Effect of laser illumination conditions on focusing performance of super-oscillatory lens

Liu Kang; He Tao; Liu Tao; Li Guo-Qing; Tian Bo; Wang Jia-Yi; Yang Shu-Ming

doi:10.7498/aps.69.20200577

Abstract

Super-oscillatory lens (SOL), a new type of planar optical element developed in recent years, may play an important role in the integrated optics, microscopy, advanced sensor, and astronomy. Based on the vectorial angular spectrum theory and genetic algorithm, both binary amplitude-type and phase-type SOLs are designed. Various sub-diffraction focusing properties can be realized by optimizing the design procedure. In order to investigate the focusing characteristics of SOLs under different illumination conditions, rigorous electromagnetic simulation calculations of the diffracted focusing light field are implemented by the finite-difference time-domain method. The results show that when the beam waist radius w₀ of the illuminating laser is less than the SOL radius a, not only the capability of super-diffraction limit focusing will decrease significantly, but also the intensity of the focal spot will attenuate by more than 50%. Comparing with the amplitude-type SOL, the waist radius w₀ has a strong effect on the phase-type SOL and causes a significant focus to shift in the positive direction. However, if w₀ is larger than 2a, the ideal focusing characteristics of SOL can be maintained. Under the condition of oblique illumination, the high numerical aperture amplitude-type SOL generally only allows a small inclination angle of less than 10°, while the phase-type SOL has a wide inclination adaptability (can exceed 40°) regardless of the numerical aperture. For the latter, the focal spot will expand laterally and the intensity will decrease sharply with the increase of inclination angle. As for low numerical aperture phase-type SOL, the focusing characteristics, including focal spot size, focusing intensity and the angular position of the focus, can keep stable within an inclination angle of 18°. For imaging infinitely distant objects, the oblique illumination will produce a fluctuating field curvature and significant negative distortion for high numerical aperture SOLs, while for the low numerical aperture SOLs, the field curvature increases with inclination angle increasing and the distortion disappears almost. The research results of this paper provide an important theoretical basis for practical applications of super-oscillatory lens in the fields of sub-diffraction light focusing, super-resolution microscopic imaging, and micro-nano processing of femtosecond laser direct writing.

Keywords:

Authors and contacts

State Key Labortory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Corresponding author: Liu Tao, liu8483@xjtu.edu.cn ; Yang Shu-Ming, shuming.yang@mail.xjtu.edu.cn

Funds: Project supported by the Excellent Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 51722509), the National Key R&D Program of China (Grant No. 2017YFB1104700), and the Natural Science Foundation for Basic Research of Shaanxi Province, China (Grant No. 2020SF-170)

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References

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[22]	Chen G, Wu Z X, Yu A P, et al. 2016 Sci. Rep. 6 37776 Google Scholar
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[24]	Rogers E T F, Savo S, Lindberg J, Roy T, Dennis M R, Zheludev N 2013 Appl. Phys. Lett. 102 031108 Google Scholar
[25]	郁道银, 谈恒英 2011 工程光学 (北京: 机械工业出版社) 第118页 Yu D Y, Tan H Y 2011 Engineering Optics (Beijing: China Machine Press) p118 (in Chinese)

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图 1 SOL衍射聚焦示意图

Figure 1. Schematic diagram of diffraction focusing of SOL.

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图 2 SOL₁的聚焦光场FDTD计算结果　(a) 轴向光强分布; (b) 焦平面内横向光强分布(Y方向)

Figure 2. Focused light field of SOL₁ by FDTD simulation: (a) On-axis intensity distribution; (b) transverse intensity distribution in the focal plane (Y-direction).

DownLoad: Full-Size Img PowerPoint

图 3 SOL聚焦性能变化(FDTD)　(a) 光斑横向尺寸; (b) 光斑中心强度

Figure 3. Focusing performance of SOL (FDTD): (a) Transverse size; (b) central intensity.

DownLoad: Full-Size Img PowerPoint

图 4 SOL₂的聚焦光场FDTD计算结果　(a) 轴向光强分布; (b) 焦平面内横向光强分布(Y方向)

Figure 4. Focused light field of SOL₂ by FDTD simulation: (a) On-axis intensity distribution; (b) transverse intensity distribution in the focal plane (Y-direction).

DownLoad: Full-Size Img PowerPoint

图 5 SOL₁聚焦光斑电场分布的FDTD结果　(a) 均匀平面波照明; (b) w₀ = a/2的高斯光束照明

Figure 5. Electric field distribution of SOL₁ by FDTD simulation: (a) Uniform plane beam illumination; (b) w₀ = a/2 Gaussian beam illumination.

DownLoad: Full-Size Img PowerPoint

图 6 BG径向偏振光束照明条件下SOL聚焦光场的VAS计算结果　(a), (b) SOL₁; (c), (d) SOL₂

Figure 6. Focused light intensity of SOL by VAS calculation under BG radially polarized illumination: (a), (b) SOL₁; (c), (d) SOL₂.

DownLoad: Full-Size Img PowerPoint

图 7 SOL对倾斜平行光的衍射聚焦成像示意图

Figure 7. Schematic diagram of diffraction focused imaging by SOL under oblique illumination.

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图 8 倾斜平行光照明的SOL衍射聚焦示意图

Figure 8. Schematic diagram of diffraction focusing by SOL under oblique illumination.

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图 9 振幅型SOL的倾斜照明FDTD聚焦计算结果　(a), (c), (e) SOL₅; (b), (d), (f) SOL₆

Figure 9. Focusing results of amplitude-type SOL under oblique illumination: (a), (c), (e) SOL₅; (b), (d), (f) SOL₆.

DownLoad: Full-Size Img PowerPoint

图 10 倾斜照明SOL的聚焦光斑尺寸　(a) FWHM_x ; (b) FWHM_z

Figure 10. Focal spot size of SOL under oblique illumination: (a) FWHM_x ; (b) FWHM_z .

DownLoad: Full-Size Img PowerPoint

图 11 不同照明角度下SOL的聚焦光斑强度

Figure 11. Focal intensity of SOL varying with oblique illumination angle.

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图 12 聚焦光斑角位置变化　(a) $\omega $-$\omega '$关系曲线; (b)$\omega $-$\varGamma $关系曲线

Figure 12. Angle position of focusing spot: (a) $\omega $-$\omega '$; (b) $\omega $-$\varGamma $.

DownLoad: Full-Size Img PowerPoint

图 13 SOL的场曲　(a) SOL₃; (b) SOL₄

Figure 13. Field curvature of SOL: (a) SOL₃; (b) SOL₄.

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表 1 优化设计的SOL结构参数

Table 1. Structural parameters of optimized SOL.

SOL	类型	∆r/μm	D/μm	f_sol/μm	NA	r_i /μm	t_i
SOL₁	振幅	0.2	16	3.5	0.92	[0.2, 0.4, 1.0, 1.2, 2.0, 2.2, 2.4, 2.6, 3.2, 3.6, 3.8, 4.0, 4.2, 4.6, 4.8, 5.2, 5.4, 6.0, 6.4, 6.8, 7.0, 7.4, 7.6, 8.0]	[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
SOL₂	相位	0.2	16	3.5	0.92	[0.2, 0.6, 1.0, 1.2, 1.6, 1.8, 2.0, 2.4, 3.0, 3.2, 4.8, 5.0, 5.4, 5.8, 6.2, 6.6, 6.8, 7.2, 7.4, 8.0]	[–1, 1, –1, 1, –1, 1, –1, 1, –1, 1, –1, 1, –1, 1, –1, 1, –1, 1, –1, 1]
SOL₃	相位	0.2	10	2.0	0.93	[0.473, 0.885, 1.458, 1.969, 2.382, 2.842, 3.044, 4.239, 4.745, 5.000]	[1, j, 1, j, 1, j, 1, j, 1, j]
SOL₄	相位	0.2	10	9.9	0.45	[0.268, 0.468, 1.165, 1.365, 1.734, 3.189, 3.399, 4.278, 4.792, 5.000]	[j, 1, j, 1, j, 1, j, 1, j, 1]
SOL₅	振幅	0.2	10	1.9	0.93	[0.600, 0.899, 1.915, 2.190, 2.440, 3.756, 4.076, 4.357, 4.769, 5.000]	[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
SOL₆	振幅	0.2	10	10.1	0.44	[0.300, 0.506, 1.248, 1.460, 1.660, 2.885, 3.085, 3.335, 3.755, 4.094, 4.294, 5.000]	[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

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[1]	Rogers E T F, Lindberg J, Roy T, Savo S, Chad J E, Dennis M R, Zhelidev N I 2012 Nat. Mater. 11 432 Google Scholar
[2]	刘涛 2013 博士学位论文 (哈尔滨: 哈尔滨工业大学) Liu T 2013 Ph. D. Dissertation (Harbin: Harbin Institute of Technology) (in Chinese)
[3]	Francia G T D 1952 IL Nuovo Cimento 9 426 Google Scholar
[4]	邱丽荣 2005 博士学位论文 (哈尔滨: 哈尔滨工业大学) Qiu L R 2005 Ph. D. Dissertation (Harbin: Harbin Institute of Technology) (in Chinese)
[5]	Diao J S, Yuan W Z, Yu Y T, Zhu Y C, Wu Y 2016 Opt. Express 24 1924 Google Scholar
[6]	Wu J, Wu Z X, He Y G, Yu A P, Zhang Z H, Wen Z Q, Chen G 2017 Opt. Express 25 6274 Google Scholar
[7]	Liu T, Tan J B, Liu J, Wang H T 2013 Opt. Lett. 38 2742 Google Scholar
[8]	Liu T, Liu Q, Yang S M, Jiang Z D, Wang T, Zhang G F 2017 Appl. Opt. 56 3725 Google Scholar
[9]	Liu T, Tan J B, Liu J, Wang H T 2013 Opt. Express 21 15090 Google Scholar
[10]	Liu T, Shen T, Yang S M, Jiang Z D 2015 J. Opt. 17 035610 Google Scholar
[11]	Liu T, Liu Q, Yang S M, Jiang Z D, Wang T, Yang X K 2017 Opt. Commun. 393 72 Google Scholar
[12]	Yuan G H, Rogers Edward T F, Zheludev N I 2017 Light: Sci. Appl. 6 e17036 Google Scholar
[13]	Qin F, Huang K, Wu J F, Teng J H, Qiu C W, Hong M H 2017 Adv. Mater. 29 1602721 Google Scholar
[14]	武靖 2018 硕士学位论文 (重庆: 重庆大学) Wu J 2018 M. S. Thesis (Chongqing: Chongqing university) (in Chinese)
[15]	Nagarajan A, Stoevelaar L P, Silvestri F, et al. 2019 Opt. Express 27 20012 Google Scholar
[16]	Liu T, Wang T, Yang S M, Sun L, Jiang Z D 2015 Opt. Express 23 32139 Google Scholar
[17]	Yang S M, Wang T, Liu T, Jiang Z D 2016 Opt. Commun. 372 166 Google Scholar
[18]	Yu Y T, Li W L, Li H Y, Li M Y, Yuan W Z 2018 Nanomaterials 8 185 Google Scholar
[19]	Ni H B, Yuan G H, Sun L D, Chang N, Zhang D, Chen R P, Jiang L Y, Chen H Y, Gu Z Z, Zhao X W 2018 RSC Adv. 8 20117 Google Scholar
[20]	Luneburg R K 1966 Mathematical Theory of Optics (Berkeley: University of California Press) p305
[21]	Liu T, Yang S M, Jiang Z D 2016 Opt. Express 24 16297 Google Scholar
[22]	Chen G, Wu Z X, Yu A P, et al. 2016 Sci. Rep. 6 37776 Google Scholar
[23]	张蕾, 蔡阳健, 陆漩辉 2004 物理学报 53 1777 Google Scholar Zhang L, Cai Y J, Lu X H 2004 Acta Phys. Sin. 53 1777 Google Scholar
[24]	Rogers E T F, Savo S, Lindberg J, Roy T, Dennis M R, Zheludev N 2013 Appl. Phys. Lett. 102 031108 Google Scholar
[25]	郁道银, 谈恒英 2011 工程光学 (北京: 机械工业出版社) 第118页 Yu D Y, Tan H Y 2011 Engineering Optics (Beijing: China Machine Press) p118 (in Chinese)

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Vol. 69, Issue 18 - 2020-09-20

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