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准确评估与灵活调控表面等离激元(surface plasmon polaritons, SPP)的传输特性对基于等离激元效应设计的高速-小型化器件具有重要意义. 本文从理论上推导了SPP在不同厚度金膜表面传播的群速度色散. 当金膜厚度小于40 nm时, 随着膜厚度的增大, SPP的群速度色散显著减小; 金膜厚度为40—60 nm时, SPP的群速度色散下降趋势变缓, 并在厚度大于60 nm后保持恒定. 利用时域有限差分方法, 数值模拟了不同传播距离下SPP的电场时间演化. 通过比较不同传输距离下SPP近场强度与入射光色散量的关系确定SPP的群速度色散, 并得到接近理论推导的结果. 进而提出利用负啁啾脉冲激发SPP来补偿传输中产生的群延迟色散, 实现不同传输距离SPP场振幅、脉宽的调控. 此外, 利用定制的SPP激发金属纳米天线, 通过调控入射脉冲色散量与SPP在金膜中传播产生群延迟色散的传播特性相结合的手段, 实现在纳米信号接收器中热点位置的fs时间灵活时空相干控制. 这对于利用等离激元效应设计和控制微型片上集成系统具有重要意义.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
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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
Fig. 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 $ {{\mathrm{f}}{\mathrm{s}}}^{2} $(绿色)的飞秒激光脉冲激发下, 传输距离分别为1 μm (b), 2 μm (c), 3 μm (d), 6 μm (e)的SPP电场演化曲线
Fig. 2. (a) Schematic diagram of the simulation setup, W = 350 nm, H = 10 nm; temporal evolution of electric field of SPP with the propagation distances of 1 μm (b), 2 μm (c), 3 μm (d), 6 μm (e) under the excitation of femtosecond laser pulse with the negative dispersion of 0 (black), –10 (red), -20 (blue), –40 fs2 (green), respectively.
图 3 (a) SPP在厚度分别为20, 30和40 nm的金膜上传播的波矢与频率的多项式拟合; (b)不同频率下SPP在20, 30和40 nm金膜表面传播的群速度色散
Fig. 3. Dispersion relationship of GVD for SPP propagating on Au film with the thicknesses of 20, 30, and 40 nm; (b) group velocity dispersion of SPP propagating on the surface of 20, 30, and 40 nm gold film at different frequencies.
图 4 (a)中心频率为375 THz (800 nm)时$ {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)
Fig. 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
Fig. 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 nm (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
Fig. 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
Fig. 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$ \text{μm} $的电场演化曲线. 当金膜厚度为20 nm (b), 30 nm (c)与40 nm (d)时, –10$ {{\mathrm{f}}{\mathrm{s}}}^{2} $飞秒激光脉冲激发的SPP在传播3$ \text{μm} $后电场强度与0 $ {{\mathrm{f}}{\mathrm{s}}}^{2} $飞秒激光脉冲激发的SPP传播3 $ \;\text{μm} $后的电场演化曲线对比.
Fig. A1. (a) Electric field evolution curves for SPP propagation 3$ \text{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 $ {{\mathrm{f}}{\mathrm{s}}}^{2} $ femtosecond laser pulse after propagation of 3 $ \text{μm} $ with the electric field evolution curves of SPP excited by a 0 $ {{\mathrm{f}}{\mathrm{s}}}^{2} $ femtosecond laser pulse after propagation of 3 $ \text{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/$ {{\mathrm{f}}{\mathrm{s}}}^{2} $ 传播长度/$ \text{μm} $ –10 0.6 –20 1.3 –30 2.0 –40 2.7 –60 4.1 
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