Quasinormal mode analysis of extremely localized optical field in body-of-revolution plasmonic structures

Zhou Qiang; Lin Shu-Pei; Zhang Pu; Chen Xue-Wen

doi:10.7498/aps.68.20190434

Abstract

Surface plasmons in metallic nanostructures can confine the optical field within the region of subwavelength, even nanometer scale, and thus enhance the light-matter interaction and other physical processes, which will lead the plasmon optics to possess attractive applications in many areas. However, the " mode volume” often used to characterize field confinement in plasmonic structures is only defined phe-nomenologically and suffers ambiguity when applied to complex structures. In this work, we develop a theoretical method to characterize the field confinement based on quasi-normal mode analysis. We recognize the fact that a plasmonic resonance may result from many eigen-modes, which together contribute to the observed field confinement. An effective mode volume is introduced for quasi-normal modes and used to characterize the field confinement when the plasmonic resonance is dominated by a single quasi-normal mode. Two typical kinds of plasmonic structures are systematically examined, and the field confinement on the order of 10 nm³–100 nm³ is confirmed. In pursuit of the ultimate field confinement, we revisit the so-called " pico-cavity” formed by an atomistic protrusion in the nano gap of the particle-on-mirror configuration. The apparent hot spot is shown to have contributions from several quasi-normal modes. The dominant one exhibits a further squeezed mode volume compared with the scenario without the protrusion, but is still well above 10 nm³.

Keywords:

Authors and contacts

School of Physics and Center for Quantum Optical Science, Huazhong University of Science and Technology, Wuhan 430074, China

Corresponding author: Zhang Pu, puzhang0702@hust.edu.cn ; Chen Xue-Wen, xuewen_chen@hust.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11874166) and the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11604109).

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References

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Cited By

图 1 半径40 nm金球的准正则模式分析　(a)旋转对称结构建模示意; (b)按指标l, m排列的归一化准正则模场分布, 中心线左右两侧分别为E_r和E_θ的实部; (c)准正则模式谱分布, 蓝色叉为模式聚点; (d)用全波仿真(full wave)、解析公式(analytical)和准正则模式重建(QNM)三种方法得到的距金纳米球表面10 nm处沿轴向的Purcell因子

Figure 1. 2D axisymmetric quasinormal mode (QNM) analysis for a spherical gold nanoparticle with r_s = 40 nm: (a) Schematic of 2D axisymmetric modeling; (b) shows a table of normalized QNM profiles arranged according to indices l and m, where on the left and right side of the axis shows the real part of E_r and E_θ, respectively; (c) spectral distribution of QNMs with an accumulation point indicated by the blue cross; (d) at a position 10 nm away from the metal surface, Purcell factor in radial direction is calculated with full wave simulation, analytical methods and QNM reconstruction.

DownLoad: Full-Size Img PowerPoint

图 2 二聚体结构准正则模式及场局域分析　二聚体由r_a = 40 nm和r_b = 5, 20, 40 nm的两个金纳米球组成, 间隙宽度d = 1, 5, 10 nm; 图中展示了: 准正则模式谱分布; 纳米间隙中心处轴向Purcell因子的准正则模式重建(黑色实线), 红色圆圈为全波仿真结果; 纳米间隙附近BDP模式的等效模体积分布

Figure 2. QNM modal analysis for dimers and the field localization of the BDP modes. The dimers consist of two spherical gold nanoparticles with r_a = 40 nm and r_b = 5, 20, 40 nm. The gap size d = 1, 5, 10 nm. In the figure are illustrated: the spectral distributions of the QNMs; the QNM reconstruction (black solid lines) of the Purcell factor at the gap center and in the axial direction, the full wave simulation results are indicated by red circles; the mode volume profiles of the BDP modes in the gap region.

DownLoad: Full-Size Img PowerPoint

图 3 PoM的准正则模式及场局域分析　PoM中金颗粒为半长轴a = 40 nm的旋转椭球, 镜面为相同材料无穷大半空间, 椭球半短轴b = 5, 20, 40 nm, 与镜面间距d = 1, 5, 10 nm; 图中展示了: 准正则模式谱分布; 纳米间隙中心处轴向Purcell因子的准正则模式重建(黑色实线), 红色圆圈为全波仿真结果; 纳米间隙附近BDP模式的等效模体积分布

Figure 3. QNM modal analysis for particle-on-mirror (PoM) structures and the field localization of the BDP modes. The gold particles are ellipsoids of revolution with semi-axes a = 40 nm and b = 5, 20, 40 nm. The gap formed with gold mirrors has d = 1, 5, 10 nm. In the figure are illustrated: the spectral distributions of the QNMs; the QNM reconstruction (black solid lines) of the Purcell factor at the gap center and in the axial direction, the full wave simulation results are indicated by red circles; the mode volume profiles of the BDP modes in the gap region.

DownLoad: Full-Size Img PowerPoint

图 4 PoM纳米间隙中原子凸起结构对光场局域的影响和准正则模式分析　(a) 存在与不存在原子凸起时的结构示意图; (b) 纳米间隙中心高度位置的局域场增强效应, 激励源为镜面上方716 nm处轴向偶极子; (c) BQP谐振波长处纳米间隙中心高度的局域场增强效应; (d) PoM结构的准正则模式谱分布以及纳米间隙中心处轴向Purcell因子的准正则模式重建; (e) BQP模式的归一化模场和等效模式体积

Figure 4. The effects of the atomistic protrusion on the field localization in the nano-gap of the particle-on-mirror (PoM) structure and the QNM modal analysis: (a) Schematic diagrams of the PoM structures with and without the atomistic protrusion; (b) the local field enhancement at the half height in the gap when excited by an axially polarized dipole 716 nm above the mirror; (c) the cross section of the local field enhancement in (b) at the resonance wavelength of the BQP mode; (d) the spectral distributions of the QNMs and the QNM reconstruction of the Purcell factor at the gap center and in the axial direction; (e) the normalized modal profile of the BQP mode and its mode volume map in the gap region.

DownLoad: Full-Size Img PowerPoint

[1]	Alvarez Puebla R, Liz Marzán L M, García de Abajo F J 2010 J. Phys. Chem. Lett. 1 2428 Google Scholar
[2]	Kuznetsov A I, Miroshnichenko A E, Brongersma M L, Kivshar Y S, Luk’yanchuk B 2016 Science 354 aag2472 Google Scholar
[3]	Schuller J A, Barnard E S, Cai W, Jun Y C, White J S, Brongersma M L 2010 Nat. Mater. 9 193 Google Scholar
[4]	Polman A 2008 Science 322 868 Google Scholar
[5]	Benz F, Schmidt M K, Dreismann A, Chikkaraddy R, Zhang Y, Demetriadou A, Carnegie C, Ohadi H, de Nijs B, Esteban R, Aizpurua J, Baumberg J J 2016 Science 354 726 Google Scholar
[6]	Kern J, Großmann S, Tarakina N V, Hackel T, Emmerling M, Kamp M, Huang J S, Biagioni P, Prangsma J C, Hecht B 2012 Nano Lett. 12 5504 Google Scholar
[7]	Ciracì C, Hill R, Mock J, Urzhumov Y, Fernández Domínguez A, Maier S, Pendry J, Chilkoti A, Smith D 2012 Science 337 1072 Google Scholar
[8]	Wei H, Xu H 2013 Nanoscale 5 10794 Google Scholar
[9]	Oulton R F, Sorger V J, Genov D, Pile D, Zhang X 2008 Nat. Photon. 2 496 Google Scholar
[10]	Kleinman S L, Frontiera R R, Henry A I, Dieringer J A, van Duyne R P 2013 Phys. Chem. Chem. Phys. 15 21 Google Scholar
[11]	Kneipp K, Wang Y, Kneipp H, Perelman L T, Itzkan I, Dasari R R, Feld M S 1997 Phys. Rev. Lett. 78 1667 Google Scholar
[12]	Stranahan S M, Willets K A 2010 Nano Lett. 10 3777 Google Scholar
[13]	Willets K A, Wilson A J, Sundaresan V, Joshi P B 2017 Chem. Rev. 117 7538 Google Scholar
[14]	Chen W, Zhang S, Deng Q, Xu H 2018 Nat. Commun. 9 801 Google Scholar
[15]	Loo C, Lowery A, Halas N, West J, Drezek R 2005 Nano Lett. 5 709 Google Scholar
[16]	Huang X, Jain P K, Sayed I H E, Sayed M A E 2007 Nanomedicine 2 681 Google Scholar
[17]	Juan M L, Righini M, Quidant R 2011 Nat. Photon. 5 349 Google Scholar
[18]	Maragò O M, Jones P H, Gucciardi P G, Volpe G, Ferrari A C 2013 Nat. Nanotechnol. 8 807 Google Scholar
[19]	Pelton M 2015 Nat. Photon. 9 427 Google Scholar
[20]	Tame M S, McEnery K R, Özdemir Ş K, Lee J, Maier S A, Kim M S 2013 Nat. Phys. 9 329 Google Scholar
[21]	Reiserer A, Rempe G 2015 Rev. Mod. Phys. 87 1379 Google Scholar
[22]	Wen J, Wang H, Chen H, Deng S, Xu N 2018 Chin. Phys. B 27 096101 Google Scholar
[23]	Maier S A 2006 Opt. Express 14 1957 Google Scholar
[24]	Koenderink A F 2010 Opt. Lett. 35 4208 Google Scholar
[25]	Esteban R, Aizpurua J, Bryant G W 2014 New J. Phys. 16 013052 Google Scholar
[26]	Muljarov E, Langbein W 2016 Phys. Rev. B 94 235438 Google Scholar
[27]	Sauvan C, Hugonin J P, Maksymov I S, Lalanne P 2013 Phys. Rev. Lett. 110 237401 Google Scholar
[28]	Leung P, Liu S, Young K 1994 Phys. Rev. A 49 3057 Google Scholar
[29]	Lalanne P, Yan W, Vynck K, Sauvan C, Hugonin J P 2018 Laser Photon. Rev. 12 1700113 Google Scholar
[30]	Yan W, Faggiani R, Lalanne P 2018 Phys. Rev. B 97 205422 Google Scholar
[31]	Muljarov E, Weiss T 2018 Opt. Lett. 43 1978 Google Scholar
[32]	Franke S, Hughes S, Dezfouli M K, Kristensen P T, Busch K, Knorr A, Richter M 2018 arXiv: 1808.06392
[33]	Hughes S, Richter M, Knorr A 2018 Opt. Lett. 43 1834 Google Scholar
[34]	Novotny L, Hecht B 2012 Principles of Nano-Optics (Cambridge: Cambridge University Press) pp350–359
[35]	Dung H T, Knöll L, Welsch D G 1998 Phys. Rev. A 57 3931 Google Scholar
[36]	Lalanne P, Yan W, Gras A, Sauvan C, Hugonin J P, Besbes M, Demésy G, Truong M, Gralak B, Zolla F 2018 arXiv: 1811.11751
[37]	Muljarov E, Langbein W 2016 Phys. Rev. B 93 075417 Google Scholar
[38]	Bai Q, Perrin M, Sauvan C, Hugonin J P, Lalanne P 2013 Opt. Express 21 27371 Google Scholar
[39]	Dezfouli M K, Hughes S 2018 Phys. Rev. B 97 115302 Google Scholar
[40]	Dezfouli M K, Tserkezis C, Mortensen N A, Hughes S 2017 Optica 4 1503 Google Scholar
[41]	Raman A, Fan S 2010 Phys. Rev. Lett. 104 087401 Google Scholar
[42]	Demésy G, Nicolet A, Gralak B, Geuzaine C, Campos C, Roman J E 2018 arXiv: 1802.02363
[43]	Tisseur F, Meerbergen K 2001 SIAM Rev. 43 235 Google Scholar
[44]	Taflove A, Hagness S C 2005 Computational Electrodynamics: the Fnite-Difference Time-Domain Method (Norwood: Artech House) pp517–552
[45]	Oxborrow M 2007 IEEE Trans. Microw. Theory Tech. 55 1209 Google Scholar
[46]	Chinellato O, Arbenz P, Streiff M, Witzig A 2005 Future Gener. Comput. Syst. 21 1263 Google Scholar
[47]	Jin J 2014 The Finite Element Method in Electromagnetics (Hoboken: Wiley-IEEE Press) pp315–371
[48]	Ho K, Leung P, van Den Brink A M, Young K 1998 Phys. Rev. E 58 2965 Google Scholar
[49]	Ge R C, Kristensen P T, Young J F, Hughes S 2014 New J. Phys. 16 113048 Google Scholar
[50]	Kristensen P T, Hughes S 2013 ACS Photon. 1 2
[51]	Iranzo D A, Nanot S, Dias E J, Epstein I, Peng C, Efetov D K, Lundeberg M B, Parret R, Osmond J, Hong J Y 2018 Science 360 291 Google Scholar
[52]	Amendola V, Pilot R, Frasconi M, Marago O M, Iati M A 2017 J. Phys. Condens. Matter 29 203002 Google Scholar
[53]	Baranov D G, Wersall M, Cuadra J, Antosiewicz T J, Shegai T 2017 ACS Photon. 5 24
[54]	Prodan E, Radloff C, Halas N J, Nordlander P 2003 Science 302 419 Google Scholar
[55]	Urbieta M, Barbry M, Zhang Y, Koval P, Sánchez Portal D, Zabala N, Aizpurua J 2018 ACS Nano 12 585 Google Scholar
[56]	Ciracì C 2017 Phys. Rev. B 95 245434 Google Scholar
[57]	Savage K J, Hawkeye M M, Esteban R, Borisov A G, Aizpurua J, Baumberg J J 2012 Nature 491 574 Google Scholar
[58]	Esteban R, Borisov A G, Nordlander P, Aizpurua J 2012 Nat. Commun. 3 825 Google Scholar
[59]	Chikkaraddy R, de Nijs B, Benz F, Barrow S J, Scherman O A, Rosta E, Demetriadou A, Fox P, Hess O, Baumberg J J 2016 Nature 535 127 Google Scholar
[60]	Barbry M, Koval P, Marchesin F, Esteban R, Borisov A G, Aizpurua J, Sánchez Portal D 2015 Nano Lett. 15 3410 Google Scholar
[61]	Trautmann S, Aizpurua J, Götz I, Undisz A, Dellith J, Schneidewind H, Rettenmayr M, Deckert V 2017 Nanoscale 9 391 Google Scholar
[62]	Weaver J, Frederikse H 1977 CRC Handbook of Chemistry and Physics (Boca Raton: CRC Press) pp12–156
[63]	Bohren C F, Huffman D R 2008 Absorption and Scattering of Light by Small Particles (Hoboken: John Wiley & Sons) pp82–129

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