臺灣博碩士論文加值系統

本論文採用以 Z-轉換輔助的有限時域差分法 (Z-transform assisted finite-difference time-domain method 或 Z-transform assisted FDTD method) 來模擬電磁波包在細金屬線陣列 (Wire Array) 和裂環共振器陣列 (Split-Ring Resonator Array) 所構成的複合式人工構造 (Wire-SRR Array) 中的傳播行為。此種複合介質具有類似於 Drude 模型形式的色散型介電係數 (permittivity) 以及類似於 Lorentz 模型的色散型磁導率 (permeability)，並在某一段頻率範圍內具有負折射率 (negative refractive index)。本研究主要探討電磁波在負折射頻段於介質中的傳播行為，具體內容包括：波包在介質中的移動速度、群速度、能量速度在介質強吸收以及弱吸收的情況下的行為比較，以及波包在空氣與Wire-SRR介質介面處的能量穿透情形。最後探討波包在Wire-SRR進入介質中的折射行為，並將其結果和單頻的折射定率做比較。

In this thesis we adopt the Z-transform assisted Finite-difference time domain (FDTD) method to simulate the wave packet propagating behavior in the artificial structure which composed with periodically arranged thin metallic wires and spilt-ring resonators (SRRs). This artificial medium has dispersive permittivity like that of Drude model and dispersive permeability like that of Lorentz model, and has negative refractive index in a range of frequency. The main theme of our study is about the propagating behaviors of electromagnetic waves in this medium in the negative refraction frequency range. We study and discuss the differences among wave packet velocity, group velocity, and energy velocity in lossless and absorptive medium respectively. Finally, we compare the energy transmittance and refractive performance of wave packet with that of the monochromatic waves.

摘要 I
誌謝 III
目次 IV
圖目錄 VI
表目錄 VIII
一、緒論 1
1-1 負折射與等效介質 1
1-2 波包在等效介質中的傳播 2
二、負折射與色散性的等效介質 3
2-1 色散模型 3
2-1-1 Drude Model 3
2-1-2 Lorenz Model 4
2-1-3 色散介質中的能量密度 5
2-1-4 吸收性色散介質中的能量密度 7
2-2 負折射材料的實現與等效模型 10
2-2-1 細金屬線陣列-等效電漿模型 10
2-2-2 裂環共振器(split-ring resonator , SRR) 14
三、時域有限差分法（Finite Difference Time Domain，FDTD）於電磁波之應用 17
3-1　FDTD架構 17
3-1-1　Maxwell's Equation 19
3-1-2　FDTD中電磁場的遞迴關係 20
3-2　完美匹配的吸收邊界 22
3-3 Z轉換(Z-Transform)在色散介質中的應用 26
3-3-1 Z轉換(Z-Transform)的基本定義 26
3-3-2 FDTD中Z 轉換於Drude Model之應用 27
3-4　FDTD數值模擬的穩定性 31
四、波包在等效介質中的模擬與分析 32
4-1　模擬參數設定 32
4-2　金屬線-裂環共振器陣列(Wire-SRR ) 35
4-2-1　波包正向入射之群速度 35
4-2-2　波包正向入射之穿透率以及反射率 40
4-2-3　波包斜向入射之折射角 46
4-3　波包的能量耗散以及介質的吸收 53
4-4 數值積分的時間間隔對於誤差之影響 57
五、結論與未來展望 59
5-1 結論 59
5-2 未來展望 60
參考文獻 61

[1] V. G. Veselago,"The Electrodynamics of Substances with Simultaneously Negative Value of and ", Sov. Phys. USPEKHI, Vol. 10, Num. 4, January,1968.
[2] J. B. Pendry," Negative Refraction Makes a Perfect Lens", Phys. Rev. Lett., Vol. 85, Num. 18, October,2000.
[3] R. A. Shelby, D. R. Smith, S. Schultz, " Experimental Verification of a
Negative Index of Refraction", Science, Vol. 292, pp. 77-79, 2001.
[4] David J. Griffiths, Introduction to Electrodynamics , third edition, pp.398-404.
[5] David K. Cheng, Field and Wave Electromagnetics , second edition, pp.373-374.
[6] L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media , Rober Maxwell, M.C., second edition, 1985, pp.272-275.
[7] R. Ruppin, "Electromagnetic energy density in a dispersive and absorptive material" , Phys. Lett. A., Vol. 299, pp. 309-312, July 2002.
[8] Pi-Gang Luan, "Power loss and electromagnetic energy density in a dispersive metamaterial medium" , Phys. Rev. E , Vol. 80, October 2009.
[9] 欒丕鋼，陳啟昌，光子晶體-從蝴蝶翅膀到奈米光子學，pp.272-282，五南圖書出版股份有限公司，2010年10月。
[10] J. B. Pendry, "Extremely Low Frequency Plasmons in Metallic Mesostructures" , Phys. Rev. Lett., Vol. 76, Num. 25, June,1996.
[11] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Nonlinear Phenomena", IEEE. T. Microw. Theory., Vol. 47, Num. 11, November,1999.
[12] Hongsheng Chen, Lixin Ran, and Jiangtao Huangfu, "Equivalent circuit model for left-handed metamaterials", J. Appl. Phys., Vol. 100, pp. 024915, July,2006.
[13] Kane S. YEE, " Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media" , IEEE .T.Antennas.Propag. ,Vol. 3, pp. 302-307,1966.
[14] Atef Z. Elsherbeni,Veysel Demir, The Finite-Difference Time-Domain Method for Electromagnetics wih MATLAB Simulations., SCITECH. ,2009.
[15] Jean-Pierre Berenger, "A Perfectly Matched Layer for the Absorption of Electromagnetic Waves" , J.Comput.Phys.,Vol.114, pp. 185-200,1994.
[16] R. Luebbers, F. Hunsberger, K. Kunz, R.Standler,and M.Schneider, "A Frequency-Dependent Finite-Difference Time-Domain Formulation for Dispersive Materials" , IEEE .T.Electromagn.C., Vol. 32, NO, 3, August 1990.
[17] Raymond J. Luebbers, Forrest Hunsberger, "FDTD for Nth-Order Dispersive Media" , IEEE .T.Antennas.Propag. ,Vol. 40, NO. 11,November 1992.
[18] Mohammad A. Alsunaidi, Ahmad A. Al-Jabr, " A General ADE-FDTD Algorithm for the Simulation of Dispersive Structures" , IEEE.Photonic.Tech.L. ,Vol. 21, pp. 817-819,June,2009.
[19] Soon-Cheol Kong, Jamesina J.Simpson, Vadim Backman. "ADE-FDTD Scattered-Field Formulation for Dispersive Materials" , IEEE.Microw.Wirel.Co. ,Vol. 18, No. 1,January,2008.
[20] M. Okoniewski, E. Okoniewska, "Drude dispersion in ADE FDTD revisited" , Electron.Lett. ,Vol. 42, No. 9,April,2006.
[21] Dennis M. Sullivan, "Z-Transform Theory and the FDTD Method" , IEEE .T.Antennas.Propag. ,Vol. 44,No. 1, pp. 28-34, January,1996.
[22] J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, J. A. Kong, "Power Propagation in Homogeneous Isotropic Frequency-Dispersive Left-Handed Media", Phys. Rev. Lett. ,Vol. 89, No, 25, December,2002.