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Optical nanoantennas support surface plasmon polariton (SPP) with a confinement of light breaking through the diffraction limit, and thereby achieving an enhancement and regulation of electromagnetic field on a deep-subwavelength scale. In this paper, a periodic array of optical nanoantennas on a metallic mirror is proposed, where the antennas are gold nanocubes forming a two-dimensional periodic array, and a single point emission source is located in the nanogap between the antenna of gold nanocube and the gold mirror. The nanogap between the antenna and mirror can support gap surface plasmon, which results in an enhanced spontaneous emission rate. Meanwhile, the periodic array of nanoantennas can support the surface lattice resonance (SLR), which can achieve directional far-field radiation perpendicular to the substrate or in a specified direction by properly designing the array period. To design the antenna that can simultaneously achieve an enhancement of spontaneous emission rate and a directional radiation of far field, the calculation of the radiation field of a single point source in a periodic structure is transformed into the calculation of the radiation fields of a set of pseudoperiodic point-source arrays by combining the array scanning method (ASM) and full-wave rigorous numerical method, thus giving the spontaneous emission rate of the emitter and the near-field distribution of the antenna. Concerning the calculation of the angular distribution of far-field radiation intensity, we start from the Maxwell’s equations and provide a rigorous formulation and proof of the reciprocity-theorem method. This proof is different from those reported in existing literature and has a more rigorous applicability for infinite-extent periodic structures or has a lower amount of computational work. Based on the reciprocity-theorem method, the antenna parameters are designed so that the enhancement factor of far-field radiation intensity reaches a maximum value of 2756 in the direction perpendicular to the substrate, and simultaneously, the enhancement factors of total spontaneous emission rate and far-field spontaneous emission rate of the point source reach 1097 and 55.50, respectively. The proposed antenna has a simple structure that is easy to design and fabricate, and the proposed design method is intuitive and easy to implement, which can be used to guide the design of high-speed, high-brightness and directional-radiation light sources.
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Keywords:
- optical nanoantenna arrays on a metallic mirror /
- spontaneous-emission enhancement /
- far-field directional radiation /
- reciprocity theorem
[1] Mühlschlegel P, Eisler H J, Martin O J F, Hecht B, Pohl D W 2005 Science 308 1607
Google Scholar
[2] Aizpurua J, Bryant G W, Richter L J, García de Abajo F J, Kelley B K, Mallouk T 2005 Phys. Rev. B 71 235420
Google Scholar
[3] Pelton M 2015 Nat. Photonics 9 427
Google Scholar
[4] Atwater H A, Polman A 2010 Nat. Mater. 9 205
Google Scholar
[5] Fang N, Lee H, Sun C, Zhang X 2005 Science 308 534
Google Scholar
[6] Traverso A J, Huang J, Peyronel T, Yang G, Tiecke T G, Mikkelsen M H 2021 Optica 8 202
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[7] Rycenga M, Xia X, Moran C H, Zhou F, Qin D, Li Z Y, Xia Y 2011 Angew. Chem. Int. Ed. 50 5473
Google Scholar
[8] Xin H, Namgung B, Lee L P 2018 Nat. Rev. Mater. 3 228
Google Scholar
[9] Gubin M Yu, Shesterikov A V, Prokhorov A V, Volkov V S 2020 Laser Photonics Rev. 14 2000237
Google Scholar
[10] Shan H, Liu Z, Wang X, Lin F, Liu Z, Shen B, Lou J, Zhu X, Ajayan P M, Fang Z 2020 Laser Photonics Rev. 14 2000233
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[11] Tyagi D, Chen T, Huang C 2020 Laser Photonics Rev. 14 2000076
Google Scholar
[12] Wang J, Jia X, Guan Y, Ren K, Yu H, Wang Z, Qu S, Yang Q, Lin J, Wang Z, Jin P 2021 Laser Photonics Rev. 15 2000512
Google Scholar
[13] Lumdee C, Yun B, Kik P G 2014 ACS Photonics 1 1224
Google Scholar
[14] Baumberg J J, Aizpurua J, Mikkelsen M H, Smith D R 2019 Nat. Mater. 18 668
Google Scholar
[15] Purcell E M 1946 Phys. Rev. 69 674
Google Scholar
[16] Huang S, Ming T, Lin Y, Ling X, Ruan Q, Palacios T, Wang J, Dresselhaus M, Kong J 2016 Small 12 5190
Google Scholar
[17] Novotny L, van Hulst N 2011 Nat. Photonics 5 83
Google Scholar
[18] Lodahl P, Mahmoodian S, Stobbe S 2015 Rev. Mod. Phys. 87 347
Google Scholar
[19] Ma R M, Oulton R F, Sorger V J, Zhang X 2013 Laser Photonics Rev. 7 1
Google Scholar
[20] Lakowicz J r., Fu Y 2009 Laser Photonics Rev. 3 221
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[21] Agio M 2012 Nanoscale 4 692
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[22] Tsakmakidis K L, Boyd R W, Yablonovitch E, Zhang X 2016 Opt. Express 24 17916
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[23] Li L, Hutter T, Steiner U, Mahajan S 2013 Analyst 138 4574
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[24] Kelly K L, Coronado E, Zhao L L, Schatz G C 2003 J. Phys. Chem. B 107 668
Google Scholar
[25] Lozano G, Louwers D J, Rodríguez S R, Murai S, Jansen O T, Verschuuren M A, Gómez Rivas J 2013 Light-Sci. Appl. 2 e66
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Google Scholar
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Google Scholar
[31] Ferry V E, Verschuuren M A, Li H B T, Verhagen E, Walters R J, Schropp R E I, Atwater H A, Polman A 2010 Opt. Express 18 A237
Google Scholar
[32] Lozano G, Rodriguez S R, Verschuuren M A, Gómez Rivas J 2016 Light-Sci. Appl 5 e16080
Google Scholar
[33] Capolino F, Jackson D R, Wilton D R, Felsen L B 2007 IEEE Trans. Antennas Propag. 55 1644
Google Scholar
[34] Kahl M, Voges E 2000 Phys. Rev. B 61 14078
Google Scholar
[35] Janssen O T A, Wachters A J H, Urbach H P 2010 Opt. Express 18 24522
Google Scholar
[36] Zhang S, Martins E R, Diyaf A G, Wilson J I B, Turnbull G A, Samuel I D W 2015 Synthetic Met. 205 127
Google Scholar
[37] Jia H, Liu H, Zhong Y 2015 Sci. Rep. 5 8456
Google Scholar
[38] Lecamp G, Hugonin J P, Lalanne P 2007 Opt. Express 15 11042
Google Scholar
[39] Born M, Wolf E 1999 Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7 th Ed. ) (Cambridge: Cambridge University)
[40] Wang N, Zhong Y, Liu H 2022 Opt. Express 30 12797
Google Scholar
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Google Scholar
[42] Palik E D 1998 Handbook of Optical Constants of Solids (Sydney: Academic Press)
[43] de Boer D K G, Verschuuren M A, Guo K, Koenderink A F, Rivas J G, Rodriguez S R K 2016 Opt. Express 24 A388
Google Scholar
[44] Aouani H, Mahboub O, Bonod N, Devaux E, Popov E, Rigneault H, Ebbesen T W, Wenger J 2011 Nano Lett. 11 637
Google Scholar
[45] Shegai T, Johansson P, Langhammer C, Käll M 2012 Nano Lett. 12 2464
Google Scholar
[46] Lechago S, García-Meca C, Griol A, Kovylina M, Bellieres L, Martí J 2019 ACS Photonics 6 1094
Google Scholar
[47] Dregely D, Lindfors K, Lippitz M, Engheta N, Totzeck M, Giessen H 2014 Nat. Commun. 5 4354
Google Scholar
[48] Li L 1997 J. Opt. Soc. Am. A 14 2758
Google Scholar
[49] Liu H T 2010 DIF CODE for Modeling Light Diffraction in Nanostructures (Tianjin: Nankai University)
[50] Zhang X, Liu H, Zhong Y 2013 Opt. Express 21 24139
Google Scholar
[51] Vecchi G, Giannini V, Gómez Rivas J 2009 Phys. Rev. Lett. 102 146807
Google Scholar
[52] Koenderink A F 2017 ACS Photonics 4 710
Google Scholar
[53] Glass A M, Liao P F, Bergman J G, Olson D H 1980 Opt. Lett. 5 368
Google Scholar
[54] Hoang T B, Akselrod G M, Argyropoulos C, Huang J, Smith D R, Mikkelsen M H 2015 Nat. Commun. 6 7788
Google Scholar
[55] Tam F, Goodrich G P, Johnson B R, Halas N J 2007 Nano Lett. 7 496
Google Scholar
[56] Wang H, Lin Y, Ma P, Zhong Y, Liu H 2019 J. Mater. Chem. C 7 13526
Google Scholar
[57] Muskens O L, Giannini V, Sánchez-Gil J A, Gómez Rivas J 2007 Nano Lett. 7 2871
Google Scholar
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Google Scholar
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Google Scholar
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图 1 周期结构中单个点电流源远场辐射强度角分布互易定理计算方法示意图. 图中所示为金属镜面上金属纳米天线(金纳米立方体)阵列结构 (a) 原问题示意图, 点源位于周期结构中, 辐射出上行远场平面波模式; (b) 互易问题示意图, 下行远场平面波模式激励周期结构. 图(a)和图(b)的电磁场分别表示为ψ1和ψ2; (c) 结构侧视图, 自下而上依次为半无限金基底、PMMA纳米间隙(厚度为h)、金纳米立方体天线(边长为D)阵列, 沿x, y方向阵列周期分别为a, b
Figure 1. Schematic diagrams of the numerical method by using the reciprocity theorem for calculating the angular distribution of far-field radiation intensity of a single electric-current point source in a periodic structure. The depicted structure is a periodic array of metallic nanoantennas (gold nanocubes) on a metallic mirror: (a) The original problem where a point source is located in a periodic structure and radiates up-going far-field plane-wave modes; (b) the reciprocity problem where a down-going far-field plane-wave mode excites the periodic structure. The electromagnetic fields of Figure (a) and Figure (b) are denoted by ψ1 and ψ2, respectively; (c) side view of the structure. From bottom to top, the structure is composed of a semi-infinite gold substrate, a PMMA nanogap (with a thickness h) and a gold-nanocube antenna array (with a side length D, and periods a and b along x and y directions, respectively)
图 2 ASM示意图 (a) 周期结构中(沿x, y方向周期分别为a, b)单个点电流源J(红色星号)的辐射问题; (b1)—(b3) 周期结构中点源赝周期阵列J∞(对应不同的kx, ky, 绿色星号)的辐射问题. 利用ASM, 图(a)中辐射场可表达为图(b)中辐射场的线性叠加
Figure 2. Schematic diagram of ASM: (a) Radiation problem of a single electric-current point source J (red asterisk) in a periodic structure (the periods along the x and y directions being a and b, respectively); (b1)–(b3) radiation problem of pseudoperiodic point-source arrays J∞ (corresponding to different kx and ky, green asterisks) in the periodic structure. By using the ASM, the radiation field in Figure (a) can be expressed as a linear superposition of the radiation fields in Figure (b).
图 3 (a) NPoM天线周期阵列的0级平面波反射率R(左轴, 实线)、电场增强因子FE(右轴, 虚线)随阵列周期a变化的曲线, 不同颜色对应不同的天线尺寸D; (b1), (b2) y = 0和z = h/2截面上电场|Ez|的分布(用入射平面波电场|Einc|作了归一化), 对应D = 200 nm, a = aSLR=989 nm
Figure 3. (a) Reflectance R (left axis, solid curves) of the zeroth-order plane wave and electric-field enhancement factor FE (right axis, dashed curves) plotted as functions of the antenna array period a, where different colors correspond to different antenna sizes D; (b1), (b2) distributions of |Ez| (normalized by the electric field |Einc| of the incident plane wave) on the cross-sections y = 0 and z = h/2 for D = 200 nm and a = aSLR = 989 nm.
图 4 (a1)—(a6) NPoM天线周期阵列中点电流源的远场辐射强度角分布P(θ, ϕ), 依次对应阵列周期a = 600, 700, 800, 900, 989, 1100 nm. 计算取天线边长D = 200 nm, 辐射波长λ = 1 μm. 设点源沿z方向偏振, 位于天线纳米间隙内rs = (D/2–10 nm, 0, h/2)位置; (b1)—(b2) TE, TM偏振零级平面波反射率R(θ, ϕ), 对应阵列周期a = aSLR = 989 nm. 图中颜色代表1–R(θ, ϕ)的数值, θ, ϕ分别为入射平面波波矢(kx,0, ky,0, –kz,0,0) = k0na(sinθcosϕ, sinθsinϕ, –cosθ)的极角、方位角. 图(a)和图(b)中显示了黑色圆、白色虚线位置极角θ、方位角ϕ的数值; (c) a = aSLR时RA的位置, 由方程(19)给出(标出了(m, n)的数值)
Figure 4. (a1)—(a6) Far-field radiation-intensity angular distributions P(θ, ϕ) of an electric-current point source in the periodic NPoM antenna array with a period a = 600, 700, 800, 900, 989, and 1100 nm, respectively. The side length D of the antenna is 200 nm and the radiation wavelength λ is 1 μm. The point source is polarized along the z direction and located at rs = (D/2–10 nm, 0, h/2) in the nanogap. (b1), (b2) Reflectance R(θ, ϕ) of the zeroth-order plane wave under TE and TM polarizations, respectively, which is obtained for array period a = aSLR = 989 nm. The θ and ϕ are respectively the polar angle and azimuth angle of the incident plane-wave wave vector (kx,0, ky,0, –kz,0,0) = k0na(sinθcosϕ, sinθsinϕ, –cosθ). The color in Figure (b1) and Figure (b2) represents the value of 1–R(θ, ϕ). Several typical values of θ and ϕ respectively at the black circles and white dashed lines in Figure (a) and Figure (b) are shown. (c) Positions of RA for a = aSLR given by Eq. (19) [with the values of (m, n) being marked].
图 5 方程(27a)给出的Am(蓝色圆圈) (a)—(d) 依次取P=10, 15, 20, 25. 两条竖直红色虚线显示了赝周期点源阵列中, 位于中央周期之外的点源抵消为0的范围–(P–1) ≤ m ≤ P–1
Figure 5. The Am given by Eq. (27a) (blue circles): (a)–(d) For P = 10, 15, 20, 25, respectively. The two vertical red-dashed lines show the range –(P–1) ≤ m ≤ P–1, where the point sources contained in the pseudoperiodic point-source arrays and out of the central period cancel to be null.
图 6 点源辐射电场在天线阵列中央周期(–a/2 ≤ x ≤ a/2, –b/2 ≤ y ≤ b/2)范围内的分布(用Γvac作了归一化). 结果采用ASM计算得到, 天线参数与图4(a5)相同 (a), (b) 对应点源所在截面y = 0, z = h/2. 从左到右分别显示了电场分量|Ex|, |Ey|, |Ez|. 白色虚线显示了界面位置
Figure 6. Distributions of electric field (normalized by Γvac) in the central period (–a/2 ≤ x ≤ a/2, –b/2 ≤ y ≤ b/2) of the antenna array radiated by a point source. The results are calculated by the ASM with the same antenna parameters as those in Fig. 4(a5): (a), (b) The field distributions on the cross sections y = 0 and z = h/2 where the point is located. The electric field components |Ex|, |Ey| and |Ez| are displayed from left to right. The white dashed lines show the interface positions.
表 1 采用不同的采样点数P = Q时, ASM计算得到的γtot和γrad
Table 1. γtot and γrad obtained with the ASM for various number P = Q of sampling points.
P = Q 10 20 40 80 γtot 1095.66 1098.05 1097.47 1098.23 γrad 415.31 54.04 55.50 55.81 
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[1] Mühlschlegel P, Eisler H J, Martin O J F, Hecht B, Pohl D W 2005 Science 308 1607
Google Scholar
[2] Aizpurua J, Bryant G W, Richter L J, García de Abajo F J, Kelley B K, Mallouk T 2005 Phys. Rev. B 71 235420
Google Scholar
[3] Pelton M 2015 Nat. Photonics 9 427
Google Scholar
[4] Atwater H A, Polman A 2010 Nat. Mater. 9 205
Google Scholar
[5] Fang N, Lee H, Sun C, Zhang X 2005 Science 308 534
Google Scholar
[6] Traverso A J, Huang J, Peyronel T, Yang G, Tiecke T G, Mikkelsen M H 2021 Optica 8 202
Google Scholar
[7] Rycenga M, Xia X, Moran C H, Zhou F, Qin D, Li Z Y, Xia Y 2011 Angew. Chem. Int. Ed. 50 5473
Google Scholar
[8] Xin H, Namgung B, Lee L P 2018 Nat. Rev. Mater. 3 228
Google Scholar
[9] Gubin M Yu, Shesterikov A V, Prokhorov A V, Volkov V S 2020 Laser Photonics Rev. 14 2000237
Google Scholar
[10] Shan H, Liu Z, Wang X, Lin F, Liu Z, Shen B, Lou J, Zhu X, Ajayan P M, Fang Z 2020 Laser Photonics Rev. 14 2000233
Google Scholar
[11] Tyagi D, Chen T, Huang C 2020 Laser Photonics Rev. 14 2000076
Google Scholar
[12] Wang J, Jia X, Guan Y, Ren K, Yu H, Wang Z, Qu S, Yang Q, Lin J, Wang Z, Jin P 2021 Laser Photonics Rev. 15 2000512
Google Scholar
[13] Lumdee C, Yun B, Kik P G 2014 ACS Photonics 1 1224
Google Scholar
[14] Baumberg J J, Aizpurua J, Mikkelsen M H, Smith D R 2019 Nat. Mater. 18 668
Google Scholar
[15] Purcell E M 1946 Phys. Rev. 69 674
Google Scholar
[16] Huang S, Ming T, Lin Y, Ling X, Ruan Q, Palacios T, Wang J, Dresselhaus M, Kong J 2016 Small 12 5190
Google Scholar
[17] Novotny L, van Hulst N 2011 Nat. Photonics 5 83
Google Scholar
[18] Lodahl P, Mahmoodian S, Stobbe S 2015 Rev. Mod. Phys. 87 347
Google Scholar
[19] Ma R M, Oulton R F, Sorger V J, Zhang X 2013 Laser Photonics Rev. 7 1
Google Scholar
[20] Lakowicz J r., Fu Y 2009 Laser Photonics Rev. 3 221
Google Scholar
[21] Agio M 2012 Nanoscale 4 692
Google Scholar
[22] Tsakmakidis K L, Boyd R W, Yablonovitch E, Zhang X 2016 Opt. Express 24 17916
Google Scholar
[23] Li L, Hutter T, Steiner U, Mahajan S 2013 Analyst 138 4574
Google Scholar
[24] Kelly K L, Coronado E, Zhao L L, Schatz G C 2003 J. Phys. Chem. B 107 668
Google Scholar
[25] Lozano G, Louwers D J, Rodríguez S R, Murai S, Jansen O T, Verschuuren M A, Gómez Rivas J 2013 Light-Sci. Appl. 2 e66
Google Scholar
[26] Lozano G, Grzela G, Verschuuren M A, Ramezani M, Rivas J G 2014 Nanoscale 6 9223
Google Scholar
[27] Vaskin A, Bohn J, Chong K E, Bucher T, Zilk M, Choi D Y, Neshev D N, Kivshar Y S, Pertsch T, Staude I 2018 ACS Photonics 5 1359
Google Scholar
[28] Liu S, Vaskin A, Addamane S, Leung B, Tsai M C, Yang Y, Vabishchevich P P, Keeler G A, Wang G, He X, Kim Y, Hartmann N F, Htoon H, Doorn S K, Zilk M, Pertsch T, Balakrishnan G, Sinclair M B, Staude I, Brener I 2018 Nano Lett. 18 6906
Google Scholar
[29] Väkeväinen A I, Moerland R J, Rekola H T, Eskelinen A P, Martikainen J P, Kim D H, Törmä P 2014 Nano Lett. 14 1721
Google Scholar
[30] van Lare M, Lenzmann F, Verschuuren M A, Polman A 2012 Appl. Phys. Lett. 101 221110
Google Scholar
[31] Ferry V E, Verschuuren M A, Li H B T, Verhagen E, Walters R J, Schropp R E I, Atwater H A, Polman A 2010 Opt. Express 18 A237
Google Scholar
[32] Lozano G, Rodriguez S R, Verschuuren M A, Gómez Rivas J 2016 Light-Sci. Appl 5 e16080
Google Scholar
[33] Capolino F, Jackson D R, Wilton D R, Felsen L B 2007 IEEE Trans. Antennas Propag. 55 1644
Google Scholar
[34] Kahl M, Voges E 2000 Phys. Rev. B 61 14078
Google Scholar
[35] Janssen O T A, Wachters A J H, Urbach H P 2010 Opt. Express 18 24522
Google Scholar
[36] Zhang S, Martins E R, Diyaf A G, Wilson J I B, Turnbull G A, Samuel I D W 2015 Synthetic Met. 205 127
Google Scholar
[37] Jia H, Liu H, Zhong Y 2015 Sci. Rep. 5 8456
Google Scholar
[38] Lecamp G, Hugonin J P, Lalanne P 2007 Opt. Express 15 11042
Google Scholar
[39] Born M, Wolf E 1999 Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7 th Ed. ) (Cambridge: Cambridge University)
[40] Wang N, Zhong Y, Liu H 2022 Opt. Express 30 12797
Google Scholar
[41] Chen Y, Zhang Y, Femius Koenderink A 2017 Opt. Express 25 21358
Google Scholar
[42] Palik E D 1998 Handbook of Optical Constants of Solids (Sydney: Academic Press)
[43] de Boer D K G, Verschuuren M A, Guo K, Koenderink A F, Rivas J G, Rodriguez S R K 2016 Opt. Express 24 A388
Google Scholar
[44] Aouani H, Mahboub O, Bonod N, Devaux E, Popov E, Rigneault H, Ebbesen T W, Wenger J 2011 Nano Lett. 11 637
Google Scholar
[45] Shegai T, Johansson P, Langhammer C, Käll M 2012 Nano Lett. 12 2464
Google Scholar
[46] Lechago S, García-Meca C, Griol A, Kovylina M, Bellieres L, Martí J 2019 ACS Photonics 6 1094
Google Scholar
[47] Dregely D, Lindfors K, Lippitz M, Engheta N, Totzeck M, Giessen H 2014 Nat. Commun. 5 4354
Google Scholar
[48] Li L 1997 J. Opt. Soc. Am. A 14 2758
Google Scholar
[49] Liu H T 2010 DIF CODE for Modeling Light Diffraction in Nanostructures (Tianjin: Nankai University)
[50] Zhang X, Liu H, Zhong Y 2013 Opt. Express 21 24139
Google Scholar
[51] Vecchi G, Giannini V, Gómez Rivas J 2009 Phys. Rev. Lett. 102 146807
Google Scholar
[52] Koenderink A F 2017 ACS Photonics 4 710
Google Scholar
[53] Glass A M, Liao P F, Bergman J G, Olson D H 1980 Opt. Lett. 5 368
Google Scholar
[54] Hoang T B, Akselrod G M, Argyropoulos C, Huang J, Smith D R, Mikkelsen M H 2015 Nat. Commun. 6 7788
Google Scholar
[55] Tam F, Goodrich G P, Johnson B R, Halas N J 2007 Nano Lett. 7 496
Google Scholar
[56] Wang H, Lin Y, Ma P, Zhong Y, Liu H 2019 J. Mater. Chem. C 7 13526
Google Scholar
[57] Muskens O L, Giannini V, Sánchez-Gil J A, Gómez Rivas J 2007 Nano Lett. 7 2871
Google Scholar
[58] Zhou W, Dridi M, Suh J Y, Kim C H, Co D T, Wasielewski M R, Schatz G C, Odom T W 2013 Nat. Nanotechnol. 8 506
Google Scholar
[59] Sreekanth K V, Krishna K H, De Luca A, Strangi G 2014 Sci. Rep. 4 6340
Google Scholar
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