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The tight focusing characteristics of azimuthally polarized vortex beams are systematically investigated in this work. The azimuthally polarized vortex beam can be decomposed into left-handed circularly polarized (LHCP) wave and right-handed circularly polarized ( RHCP) wave. It is found that the longitudinal component of LHCP and RHCP at the focal plane are equal in magnitude but opposite in phase. Thus, the total longitudinal field disappears because of the completely destructive interference. In contrast, there is almost no interference between the transverse component of LHCP and RHCP. Thus, the total transverse field is the incoherent superposition of them. Since the absolute value of the topological charge of LHCP component and RHCP component are not equal, the transverse component of LHCP and RHCP will be concentrated in the different areas on the focal plane. It is the reason for the orbit-induced SAM to be localized on the focal plane. Then, we compare the focal spot characteristics of the radially polarized beam and the azimuthally polarized beam with a first-order vortex. The advantages and disadvantages of them are discussed in detail, respectively. For the radially polarized beam, the central focal spot is mainly longitudinal component, and the sidelobe is mainly transverse component. For the azimuthally polarized vortex beam with $l = 1$ , the central focal spot is mainly LHCP component, and the sidelobe is mainly RHCP component. In both cases, the field distributions of the central spots are the same, and both show a distribution similar to the zero-order Bessel function. The situation of the sidelobe is different. The sidelobe of the radially polarized beam shows a distribution similar to the first-order Bessel function and the sidelobe of the azimuthally polarized vortex beam indicates a distribution similar to the second-order Bessel function. Therefore, the sidelobe of the radially polarized beam is closer to that of the optical axis, resulting in a larger central focal spot size. On the other hand, the sidelobe of the radially polarized beam accounts for a much smaller proportion of the total energy than that of the azimuthally polarized vortex beam. So the sidelobe peak intensity of the radially polarized beam is lower. Finally, an optimal binary phase element is designed to obtain an ultra-long super-resolution optical needle. The transverse full weight of half maximum (FWHM) can achieve$0.391\lambda $ and the longitudinal FWHM can reach to$25.5\lambda $ by using only 6 belts.-
Keywords:
- vector beam /
- vortex beam /
- super-resolution needle
[1] Li Y C, Xin H B, Lei H X, Liu L L, Li Y Z, Zhang Y, Li B J 2016 Light Sci. Appl. 5 e16176
Google Scholar
[2] Gong Z, Pan Y L, Videen G, Wang C 2018 J. Quant. Spectrosc. Ra. 214 94
Google Scholar
[3] Wetzstein G, Ozcan A, Gigan S, Fan S, Englund D, Soljačić M, Denz C, Miller D A B, Psaltis D 2020 Nature 588 39
Google Scholar
[4] 许琳茜, 朱榕琪, 朱竹青, 贡丽萍, 顾兵 2022 物理学报 71 147801
Google Scholar
Xu L X, Zhu R Q, Zhu Z Q, Gong L P, Gu B 2022 Acta Phys. Sin. 71 147801
Google Scholar
[5] Zhang L, Qiu X D, Zeng L W, Chen L X 2019 Chin. Phys. B 28 094202
Google Scholar
[6] Zhao J, Winetraub Y, Du L, Van Vleck A, Ichimura K, Huang C, Aasi S Z, Sarin K Y, de la Zerda A 2022 Optica 9 859
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[7] Cao R, Zhao J J, Li L, Du L, Zhang Y D, Luo Y L, Jiang L M, Davis S, Zhou Q F, de la Zerda A, Wang L V 2022 Nat. Photon. 17 89
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[8] Wang H, Shi L, Lukyanchuk B, Sheppard C, Chong C T 2008 Nat. Photon. 2 501
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[9] Huang K, Shi P, Kang X L, Zhang X, Li Y P 2010 Opt. Lett. 35 965
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[11] Zhang T, Li M, Ye H, Shi C 2020 Opt. Commun. 460 125178
Google Scholar
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[13] Hao X, Kuang C F, Wang T T, Liu X 2010 Opt. Lett. 35 3928
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[14] Yuan G H, Wei S B, Yuan X C 2011 Opt. Lett. 36 3479
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[15] Wang S C, Li X P, Zhou J Y, Gu M 2014 Opt. Lett. 39 5022
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[18] Zhan Q 2009 Adv. Opt. Photon. 1 1
Google Scholar
[19] Kozawa Y, Sato S 2021 Prog. Opt. 66 35
Google Scholar
[20] Sato S, Kozawa Y 2009 J. Opt. Soc. Am. A Opt. Image Sci. Vis. 26 142
Google Scholar
[21] Li M M, Cai Y N, Yan S H, Liang Y S, Zhang P, Yao B L 2018 Phys. Rev. A 97 053842
Google Scholar
[22] Monteiro P B, Neto P A M, Nussenzveig H M 2009 Phys. Rev. A 79 033830
Google Scholar
[23] Ostrovsky A S, Rickenstorff-Parrao C, Arrizón V 2013 Opt. Lett. 38 534
Google Scholar
[24] Grosjean T, Courjon D 2007 Opt. Commun. 272 314
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[25] Vaity P, Rusch L 2015 Opt. Lett. 40 597
Google Scholar
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图 1
$l = 1$ 时, 焦平面的(a)光强和偏振分布; (b)斯托克斯分量S3的分布. 图尺寸为$1.5\lambda \times 1.5\lambda $ Figure 1. (a) Intensity and polarization distributions of the beam at the focal plane with
$l = 1$ ; (b) calculation results of Stokes parameters S3 at the focal plane with$l = 1$ .
图 2 (a)
$l = 2$ 和(b)$l = 0$ 时, 焦平面的左旋分量${I_{{\text{EL}}}}$ 、右旋分量${I_{{\text{ER}}}}$ 、总光强${I_0}$ 以及斯托克斯分量S3分布Figure 2. Intensity of the left-handed component, intensity of the right-handed component, total intensity and the Stokes parameters S3 at the focal plane with (a)
$l = 2$ and (b)$l = 0$ .
图 3
$l = 1$ 时,${E_{{\text{0 L}}}}$ 和${E_{{\text{0 R}}}}$ 的横向分量在焦平面的相干叠加和非相干叠加的光强计算结果Figure 3. Calculation results of coherent superposition and incoherent superposition of the transverse components of
${E_{{\text{0 L}}}}$ and${E_{{\text{0 R}}}}$ with$l = 1$ .
图 4 不同情况下,
${I_{\text{A}}}$ 和${I_{\text{R}}}$ 的计算结果(图中同时给出了各自的第2项计算结果以便于比较) (a)$n = 1$ ,${\text{NA}} = 1$ ; (b)$n = 1$ ,${\text{NA}} = 0.95$ ; (c)$n = 1.52$ ,${\text{NA}} = 1.4$ Figure 4. Calculation results of
${I_{\text{A}}}$ and${I_{\text{R}}}$ under different conditions: (a)$n = 1$ ,${\text{NA}} = 1$ ; (b)$n = 1$ ,${\text{NA}} = 0.95$ ; (c)$n = 1.52$ ,${\text{NA}} = 1.4$ . The second terms of${I_{\text{A}}}$ and${I_{\text{R}}}$ are also shown here for the convenience of comparing
图 5
$n = 1$ ,${\text{NA}} = 0.95$ 时, 径向偏振光和1阶角向矢量涡旋光的焦斑FWHM随$R/T$ 的变化曲线Figure 5. Focus FWHM of different beams as a function of
$R/T$ with$n = 1$ ,${\text{NA}} = 0.95$ .
图 6 (a)六环带相位板示意图; (b)聚焦过程示意图
Figure 6. (a) Phase structure of a six-belt binary element; (b) the focusing setup.
图 7
$n = 1$ ,${\text{NA}} = 0.95$ ,$R/T = 25$ 时, (a)相位调制前后的焦平面光强分布; (b)相位调制前后的光轴光强分布; (c)调制前焦点区域$\rho {\text{-}}z$ 面的二维光强分布; (d)调制后焦点区域$\rho {\text{-}}z$ 面的二维光强分布Figure 7. Under the condition of
$n = 1$ ,${\text{NA}} = 0.95$ ,$R/T = 25$ , (a) the intensity distributions at the focal plane with and without phase modulation, (b) the intensity distributions at the optic axis with and without phase modulation, (c) the two-dimensional (2D) intensity distribution in the$\rho {\text{-}}z$ plane without phase modulation, and (d) the 2D intensity distribution in the$\rho {\text{-}}z$ plane with phase modulation.表 1 不同情况下, 由(30)式和(31)式计算获得的中心焦斑的FWHM (
${\lambda _n} = \lambda /n$ 为介质中波长)Table 1. Calculation results of FWHM of the focal spot by using Eqs. (30) and (31) under different conditions. Here,
${\lambda _n} = \lambda /n$ is the wavelength in the medium.$n = 1$, ${\text{NA}} = 1$ $n = 1$, ${\text{NA}} = 0.95$ $n = 1.52$, ${\text{NA}} = 1.4$ ${I_{\rm A}}$的FWHM $0.371{\lambda _n}$ $0.389{\lambda _n}$ $0.403{\lambda _n}$ ${I_{\text{R}}}$的FWHM $0.359{\lambda _n}$ $0.391{\lambda _n}$ $0.412{\lambda _n}$ 
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[1] Li Y C, Xin H B, Lei H X, Liu L L, Li Y Z, Zhang Y, Li B J 2016 Light Sci. Appl. 5 e16176
Google Scholar
[2] Gong Z, Pan Y L, Videen G, Wang C 2018 J. Quant. Spectrosc. Ra. 214 94
Google Scholar
[3] Wetzstein G, Ozcan A, Gigan S, Fan S, Englund D, Soljačić M, Denz C, Miller D A B, Psaltis D 2020 Nature 588 39
Google Scholar
[4] 许琳茜, 朱榕琪, 朱竹青, 贡丽萍, 顾兵 2022 物理学报 71 147801
Google Scholar
Xu L X, Zhu R Q, Zhu Z Q, Gong L P, Gu B 2022 Acta Phys. Sin. 71 147801
Google Scholar
[5] Zhang L, Qiu X D, Zeng L W, Chen L X 2019 Chin. Phys. B 28 094202
Google Scholar
[6] Zhao J, Winetraub Y, Du L, Van Vleck A, Ichimura K, Huang C, Aasi S Z, Sarin K Y, de la Zerda A 2022 Optica 9 859
Google Scholar
[7] Cao R, Zhao J J, Li L, Du L, Zhang Y D, Luo Y L, Jiang L M, Davis S, Zhou Q F, de la Zerda A, Wang L V 2022 Nat. Photon. 17 89
Google Scholar
[8] Wang H, Shi L, Lukyanchuk B, Sheppard C, Chong C T 2008 Nat. Photon. 2 501
Google Scholar
[9] Huang K, Shi P, Kang X L, Zhang X, Li Y P 2010 Opt. Lett. 35 965
Google Scholar
[10] Guo H M, Weng X Y, Jiang M, Zhao Y H, Sui G R, Hu Q, Wang Y, Zhuang S L 2013 Opt. Express 21 5363
Google Scholar
[11] Zhang T, Li M, Ye H, Shi C 2020 Opt. Commun. 460 125178
Google Scholar
[12] He J, Zhuang J C, Ding L, Huang K 2021 Appl. Opt. 60 3081
Google Scholar
[13] Hao X, Kuang C F, Wang T T, Liu X 2010 Opt. Lett. 35 3928
Google Scholar
[14] Yuan G H, Wei S B, Yuan X C 2011 Opt. Lett. 36 3479
Google Scholar
[15] Wang S C, Li X P, Zhou J Y, Gu M 2014 Opt. Lett. 39 5022
Google Scholar
[16] Liu L, Shi L, Li F, Yu S, Wang S, Du J, Liu M, Qi B, Yan W 2021 IEEE Photon. J. 13 1
Google Scholar
[17] Gao X Z, Zhao P C, Zhao J H, Sun X F, Liu J J, Yang F, Pan Y 2022 Opt. Express 30 26275
Google Scholar
[18] Zhan Q 2009 Adv. Opt. Photon. 1 1
Google Scholar
[19] Kozawa Y, Sato S 2021 Prog. Opt. 66 35
Google Scholar
[20] Sato S, Kozawa Y 2009 J. Opt. Soc. Am. A Opt. Image Sci. Vis. 26 142
Google Scholar
[21] Li M M, Cai Y N, Yan S H, Liang Y S, Zhang P, Yao B L 2018 Phys. Rev. A 97 053842
Google Scholar
[22] Monteiro P B, Neto P A M, Nussenzveig H M 2009 Phys. Rev. A 79 033830
Google Scholar
[23] Ostrovsky A S, Rickenstorff-Parrao C, Arrizón V 2013 Opt. Lett. 38 534
Google Scholar
[24] Grosjean T, Courjon D 2007 Opt. Commun. 272 314
Google Scholar
[25] Vaity P, Rusch L 2015 Opt. Lett. 40 597
Google Scholar
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