College, London
Janos Perczel, Invisibility cloaking without superluminal propagation
Steven G. Johnson, Coordinate Transformation & Invariance in Electromagnetism, notes for the course 18.369 at MIT
Maxwell Equation http://www.ece.rutgers.edu/~orfanidi/ewa/ch01.pdf
[44]http://www.eecs.ucf.edu/~tomwu/course/eel6482/notes/07%20Maxwell's%2
0Equations.pdf
Dr. Raymond RumpfR, Transformation Electromagnetics, ECE5390 Special Topics:
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