本論文主要討論三維介電質光子晶體，並希望藉由適當的形狀設計來得到具有手徵性質的光子晶體，以介電質取代金屬來克服一般手徵超材料的缺點。在本論文中設計了三種形狀的光子晶體柱並且以兩個方面討論它們的特性，第一個方面是以平面波展開法來計算光子晶體的頻帶結構，並且檢視本徵模態的場向量是否具有圓偏振的特徵來驗證光子晶體的手徵性。我們也發展了一套方法來對模態分類，最後我們取頻帶結構中的前兩個頻帶來計算出其等效手徵參數。

第二個方面我們將光子晶體做成有限厚度平板並以線偏振平面波正向入射，以RCWA (Rigorous Coupled-Wave Analysis)方法計算此平面波通過平板後的偏振態與原先偏振態的夾角以分析光子晶體作為偏振旋轉器的效果，並且我們也發現特定頻率的光會產生leaky guided resonance，這種共振會增強光子晶體的圓二色性進而使得在該頻率的偏振旋轉角度會特別大。

This thesis concerns the effective chirality of some three-dimensional dielectric photonic crystals consisting of two-dimensional-periodically arranged twisted rods. Through properly choosing the dielectric rod’s shape in a 2D unit cell, we can obtain a photonic crystal with nonzero chirality, overcoming the disadvantage of absorption of the chiral metamaterial consisting of metallic elements.

In our work we design three different types of photonic crystal and discuss their properties in two aspects. The first is calculating their band structures using the plane wave expansion method and check if their field vectors have the feature of circular polarization to confirm their chirality. We also developed a way to classify these propagating modes, and extracted the effective chirality parameters for the first two bands in the band structure.

In another aspect, we consider a photonic crystal slab with finite thickness and being illuminated by a normally incident linearly polarized plane EM wave, and use the RCWA (Rigorous Coupled-Wave Analysis) method to calculate the field distribution. We then compare the polarization states of the outgoing wave with the incident wave and analyze the effectiveness of the photonic crystal slab as a polarization rotator. We also found that at certain frequencies there occur resonances called leaky guided resonances. Such resonances enhance the circular dichroism of the photonic crystal slab, result in large polarization rotation angles.

[1] E.Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics", Phys. Rev. Lett., Vol. 58 2059 (1987)
[2] S. John, "Strong Localization of Photons in Certain Disordered Dielectric Superlattices", Phys. Rev. Lett. 58, 2486 (1987)
[3] P.W. Anderson, "Absence of Diffusion in Certain Random Lattice", Phys. Rev. 109,1492 (1958)
[4] J. D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, Robert D. Meade, "Photonic Crystals: Molding the Flow of Light", Princeton University Press, 2nd edition (2008)
[5] Joshua N. Winn, Yoel Fink, Shanhui Fan, J.D. Joannopoulos, "Omnidirectional reflection from a one-dimensional photonic crystal", Optics Letters, Vol. 23, No.20, 1573 (1998)
[6] J. B. Pendry, "Negative Refraction Makes a Perfect Lens", Phys. Rev. Lett., Vol. 85, No. 18, 3966 (2000)
[7] Weiguo Yang "Negative Refraction Does not make Perfect Lenses" arXiv:1306.2351 (2013)
[8] Pi-Guan Luan "Proper boundary conditions for the imaging problem of a negative-refraction slab lens" WSEAS transcations on electronics, Vol. 1,p236 (2004)
[9] D.R. Smith, Willie J.Padilla, D.C. Vier, S.C. Nemat_Nasser, S. Schultz "Composite Medium with Simultaneously Negative Permeability and Permittivity" Phys. Rev. Lett., Vol. 84, No. 14 (2000)
[10] Zhaowei Liu, Hyesog Lee, Yi Xiong, Cheng Sun and Xiang Zhang "Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects"Science, 315, 1686 (2007)
[11] J. B. Pendry "A chiral route to negative refraction" Science", 306, 1353 (2004)
[12] Benfeng Bai, Yuri Svirko, Jari Turuen, Tuomas Vallius "Optical activity in planar chiral metamaterials: Theoreitical study" Phys. Rev. A, 76 ,0238111 (1958)
[13] Yuqiang Ye, Sailing He "90o polarization rotator using a bilayered chiral metamaterial with giant optical activity" Appl. Phys. Lett., 96, 203501 (2010)
[14] Caiping Xi "Terahertz angle-insensitive 90o polarization rotator using chiral metamaterial" Physica B, 422, 83 (2013)
[15] A. Lahtakia, V. K. Varadan, V. V. Vardan "Time-Harmonic Electromagnetic Fields in Chiral Media (Lecture Notes in Physics )" Springer, 1st edition (1989)
[16] Amnon Yariv, Pochi Yeh "Optical Waves in Crystals" Wiley-Interscience (1983)
[17] Warren N. Herman "Polarization eccentricity of the transverse field for modes in chiral core planar waveguides" J. Opt. Soc. Am. A, Vol. 18,No. 11,2806 (2001)
[18] David Rosenblatt, Avner Sharon, Asher A. Friesem "Resonant Grating Waveguide Structures" IEEE Vol. 33,No. 11,1492 (1997)
[19] 欒丕綱，陳啟昌，光子晶體-從蝴蝶翅膀到奈米光子學，第四章，五南圖書出版股份有限公司 (2010)
[20] D. M. Whittaker, I.S. Culshaw "Scattering-matrix treatment of patterned multilayer photonic structures" Phys. Rev. B, Vol. 60, No. 4,2610 (1999)
[21] Daivd Yuk Kei Ko, J. C. Inkson "Matrix method for tunneling in heterostructures: Resonant tunneling in multilayers systems" Phys. Rev. B, 38, 9945 (1988)
[22] Chao Wu, Hongqiang Li, Zeyong Wei, Xiaotong Yu, C. T. Chan " Theory and Experimental Realization of Negative Refraction in a Metallic Helix Array" Phys. Rev. Lett., 105, 247401 (2010)
[23] Xiangfeng Meng, Benfeng Bai, Petri Karvinen, Kuniaki Konishi, Jari Turunen, Yuri Svirko, Makoto Kuwata-Gonokami "Experimental realization of all-dielectric planar chiral metamaterials with large optical activity in direct transmission" Thin Solid Films, 516, 8745 (2008)
[24] Xiangfeng Meng, Benfeng Bai, Petri Karvinen, Kuniaki Konishi, Jari Turunen, Yuri Svirko, Makoto Kuwata-Gonokami, Anni Lehmuskero "Mechanism of the large polariztion rotation effect in the all-dielectric artificially chiral nanogratings" Optics Express , Vol. 17, No. 2, 688 (2009)
[25] Lin Wu, ZhenYu Yang, Ming Zhao, Yang Yu, ShenXi Li, QianPeng Zhang, XinHua Yuan " Polarization characteristics of the metallicstructure with elliptically helical metamaterials" Optics Express, Vol. 19, No. 18, 17539 (2011)