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Understanding the propagation characteristics of surface plasmon polaritons (SPPs) is of great significance in designing and constructing on-chip integrated systems utilizing plasmonic effect. Accurately characterizing and flexibly controlling SPP on thin metal film are indispensable. Here, we theoretically derive the group velocity dispersion of SPP propagation on the surface of Au films with various thicknesses. The results obtained in this work indicate that when the thickness of the Au film is less than 40 nm, group velocity dispersion of SPP decreases significantly as the film thickness increases. The decrease of group velocity dispersion becomes mild with the thickness increasing from 40 nm to 60 nm, then the dispersion keeps a very low constant value for the film thicker than 60 nm. Using the finite-difference time-domain method, temporal evolution of localized electric field of SPP is numerically simulated for various propagation distances. By comparing the field amplitudes and the dispersions of SPP which are excited by incident light pulses with different dispersions, group velocity dispersions of SPP on the Au films are obtained, showing a good consistence with the theoretical results. Moreover, we demonstrate that by utilizing the tailored SPP to excite metal nanoantenna, selective excitations at different frequencies on a femtosecond temporal scale can be achieved through localized surface plasmonic resonant effect. Manipulating the sign and amount of the dispersion from the incident pulse, the active control of the switching sequence and switching time of electric field between the Au cylinders can be achieved. Manipulating the propagation distance of SPP, the active control of the switching time of electric field between the Au cylinders can be achieved. Therefore, those results provide a promising avenue for realizing functions such as signal propagation, reception, adjustment, and encoding in on-chip interconnect circuit systems based on SPP. This work shows that the dispersion can be used as degree of freedom for controlling the amplitude, phase and pulse width of SPP propagating on thin film, and it is of great importance in designing and controlling on-chip integrated systems through utilizing plasmonic effect, such as ultrafast frequency demodulators and nanoantennas in on-chip interconnect optical circuits.
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Keywords:
- surface plasmon polaritons /
- femtosecond laser pulse /
- group velocity dispersion /
- group delay dispersion
[1] Komatsu K, Pápa Z 2024 Nano Lett. 24 2637
Google Scholar
[2] Sandtke M 2007 Ph. D. Dissertation (Enschede: University of Twente
[3] Zayats A V, Smolyaninov I I 2003 J. Opt. A: Pure Appl. Opt. 5 S16
Google Scholar
[4] Maier S A 2007 Plasmonics: Fundamentals and Applications (Vol. 1) (New York: Springer) pp39–50
[5] Barnes W L, Dereux A, Ebbesen T W 2003 Nature 424 824
Google Scholar
[6] Pitarke J M, Silkin V M, Chulkov E V, Echenique P M 2007 Rep. Prog. Phys. 70 1
Google Scholar
[7] Song H B, Lang P S, Ji B Y, Xu Y, Peng S Y, Song X W, Lin J Q 2024 J. Phys. Chem. Lett. 15 7924
Google Scholar
[8] Goerlitzer E S A, Mohammadi R, Nechayev S, Volk K, Rey M, Banzer P, Karg M, Vogel N 2020 Adv. Mater. 32 2001330
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[9] Joly A G, Gong Y, El-Khoury P Z, Hess W P 2018 J. Phys. Chem. Lett. 9 6164
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[10] Sumimura A, Ota M 2016 IEEE Photonics Technol. Lett. 28 2419
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[11] Razinskas G, Kilbane D, Melchior P, Geisler P, Krauss E, Mathias S, Hecht B, Aeschlimann M 2016 Nano Lett. 16 6832
Google Scholar
[12] Gramotnev D K, Bozhevolnyi S I 2010 Nat. Photonics 4 83
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[13] Jin J J, Li X, Guo Y H, Pu M B, Gao P, Ma X L, Luo X G 2019 Nanoscale 11 3952
Google Scholar
[14] Pors A, Nielsen M G, Bernardin T, Weeber J C, Bozhevolnyi S I 2014 Light Sci. Appl. 3 e197
Google Scholar
[15] Lin J, Mueller J P B, Wang Q, Yuan G H, Antoniou N, Yuan X C, Capasso F 2013 Science 340 331
Google Scholar
[16] Rockstuhl C, Herzig H P 2004 Opt. Lett. 29 2181
Google Scholar
[17] Bernatová S, Donato M G, Ježek J, Pilát Z, SamekO, Magazzù A, Maragò O M, Zemánek P, Gucciardi P G 2019 J. Phys. Chem. C 123 5608
[18] Yao W J, Liu S, Liao H M, Li Z, Sun C W, Chen J J, Gong Q H 2015 Nano Lett. 15 3115
Google Scholar
[19] Qin Y L, Song X W, Ji B Y, Xu Y, Lin J Q 2019 Opt. Lett. 44 2935
Google Scholar
[20] Buckanie N M, Kirschbaum P, Sindermann S, Meyer zu Heringdorf F J 2013 Ultramicroscopy 130 49
Google Scholar
[21] Weeber J C, Lacroute Y, Dereux A, Devaux E, Ebbesen T, Girard C, González M U, Baudrion A L 2004 Phys. Rev. B 70 235406
Google Scholar
[22] Leißner T, Lemke C, Jauernik, S, Müller M, Fiutowski J, Tavares L, Thilsing-Hansen K, Kjelstrup-Hansen J, Magnussen O, Rubahn H G, Bauer M 2013 Opt. Express 21 8251
Google Scholar
[23] Lepetit L, Chériaux G 1995 J. Opt. Soc. Am. B 12 2467
Google Scholar
[24] Iaconis C, Walmsley I A 1998 Opt. Lett. 23 792
Google Scholar
[25] Yi J M, Hou D 2017 ACS Photonics 4 347
Google Scholar
[26] 虞华康, 刘伯东, 吴婉玲, 李志远 2019 物理学报 68 149101
Google Scholar
Yu H K, Liu B D, Wu W L, Li Z Y 2019 Acta Phys. Sin. 68 149101
Google Scholar
[27] 张文君, 高龙, 魏红, 徐红星 2019 物理学报 68 147302
Google Scholar
Zhang W J, Gao L, Wei H, Xu H X 2019 Acta Phys. Sin. 68 147302
Google Scholar
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Google Scholar
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Google Scholar
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图 1 (a) SPP波矢的实部与频率的多项式拟合; (b)不同频率下SPP在10 nm金膜表面传播的群速度色散
Figure 1. (a) Polynomial fitting of the real part of SPP wave vector versus frequency; (b) group velocity dispersion of SPP propagating on the surface of 10 nm gold film at different frequencies.
图 2 (a)模拟设置示意图, W = 350 nm, H = 10 nm; 在色散量分别为0 (黑色)、–10 (红色)、–20 (蓝色)与–40 fs2 (绿色)的飞秒激光脉冲激发下, 传输距离分别为1 μm (b), 2 μm (c), 3 μm (d), 6 μm (e)的SPP电场演化曲线
Figure 2. (a) Schematic diagram of the simulation setup, W = 350 nm, H = 10 nm; temporal evolution of electric field of SPP with the propagation distances of 1 μm (b), 2 μm (c), 3 μm (d), 6 μm (e) under the excitation of femtosecond laser pulse with the negative dispersion of 0 (black), –10 (red), -20 (blue), –40 fs2 (green), respectively.
图 3 (a) SPP在厚度分别为20, 30和40 nm的金膜上传播的波矢与频率的多项式拟合; (b)不同频率下SPP在20, 30和40 nm金膜表面传播的群速度色散
Figure 3. Dispersion relationship of GVD for SPP propagating on Au film with the thicknesses of 20, 30, and 40 nm; (b) group velocity dispersion of SPP propagating on the surface of 20, 30, and 40 nm gold film at different frequencies.
图 4 (a)中心频率为375 THz (800 nm)时$ {k}_{{\mathrm{s}}{\mathrm{p}}{\mathrm{p}}} $的实部随金膜厚度的变化; (b)中心波长为800 nm的入射光激发的SPP, 在不同厚度的金膜表面传播产生的GVD(黑色为理论拟合结果, 红色为FDTD模拟结果)
Figure 4. (a) Real part of $ {k}_{{\mathrm{s}}{\mathrm{p}}{\mathrm{p}}} $ versus the thickness of Au film at 375 THz (800 nm); (b) GVD versus the thickness of Au film obtained from theoretical calculation (black) and simulation calculation (red).
图 5 (a) SPP激发下的金圆柱结构示意图; (b)两个纳米圆柱结构热点位置的近场谱; (c) 315.8 THz频率激发下金纳米圆柱结构的近场分布图; (d) 370.8 THz频率激发下金纳米圆柱结构的近场分布图
Figure 5. (a) Schematic diagram of Au cylinders with the excitation of SPP; (b) near-field spectrum at the hot spots of two Au nano-cylinders; field profiles of the two Au nano-cylinders at 315.8 THz (c) and 370.8 THz (d) of excitation frequency.
图 6 色散量分别为0 fs2 (a), 40 fs2 (b), 80 fs2 (c)与 –80 fs2 (d)的激光脉冲激发下A1 与 A2 纳米圆柱结构热点处的电场演化曲线
Figure 6. Temporal evolution of electric field at the hotspot from A1 and A2 nano-cylinders excited by 0 fs2 (a), 40 fs2 (b), 80 fs2 (c) and –80 fs2 (d) laser pulse.
图 7 色散量为40 fs2的脉冲激发下, 传播距离距凹槽边缘2 μm (a)与12 μm (b)时, A1 与 A2 纳米圆柱结构热点处的电场演化曲线; (c)色散量为0 fs2的脉冲激发下, 传播距离距凹槽边缘12 μm时, A1 与 A2 纳米圆柱结构热点处的电场演化曲线
Figure 7. Temporal evolution of electric field at the hotspot from A1 and A2 nano-cylinders when the propagation distance is 2 μm (a) and 12 μm (b) from the edge of the groove under the excitation of the incident laser pulse with a dispersion of 40 fs2; (c) temporal evolution of electric field at the hotspot from A1 and A2 nano-cylinders when the propagation distance is 12 μm from the edge of the groove under the excitation of the incident laser pulse with a dispersion of 0 fs2.
图 A1 (a)金膜厚度分别为10, 20, 30与40 nm时相同变换极限脉冲激发的SPP传播3 μm的电场演化曲线. 当金膜厚度为20 nm (b), 30 nm (c)与40 nm (d)时, –10 fs2飞秒激光脉冲激发的SPP在传播3 μm后电场强度与0 fs2飞秒激光脉冲激发的SPP传播3 μm后的电场演化曲线对比
Figure A1. (a) Electric field evolution curves for SPP propagation 3 μm excited by the same propagation limit pulse for Au film thicknesses of 10, 20, 30 and 40 nm. Comparison of the electric field strength of SPP excited by a –10 fs2 femtosecond laser pulse after propagation of 3 μm with the electric field evolution curves of SPP excited by a 0 fs2 femtosecond laser pulse after propagation of 3 μm when the Au film thickness is 20 nm (b), 30 nm (c) and 40 nm (d).
表 1 入射激光脉冲的色散量与 SPP 的传播长度
Table 1. Dispersion of the incident laser pulse and propagation lengths of SPP.
入射激光脉冲的色散量/fs2 传播长度/μm –10 0.6 –20 1.3 –30 2.0 –40 2.7 –60 4.1 
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[1] Komatsu K, Pápa Z 2024 Nano Lett. 24 2637
Google Scholar
[2] Sandtke M 2007 Ph. D. Dissertation (Enschede: University of Twente
[3] Zayats A V, Smolyaninov I I 2003 J. Opt. A: Pure Appl. Opt. 5 S16
Google Scholar
[4] Maier S A 2007 Plasmonics: Fundamentals and Applications (Vol. 1) (New York: Springer) pp39–50
[5] Barnes W L, Dereux A, Ebbesen T W 2003 Nature 424 824
Google Scholar
[6] Pitarke J M, Silkin V M, Chulkov E V, Echenique P M 2007 Rep. Prog. Phys. 70 1
Google Scholar
[7] Song H B, Lang P S, Ji B Y, Xu Y, Peng S Y, Song X W, Lin J Q 2024 J. Phys. Chem. Lett. 15 7924
Google Scholar
[8] Goerlitzer E S A, Mohammadi R, Nechayev S, Volk K, Rey M, Banzer P, Karg M, Vogel N 2020 Adv. Mater. 32 2001330
Google Scholar
[9] Joly A G, Gong Y, El-Khoury P Z, Hess W P 2018 J. Phys. Chem. Lett. 9 6164
Google Scholar
[10] Sumimura A, Ota M 2016 IEEE Photonics Technol. Lett. 28 2419
Google Scholar
[11] Razinskas G, Kilbane D, Melchior P, Geisler P, Krauss E, Mathias S, Hecht B, Aeschlimann M 2016 Nano Lett. 16 6832
Google Scholar
[12] Gramotnev D K, Bozhevolnyi S I 2010 Nat. Photonics 4 83
Google Scholar
[13] Jin J J, Li X, Guo Y H, Pu M B, Gao P, Ma X L, Luo X G 2019 Nanoscale 11 3952
Google Scholar
[14] Pors A, Nielsen M G, Bernardin T, Weeber J C, Bozhevolnyi S I 2014 Light Sci. Appl. 3 e197
Google Scholar
[15] Lin J, Mueller J P B, Wang Q, Yuan G H, Antoniou N, Yuan X C, Capasso F 2013 Science 340 331
Google Scholar
[16] Rockstuhl C, Herzig H P 2004 Opt. Lett. 29 2181
Google Scholar
[17] Bernatová S, Donato M G, Ježek J, Pilát Z, SamekO, Magazzù A, Maragò O M, Zemánek P, Gucciardi P G 2019 J. Phys. Chem. C 123 5608
[18] Yao W J, Liu S, Liao H M, Li Z, Sun C W, Chen J J, Gong Q H 2015 Nano Lett. 15 3115
Google Scholar
[19] Qin Y L, Song X W, Ji B Y, Xu Y, Lin J Q 2019 Opt. Lett. 44 2935
Google Scholar
[20] Buckanie N M, Kirschbaum P, Sindermann S, Meyer zu Heringdorf F J 2013 Ultramicroscopy 130 49
Google Scholar
[21] Weeber J C, Lacroute Y, Dereux A, Devaux E, Ebbesen T, Girard C, González M U, Baudrion A L 2004 Phys. Rev. B 70 235406
Google Scholar
[22] Leißner T, Lemke C, Jauernik, S, Müller M, Fiutowski J, Tavares L, Thilsing-Hansen K, Kjelstrup-Hansen J, Magnussen O, Rubahn H G, Bauer M 2013 Opt. Express 21 8251
Google Scholar
[23] Lepetit L, Chériaux G 1995 J. Opt. Soc. Am. B 12 2467
Google Scholar
[24] Iaconis C, Walmsley I A 1998 Opt. Lett. 23 792
Google Scholar
[25] Yi J M, Hou D 2017 ACS Photonics 4 347
Google Scholar
[26] 虞华康, 刘伯东, 吴婉玲, 李志远 2019 物理学报 68 149101
Google Scholar
Yu H K, Liu B D, Wu W L, Li Z Y 2019 Acta Phys. Sin. 68 149101
Google Scholar
[27] 张文君, 高龙, 魏红, 徐红星 2019 物理学报 68 147302
Google Scholar
Zhang W J, Gao L, Wei H, Xu H X 2019 Acta Phys. Sin. 68 147302
Google Scholar
[28] Kiani F, Tagliabue G 2022 Chem. Mater. 34 1278
Google Scholar
[29] Qin H L, Wang D, Huang Z L, Wu D M, Zeng Z C, Ren B, Xu K, Jin J 2013 J. Am. Chem. Soc. 135 12544
Google Scholar
[30] Maniyara R A, Rodrigo D, Yu R W, Canet-Ferrer J, Ghosh D S, Yongsunthon R, Baker D E, Rezikyan A, de Abajo F J G, Pruneri V 2019 Nat. Photonics 13 328
Google Scholar
[31] Zhang Z 2011 Femtosecond Laser Technology (Vol. 1) (Beijing: China Science Publishing & Media LTD) pp6–12
[32] Burke J J, Stegeman G I, Tamir T 1986 Phys. Rev. B 33 5186
Google Scholar
[33] Johnson P B, Christy R W 1972 Phys. Rev. B 6 4370
Google Scholar
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