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Generation of surface plasmon vortices based on double-layer Archimedes spirals

Wu Li-Xiang Li Xin Yang Yuan-Jie

Generation of surface plasmon vortices based on double-layer Archimedes spirals

Wu Li-Xiang, Li Xin, Yang Yuan-Jie
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  • Surface plasmon polariton (SPP) is an electromagnetic oscillation which is produced by the interaction of free electrons on metal surface with incident light field. Using some specific metal metasurface structures, plasmonic vortices can be easily obtained. The SPP can well confine the light field within the interface between metal and medium. The SPP has two excellent characteristics: the ability to break through the diffraction limit and the strong field enhancing effect. Consequently, it has unique advantages in many applications, such as nanoparticle trapping, near-field imaging and sensing.The key point lies in selecting the incident light and the construction of metasurface structure when we want to produce SPP vortices. Especially the construction of metasurface structure has great influence on the generation of SPP vortices. So far, many structures have been proposed to produce SPP vortices. In 2015, Byoungho Lee research group (Lee S Y, Kim S J, Kwon H, Lee B 2015 IEEE Photonics Technol. Lett. 27 705) designed a structure of double-ring distributed nanoslits which can produce high-order plasmonic vortices by circularly polarized light. In addition, the use of Archimedes slit structure to generate optical vortices is currently a more common method. More recently, a novel plasmonic vortex lens was proposed to produce SPP vortices with arbitrary topological charges. The plasmonic vortex lens consists of an array of gold film nanoslits, and the superposing of the SPP field excited by each nanoslit can produce a composite plasmonic vortex. Here, we propose a novel spiral metasurface structure for generating surface plasmonic vortices. Using the combination of theoretical analysis and finite difference time domain (FDTD) simulation, we find that the topological charge quantity of surface plasmon vortices can be changed by adjusting five parameters, namely, the chirality of incident circularly polarized light, the difference in radius between inner and outer nanocavity array, the rotational velocity factor of nanocavity array, the number of segments of spiral, and the pitch of spiral. This metasurface structure has many adjustable parameters and each parameter can influence the final surface plasmon vortices. Therefore, the plasmonic vortices with an arbitrary topological charge quantity can be generated and manipulated simply by using such a metasurface structure. This work can be expected to have a very broad prospect of applications in super resolution microscopy, quantum cryptography, nanoparticle manipulation, optical data storage, and optical communication.
      Corresponding author: Yang Yuan-Jie, dr.yang2003@uestc.edu.cn
    [1]

    Maier S A 2007 Plasmonics: Fundamentals and applications (Berlin: Springer) pp49–52

    [2]

    Fang N, Lee H, Sun C, Zhang X 2005 Science 308 534

    [3]

    Luo X G, Ishihara T 2004 Opt. Express 12 3055

    [4]

    Langhammer C, Larsson E M, Kasemo B, Zoric I 2010 Nano Lett. 10 3529

    [5]

    Nie S, Emory S R 1997 Science 275 1102

    [6]

    Wurtz G A, Pollard R, Zayats A V 2006 Phys. Rev. Lett. 97 057402

    [7]

    Allen L, Beijersbergen M W, Spreeuw R J C, Woerdman J P 1992 Phys. Rev. A 45 8185

    [8]

    Yang Y J, Thirunavukkarasu G, Babiker M, Yuan J 2017 Phys. Rev. Lett. 119 094802

    [9]

    Chen C F, Ku C T, Tai Y H, Wei P K, Lin H N, Huang C B 2015 Nano Lett. 15 2746

    [10]

    Yu N F, Genevet P, Kats M A, Aieta F, Tetienne J P, Capasso F, Gaburro Z 2011 Science 334 333

    [11]

    Vedantam S, Lee H, Tang J, Conway J, Staffaroni M, Yablonovitch E 2009 Nano Lett. 9 3447

    [12]

    Gorodetski Y, Niv A, Kleiner V, Hasman E 2008 Phys. Rev. Lett. 101 043903

    [13]

    李嘉明, 唐鹏, 王佳见, 黄涛, 林峰, 方哲宇, 朱星 2015 物理学报 64 194201

    Li J M, Tang P, Wang J J, Huang T, Lin F, Fang Z Y, Zhu X 2015 Acta Phys. Sin. 64 194201

    [14]

    Lee S Y, Kim S J, Kwon H, Lee B 2015 IEEE Photonics Technol. Lett. 27 705

    [15]

    Tang B J, Zhang B F, Ding J P 2019 Appl. Opt. 58 833

    [16]

    Hell S W, Wichmann J 1994 Opt. Lett. 19 780

    [17]

    Vallone G, D’AmbrosioV, Sponselli A, Slussarenko S, Marrucci L, Sciarrino F, Villoresi P 2014 Phys. Rev. Lett. 113 060503

    [18]

    Hong Z Y, Zhang J, Drinkwater B W 2015 Phys. Rev. Lett. 114 214301

    [19]

    Marzo A, Caleap M, Drinkwater B W 2018 Phys. Rev. Lett. 120 044301

    [20]

    Tsai W Y, Huang J S, Huang C B 2014 Nano Lett. 14 547

    [21]

    Ding D S, Zhou Z Y, Shi B S, Guo G C 2013 Nat. Commun. 4 2527

    [22]

    Nicolas A, Veissier L, Giner L, Giacobino E 2014 Nat. Photonics 8 234

    [23]

    Wang J, Yang J Y, Fazal I M, Ahmed N 2012 Nat. Photonics 6 488

    [24]

    Bozinovic N, Yue Y, Ren Y X, Tur M 2013 Science 340 1545

    [25]

    Rui G H, Zhan Q W, Cui Y P 2015 Sci. Rep. 5 13732

    [26]

    Yang S Y, Chen W B, Nelson R L, Zhan Q W 2009 Opt. Lett. 34 3048

  • 图 1  螺线分布的超表面结构示意图 (a)螺线的段数$q=1$; (b)螺线的段数$q=2$

    Figure 1.  Schematic diagram of super-surface structure of spiral distribution: (a) Number of segments of helix is equal to 1; (b) number of segments of helix is equal to 2.

    图 2  内外两层螺线之间的距离$d = 0$时的结构示意图、电场强度以及相位分布图 (a)−(c) $\sigma=- 1$; (d)−(f) $\sigma=1$; (g)−(i) $\sigma=1$; (j)−(l) $\sigma=1$

    Figure 2.  The obtained electric field intensity and phase distribution when distance between inner and outer spirals is equal to 0: (a)−(c) $\sigma=- 1$; (d)−(f) $\sigma=1$; (g)−(i) $\sigma=1$; (j)−(l) $\sigma=1$.

    图 3  内外两层螺线之间的距离$d = {\lambda _{\rm SPP}}/2$时的结构示意图、电场强度以及相位分布图 (a)−(c) $\sigma = - 1$; (d)−(f) $\sigma = - 1$; (g)−(i) $\sigma = 1$; (j)−(l) $\sigma = 1$; (m)−(o) $\sigma = 1$

    Figure 3.  The obtained electric field intensity and phase distribution when distance between inner and outer spirals is equal to ${\lambda _{\rm SPP}}/2$: (a)−(c) $\sigma = - 1$; (d)−(f) $\sigma = - 1$; (g)−(i) $\sigma = 1$; (j)−(l) $\sigma = 1$; (m)−(o) $\sigma = 1$.

  • [1]

    Maier S A 2007 Plasmonics: Fundamentals and applications (Berlin: Springer) pp49–52

    [2]

    Fang N, Lee H, Sun C, Zhang X 2005 Science 308 534

    [3]

    Luo X G, Ishihara T 2004 Opt. Express 12 3055

    [4]

    Langhammer C, Larsson E M, Kasemo B, Zoric I 2010 Nano Lett. 10 3529

    [5]

    Nie S, Emory S R 1997 Science 275 1102

    [6]

    Wurtz G A, Pollard R, Zayats A V 2006 Phys. Rev. Lett. 97 057402

    [7]

    Allen L, Beijersbergen M W, Spreeuw R J C, Woerdman J P 1992 Phys. Rev. A 45 8185

    [8]

    Yang Y J, Thirunavukkarasu G, Babiker M, Yuan J 2017 Phys. Rev. Lett. 119 094802

    [9]

    Chen C F, Ku C T, Tai Y H, Wei P K, Lin H N, Huang C B 2015 Nano Lett. 15 2746

    [10]

    Yu N F, Genevet P, Kats M A, Aieta F, Tetienne J P, Capasso F, Gaburro Z 2011 Science 334 333

    [11]

    Vedantam S, Lee H, Tang J, Conway J, Staffaroni M, Yablonovitch E 2009 Nano Lett. 9 3447

    [12]

    Gorodetski Y, Niv A, Kleiner V, Hasman E 2008 Phys. Rev. Lett. 101 043903

    [13]

    李嘉明, 唐鹏, 王佳见, 黄涛, 林峰, 方哲宇, 朱星 2015 物理学报 64 194201

    Li J M, Tang P, Wang J J, Huang T, Lin F, Fang Z Y, Zhu X 2015 Acta Phys. Sin. 64 194201

    [14]

    Lee S Y, Kim S J, Kwon H, Lee B 2015 IEEE Photonics Technol. Lett. 27 705

    [15]

    Tang B J, Zhang B F, Ding J P 2019 Appl. Opt. 58 833

    [16]

    Hell S W, Wichmann J 1994 Opt. Lett. 19 780

    [17]

    Vallone G, D’AmbrosioV, Sponselli A, Slussarenko S, Marrucci L, Sciarrino F, Villoresi P 2014 Phys. Rev. Lett. 113 060503

    [18]

    Hong Z Y, Zhang J, Drinkwater B W 2015 Phys. Rev. Lett. 114 214301

    [19]

    Marzo A, Caleap M, Drinkwater B W 2018 Phys. Rev. Lett. 120 044301

    [20]

    Tsai W Y, Huang J S, Huang C B 2014 Nano Lett. 14 547

    [21]

    Ding D S, Zhou Z Y, Shi B S, Guo G C 2013 Nat. Commun. 4 2527

    [22]

    Nicolas A, Veissier L, Giner L, Giacobino E 2014 Nat. Photonics 8 234

    [23]

    Wang J, Yang J Y, Fazal I M, Ahmed N 2012 Nat. Photonics 6 488

    [24]

    Bozinovic N, Yue Y, Ren Y X, Tur M 2013 Science 340 1545

    [25]

    Rui G H, Zhan Q W, Cui Y P 2015 Sci. Rep. 5 13732

    [26]

    Yang S Y, Chen W B, Nelson R L, Zhan Q W 2009 Opt. Lett. 34 3048

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  • Received Date:  17 May 2019
  • Accepted Date:  11 August 2019
  • Available Online:  27 November 2019
  • Published Online:  01 December 2019

Generation of surface plasmon vortices based on double-layer Archimedes spirals

    Corresponding author: Yang Yuan-Jie, dr.yang2003@uestc.edu.cn
  • School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China

Abstract: Surface plasmon polariton (SPP) is an electromagnetic oscillation which is produced by the interaction of free electrons on metal surface with incident light field. Using some specific metal metasurface structures, plasmonic vortices can be easily obtained. The SPP can well confine the light field within the interface between metal and medium. The SPP has two excellent characteristics: the ability to break through the diffraction limit and the strong field enhancing effect. Consequently, it has unique advantages in many applications, such as nanoparticle trapping, near-field imaging and sensing.The key point lies in selecting the incident light and the construction of metasurface structure when we want to produce SPP vortices. Especially the construction of metasurface structure has great influence on the generation of SPP vortices. So far, many structures have been proposed to produce SPP vortices. In 2015, Byoungho Lee research group (Lee S Y, Kim S J, Kwon H, Lee B 2015 IEEE Photonics Technol. Lett. 27 705) designed a structure of double-ring distributed nanoslits which can produce high-order plasmonic vortices by circularly polarized light. In addition, the use of Archimedes slit structure to generate optical vortices is currently a more common method. More recently, a novel plasmonic vortex lens was proposed to produce SPP vortices with arbitrary topological charges. The plasmonic vortex lens consists of an array of gold film nanoslits, and the superposing of the SPP field excited by each nanoslit can produce a composite plasmonic vortex. Here, we propose a novel spiral metasurface structure for generating surface plasmonic vortices. Using the combination of theoretical analysis and finite difference time domain (FDTD) simulation, we find that the topological charge quantity of surface plasmon vortices can be changed by adjusting five parameters, namely, the chirality of incident circularly polarized light, the difference in radius between inner and outer nanocavity array, the rotational velocity factor of nanocavity array, the number of segments of spiral, and the pitch of spiral. This metasurface structure has many adjustable parameters and each parameter can influence the final surface plasmon vortices. Therefore, the plasmonic vortices with an arbitrary topological charge quantity can be generated and manipulated simply by using such a metasurface structure. This work can be expected to have a very broad prospect of applications in super resolution microscopy, quantum cryptography, nanoparticle manipulation, optical data storage, and optical communication.

    • 表面等离激元(surface plasmon polariton, SPP)[1]是由金属表面的自由电子与入射光场发生相互作用而产生的一种电磁振荡. 由于SPP能够将光场很好地束缚在金属和介质的分界面上, 同时还能突破衍射极限[2,3]并产生较高的场增强效应[4-6], 因而在产生光学旋涡[7,8]方面具有独特的优越性. 如果要利用激发SPP的方式来产生涡旋光, 关键在于入射光的选择以及亚波长结构的构造. 尤其是亚波长结构的构造, 对于激发SPP来产生涡旋光的影响较大. 到目前为止, 对于如何通过激发SPP来产生涡旋光的方式研究已经引起了很多研究者的兴趣, 利用线偏振光照射十字形或V字形金属纳米结构[9,10], 利用径向偏振光照射均匀对称分布的纳米圆环结构[11], 或者是使用圆偏振光照射螺旋金属纳米狭缝结构[12,13]都能用来产生SPP涡旋场.

      2015年, Lee等[14]设计了一种由双层圆环分布的纳米孔阵列构成的表面等离激元透镜, 这种结构的表面等离激元透镜可以在左旋或右旋圆偏振光照射时产生高阶的涡旋光, 而所产生的涡旋光的阶数为$ \sigma (2 n - 1) $, 其中n表示圆环分布中的矩形纳米孔旋转的角度与方位角的比值. 最近, Tang等[15]设计了一种新型的结构, 改进后的结构可以使得当左旋或右旋圆偏振光照射时产生的不再仅仅是符号相反的涡旋光, 而是根据结构参数设计得到的不同数值的涡旋光. 本文则将双层圆环结构与阿基米德狭缝结构相结合, 提出了一种双层螺线分布的矩形纳米孔阵列. 与上述两者不同的是, 本文的结构可供调控的参数更多, 同时还研究了内外两层螺线之间的距离对形成的涡旋场的影响. 本文先采用解析的方法, 计算相应空间内电场的分布情况, 得到了SPP涡旋场的拓扑荷数与入射圆偏振光的手性、内外两层螺线之间的距离、纳米孔阵列的旋转变化速度因子、螺线的段数以及螺距这五个参数的解析关系式. 随后使用三维有限时域差分法来对理论推导得到的关系式进行验证, 并将模拟仿真得到的电场分布情况与理论公式所描述的电场特性进行对比, 最终发现理论与仿真的结果符合得十分完美. 本文的研究将在超分辨显微镜[16]、量子信息技术[17]、粒子操纵[18-20]、光信息处理[21,22]与光通信[23,24]等领域具有一定的应用前景.

    2.   理论分析
    • 本文采用的螺线结构为阿基米德螺线结构. 阿基米德螺线是一种比较常见的螺线结构, 它的结构可以根据结构旋性分成左旋阿基米德螺线结构及右旋阿基米德螺线结构, 在柱坐标系下, 完整的一圈阿基米德螺线可以用公式表示为[9]

      其中 ± 分别对应右旋与左旋阿基米德螺线结构, $ r_0 $为阿基米德螺线的初始半径, r为阿基米德螺线上任意一点到结构中心的距离, $ {\lambda _{\rm SPP}} $为产生的SPP波长, ϕ为方位角, 而m则表示阿基米德螺线的螺距与SPP波长的绝对值倍数(理论上可选取任意正数).

      常见的阿基米德螺线的段数$ q = 1 $, 而当螺线的段数$ q > 1 $时, 右旋阿基米德螺线结构表达形式为[25]

      其中 Λ表示的是阿基米德螺线的螺距, q为构成SPP透镜的阿基米德螺线的段数, $ \Delta \phi = 2{\text{π}}/q $, a为一个正整数, 用来描述每一段阿基米德螺线, 满足关系$ 1 \leqslant a \leqslant q $.

      实际的超表面结构示意图如图1所示, 该超表面结构是由多个按照螺线分布的矩形纳米孔构成, 并且还分为内外两层, 将内外两层之间的距离用d来表示. 图1(a)所展示的是螺线的段数$ q = 1 $时的结构示意图, 而图1(b)所展示的则是螺线的段数$ q = 2 $时的结构示意图.

      Figure 1.  Schematic diagram of super-surface structure of spiral distribution: (a) Number of segments of helix is equal to 1; (b) number of segments of helix is equal to 2.

      当圆偏振光入射到该结构上时, 处于不同位置的矩形纳米腔均会激发SPP, 随后传播至中心位置发生干涉, 并产生相应的干涉场. 假设观测点为$ ( {\rho, \theta, z} ) $, 那么得到的光场将会是由每一段阿基米德螺线分布的矩形纳米腔作为激励源而产生的光场叠加和, 具体公式为

      其中ρ, θ, z分别表示观测点到中心点的距离、方位角和沿z轴方向的坐标值; 而r, ϕ则分别表示激励源到中心点的半径与方位角; $ {k_z} $为SPP沿z轴方向在空气中的衰减系数; $ {k_{\rm SPP}} $为激发的SPP的波数; $ {A_0} $为在每个激励源位置的最大振幅; $ \varphi \left( \phi \right) $为激发的激励源相对于x轴方向的方位角分布函数; σ表示入射圆偏振光的手性($ \sigma = - 1 $$ \sigma = +1 $分别对应左旋与右旋圆偏振光). 入射圆偏振光在柱坐标系下可描述为[26]

      结构中设计外层排列的矩形纳米孔旋转的角度$ {\varphi _{\rm out}} $垂直于内层排列的矩形纳米孔旋转的角度$ {\varphi _{\rm in}} $, 由此可得到关系式

      其中n为矩形纳米孔阵列的旋转变化速度因子, 表示的是一段螺线起始位置的矩形纳米孔与末尾位置的矩形纳米孔在过程中一共转过了$ 2 n{\text{π}}$的角度.

      假设由外层圆环带来的相位扰动可以忽略不计, 此时总的光场是外层与内层排列的矩形纳米孔所形成的光场的叠加,

      根据内外两层螺线之间的距离的不同, 对(6)式进行进一步的推导, 得到关系式

      (7)式即为本文的最终理论结论, 可以看出, 当$ d = 0 $$ d = {\lambda _{\rm SPP}}/2 $时, 结构中心处将会产生符合贝塞尔函数分布的隐失波形式的电场. 当$ d = 0 $时产生的是一个拓扑荷数为$ \sigma + mq $的SPP涡旋光; 而当$ d = {\lambda _{\rm SPP}}/2 $时产生的则是一个拓扑荷数为$ \sigma \left( {2 nq - 1} \right) + mq $的SPP涡旋光.

    3.   有限时域差分仿真与讨论
    • 借助软件工具来对上述理论结果进行验证, 这里使用三维有限时域差分法(finite difference time domain, FDTD)来进行模拟仿真, 具体的参数设置为: 入射光选用波长为980 nm的圆偏振光, 相应产生的SPP的波长为$ {\lambda _{\rm SPP}} = 967 $ nm; 金膜的厚度设置为100 nm, 金膜的介电常数为$ \varepsilon = $ –38.31 + i1.16, 金膜下表面的介质为石英, 石英衬底的折射率为1.45; 在金膜上刻蚀的矩形纳米孔的深度为100 nm, 长度为400 nm, 宽度为100 nm; 内层排列的矩形纳米孔的半径为4 μm, 外层排列的矩形纳米孔的半径为4.483 μm, 进行结果观察的观测平面的位置在金膜上方20 nm处. 进行FDTD仿真的网格参数设置为20 nm × 20 nm × 50 nm, 边界条件选择完美匹配层(perfectly matched layer, PML).

      图2给出了内外两层螺线之间的距离$ d = 0 $时的超表面结构示意图、最终的电场强度以及相位分布图. 首先可以确认, 该超表面结构能够成功地激发SPP涡旋光. 图2(a)图2(c)表明: 当阿基米德螺线的螺距与SPP波长的绝对值倍数$ m = 1 $, 螺线的段数$ q = 1 $, 左旋圆偏振光入射时, 可以在超表面的结构中心处观察到聚焦现象. 图2(d)图2(f)改为右旋圆偏振光入射, 其他参数不变, 在结构中心处可以观察到中心为暗斑的交叉明暗圆环, 根据其中的相位分布图可知此时产生的是拓扑荷数为2的SPP涡旋光; 如图2(g)图2(i)所示, 将螺线的段数设定为$ q = 2 $时, 螺距与SPP波长的绝对值倍数$ m = 1 $, 入射光为右旋圆偏振光时, 得到的将会是拓扑荷数为3的SPP涡旋光; 图2(j)图2(l)描述了螺线的段数$ q = 2 $, 螺距与SPP波长的绝对值倍数$ m = 2 $, 入射光为右旋圆偏振光时的结构示意图、电场强度图以及相位分布图, 结果产生的是拓扑荷数为5的SPP涡旋光. 上述四种情况所得到的SPP涡旋光的拓扑荷数满足(7)式中$ d = 0 $时的关系式, 理论与仿真结果一致.

      Figure 2.  The obtained electric field intensity and phase distribution when distance between inner and outer spirals is equal to 0: (a)−(c) $\sigma=- 1$; (d)−(f) $\sigma=1$; (g)−(i) $\sigma=1$; (j)−(l) $\sigma=1$.

      图3给出的则是当内外两层螺线之间的距离$ d = {\lambda _{\rm SPP}}/2 $时的超表面结构示意图、最终的电场强度以及相位分布图. 根据(7)式中的关系, 此时还需要考虑矩形纳米孔阵列的旋转变化速度因子n可能对结构中心的电场带来的影响.

      Figure 3.  The obtained electric field intensity and phase distribution when distance between inner and outer spirals is equal to ${\lambda _{\rm SPP}}/2$: (a)−(c) $\sigma = - 1$; (d)−(f) $\sigma = - 1$; (g)−(i) $\sigma = 1$; (j)−(l) $\sigma = 1$; (m)−(o) $\sigma = 1$.

      图3使用了控制变量的方法分别对理论结果中所描述的入射光的手性σ、矩形纳米孔阵列的旋转变化速度因子n、螺线的段数q以及螺距与SPP波长的绝对值倍数m这四个变量进行了仿真. 若按照理论推导的结果进行计算, 此刻所激发的应该为拓扑荷数为$ \sigma \left( {2 nq - 1} \right) + mq $的SPP涡旋光. 图3(a)图3(c)说明当左旋圆偏振光入射到结构参数为$ m = n = q = 1 $的超表面上时, 激发了聚焦电场; 当改为右旋圆偏振光入射到相同的结构上时, 就会如图3(g)图3(i)所示激发出2阶的SPP涡旋光; 若此时分别将速度因子n以及螺线的段数q的取值增大为2, 则得到图3(j)(l)图3(m)(o)所示的仿真结果, 分别激发4阶和5阶的SPP涡旋光. 图3(d)图3(f)则是在图3(a)图3(c)的基础上改变了螺距与SPP波长的绝对值倍数m, 然后得到的是1阶的SPP涡旋光. 经过对比可以看出, 当内外两层螺线之间的距离$ d = {\lambda _{\rm SPP}}/2 $时仿真结果与理论结果也符合得很好.

    4.   结 论
    • 研究了一种可以用来激发SPP涡旋光的双层阿基米德螺线分布的超表面结构. 理论分析与推导后的方程表明, 内外两层螺线之间的距离d的取值不同会极大地影响所得到的SPP涡旋光的表达式. 理论上可以通过调节入射光的手性σ、内外两层螺线之间的距离d、纳米孔阵列的旋转变化速度因子n、螺线的段数q以及螺距与SPP长的绝对值倍数m这五个参数来实现对产生的SPP涡旋光的操控. 随后, 利用FDTD仿真软件对该超表面结构进行了模拟, 仿真结果与理论结果完全符合, 证明了使用这种螺线分布的超表面结构激发SPP涡旋光的可行性. 这样可以通过精确控制与组合相关参数十分简单地产生任意拓扑荷数的SPP涡旋光, 由于可调控的参数多, 自由度大, 可以根据需求灵活地进行设计, 因而在微纳级别的光学芯片方面具备很大的应用潜力.

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