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为生成得到偏振奇异光场,采用了生成全庞加莱光束的方法[10],即叠加两束正交偏振的圆偏振光。位于初始平面z=0的叠加光场可以表示为[10]:
$ \boldsymbol{E}(\boldsymbol{r} ; \gamma)=\cos \gamma \cdot \boldsymbol{e}_1 U_1(\boldsymbol{r})+\sin \gamma \cdot \boldsymbol{e}_{\mathrm{r}} U_2(\boldsymbol{r}) $
(1) 式中: r为空间向量; γ为控制光束强度的常量参数; el和er分别为左旋和右旋圆偏振基矢量; U1(r)和U2(r)分别为携带或不携带涡旋相位的圆艾里光束。位于初始平面的携带涡旋相位的圆艾里光束,其电场在柱坐标系(r, φ, z)中可表示[26]:
$ \begin{aligned} U(r, \varphi, z=0)= & A_0 \cdot A\left(\frac{r_0-r}{w_0}\right) \exp \left(a \frac{r_0-r}{w_0}\right) \cdot \\ & {[r \exp (\mathrm{i} \varphi)]^l } \end{aligned} $
(2) 式中: A0为振幅常量; A(·)为艾里函数; r0为主艾里光环的半径; w0为径向缩放; a为决定传播距离的衰变参数; l为涡旋相位的拓扑荷。对于l=0,即代表圆艾里光束不携带涡旋相位。
为生成分数偏振奇异光场,将U1设置为携带涡旋相位且拓扑荷l为分数数值的圆艾里光束,U2为不携带涡旋相位的圆艾里光束。设定光场U1中l的数值为0.1、0.3、0.5、0.7、0.9和1.0,设定γ=π/4,r0=1 mm,w0=0.08 mm,a=0.1,光场尺寸为4 mm×4 mm。将上述参数结合式(1)和式(2),即U1为左旋圆偏振、U2为右旋圆偏振时,初始平面光场的偏振态分布如图 1所示。图中偏振椭圆的不同颜色代表了不同的旋向,绿色为右旋偏振,红色为左旋偏振。图 1展示了左旋分量U1中的拓扑荷不同时,光场偏振态分布的演化过程。当l=0.1时,光场偏振态接近于线偏振;随着l数值的不断变大,光场中偏振椭圆的椭圆率也随之变大;当l=1.0时,光场中出现了一个圆偏振点,即C点,此时光场的偏振态分布是一个典型的“柠檬”结构;当l取其它数值时,虽然其偏振态分布与“柠檬”结构相似,但是其光场中并没有圆偏振点,可将其称为“准柠檬”结构。
图 1 左旋圆偏振分量U1中不同拓扑荷(l≤1.0)时初始平面光场的偏振态分布
Figure 1. Polarization states of light fields in the initial plane for different values of topological charge (l≤1.0) in left-handed circularly polarized component U1
与之类似,将U1设置为右旋圆偏振、U2为左旋圆偏振时,初始平面光场的偏振态分布如图 2所示。此时初始平面的偏振态从接近于线偏振逐渐变化到“星”结构;当l取其它数值时,其偏振态分布可称为“准星”结构。
图 2 右旋圆偏振分量U1中不同拓扑荷(l≤1.0)时初始平面光场的偏振态分布
Figure 2. Polarization states of light fields in the initial plane for different values of topological charge (l≤1.0) in right-handed circularly polarized component U1
l>1.0时的情形同样值得关注,以U1为左旋圆偏振为例,当拓扑荷l取值为1.1、1.3、1.5、1.7、1.9和2.0时,初始平面的偏振态分布如图 3所示。结合图 1中l=1.0的情形,图 3展示了光场从“柠檬”到高阶偏振奇异光场的演化过程。
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为研究光场的拓扑结构,斯托克斯相位是一个非常重要的参量[30]。斯托克斯相位场中的相位奇异点对应矢量光场中的偏振奇异点,其表达式为[30]:
$ S_{12}=S_1+\mathrm{i} S_2 $
(3) 式中: S1、S2分别为斯托克斯参量; i为虚数单位。斯托克斯相位ϕ=arctan(S2/S1),对于U1, l=0.5+U2, l=0, 在不同传输距离z处的斯托克斯相位场计算如图 6所示。在初始平面,斯托克斯相位从-π/2变化到π/2;而在其传输过程中,其相位的变化均为从-π到π。在传输过程的初段,可以看到其中心涡旋结构遭到破坏,经过一段距离之后,如z为464 mm和500 mm处,发现相位场中又至少出现了一个相位奇异点,如图中红色圆圈所示。
图 6 叠加光场U1, l=0.5+U2, l=0在不同传输距离z处的斯托克斯相位场
Figure 6. Stokes phase fields of superposed light fields U1, l=0.5+U2, l=0 at different propagation distances z
如果将传输距离z=464 mm和z=500 mm两处的光场尺寸缩放到0.2 mm×0.2 mm,这个局部区域的光场偏振态分布如图 7所示。图中黄线表示S1=0,蓝线表示S2=0,两者交点,即图中实心红圆即为偏振奇异点。也就是说在初始平面不存在偏振奇异点的情况下,经过一定距离的传输,光场中出现了偏振奇异点。
图 7 U1, l=0.5+U2, l=0中心为0.2 mm×0.2 mm的区域在不同传输距离z的偏振态分布
Figure 7. Polarization states of the center section 0.2 mm×0.2 mm of U1, l=0.5+ U2, l=0 at different propagation distances z
对于U1, l=1.5+U2, l=0的情形,不同传输距离的斯托克斯相位场如图 8所示。由图可以看出,在初始平面,斯托克斯相位场仅包含一个相位奇异点,而在传输距离z=464 mm和z=500 mm处,至少都包含两个相位奇异点。
图 8 叠加光场U1, l=1.5+U2, l=0在不同传输距离z处的斯托克斯相位场
Figure 8. Stokes phase fields of superposed light fields U1, l=1.5+U2, l=0 at different propagation distances z
同样地,可以得出在z=464 mm和z=500 mm处中心0.2 mm×0.2 mm区域的偏振态分布,如图 9所示。结合图 8可以看出,在传输初期拓扑结构遭到破坏的情况下,经过一定的传输距离之后依旧可以重建,并且出现了更多的偏振奇异点。结合图 7可以看出,基于圆艾里光束的分数偏振奇点在空间中传播时拓扑结构的可恢复性,利用这一特性可将该偏振奇点用于自由空间光通信中。
基于分数涡旋光束的偏振奇点传输特性研究
Study of propagation characteristics of polarization singularities based on fractional vortex beams
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摘要: 为了研究基于分数涡旋光束的偏振奇点在自由空间的动态传输特性, 采用数值模拟的方法, 进行了理论分析和数据仿真, 得到基于分数圆艾里涡旋光束的偏振奇点在自由空间传输时拓扑结构的演变。结果表明, 无论在初始平面内是否含有圆偏振奇点, 基于分数圆艾里涡旋光束的偏振奇点的自聚焦特性都会在传播过程中影响光场的拓扑结构, 使得光场在传输之初并不包含圆偏振奇异点, 但是在经过其自聚焦距离传输之后, 光场中将会出现若干个圆偏振奇点; 与基于整数圆艾里涡旋光束的偏振奇点相比, 两者均具有奇异拓扑结构的自恢复特性, 而基于分数的涡旋光束会给恢复后的光场中带来更多的圆偏振奇点。该工作将加深对偏振奇点传输特性的理解, 为其应用提供理论基础, 并拓宽圆艾里光束的应用范围。Abstract: In order to study the propagation characteristics of the polarization singularity based on the fractional vortex beam in free space, the method of numerical simulation was adopted, the theoretical analysis and data simulation were carried out, and the evolution of topology of polarization singularity based on the fractional circular Airy vortex beam was obtained during propagation in free space. The results show that regardless of whether there is a circular polarization singularity in the initial plane, the self-focusing properties of the polarization singularity based on the fractional circular Airy vortex beam will affect the topology of the light field during the propagation process, making the light field doesn't contain circular polarization singularities at the beginning of transmission, but after traveling through its self-focusing distance, several circular polarization singularities will appear in the light field. Compared with polarization singularities based on integer circular Airy vortex beams, both have singular topology self-recovery properties, while fractional-based vortex beams will bring more circular polarization singularities into the recovered light field point. This work will deepen the understanding of the propagation characteristics of polarization singularities, provide a theoretical basis for its application, and broaden the application range of circular Airy beams.
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