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光在改光学元件中传播时,由于不同位置的厚度不同,光在介质中积累的相位不同,可实现波前的调控,即斯涅耳定律。以透镜为例,从中心到边缘的厚度呈梯度变化,所产生的光程差会形成一定的梯度。对于超表面,该定律可以进行推广,即广义斯涅耳定律。电磁波通过超表面,会产生一定梯度的相位突变。
根据费马原理,光从一点传播到另一点时,无论传播路径如何,其光程保持一定。如图 1a中的折射模型,有:
$ k_{0} n_{\mathrm{i}} \sin \theta_{\mathrm{i}} \mathrm{d} x+(\varphi+\mathrm{d} \varphi)=k_{0} n_{\mathrm{t}} \sin \theta_{\mathrm{t}}+\varphi $
(1) 式中,k0=2π/λd为波数,λd为设计所需的工作波长,ni和nt分别为入射和透射介质的折射率,θi和θt分别为入射和透射光路与法线的夹角,φ和φ+dφ是两条光路通过不连续界面时相位不连续产生的相位变化。设定波长为λ0,推得广义斯涅耳的折射公式[24]:
$ \sin \theta_{\mathrm{t}} n_{\mathrm{t}}-\sin \theta_{\mathrm{i}} n_{\mathrm{i}}=\frac{\lambda_{0}}{2 \pi} \frac{\mathrm{d} \varphi}{\mathrm{d} x} $
(2) 同理,根据图 1b中的反射模型,有:
$ k_{0} n_{\mathrm{i}} \sin \theta_{\mathrm{i}}+(\varphi+\mathrm{d} \varphi)=k_{0} n_{\mathrm{i}} \sin \theta_{\mathrm{r}}+\varphi $
(3) 式中,θr为反射介质折射率,广义斯涅耳定律的反射公式为[24]:
$ \sin \theta_{\mathrm{r}}-\sin \theta_{\mathrm{i}}=\frac{\lambda_{0}}{2 \pi n_{\mathrm{i}}} \frac{\mathrm{d} \varphi}{\mathrm{d} x} $
(4) (4) 式表明,只要改变每一点的相位突变dφ/dx,使其呈一定的梯度变化,即可实现对波前的灵活调控。若想实现聚焦功能,则需要将超表面上的相位梯度设计成抛物线分布,每一点相位分布应满足:
$ \varphi\left(r_{\mathrm{NF}}\right)=k_{0}\left(\sqrt{f^{2}+r_{\mathrm{NF}}^{2}}-f\right) $
(5) 式中,f为焦距,rNF为微结构所在位置到超表面中心的距离。在实际设计时,设工作波长为λd,表面上位于(x,y)的纳米微粒所产生的相位突变φNF为[5]:
$ \varphi_{\mathrm{NF}}(x, y)=\frac{2 \pi}{\lambda_{\mathrm{d}}}\left(f-\sqrt{x^{2}+y^{2}+f^{2}}\right) $
(6) 根据相位突变值为超表面纳米结构指向角的2倍,位于(x,y)的纳米结构沿轴向旋转角度θNF为[5]:
$ \theta_{\mathrm{NF}}(x, y)=\frac{\pi}{\lambda_{\mathrm{d}}}\left(f-\sqrt{x^{2}+y^{2}+f^{2}}\right) $
(7) -
首先结合线偏振光,设计并优化初始的聚焦超表面结构。聚焦波长532nm的线偏振光,以长方体TiO2为微结构,长250nm,宽95nm,高600nm,在SiO2的衬底上以325nm为周期按四边形排成晶胞,如图 2a所示。根据聚焦公式(7)式,确定不同位置微元的特定旋转角度,设计焦距为4μm。图 2b为532nm的线偏振光通过该原始超表面(见图 2a)后,焦点位置的xy横截面的光强分布,图 2c为轴向xz面的光强分布。可以看出,该结构对线偏振光有初步的聚焦功能,但聚焦特性不理想。进一步优化TiO2的结构,将长方体改为椭圆柱,其它参量不变。图 2d为初始结构经过第1次优化后的微结构为椭圆柱结构,椭圆的长轴为250nm,短轴为95nm,椭圆柱高600nm。在以SiO2的衬底上,4个椭圆柱为一组,以正方形晶胞排布成阵列。以532nm的线偏振光入射,横向和轴向的聚焦特性如图 2e和图 2f所示。相比于初始结构,该优化结构能较好地实现紧聚焦,表明聚焦特性有了显著的提升。需要注意的是,聚焦场沿轴向为约2000nm的长焦深,也意味着透过超表面轴向聚焦特性有待提升。
为进一步优化聚焦特性,第2次优化过程在保持图 2d中微元结构和材料不变的基础上,调整微粒的排布使其更趋近于环状分布,将原有的4个微元组成的正方形晶胞优化为7个微元组成的正六边形晶胞,如图 2g所示。以532nm的线偏振光照射该超表面,得到的图 2h、图 2i分别为焦点位置横向和轴向的光强分布图。从图 2h可以看出,相比于优化前的聚焦场,有着更少的杂散光和旁瓣。图 2i中,沿轴向光场聚焦也更集中,表明优化后的整体结构可以在设计好的焦距位置有效汇聚光场。
经过两次优化后得到的聚焦超表面结构如图 3所示。图 3a中以SiO2为衬底,厚度H=600nm,上方微元呈椭圆柱结构,由TiO2制成,长轴R1=250nm,短轴R2=95nm,高度H=600nm; 每个椭圆柱之间的间距D=325nm,如图 3b所示,7个微元为一组构成正六边形晶胞; 图 3c为线偏振聚焦超表面结构。
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沿z轴传播的傍轴场为U=Ψ(ξ, η, z)eik0z,其中(ξ, η)是轴上位于z处的光束横截面上的坐标,Ψ满足近轴波动方程。在椭圆坐标系中,得到波动方程的解为高斯光束调制形式并在右侧插入因斯多项式的乘积,得到奇偶IG模式的一般表达式[15]:
$ G_{\mathrm{e}, p, m}(r, \varepsilon)=\frac{C w_{0}}{w(z)} C_{m, p}(\mathrm{i} \xi, \varepsilon) C_{m, p}(\eta, \varepsilon) \exp \left[k_{0} z+\frac{k_{0} r^{2}}{2 R(z)}-(p+1) \Psi_{\mathrm{GS}}(z)\right] \mathrm{i} $
(8) $ G_{o, p, m}(r, \varepsilon)=\frac{S w_{0}}{w(z)} S_{m, p}(\mathrm{i} \xi, \varepsilon) S_{m, p}(\eta, \varepsilon) \exp \left[\frac{-r^{2}}{w^{2}(z)}\right] \exp \left[k_{0} z+\frac{k_{0} r^{2}}{2 R(z)}-(p+1) \Psi_{\mathrm{GS}}(z)\right] \mathrm{i} $
(9) 式中,下标e和o分别表示偶模和奇模,r为半径,ε=2f02/w02是由束腰半径w0和束腰面半焦距f0决定的椭圆参量,表示椭圆率的变化程度; C和S为归一化常数,激光在轴上z处的横截宽度是w(z)=w0×$\sqrt{1+z^{2} / z_{\mathrm{R}}^{2}} $,其中zR=k0w02/2为瑞利长度; Cm, p(η, ξ)和Sm, p(η, ξ)分别表示带有阶数p和级数m的偶次和奇次因斯多项式,p和m满足(-1)p-m=1,始终具有相同的奇偶性; 光波前的曲率半径为R(z)=z+zR2/z,阶数是p的IG模的Gouy相移为-(p+1)ΨGS(z)=-(p+1)arctan(z/zR)。
而IG矢量光场可以由两个具有偶模和奇模IG模式的正交偏振分量叠加构成,用琼斯矢量表示为:
$ \boldsymbol{E}_{\mathrm{IGV}}(x, y)=\left[\begin{array}{c} G_{\mathrm{e}, p_{x}, m_{x}, \varepsilon_{x}} \\ G_{\mathrm{o}, p_{y}, m_{y}, \varepsilon_{y}} \exp (\mathrm{i} \delta) \end{array}\right] $
(10) 式中,Ge, px, mx, εx和Go, py, my, εy分别为x、y偏振分量,下标px, mx, εx和py, my, εy分别是光场x分量和y分量IG模式的阶数、级数、椭圆参量,δ是x、y分量之间的相位延迟。
为了适用于IG矢量光,这里仍采用线偏振聚焦的正六边形超表面结构,因该结构对不同偏振态都有相同的汇聚能力,所以理论上并不影响聚焦效果。选取4阶的IG矢量光为入射光场[23],其入射光场强度分布及聚焦光场如图 4所示。其中,图 4a为波长532nm的4阶IG矢量光场强度分布; 图 4b经过正六边形聚焦超表面后焦平面处的光场强度分布。可以发现,该结构虽然能够起到汇聚的作用,但对于IG复杂矢量光场,内部强度信息并不能很好在聚焦场中体现,所以需要进一步优化结构。
对整体结构分析,能够影响聚焦特性的因素之一是正六边形的空间方位角。此处,作者尝试将具有六边形晶胞的超表面结构整体逆时针旋转30°,仿真结果如图 4c所示。与优化前相比,焦平面位置的光场强度信息更完善、详细。进一步对比入射光场与聚焦光场,提取电场的x方向分量Ex和y方向分量Ey,如图 4d所示,从左到右分别依次为:4阶IG矢量光场入射光(见图 4a)的Ex分量、Ey分量; 经过优化超表面结构后的聚焦场(见图 4c)中的Ex分量、Ey分量。对比分析可以发现,经优化后,主要空间强度结构信息在聚焦过程中能够实现较好的传递。
3阶IG矢量光场的聚焦场空间结构较4阶聚焦场更为简单。进一步对3阶IG矢量光场的超表面聚焦进行研究,如图 5所示。其中,图 5a为波长532nm的3阶IG矢量光场,图 5b为3阶IG矢量光场的超表面聚焦场横向平面的光强分布,图 5c为3阶IG矢量光场的超表面聚焦场在轴向平面的光强分布。从图中可以看出,3阶IG矢量光场聚焦后内部强度信息同样有缺失,但IG模式的基本结构仍然可以保持。分析图 4b与图 5b,观察到经超表面得到的不同阶聚焦场均呈现一定的非轴对称性,表现为一、三象限与二、四象限的聚焦光斑强弱不同。这主要是受具有六边形晶胞的超表面结构的整体旋转角度所影响。结构设计时,通过中心旋转超表面至适当的角度可减弱这种非轴对称性,从而实现优化。
基于几何相位超表面的Ince-Gaussian矢量涡旋光场聚焦
Focusing of Ince-Gaussian vector vortex optical field based on geometric phase metasurface
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摘要: 为了实现电介质超表面的聚焦功能和对光场相位的调控,采用几何相位调制原理设计微元结构及空间分布,以SiO2为基底、亚波长TiO2椭圆柱的六边形晶胞为基本结构,设计了一种相位突变呈抛物线梯度分布的聚焦超表面,适用于480nm~580nm波段。基于此结构进行了理论分析和实验验证,发现该结构对线偏振光聚焦,其归一化后的半峰全宽约为428nm,而对矢量光聚焦约为258nm,获得了更出色的聚焦效果。研究了3阶和4阶Ince-Gaussian矢量光场通过该超表面后的聚焦特性,得到了聚焦场能保持入射矢量光场的基本空间结构,但中心结构信息会有损失的结果,即Ince-Gaussian矢量涡旋光场由于涡旋相位的存在,聚焦后会呈现破缺的空间结构。结果表明,超表面结构和入射光场矢量结构之间的匹配程度是影响聚焦特性的重要因素。该研究为理解复杂矢量光场的超表面聚焦机理提供了参考。Abstract: In order to realize the focusing function of the dielectric metasurface and the adjustment of the phase of the light field, the geometric phase modulation principle was used to design the micro-element structure and spatial distribution. Using SiO2 as the substrate and the hexagonal unit cell of sub-wavelength TiO2 elliptical cylinder as the basic structure, a metalens with a parabolic gradient distribution of phase mutation was designed, which was suitable for the wavelength range of 480nm to 580nm. Based on this structure, theoretical analysis and experimental verification were carried out. It is found that for the linearly polarized light with this structure, the normalized full width at half maximum of the focus is about 428nm, and the full width at half maximum obtained by focusing on vector light is about 258nm, which is better than that of the linearly polarized light. The focusing characteristics of the 3rd-order and 4th-order Ince-Gaussian vector light field after passing through the metasurface were studied, and the basic spatial structure of the focusing field can maintain the incident vector light field, but the center structure information will be lost. That is, the Ince-Gaussian vector vortex light field will show a broken spatial structure after focusing due to the existence of the vortex phase. The results show that the matching degree between the metasurface structure and the incident light field vector structure is an important factor affecting the focusing characteristics. This research provides a reference for understanding the metasurface focusing mechanism of complex vector light fields.
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Key words:
- diffraction /
- metasurface /
- geometric phase /
- vector optical field /
- Ince-Gaussian /
- vortex phase
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