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理论上,杂化矢量光通常由两种携带不同轨道角动量的正交偏振光叠加构成。普遍意义的杂化矢量光可以表示为:
$$\vec E(\rho ,\varphi ) = A{E_{{l_1}}}(\rho ,\varphi ){\hat e_1} + B{E_{{l_2}}}(\rho ,\varphi ){\hat e_2}$$ (1) 式中:ρ, φ, z分别代表柱坐标系的坐标;
${\hat e_1}$ 和${\hat e_2}$ 分别代表两个正交偏振的单位矢量;A和B为光场轮廓参数;${E_{{l_1}}}(\rho ,\varphi )$ 和${E_{{l_2}}}(\rho ,\varphi )$ 分别代表携带轨道角动量为l1和l2的光束,其形式取决于具体实现方式。2010年,王慧田课题组[17]采用两束携带±1的轨道角动量涡旋光场实现杂化矢量光场,公式(1)写为:
$$\vec E(\rho ,\varphi ) = \exp (i\varphi ){\hat e_1} + \exp ( - i\varphi ){\hat e_2}$$ (2) 式中:
${\hat e_1} = (\cos \phi ,\sin \phi )$ 与${\hat e_2} = ( - \sin \phi ,\cos \phi )$ 由水平和垂直基矢旋转得到,$\phi $ 为其旋转角度。不同$\phi $ 对应的偏振分布如图1所示。同一年,Brown课题组[18]采用高阶Laguerre-Gauss(LG)光束与基模高斯光叠加实现另一种杂化矢量光——FP光,相应地,公式(1)可以写为:$$\vec E(\rho ,\varphi ) = {{LG}}_0^0(\rho ,\varphi ){\hat e_1} + {{LG}}_0^l(\rho ,\varphi ){\hat e_2}$$ (3) 其中,LG光束可以描述为:
$$LG_0^l(r,\varphi ) = {E_0}{r^l}L_0^l\left(\frac{{2r}}{{{w^2}}}\right)\exp \left( - \frac{{{r^2}}}{{{w^2}}}\right)\exp (il\varphi )$$ (4) 式中:φ为方位角;w为束腰;
$ L{G}_{0}^{l}(·)$ 为相关的拉盖尔多项式。不同${\hat e_1}$ 和${\hat e_2}$ 组合下全庞加莱光的偏振分布如图2所示。图 2 (a)
${\hat e_1} = (1,i)$ 和${\hat e_2} = (1, - i)$ 、(b)${\hat e_1} = (1, - i)$ 和${\hat e_2} = (1,i)$ 、(c)${\hat e_1} = (1,0)$ 和${\hat e_2} = (0,1)$ 时,FP光束的偏振分布[18]Figure 2. Distribution of polarization for FP beams where
${\hat e_1}$ and${\hat e_2}$ are, respectively, (a)${\hat e_1} = (1,i)$ ,${\hat e_2} = (1, - i)$ ; (b)${\hat e_1} = (1, - i)$ ,${\hat e_2} = (1,i)$ ; (c)${\hat e_1} = (1,0)$ ,${\hat e_2} = (0,1)$ [18]除了LG光束,其他携带轨道角动量的光束叠加也可以实现杂化矢量光,例如2013年,Cardano等[29]通过超几何高斯光产生的杂化矢量光场可以表示为:
$$\vec E\left( {\rho ,\varphi } \right) = HyG{G_{ - |l|,l}}\left( \rho \right)[A{\hat e_{{L}}} + B\exp \left( {2il\varphi } \right){\hat e_R}]$$ (5) 这里,HyGGpm为超几何高斯光束:
$$HyG{G_{pm}} = {C_{pm}}\frac{{\varGamma (1 + |m| + p/2)}}{{\varGamma (|m| + 1)}}{i^{|m| + 1}}{\left(\frac{{\textit{z}}}{{{{\textit{z}}_R}}}\right)^{p/2}}{\left(\frac{{\textit{z}}}{{{{\textit{z}}_R}}} + i\right)^{ - [1 + |m| + p/2]}}{\rho ^{|m|}}{{\rm{e}}^{ - i{r^2}/{w^2}(\zeta + i)}}{{\rm{e}}^{im\varphi }}{}_1{F_1}\left( - \frac{p}{2},|m| + 1;\dfrac{{{\textit{z}}_R^2{r^2}}}{{{w^2}{\textit{z}}({\textit{z}} + i{{\textit{z}}_R})}}\right)$$ (6) 式中:
$\varGamma ()$ 为伽马函数;$_1{F_1}()$ 为合流超几何函数;w为束腰;${{\textit{z}}_R}$ 为瑞利范围。2015年,Shvedov等[30]基于Bessel光束产生杂化矢量光场:$$\vec E\left( {\rho ,\varphi } \right) = A{J_{|{l_1}|}}\left( {{k_\rho }\rho } \right)\exp \left( {i{l_1}\varphi } \right){\hat e_{\rm{L}}} + B{J_{|{l_2}|}}\left( {{k_\rho }\rho } \right)\exp \left( {i{l_2}\varphi } \right){\hat e_R}$$ (7) 式中:
$ {J}_{\left|l\right|}(·)$ 为一类Bessel函数。2016年,Garcia-Gracia等[31]通过Mathieu光束产生杂化矢量光:$$\vec E\left( {\rho ,\varphi } \right) = {\rm{MB}}{}_0^e\left( {\rho ,\varphi } \right){\hat e_L} + {\rm{HMB}}_l^ \pm \left( {\rho ,\varphi } \right){\hat e_R}$$ (8) 式中:MB和HM代表不同的Mathieu光束。
尽管杂化矢量光可以通过不同形式的光束实现,但是其偏振存在一定的相似性。这种相似性反应在偏振奇异性上。Nye[32]在1983年提出偏振奇异性的概念,Dennis[33]在2002年对光场的奇异性做了详细的综述。偏振奇异性包括偏振取向的不确定性——C点和偏振旋向的不确定性——L线。C点偏振奇异性一般用C点指数IC描述。IC定义为线积分[29]:
$${I_{\rm{C}}} = \frac{1}{{2\pi }}\oint_L {{\rm{d}}R \cdot \nabla {\gamma _0}} $$ (9) 式中:γ0
为相位分布;L为绕C点的闭合曲线。杂化矢量光比柱矢量光场具有更为丰富的C点奇异性。一般柱矢量光场C点指数的大小为±1/2,而杂化矢量光为1/2的整数倍。杂化矢量光的C点附近偏振态椭圆半长轴构成一簇围绕C点的流线。不同的C点指数对应不同形状流线:对于IC=1/2的C点,构成星形流线;对于IC=−1/2的C点,构成分柠檬形或变种星形流线;对于IC =1的C点,构成螺旋形流线,如图3所示[29]。杂化矢量光的IC 只取决于基矢光携带的轨道角动量,与具体光束类型无关。从某种意义上讲,这是因为携带相同角动量的杂化矢量光的Stokes参数在统计上是不变的,与具体光束形式无关[34]。 与C点丰富的拓扑结构相比,L线的拓扑性比较单一,没有复杂的类型划分。如图3所示[29],通常杂化矢量光的L线是闭合同心圆环,其半径与光场轮廓参数A、B有关,与携带轨道角动量和具体光束类型无关。
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矢量光的聚焦特性可以用Richards-Wolf矢量衍射方法进行数值分析。对于沿着z轴传播的光束而言,焦平面附近的光场可以表示为[9]:
$$ {\vec E(r,\phi ,{\textit{z}}) = \dfrac{{ikf{{\rm{e}}^{ - ikf}}}}{{2\pi }}\displaystyle\int\limits_0^{{\theta _{\max }}} {{\rm{d}}\theta \displaystyle\int\limits_0^{2\pi } {{\rm{d}}\varphi [} } ({E_\varphi }{{\overset{\frown} {\rm{e}}} _\varphi } + {E_\rho }{{\overset{\frown} {\rm{e}}} _\rho }){ {\rm e}^{i\overrightarrow k \cdot \overrightarrow r }}\sin \theta \sqrt {\cos \theta }]} $$ (10) 式中:Eφ和Eρ为入瞳处矢量光的角向和径向分量;θmax是由透镜数值孔径决定的最大角度;k是波矢。
Brown研究组[7]发现相对于均匀偏振分布的光束,径向偏振的柱矢量在紧聚焦过程中产生更强的纵向电场,而角向偏振的柱矢量光在紧聚焦区域则不存在纵向电场。Wang等[55]利用二元环状光学器件对径向柱矢量光进行紧聚焦,产生一个超长“光针”聚焦分布,经过参数优化后获得超衍射极限的光针。除此之外,利用紧聚焦柱矢量光束还可以激发纳米天线中等离激元,在定制等离子纳米天线的光散射方面具有巨大的应用潜力[56]。
杂化矢量光的紧聚焦场比柱矢量光场的行为更为丰富。在聚焦场中心附近,偏振椭圆的取向不再局限于x-y平面,大部分沿z轴方向,场强存在较强z分量,如图11所示[24]。沿z轴方向的偏振椭圆导致其自旋方向垂直于z轴,即横向自旋[57-58]。横向自旋是自旋—轨道角动量守恒的结论,它的大小取决于入瞳处偏振椭圆的椭圆率和椭圆取向。
此外,聚焦杂化矢量光场的纵向光强分布也十分有趣。2017年,满忠胜课题组[59]对任意偏振态的光束聚焦特性进行分析,提出了通过控制正交线偏振基模实现聚焦整形的方法。2018年,詹其文团队[60]指出低数值孔径聚焦的杂化矢量光场可以在焦平面上形成一个柱对称的阶跃光强分布,通过对杂化矢量光参数的控制可以实现阶跃光场定制化。同年,李艳秋课题组[61]分析了FP光束在紧聚焦条件下的光强与纵向电场相位分布,证明了纵向电场具有轨道角动量分布。2019年,笔者与陈理想课题组[27]分析了紧聚焦下杂化矢量纵向光强对称性与其阶次之间的关系,给出光强分布图样具有m−n+2重轴对称性,m和n代表轨道角动量,具体如图12所示。
杂化矢量光场独特的聚焦特性使其在光学捕获方面更具优势。2012年,Wang[62]发现在光束截面,FP光对瑞利粒子表现为指向光束中心的力场,能将微粒束缚在光束中心。2018年,Xue等[63]指出,聚焦的杂化矢量光在束腰处的光力更为稳定,而且可以克服热运动的影响。2019年,笔者与陈理想课题组[27]研究了强聚焦情况下杂化矢量光的光力行为,发现微粒可在多个位置被捕获,捕获位置的数量取决于携带轨道角动量的值,这恰好与聚焦光场光强分布对应。除此之外,聚焦杂化矢量光场还可以将微粒限制在焦平面附近,有效地实现三维捕获。
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在实际大气中传播时,由于大气不均匀分布、气流、湍流等对激光光强产生负面影响,引起光强闪烁。2015年,Wei等[28]通过调节杂化矢量光光场轮廓参数的比值大幅降低了光束经过湍流时的闪烁现象。相比于高斯光束或LG光束,杂化矢量光在减弱湍流引起的闪烁具有明显优势,为大气激光通讯、激光武器、激光遥感等应用奠定基础。
实际上,杂化矢量光在自由空间传输时,偏振模式和Gouy相位相关。对于正交圆偏振LG光产生的杂化矢量光而言,Gouy相位为φ=(lR-lL)arctanz/ZR,其中,lR和lL分别为右旋LG光和左旋LG的轨道角动量。随着传输距离z的增加,Gouy相位发生改变,导致偏振产生一个正比于Gouy相位的旋转[64]。旋转方向取决于(lR-lL)的正负,若为正值时,旋转方向是逆时针;若为负值时,旋转方向为顺时针。因此,Arora等[20]提出可以通过杂化矢量光传输时的旋转方向判断轨道角动量的正负。
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杂化矢量光的非线性光学效应亦是一个非常有意义的课题。早期关于非线性光学效应的工作不涉及频率转换。例如2014年,陈理想课题组[65]报道了杂化矢量光的电光效应。经过电光效应的杂化矢量光在自由空间传播过程中,其C点和L线会产生分裂,分裂行为与细胞核和细胞质的分裂相似。直到2018年,涉及频率转换的非线性光学效应研究出现。笔者与陈理想课题组[26]利用相位匹配导致倍频光束输出单一偏振这一特性,通过倍频将杂化矢量光不可见的偏振拓扑结构可视化。由于大多数非线性过程具有偏振敏感性,因此杂化矢量光的非线性频率转换具有挑战性。2019年,笔者[25]从相位匹配输出单一线偏振光出发,提出利用正交级连晶体装置对FP光进行倍频,不仅在频率加倍的同时保留了光束的偏振特性,而且实现了拓扑荷的加倍,使其呈现出新的流线形状,图13给出了星形流线的FP光在倍频后转化为螺旋形流线。同年,Yang等[66]引入Sagnac干涉仪实现正交偏振倍频矢量光叠加,从而得到倍频的杂化矢量光。接着,基于Yang等人的装置,朱智涵课题组[67]提出自锁干涉仪,得到倍频后的杂化矢量光,同时发现倍频矢量光的C点在衍射传播中具有旋转效应。2020年,基于同样装置,他们[68]在II类相位匹配的长周期极化晶体PPKTP中实现了参量上转换,在频率转换的同时,保证杂化矢量光的偏振模式不变性。他们的结果为基于矢量模式的高维量子或高容量经典信道接口奠定了基础。2021年,Silva等[69]基于正交级连晶体的装置研究了杂化矢量光的参量下转换。泵浦光和信号光中有一束为杂化矢量光而另外一束为线偏振光时,可以输出与输入偏振一致的杂化矢量光。
除了频率转换,另一个有趣的非线性光场效应是偏振旋转。由于Gouy相位的影响,杂化矢量光在自由空间传播时存在一定的偏振旋转,但是这个旋转角度有限。2020年,东南大学顾兵课题组[70]发现FP光束经过各向同性的Kerr介质会出现整体偏振旋转。对于各向异性的Kerr介质,FP光束在传播过程中,光强图样由中心亮斑围绕的衍射环变成椭圆结构并伴随着两段衍射环。这主要是由于各向异性介质导致柠檬形FP光具有空间结构的非线性相移,这种相移来自于两个幅度不相等的左旋和右旋分量,最终引起光强图样和偏振分布出现了对称性破缺。除了Kerr介质,西安交通大学李福利课题组[71-72]发现杂化矢量光在铷原子蒸汽中也会发生偏振旋转。由于原子蒸汽对光子的吸收和激发,在原子蒸汽中传播时,杂化矢量光偏振旋转角则可以突破Gouy相位导致偏振旋转的极限。通过调节原子的相关参数可以实现任意偏振旋转角的控制,这对于偏振控制具有潜在的应用价值。
Research progress of hybrid vector beams (Invited)
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摘要: 偏振是光场的一个重要矢量属性。依据空间偏振分布的不同,可以将光场分为标量光场和矢量光场。其中,非均匀偏振分布的矢量光场在光场传播、聚焦、非线性效应等方面表现出一系列有趣的行为,对其进行深入研究具有重要的科学意义和应用价值。一种在同一波阵面上包含不同偏振态的杂化矢量光场,因其比普通矢量光场具有更丰富的偏振特性,自2010年被发现以来,在光通信、光学操控、量子通信等领域中展现诸多诱人的前景,目前已成为光学领域的一个研究热点。综述着重介绍了杂化矢量光的制备以及其在聚焦、传播和非线性光学等方面中的特性和应用。Abstract: Polarization is an important property of light field. According to the spatial polarization distribution, optical field can be divided into the scalar field and the vector field. Since the vector field with inhomogeneous polarization distribution performs some interesting behaviors in light propagation, studying the vector beam is of either scientific significance or engineering applied importance. The hybrid vector beams, which contains all kinds of polarization state in the same wave front, has attracted a lot of interests since 2010. Due to the richer polarization states than the normal vector beam, the hybrid vector beam shows attractive prospect on the optical communication, optical manipulation and quantum communication fields. Here, the generation, focusing, propagating and nonlinear optics of the hybrid vector beam were summarized.
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Key words:
- hybrid vector beams /
- polarization singularity /
- full Poincaré beams /
- Poincaré sphere
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图 2 (a)
${\hat e_1} = (1,i)$ 和${\hat e_2} = (1, - i)$ 、(b)${\hat e_1} = (1, - i)$ 和${\hat e_2} = (1,i)$ 、(c)${\hat e_1} = (1,0)$ 和${\hat e_2} = (0,1)$ 时,FP光束的偏振分布[18]Figure 2. Distribution of polarization for FP beams where
${\hat e_1}$ and${\hat e_2}$ are, respectively, (a)${\hat e_1} = (1,i)$ ,${\hat e_2} = (1, - i)$ ; (b)${\hat e_1} = (1, - i)$ ,${\hat e_2} = (1,i)$ ; (c)${\hat e_1} = (1,0)$ ,${\hat e_2} = (0,1)$ [18] -
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