近年來拓樸學的概念引入物理學門，其中在凝態物理及光子學中的對於能帶理論及晶體對稱性上的理解幫助甚多，進而形成一研究學門，即拓樸光子學。透過表面電漿子的色散關係可以得知，表面電漿子的波向量可以在比光子波向量大的前提以相同的頻率存在(波長與向量成反比)，因此相較光子，其表面電漿子則具有突破繞射極限的淺能，因此本篇論文有別於傳統光子晶體以光子作為媒介，改用表面電漿子在近場傳播並與表面電漿子晶體作用，而形成能帶。在這種電漿子能帶中可以觀察許多有趣的現象，如:拉比分裂(Rabi splitting)、連續譜束縛態(Bound states in the continuum)和偏振渦漩(Polarization vortex)。其中我們對偏振渦漩與表面電漿子晶體中對稱性的連結較感興趣，進而對於拓樸電荷(Topological charges)的形成與其中的拓樸不變性進行理論模擬計算以及實驗驗證。
文中，首先我們將會介紹表面電漿極化子的背景知識，並且給出品質因子的重要性，其可以讓我們在有興趣的波段選用正確的電漿子材料。從計算結果可以得出，在可見光範圍下，銀為光學損耗最低、表面電漿子傳播長度最長的金屬，因此我們選用銀為此項研究的主要材料。接著我們使用熱蒸鍍的方式在藍寶石基板上成長高品質金屬銀磊晶薄膜，後續利用聚焦離子束蝕刻出具有高保真性的表面電漿子晶體。而在理論模擬方面，利用有限時域差分法(Finite-difference time-domain)針對不同對稱性的表面電漿子晶體進行模擬，並取得該晶體中的本徵模態及其場強分布，從模擬結果可得知本徵模態與晶體的旋轉對稱性相關，並且在不同對稱性的週期結構皆會在高對稱點(Γ)上具有連續譜束縛態的存在，倘若我們對高對稱點(Γ)周圍的電場做空間平均積分，其圍繞高對稱點(Γ)的封閉路徑上，即可得連續變化的偏振態，我們稱其為偏振渦旋，並且在封閉路徑上的偏振態總體變化的結果皆以2π×q形式呈現，其中q為整數，因此可稱其為拓樸電荷(q)。這種拓樸電荷的性質是由晶體的旋轉對稱性決定的，其不會因為晶胞的不同或者製程上的缺陷而改變，這也是正所謂的對稱保護。
接著，我們利用實驗室自行搭建的微觀角度分辨顯微鏡量測表面電漿子晶體中的能帶關係，並且使用MATLAB進行數據分析。從實驗結果可得出，連續譜束縛態皆存在表面電漿子晶體中，並對其做偏振分析後得出的拓樸電荷數皆與模擬相吻合。因此我們得以驗證，旋轉對稱性決定了拓樸電荷的產生。本篇論文節錄了對於拓樸電荷與對稱性的模擬及實驗分析，希望對於對拓樸光子學有興趣的讀者有所啟發。

Recently, the concepts of topology have been introduced into the physics discipline, which brought many benefits to comprehending band theory and crystal symmetry in condensed matter physics and photonics, and then formed a discipline, that is, topological photonics.Through the dispersion relations of surface plasmons, we can tell that under the same frequency, the wave vector of surface plasmons is larger than the photon wave vector (wavelength is inversely proportional to the wave vector) , since surface plasmons have the potential to break through the diffraction limit. Therefore, in this thesis is different from the traditional research, which uses photons as the propagating medium for photonic crystal, and instead uses surface plasmons to propagate in the near field and interact with surface plasmonic crystals to form energy bands. Many interesting phenomena can be observed in this plasmon band, such as Rabi splitting, bound states in the continuum, and polarization vortex. Among them, we are more interested in the connection between polarization vortex and rotational symmetry in surface plasmonic crystals, and then theoretical simulations and experimental verifications of the topological charges formation and the topological invariance kicks in. In this article, we will first introduce the background knowledge of surface plasmon polaritons and give the importance of quality factors, which allow us to select the right plasmonic material in the wavelength range we interested in. From the calculation results, it can be concluded that silver is the metal with the lowest optical loss and the longest propagation length in the visible light range, so we chose silver as the main material for our research. Next, we use thermal evaporator to epitaxy growth high-quality epitaxial silver films on sapphire substrates, and then use focused ion beams to etch surface plasmonic crystals with high fidelity. In terms of theoretical simulation, the finite-difference time-domain method is used to simulate surface plasmonic crystals with different symmetries, and the specific eigenmodes and their field distributions in the crystal are obtained, with the simulation result we can observe that eigenmodes are related to the rotational symmetry of the crystals, and periodic structures of different symmetries will all have a bound state in the continuum on the high symmetry point (Γ) , if we use the spatial average integral of the electric field around the high symmetry point (Γ) , which revolves around the closed contour of the high symmetry point (Γ) , you can obtain a continuously changing polarization state, which is a polarization vortex, and the results of the overall change in the polarization state on the closed contour are presented in the 2π×q form, which q is an integer, so it can be called as topological charge (q). The properties of the topological charges are determined by the rotational symmetry of the crystal, which do not change due to differences in cells or process defects, which is the so-called symmetry protection. Next, we used a homemade angle-resolved microscope to measure the band relation in the surface plasmonic crystals and use MATLAB program for data analysis. From the experimental results, it can be concluded that the bound state in the continuums are all present in the surface plasmonic crystals, and the topological charges which obtain from the polarization analysis are consistent with the simulation results. Therefore, we can verify that rotational symmetry determines the formation of topological charges. This thesis excerpts the simulation and experimental analysis of topological charges and rotational symmetry, hoping to inspire readers who interested in topological photonics.

一、 摘要 I
二、 Abstract III
三、 致謝 V
四、 目錄 VI
五、 圖目錄 VIII
六、 表目錄 I
一、 簡介以及原理 1
1.1 光子晶體與超穎材料 1
1.2 表面電漿子的背景 2
1.2.1 表面電漿極化子(SPPs) 3
1.2.2 局域性表面電漿子(Localized Surface Plasmons, LSPs) 6
1.2.3 表面電漿子的品質因子(Quality factor, Q) 7
1.3 表面電漿子晶體 (Plasmonic crystals) 9
1.3.1 表面電漿子晶體中的連續譜束縛態 (Bound states in the Continuum) 11
1.4 表面電漿子晶體中的拓樸光子學 14
1.4.1 偏振渦漩 (Polarization vortex) 14
1.4.2 二維晶體中的對稱性 18
二、 儀器介紹及原理 23
2.1 生長和蝕刻技術 23
2.1.1 熱蒸鍍(Thermal evaporator) 23
2.1.2 原子層沉積(Atomic layer deposition, ALD) 24
2.1.3 聚焦離子束(Focused ion beam, FIB)蝕刻 25
2.2 光學量測 25
2.2.1 橫向電場(TE)和橫向磁場(TM)模態的定義 26
2.2.2 傅立葉光學介紹 26
2.2.3 微觀角度分辨顯微鏡和光譜學 29
2.3 模擬介紹及原理 31
2.3.1 時域有限差分法 (Finite-Difference Time-Domain, FDTD) 31
三、 實驗流程及分析 32
3.1 樣品製備 32
3.2 量測系統校正及數據分析 34
3.2.1 量測系統校正 34
3.2.2 數據分析 36
3.3 偏振渦旋的理論模擬 38
3.3.1 方形晶格 38
3.3.2 六方晶格 41
3.3.3 蜂巢晶格 44
3.4 偏振渦旋的實驗量測 48
3.4.1 方形晶格 48
3.4.2 六方晶格 49
3.4.3 蜂巢晶格 50
四、 結論 52
五、 附錄 53
六、 參考文獻 55

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