源自於凝聚態物理中量子霍爾效應(quantum Hall effect)在1980年被Klitzing發現，與Haldane在1988年提出有關於物質的拓樸相變和拓樸相領域的模型，使拓樸絕緣體(topological insulator)引起學界的廣泛注目。而2005年時，Kane與Mele又提出量子自旋霍爾效應(quantum spin Hall effect)，使電子利用自旋的特性，能不用打破時間反演對稱來得到拓樸絕緣體。在2015年時，Wu利用光子晶體內的電場的波函數創造出來的偽自旋，進一步使光子去取代電子的行為而得到拓樸絕緣體。而拓樸絕緣體有個特別的邊緣態，是未來計算速度提升與傳播效率提升很重要的一部分。
此篇論文即是利用FDTD數值模擬法去模擬出利用光子晶體所創造出來的拓樸絕緣體中的邊緣態。首先我們模擬利用介電質材料置於空氣中去創造出六角形蜂窩狀晶格的光子晶體結構，接著將結構變形，並利用Order N方法去計算能帶結構，能帶結構中會產生一般能隙與拓樸能隙，當中會發現能帶反轉的現象，於是我們知道過程中有經過一個拓樸過渡。接著將有一般能隙與拓樸能隙的結構合併並一樣利用Order N方法去計算能帶結構。在此能帶結構的能隙中會有兩條曲線的產生，這兩條被稱為拓樸邊緣態。最後利用不同偽自旋的模態在邊界處激發，並將激發的磁場分量做一些格子點上的修正，可以使電磁波在邊界只朝單一方向傳播的邊緣態，且此邊緣態受拓樸保護，因此可以不受晶體缺陷與障礙物的影響。

With the discovery of the quantum Hall effect, and the model of the topological phase transition proposed by Haldane, the topological insulator has aroused attention in academia. After that, Kane and Mele proposed the quantum spin Hall effect which use the spin properties of the electrons to obtain topological insulator without breaking the time reversal symmetry. Afterward, Wu used the wave function of the electric field in photonic crystals to create the pseudospin. Hence, we enable to use photonics to replace electrons to obtain a photonic crystals topological insulator. Topological insulator has a special edge states which is an important part to speed up the calculation speed and elevated the efficiency of the propagation.
In this thesis, we use FDTD to simulate the topological insulator by photonic crystals. First, we use dielectric materials to create a hexagonal honeycomb lattice. Then deform the structure and use Order N to calculate the band structure respectively. The band structure one is a trivial band gap and another is a topological band gap. We noticed it will have the band inversion, so we know that there is a topological transition in the process. Finial, we use different pseudospin mode to excited, and the excited magnetic field components need to modify on the grid, after then, we can find the one-way propagation at the boundary. And the edge states are protected by topological, so it can immunity the defect.

[1] Hasan et al, “Colloquium: Topological Insulators,” Rev. Mod. Phys. 82, 3045 (2010)
[2] Mei et al, “Pseudo-Time-Reversal Symmetry and Topological Edge States in Two-Dimensional Acoustic Crystals,” Scientific Reports volume 6, Article number: 32752 (2016)
[3] Long-Hua Wu, Xiao Hu, “Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material,” Phys. Rev. Lett. 114, 223901 (2015)
[4] Qiu et al, “Pseudospin Dependent One-Way Transmission in Graphene-Based Topological Plasmonic Crystals,” Nanoscale Research Letters 13:113 (2018)
[5] Barik et al, “Two-Dimensionally Confined Topological Edge States in Photonic Crystals,” New Journal of Physics 18, 113013 (2016)
[6] Chen et al, “Self-Ordering Induces Multiple Topological Transitions for In-Plane Bulk Waves in Solid Phononic Crystals,” Physical review B 98, 014302 (2018)
[7] Zhang et al., “Topological Creation of Acoustic Pseudospin Multipoles in a Flow-Free Symmetry-Broken Metamaterial Lattice,” Phys. Rev. Lett. 118, 084303 (2017)
[8] Wang et al, “Observation of Unidirectional Backscattering-Immune Topological Electromagnetic States,” nature, 461, pages772–775 (2009)
[9] Khanikaev et al, “Photonic Topological Insulators,” nature materials, 12, pages233–239 (2013)
[10] Hafezi et al, “Imaging Topological Edge States in Silicon Photonics,” Nature Photonics volume 7, pages1001–1005 (2013)
[11] Haldane et al, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100, 013904 (2008)
[12] Chen et al, “Multiple Topological Phase Transitions in a Gyromagnetic Photonic Crystal,” Phys. Rev. A 95, 043827 (2017)
[13] John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattice,” Phys. Rev. Lett., 58, p.2486(1987)
[14] Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett.,58, p.2059(1987)
[15] J. D. Joannopoulos, R. D. Meade, J. N. Winn, “Photonic Crystals: Molding the Flow of Light,” Princeton University Press, 41, William Street, Princeton, New Jersey 08540, p.6 (1995)
[16] Igor A. Sukhoivanov, Igor V. Guryev, “Photonic Crystals,” Springer Series in Optical Sciences, Chapter4.3, p.45
[17] J.D. Joannopoulos, R.D. Meade, J.N. Winn, “Photonic Crystals: Molding the Flow of Light,” 2nd, Princeton University Press, Princeton (1995)
[18] Charles Kittel, “Introduction to Solid State Physics.” John Wiley & Sons, Inc.,7th (1996)
[19] K. Laasonen and R. M. Nieminen, “Molecular Dynamics Using The Tight-Binding Approximation,” Journal of Physics: Condensed Matter 2, 1509 (1990)
[20] Igor A. Sukhoivanov, Igor V. Guryev, “Photonic Crystals,” Springer Series in Optical Sciences, Chapter4.6, p.50
[21] Pickover, Clifford A., “The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology,” Thunder's Mouth Press (2006)
[22] L. S. Pontryagin, Arlen Brown, P. S. V. Naidu, “Topological Groups,” Routledge, 3rd Edition, 14 December (2018)
[23] David J. Griffiths, “Introduction to Electrodynamics,” Pearson Education Limited, 4th, 2014, Chapter5, p.214
[24] E. H. Hall, “On a New Action of the Magnet on Electric Currents,” American Journal of Mathematics vol. 2, p.287-292 (1879)
[25] Klaus von Klitzing, “The quantized Hall effect,” Rev. Mod. Phys. 58, 519 (1986)
[26] M. V. Berry, “Quantal Phase Factors Accompanying Adiabatic Changes,” Proc. R. Soc. A 392, 45–57 (1984)
[27] David J. Griffiths, “Introduction to Electrodynamics,” Pearson Education Limited, 4th, Chapter1, p.34 (2014)
[28] Tomoki Ozawa et al., “Topological photonics,” Reviews Of Modern Physics, Volume 91, January–March (2019)
[29] Haldane, “Model for a Quantum Hall Effect Without Landau Levels: Condensed-Matter Realization of the “ Parity Anomaly”,” Phys. Rev. Lett., 61 (1988)
[30] C. L. Kane and E. J. Mele, “Quantum Spin Hall Effect in Graphene,” Phys. Rev. Lett. 95, 226801 (2005)
[31] A. Taflove, S. Hagnes, “Computational Electrodynamics：The Finite Difference Time Domain Method Artech,” Boston, MA (2000)
[32] Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Transactions on antennas and propagation 14(3), 302 (1966)
[33] Berenger, Jean Pierre, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” Journal of computational physics 114(2), 185 (1994)
[34] Roden, J. Alan, and Stephen D. Gedney, “Convolutional PML (CPML): An Efficient FDTD Implementation of the CFS-PML for Arbitrary Media,” Microwave and optical technology letters, 27(5), 334-338 (2000)