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Design and application of flat spiral phase plate

Wu Wen-Bing Sheng Zong-Qiang Wu Hong-Wei

Design and application of flat spiral phase plate

Wu Wen-Bing, Sheng Zong-Qiang, Wu Hong-Wei
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  • Phase is an important characteristic of electromagnetic waves. It is well known that a beam with a helical wave front characterized by a phase of $\exp({\rm{i}}l\theta )$ (which depends on azimuthal angle $\theta$ and topological charge l), has a momentum component along the azimuthal direction, resulting in an orbital angular momentum of per photon along the beam axis. Owing to its fascinating properties, the beam has received a great deal of attention and has provided novel applications in manipulation of particles or atoms, optical communication, optical data storage. In order to meet the needs of various applications, techniques for efficiently generating optical beams carrying orbital angular momentum are always required. Current schemes for generating the beams carrying orbital angular momentum include computer-generated holograms, spiral phase plates, spatial light modulators, and silicon integrated optical vortex emitters. Among the usual methods to produce helical beams, the traditional spiral phase plate is an optical device that utilizes the progressive increasing of height of a dielectric material along an azimuthal direction to produce a vortex beam for beam phase modulation with a high conversion efficiency. However, it is difficult to regulate the topological charge l of the outgoing beam through the superposition of the phase plates due to the special geometric feature. In this paper, the flat spiral phase plate is designed by compressing the height of traditional spiral phase plate, and inducing the refractive index to increase in the azimuthal direction based on coordinate transformation. By means of theoretical analysis and numerical simulation, it is found that the flat spiral phase plate can produce high quality vortex beams just as the traditional spiral phase plate can do. Particularly, the height of the flat spiral phase plate and the topological charge l carried by the vortex beams can be arbitrarily adjusted according to the refractive index selection of the dielectric material. In order to meet the needs of practical applications, the vortex beams with different topological charges can be obtained by stacking multiple layers of flat spiral phase plates. The flat spiral phase plate has broad potential applications in the fields of optical transmission and optical communication.
      Corresponding author: Sheng Zong-Qiang, zqsheng@aust.edu.cn ; Wu Hong-Wei, hwwu@aust.edu.cn
    [1]

    Padgett M, Courtial J, Allen L 2004 Phys. Today 57 35

    [2]

    苏志锟, 王发强, 路轶群, 金锐博, 梁瑞生, 刘颂豪 2008 物理学报 57 3016

    Su Z K, Wang F Q, Lu Y Q, Jin R B, Liang R S, Liu S H 2008 Acta Phys. Sin. 57 3016

    [3]

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

    [4]

    陈理想, 张远颖 2015 物理学报 64 164210

    Chen L X, Zhang Y Y 2015 Acta Phys. Sin. 64 164210

    [5]

    Grier D G 2003 Nature 424 810

    [6]

    Andersen M F, Ryu C, Clade P, Natarajan V, Vaziri A, Helmerson K, Phillips W D 2006 Phys. Rev. Lett. 97 170406

    [7]

    Gibson G, Courtial J, Padgett M J, Vasnetsov M, Pas'ko V, Barnett S M, Franke-Arnold S 2004 Opt. Express 12 5448

    [8]

    Molina-Terriza G, Torres J P, Torner L 2007 Nature Phys. 3 305

    [9]

    Torner L, Torres L P, Carrasco S 2005 Opt. Express 13 873

    [10]

    Dholakia K, Cizmar T 2011 Nat. Photonics 5 335

    [11]

    Bazhenov V Y, Vasnetsov M V, Soskin M S 1990 JETP Lett. 52 429

    [12]

    Heckenberg N R, McDuff R, Smith C P, White A G 1992 Opt. Lett. 17 221

    [13]

    Beijersbergen M W, Coerwinkel R P C, Kristensen M, Woerdman J P 1994 Opt. Commun. 112 321

    [14]

    范庆斌, 徐挺 2017 物理学报 66 144208

    Fan Q B, Xu T 2017 Acta Phys. Sin. 66 144208

    [15]

    李明, 陈阳, 郭光灿, 任希峰 2017 物理学报 66 144202

    Li M, Chen Y, Guo G C, Ren X F 2017 Acta Phys. Sin. 66 144202

    [16]

    Chen L X, She W L 2009 Opt. Lett. 34 178

    [17]

    Oemrawsingh S S R, van Houwelingen J A W, Eliel E R, Woerdman J P, Verstegen E J K, Kloosterboer J G, Hooft G W 't 2004 Appl. Opt. 43 688

    [18]

    Kotlyar V V, Khonina S N, Kovalev A A, Soifer V A 2006 Opt. Lett. 31 1597

    [19]

    Lee W M, Yuan X C, Cheong W C 2004 Opt. Lett. 29 1796

    [20]

    Rotschild C, Zommer S, Moed S, Hershcovitz O, Lipson S G 2004 Appl. Opt. 43 2397

    [21]

    刘国昌, 李超, 邵金进, 方广有 2014 物理学报 63 154102

    Liu G C, Li C, Shao J J, Fang G Y 2014 Acta Phys. Sin. 63 154102

    [22]

    汪会波, 罗孝阳, 董建峰 2015 物理学报 64 154102

    Wang H B, Luo X Y, Dong J F 2015 Acta Phys. Sin. 64 154102

    [23]

    Pendry J B, Schurig D, Smith D R 2006 Science 312 1780

    [24]

    Lai Y, Chen H Y, Zhang Z Q, Chan C T 2009 Phys. Rev. Lett. 102 093901

    [25]

    Li J, Pendry J B 2008 Phys. Rev. Lett. 101 203901

    [26]

    Zhao J Z, Wang D L, Peng R W, Hu Q, Wang M 2011 Phys. Rev. E 84 046607

    [27]

    Lai Y, Chen H, Zhang Z Q, Chan C T 2009 Phys. Rev. Lett. 102 253902

    [28]

    Jiang W X, Ma H F, Cheng Q, Cui T J 2010 Appl. Phys. Lett. 96 121910

    [29]

    Rahm M, Schurig D, Roberts D A, Cummer S A, Smith D R, Pendry J B 2008 Photonics Nanostruct. Fundam. Appl. 6 87

    [30]

    Ma H F, Cui T J 2010 Nat. Commun. 1 124

    [31]

    Smith D R, Mock J J, Starr A F, Schurig D 2005 Phys. Rev. E 71 036609

    [32]

    Mei Z L, Bai J, Cui T J 2010 Appl. Phys. 43 055404

    [33]

    Ma H F, Cai B J, Zhang T X, Yang Y, Jiang W X, Cui T J 2013 IEEE Trans. Antennas Propag. 61 2561

  • 图 1  传统螺旋相位板和平板式螺旋相位板的结构示意图 (a)传统螺旋相位板; (b)平板式螺旋相位板(颜色深浅表示折射率的大小)

    Figure 1.  Schematic diagram of a conventional spiral phase plate and a flat-plate spiral phase plate: (a) A conventional spiral phase plate; (b) a flat-plate spiral phase plate (the color depth indicates the size of the refractive index).

    图 2  数值模拟结果 (a)平板式螺旋相位板产生的光束横截面场分布图; (b)平板式螺旋相位板横截面相位分布图; (c)设计的平板式螺旋相位板空间材料分布图; (d)高斯光束入射平板式螺旋相位板产生涡旋光束的xz截面图

    Figure 2.  The simulation results: (a) Cross-sectional field distribution of the beam produced by the flat-plate spiral phase plate; (b) phase distribution of cross section of flat-plates piral phase plate; (c) designed flat-plate spiral phase plate space material distribution; (d) the Gaussian beam incident on the flat-plate spiral phase plate produces a xz cross-sectional view of the vortex beam.

    图 3  数值模拟结果 (a)平板式螺旋相位板产生的光束横截面场分布图; (b)平板式螺旋相位板横截面相位分布图; (c)平板式螺旋相位板光强分布图; (d)设计的平板式螺旋相位板空间材料分布图; (e)传统的螺旋相位板产生的光束横截面场分布图

    Figure 3.  The simulation results: (a) Cross-sectional field distribution of the beam produced by the flat-plate spiral phase plate; (b) phase distribution of cross section of flat-plates piral phase plate; (c) light intensity distribution of flat-plates piral phase plate; (d) designed flat-plate spiral phase plate space material distribution; (e) a cross-sectional field distribution diagram of a beam produced by a conventional spiral phase plate.

    图 4  (a) 双层l = –1的平板式螺旋相位板叠加产生的光束横截面场分布图; (b)由三层l = –1的平板式螺旋相位板叠加产生的光束横截面场分布图

    Figure 4.  (a) Cross-sectional field distribution of the beam produced by the superposition of a flat-plate spiral phase plate with a double layer l = –1; (b) the cross-sectional field distribution of the beam produced by the superposition of three layers of flat spiral phase plates with l = –1.

  • [1]

    Padgett M, Courtial J, Allen L 2004 Phys. Today 57 35

    [2]

    苏志锟, 王发强, 路轶群, 金锐博, 梁瑞生, 刘颂豪 2008 物理学报 57 3016

    Su Z K, Wang F Q, Lu Y Q, Jin R B, Liang R S, Liu S H 2008 Acta Phys. Sin. 57 3016

    [3]

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

    [4]

    陈理想, 张远颖 2015 物理学报 64 164210

    Chen L X, Zhang Y Y 2015 Acta Phys. Sin. 64 164210

    [5]

    Grier D G 2003 Nature 424 810

    [6]

    Andersen M F, Ryu C, Clade P, Natarajan V, Vaziri A, Helmerson K, Phillips W D 2006 Phys. Rev. Lett. 97 170406

    [7]

    Gibson G, Courtial J, Padgett M J, Vasnetsov M, Pas'ko V, Barnett S M, Franke-Arnold S 2004 Opt. Express 12 5448

    [8]

    Molina-Terriza G, Torres J P, Torner L 2007 Nature Phys. 3 305

    [9]

    Torner L, Torres L P, Carrasco S 2005 Opt. Express 13 873

    [10]

    Dholakia K, Cizmar T 2011 Nat. Photonics 5 335

    [11]

    Bazhenov V Y, Vasnetsov M V, Soskin M S 1990 JETP Lett. 52 429

    [12]

    Heckenberg N R, McDuff R, Smith C P, White A G 1992 Opt. Lett. 17 221

    [13]

    Beijersbergen M W, Coerwinkel R P C, Kristensen M, Woerdman J P 1994 Opt. Commun. 112 321

    [14]

    范庆斌, 徐挺 2017 物理学报 66 144208

    Fan Q B, Xu T 2017 Acta Phys. Sin. 66 144208

    [15]

    李明, 陈阳, 郭光灿, 任希峰 2017 物理学报 66 144202

    Li M, Chen Y, Guo G C, Ren X F 2017 Acta Phys. Sin. 66 144202

    [16]

    Chen L X, She W L 2009 Opt. Lett. 34 178

    [17]

    Oemrawsingh S S R, van Houwelingen J A W, Eliel E R, Woerdman J P, Verstegen E J K, Kloosterboer J G, Hooft G W 't 2004 Appl. Opt. 43 688

    [18]

    Kotlyar V V, Khonina S N, Kovalev A A, Soifer V A 2006 Opt. Lett. 31 1597

    [19]

    Lee W M, Yuan X C, Cheong W C 2004 Opt. Lett. 29 1796

    [20]

    Rotschild C, Zommer S, Moed S, Hershcovitz O, Lipson S G 2004 Appl. Opt. 43 2397

    [21]

    刘国昌, 李超, 邵金进, 方广有 2014 物理学报 63 154102

    Liu G C, Li C, Shao J J, Fang G Y 2014 Acta Phys. Sin. 63 154102

    [22]

    汪会波, 罗孝阳, 董建峰 2015 物理学报 64 154102

    Wang H B, Luo X Y, Dong J F 2015 Acta Phys. Sin. 64 154102

    [23]

    Pendry J B, Schurig D, Smith D R 2006 Science 312 1780

    [24]

    Lai Y, Chen H Y, Zhang Z Q, Chan C T 2009 Phys. Rev. Lett. 102 093901

    [25]

    Li J, Pendry J B 2008 Phys. Rev. Lett. 101 203901

    [26]

    Zhao J Z, Wang D L, Peng R W, Hu Q, Wang M 2011 Phys. Rev. E 84 046607

    [27]

    Lai Y, Chen H, Zhang Z Q, Chan C T 2009 Phys. Rev. Lett. 102 253902

    [28]

    Jiang W X, Ma H F, Cheng Q, Cui T J 2010 Appl. Phys. Lett. 96 121910

    [29]

    Rahm M, Schurig D, Roberts D A, Cummer S A, Smith D R, Pendry J B 2008 Photonics Nanostruct. Fundam. Appl. 6 87

    [30]

    Ma H F, Cui T J 2010 Nat. Commun. 1 124

    [31]

    Smith D R, Mock J J, Starr A F, Schurig D 2005 Phys. Rev. E 71 036609

    [32]

    Mei Z L, Bai J, Cui T J 2010 Appl. Phys. 43 055404

    [33]

    Ma H F, Cai B J, Zhang T X, Yang Y, Jiang W X, Cui T J 2013 IEEE Trans. Antennas Propag. 61 2561

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  • Received Date:  09 September 2018
  • Accepted Date:  03 November 2018
  • Available Online:  23 March 2019
  • Published Online:  01 March 2019

Design and application of flat spiral phase plate

    Corresponding author: Sheng Zong-Qiang, zqsheng@aust.edu.cn
    Corresponding author: Wu Hong-Wei, hwwu@aust.edu.cn
  • 1. School of Mechanics and Photoelectric Physics, Anhui University of Science and Technology, Huainan 232001, China
  • 2. National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

Abstract: Phase is an important characteristic of electromagnetic waves. It is well known that a beam with a helical wave front characterized by a phase of $\exp({\rm{i}}l\theta )$ (which depends on azimuthal angle $\theta$ and topological charge l), has a momentum component along the azimuthal direction, resulting in an orbital angular momentum of per photon along the beam axis. Owing to its fascinating properties, the beam has received a great deal of attention and has provided novel applications in manipulation of particles or atoms, optical communication, optical data storage. In order to meet the needs of various applications, techniques for efficiently generating optical beams carrying orbital angular momentum are always required. Current schemes for generating the beams carrying orbital angular momentum include computer-generated holograms, spiral phase plates, spatial light modulators, and silicon integrated optical vortex emitters. Among the usual methods to produce helical beams, the traditional spiral phase plate is an optical device that utilizes the progressive increasing of height of a dielectric material along an azimuthal direction to produce a vortex beam for beam phase modulation with a high conversion efficiency. However, it is difficult to regulate the topological charge l of the outgoing beam through the superposition of the phase plates due to the special geometric feature. In this paper, the flat spiral phase plate is designed by compressing the height of traditional spiral phase plate, and inducing the refractive index to increase in the azimuthal direction based on coordinate transformation. By means of theoretical analysis and numerical simulation, it is found that the flat spiral phase plate can produce high quality vortex beams just as the traditional spiral phase plate can do. Particularly, the height of the flat spiral phase plate and the topological charge l carried by the vortex beams can be arbitrarily adjusted according to the refractive index selection of the dielectric material. In order to meet the needs of practical applications, the vortex beams with different topological charges can be obtained by stacking multiple layers of flat spiral phase plates. The flat spiral phase plate has broad potential applications in the fields of optical transmission and optical communication.

    • 近年来, 随着电磁波材料的研究和加工技术的发展, 人们已经可以有效地调节电磁波的振幅、偏振以及相位等物理量进而操控电磁波的传输. 众所周知, 当一束光具有相位因子$\exp({\rm{i}}l\theta )$ (l为拓扑荷数, $\theta $为方位角)时[1,2], 其将具有螺旋相位波前, 最常见的涡旋光束是拉盖尔-高斯光束. 由于这类光束的螺旋相位波前特征使得涡旋光束在方位角方向上有线动量分量, 进而使得其所包含的每一个光子均携带有值为$l\hbar $的轨道角动量. 自从Allen等[3]发现涡旋光束携带光学轨道角动量性质之后, 涡旋光束迅速成为人们研究的热点并广泛应用于原子和小粒子的捕获、经典和量子通讯技术、光学成像以及生物物理等领域[4-10]. 为了能够满足涡旋光束的实际应用, 人们已经在不同波段设计了产生涡旋光束的光学器件, 常用的有螺旋相位板、空间光学调制器、计算全息图、超表面以及人工表面等离激元结构等一系列的方法[11-20]. 相比于其他产生涡旋光束的光学器件, 传统的螺旋相位板具有高效率、宽频带产生涡旋光束的优点. 然而, 由于传统螺旋相位板的特殊结构特征(阶梯型结构), 使其不能简单、便捷地直接使用多个螺旋相位板叠加调节出射涡旋光束的角量子数, 从而导致每一个传统的螺旋相位板只能直接产生一束携带固定光学角动量的涡旋光束.

      变换光学作为设计新型功能材料的有力工具, 已经被人们广泛关注. 坐标变换的理论基础是麦克斯韦方程组在空间坐标变换中形式不变性[21], 其基本思想是将扭曲空间中的几何变形转化为平直空间中的材料特殊分布, 再通过设计超构材料来实现变换光学计算的材料分布, 从而控制电磁波的传播[22]. 基于变换光学方法, 许多新奇的光学器件已经被设计. 例如, 不可见隐身斗篷、光学幻象器件、光集中器、平直的Luneburg透镜等器件[23-30].

      在本文中, 基于z轴坐标变换将沿方位角方向高度螺旋增加的传统螺旋相位板变换为高度不变、材料的折射率沿方位角方向逐渐增大的平板式螺旋相位板. 理论分析与数值模拟计算结果表明, 这种平板式螺旋相位板与传统螺旋相位板一样能够产生高质量的涡旋光束, 并且这种平板式螺旋相位板的高度和涡旋光束携带的轨道角动量指数可以根据介质材料的折射率选取而任意调节. 特别地, 这种平板式螺旋相位板可以通过增加或者减少相位板的数量直接调控出射涡旋光束的角量子数. 另外, 也可以通过直接翻转平板式螺旋相位板调节出射涡旋光束的角量子数的符号. 平板式螺旋相位板具有方便调节角量子数的特点可能在实际产生和调节涡旋光束方面具有潜在的应用.

    2.   平板式螺旋相位板设计与分析
    • 传统的螺旋相位板是一块沿方位角方向折射率恒定、高度递增的介质材料板, 如图1(a)所示. 螺旋相位板的厚度h从最低厚度h0沿方位角方向增加到h0 + hs, 介质材料厚度满足关系式: h = ${h_0} + {h_{\rm{s}}}\theta /(2{\text{π}} )$. 图1(a)中, r是螺旋相位板的半径, $\theta $是方位角. 为了能实现通过增加或者减少螺旋相位板的叠加数量, 基于z轴坐标变换将沿方位角材料相同、高度递增的传统螺旋相位板变换为沿方位角高度相同、折射率递增的平板式螺旋相位板, 如图1(b)所示, 其中颜色的深浅表示材料折射率的分布不同. 平板式螺旋相位板中, $r' $表示结构的半径, $\theta '$是方位角.

      Figure 1.  Schematic diagram of a conventional spiral phase plate and a flat-plate spiral phase plate: (a) A conventional spiral phase plate; (b) a flat-plate spiral phase plate (the color depth indicates the size of the refractive index).

      根据坐标变换思想, 从传统的螺旋相位板空间变换到平板式螺旋相位板空间可以通过z轴变换得到. 传统螺旋相位板的空间坐标$\left( {r,\theta ,z} \right)$与平板式螺旋相位板的空间坐标$\left( {r',\theta ',z'} \right)$之间的关系可以写为

      上面坐标变换的操作为螺旋相位板的半径以及方位角不变, 但是沿着方位角方向压缩传统螺旋相位板的高度到基础厚度h0. 基于变换光学方法, 可以计算出平板式螺旋相位板的空间材料分布(介电常数$\varepsilon '$与磁导率$\mu '$分布). 推导出的$r{\text{-}} \theta $平面内的介电常数与磁导率对角分量分别为

      $r{\text{-}}\theta $平面内, 材料参数矩阵的非对角元为0. 由于入射光束是沿着z方向传播, 因此光束的相位分布由折射率nz决定. 因此, 平板式螺旋相位板的空间折射率可以写为

      为了计算的方便, 这里取${\varepsilon _{rr}}{\mu _{\theta \theta }} = 1$. 入射高斯光束的波长为$\lambda $, 这样通过设计的螺旋相位板产生的光学轨道角动量阶数则为$l = {h_{\rm{s}}}/\lambda $. 相应地, 光束沿着方位角方向被螺旋相位板所附加的相位延迟因子为$\Delta \varPhi = {h_{\rm{s}}}\theta /\lambda $. 其中平板式螺旋相位板的半径为$4\lambda $, 高度${h_0} = \lambda $, 折射率参数中${h_{\rm{s}}} = l\lambda $.

    3.   数值模拟结果与讨论
    • 传统螺旋相位板使入射光束的相位延迟为[17]

      式中n, n0分别为传统螺旋相位板的折射率以及周围介质的折射率; $\lambda$为入射光波长; $\Delta \varPhi$的大小取决于旋转方位角$\theta$. 取传统螺旋相位板的拓扑荷为l = hs(nn0)/$\lambda$, 则当$\theta$由0增加到$2{\text{π}}$时, $\Delta \varPhi$的增量(相移量)为$2l{\text{π}} $. 而本文设计的平板式螺旋相位板的增加高度hs的值为0, 平板式螺旋相位板使入射光束的相位延迟为

      式中n'为平板式螺旋相位板方位角方向的折射率, $\Delta \varPhi$的大小取决于平板式螺旋相位板的折射率$n' $. 与传统螺旋相位板相比, 只需要不断地改变平板式螺旋相位板的折射率$n' $, 就能达到与传统螺旋相位板相同的效果.

      图2中, 为了产生出射涡旋光束携带角量子数为1, 本文设置了折射率参数中${h_{\rm{s}}} = \lambda $, 波长$\lambda = 1$ mm的微波. 图2(a)给出了高斯波入射平板式螺旋相位板之后, 出射光束的横截面场分布, 从场分布中可以看到有两个沿着逆时针旋转的机翼. 由于设置了角量子数为l = 1, 因此空间相位分布也出现了旋转一周相位变换为$2{\text{π}}$, 如图2(b)所示. 在图中的原点处出现了相位奇点现象. 在这一点处相位具有不确定性, 并且此点的光场强度为零. 图2(c)是平板式螺旋相位板的空间材料折射率分布, 沿着逆时针方向, 螺旋相位板的材料折射率从$ n'$= 1增加到$ n'$= 2. 图2(d)为光场分布图, 所呈现的是高斯光束沿着z方向入射平板式螺旋相位板并且产生具有轨道角动量的涡旋光束的整个过程. 图中白色虚线区域为设计的平板式螺旋相位板位置. 由图中可以看出入射的高斯光束具有平面波式的相位分布, 然而当经过平板式螺旋相位板之后, 光束被折射出两束出射光. 由光场分布图可以对两束出射光的相位进行比较, 发现两束光束的相位相差$2{\text{π}}$. 由图2(d)也可以看出, 出射光束的中心线上相位具有不确定性, 而且光束的强度为零, 所以出射光束横截面上的强度分布呈现出面包圈式的分布.

      Figure 2.  The simulation results: (a) Cross-sectional field distribution of the beam produced by the flat-plate spiral phase plate; (b) phase distribution of cross section of flat-plates piral phase plate; (c) designed flat-plate spiral phase plate space material distribution; (d) the Gaussian beam incident on the flat-plate spiral phase plate produces a xz cross-sectional view of the vortex beam.

      除了能够设计这种沿着逆时针增加光束相位以赋予光束l的光学轨道角动量以外, 还可以沿着y轴将平板式螺旋相位板翻转, 使材料折射率沿着顺时针增加, 以便实现出射光束的附加相位沿着顺时针增加, 进而获得光学轨道角动量为负值的涡旋光束.

      图3(a)所示为高斯光束入射经翻转后的平板式螺旋相位板得到的出射光束在横截面上的场分布图. 从图中可以看到, 场分布呈现出两个螺旋翼状的分布并且沿顺时针旋转. 在顺时针旋转一周时, 相位发生$2{\text{π}}$的改变, 如图3(b)所示. 图3(c)是平板式螺旋相位板的光强分布图, 由图可以看出光强分布为面包圈式的分布. 图3(d)为翻转后的平板式螺旋相位板折射率分布图. 可以看到, 平板式螺旋相位板的折射率沿着方位角方向顺时针从$n' $ = 1增加到$n' $= 2. 为了比较这种通过翻转操作得到的平板式螺旋相位板与传统的螺旋相位板产生涡旋光束的性能, 我们计算了传统螺旋相位板的出射光束横截面场分布, 如图3(e)所示. 由图可以发现, 平板式螺旋相位板能够产生与传统螺旋相位板几乎相同的涡旋光束.

      传统螺旋相位板虽然具有高光束转化效率和承载高功率激光的能力, 但是由于其特殊结构特征(阶梯型结构), 使其不能简单、便捷地直接使用多个螺旋相位板叠加调节出射涡旋光束的角量子数, 从而导致每一个传统的螺旋相位板只能直接产生一束携带固定光学角动量的涡旋光束. 平板式螺旋相位板可以通过增加或者减少相位板的数量直接调控出射涡旋光束的角量子数, 进而克服传统螺旋相位板的缺点.

      Figure 3.  The simulation results: (a) Cross-sectional field distribution of the beam produced by the flat-plate spiral phase plate; (b) phase distribution of cross section of flat-plates piral phase plate; (c) light intensity distribution of flat-plates piral phase plate; (d) designed flat-plate spiral phase plate space material distribution; (e) a cross-sectional field distribution diagram of a beam produced by a conventional spiral phase plate.

      图4(a)为高斯光束通过二层l = –1的平板式螺旋相位板(如图3(a)所示)后产生的光束横截面场分布图. 在图中可以发现光束的横截面上有四个螺旋翼顺时针旋转. 也就是说, 不具有纵向光学轨道角动量的高斯光束通过二层平板式螺旋相位板之后, 产生了轨道角动量为l = –2的涡旋光束. 同理, 如果用高斯光束入射两层l = 1的平板式螺旋相位板, 那么产生的出射光束横截面上也将出现四个螺旋翼, 但是四翼的旋转方向将是逆时针旋转. 同样地, 如果继续增加平板式螺旋相位板的数量到三层时, 用同样的高斯光束入射三层l = –1的平板式螺旋相位板, 出射光束的横截面上场分布呈现了6个顺时针旋转的螺旋翼. 此时, 出射光束携带的纵向角量子数为l = –3. 以此类推, 可以通过插入不同数量的轨道角动量为l = –1的螺旋相位板以获得不同角量子数的涡旋光束. 原则上, 也可以通过设置${h_{\rm{s}}} = \lambda /m$, $m=1,\;2,\;3\cdots $, 对应于不同的m, 设计出不同的分数阶轨道角动量的螺旋相位板(这里没有给出分数阶螺旋相位板产生光束的横截面场分布), 但是在螺旋相位板的叠加过程中必须保证折射率高的部分与折射率高的部分对齐, 折射率低的部分与折射率低的部分对齐, 这样才能产生以上所述结果.

      Figure 4.  (a) Cross-sectional field distribution of the beam produced by the superposition of a flat-plate spiral phase plate with a double layer l = –1; (b) the cross-sectional field distribution of the beam produced by the superposition of three layers of flat spiral phase plates with l = –1.

    4.   总 结
    • 本文基于z 轴坐标变换将沿方位角方向高度螺旋增加的传统螺旋相位板变换为高度不变、材料的折射率沿方位角方向逐渐增大的平板式螺旋相位板. 研究发现这种平板式螺旋相位板不但可以像传统螺旋相位板一样能够产生高质量的涡旋光束, 而且它的高度和涡旋光束携带的角量子数可以根据介质材料的折射率选取而任意调节. 有趣的是, 这种平板式螺旋相位板可以通过直接翻转平板式螺旋相位板调节出射涡旋光束的角量子数的符号, 也可以通过增加或者减少相位板的数量直接调控出射涡旋光束的角量子数. 在制作平板式螺旋相位板时可以使用超构材料(metamaterial)[31], 如可以采用在材料上钻孔的方式形成孔阵列, 通过调节孔阵列中孔径的大小来实现折射率的变化, 并且这种方法在印刷电路板(printed circuit board)上即可实现, 制作方便、便捷[32,33]. 平板式螺旋相位板可以方便地调节光学角量子数的特点可能在实际产生和调节涡旋光束方面具有潜在的应用.

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