臺灣博碩士論文加值系統

本文以有限元素法來探討方形晶格光子晶體加入磁光材料後，是否具有由拓樸性質所產生單一行進表面態，藉由分析此結構之能帶結構和在不同材質介面產生之能帶，我們確定其在頻率大約在512太赫茲至534太赫茲時，具有一個單一行進表面態，除此之外，在頻率為516太赫茲時，我們使用線波源激發表面態和用理想導體作為阻礙展示表面態不受阻礙繞過雜質的特性。最後我們分別利用改變光子晶體柱中的介電常數和半徑來在不同的特徵模態上產生能帶對調，並利用貝里曲率和陳數字的計算，發現能帶對調會造成貝里曲率在各個被翻轉的能帶上改變，而光學量子反常霍爾態則沒有發生改變。

In this thesis, we use the finite element method to investigate whether there is a chiral edge state protected by the property of topology after adding a magneto-optical medium into a photonic crystal. We analyze the band structure of this system. We discover that there will be a chiral edge state during the frequency is set at the range from 512 THz to 534 THz. Furthermore, we use a line source as a stimulation of the chiral edge state and a perfect electric conductor as an obstacle to demonstrate the property of circumventing the obstacle at 516 THz. We also change the square-lattice rod’s permittivity and radius to generate a band inversion in different eigenmode respectively. We find that the Berry curvature will change under the band inversion by calculating the Chern number and the Berry curvature distribution. However, the photonic quantum anomalous Hall phase for our system remains the same.

口試委員會審定書 #
誌謝 i
中文摘要 ii
ABSTRACT iii
CONTENTS iv
LIST OF TABLES vi
Chapter 1 Introduction 1
1.1 Photonic Crystal 1
1.2 Photonic Topological Insulator 6
Chapter 2 The Simulation Method 12
2.1 Finite Element Method 12
2.2 COMSOL Package 16
2.3 Practices on FEM Calculation 19
2.3.1 Band Structure 19
2.3.2 Edge State 20
2.3.3 Chiral Edge State 21
Chapter 3 Calculation of the Chern Number 24
3.1 Introduction to Berry Phase 24
3.2 Adiabatic Evolution 24
3.3 Berry Phase in Bloch Bands 26
3.4 Calculation of the Chern Number 27
Chapter 4 Chiral Edge State in Magneto-Optical Square-Lattice Photonic Crystals 29
4.1 Calculation of the Chern number 29
4.2 Edge State 35
4.3 Chiral Edge State 36
Chapter 5 Manipulation of Photonic Quantum Anomalous Hall Phase in a Square Lattice of Magneto-Optical Rods 39
5.1 Introduction 39
5.1.1 Motivations 42
5.2 Results and Discussion 44
5.2.1 Band Inversion due to the Change of Rod’s Permittivity 48
5.2.2 Band Inversion due to the Change of Rod’s Radius 55
5.3 Summary 60
Chapter 6 Conclusions 61
References 62

[1]E. Yablonovitch. "Inhibited Spontaneous Emission in Solid-State Physics and Electronics." Physical Review Letters 58, 2059(1987).
[2]S. John. "Strong localization of photons in certain disordered dielectric super lattices." Physical Review Letters 58, 2486 (1987).
[3]E. Yablonovitch, T. J. Gmitter, and K. M. Leung. "Photonic band structure: The face-centered-cubic case employing non-spherical atoms." Physical Review Letters 67, 2295(1991)
[4]J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade. Photonic Crystals Molding the Flow of Light 2rd Edition(Princeton University Press,2011)
[5]S. Raghu and F. D. M. Haldane. "Analogs of quantum-Hall-effect edge states in photonic crystals." Physical Review A 78,033834 (2008)
[6] F. D. M. Haldane and S. Raghu. "Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry." Physical Review Letters 100, 013904(2008)
[7]Z. Wang, Y. D. Chong, J. D. Joannopoulos and M. Soljačić. "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal." Physical Review Letters 100, 013905(2008).
[8]Z. Wang, Y. D. Chong, J. D. Joannopoulos and M. Soljačić. "Observation of unidirectional backscattering-immune topological electromagnetic states." Nature 461, 7265(2009).
[9]T. Ochiai, and M. Onoda. "Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states." Physical Review B 80, 155103 (2009).
[10]C. Day. "Quantum spin Hall effect shows up in a quantum well insulator, just as predicted," Physics Today 61, 19 (2008).
[11]A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets. "Photonic topological insulators." Nature Materials 12,233 (2013).
[12]R. L. Courant, "Variational Methods for the Solution of Problems of Equilibrium and Vibration," Bulletin of the American Mathematical Society 49, 1 (1943).
[13]https://en.wikipedia.org/wiki/Topological_insulator
[14]C. J. Reddy, M. D. Deshpande, C. R. Cockrell, and F. B. Beck, "Finite Element Method for Eigenvalue Problems in Electromagnetics," NASA Technical Paper 3485 (1994).
[15]P. Silvester. “Construction of triangular finite element universal matrices”, International Journal for Numerical Methods in Engineering, Vol. 12, 237. (1978)
[16]Society for Industrial and Applied Mathematics. (2015)
[17]https://www.comsol.com/company
[18]http://en.wikipedia.org/wiki/COMSOL_Multiphysics3
[19]W. Y. He and C. T. Chan. "The emergence of Dirac points in photonic crystals with mirror symmetry." Scientific Reports 5, 8186(2015).
[20]M. V. Berry. "Quantal Phase Factors Accompanying Adiabatic Changes." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 392, 1802 (1984).
[21]Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom. “Observation of the Spin Hall Effect in Semiconductors”, Science 306, 1910(2004).
[22]A. Messiah, Quantum Mechanics, Vol. 2. (North-Holland Publishing company Amsterdam, 1962)
[23]T. Fukui, Y. Hatsugai and H. Suzuki. "Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductance." Journal of the Physical Society of Japan 74, 1674 (2005).
[24]S. A. Skirlo, L. Lu and M. Soljačić. "Multimode One-Way Waveguides of Large Chern Numbers." Physical Review Letters 113,113904 (2014).
[25]X. Huang, Y. Lai, Z. H. Hang, H. Zheng and C. T. Chan. "Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials." Nature Materials 10, 582(2011).
[26]S. A. Skirlo, L. Lu, Y. Igarashi, Q. Yan, J. D. Joannopoulos and M. Soljacic. "Experimental Observation of Large Chern Numbers in Photonic Crystals." Physical Review Letters 115,253901 (2015).