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Theoretical analysis of tunable double plasmon induced transparency in three-ellipse-shaped resonator coupled waveguide

Gu Xin Zhang Hui-Fang Li Ming-Yu Chen Jun-Ya He Ying

Gu Xin, Zhang Hui-Fang, Li Ming-Yu, Chen Jun-Ya, He Ying. Theoretical analysis of tunable double plasmon induced transparency in three-ellipse-shaped resonator coupled waveguide. Acta Phys. Sin., 2022, 71(24): 247301. doi: 10.7498/aps.71.20221365
Citation: Gu Xin, Zhang Hui-Fang, Li Ming-Yu, Chen Jun-Ya, He Ying. Theoretical analysis of tunable double plasmon induced transparency in three-ellipse-shaped resonator coupled waveguide. Acta Phys. Sin., 2022, 71(24): 247301. doi: 10.7498/aps.71.20221365

Theoretical analysis of tunable double plasmon induced transparency in three-ellipse-shaped resonator coupled waveguide

Gu Xin, Zhang Hui-Fang, Li Ming-Yu, Chen Jun-Ya, He Ying
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  • The tunable double plasmon-induced transparency (PIT) effects are investigated in a waveguide coupled by the three ellipse-shaped resonators. By the finite element method, we study the influences of coupling modes of the three ellipse-shaped resonators, waveguide structure parameters and the refractive indices of dielectric in three ellipse-shaped resonators on double PIT effects. The waveguide structure consists of three ellipse-shaped resonators, and is similar to a four-level structure of the atomic system. The bottom ellipse-shaped resonator can be named a bright mode, the middle and top ellipse-shaped resonators each can be seen as a dark mode. In order to obtain an ideal double PIT transparency window, we also numerically analyze the optical transmission characteristics of structures of several three-ellipse-shaped resonator coupled waveguides. Furthermore, we mainly discuss the transmission spectra in the better three-ellipse-shaped resonator coupled waveguide structure as a function of the radii of the long axis in ellipse-shaped resonators, the coupling distance between the bottom ellipse-shaped resonator and the bus waveguide, the coupling distance between ellipse-shaped resonators, and the symmetry broken degree. In addition, we also consider the effect of the refractive indices of dielectric in three ellipse-shaped resonators on double PIT spectra. It is found that the transmission spectra in the three-ellipse-shaped resonator coupled waveguide have obvious red shift when the refractive indices of dielectric in the three ellipse-shaped resonators increase. All the simulation results may provide the theoretical basis for the potential application of multiple PIT in plasma switches and sensors.
      PACS:
      73.20.Mf(Collective excitations (including excitons, polarons, plasmons and other charge-density excitations))
      73.40.Rw(Metal-insulator-metal structures)
      42.50.Gy(Effects of atomic coherence on propagation, absorption, and Amplification of light; electromagnetically induced transparency and Absorption)
      42.79.Gn(Optical waveguides and couplers)
      Corresponding author: Zhang Hui-Fang, hfzhang1967@shu.edu.cn
    • Funds: Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11804219).

    表面等离极化激元(surface plasmon polaritons, SPPs)是局域在金属表面的一种自由电子和光子相互作用形成的电磁振荡, 其振幅在垂直于金属界面向两侧呈指数衰减[1,2]. SPPs能很好地突破衍射极限的限制, 在纳米级尺度上实现对光的调控. 为了传输SPPs, 提出了许多波导结构, 其中金属-介质-金属(metal-insulator-metal, MIM)型波导结构具有结构简单、易于加工、集成度高等优点, 被认为是最有前景的纳米集成光电路波导结构之一[3-5]. 目前, 已有众多基于MIM波导结构的光学器件被相继提出, 如滤波器[6-8]、传感器[9-11]以及解复用器[12,13]等.

    电磁诱导透明(electromagnetically induced transparency, EIT)是三能级原子系统与外电磁场之间相互作用产生的一种干涉光学现象, 使得材料在原本的吸收带上形成一个窄带的传输窗口[14]. 原子系统 EIT 效应能够产生慢光效应, 减慢光的传播速度, 在制造光存储器件等方面有着十分重要的意义. 长期以来, 人们一直关注原子气体、固体以及冷原子中的EIT现象. 最近, 许多经典体系如超材料[15,16]、超表面[17,18]和波导[19-21]等体系被用来实现类 EIT 效应. 等离子体诱导透明(plasmon-induced transparency, PIT)是一种基于SPPs与外界电磁场之间相互作用[22], 从而产生和原子EIT 相似的现象. 目前, PIT已广泛用于光学存储、非线性光学和光信号处理中. PIT基于MIM型表面等离激元光波导结构而产生的光学效应已成为人们关注的焦点. Han等[23]设计含有圆盘谐振器的表面等离子体光波导系统以实现动态可调的 PIT 效应, 并研究了慢光效应. Niu等[24]提出一种基于MIM波导的齿轮形环形谐振器结构, 以实现双重和三重PIT效应. Wang等[25]使用一个周期阵列的两个失谐谐振腔孔径侧耦合MIM波导以拓宽PIT窗口. Wen等[26]在T形腔、Li等[22]在U形腔、Cao等[27]在stubs腔中数值模拟单重及多重PIT效应; Wang等[28]在椭圆谐振腔中提出了可调谐对称线形单重PIT窗口. 多个SPPs谐振腔之间的相互作用可以产生更多的透明窗口, 提供更多的灵活性来调整等离子体系统的光学性能.

    本文主要研究三椭圆谐振腔耦合波导中可调谐双重PIT效应, 利用有限元法(finite element method, FEM), 分别讨论三椭圆谐振腔耦合方式、波导结构参数及椭圆腔填充材料有效折射率对双重PIT效应的影响. 通过数值分析多种三椭圆谐振腔耦合波导结构PIT光谱和电场分布, 找到三个椭圆腔较佳的耦合方式, 以获得较理想的双重PIT透明窗口. 针对此波导结构, 讨论双PIT光谱随椭圆腔长轴半径、耦合距离、对称破缺度, 以及椭圆腔填充材料有效折射率的变化规律. 结果表明, 可通过灵活改变波导结构几何参数以及填充材料有效折射率等来调控双PIT效应. 多重透明窗口的数值模拟为PIT在等离子体开关及传感器方面潜在应用提供理论基础.

    三椭圆谐振腔耦合波导结构由MIM主波导和3个椭圆谐振腔组成, 如图1(a)所示. 相对于圆形腔来说, 椭圆腔与主波导间耦合效果更显著, 较多的能量可直接耦合到底部椭圆腔, 则底部椭圆腔称为亮模. 顶部椭圆谐振腔可以被亮模激发, 而不能被主波导直接耦合, 所以顶部椭圆腔称为暗模. 图1(a)中部椭圆谐振腔称为亮模还是暗模, 由其所在的位置决定. 当椭圆腔和主波导之间的耦合距离远大于SPPs在金属中的穿透深度(在银中约为25—30 nm)[29] 时, 主波导不能直接耦合椭圆腔, 称为暗模. 如果耦合距离小于穿透深度时, 称为亮模. 左侧中部椭圆腔与主波导之间的距离大于SPPs在银中的穿透深度, 称为暗模, 则此波导结构是由1个亮模和2个暗模组成.

    图 1 (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s1 = s2); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为849, 855, 860, 866, 883 nm时的电场分布\r\nFig. 1. (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s1 = s2); (b) transmission spectra of the two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 849, 855, 860, 866, 883 nm, respectively.
    图 1  (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s1 = s2); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为849, 855, 860, 866, 883 nm时的电场分布
    Fig. 1.  (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s1 = s2); (b) transmission spectra of the two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 849, 855, 860, 866, 883 nm, respectively.

    图1(a)主波导宽度W设为50 nm, 以确保只有基模(TM0)可以在此波导结构中传播[30]. 底部椭圆腔与主波导间耦合距离为H, 底部与顶部椭圆腔间耦合距离为h, 皆为10 nm. 左侧中部椭圆腔和右侧顶部及右侧底部椭圆腔间纵向距离分别为s1s2 (此波导中s1 = s2). 左侧中部椭圆腔和右侧椭圆腔间耦合距离d为20 nm. 左侧中部、右侧顶部和底部三个椭圆腔的长轴半径分别为r1, r2r3, 短轴半径都为80 nm. 若无特别说明, 本文中三椭圆腔长轴半径皆选取为r1 = r2 = r3 = 200 nm.

    主波导和三椭圆腔内的填充介质设为空气, 其相对介电常数等于1. 而其中的基底金属为银, 银的相对介电常数通常可用 Drude 模型表示[31]:

    ε(ω)=εω2p(ω2+iωγ)
    (1)

    其中, 角频率无限大时的介电常数ε= 3.7, 等离子体频率ωp= 1.38×1016 Hz, γ = 2.73×1013 Hz为自由电子阻尼衰减频率, ω为入射光的角频率. 使模型处于x-y平面, 用FEM仿真软件COMSOL Multiphysics进行计算和模拟, 在仿真过程中, 设置了完美匹配层吸收边界条件. 网格剖分时, 一般要求网格最大不超过入射光波长的1/5.

    图1(b)是波导结构透射光谱的数值模拟结果, 当只有右侧顶部和底部两个椭圆腔时, 其透射谱如图1(b)黑色虚线所示[28]. 透射谱中有1个透射峰和2个透射谷, 峰值波长约为863 nm, 谷值波长分别约为851 nm和883 nm. 当在左侧中部再放置一个椭圆腔, 构成三椭圆谐振腔波导结构, 其透射谱如图1(b)红色实线所示. 从图1(b)中观察到, 在波长为863 nm处透射峰分裂为两个透射峰, 一个蓝移到855 nm左右, 一个红移到866 nm左右, 相应透射率分别约为0.777和0.817, 中间形成了一个窄的透射谷, 所对应波长约为860 nm, 透射率约为0.495, 出现了双重PIT效应. 新加的左侧中部椭圆腔是一个暗模, 其共振波长位于原透射峰波长位置附近, 接近右侧顶部椭圆腔共振波长, 二者之间的耦合效应将明显影响波导结构透射谱. 在波长为851 nm处透射谷位置稍微蓝移到849 nm且透射率略微升高, 在波长为883 nm处透射谷的位置和透射率基本不变. 这说明左侧中部椭圆腔与右侧底部椭圆腔之间耦合效应在此波长处非常小, 波导结构频谱基本没有发生改变. 为了更好地理解PIT透射特性的物理机制, 进一步讨论三椭圆腔波导结构中的电场分布. 图1(c), (e), (g)分别显示三个透射谷波长位置处电场分布, 可以看出, 在波长849, 883 nm处右侧底部椭圆腔受到一定的激励, 在右侧几乎没有光场通过, 在波长860 nm处右侧底部椭圆腔受到较小激励, 因此有少部分的SPPs通过. 图1(d)图1(f)分别显示了波长855 nm和866 nm处的电场分布, 左侧中部和右侧顶部两个椭圆腔被有效激发, 电场大多局域在这两个椭圆腔中, 右侧底部椭圆腔电场较弱, 从而在透射谱中形成明显的透明窗口.

    图2(a)显示的三椭圆谐振腔波导结构也由1个亮模和2个暗模组成, 新加的左侧椭圆腔与右侧顶部椭圆腔等高. 保持其他参数不变, 此波导结构透射谱如图2(b)所示, 其中黑色虚线是图1(b)中黑色虚线的重复, 红色实线则表示此三椭圆谐振腔波导结构的透射谱. 从图2(b)发现, 在原透射谷波长为851 nm处出现一个新的透射峰, 相应透射率约为0.809, 产生明显的PIT效应, 导致原来的透射谷分裂为两个透射谷, 一个蓝移到844 nm左右, 一个红移到867 nm左右, 相应透射率约为0.223和0.098, 原透明窗口则红移到波长约为877 nm处, 透射率为0.776. 新加的左侧顶部椭圆腔是一个暗模, 其共振波长在原透射谷附近, 且与右侧顶部椭圆腔大致相同, 两个暗模之间强耦合效应将影响此波导结构透射谱, 并产生显著的双重PIT效应. 而在波长为883 nm处透射谷位置红移到888 nm, 且相应的透射率约为0.158, 这说明新加的左侧顶部椭圆腔共振波长不在此透射谷波长位置附近, 且与右侧底部椭圆腔之间的耦合效应较弱, 故对此波长附近处波导结构的透射谱影响不大. 图2(c), (e), (g)分别是波长为844, 867, 888 nm处的电场分布, 底部椭圆腔受到较强激励, 几乎没有SPPs从右侧输出, 因此产生明显的透射谷. 图2(d)图2(f)分别显示波长为851, 877 nm处的电场分布, 两顶部椭圆腔被有效地激发, 内部形成增强的场束缚, 而右侧底部椭圆腔内部由于三条传播路径之间SPPs的相消干涉而被抑制, 受到较弱激励, 大量的SPPs可传输到输出端口, 从而在透射谱中形成明显的透射峰.

    图 2 (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s1 = 0); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为844, 851, 867, 877, 888 nm时的电场分布\r\nFig. 2. (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s1 = 0); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 844, 851, 867, 877, 888 nm, respectively.
    图 2  (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s1 = 0); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为844, 851, 867, 877, 888 nm时的电场分布
    Fig. 2.  (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s1 = 0); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 844, 851, 867, 877, 888 nm, respectively.

    图3(a)所示的波导结构由2个亮模和1个暗模组成, 新加的左侧椭圆腔与右侧底部椭圆腔等高. 在图3(b)中, 黑色虚线是图1(b)中黑色虚线的重复, 红色实线表示保持其他参数不变的情况下, 此三椭圆谐振腔波导结构的透射谱. 从图3(b)中观察到, 原透射谷从波长851 nm蓝移到846 nm处, 且相应透射率约为0.028, 这说明新加的左侧底部椭圆腔与右侧顶部椭圆腔之间耦合效应较弱, 故对此波长位置附近处波导结构的透射谱影响不大. 但在波长为883 nm透射谷处出现一个新的透射峰, 产生明显的PIT效应, 导致原来的透射谷分裂为两个透射谷, 一个蓝移到866 nm左右, 一个红移到897 nm左右, 相应透射率约为0.078和0.151. 二个透明窗口相对于原透明窗口一个蓝移到波长约858 nm处, 一个红移到波长约883 nm处, 相应透射率约为0.643和0.653. 新加的左侧底部椭圆腔是一个亮模, 其共振波长与右侧底部椭圆腔一致, 两个底部椭圆腔之间强耦合效应将明显影响此波导结构透射谱, 并产生双重PIT效应. 图3(c), (e), (g)分别显示波长为846, 866, 897 nm处电场分布, 电场大多局域在底部椭圆腔, 几乎没有SPPs从右侧输出, 产生明显的透射谷. 图3(d)图3(f)分别显示波长为858, 883 nm处电场分布, 电场大多被局域在左侧和右侧顶部椭圆腔中, 在透射谱中形成明显的透射峰.

    图 3 (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s2 = 0); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为846, 858, 866, 883, 897 nm时的电场分布\r\nFig. 3. (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s2 = 0); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 846, 858, 866, 883, 897 nm, respectively.
    图 3  (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s2 = 0); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为846, 858, 866, 883, 897 nm时的电场分布
    Fig. 3.  (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s2 = 0); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 846, 858, 866, 883, 897 nm, respectively.

    图4(a)是亮模-暗模-暗模三椭圆谐振腔波导结构, 第三个椭圆腔放在右侧双椭圆腔上方, 中部与顶部椭圆腔之间耦合距离c设为10 nm, 中部和顶部椭圆腔与底部椭圆腔之间横向距离(即对称破缺度)分别为x1x2, 都取为0. 在图4(b)中, 黑色虚线是图1(b)中黑色虚线的再现, 红色实线表示此三椭圆谐振腔波导结构的透射谱. 从图4(b)看出, 原透射谷波长为851 nm处出现一个新的透射峰, 相应透射率约为0.807, 产生明显的PIT效应, 导致原来的透射谷分裂为两个透射谷, 一个蓝移到845 nm左右, 一个红移到867 nm左右, 相应透射率约为0.234和0.099, 原透明窗口则红移到波长约为878 nm处, 透射率为0.782. 顶部椭圆腔是一个暗模, 其与中部椭圆腔之间强耦合效应将明显影响此波导结构透射谱, 并产生双重PIT效应. 在波长为883 nm处透射谷位置红移到889 nm, 透射率约为0.176. 顶部椭圆腔与底部椭圆腔之间耦合效应较弱, 故对此波长附近处波导结构的透射谱影响不大. 图4(c), (e), (g)分别显示波长为845, 867, 889 nm处的电场分布, 可看出底部椭圆腔受到较大的激励, 几乎没有SPPs从右侧输出, 产生明显的透射谷. 图4(d)图4(f)分别显示了波长为851, 878 nm处的电场分布, 电场大多被局域在顶部和中部两个椭圆腔中, 这两个椭圆腔被有效地激发, 底部椭圆腔内部电场因SPPs的相消干涉而被抑制, 在透射谱中形成明显的透射峰.

    图 4 (a) 三椭圆谐振腔耦合波导结构(三椭圆腔在一条直线上竖直放置); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱; (c)−(g) 三椭圆腔波导结构中波长分别为845, 851, 867, 878, 889 nm时的电场分布\r\nFig. 4. (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed vertically in a straight line); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 845, 851, 867, 878, 889 nm, respectively.
    图 4  (a) 三椭圆谐振腔耦合波导结构(三椭圆腔在一条直线上竖直放置); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱; (c)−(g) 三椭圆腔波导结构中波长分别为845, 851, 867, 878, 889 nm时的电场分布
    Fig. 4.  (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed vertically in a straight line); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 845, 851, 867, 878, 889 nm, respectively.

    图5(a)是一种轴对称三椭圆腔波导结构. 在图2(a)中, 左移右底部椭圆腔直至其几何中心O3位于两个顶部椭圆腔几何中心连线O1O2的中线上, O1O3O2 三点构成倒等腰三角形, 波导结构呈现轴对称性, 由1个亮模和2个暗模组成. 图5(b)中黑色虚线是图2(a)所示三椭圆腔波导结构透射谱, 红色实线为此轴对称三椭圆腔波导结构透射谱. 由图5(b)看出, 轴对称波导结构透射谱中只有一个较窄的透射峰, 峰值波长约为876 nm, 透射率约为0.719. 两个透射谷谷值波长分别约为865 nm和883 nm, 透射率约为0.052和0.190. 在图5(a)中, 底部椭圆腔与两个顶部椭圆腔构成对称分布, 它与两个顶部椭圆腔耦合作用相同, 两个顶部椭圆腔具有相同谐振情况, 此轴对称三椭圆腔耦合波导结构形成双共振频率系统, 退化成三能级原子系统, 故只观察到一个透射峰. 但与图1(b)中双椭圆腔波导结构透射谱相比, 此透射谱明显变窄并出现失谐态. 这是因为底部椭圆腔与两个顶部椭圆腔有较大的对称破缺度, 导致底部椭圆腔与两个顶部椭圆腔耦合作用变弱. 两个顶部椭圆腔都为暗模, 彼此之间耦合效应较弱, 透射峰变窄. 图5(c)图5(e)分别显示波长为865 nm和883 nm处的电场分布,可看到底部椭圆腔受到强烈激励, 几乎没有SPPs从右侧输出, 产生明显的透射谷. 图5(d)显示波长为876 nm处电场分布, 电场大多被局域在顶部椭圆腔中, 形成明显的透射峰.

    图 5 (a) 轴对称三椭圆谐振腔耦合波导结构(三椭圆腔倒等腰三角形放置且O3O1 = O3O2); (b) 非轴对称(黑色虚线)和轴对称(红色实线)三椭圆腔波导结构透射谱; (c)−(e) 轴对称波导结构中波长分别为865, 876, 883 nm时的电场分布\r\nFig. 5. (a) Schematic diagram of the axisymmetric three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed in an inverted isosceles triangle and O3O1 = O3O2); (b) transmission spectra of the non-axisymmetric (black dash) and the axisymmetric (red solid) three ellipse-shaped resonators waveguide structure; (c)−(e) electric field distribution of the axisymmetric three ellipse-shaped resonators waveguide structure at wavelength of 865, 876, 883 nm, respectively.
    图 5  (a) 轴对称三椭圆谐振腔耦合波导结构(三椭圆腔倒等腰三角形放置且O3O1 = O3O2); (b) 非轴对称(黑色虚线)和轴对称(红色实线)三椭圆腔波导结构透射谱; (c)−(e) 轴对称波导结构中波长分别为865, 876, 883 nm时的电场分布
    Fig. 5.  (a) Schematic diagram of the axisymmetric three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed in an inverted isosceles triangle and O3O1 = O3O2); (b) transmission spectra of the non-axisymmetric (black dash) and the axisymmetric (red solid) three ellipse-shaped resonators waveguide structure; (c)−(e) electric field distribution of the axisymmetric three ellipse-shaped resonators waveguide structure at wavelength of 865, 876, 883 nm, respectively.

    图6(a)是另一种轴对称三椭圆腔波导结构. 在图3(a)中, 左移顶部椭圆腔直至其几何中心O2位于两个底部椭圆腔几何中心连线O1O3的中线上, O1O2O3三点构成等腰三角形, 波导结构呈现轴对称性, 由2个亮模和1个暗模组成. 在图6(b)中, 黑色虚线是图3(a)所示三椭圆腔波导结构透射谱, 红色实线表示此轴对称三椭圆谐振腔波导结构透射谱. 与黑色虚线不同的是, 此轴对称波导结构透射谱中只有一个较宽的透明窗口, 峰值波长约为879 nm, 透射率约为0.682. 两个透射谷谷值波长分别约为853 nm和895 nm, 透射率约为0.010和0.154. 在图6(a)中, 由于波导结构呈轴对称分布, 顶部椭圆腔与两个底部椭圆腔耦合作用相同, 两个底部椭圆腔具有相同谐振情况, 此对称三椭圆腔耦合波导结构形成双共振频率系统, 退化成三能级原子系统, 只观察到一个明显变宽的透明窗口. 这是因为能量不仅可通过两个并行通道从两个底部椭圆腔耦合到顶部椭圆腔, 而且两个底部椭圆腔之间也有强烈的耦合效应, 也可产生PIT效应, 这不是原始PIT效应的简单叠加, 从而导致PIT效应的透明窗口变宽[32,33]. 图6(c)图6(e)分别显示两个透射谷波长处电场分布, 可看出底部两椭圆腔受到较大的激励, 几乎没有SPPs从右侧输出. 图6(d)显示透射峰波长处电场分布, 这时底部椭圆腔受到的激励变小, 则SPPs可以从右侧输出, 出现单重PIT效应.

    图 6 (a) 轴对称三椭圆谐振腔耦合波导结构(三椭圆腔正等腰三角形放置且O2O1 = O2O3); (b) 非轴对称(黑色虚线)和轴对称(红色实线)三椭圆腔波导结构透射谱; (c)−(e) 轴对称波导结构中波长分别为853, 879, 895 nm时的电场分布\r\nFig. 6. (a) Schematic diagram of the axisymmetric three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed in a positive isosceles triangle and O2O1 = O2O3); (b) transmission spectra of the non-axisymmetric (black dash) and the axisymmetric (red solid) three ellipse-shaped resonators waveguide structure; (c)−(e) electric field distribution of the axisymmetric three ellipse-shaped resonators waveguide structure at wavelength of 853, 879, 895 nm, respectively.
    图 6  (a) 轴对称三椭圆谐振腔耦合波导结构(三椭圆腔正等腰三角形放置且O2O1 = O2O3); (b) 非轴对称(黑色虚线)和轴对称(红色实线)三椭圆腔波导结构透射谱; (c)−(e) 轴对称波导结构中波长分别为853, 879, 895 nm时的电场分布
    Fig. 6.  (a) Schematic diagram of the axisymmetric three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed in a positive isosceles triangle and O2O1 = O2O3); (b) transmission spectra of the non-axisymmetric (black dash) and the axisymmetric (red solid) three ellipse-shaped resonators waveguide structure; (c)−(e) electric field distribution of the axisymmetric three ellipse-shaped resonators waveguide structure at wavelength of 853, 879, 895 nm, respectively.

    除此之外, 还可以把三椭圆腔分别放置在主波导两侧, 发现其透射谱的数值模拟结果不太理想, 则不再赘述. 综上所述, 这6种波导结构中三椭圆腔耦合方式不同, 使得其透射特性都不同. 对于轴对称三椭圆腔波导结构, 只观察到单重PIT效应, 而其余4种波导结构, 都出现双重PIT效应. 相比之下, 图2图4这两种波导结构双PIT窗口透射率较高, 半高全宽较窄, 透射特性较理想.

    以上都是关于三椭圆腔耦合波导结构光学透射特性的理论分析, 为了实际应用价值, 下面简要讨论其实验可行性. 根据科研工作者所设计的关于类EIT现象实验装置图[34-36], 我们尝试提出三椭圆腔波导结构双PIT效应的实验构思示意简图, 如图7所示. 利用平面镜使外部光源进入显微镜, 将光波注入到三椭圆腔波导结构模型样品上. 输出光通过显微镜后与透镜光纤耦合, 通过电荷耦合器件(CCD)进入显示器. 再用光功率计采集透射光束, 在光谱仪上得到器件结构的光学特性.

    图 7 三椭圆腔耦合波导结构双PIT效应实验设计示意简图\r\nFig. 7. Schematic diagram of experimental design of double PIT effects for the three ellipse-shaped resonators coupled waveguide structure.
    图 7  三椭圆腔耦合波导结构双PIT效应实验设计示意简图
    Fig. 7.  Schematic diagram of experimental design of double PIT effects for the three ellipse-shaped resonators coupled waveguide structure.

    根据第2节的讨论, 图2(a)图4(a)所示波导结构的透射特性都较理想. 这里想要说明的是, 这两种波导结构透射特性随椭圆腔长轴半径、耦合距离及椭圆腔填充材料有效折射率等结构参数的变化规律大致相同. 但图4(a)的亮模-暗模-暗模三椭圆腔波导结构更具有代表性, 且具有同向和反向对称破缺度等更多的结构参数来调谐双PIT效应. 为避免赘述, 图2(a)波导结构的数值模拟结果在此不一一阐明, 故只详细讨论图4(a)波导结构透射特性随结构参数的变化规律.

    针对图4(a)所示的波导结构, 保持x1 = x2 = 0,H = h = c = 10 nm不变, 研究椭圆腔长轴半径对波导结构透射谱的影响, 如图8所示. 当r1 = r2 = r3 = 200 nm时, 波导结构透射光谱显示为相对对称的双PIT光谱线型, 正如图8(b), (e), (h), (k)所示. 在图8(a)(c)中, 保持r2 = r3 = 200 nm不变, 讨论双PIT现象随r1的变化情况. 当r1 = 190 nm或210 nm, 在图8(a)图8(c)观察到明显的非对称谱线, 这些非对称谱线被称为Fano共振, 图8(a)图8(c)中光谱线型显示为镜像对称性. 顶部椭圆腔为一个暗模, 其长轴半径的变化明显影响顶部椭圆腔谐振波长[28]. 当r1减小时, 顶部椭圆腔谐振波长蓝移, 偏离且小于中部和底部椭圆腔谐振波长, 此暗模在波长较小处出现明显的失谐态且透射率增大; 当r1增大时, 顶部椭圆腔谐振波长红移, 偏离且大于底部和中部椭圆腔谐振波长, 此暗模在波长较大处出现明显的失谐态且透射率增大. 总之, 在亮模-暗模双椭圆腔波导结构中, 又加了一个顶部椭圆腔(暗模), 形成亮模-暗模-暗模结构. 当r1减小或增大时, 新增暗模的谐振波长偏离亮模-暗模波导结构谐振波长较远, 产生失谐现象, 但对亮模-暗模波导结构共振耦合透明窗口影响不大, 耦合透明谱线相对对称.

    图 8 当改变椭圆腔长轴半径时, 三椭圆谐振腔波导结构的透射谱 (a)−(c) 改变顶部椭圆腔长轴半径r1;  (d)−(f) 改变中部椭圆腔长轴半径r2; (g)−(i) 改变底部椭圆腔长轴半径r3; (j)—(l) 改变r1和r3\r\nFig. 8. Transmission spectra in the three ellipse-shaped resonators coupled waveguide structure when changing the long-axis radius of the elliptical cavity: (a)−(c) Change radius of the long axis r1 in the top ellipse-shaped resonator; (d)−(f) change r2 in the middle ellipse-shaped resonator; (g)−(i) change r3 in the bottom ellipse-shaped resonator; (j)−(l) change r1 and r3.
    图 8  当改变椭圆腔长轴半径时, 三椭圆谐振腔波导结构的透射谱 (a)−(c) 改变顶部椭圆腔长轴半径r1; (d)−(f) 改变中部椭圆腔长轴半径r2; (g)−(i) 改变底部椭圆腔长轴半径r3; (j)—(l) 改变r1r3
    Fig. 8.  Transmission spectra in the three ellipse-shaped resonators coupled waveguide structure when changing the long-axis radius of the elliptical cavity: (a)−(c) Change radius of the long axis r1 in the top ellipse-shaped resonator; (d)−(f) change r2 in the middle ellipse-shaped resonator; (g)−(i) change r3 in the bottom ellipse-shaped resonator; (j)−(l) change r1 and r3.

    图8(d)(f)中, 保持r1 = r3 = 200 nm不变, 研究r2对双PIT现象的影响. 当r2 = 190 nm或210 nm时, 在图8(d)图8(f)仍可以观察到明显的非对称谱线, 中部椭圆腔长轴半径的变化使得谱线线型的镜像对称性消失. 中部椭圆腔与顶部椭圆腔和底部椭圆腔之间都有较强的耦合作用, 三个椭圆腔之间相消干涉将明显影响透射谱的变化. 当r2减小时, 中部椭圆腔谐振波长蓝移, 且小于顶部椭圆腔谐振波长小于底部椭圆腔谐振波长, 两个暗模出现明显的失谐态且吸收减少, 如图8(d)所示; 当r2增大时, 中部椭圆腔谐振波长红移, 且大于底部椭圆腔谐振波长大于顶部椭圆腔谐振波长, 中部和顶部椭圆腔出现明显的失谐态且吸收减少, 如图8(f)所示. 底部和顶部椭圆腔长轴半径相等, 则其谐振波长相近, 二者共振耦合产生透明窗口, 由于中部椭圆腔对底部和顶部椭圆腔都有耦合作用, 且都有失谐, 从而影响透明窗口也产生失谐现象, 耦合透明谱线不再对称.

    图8(g)(i)中, 保持r1 = r2 = 200 nm不变,研究r3对双PIT现象的影响. 当r3 = 190 nm或210 nm时, 在图8(g)图8(i)可以观察到非对称的透射谱线, 图8(g)图8(i)中光谱线型大致呈镜像对称性. 底部椭圆腔为一个亮模, 其长轴半径的变化影响底部椭圆腔谐振波长. 当r3减小时, 底部椭圆腔共振波长减小, 亮模移动到波长较小处, 出现失谐现象且吸收增强, 两个暗模吸收减小, 如图8(g)所示; 当r3增大时, 底部椭圆腔共振波长增大, 亮模移动到波长较大处, 出现失谐现象且吸收增强, 两个暗模吸收减少, 如图8(i)所示. 中部和顶部椭圆腔长轴半径相等, 谐振波长也相近, 二者共振耦合产生透明窗口, 由于底部椭圆腔和中部椭圆腔间耦合作用且有失谐, 从而影响透明窗口也产生失谐现象, 耦合透明谱线不再对称.

    图8(j)(l)中, 保持r2 = 200 nm不变, 研究r1r3同时变化对双PIT现象的影响. 在图8(j)图8(l)可以观察到非对称的透射谱线, 图8(j)图8(l)中光谱线型大致呈镜像对称性. 在图8(j)中, 三个椭圆腔的长轴半径都不相同(r1 = 190 nm, r2 = 200 nm, r3 = 210 nm), 则三个腔的谐振波长也都不相同, 与图8(a)相比, 失谐现象更加显著, 图8(l)相比于图8(c)也是如此. 综上所述, 三个椭圆腔长轴半径的变化显著影响双PIT透射谱, 即可通过改变长轴半径来调控类EIT现象.

    针对图4(a)所示的波导结构, 保持r1 = r2 = r3 = 200 nm不变, 分别讨论双PIT效应随耦合距离H, h, c, 对称破缺度x1x2, 以及三椭圆腔内填充材料有效折射率n的变化情况, 见图9. 为描述类EIT现象, 引入参数品质因子Q = λ0/λFWHM, 其中, λ0λFWHM分别是透射峰峰值波长和半高全宽. 在图9(a)(e)中, 设n为1.

    图 9 三椭圆谐振腔波导结构透射谱随结构参数的变化 (a) 当x1 = x2 = 0, h = c = 10 nm时, 透射谱随H的变化; (b) 当x1 = x2 = 0, H = c = 10 nm时, 透射谱随h的变化; (c) 当x1 = x2 = 0, H = h = 10 nm时, 透射谱随c的变化; (d), (e) 当H = h = c = 10 nm时, 透射谱随x1和x2的变化; (f) 当x1 = x2 = 0, H = h = 10 nm时, 透射谱随n的变化\r\nFig. 9. Transmission spectra in the three ellipse-shaped resonators coupled waveguide structure with different parameters: (a) With H when x1 = x2 = 0, h = c = 10 nm; (b) with h when x1 = x2 = 0, H = c =10 nm; (c) with c when x1 = x2 = 0, H = h = 10 nm; (d), (e) with x1 and x2 when H = h = c = 10 nm; (f) with n when x1 = x2 = 0, H = h = 10 nm.
    图 9  三椭圆谐振腔波导结构透射谱随结构参数的变化 (a) 当x1 = x2 = 0, h = c = 10 nm时, 透射谱随H的变化; (b) 当x1 = x2 = 0, H = c = 10 nm时, 透射谱随h的变化; (c) 当x1 = x2 = 0, H = h = 10 nm时, 透射谱随c的变化; (d), (e) 当H = h = c = 10 nm时, 透射谱随x1x2的变化; (f) 当x1 = x2 = 0, H = h = 10 nm时, 透射谱随n的变化
    Fig. 9.  Transmission spectra in the three ellipse-shaped resonators coupled waveguide structure with different parameters: (a) With H when x1 = x2 = 0, h = c = 10 nm; (b) with h when x1 = x2 = 0, H = c =10 nm; (c) with c when x1 = x2 = 0, H = h = 10 nm; (d), (e) with x1 and x2 when H = h = c = 10 nm; (f) with n when x1 = x2 = 0, H = h = 10 nm.

    图9(a)中, 当x1 = x2 = 0, h = c = 10 nm时, 讨论H对双PIT透射谱的影响. 随着H从10 nm增加到40 nm, 仍然可观察到双PIT现象, 但双PIT峰值即透射率逐渐增大, 峰值波长变化不大, 透射峰半高全宽增大, 则两个透明窗口的Q值减小. 在图9(b)中, 在x1 = x2 = 0, H = c = 10 nm的情况下, 讨论h对双PIT透射谱的影响. 当h从10 nm增加到25 nm时, 双透明窗口的透射率逐渐减小, PIT效应逐渐减弱. 耦合距离越小, 耦合强度越强. 随着h增加, 左侧窗口峰值波长略微蓝移, 半高全宽变化甚微, Q值略微减小; 右侧窗口峰值波长蓝移, 半高全宽明显减少, Q值增大, 即耦合距离h对左侧窗口影响较小, 对右侧窗口影响显著. 在图9(c)中, x1 = x2 = 0, H = h = 10 nm时, 讨论c对双PIT透射谱的影响. 当c从10 nm增加到40 nm时, 左侧透射峰红移, 峰值下降, 半高全宽减少, Q值增大; 右侧透射峰蓝移, 峰值略微上升, 半高全宽明显增大, Q值减少. 在图9(d)(e)中, H = h = c = 10 nm时, 讨论x1x2对双PIT透射谱的影响. 设椭圆腔相对底部椭圆腔向右偏为正, 向左偏为负. 在图9(d)中, x1x2同方向变化, x1为0, 20, 40, 60 nm, x2为0, 40, 80, 120 nm. 随着x1x2的增大, 两透射峰峰值都明显降低. 左侧透射峰出现明显的失谐态, 峰值波长红移, 半高全宽减少, Q值增大; 右侧透射峰没有出现失谐态, 峰值波长蓝移, 半高全宽明显减小, Q值略微增大. 在图9(e)中, x1x2呈反方向变化, x1为0, 20, 40, 60 nm, x2为0, –20, –40, –60 nm. 随着x1x2的变化, 底部与中部椭圆腔横向距离增大, 耦合作用变弱, 中部与顶部椭圆腔之间横向距离加倍增大, 耦合作用更弱, 左侧透射峰红移, 峰值明显减小, 且逐渐消失; 右侧透射峰蓝移, 峰值略微减小, 出现单重PIT效应. 在图9(f)中, 在x1 = x2 = 0, H = h = c = 10 nm的情况下, 研究n对双PIT透射谱的影响. 随着n的增大, 透射谱整体发生明显的红移. 综上所述, 可通过灵活改变三椭圆谐振腔耦合波导结构几何参数以及椭圆腔内填充材料有效折射率等来调控类EIT现象.

    本文利用有限元法, 研究了三椭圆腔耦合波导结构中可调谐双重PIT效应. 为了获得较理想的透明窗口, 数值分析了多种三椭圆腔耦合波导结构光学透射特性. 针对最佳波导结构, 分别讨论椭圆腔长轴半径、耦合距离、对称破缺度、及椭圆腔填充材料有效折射率等波导结构参数对双PIT光谱的影响. 结果表明, 当分别改变三个椭圆腔长轴半径时, 双PIT光谱出现失谐现象, 甚至透射光谱线型还显示了镜像对称性. 当增大底部椭圆腔与主波导之间耦合距离时, 双PIT效应明显减弱, Q值减小. 当增大三个椭圆腔之间耦合距离时, 双PIT效应减弱, 一个窗口Q值减小, 另一个Q值增大, 双窗口逐渐靠近. 总之, 耦合距离越小, 耦合强度越强, 双PIT效应愈显著. 当对称破缺度同方向增大时, 一个窗口出现明显的失谐态, 另一个却没有. 双窗口Q值增大, 且逐渐靠近. 当对称破缺度反方向增大, 双PIT效应减弱, 两个窗口逐渐靠近, 出现单重PIT效应. 当椭圆腔填充材料有效折射率增大时, 透射谱发生明显红移. 多重类EIT现象的数值模拟结果可为PIT谱的调谐及在等离子体开关和传感器方面潜在应用提供理论基础.

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  • 图 1  (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s1 = s2); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为849, 855, 860, 866, 883 nm时的电场分布

    Figure 1.  (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s1 = s2); (b) transmission spectra of the two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 849, 855, 860, 866, 883 nm, respectively.

    图 2  (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s1 = 0); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为844, 851, 867, 877, 888 nm时的电场分布

    Figure 2.  (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s1 = 0); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 844, 851, 867, 877, 888 nm, respectively.

    图 3  (a) 三椭圆谐振腔耦合波导结构(三椭圆腔左右放置且s2 = 0); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱;(c)−(g) 三椭圆腔波导结构中波长分别为846, 858, 866, 883, 897 nm时的电场分布

    Figure 3.  (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed left and right and s2 = 0); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 846, 858, 866, 883, 897 nm, respectively.

    图 4  (a) 三椭圆谐振腔耦合波导结构(三椭圆腔在一条直线上竖直放置); (b) 双椭圆(黑色虚线)和三椭圆(红色实线)波导结构透射谱; (c)−(g) 三椭圆腔波导结构中波长分别为845, 851, 867, 878, 889 nm时的电场分布

    Figure 4.  (a) Schematic diagram of three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed vertically in a straight line); (b) transmission spectra of two (black dash) and three (red solid) ellipse-shaped resonators waveguide structure; (c)−(g) electric field distribution of three ellipse-shaped resonators waveguide structure at wavelength of 845, 851, 867, 878, 889 nm, respectively.

    图 5  (a) 轴对称三椭圆谐振腔耦合波导结构(三椭圆腔倒等腰三角形放置且O3O1 = O3O2); (b) 非轴对称(黑色虚线)和轴对称(红色实线)三椭圆腔波导结构透射谱; (c)−(e) 轴对称波导结构中波长分别为865, 876, 883 nm时的电场分布

    Figure 5.  (a) Schematic diagram of the axisymmetric three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed in an inverted isosceles triangle and O3O1 = O3O2); (b) transmission spectra of the non-axisymmetric (black dash) and the axisymmetric (red solid) three ellipse-shaped resonators waveguide structure; (c)−(e) electric field distribution of the axisymmetric three ellipse-shaped resonators waveguide structure at wavelength of 865, 876, 883 nm, respectively.

    图 6  (a) 轴对称三椭圆谐振腔耦合波导结构(三椭圆腔正等腰三角形放置且O2O1 = O2O3); (b) 非轴对称(黑色虚线)和轴对称(红色实线)三椭圆腔波导结构透射谱; (c)−(e) 轴对称波导结构中波长分别为853, 879, 895 nm时的电场分布

    Figure 6.  (a) Schematic diagram of the axisymmetric three ellipse-shaped resonators coupled waveguide structure (three ellipse-shaped resonators are placed in a positive isosceles triangle and O2O1 = O2O3); (b) transmission spectra of the non-axisymmetric (black dash) and the axisymmetric (red solid) three ellipse-shaped resonators waveguide structure; (c)−(e) electric field distribution of the axisymmetric three ellipse-shaped resonators waveguide structure at wavelength of 853, 879, 895 nm, respectively.

    图 7  三椭圆腔耦合波导结构双PIT效应实验设计示意简图

    Figure 7.  Schematic diagram of experimental design of double PIT effects for the three ellipse-shaped resonators coupled waveguide structure.

    图 8  当改变椭圆腔长轴半径时, 三椭圆谐振腔波导结构的透射谱 (a)−(c) 改变顶部椭圆腔长轴半径r1; (d)−(f) 改变中部椭圆腔长轴半径r2; (g)−(i) 改变底部椭圆腔长轴半径r3; (j)—(l) 改变r1r3

    Figure 8.  Transmission spectra in the three ellipse-shaped resonators coupled waveguide structure when changing the long-axis radius of the elliptical cavity: (a)−(c) Change radius of the long axis r1 in the top ellipse-shaped resonator; (d)−(f) change r2 in the middle ellipse-shaped resonator; (g)−(i) change r3 in the bottom ellipse-shaped resonator; (j)−(l) change r1 and r3.

    图 9  三椭圆谐振腔波导结构透射谱随结构参数的变化 (a) 当x1 = x2 = 0, h = c = 10 nm时, 透射谱随H的变化; (b) 当x1 = x2 = 0, H = c = 10 nm时, 透射谱随h的变化; (c) 当x1 = x2 = 0, H = h = 10 nm时, 透射谱随c的变化; (d), (e) 当H = h = c = 10 nm时, 透射谱随x1x2的变化; (f) 当x1 = x2 = 0, H = h = 10 nm时, 透射谱随n的变化

    Figure 9.  Transmission spectra in the three ellipse-shaped resonators coupled waveguide structure with different parameters: (a) With H when x1 = x2 = 0, h = c = 10 nm; (b) with h when x1 = x2 = 0, H = c =10 nm; (c) with c when x1 = x2 = 0, H = h = 10 nm; (d), (e) with x1 and x2 when H = h = c = 10 nm; (f) with n when x1 = x2 = 0, H = h = 10 nm.

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  • Abstract views:  4723
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Publishing process
  • Received Date:  09 July 2022
  • Accepted Date:  05 September 2022
  • Available Online:  12 December 2022
  • Published Online:  24 December 2022

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