在本論文中，我們用有限時域差分法(FDTD) 模擬單層石墨烯的局域表面電漿共振模態，分別對兩種不同形狀(圓形、正六角形)的石墨烯進行模態分析。選擇六角形的原因是，六角形是石墨烯的單位晶格形狀，因此不管是任何形狀的石墨烯一定包含有六角形的對稱性質；至於圓形則是最對稱的圖形，不管從任何角度來看都是對稱的，先從圓形開始觀察，可以簡化分析的工作。
對於圓形的石墨烯，我們用平面波去激發不同特徵波長下的模態，並用第一類貝索函數的記號做分類。對於六角形的石墨烯，我們用點波源去激發不同波長下的模態。由於六角形的對稱性相對複雜許多，共有八種情況，因此，需要對不同的對稱性，去放置不同位置的點波源，才能正確激發出模態。
本篇論文的第三章、第四章分別呈現石墨烯圓形、六角形的模態分析，列出從最低階模態到最高階模態的對應電場場強。可以幫助人們更認識石墨烯的表面電漿共振性質。

In this thesis, we demonstrate the localized surface plasmon resonance of single layer graphene flake in circular/ hexagonal shape.
The reason we survey these two geometries is that hexagon is the natural bound-ary of graphene crystal lattice. We are curious about what special modes will form in the hexagonal cavity. For the circular shape, it has the highest symmetry, and we only have to consider the azimuthal symmetry and radial mode. This work helps us to un-derstand the fundamental mode in graphene cavity.
For the case of circular flake, we launch x-polarized plane wave to excite the field pattern with nonzero dipole moment. And, we designate these modes by nota-tion with the first kind of Bessel function.
For the case of hexagonal flake, we launch point source to excite the field pattern with zero net dipole moment. According the eight symmetry classes of hexagon, we launch point sources on corresponding location. And, we designate these modes from the fundamental mode to the highest mode.
This thesis will help people to understand the LSPR mode in graphene flake.

[1]J. M. Marmolejo-Tejada and J. Velasco-Medina, "Review on graphene nanoribbon devices for logic applications" Microelectronics Journal, 48, 18-38 (2016).
[2]W. H. Wang, T. Christensen, A. P. Jauho, K. S. Thygesen, M. Wubs, and N. A. Mortensen, "Plasmonic eigenmodes in individual and bow-tie graphene nanotri-angles” Supplementary information, Scientific Reports, 5, 6 (2015).
[3]T. D. Rossing, "Chaldni law for vibrating plates" American Journal of Physics, 50, 271-274 (1982).
[4]S. Errede, “Mathematical Musical Physics of the Wave Equation” UIUC Physics 406 Acoustical Physics of Music. Illinois, 22 (2002).
[5]L. T. Cureton and J. R. Kuttler, "Eigenvalues of the Laplacian on regular poly-gons and polygons resulting from their dissection" Journal of Sound and Vibra-tion, 220, 83-98 (1999).
[6]T. Nobis and M. Grundmann, "Low-order optical whispering-gallery modes in hexagonal nanocavities" Physical Review A, 72, 063806 (2005).
[7]A. Otto, "Excitation of nonradiative surface plasma waves in silver by method of frustrated total reflection" Zeitschrift Fur Physik, 216, 398-410 (1968).
[8]K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media" IEEE Transactions on Antennas and Propagation, 14, 302-307 (1966).
[9]C. M. Furse and O. P. Gandhi, "Why the DFT is faster than the FFT for FDTD time-to-frequency domain conversions" IEEE Microwave and Guided Wave Letters, 5, 326-328 (1995).
[10]K. Ziegler, "Robust transport properties in graphene" Physical Review Letters, 97, 4 (2006).
[11]G. W. Hanson, "Dyadic Green's functions and guided surface waves for a sur-face conductivity model of
graphene" Journal of Applied Physics, 103, 8 (2008).
[12]J. S. Wang, T. H. Huang, S. H. Chang, and C. C. Kaun, "Modified Pade analyti-cal approximation of graphene
interband conductivity," presented at the The 8th
International Conference on Surface Plasmon Photonics,
Taipei (2017)
[13]A. Mock, "Pade approximant spectral fit for FDTD simulation of graphene in the near infrared," Optical Materials Express, 2, 771-781 (2012).
[14]R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, et al., "Fine structure constant defines visual transparency of graphene," Science, 320, 1308-1308 (2008).
[15]R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, et al., "Supporting Online Material," Science, 320, 1308 (2008).
[16]C. P. Yu and H. C. Chang, "Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers" Optics Express, 12, 6165-6177 (2004).