負折射現象最早於1958年由蘇聯物理學家Veselago提出。Veselago提出了左手材料的概念，顛覆了物理學的傳統知識。左手材料從此成為近十餘年來物理學界熱門的研究話題。最近(2013年)，Fedorov 和 Nakajima研究得出使用非左手材料如金屬材質也可得到負折射現象，這篇研究對於不均勻波的敘述以及材料結構有更深一層的探討。本篇論文將金屬取代為半導體，使用InSb作為半導體材料。我們參考Sánchez 和 Halevi 於2003年發表的研究報告，當中對於InSb材料參數講述詳細。
基於先前的研究成果，我們根據真實實驗數據進行理論數據模擬，成功找出發生負折射現象的最佳條件。並且，對於參數間的相互影響也進行了詳盡的分析。半導體性質受三種參數影響：溫度、參雜濃度、入射波頻率。在此，利用Snell's law進行相關推導，找出折射角與材料參數之間的關係，此關係式讓我們判斷負折射現象的發生。接著，我們將半導體材料以及介電材料進行週期性排列，形成一維光子晶體。光子晶體有獨特的特性，除了半導體參數影響外，厚度比例也是影響的因素。我們設定光波沿x-z正向傳播，使用群速度作為判斷負折射現象發生條件。當x方向的群速度沿相反方向行進，可判斷負折射現象發生。
本論文架構如下：第一章講述InSb基本參數、不均勻波理論以及光子晶體概論等基本介紹；第二章講述本論文重點理論-負折射現象的基本概念；第三章推論出在不均勻介質中，負折射理論的關係式；第四章，進入本篇論文主要研究，分別闡述理論以及展現數值分析結果，詳盡的描述材料參數的相互關係；最後，在第五章，我們從半導體光子晶體基本理論出發，完整推導在光子晶體中，群速度的表達式，並且改變模擬的操作變因，多面向地分析負折射發生條件以及負折射發生之明顯程度。

This thesis studies the phenomenon of negative refraction (NR) in semiconductor (InSb)-dielectric layered structures. Using the complex wave vector, we analyze the propagation direction of light energy in semiconductor-dielectric interface, and show that the direction of P-polarized light is on the same side of the incident light. In semiconductor material, propagation of an electromagnetic wave is inhomogeneous described in Chapter 3. Negative refraction can happen in semiconductor-dielectric interface under certain appropriate conditions, which will be discussed in detail in Chapter 4.
In second part, we have theoretically explored the NR in one-dimensional semiconductor-dielectric photonic crystals (SDPC) consisting of n-type InSb and dielectric materials. By using a theoretical analysis in group velocity, we can investigate phenomenon of NR in SDPC. It is also shown that the NR is closely related to the thicknesses of constituents of SDPC. In addition, we also investigate the tunable feature in NR by varying the temperature and the doping concentration. This study will be presented in Chapter 5.

Keywords: Negative Refraction, Semiconductor, Photonic Crystals, Group Velocity, Inhomogeneous Waves.

[1] V. G. Veselago, The electrodynamics of substances with simultaneously negative values of permittivity and permeability, Sov. Phys. USPEKHI 10, 509 (1968).
[2] R. A. Shelby, D. R. Smith, S. Schultz, Experimental verification of a negative index of refraction, Science 292, 77 (2001).
[3] J. L. Garcia-Pomar, M. Nieto-Vesperinas, Transmission study of prisms and slabsof lossy negative index media, Opt. Express 12 (10), 2081 (2004).
[4] M. Wegener, G. Dolling, S. Linden, Backward waves moving forward, Nat. Mater. 6, 475 (2007).
[5] W. Gu, Y.-H. Wu, Y.-R. Chen, Z.-H. Dai, W.-X. Zhou, Y.-X. Zheng, L.-Y. Chen, Study on the properties of light propagation at the metal interface, J. Infrared Mil-limeter Waves 28 (1), 31 (2009).
[6] F. Zhang, S.M. Feng, Y. H. Shan, Research of negative refractive direction of P-light in metal, Optik 125, 338 (2013).
[7] T. Xu, A. Agrawal, M. Abashin, K. J. Chau, H. J. Lezec, All-angle negative refraction and active flat lensing of ultraviolet light, Nature 497, 470 (2013).
[8] W. Cai, V. Shalaev, Optical Metamaterials (New York: Springer) (2010).
[9] H. C. Chen, Theory of Electromagnetic Waves (New York: McGraw-Hill) (1983).
[10] V. Yu Fedorov, T. Nakajima, Negative refraction of inhomogeneous waves in lossy isotropic metamaterials, arXiv:1305.6393v3 [physics.optics] (2013).
[11] R. A. Shelby, Microwave experiments with left-handed materials, Ph. D Dissertation of UCSD, 4058 (2002).
[12] 崔萬照、馬偉、邱樂德、張洪太。 《電磁超介質及其應用》 。北京：國防工業出版社，2008。
[13] P. Halevi, F. Ramos-Mendieta, Tunable Photonic Crystals with Semiconducting Constituents, Phys. Rev. Lett. 85, 1875 (2000).
[14] A. S. Sa´nchez, P. Halevi, Simulation of tuning of one-dimensional photonic crystals in the presence of free electrons and holes, J. Appl. Phys. 94, 797 (2003).
[15] M. Notomi, Theory of light propagation in strongly modulated photonic crystals: Refraction like behavior in the vicinity of the photonic band gap, Phys. Rev. B 62, 10696 (2000).
[16] C. Luo, S. G. Johnson, J. D. Joannopoulos, All-angle negative refraction without negative effective index, Phys. Rev. B 65, 201104 (2002).
[17] P. M. Valanju, R. M. Walser, A. P. Valanju, Wave refraction in negative-Index media: Always positive and very inhomogeneous, Phys. Rev. Lett. 85, 3966 (2000).
[18] 欒丕綱。 《光子晶體-從蝴蝶翅膀到奈米光子學》 。台北：五南出版社，2005。
[19] M. Gerken, D. A .B. Miller, Multilayer thin-film structures with high spatial dispersion, Appl. Opt. 42, 1330 (2003).
[20] R. Srivastava, K. B. Thapa, S. Pati, S.P. Ojha, Negative refraction in 1D photonic crystals, Solid State Commun. 147, 157 (2008).
[21] Y.Y. Chen, Z.M. Huang, J.L. Shi, C.F. Li, Q. Wang, Frequency bands of negativerefraction in finite one-dimensional photonic crystals, Chinese Phys. 16, 173 (2007).
[22] P. V. Parimi, W.T. Lu, P. Vodo, J. Sokolo, J.S. Derov, S. Sridhar, Negative refractionand left-hand electromagnetism in microwave photonic crystals, Phys. Rev. Lett. 92, 127401-1 (2004).
[23] R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, C. M. Soukoulis, Negative refraction and superlens behavior in a two-dimensional photonic crystal, Phys. Rev. B 71, 085106-1 (2005).
[24] C. Luo, S. G. Johnson, J. D. Joannopoulos, J. B. Pendry, Negative refraction without negative index in metallic photonic crystals, Opt. Exp. 11, 746 (2003).
[25] B. Guo, M.-Q. Xie, Negative refraction in one-dimensional plasma photonic crystals, Optik - Int. J. Light Electron Opt. (2014) .
[26] C. Liu, J. Ye, Y. Zhang, Thermally tunable THz filter made of semiconductors, Opt. Comm. 283, 865 (2010).
[27] X. Dai, Y. Xiang, S. Wen, H. He, Thermally tunable and omnidirectional terahertz photonic bandgap in the one-dimensional photonic crystals containing semiconductor InSb, J. Appl. Phys. 109, 053104 (2011)
[28] T.-W. Chang, J.-J. Wu, C.-J. Wu, Complex Photonic Band Structures In A Photonic Crystal Containing Lossy Semiconductor InSb, Prog. Electromagn. Res. 131, 153 (2012).
[29] C.-C. Liu, C.-J. Wu, Transmission Properties in a Finite Extrinsic Semiconductor Photonic Crystal, Opt. Rev. to be appeared (2014).
[30] 溫熙森。 《光子/聲子晶體理論與技術》 。北京：科學出版社，2006。
[31] H. M. Barlow, A. L. Cullen, Surface Waves, Proc. IEE, Pt. III. 100, 329 (1953).
[32] S. A. Schelkunoff, Anatomy of Surface Waves, IRE Trans. Antennas Propagat, AP-7, 133 (1959).
[33] T. Tamir, F. Y. Koo, Varieties of Leaky Waves and Their Excitation Along Multilayer Structures, IEEE J. Quant. Electr. 22, 544 (1986).
[34] J. Zenneck, Uber die Fortpflanzung ebener electromagnetische Wellen langs einer ebenen Leiterflache und ihre Beziehung zur drachtlosen Telegraphie, Annalen der Physik. 23, 846 (1907).
[35] S. J. Orfanidis, Electromagnetic Waves and Antennas, ECE Department Rutgers University. (2008).
[36] P. Halevi, A. Mendoza-Herndndez, Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium, J. Opt. 71, 1238 (1981).
[37] V. Yu. Fedorov, T. Nakajima, All-angle collimation of incident light in µ-near-zero metamaterials, Opt. Express. 21, 27789 (2013)