Longitudinal super-resolution spherical   multi-focus array based on column vector  light modulation

Xia Xiaolan; Zeng Xianzhi; Song Shichao; Liu Xiaowei; Cao Yaoyu

doi:10.12086/oee.2022.220109

Abstract

Featured by the capability of multi degree-of-freedom light-field manipulations while reserving high spatial resolution, multifocal laser arrays have been widely applied in femtosecond laser micro/nanofabrication, optical trapping, and so forth. Yet, due to the relatively lower axial resolution of single focuses within the array in comparison with the lateral resolution of their own, multifocal laser array has been refrained from isotropic 3D nanofabrication. Herein, we propose a feasible method for generation of axially super-resolved multifocal array with quasi-spherical focal spots. In particular, quasi-spherical multifocal array is optically synthesized via precise modulation on the coherent superposition of the orthogonal radially polarized beam (RPB) and azimuthally polarized beam (APB) states in the focal region based on annular amplitude modulation. We show theoretically the generation of quasi-spherical multifocal array with a high uniformity up to 99%. The average axial and lateral full-width-half maximum (FWHM) of the focal array are measured to be 0.76λ with the standard deviations in the axial and lateral directions being 0.005λ and 0.019λ, respectively. The presented strategy for synthesis of quasi-spherical multifocal array with high uniformity paves the way for high-precision laser fabrication of 3D micro/nano devices.

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1. 引　言

因此，本文提出了一种生成纵向超分辨率的准球形激光多焦点阵列的方法，通过对矢量光场调控方法的设计和调控效果的计算，在理论上验证了其有效性和可行性。该方法同时采用了径向偏振光和角向偏振光两种正交偏振光作为入射光场，通过设计和计算空间光调制器的相位图，结合环形光衰减片的振幅调制过程，分别对径向和角向两种偏振光场的波前进行调控，从而得到纵向(Z方向)上超分辨的多焦点阵列，再根据横向(X-Y方向)与Z方向上尺寸的差异，将径向偏振和角向偏振两条光束按照一定的强度比例叠加，最终生成焦斑单元各向同性、高均匀性的准球形激光多焦点阵列，以满足激光微纳加工制备各向同性微纳器件的需求。

柱矢量光是一种波面内各点的偏振呈轴对称分布、偏振方向随传播位置不断变化的光束，主要包括径向偏振光(radially polarized beam，RPB)和角向偏振光(azimuthal polarized beam，APB)，在激光诱捕^[13]、光学显微成像^[14-15]、光存储^[16-19]、超分辨激光加工^[20-22] 等领域的广阔应用前景引起了广泛关注。2000年，Youngworthh和Brown根据Richards-Worf提出的柱矢量光场衍射理论，分别对径向偏振光场和角向偏振光场进行聚焦光场的理论计算^[23]。计算结果表明，在高数值孔径物镜聚焦下，径向偏振光在经过高数值孔径紧聚焦之后，具有很强的纵向场分量，可以生成超越衍射极限的聚焦光斑，得到比标量光场更小尺寸的焦点。同时，利用光场入瞳函数振幅调控，可进一步压缩光斑的纵向尺寸^[24]。然而如何通过调控柱矢量光场生成球形激光多焦点阵列仍然面临设计上的巨大挑战。

激光多焦点阵列以兼具并行光场处理和焦斑单元高空间分辨率的特点，被广泛应用在光学诱捕和操纵以及飞秒激光微纳制造等领域^[1-4]。利用计算全息(computer generate hologram，CGH)技术，通过在空间光调制器(spatial light modulator，SLM)上生成相位图对入射光场的振幅和相位信息进行调制，可根据应用需求同时复制合成数十个甚至上百个相同的激光焦点，构成多焦点阵列。利用平行处理过程，在应用中，激光多焦点阵列不仅可以使光场一次捕获和控制多个对象^[5-7] ，而且可以极大提升光场在微纳结构加工中的效率^[8-12]。然而，受光学系统设计和光学衍射性质限制，目前通过计算全息生成的激光多焦点阵列普遍存在焦斑单元纵向分辨率弱于横向分辨率、光强分布各向异性的特点。长轴为纵轴的椭球形焦斑单元形状给激光多焦点的应用带来极大局限性。如何提升焦点纵向分辨率，实现光强分布各向同性的焦斑单元成为亟需解决的问题。

2. 理论模型

Figure 1. Schematic diagram of the principle of synthesizing longitudinal super-resolution quasi-spherical multifocal arrays based on the superposition principle of cylindrical vector light modulation

Figure 1. Full-Size Img PowerPoint

Schematic diagram of the principle of synthesizing longitudinal super-resolution quasi-spherical multifocal arrays based on the superposition principle of cylindrical vector light modulation

研究表明，在光学系统的入瞳平面引入光场调制可以压缩焦点的纵向尺寸，得到纵向超分辨率的焦点^[25-27]。目前已经针对不同的应用场景提出了不同种类的入瞳调制函数^[28-32]。例如环形光衰减片(shaded-ring filter)，二元相位调制(0-π phase plate)，全衰减型环形调制片(dark ring filter)。由于环形光衰减片可以认为是二元相位环形调制片或全衰减型环形调制片与一个环形透过区域的叠加，而环形透过区域的焦点对主瓣的贡献大于旁瓣，所以会使环形光衰减片相比于其它两类具有更低的旁瓣强度。因此本文研究中基于该调制方式实现柱矢量光的振幅调制。将柱矢量光分解为径向偏振光与角向偏振光分别进行振幅调制，通过优化环形光衰减片的调制环半径参数、衰减系数以及两种基的叠加比例，进而可实现纵向超分辨率的准球形多焦点阵列。

径向光焦点的径向分量为空心光斑，纵向分量为实心光斑，纵向分辨率主要取决于纵向分量。因此，径向光的环形光衰减片的参数优化将针对式(1)所计算的纵向分量进行。角向偏振光的焦点分布为空心光斑，其环形光衰减片的参数优化将针对空心光斑强度最强位置处的纵向分辨率进行。

当径向偏振单色平面波被无像差高数值物镜聚焦时，根据德拜矢量衍射理论^[33]，焦区的纵向分量E_z和径向分量E_r的电场分别为

仅使用一束径向偏振光生成的多焦点阵列，难以保证每个焦点尺寸的均一性。为了实现纵、横尺寸相等且均一的多焦点阵列，将角向偏振光的纵向超分辨多焦点阵列与径向偏振光纵向超分辨多焦点阵列进行同位叠加为有效的解决方案。是基于柱矢量光调控实现纵向超分辨率准球形多焦点阵列的原理示意图。从可以看出，径向偏振光(RPB)和角向偏振光(APB)分别通过空间光调制器(SLM1)和空间光调制器(SLM2)，以及环形光衰减片(pupil mask)后进行叠加。环形光衰减片的直径大小等于物镜入瞳尺寸， ${R}_{1}$ 是环形光衰减片的阴影环形区域的归一化内径(内径真实值除以物镜入瞳)， ${R}_{2}$ 是环形光衰减片的阴影环形区域的归一化外径(外径真实值除以物镜入瞳)；阴影环形区域的内径 ${R}_{1}$ 与外径 ${R}_{2}$ 之间的区域为振幅衰减区域，振幅衰减系数α被定义为 $\alpha = 1-\mathrm{s}\mathrm{q}\mathrm{r}\mathrm{t}(1-(T_{\rm{eff}})/({R}_{2}^{2}-{R}_{1}^{2}))$ ，T_eff是环形光衰减片的激光能量透过率。空间光调制器实现相位调制，用于调控多焦点阵列中的焦点个数与各焦点中心位置，产生高均一性的多焦点阵列；环形光衰减片可以压缩焦点的纵向尺寸实现纵向超分辨率。由于PLUTO型SLM的衍射效率为60%，另外，激光能量的衰减与环形光衰减片中阴影环形区域的振幅衰减系数也有关，实验中环形光衰减片的光透过率为35%，所以，SLM与环形光衰减片相叠加会使激光能量有较大的衰减。

当角向偏振单色平面波被无像差高数值物镜聚焦时，根据德拜矢量衍射理论^[33]，焦区的电场为

其中： ${P}\left(\theta \right)={P}\left(R\right)\sqrt{\mathrm{c}\mathrm{o}\mathrm{s}\theta }$ 是聚焦物镜的切趾函数， ${P}\left(R\right)$ 是物镜孔径的振幅分布，R是孔径面的坐标。 ${\mathrm{\theta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}={a}\mathrm{*}\mathrm{s}\mathrm{i}\mathrm{n}({N}{A}/{n})$ 是会聚角 $\theta$ 的最大值。k是波数，r和z是焦点区域的坐标，J₀和J₁是第一类贝塞尔函数的零阶和一阶函数。

3. 准球形单焦点

首先基于环形光衰减片和柱矢量光场实现准球形单焦点。径向偏振光经过环形光衰减片后(调制参数为R₁ = 0.5217, R₂ = 0.9629，环形光衰减片的振幅衰减系数是 0.9121)，焦点的纵向半高全宽从1.03λ被压缩为0.71λ，横向半高全宽为0.5481λ。角向偏振光经过环形光衰减片后(调制参数为R₁ = 0.4937, R₂ = 0.9483，以及环形光衰减片的振幅衰减系数是 0.0912)，在焦点强度最强处的纵向半高全宽从1.02λ被压缩为0.71λ。为了使纵向尺寸和横向尺寸相等，通过优化角向偏振光和径向偏振光叠加时的振幅比例系数，可以实现横向、纵向尺寸接近的球形焦点，如图2(a)所示。我们利用合成焦点不同方向上的半高全宽的标准差来判断球面形状的质量。当角向偏振光和径向偏振光的幅值比为0.35:1时，合成焦点沿0°到90°之间的五个方向上的半高全宽标准差最小，为0.022λ。图2(b)是合成焦点沿x轴、z轴、x=z三个方向(如图2(e)虚线所示, 虚线1, 2, 3分别代表z，x，x=z三个方向)的强度分布曲线，三者具有一致的半高全宽值，均为0.71λ，因此可以认为焦点的几何形状为球形。图2(c)~2(e)显示两组基及合成焦点在x-z面上的二维光场强度分布。

Figure 2. (a) The standard deviation of the full width at half maximum in the five directions of the composite focus under different amplitude ratios of angularly polarized beam and radially polarized beam; (b) Field strength curves of the intensity distribution of the synthetic focus along different directions; (c)~(e) Azimuthal polarized beam superimposed on radially polarized beam to generate a two-dimensional intensity distribution of the composite focus

Figure 2. Full-Size Img PowerPoint

(a) The standard deviation of the full width at half maximum in the five directions of the composite focus under different amplitude ratios of angularly polarized beam and radially polarized beam; (b) Field strength curves of the intensity distribution of the synthetic focus along different directions; (c)~(e) Azimuthal polarized beam superimposed on radially polarized beam to generate a two-dimensional intensity distribution of the composite focus

Figure 3. (a)~(b) Two-dimensional light field intensity distributions of spherical foci generated by superposition in the x-z plane, as well as lateral and longitudinal dimensions at different thresholds and different objective NAs, where the dotted line is the longitudinal dimension and the solid line is the transverse dimension; (c) The ratio of the longitudinal and transverse full width at half maximum of the focal point for angularly polarized light to radially polarized light at different threshold intensities and numerical apertures

Figure 3. Full-Size Img PowerPoint

(a)~(b) Two-dimensional light field intensity distributions of spherical foci generated by superposition in the x-z plane, as well as lateral and longitudinal dimensions at different thresholds and different objective NAs, where the dotted line is the longitudinal dimension and the solid line is the transverse dimension; (c) The ratio of the longitudinal and transverse full width at half maximum of the focal point for angularly polarized light to radially polarized light at different threshold intensities and numerical apertures

在实际应用中，光场作用区间的大小离不开材料的响应特性，如激光微纳制造技术中通常可以采用阈值效应来实现高分辨率微纳结构加工。通过将阈值效应与基于柱矢量光调控实现纵向超分辨球形焦点的技术进行结合，可以进一步提高球形焦点的横向和纵向分辨率。通过设置高于旁瓣强度的阈值可以抑制旁瓣对加工的影响，选用越高的阈值，则可以容忍越高的旁瓣强度，从而实现更高的分辨率。图3(a)表示在干镜(NA=0.9)、水镜(NA=1.2)、油镜(NA=1.4)情况下，阈值强度分别为50%、70%、80%的合成焦点的二维光场强度分布示意图。当阈值设置为70%，则优化角向偏振光和径向偏振光的振幅调制，将归一化旁瓣强度设置为70%，并以合成焦点在70%强度处半高全宽的标准差最小为标准来选择角向与径向光的叠加比例。从图3(b)中可以看出，随着数值孔径(NA)的增加，合成焦点的纵向尺寸和横向尺寸逐渐减小，虚线代表纵向尺寸，实线代表横向尺寸，纵向尺寸与横向尺寸非常接近。图3(c)显示了各个阈值和NA下，所实现的焦点横向尺寸与纵向尺寸之比，其中当阈值强度为80%，NA为1.4时，合成焦点的纵横尺寸比例是1，可以视为准球形焦点，且具有0.4λ的各向同性分辨率。

4. 准球形多焦点阵列

Figure 5. Full-Size Img PowerPoint

(a), (d) The x-y cross-section (g) of the quasi-spherical multifocal array obtained by focusing and stacking the modulated angularly polarized light (a) and the modulated radially polarized light (d); (b), (e) The x-z profile (h) of the quasi-spherical multifocal array obtained by focusing the modulated angularly polarized light (b) and the modulated radially polarized light (e); (c), (f), (i) are magnifications of the marked foci in (b), (e), (h), respectively

图6(a)表示合成多焦点阵列的x-y面二维光场强度分布，经过计算得出合成多焦点阵列的横向半高全宽的平均值为0.76λ，标准差是0.019λ。为了计算10×10合成多焦点阵列中横向尺寸的均一性，我们分别找出合成多焦点阵列中的尺寸最小与尺寸最大的焦点，并且对比了这两个焦点的横向半高全宽。图6(b)是x-y面合成多焦点阵列中最小焦点与最大焦点的光场强度曲线对比图，这两个焦点的半高全宽相差0.08λ，仅为半高全宽的10.5%，横向尺寸的均一性高达95%。图6(c)是x-z面的最大焦点与最小焦点的强度分布沿不同方向的场强曲线，最大焦点的半高全宽为0.76λ，最小焦点的半高全宽为0.75λ，纵向尺寸的均一性高达99%。合成的100个焦点阵列的纵向半高全宽的平均值是0.76λ、标准差是0.005λ；横向半高全宽的平均值是0.76λ、标准差是0.019λ。调幅径向偏振光叠加调幅角向偏振光的合成多焦点阵列的横向与纵向的尺寸比例为1，且纵向半高全为0.76λ，可视为纵向超分辨率的准球形多焦点阵列。图6(d)是5×5×5纵向超分辨率准球形多焦点阵列的三维光场强度分布，多焦点阵列焦点周期都设置为5λ。通过加载纵向移动的球面波相位可以实现二维多焦点阵列在纵向上移动，把不同z层的二维多焦点阵列叠加起来，从而生成三维的纵向超分辨率的准球形多焦点阵列。目前利用SLM调制技术和环形光衰减片压缩光学焦点的纵向分辨率的方法已经在光学显微成像和光存储等领域得到验证^[38-39]，也为本文在理论上提出的并行准球形多焦点的方法用于微纳加工提供了可行性支持。

基于德拜矢量理论迭代算法^[34]和快速傅里叶计算方法^[35-37]可以得到精确的相位调制全息图，加载到空间光调制器上，在高数值孔径物镜的焦平面可生成高均一性的多焦点阵列。多焦点阵列的均一性被定义为U=1−(I_max−I_min)/(I_max+I_min)，其中I_max和I_min分别表示多焦点阵列中的最大和最小光强。为了探究多焦点阵列中的焦点个数对均一性和迭代次数的影响，实验中通过计算不同的多焦点阵列来观察迭代次数对均一性的影响。图4(a)分别为3×3、5×5、7×7、11×11、21×21和51×51径向光多焦点阵列的二维场强分布。多焦点阵列焦点周期都设置为5λ。从图4(b)中可以看出，5×5多焦点阵列对应的收敛速度最快，在迭代次数小于15时，可达到95%；而51×51多焦点阵列需要40次迭代，均一性才能达到95%。可以看到，随着焦点个数的增加，迭代次数的收敛速度在下降，当均一性达到95%以上时，增加迭代次数对均一性的提升逐渐减弱；当迭代次数增加到20次以上时，所有多焦点阵列的均一性都能达到95%以上。

Figure 6. (a) The lateral light field intensity distribution at the focal point of the synthetic quasi-spherical multifocal array; (b) Comparison of the light field intensity curves of the maximum focus and the minimum focus on the x-y plane; (c) The field intensity curves of the light intensity distribution of the maximum focus and the minimum focus in the x-z plane along different directions; (d) Three-dimensional light intensity distribution of synthetic longitudinal super-resolution quasi-spherical multifocal arrays

Figure 6. Full-Size Img PowerPoint

(a) The lateral light field intensity distribution at the focal point of the synthetic quasi-spherical multifocal array; (b) Comparison of the light field intensity curves of the maximum focus and the minimum focus on the x-y plane; (c) The field intensity curves of the light intensity distribution of the maximum focus and the minimum focus in the x-z plane along different directions; (d) Three-dimensional light intensity distribution of synthetic longitudinal super-resolution quasi-spherical multifocal arrays

Figure 4. (a) Radial polarized beam multifocal array with different number of focal points; (b) The variation of uniformity and iteration number with the number of different foci

Figure 4. Full-Size Img PowerPoint

(a) Radial polarized beam multifocal array with different number of focal points; (b) The variation of uniformity and iteration number with the number of different foci

将上述基于柱矢量光调控生成准球形单焦点的理论基础与紧聚焦矢量多焦点阵列技术进行结合，可实现纵向超分辨率的准球形多焦点阵列。图5是10×10纵向超分辨率的准球形多焦点阵列的二维光场强度分布。图5(a)~5(b)分别是调幅角向偏振光多焦点阵列在x-y、x-z面上的二维光场强度分布，图5(c)显示了其中一个焦点在x-z面光场分布的放大图。图5(d)~(e)分别是调幅径向偏振光多焦点阵列在x-y、x-z面上的二维光场强度分布, 图5(f)显示了其中一个焦点在x-z面上光场分布的放大图。图5(g)~(h)展示的是径向与角向偏振光叠加得到的准球形多焦点阵列在x-y、x-z面上的二维光场强度分布。图5(i)显示了其中一个准球形焦点在x-z面上光场分布的放大图。径向偏振光经过空间光调制器和环形光衰减片(物镜NA=1.4，n=1.514)，生成了纵向尺寸是0.76λ的高均一性10×10多焦点阵列。纵向分辨率相比于受衍射极限的焦点纵向尺寸1.03λ，实现了26.2%的提升。角向偏振光经过空间光调制器和环形光衰减片(物镜NA=1.4，n=1.514)，生成纵向尺寸是0.76λ的高均一性的10×10多焦点阵列，纵向分辨率相比于受衍射极限的焦点纵向尺寸1.02λ，实现了25.5%的提升。当角向偏振光和径向偏振光的幅值比为0.55∶1时，多焦点阵列的横向尺寸和纵向尺寸均为0.76λ，最终可以得到各向同性且高均一性的纵向超分辨率的准球形多焦点阵列。

5. 结　论

本文通过对矢量光场波前的设计和调制，在理论上基于数值仿真展示了如何生成均一性达到 99% 的准球形激光多焦点阵列。在利用柱矢量光场两个偏振分量聚焦特性基础上，提出了一种基于柱矢量光场相位和振幅调控生成纵向超分辨率准球形激光多焦点阵列的方法。通过对柱矢量光的径向偏振光(RPB)光束和角向偏振光(APB)光束分别进行相位和振幅调控，利用环形光衰减片的振幅调制能力形成纵向超分辨焦斑，并将两种偏振光束以适当的光强比例在焦区叠加，从而合成具有准球形多焦点阵列。10×10合成多焦点阵列纵向半高全宽的平均值为0.76λ、标准差为0.005λ，横向半高全宽的平均值为0.76λ、标准差为0.019λ。在此情况下，标准差远小于半高全宽的平均值，横向与纵向尺寸可视为相等。该具有高尺寸均一性的准球形激光多焦点阵列可为激光微纳加工精准制备微纳器件提供新的途径。

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- Xia Xiaolan, 1432840032@qq.com On this Site On Google Scholar
  - Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Institute of Photonics Technology, Jinan University, Guangzhou, Guangdong 511443, China
- Zeng Xianzhi On this Site On Google Scholar
  - Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Institute of Photonics Technology, Jinan University, Guangzhou, Guangdong 511443, China
- Song Shichao On this Site On Google Scholar
  - Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Institute of Photonics Technology, Jinan University, Guangzhou, Guangdong 511443, China
- Corresponding author: Liu Xiaowei, liuxiaowei@zhejianglab.com On this Site On Google Scholar
  liuxiaowei@zhejianglab.com
  - Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Institute of Photonics Technology, Jinan University, Guangzhou, Guangdong 511443, China
  - Research Center for Humanoid Sensing, Zhejiang Lab, Hangzhou, Zhejiang 311121, China
- Corresponding author: Cao Yaoyu, yaoyucao@jnu.edu.cn On this Site On Google Scholar
  yaoyucao@jnu.edu.cn
  - Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Institute of Photonics Technology, Jinan University, Guangzhou, Guangdong 511443, China

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DOI: 10.12086/oee.2022.220109

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Xia Xiaolan, Zeng Xianzhi, Song Shichao, Liu Xiaowei, Cao Yaoyu. Longitudinal super-resolution spherical multi-focus array based on column vector light modulation. Opto-Electronic Engineering 49, 220109 (2022). DOI: 10.12086/oee.2022.220109

Xia Xiaolan, Zeng Xianzhi, Song Shichao, Liu Xiaowei, Cao Yaoyu. Longitudinal super-resolution spherical multi-focus array based on column vector light modulation. Opto-Electronic Engineering 49, 220109 (2022). DOI: 10.12086/oee.2022.220109

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- Received Date June 02, 2022
- Revised Date July 15, 2022
- Accepted Date July 15, 2022
- Available Online September 01, 2022
- Published Date November 24, 2022
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[1]	Kato J I, Takeyasu N, Adachi Y, et al. Multiple-spot parallel processing for laser micronanofabrication[J]. Appl Phys Lett, 2005, 86(4): 044102. DOI: 10.1063/1.1855404 CrossRef Google Scholar
[2]	Yamaji M, Kawashima H, Suzuki J I, et al. Three dimensional micromachining inside a transparent material by single pulse femtosecond laser through a hologram[J]. Appl Phys Lett, 2008, 93(4): 041116. DOI: 10.1063/1.2965451 CrossRef Google Scholar
[3]	Zhang Y L, Chen Q D, Xia H, et al. Designable 3D nanofabrication by femtosecond laser direct writing[J]. Nano Today, 2010, 5(5): 435−448. DOI: 10.1016/j.nantod.2010.08.007 CrossRef Google Scholar
[4]	聂芳松, 姜美玲, 张明偲, 等. 基于迂回相位的轨道角动量Talbot阵列照明器[J]. 光电工程, 2020, 47(6): 200093. DOI: 10.12086/oee.2020.200093 Nie F S, Jiang M L, Zhang M S, et al. Orbital angular momentum Talbot array illuminator based on detour phase encoding[J]. Opto-Electron Eng, 2020, 47(6): 200093. DOI: 10.12086/oee.2020.200093 CrossRef Google Scholar
[5]	Dufresne E R, Spalding G C, Dearing M T, et al. Computer-generated holographic optical tweezer arrays[J]. Rev Sci Instrum, 2001, 72(3): 1810. DOI: 10.1063/1.1344176 CrossRef Google Scholar
[6]	Di Leonardo R, Ianni F, Ruocco G. Computer generation of optimal holograms for optical trap arrays[J]. Opt Express, 2007, 15(4): 1913−1922. DOI: 10.1364/OE.15.001913 CrossRef Google Scholar
[7]	Cai Y N, Yan S H, Wang Z J, et al. Rapid tilted-plane Gerchberg-Saxton algorithm for holographic optical tweezers[J]. Opt Express, 2020, 28(9): 12729−12739. DOI: 10.1364/OE.389897 CrossRef Google Scholar
[8]	Hasegawa S, Hayasaki Y, Nishida N. Holographic femtosecond laser processing with multiplexed phase Fresnel lenses[J]. Opt Lett, 2006, 31(11): 1705−1707. DOI: 10.1364/OL.31.001705 CrossRef Google Scholar
[9]	Hasegawa S, Hayasaki Y. Holographic femtosecond laser processing with multiplexed phase Fresnel lenses displayed on a liquid crystal spatial light modulator[J]. Opt Rev, 2007, 14(4): 208−213. DOI: 10.1007/s10043-007-0208-9 CrossRef Google Scholar
[10]	Salter P S, Booth M J. Addressable microlens array for parallel laser microfabrication[J]. Opt Lett, 2011, 36(12): 2302−2304. DOI: 10.1364/OL.36.002302 CrossRef Google Scholar
[11]	Zandrini T, Shan O M, Parodi V, et al. Multi-foci laser microfabrication of 3D polymeric scaffolds for stem cell expansion in regenerative medicine[J]. Sci Rep, 2019, 9(1): 11761. DOI: 10.1038/s41598-019-48080-w CrossRef Google Scholar
[12]	Geng Q, Wang D E, Chen P F, et al. Ultrafast multi-focus 3-D nano-fabrication based on two-photon polymerization[J]. Nat Commun, 2019, 10(1): 2179. DOI: 10.1038/s41467-019-10249-2 CrossRef Google Scholar
[13]	Skelton S E, Sergides M, Saija R, et al. Trapping volume control in optical tweezers using cylindrical vector beams[J]. Opt Lett, 2013, 38(1): 28−30. DOI: 10.1364/OL.38.000028 CrossRef Google Scholar
[14]	Biss D P, Youngworth K S, Brown T G. Dark-field imaging with cylindrical-vector beams[J]. Appl Opt, 2006, 45(3): 470−479. DOI: 10.1364/AO.45.000470 CrossRef Google Scholar
[15]	Feng Z X, Cheng D W, Wang Y T. Iterative freeform lens design for prescribed irradiance on curved target[J]. Opto-Electron Adv, 2020, 3(7): 200010. DOI: 10.29026/oea.2020.200010 CrossRef Google Scholar
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Longitudinal super-resolution spherical multi-focus array based on column vector light modulation

Abstract

Keywords

1. 引　言

2. 理论模型

3. 准球形单焦点

4. 准球形多焦点阵列

5. 结　论

利益冲突

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Xia Xiaolan, 1432840032@qq.com On this Site On Google Scholar

Zeng Xianzhi On this Site On Google Scholar

Song Shichao On this Site On Google Scholar

Corresponding author: Liu Xiaowei, liuxiaowei@zhejianglab.com On this Site On Google Scholar

Corresponding author: Cao Yaoyu, yaoyucao@jnu.edu.cn On this Site On Google Scholar

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Longitudinal super-resolution spherical multi-focus array based on column vector light modulation

Abstract

Keywords

1. 引 言

2. 理论模型

3. 准球形单焦点

4. 准球形多焦点阵列

5. 结 论

利益冲突

References

Cited by

Periodical cited type(6)

Other cited types(0)

Author Information

Xia Xiaolan, 1432840032@qq.com On this SiteOn Google Scholar

Zeng Xianzhi On this SiteOn Google Scholar

Song Shichao On this SiteOn Google Scholar

Corresponding author: Liu Xiaowei, liuxiaowei@zhejianglab.com On this SiteOn Google Scholar

Corresponding author: Cao Yaoyu, yaoyucao@jnu.edu.cn On this SiteOn Google Scholar

Copyright

About this Article

Cite this Article

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Corresponding author: Cao Yaoyu, yaoyucao@jnu.edu.cn

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1. 引　言

5. 结　论

Xia Xiaolan, 1432840032@qq.com On this Site On Google Scholar

Zeng Xianzhi On this Site On Google Scholar

Song Shichao On this Site On Google Scholar

Corresponding author: Liu Xiaowei, liuxiaowei@zhejianglab.com On this Site On Google Scholar

Corresponding author: Cao Yaoyu, yaoyucao@jnu.edu.cn On this Site On Google Scholar