本文提出一種結合自動分群(clustering)的反向設計方法，用於優化高性能光束偏轉超穎表面(beam deflection metasurfaces)，命名為基因型態樹狀搜尋法(genetic-type tree search)。不同於以往設計光束偏轉元件時，需先建立一組由單元結構在目標波長下基本特性(相位、振幅、群速度等等)的資料庫。其中，對相位的覆蓋量更是嚴格要求，以做出符合目標相位輪廓的超穎表面。
我們受到蒙地卡羅樹狀搜尋法(Monte Carlo tree search)和基因演算法(genetic algorithm)的啟發，透過同時優化超穎表面上振幅與相位兩項光學特性，來獲得高指向性的光束偏轉超穎表面的結構排列方式。經過優化後的光束偏轉超穎表面，其非直覺的反射率分布和非線性的相位輪廓有助於實現高程度的光束偏轉控制。此方法利用振幅彌補在設計光束偏轉元件時，對相位調製範圍的侷限，同時大大改善了正向設計方法限制空間分辨率的問題，並搭配分群的機制讓資料庫進行分類以降低計算量，使我們可通過一般個人電腦進行一定程度的優化。
我們的方法減少了對散射光特性的要求，結合演算法大幅降低計算的複雜度，從而為高指向性光束偏轉超穎表面開拓了一條道路，希望在未來能延伸至衛星通訊、無線系統或光檢測與測距系統等等。

We introduce a genetic-type tree search (GTTS) method combined with clustering to optimize high-performance beam deflection metasurfaces. Inspired by the Monte Carlo tree search and genetic algorithm, we realize highly-directive beam steering metasurfaces with co-optimization of amplitude and phase on metasurfaces. The optimized non-intuitive reflectivity distribution and nonlinear phase profile across the beam deflection metasurface help to achieve highly-directive beam deflection. The developed GTTS is capable of compensating the limitation of phase modulation range via amplitude variation. Thus, the spatial resolution as well as deflected angular range are improved through the forward design method. With the clustering, the database can be classified to reduce the amount of calculation, so that we can perform a certain optimization through personal computers. Our approach reduce the requirements of optical conditions, and combined with the algorithm to reduce the complexity of the calculation, opening a way to realize high-performance metasurfaces that could be used in a wide range of advanced nanophotonic applications.

[1] X. Luo, D. Tsai, M. Gu, and M. Hong, "Subwavelength interference of light on atructured surfaces," Adv. Opt. Photonics 10, 757-842 (2018).
[2] Z. Wu, L. Li, Y. Li, and X. Chen, "Metasurface superstrate antenna with wideband circular polarization for satellite communication application," IEEE Antennas Wirel. Propag. Lett. 15, 374-377 (2016).
[3] F. Yesilkoy, E. R. Arvelo, Y. Jahani, M. Liu, A. Tittl, V. Cevher, Y. Kivshar, and H. Altug, "Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces," Nat. Photonics 13, 390-396 (2019).
[4] G.-Y. Lee, J.-Y. Hong, S. Hwang, S. Moon, H. Kang, S. Jeon, H. Kim, J.-H. Jeong, and B. Lee, "Metasurface eyepiece for augmented reality," Nat. Commun. 9, 4562 (2018).
[5] G. K. Shirmanesh, R. Sokhoyan, P. C. Wu, and H. A. Atwater, "Electro-optically tunable multifunctional metasurfaces," ACS Nano 14, 6912-6920 (2020).
[6] A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, "Spatiotemporal light control with active metasurfaces," Science 364, eaat3100 (2019).
[7] L. Li, Z. Liu, X. Ren, S. Wang, V.-C. Su, M.-K. Chen, C. H. Chu, H. Y. Kuo, B. Liu, W. Zang, G. Guo, L. Zhang, Z. Wang, S. Zhu, and D. P. Tsai, "Metalens-array–based high-dimensional and multiphoton quantum source," Science 368, 1487 (2020).
[8] T. Stav, A. Faerman, E. Maguid, D. Oren, V. Kleiner, E. Hasman, and M. Segev, "Quantum entanglement of the spin and orbital angular momentum of photons using metamaterials," Science 361, 1101 (2018).
[9] T. Hao, W. Zheng, W. He, and K. Lin, "Air-Ground impedance matching by depositing metasurfaces for enhanced GPR detection," IEEE Trans. Geosci. Remote Sens. 58, 4061-4075 (2020).
[10] H. Hashida, Y. Kawamoto, and N. Kato, "Intelligent reflecting surface placement optimization in air-ground communication networks toward 6G," IEEE Wireless Commun. 27, 146-151 (2020).
[11] D. F. Sievenpiper, J. H. Schaffner, H. J. Song, R. Y. Loo, and G. Tangonan, "Two-dimensional beam steering using an electrically tunable impedance surface," IEEE Trans. Antennas Propag. 51, 2713-2722 (2003).
[12] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788 (2004).
[13] M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, "Chiral plasmonics," Sci. Adv. 3, e1602735 (2017).
[14] E. Hutter, and J. H. Fendler, "Exploitation of localized surface plasmon resonance," Adv. Mater. 16, 1685-1706 (2004).
[15] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, "Light propagation with phase discontinuities: generalized laws of reflection and refraction," Science 334, 333 (2011).
[16] S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, C. Hung Chu, J.-W. Chen, S.-H. Lu, J. Chen, B. Xu, C.-H. Kuan, T. Li, S. Zhu, and D. P. Tsai, "Broadband achromatic optical metasurface devices," Nat. Commun. 8, 187 (2017).
[17] C. A. Balanis, Antenna theory: analysis and design (John wiley & sons, 2015).
[18] P. C. Wu, J.-W. Chen, C.-W. Yin, Y.-C. Lai, T. L. Chung, C. Y. Liao, B. H. Chen, K.-W. Lee, C.-J. Chuang, C.-M. Wang, and D. P. Tsai, "Visible metasurfaces for on-chip polarimetry," ACS Photonics 5, 2568-2573 (2018).
[19] N. Yu, and F. Capasso, "Flat optics with designer metasurfaces," Nat. Mater. 13, 139-150 (2014).
[20] F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, "Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities," Nano Lett. 12, 1702-1706 (2012).
[21] S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, "Inverse design in nanophotonics," Nat. Photonics 12, 659-670 (2018).
[22] E. Bayati, R. Pestourie, S. Colburn, Z. Lin, S. G. Johnson, and A. Majumdar, "Inverse designed metalenses with extended depth of focus," ACS Photonics 7, 873-878 (2020).
[23] W. T. Chen, A. Y. Zhu, and F. Capasso, "Flat optics with dispersion-engineered metasurfaces," Nat. Rev. Mater. 5, 604-620 (2020).
[24] A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, "Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer," Nat. Photonics 9, 374-377 (2015).
[25] R. Pestourie, C. Pérez-Arancibia, Z. Lin, W. Shin, F. Capasso, and S. G. Johnson, "Inverse design of large-area metasurfaces," Opt. Express 26, 33732-33747 (2018).
[26] Z. Lin, V. Liu, R. Pestourie, and S. G. Johnson, "Topology optimization of freeform large-area metasurfaces," Opt. Express 27, 15765-15775 (2019).
[27] T. P. Xiao, O. S. Cifci, S. Bhargava, H. Chen, T. Gissibl, W. Zhou, H. Giessen, K. C. Toussaint, E. Yablonovitch, and P. V. Braun, "Diffractive spectral-splitting optical element designed by adjoint-based electromagnetic optimization and fabricated by femtosecond 3D direct laser writing," ACS Photonics 3, 886-894 (2016).
[28] D. Sell, J. Yang, S. Doshay, and J. A. Fan, "Periodic dielectric metasurfaces with high-efficiency, multiwavelength functionalities," Adv. Opt. Mater. 5, 1700645 (2017).
[29] D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, "Ultra-high-efficiency anomalous refraction with dielectric metasurfaces," ACS Photonics 5, 2402-2407 (2018).
[30] T. Phan, D. Sell, E. W. Wang, S. Doshay, K. Edee, J. Yang, and J. A. Fan, "High-efficiency, large-area, topology-optimized metasurfaces," Light Sci. Appl. 8, 48 (2019).
[31] D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, "Large-angle, multifunctional metagratings based on freeform multimode geometries," Nano Lett. 17, 3752-3757 (2017).
[32] Z. Liu, D. Zhu, S. P. Rodrigues, K.-T. Lee, and W. Cai, "Generative model for the inverse design of metasurfaces," Nano Lett. 18, 6570-6576 (2018).
[33] J. Jiang, and J. A. Fan, "Global Optimization of dielectric metasurfaces using a physics-driven neural network," Nano Lett. 19, 5366-5372 (2019).
[34] W. Ma, F. Cheng, and Y. Liu, "Deep-learning-enabled on-demand design of chiral metamaterials," ACS Nano 12, 6326-6334 (2018).
[35] P. R. Wiecha, and O. L. Muskens, "Deep learning meets nanophotonics: A generalized accurate predictor for near fields and far fields of arbitrary 3D nanostructures," Nano Lett. 20, 329-338 (2020).
[36] M. M. R. Elsawy, S. Lanteri, R. Duvigneau, G. Brière, M. S. Mohamed, and P. Genevet, "Global optimization of metasurface designs using statistical learning methods," Sci. Rep. 9, 17918 (2019).
[37] K. Yao, R. Unni, and Y. Zheng, "Intelligent nanophotonics: merging photonics and artificial intelligence at the nanoscale," Nanophotonics 8, 339-366 (2019).
[38] M. M. Elsawy, S. Lanteri, R. Duvigneau, J. A. Fan, and P. Genevet, "Numerical optimization methods for metasurfaces," Laser Photonics Rev. 14, 1900445 (2020).
[39] S. D. Campbell, D. Sell, R. P. Jenkins, E. B. Whiting, J. A. Fan, and D. H. Werner, "Review of numerical optimization techniques for meta-device design," Opt. Mater. Express 9, 1842-1863 (2019).
[40] J. S. Jensen, and O. Sigmund, "Topology optimization for nano‐photonics," Laser Photonics Rev. 5, 308-321 (2011).
[41] C.-H. Lin, Y.-S. Chen, J.-T. Lin, H. C. Wu, H.-T. Kuo, C.-F. Lin, P. Chen, and P. C. Wu, "Automatic inverse design of high-performance beam-steering metasurfaces via genetic-type tree optimization," Nano Lett. 21, 4981-4989 (2021).
[42] H. Chung, and O. D. Miller, "Tunable metasurface inverse design for 80% switching efficiencies and 144° angular deflection," ACS Photonics 7, 2236-2243 (2020).
[43] C. B. Browne, E. Powley, D. Whitehouse, S. M. Lucas, P. I. Cowling, P. Rohlfshagen, S. Tavener, D. Perez, S. Samothrakis, and S. Colton, "A survey of monte carlo tree search methods," IEEE Trans. Comput. Intell. AI Games 4, 1-43 (2012).
[44] S. Vassilvitskii, and D. Arthur, "k-means++: The advantages of careful seeding," in Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms(2006), pp. 1027-1035.
[45] H. Chung, and O. D. Miller, "Tunable metasurface inverse design for 80% switching efficiencies and 144 angular deflection," ACS Photonics 7, 2236-2243 (2020).
[46] I. Haller, M. Hatzakis, and R. Srinivasan, "High-resolution positive resists for electron-beam exposure," IBM J. Res. Dev. 12, 251-256 (1968).
[47] M. Quirk, and J. Serda, Semiconductor manufacturing technology (Prentice Hall Upper Saddle River, NJ, 2001).
[48] L. Igor, W. L. Philip, S. L. Craig, S. K. Nikolai, B. Vladimir, and F. Andrew, "Intracavity mode competition between classes of flat-top beams," in Proc.SPIE(2008).
[49] D. Engström, J. Bengtsson, E. Eriksson, and M. Goksör, "Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator," Opt. Express 16, 18275-18287 (2008).
[50] R. J. Mailloux, Phased array antenna handbook (Artech house, 2017).
[51] 賴耿楊，IC製程之濺射技術，復漢出版社，19895
[52] https://zh.wikipedia.org/wiki/機器學習
[53] https://kopu.chat/2017/07/28/機器是怎麼從資料中「學」到東西的呢?/
[54] http://www.zeon.co.jp/
[55] http://www.fortechgrps.com/