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任意的矢量光束均可被分解为具有不同复振幅分布的正交偏振分量 [17]。以具有横向非均匀偏振分布的矢量光场为例,其通常可表示为:
$$ \begin{split} \boldsymbol{E}\left(r,\varphi \right)=&{{\boldsymbol{E}}}_{1}\left(r,\varphi \right)\mathrm{e}\mathrm{x}\mathrm{p}\left[i{\delta }_{1}\left(r,\varphi \right)\right]{\boldsymbol{e}}_{1}+\\ & {{\boldsymbol{E}}}_{2}\left(r,\varphi \right)\mathrm{e}\mathrm{x}\mathrm{p}\left[i{\delta }_{2}\left(r,\varphi \right)\right]{\boldsymbol{e}}_{2} \end{split}$$ (1) 式中:r和ϕ分别代表径向和旋向坐标;E1和E2为两束光的振幅分布;δ1和δ2为位相分布;e1和e2代表正交偏振基矢。从另一个角度来看,矢量光束偏振态的复杂空间分布源于两个具有不同复振幅分布的正交偏振光束的线性叠加,偏振态的空间变化是由振幅差和位相差的空间变化导致。若将两束具有相反旋向变化位相分布δ1 = δ2 = –ϕ的左右旋圆偏振光等幅叠加,则可产生局域线偏振矢量光场;相反,若将两束振幅分别为E1=cosϕ,E2=sinϕ的正交线偏振平面光波进行叠加 ,同样可以产生局域线偏振矢量光场。对于偏振态纵向变化矢量光束来说,其生成方法同样遵循上述原理,即可通过在正交偏振分量间构造随传播变化的位相差或振幅差,达到偏振态纵向空域调控的效果。偏振态纵向变化示意图如图1所示。
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参照基于位相调制产生矢量光束的方法,可以通过构造纵向变化的位相差以实现偏振态纵向变化的矢量光束。为了实现上述要求,通常采用基于径向位相调制的贝塞尔光束。一般情况下,贝塞尔光可通过轴棱镜[33-35]或计算全息图衍射[36-39]实现在初始平面的径向位相调制,此位相调制可表示为:
$$ \phi \left(r\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-i{k}_{r}r\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(\pm il\varphi \right) $$ (2) 式中:kr为径向波矢。上述位相调制会使光束在传播过程中逐渐收敛并产生一个近似的无衍射区zmax=rmaxk/kr,其中rmax为位相掩模的最大半径,k为波数。如果l = 0,则会在无衍射区内产生近似的零阶贝塞尔光束。当波数k固定时,波矢的纵向分量kz则随着横向分量变化而变化。如果两个正交偏振的贝塞尔光束E1和E2由具有不同的初始位相调制即径向波数时,则在自由空间中传播时在正交偏振分量间将出现动态变化的位相差,其可表示为:
$$ \delta \left(z\right)=\left({k}_{z2}-{k}_{z1}\right)z $$ (3) 那么偏振态随传播变化的矢量贝塞尔光束可表示为:
$$\begin{split} \boldsymbol{E}={\boldsymbol{E}}_{1}+{\boldsymbol{E}}_{2}={{\boldsymbol{E}}}_{1}{\boldsymbol{e}}_{1}+{E}_{2}\mathrm{e}\mathrm{x}\mathrm{p}\left[i\delta \left(z\right)\right]{\boldsymbol{e}}_{2} \end{split} $$ (4) 因此,可以通过构造纵向变化的位相差来实现偏振态纵向变化矢量光束。
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除了纵向变化的位相调制,构造纵向变化的振幅差异同样是产生偏振态纵向变化矢量光束的有效方法。考虑两个具有正交偏振的零阶贝塞尔高斯光束,其电场分布E可表示为[40]:
$$\begin{split} \boldsymbol{E}\left(r,\varphi \right)=& \mathrm{e}\mathrm{x}\mathrm{p}\left(-{r}^{2}/{\omega }_{0}^{2}\right){J}_{0}\left({k}_{r}r\right)\mathrm{ex}\mathrm{p}\left(i{k}_{z}z\right) \cdot\\ & \left[{E}_{1}\left(z\right){{\boldsymbol{e}}}^{i{\delta }_{1}\left(z\right)}{\boldsymbol{e}}_{1}+{E}_{2}\left(z\right){{\boldsymbol{e}}}^{i{\delta }_{2}\left(z\right)}{\boldsymbol{e}}_{2}\right] \end{split} $$ (5) 式中:E1,2(z)为轴向强度分布;kr和kz分别为径向和纵向波矢;δ1,2(z)为独立于动态位相的位相延迟。当δ1 – δ2为固定值时,则由轴向振幅来影响纵向变化的偏振态,这种振幅调制可通过空间频谱来实现[41-42]。根据修正的傅里叶变换,具有重塑轴向包络的准贝塞尔光在柱坐标系下可表示为[40]:
$$ {E}_{\mathrm{1,2}}\left(z\right)={\int }_{0}^{k}{U}_{\mathrm{1,2}}\left(\sqrt{{k}^{2}-{k}_{z}^{2}},z=0\right) {\cdot {\rm{e}}}^{i{k}_{z}z}{k}_{z}{\rm{d}}{k}_{z} $$ (6) 式中:U1,2代表空间频谱。因此,对纵向振幅分布的操纵可以转化为对空间频谱的调制。将放置在傅里叶平面上的定制掩模作为空间频谱滤波器,即可实现预期的振幅调制。基于傅里叶逆变换,由期望振幅反演的空间频谱可表示为[40]:
$$ {U}_{\mathrm{1,2}}\left(\sqrt{{k}^{2}-{k}_{z}^{2}},z=0\right)=\frac{1}{2\pi {k}_{z}}F\left[{E}_{\mathrm{1,2}}\left(z\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(i{k}_{z0}z\right)\right] $$ (7) 不同的空间频谱滤波器可以定制两个正交偏振分量间的振幅关系,从而确保传播过程中偏振的精确调控。
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虽然在线性或非线性传播过程中,矢量光束偏振态随传播变化的现象并不罕见,但偏振态纵向变化矢量光束更加强调偏振态纵向空域调控的可控性。这种可控性一方面体现在偏振态调控的准确性上,即保证偏振态的长轴取向和椭偏率准确符合理论设计。另一方面,这种可控性体现在偏振态变化形式的灵活性上。相比于矢量光场偏振态的横向调控,偏振态纵向调控并不能那么直接,即通过单一平面光场的调制,影响光束在传播过程中的振幅、位相和偏振,这通常具有更高的实现难度。随着光场调控技术的发展,偏振态纵向变化矢量光束的生成方法也在不断发展,其产生方法可以大致分为两类:即相位调制法和空间频谱滤波法。
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复振幅调制法是指通过调控初始横向平面的位相分布进而影响纵向方向的偏振态分布。2015年,Moreno等人[43]通过在两个正交线偏振分量之间引入一个额外的径向位相延迟,从而产生偏振态沿纵向方向变化的零阶和高阶贝塞尔光束。原理上利用径向衍射光栅来使光束产生轴锥镜位相剖面,通过在衍射光栅前添加线性径向位相延迟,偏振态将会沿着传播方向连续变化,其原理图如图2所示。
实验上采用空间光调制器分屏的方式结合4f系统所组成的实验系统,如图3所示。首先45°线偏振光入射到空间光调制器上,空间光调制器被分为左右两部分,每部分加载具有不同倾角的轴锥位相灰度图,其中只有水平线偏振分量才会得到调制。接着通过透镜和反射镜,光束被反射到空间光调制器的另一部分。在四分之一波片的作用下,偏振态发生了反转,使得水平偏振分量变成竖直偏振分量,从而使得未调制的偏振分量通过空间光调制器的另一部分实现位相调制,最后将两个正交偏振分量合成,从而实现偏振态纵向变化贝塞尔光束的生成。
2016年,Moreno等人在上述研究的基础上,通过在两个正交偏振分量间加入具有相反拓扑荷的螺旋位相以及径向位相,从而实现了偏振态及拓扑荷随传播变化的矢量贝塞尔光束[44]。在实验上利用空间光调制器通过改变加载其上的位相灰度图实现轴锥镜和可变螺旋板的功能。由于可变螺旋板可将涡旋位相合并到圆偏振分量上,因此需要在入射端加入45°放置的1/4波片,在出射端加入以–45°放置的1/4波片和竖直放置的1/2波片,实验系统如图4所示。最终分别生成了一阶、二阶、四阶的偏振态纵向变化的矢量光束。对于一阶即拓扑荷l1 = –l2 = 1的矢量光束,经过偏振片后其强度呈现两瓣花瓣状分布,并且随着传播,两瓣的方向改变,这说明随着光束传播局域线偏振态的方向发生了变化。当传播了50 cm后,光束由径向偏振光转换为旋向偏振光。此外,Moreno等人还将光束分成5个不同径向区域,并调控不同径向区域使其具有l1 = –l2 = 1、2、3、2、1不同的涡旋拓扑荷数,从而获得了偏振态和拓扑荷均随传播变化的矢量贝塞尔光,并测得了距离空间光调制器0.5、0.85、1.2、1.55、1.9 m处的光强,经偏振片后,其光强分别呈现两瓣、四瓣、六瓣、四瓣、两瓣的分布,实现了矢量光束阶数随传播变化的效果。
除了利用空间光调制器分屏的方式来实现对正交偏振分量的独立调控,Gao等人在单一路径上利用级联空间光调制器的方式同样实现了偏振态纵向变化矢量光束的生成[45]。光束入射到空间光调制器1上,其上加载径向位相模式图,将水平偏振分量转化为锥形波;接着光束经偏振转化后入射到空间光调制器2上,从而在不同正交偏振分量间实现不同的附加径向位相调制,其实验装置图如图5所示。此外,利用空间光调制器2加入了涡旋位相调制,还实现了偏振态纵向变化的一阶和二阶矢量光束。
以上方法都是利用加载在空间光调制器上的位相掩膜实现对光束的调制。最近,Capasso等人基于超表面,可在不需要设定入射偏振态的情况下,实现沿传播方向变化的偏振响应[46],其原理图如图6(a)所示。
超表面由具有结构双折射的介电纳米柱以亚波长间隔阵列的方式排列而成。若想实现随传播变化的偏振分布,可将两个具有相同波长但偏振态和径向空间频率不同的单色波叠加。超表面是实现上述偏振响应的便捷平台,可作为是一种z轴依赖偏振器件,基于琼斯计算法[47]和双相全息术[48],通过控制介电纳米柱的组成和排列结构,超表面可以将入射光转换为不同的偏振态纵向变化矢量光束,实验装置图如图6(b)所示。
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上述的研究是在单一光路上对光束进行调制,实现了偏振态纵向变化矢量光束的生成。除此以外,也有研究小组利用基于全息光栅的双光路生成方法实现了灵活有效的偏振态纵向调控。
2016年,Zhao等人通过映射径向变化PB位相,实现偏振态纵向变化矢量光束的生成,并探索了偏振态纵向变化矢量光束的传输自愈特性[49]。实验上将Sagnac干涉仪与4f系统相结合。系统主要分为两大部分,其中一部分是由空间光调制器和透镜M1、透镜M2组成的干涉仪组成,用来调控横向的PB位相;另一部分是基于轴锥镜用来实现横向-纵向的映射,实现了偏振态沿庞加莱球赤道和经线变化的零阶和高阶矢量贝塞尔高斯光束的生成,实验装置图如图7所示。
相比于偏振态的横向空域调控,偏振态纵向调控通常不那么直接,即需要通过初始平面的横向调控,间接控制光束的传输过程,达到偏振态纵向变化的效果。偏振态纵向变化矢量光束的可控性也因此受到限制。这种可控性一方面体现在偏振态纵向调控的准确性上。基于初始径向位相调制,正交偏振分量间的径向空间频率存在差异,是偏振态纵向变化矢量光束得以产生的前提条件。但是这种类似于轴锥镜的位相调制也会对产生的类贝塞尔光束的局域振幅产生影响。考虑将两个具有不同径向空间频率kr1、kr2且偏振态正交的左右旋圆偏振基矢进行叠加。经菲涅耳衍射后,光束的振幅和光强与径向空间频率直接相关[50],则:
$$ I\left(r,z\right)=\left(\frac{2\pi z}{k}\right){E}_{0}^{2}{k}_{r}^{2}{J}_{0}^{2}\left({k}_{r}r\right) $$ (8) 对于偏振态纵向变化矢量光束来说,初始平面正交偏振分量的等幅叠加在传播过程中并不能依然保持相等,特别是对于偏振态变化频率高、径向空间频率差异大的矢量光束,这将会对偏振纵向调控的准确性产生显著的影响。为了解决这个问题,Wang等人基于二维二元位相型全息光栅和4f系统提出振幅-位相联合调控方法[50],实验装置如图8(a)所示。其中,光栅的放大图如图8(b)所示,通过调控光栅的占空比即可消除传播过程正交偏振分量间的振幅差异,从而实现利用光束调控的准确性。利用该实验系统,生成了局域线偏振随传播旋转的类贝塞尔光束,在不同的传播距离处放置了不同角度的偏振片,从而检验了实验结果的准确性。在不同的传播距离下,光束中心经过偏振器后可以完全消光,有效地消除了径向指数的影响,中心偏振态最高约为6.7 (°)/mm的旋转角速度,表明实现了偏振态沿庞加莱球赤道纵向快速变化矢量光束的准确生成。
偏振态纵向变化矢量光束的可控性还体现在偏振态纵向空域调控的灵活性上。上述介绍的偏振态纵向变化矢量光束,大多是沿着庞加莱球赤道或者经线变化的。为丰富矢量光束的形式,2022年,Lü等人由偏振态纵向变化的Stokes参量表征出发,以庞加莱球上不同圆路径定义偏振态纵向变化的形式,利用空间光调制器和4f系统实现了沿着庞加莱球任意圆路径偏振态纵向变化矢量光束的生成[51]。实验上基于二维二元位相型全息光栅及可调波片组,搭建了振幅-位相-偏振联合调控矢量光束生成系统,实验系统如图9(a)所示,532 nm连续光激光器输出的高斯光束经过扩束和准直后转换为水平线偏振光,入射到位于由透镜L1 (焦距为1000 mm)和透镜L2 (焦距为200 mm)组成的4f系统前焦面的空间光调制器上,其上加载二维二元位相型全息光栅,入射光透过光栅后被衍射为不同的级次,空间滤波器将两维+1级衍射光滤出,并分别通过位于空间频谱面上的波片组(由1/2波片和1/4波片组成)转化成所需的正交偏振基矢;再经过位于4f系统后焦面的Ronchi光栅进行合束。实验上通过调节光栅占空比和光栅结构调控了正交偏振分量的振幅比和位相分布,通过调节波片组调控两叠加光束的偏振态,从而实现了振幅-位相-偏振态的联合调控。通过设置不同的波片组角度及光栅的结构,调控不同的叠加偏振基矢以两束光的振幅和位相,实验上生成了偏振态沿庞加莱球任意圆路径纵向变化的矢量光束。
图 9 (a) 偏振态纵向变化矢量光束生成与探测系统图;(b) 偏振态沿传播方向级联变化的双模矢量光束示意图[51]
Figure 9. (a) Generation and detection system diagram of vector optical beams with longitudinally varying polarization; (b) Schematic diagram of the dual-mode vector optical beam with cascaded variations of polarization in longitudinal direction [51]
此外,为了验证实验方案的可拓展性,参考横向平面偏振非连续分布的双模矢量光场的概念,Lü等人提出了偏振态纵向变化双模矢量光束的概念。纵向双模矢量光束偏振态变化示意图如图9(b)所示,图中红色和蓝色部分偏振态的变化是沿着庞加莱球的同一条圆轨迹,但在传播过程中表现出一个跳跃,即实现了偏振态的非连续纵向调控。通过在不同径向区域引入不同的初始位相,实现了不同偏振变化的纵向级联。
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除了直接对光束初始平面的复振幅进行调控,也可以通过对光束空间频谱面的调控实现偏振态纵向变化矢量光束的生成。Zhao等人基于空间频谱优化的方法,利用Sagnac干涉系统[18,49,52-53]将两个具有互补光强和正交偏振态的光束进行叠加,从而产生轴向强度均匀并且偏振态随传播变化的准贝塞尔光束[40]。原理图如图10所示。利用逆傅里叶变换设计出正交偏振分量所对应的空间频谱,从而获得所需的偏振纵向变化。这里假设准贝塞尔光束由两个具有线性轴向强度变化的正交偏振光合成,两偏振分量的轴向强度不断变化,叠加光束的偏振态也因此随传播改变。为了获得均匀的偏振态变化,Zhao等人设计了两个分别具有正弦和余弦函数的轴向包络的光束,当两光束位相差为0时,其偏振态为线偏振。实验上分别测量了z = 0、1.9、3.8、5.6、7.5 cm处的Stokes参量,实验结果表明准贝塞尔光束的偏振态在传播过程中保持线性偏振,且呈现周期性变化形式。
利用空间频谱滤波法也可用来实现高阶偏振态纵向变化矢量光束的生成[54]。零阶贝塞尔光束可以被认为是环形狭缝的傅里叶变换[55],利用特殊设计的螺旋狭缝[55-56]可以在不同传播距离处产生具有不同拓扑荷的涡旋光束,其原理图如图11所示。
将左右旋圆偏振光分别照射拓扑荷为3和–3的螺旋狭缝,将产生的拓扑荷随传播变化的贝塞尔光束共线叠加,即可获得偏振态和阶数均随传播变化的矢量贝塞尔光束,随着传播距离的逐渐增大,矢量贝塞尔光束的拓扑荷逐渐趋于零,同时偏振分布也会随传播变化。
以上两种方法均是基于单向激光传播框架下的偏振态纵向空域调控方法。近期有研究表明,具有不同偏振态和位相分布的反向激光也能够在三维空间中实现复杂的偏振态纵向变化矢量光束[57]。
Research progress of vector optical beam with longitudinally varying polarization (invited)
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摘要: 立足于偏振态空域调控技术,矢量光束所特有的偏振态空间结构属性,使其在光学及其交叉学科领域表现出巨大的研究价值和应用潜力。以往的研究主要关注偏振态的横向空域调控,而纵向(传播)方向同样是重要的光场调控维度,偏振态纵向变化矢量光束的出现,拓展了矢量光束的调控维度,为光与物质的相互作用带来更多可能,引起了广泛关注。文中围绕着偏振态纵向空域调控技术的发展,介绍了基于纵向变化位相差和振幅差的偏振态纵向变化矢量光束生成理论,并总结归纳了利用相位调制和空间频谱滤波的两类实验生成方法。此外,文中还对偏振态纵向变化矢量光束目前存在的问题进行了讨论,并对其发展前景进行了展望。Abstract:
Significance Based on the spatial manipulation technique of polarization, the unique spatially structured properties of polarization make vector optical beam show great research values and application potentials in optics and its interdisciplinary fields. Previous studies mainly focused on the polarization manipulation in the transverse plane, but the longitudinal (propagation) direction is also an important dimension for manipulating optical field. The unique beam, with customized polarization distribution in the longitudinal direction, has attracted increasing attentions in recent years. Beyond enriching the diversity of vector beam, the variation of polarization along the direction of propagation provides increased scope for the light-matter interaction, especially in the optical nonlinear effect and spin-orbit coupling. Moreover, it also offers applied advantages in remote polarimetry, material deep processing and three-dimensional micromanipulation. Progress First, the generation principles of vector beam with longitudinally varying polarization are introduced. In order to realize the variation of polarization in the longitudinal direction in free space, the direct method to modulate the propagation environment of the polarized beam are unsuitable and the possibility of modulating the beam at the initial plane to indirectly control the longitudinal distribution of polarization should be taken into consideration. Two main methods reported to achieve longitudinally varying polarization are the construction of varying phase difference and amplitude difference in the propagation direction. The longitudinally varying phase difference is achieved by discrepant initial radial phase modulations on the orthogonally polarized components, while the construction of varying amplitude difference in the propagation direction is achieved by different spatial spectrum filters for the customized amplitude relations between orthogonally polarized components. Relevant experimental methods are summarized which can be divided into the modulation of phase and the filtering of the spatial spectrum. The phase modulation method includes the single-path generation method based on phase mask (Fig.3-6) and double-path generation method based on holographic gratings (Fig.7-9). As the most common method to generate vector beam with longitudinally varying polarization, the phase modulation method has the problem of controllability on the polarization variation. On the one hand, this controllability is reflected in the accuracy of the longitudinal manipulation of polarization. Due to the different initial phase modulation, corresponding variation occurs in the amplitude of orthogonally polarized components during propagation, which will inevitably affect accuracy of polarization manipulation especially for the high-frequency longitudinal variation of polarization. On the other hand, the controllability is reflected in the flexibility of the longitudinal spatial modulation of polarization. Except for the variation of polarization along the equator and meridian of the Poincaré sphere, the continuous longitudinal variation of polarization can also track other trajectories on the Poincaré sphere. The recent works to improve the controllability of longitudinal manipulation of polarization are discussed in detail. Conclusions and Prospects Methods to generate vector beam with longitudinally varying polarization along propagation direction have been rapidly developed in recent years. Although many approaches to achieve the longitudinal manipulation of polarization have been demonstrated, there are still some problems to be solved. On the one hand, the generation method which balances efficiency and flexibility will contribute to research and practical applications of vector beams with longitudinally varying polarization. On the other hand, based on the novel spatial manipulation dimension of polarization, the interaction between the longitudinally varying polarization and matter still need to be further studied to give full play to its longitudinal polarization "ruler" role, and the application of unique vector beam in laser depth machining, laser measurement and optical micromanipulation needs to be expanded. The research of this paper aims to provide some reference for the design and generation of vector beam with longitudinally varying polarization. The generation theories and experimental methods of vector beam with longitudinally varying polarization are summarized, and the development prospects are also forecasted, which may be helpful for the manipulation techniques of optical field and its applications in laser fabrication, laser measurement and optical micromanipulation. -
Key words:
- vector beam /
- polarization /
- Bessel beam /
- Poincaré sphere
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图 9 (a) 偏振态纵向变化矢量光束生成与探测系统图;(b) 偏振态沿传播方向级联变化的双模矢量光束示意图[51]
Figure 9. (a) Generation and detection system diagram of vector optical beams with longitudinally varying polarization; (b) Schematic diagram of the dual-mode vector optical beam with cascaded variations of polarization in longitudinal direction [51]
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