存在於金屬表面並具有傳遞性質的表面電漿子(Surface plasmon)，其相較於光子具有近場下的侷域性以及突破繞射極限(Diffraction limit)的限制等優勢，透過以上特性能夠更有效的調控及分析模態的傳遞與耦合行為，儘管表面電漿子系統中的模態共振會造成一定的光學損耗，但卻開啟了非厄米光子學(Non-Hermitian photonics)方面的研究探討，本文將透過表面電漿子晶體結構參數的調整下改變系統中模態的耦合行為，藉此來討論非厄米參數空間中的演化過程。
實驗中，利用銀金屬製備二維矩形孔洞陣列的電漿子晶體，將孔洞的寬度定義為晶體的結構參數並對其進行調控，並使用角度解析反射光譜(Angle-resolved reflectance spectrum)在可見光波段下進行系統色散關係(Dispersion relation)的分析。此結構會讓系統在動量空間高對稱點的模態產生具對稱保護性的近場場強分布導致部分模態無法由入射光所激發，因此在遠場量測下可以看到在高對稱點下不同能帶會有頻譜帶寬(Bandwidth)上的差異，藉由能帶帶寬的差異定義為暗態(Dark mode)與亮態(Bright mode)，並且透過調制晶體的結構參數，觀察高對稱點下不同能帶之間能隙的變化並進一步產生能帶反轉(Band flip)的現象。而頻譜帶寬對應模態共振與環境交互作用產生的輻射型光學損耗，因此該系統會引入非厄米物理系統的概念，建構數學模型來描述該系統對應的參數空間，並在頻譜中亮暗態的重合處定義為參數空間中的例外點(Exceptional points)，而該系統的能帶反轉現象可以看作是系統演化經過例外點產生的相變行為。該系統可以做進一步的延伸，利用具不同結構參數的表面電漿子晶體進行拼接製成的結構接面，在該接面下會產生具拓樸保護性的模態共振行為，此共振會產生垂直於晶體表面的低發散角輻射行為，此現象具有的空間局域性配合表面電漿子的近場增強效果，以及具指向性的輻射行為可以達到較佳的出光效率，藉此提升表面電漿子在光學共振腔(Optical resonator)的應用性。

The thesis discusses surface plasmons that exist on metal surfaces and their applications in optics. Surface plasmons are electronic excitations that exist on the surface of metals and possess unique optical properties, offering advantages such as near-field localization and the ability to surpass the diffraction limit when compared to photons. These characteristics enable more effective control and analysis of mode propagation and coupling behavior. While mode resonance in surface plasmon systems can lead to some optical losses, it has opened up research opportunities in non-Hermitian photonics. This article explores how adjusting the parameters of surface plasmon crystal structures can alter mode coupling behavior in non-Hermitian parameter space.
In this work, a two-dimensional rectangular hole array plasmonic crystal was fabricated using silver metal. The width of the holes was defined as a structural parameter of the crystal and was controlled. Angle-resolved reflectance spectrum was employed in the visible light range to analyze the system's dispersion relation. This structure led to the generation of modes with symmetrically protected near-field intensity distributions at high-symmetry points in momentum space, resulting in some modes that cannot be excited by incident light. Consequently, differences in bandwidth were observed in different bands at high-symmetry points when measured in the far field. By defining these differences in bandwidth as dark and bright modes and modulating the crystal's structural parameters, changes in band gaps between different bands at high-symmetry points were observed, leading to band flips. The bandwidth corresponds to optical losses generated by mode resonance and environmental interactions, introducing the concept of non-Hermitian physical systems. A mathematical model was constructed to describe the parameter space of the system, defining exceptional points in the spectrum where dark and bright modes coincide. The phenomenon of band flips in this system can be seen as a phase transition behavior resulting from the system's evolution passing through exceptional points.
This system can be further extended by creating interfaces between surface plasmonic lattice with different structural parameters. At such interfaces, mode resonance with topological protection occurs, leading to leakage radiation with low divergence angles perpendicular to the crystal surface. This phenomenon combines the spatial localization with the near-field enhancement effect of surface plasmons and directional radiation behavior, which can enhance the efficiency of light emission and improve the applicability of surface plasmons in optical resonators.

一、 摘要 I
二、 Abstract II
三、 致謝 III
四、 目錄 IV
五、 圖目錄 VI
一、 基本原理介紹 1
1.1 表面電漿子背景與原理 1
1.1.1 表面電漿極化子(Surface Plasmon Polaritons, SPPs) 3
1.1.2 局域性表面電漿子(Localized Surface Plasmons, LSPs) 5
1.1.3 電漿材料的品質因子(Quality Factor) 7
1.2 電漿子晶體(Plasmonic Crystals) 9
1.2.1 電漿子晶體中的布洛赫波(Bloch Wave) 10
1.2.2 電漿子晶體中的繞射級數(Diffraction Order) 12
1.3 非厄米物理系統(Non-Hermitian Systems) 16
1.3.1 參數空間中的例外點(Exceptional Points, EPs) 17
1.4 電漿子晶體中的能帶反轉現象(Band Flip in Plasmonic Crystal) 19
1.4.1 模態耦合機制(Coupling Regime of Guide Mode) 21
1.4.2 非厄米等效哈密頓量(Non-Hermitian Effective Hamiltonian) 23
二、 實驗儀器原理介紹 26
2.1 薄膜生長與蝕刻技術 26
2.1.1 熱蒸鍍機(Thermal Evaporator) 26
2.1.2 聚焦離子束蝕刻(Focused Ion Beam, FIB) 27
2.1.3 原子層沉積(Atomic Layer Deposition, ALD) 28
2.2 光學量測技術 30
2.2.1 光學系統的動量空間(Momentum Space) 30
2.2.2 角度解析反射光譜(Angle-Resolved Reflectivity Spectrum) 31
三、 實驗流程 34
3.1 樣品製備流程 34
3.2 光學特性量測 36
四、 量測結果分析 38
4.1 矩形陣列電漿子晶體 38
4.2 方形陣列電漿子晶體 42
4.3 拓樸介面態 (Topological Interface State) 47
五、 結論 50
六、 參考資料 52

[1] Barnes, W. L.; Dereux, A.; Ebbesen, T. W. Surface plasmon subwavelength optics. Nature 2003, 424 (6950), 824-830.
[2] Ritchie, R. H. Plasma losses by fast electrons in thin films. Phys. Rev. 1957, 106 (5), 874-881.
[3] Powell, C. J.; Swan, J. B. Origin of the characteristic electron energy losses in aluminum. Phys. Rev. 1959, 115 (4), 869-875.
[4] Kretschmann, E.; Raether, H. Radiative decay of non radiative surface plasmons excited by light. 1968, 23 (12), 2135-2136.
[5] Raether, H. Surface plasmons on smooth and rough surfaces and on gratings; Springer Tracts Mod. Phys., 1988.
[6] Ebbesen, T. W.; Lezec, H. J.; Ghaemi, H. F.; Thio, T.; Wolff, P. A. Extraordinary optical transmission through sub-wavelength hole arrays. Nature 1998, 391 (6668), 667-669.
[7] Mayer, K. M.; Hafner, J. H. Localized surface plasmon resonance sensors. Chem. Rev. 2011, 111 (6), 3828-3857.
[8] Zhang, J.; Zhang, L.; Xu, W. Surface plasmon polaritons: physics and applications. J. Phys. D: Appl. Phys. 2012, 45 (11), 113001.
[9] Gao, H.; Zhou, W.; Odom, T. W. Plasmonic crystals: A platform to catalog resonances from ultraviolet to near-infrared wavelengths in a plasmonic library. Adv. Funct. Mater. 2010, 20 (4), 529-539.
[10] Zayats, A. V.; Smolyaninov, I. I.; Maradudin, A. A. Nano-optics of surface plasmon polaritons. Phys. Rep. 2005, 408 (3), 131-314.
[11] Craig F. Bohren, D. R. H. Absorption and scattering by a sphere; 1998.
[12] Johnson, P. B.; Christy, R. W. Optical Constants of the Noble Metals. Phys. Rev. B 1972, 6 (12), 4370-4379.
[13] Wang, F.; Shen, Y. R. General properties of local plasmons in metal nanostructures. Phys. Rev. Lett. 2006, 97 (20), 206806.
[14] Cheng, C.-W.; Gwo, S. Chapter 1 - Fundamentals of plasmonic materials. In Plasmonic Materials and Metastructures, Gwo, S., Alù, A., Li, X., Shih, C.-K. Eds.; Elsevier, 2024; pp 3-33.
[15] Joannopoulos, J. D.; Johnson, S. G.; Winn, J. N.; Meade, R. D. Photonic Crystals: Molding the Flow of Light; Princeton University Press, 2008.
[16] Pockrand, I. Reflection of light from periodically corrugated silver films near the plasma frequency. Phys. Lett. A 1974, 49 (3), 259-260.
[17] Hsu, C. W.; Zhen, B.; Stone, A. D.; Joannopoulos, J. D.; Soljačić, M. Bound states in the continuum. Nat. Rev. Mater. 2016, 1 (9), 16048.
[18] Lu, L.; Le-Van, Q.; Ferrier, L.; Drouard, E.; Seassal, C.; Nguyen, H. S. Engineering a light-matter strong coupling regime in perovskite-based plasmonic metasurface: quasi-bound state in the continuum and exceptional points. Photonics Res. 2020, 8 (12), A91-A100.
[19] Özdemir, Ş. K.; Rotter, S.; Nori, F.; Yang, L. Parity–time symmetry and exceptional points in photonics. Nat. Mater 2019, 18 (8), 783-798.
[20] van Exter, M. P.; Tenner, V. T.; van Beijnum, F.; de Dood, M. J. A.; van Veldhoven, P. J.; Geluk, E. J.; ’t Hooft, G. W. Surface plasmon dispersion in metal hole array lasers. Opt. Express 2013, 21 (22), 27422-27437.
[21] Bender, C. M.; Boettcher, S. Real Spectra in Non-Hermitian Hamiltonians Having $\mathsc{P}\mathsc{T}$ Symmetry. Phys. Rev. Lett. 1998, 80 (24), 5243-5246.
[22] Ozawa, T.; Price, H. M.; Amo, A.; Goldman, N.; Hafezi, M.; Lu, L.; Rechtsman, M. C.; Schuster, D.; Simon, J.; Zilberberg, O.; et al. Topological photonics. RMP 2019, 91 (1), 015006.
[23] Sang, Y.; Wang, C.-Y.; Raja, S. S.; Cheng, C.-W.; Huang, C.-T.; Chen, C.-A.; Zhang, X.-Q.; Ahn, H.; Shih, C.-K.; Lee, Y.-H.; et al. Tuning of Two-Dimensional Plasmon–Exciton Coupling in Full Parameter Space: A Polaritonic Non-Hermitian System. Nano Lett. 2021, 21 (6), 2596-2602.
[24] Wiersig, J. Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection. Phys. Rev. Lett. 2014, 112 (20), 203901.
[25] Feng, L.; Wong, Z. J.; Ma, R.-M.; Wang, Y.; Zhang, X. Single-mode laser by parity-time symmetry breaking. Science 2014, 346 (6212), 972-975.
[26] Doppler, J.; Mailybaev, A. A.; Böhm, J.; Kuhl, U.; Girschik, A.; Libisch, F.; Milburn, T. J.; Rabl, P.; Moiseyev, N.; Rotter, S. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 2016, 537 (7618), 76-79.
[27] Lee, S.-G.; Magnusson, R. Band flips and bound-state transitions in leaky-mode photonic lattices. Phys. Rev. B 2019, 99 (4), 045304.
[28] Kazarinov, R.; Henry, C. Second-order distributed feedback lasers with mode selection provided by first-order radiation losses. IEEE J. Quantum Electron. 1985, 21 (2), 144-150.
[29] Zhang, Y.; Zhao, M.; Wang, J.; Liu, W.; Wang, B.; Hu, S.; Lu, G.; Chen, A.; Cui, J.; Zhang, W.; et al. Momentum-space imaging spectroscopy for the study of nanophotonic materials. Sci. Bull. 2021, 66 (8), 824-838.
[30] Sun, S.; Ding, Y.; Li, H.; Hu, P.; Cheng, C.-W.; Sang, Y.; Cao, F.; Hu, Y.; Alù, A.; Liu, D.; et al. Tunable plasmonic bound states in the continuum in the visible range. Phys. Rev. B 2021, 103 (4), 045416.
[31] Zhang, Y.; Chen, A.; Liu, W.; Hsu, C. W.; Wang, B.; Guan, F.; Liu, X.; Shi, L.; Lu, L.; Zi, J. Observation of Polarization Vortices in Momentum Space. Phys. Rev. Lett. 2018, 120 (18), 186103.
[32] Hsu, C. W.; Zhen, B.; Lee, J.; Chua, S.-L.; Johnson, S. G.; Joannopoulos, J. D.; Soljačić, M. Observation of trapped light within the radiation continuum. Nature 2013, 499 (7457), 188-191.
[33] Zhen, B.; Hsu, C. W.; Lu, L.; Stone, A. D.; Soljačić, M. Topological Nature of Optical Bound States in the Continuum. Phys. Rev. Lett. 2014, 113 (25), 257401.
[34] Zhen, B.; Hsu, C. W.; Igarashi, Y.; Lu, L.; Kaminer, I.; Pick, A.; Chua, S.-L.; Joannopoulos, J. D.; Soljačić, M. Spawning rings of exceptional points out of Dirac cones. Nature 2015, 525 (7569), 354-358.
[35] Lee, K. Y.; Yoon, S.; Song, S. H.; Yoon, J. W. Topological beaming of light. Sci. Adv. 8 (49), eadd8349.
[36] Su, W. P.; Schrieffer, J. R.; Heeger, A. J. Solitons in Polyacetylene. Phys. Rev. Lett. 1979, 42 (25), 1698-1701.
[37] Lee, K. Y.; Yoo, K. W.; Choi, Y.; Kim, G.; Cheon, S.; Yoon, J. W.; Song, S. H. Topological guided-mode resonances at non-Hermitian nanophotonic interfaces. 2021, 10 (7), 1853-1860.
[38] Hirose, K.; Liang, Y.; Kurosaka, Y.; Watanabe, A.; Sugiyama, T.; Noda, S. Watt-class high-power, high-beam-quality photonic-crystal lasers. Nat. Photonics 2014, 8 (5), 406-411.