在這篇論文中，我們探討手性材料的第一條與第二條能帶，並藉由推導介面條件得到波方程在週期結構的解析解。然而，由於係數矩陣的行列式過於複雜，我們無法直接從中得到手性材料頻率與波數的確切關係，因此我們提出了一個數值方法去克服這個困難。我們運用了有限差分法來解特徵值問題，得到頻率並代回係數矩陣的行列式，最後檢驗其誤差收斂性。從我們的數值結果顯示，我們得到第一條與第二條能帶的特徵值的二階收斂。除此之外，我們發現當在z方向的波數等於零的情況下，會有四階收斂現象發生。

In this thesis, we discuss the first and second band of chiral medium and derivate the interface conditions for obtaining electromagnetic plane wave solutions. Owing to the complicated determinant of coefficient matrix, we cannot obtain the explicit relation of frequency and wavenumber of chiral medium directly. Thus we propose the numerical method to overcome this difficulty. Furthermore, we apply finite difference method to solve the eigenvalue problem and substitute the frequency into the determinant of coefficient matrix. And we investigate the rate of convergences. From the numerical results, we obtain the second-order convergences of the first and second band. Besides, we discover forth-order convergences when wavenumber of z component is equal to zero.

[1] 張高德and 欒丕綱. 光子晶體中的波傳播. 物理雙月刊, Volume 28(5):844–850,2006.
[2] 欒丕綱et al. 光子晶體. 五南圖書出版股份有限公司, 2005.
[3] Biswajit Banerjee. An Introduction to Metamaterials and Waves in Composites.CRC Press, 2011.
[4] S Bassiri, C.H Papas, and N Engheta. Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab. JOSA A, Volume 5(9):1450–1459,1988.
[5] Thomas Rylander Bondeson, Anders and Pär Ingelström. Computational electromagnetics,volume Volume 51. Springer, 2005.
[6] I Chern, Yu-Chen Shu, et al. A coupling interface method for elliptic interface problems. Journal of Computational Physics, Volume 225(2):2138–2174, 2007.
[7] Ruey-Lin Chern. Wave propagation in chiral media: composite fresnel equations.Journal of Optics, Volume 15(7):075702, 2013.
[8] Ruey-Lin Chern and Po-Han Chang. Negative refraction and backward wave in chiral mediums: illustrations of gaussian beams. Journal of Applied Physics, Volume113(15):153504, 2013.
[9] Nader Engheta and Dwight L Jaggard. Electromagnetic chirality and its applications. Antennas and Propagation Society Newsletter, IEEE, Volume 30(5):6–12,1988.
[10] Ronald P. Fedkiw. The Ghost Fluid Method for Numerical Treatment of Discontinuities and Interfaces. Springer US, 2001.
[11] John D Joannopoulos, Steven G Johnson, Joshua N Winn, and Robert D Meade. Photonic crystals: molding the flow of light. Princeton university press, 2011.
[12] John Lekner. Properties of a chiral slab waveguide. Pure and Applied Optics: Journal of the European Optical Society Part A, Volume 6(3):373, 1997.
[13] Molina-Cuberos G.J Núñez M.J García-Collado A.J Margineda, J and E Martín. Electromagnetic characterization of chiral media. ISBN, pages 980–953, 2012.
[14] Dennis W Prather. Photonic Crystals, Theory, Applications and Fabrication, volume Volume 68. John Wiley & Sons, 2009.
[15] Bingnan Wang, Jiangfeng Zhou, Thomas Koschny, Maria Kafesaki, and Costas M Soukoulis. Chiral metamaterials: simulations and experiments. Journal of Optics A: Pure and Applied Optics, Volume 11(11):114003, 2009.
[16] N.G. Warren. Excel HSC Physics. Excel HSC. Pascal Press, 2003.
[17] M Yokota and Y Yamanaka. Dispersion relation and field distribution for a chiral slab waveguide. International Journal of Microwave And Optical Technology,Volume 1(1), 2006.