本文著筆研究於超潁材料所組成的一維光子晶體，此光子晶體所產生的能隙現象在本文有做深入的探討與研究，我們所謂的超穎材料是指材料參數中介電常數和導磁常數之其一常數為負值，而另一個常數為正值的材料，此單一材料參數為負值所組成的光子晶體我們簡稱SNG，接著我們考慮材料的厚度比例對於能隙的邊界和禁帶的頻寬所產生之影響，我們簡單的結合左手傳輸線理論，利用近似的方式，有效的利用厚度變化來描述能隙邊界與寬度變化的趨勢，並且討論加入損耗後所造成的影響。
除此之外，我們也利用阻抗匹配的理論去討論不同角度入射對於能隙的寬度所產生的影響，並做理論上的分析，然而一樣是用單一負值的材料參數所組成的光子晶體，我們討論這兩種極化模態的穿透光譜，當然主要的討論方向為不同的角度和不同的極化模態所產生的能隙禁帶大小變化，接著我們成功的利用(k2/k1)參數去討論厚度和角度的變化與能隙的關係，在此篇論文中我們有清楚的定義，此參數是利用不同角度在阻抗匹配狀態下所計算出的波向量比例關係，利用這個比例關係我們可以清楚的解釋在不同模態下，角度改變造成光子晶體能隙的禁帶大小跟著改變的傾向，而這個比例關係也可以用於解釋厚度變化所導致禁帶大小和能隙邊界的變化情形，此研究結果對於研究單一負值的材料所組成的光子晶體具有相當幫助。

In this thesis, the thickness-dependent photonic bandgap for a one-dimensional photonic crystal consisting of two different single-negative materials is theoretically investigated. The two single-negative (SNG) materials include one with a single-negative permittivity and the other having a single-negative permeability. It is found that the size of the bandgap and the positions of the bandedges are strongly dependent on the thickness ratio of the two constituent SNG layers. Then, we also consider about in loss case effective for bandgap. Moreover, the angular dependence of the photonic band structure of a single-negative one-dimensional photonic crystal is theoretically investigated. To use the same photonic crystal structure, we discussed two propagation mode, which are TE mode and TM mode, for different incident angle. We use the concept of impedance matching to discuss the bandgap shift, thus the gap size with incident angle had been described clearly. Additionally, the characteristic frequency determined by the condition of impedance match is closely related to the center frequency of the bandgap in TE mode and TM mode. We used the parameter by ratio of the wavenumber on two medium, which is the wavenumber ratio of the two materials, to successfully explain angle varition and thickness-dependent bandgap and the bandedges for two polarization modes.

Reference

[1] N. Ouchani, D. Bria, A. Nougaouia, B. Djafari-Rouhani, “Photonic band structure and omnidirectional band gap in anisotropic superlattice,” Solar Energy Materials & Solar Cells 90 (2006) 1445–1457
[2] T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals”, Phys. Lett. A 373, 169-172 (2008).
[3] S. Wang, C. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer”, Phys. Lett. A 351, 391-397 (2006).
[4] Haitao Jiang,1,* Hong Chen,1 and Shiyao Zhu2, “Localized gap-edge fields of one-dimensional photonic crystals with an -negative and a -negative defect,” Physical Review E 73, 046601 (2006)
[5] Juan A. Monsoriu, “Interaction between non-Bragg band gaps in 1D metamaterial photonic crystals,” Optics Express 12958, Vol. 14, No. 26, 25 December (2006)
[6] Munazza Zulfiqar Ali , Tariq Abdullah, “Properties of the angular gap in a one-dimensional photonic band gap structure containing single negative materials,” Physics Letters A 372 (2008) 1695–1700
[7] Kyoung-Youm Kim, “Properties of photon tunneling through single-negative materials, “Optics Letters, Vol. 30, No. 4, February 15, 2005
[8] P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials”, Physical Review B 77, 115332 (2008)
[9] Liwei Zhang, Yewen Zhang, ,* Yaping Yang, Hongqiang Li, Hong Chen, and Shiyao Zhu, “Experimental observation of Rabi splitting in effective near-zero-index media in the microwave regime”, Physical Review E 78, 035601® (2008)
[10] Haitao Jiang, Hong Chen, Hongqiang Li, and Yewen Zhang”, Compact high-Q filters based on one-dimensional photonic crystals containing single-negative materials”, Journal Of Applied Physics 98, 013101 (2005)
[11] Akhlesh Lakhtakia, Clifford M. Krowne, “Restricted equivalence of paired epsilon–negative and mu–negative layers to a negative phase–velocity material (alias left–handed material),” Optik 114, No. 7 (2003) 305–307
[12] Liwei Zhang, Yewen Zhang*, Li He, Hongqing Li, and Hong Chen, “Experimental study of photonic crystals consisting of -negative and -negative materials”, Physical Review E 74, 056615 (2006)
[13] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart. “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter, vol. 10, pp. 4785–4809,1998
[14] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart. “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Micr. Theory. Tech., vol. 47, no. 11, pp. 2075–1084, Nov. (1999).
[15] Atsushi Sanada, Member, IEEE, Christophe Caloz, Member, IEEE, and Tatsuo Itoh, Life Fellow, IEEE, “Characteristics of the Composite Right/Left-Handed Transmission Lines,” IEEE Microwave And Wireless Components Letters, VOL. 14, NO. 2, FEBRUARY (2004)
[16] Christophe Caloz, Member, IEEE, Atsushi Sanada, Member, IEEE, and Tatsuo Itoh, Fellow, IEEE, “A Novel Composite Right-/Left-Handed Coupled-Line Directional Coupler With Arbitrary Coupling Level and Broad Bandwidth,” IEEE Transactions On Microwave Theory And Techniques, VOL. 52, NO. 3, March (2004)
[17] HU Xin and ZHANG Pu, “A novel dual-band balun based on the dual structure of composite right/left handed transmission line,” IEEE, (2006)
[18] C. Caloz and T. Itoh, Electromagnetic Metamaterials (John Wiley & Sons, New Jersey, (2006).
[19] Dan Kuylenstierna, Student Member, IEEE, Andrei Vorobiev, Peter Linnér, Senior Member, IEEE, and Spartak Gevorgian, Senior Member, IEEE, “Composite Right/Left Handed Transmission Line Phase Shifter Using Ferroelectric Varactors,” IEEE Microwave And Wireless Components Letters, Vol. 16, No. 4, April (2006)
[20] J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmons in metallic mesostructures”, Phys. Rev. Lett. 76, 4773-4777 (1996).
[21] Y. Fang, and S. He, “Transparent structure consisting of metamaterial layers and matching layers”, Phys. Rev. A 78, 023813 (2008).
[22] Natalia M. Litchinitser,* Andrei I. Maimistov, Ildar R. Gabitov, Roald Z. Sagdeev, and Vladimir M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Optics Letters, Vol. 33, No. 20, October 15, (2008)
[23] Yuanjiang Xiang, Xiaoyu Dai, and Shuangchun Wen* “Omnidirectional gaps of one-dimensional photonic crystals containing indefinite metamaterials,” J. Opt. Soc. Am. B, Vol. 24, No. 9, September (2007)
[24] M. D Pozar. Microwave Engineering, Third Edition, John Wiley, (2005).
[25] Lihong Shi, Shuming Wang, Chaojun Tang, Lei Gao, “Omnidirectional surface guided modes from one-dimensional photonic crystal formed by single-negative materials,” Journal of Magnetism and Magnetic Materials 311 (2007) 609–613
[26] Yuanjiang Xiang, Xiaoyu Dai, Shuangchun Wen,* and Dianyuan Fan, “Omnidirectional and multiple-channeled highquality filters of photonic heterostructures containing single-negative materials,” J. Opt. Soc. Am. A, Vol. 24, No. 10, October (2007)
[27] Y. H. Chen, J. W. Dong, and H. Z. Wang, “Omnidirectional resonance modes in photonic crystal heterostructures containing single-negative materials,” J. Opt. Soc. Am. B, Vol. 23, No. 10, October (2006)
[28] Li-Gang Wang, Hong Chen,and Shi-Yao Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Physical Review B 70, 245102 (2004)
[29] L. Gao, C. J. Tang, S. M. Wang, “Photonic band gap from a stack of single-negative materials”, J. Magn. Magn. Mater. 301, 371-377 (2006).
[30] D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity “, Phys. Rev. Lett. 84, 4184-4187 (2000).
[31] A. Alu, N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency”, IEEE Trans. Antennas. Prop. 51, 2558-2571 (2003).
[32] H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials”, Phys. Rev. E 69, 066607 (2004).
[33] L. G. Wang, H. Chen, S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials “, Phys. Rev. B 70, 245102 (2004).
[34] R. Ruppin, “Surface polaritons of a left-handed material slab”, J. Phys.: Condens. Matter 13, 1811-1818 (2001).
[35] J. R. Canto, S. A. Matos, C. R. Paiva, A. M. Barbosa, “Effect of losses in a layered structure containing DPS and DNG media”, PIERS Online. 4, 546-555 (2008).
[36] Pochi Yeh, Optical Wave in Layered Media (John Wiley & Sons, New York, 1998)
[37] YI JIN and SAILING HE, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” Journal of Nonlinear Optical Physics & Materials, Vol. 17, No. 3 (2008) 349–355
[38] Sophocles J. Orfanidis, “Electromagnetic Waves and Antennas,”