近年來利用超穎材料使相位調變受到了人們關注，而相位的變化是與共振相關的，因此對於共振模態的分析相對重要的，而材料的選擇上也從吸收率高的金屬轉往介電質，所以我們一開始藉由米氏理論與FDTD去探索矽奈米球的特徵，從米氏理論得到的解析解可以知道當球的直徑在500nm時，電磁偶極與四極在頻譜上幾乎沒有與高階模態重疊，接著我們利用FDTD數值解針對不同模態去觀察電場與磁場的變化，並找出他們各自模態的特徵。我們試著模擬了金奈米V型陣列，這是利用結構設計產生雙共振模態，可以觀察相位的變化是否會受到雙共振模態的影響達到2pi的相位差。接者我們模擬了矽奈米圓盤，改變奈米圓盤的直徑觀察其電磁場模態變化。相位變化與納米圓盤的直徑成線性比例。最後我們排列不同直徑的納米圓盤入射平面波則可以有3度角的偏轉。

In recent years, the metamaterials for phase modulation has attracted attention, and the variation of phase is related to resonance. Therefore, the analysis of resonance modes is relatively important, and the selection of materials has also shifted from metals with high absorption rates to dielectric. Therefore, we first explored the characteristics of silicon nanospheres through Mie theory and FDTD. From Mie theory we can obtain the analytic solution, it can be known that when the diameter of the sphere is 500 nm, electric and magnetic dipolar and quadrupole almost don’t overlap with higher-order modes in the spectrum. Then we observe the changes of the electric field and magnetic field for different modes by the FDTD numerical solution, and find out the characteristics of their respective modes. We tried to simulate the V-shaped array of gold antenna, which uses structural design to achieve double resonance mode, and we can observe whether the phase change will be affected by the double resonance mode to achieve a phase difference of 2pi. Then we simulated the silicon nanodisk and changed the diameter of the nanodisk to observe the modal changes of the electromagnetic field. The phase change is linearly proportional to the diameter of the nanodisk. Finally, we arrange the nanodisks of different diameters to deflect the incident plane wave with an angle of 3 degrees.

[1] Shelby, R. A.; Smith D.R.; Shultz S.; Nemat-Nasser S.C. "Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial". Applied Physics Letters. 78 (4): 489 (2001)
[2] Pendry JB Negative refraction makes a perfect lens. Phys Rev Lett 85:3966–3969 (2000)
[3] Yu, N. & Capasso, F. Flat optics with designer metasurfaces. Nature Mater. 13, 139–150 (2014)
[4] Yu, N. et al. Light propagation with phase discontinuities: Generalized laws of reflection and refraction. Science 334, 333–337 (2011)
[5] Yu, N. et al. Flat optics: Controlling wavefronts with optical antenna metasurfaces. IEEE J. Sel. Top. Quant. Electron. 19, 4700423 (2013)
[6] Sun, S. et al. High-efficiency broadband anomalous reflection by gradient meta-surfaces. Nano Lett. 12, 6223–6229 (2012)
[7] Sun, S. et al. Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves. Nature Mater. 11, 426–431 (2012)
[8] Huang, L. et al. Dispersionless phase discontinuities for controlling light propagation. Nano Lett. 12, 5750–5755 (2012)
[9] C. Wenshan, C.V. Shalaev, Optical Metamaterials: Fundamentals and Applications, Springer, New York (2009)
[10] A. Afridi, J. Canet‐Ferrer, L. Philippet, J. Osmond, P. Berto, R. Quidant, ACS Photonics, 5, 4497 (2018)
[11] Veselago, V. G., “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Uspekhi, 10. 509-514. (1968)
[12] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)
[13] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)
[14] Liu, N., Liu, H., Zhu, S. N. & Giessen, H. Stereometamaterials. Nature Photon. 3, 157–162 (2009)
[15] Willets, K. A. & Van Duyne, R. P. Localized surface plasmon resonance spectroscopy and sensing. Annu. Rev. Phys. Chem. 58, 267–297 (2007)
[16] Craig F. Bohren, Donald R. Huffman, Absorption and Scattering of Light by Small Particles, ISBN-10: 0-47 1-29340-7
[17] A. García-Etxarri et al., Strong magnetic response of submicron silicon particles in the infrared. Opt. Express 19, 4815–4826 (2011)
[18] D. Tzarouchis and A. Sihvola, “Light scattering by a dielectric sphere: perspectives on the Mie resonances,” Appl. Sci. 8, 184 (2018)
[19] David J.Griffiths, Introduction to Electrodynamics, 4th Edition, United States of America: Pearson, . ISBN 978-0-321-85656-2.
[20] K.S. Yee Numerical solution of initial boundary value problems involving Maxwells Equations in isotropic media IEEE Trans. Antennas and Propagation, 14 (3), 302-307 (1966)
[21] Allen Taflove and Susan C. Hagness (2005), Computational Electrodynamics：The Finite-Difference Time-Domain Method, 3rd ed. Artech House Publishers. ISBN 978-1-58053-832-9.
[22] J. Berenger (1994). "A perfectly matched layer for the absorption of electromagnetic waves". Journal of Computational Physics. 114. (1994)
[23] 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)
[24] D.E. Merewether, R. Fisher, F.W. Smith, “On Implementing a Huygens Source Scheme in a Finite Difference Program to Illuminate Scattering Bodies”, IEEE Trans. Nucl. Sc., 27, 1829-1834, (1980)
[25] K. Umashankar, A. Taflove, “A Novel Method to Analyze Electromagnetic Scattering of Complex Objects”, IEEE Trans. Electrom. Compat., 24, 397-405, (1982)