Abstract

Materials with relatively small refractive indices (n<2), such as glass, quartz, polymers, some ceramics, etc., are the basic materials in most optical components (lenses, optical fibres, etc.). In this review, we present some of the phenomena and possible applications arising from the interaction of light with particles with a refractive index less than 2. The vast majority of the physics involved can be described with the help of the exact, analytical solution of Maxwell’s equations for spherical particles (so called Mie theory). We also discuss some other particle geometries (spheroidal, cubic, etc.) and different particle configurations (isolated or interacting) and draw an overview of the possible applications of such materials, in connection with field enhancement and super resolution nanoscopy.

© 2017 Optical Society of America

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D. G. Baranov, R. S. Savelev, S. V. Li, A. E. Krasnok, and A. Alu, “Modifying magnetic dipole spontaneous emission with nanophotonic structures,” Laser Photonics Rev. 2017, 1600268 (2017).

M. Guo, Y. H. Ye, J. Hou, B. Du, and T. Wang, “Imaging of sub-surface nanostructures by dielectric planer cavity coupled microsphere lens,” Opt. Commun. 383, 153–158 (2017).
[Crossref]

B. Yan, Z. Wang, A. I. Parker, Y.-K. Lai, P. J. Thomas, L. Yue, and J. N. Monks, “Superlensing microscope objective lens,” Appl. Opt. 56, 3142–3144 (2017).
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H. Zhu, M. Chen, S. Zhou, and L. Wu, “Synthesis of High Refractive Index and Shape Controllable Colloidal Polymer Microspheres for Super-Resolution Imaging,” Macromolecules 50(2), 660–665 (2017).
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2016 (29)

H. Zhu, W. Fan, S. Zhou, M. Chen, and L. Wu, “Polymer Colloidal Sphere-Based Hybrid Solid Immersion Lens for Optical Super-Resolution Imaging,” ACS Nano 10(10), 9755–9761 (2016).
[Crossref] [PubMed]

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016).
[Crossref] [PubMed]

F. Wang, L. Liu, H. Yu, Y. Wen, P. Yu, Z. Liu, Y. Wang, and W. J. Li, “Scanning superlens microscopy for non-invasive large field-of-view visible light nanoscale imaging,” Nat. Commun. 7, 13748 (2016).
[Crossref] [PubMed]

M. Guo, Y. H. Ye, J. Hou, and B. Du, “Size-dependent optical imaging properties of high-index immersed microsphere lens,” Appl. Phys. B 122(3), 1–7 (2016).
[Crossref]

M. S. Kim, T. Scharf, S. Mühlig, M. Fruhnert, C. Rockstuhl, R. Bitterli, W. Noell, R. Voelkel, and H. P. Herzig, “Refraction limit of miniaturized optical systems: a ball-lens example,” Opt. Express 24(7), 6996–7005 (2016).
[Crossref] [PubMed]

B. Yan, L. Yue, and Z. Wang, “Engineering near-field focusing of a microsphere lens with pupil masks,” Opt. Commun. 370, 140–144 (2016).
[Crossref]

Z. Wang, “Microsphere super-resolution imaging,” Nanoscience 3, 193–210 (2016).
[Crossref]

W. Fan, B. Yan, Z. Wang, and L. Wu, “Three-dimensional all-dielectric metamaterial solid immersion lens for subwavelength imaging at visible frequencies,” Sci. Adv. 2(8), e1600901 (2016).
[Crossref] [PubMed]

J. N. Monks, B. Yan, N. Hawkins, F. Vollrath, and Z. Wang, “Spider Silk: Mother Nature’s Bio-Superlens,” Nano Lett. 16(9), 5842–5845 (2016).
[Crossref] [PubMed]

K. W. Allen, F. Abolmaali, J. M. Duran, G. Ariawansa, N. L. Limberopoulos, A. M. Urbas, and V. N. Astratov, “Increasing sensitivity and angle-of-view of mid-way infrared detectors by integration with dielectric microspheres,” Appl. Phys. Lett. 108(24), 241108 (2016).
[Crossref]

A. V. Maslov and V. N. Astratov, “Imaging of sub-wavelength structures radiating coherently near microspheres,” Appl. Phys. Lett. 108(5), 051104 (2016).
[Crossref]

H. Yang, R. Trouillon, G. Huszka, and M. A. M. Gijs, “Super-resolution imaging of a dielectric microsphere is governed by the waist of its photonic nanojet,” Nano Lett. 16(8), 4862–4870 (2016).
[Crossref] [PubMed]

Y. Ben-Aryeh, “Increase of resolution by use of microspheres related to complex Snell’s law,” J. Opt. Soc. Am. A 33(12), 2284–2288 (2016).
[Crossref] [PubMed]

B. Du, Y. H. Ye, J. Hou, M. Guo, and T. Wang, “Sub-wavelength image stitching with removable microsphere-embedded thin film,” Appl. Phys., A Mater. Sci. Process. 122(1), 1–6 (2016).
[Crossref]

F. Wang, L. Liu, P. Yu, Z. Liu, H. Yu, Y. Wang, and W. J. Li, “Three-Dimensional Super-Resolution Morphology by Near-Field Assisted White-Light Interferometry,” Sci. Rep. 6(1), 24703 (2016).
[Crossref] [PubMed]

H. S. S. Lai, F. Wang, Y. Li, B. Jia, L. Liu, and W. J. Li, “Super-Resolution Real Imaging in Microsphere-Assisted Microscopy,” PLoS One 11(10), e0165194 (2016).
[Crossref] [PubMed]

X. Yang, H. Xie, E. Alonas, Y. Liu, X. Chen, P. J. Santangelo, Q. Ren, P. Xi, and D. Jin, “Mirror-enhanced super-resolution microscopy,” Light Sci. Appl. 5(6), e16134 (2016).
[Crossref] [PubMed]

J. Li, W. Liu, T. Li, I. Rozen, J. Zhao, B. Bahari, B. Kante, and J. Wang, “Swimming Microrobot Optical Nanoscopy,” Nano Lett. 16(10), 6604–6609 (2016).
[Crossref] [PubMed]

E. Bor, M. Turduev, and H. Kurt, “Differential evolution algorithm based photonic structure design: numerical and experimental verification of subwavelength λ/5 focusing of light,” Sci. Rep. 6(1), 30871 (2016).
[Crossref] [PubMed]

P. Y. Li, Y. Tsao, Y. J. Liu, Z. X. Lou, W. L. Lee, S. W. Chu, and C. W. Chang, “Unusual imaging properties of superresolution microspheres,” Opt. Express 24(15), 16479–16486 (2016).
[Crossref] [PubMed]

H. Zhu, W. Fan, S. Zhou, M. Chen, and L. Wu, “Polymer Colloidal Sphere-Based Hybrid Solid Immersion Lens for Optical Super-Resolution Imaging,” ACS Nano 10(10), 9755–9761 (2016).
[Crossref] [PubMed]

G. Gu, R. Zhou, H. Xu, G. Cai, and Z. Cai, “Subsurface nano-imaging with self-assembled spherical cap optical nanoscopy,” Opt. Express 24(5), 4937–4948 (2016).
[Crossref]

M. Wu, R. Chen, J. Soh, Y. Shen, L. Jiao, J. Wu, X. Chen, R. Ji, and M. Hong, “Super-focusing of center-covered engineered microsphere,” Sci. Rep. 6(1), 31637 (2016).
[Crossref] [PubMed]

C. Xu, S. Zhang, J. Shao, B. R. Lu, R. Mehfuz, S. Drakeley, F. Huang, and Y. Chen, “Photon nanojet lens: design, fabrication and characterization,” Nanotechnology 27(16), 165302 (2016).
[Crossref] [PubMed]

B. Yan, L. Y. Yue, and Z. B. Wang, “Engineering near-field focusing of a microsphere lens with pupil masks,” Opt. Commun. 370, 140–144 (2016).
[Crossref]

M. Wu, R. Chen, J. Soh, Y. Shen, L. Jiao, J. Wu, X. Chen, R. Ji, and M. Hong, “Super-focusing of center-covered engineered microsphere,” Sci. Rep. 6(1), 31637 (2016).
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L. Yue, B. Yan, and Z. Wang, “Photonic nanojet of cylindrical metalens assembled by hexagonally arranged nanofibers for breaking the diffraction limit,” Opt. Lett. 41(7), 1336–1339 (2016).
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I. V. Minin, O. V. Minin, V. Pacheco-Peña, and M. Beruete, “Subwavelength optical trap in the field of a standing wave on photonic jets,” Quantum Electron. 46, 555–557 (2016).
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C.-Y. Liu and C.-C. Li, “Photonic nanojet induced modes generated by a chain of dielectric microdisks,” Optik (Stuttg.) 127(1), 267–273 (2016).
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R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15(3), 2137–2142 (2015).
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I. V. Minin, O. V. Minin, and Y. E. Geintz, “Localized EM and photonic jets from non-spherical and non-symmetrical dielectric mesoscale objects: brief review,” Ann. Phys. 527(7–8), 491–497 (2015).
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Yu. E. Geintz, A. A. Zemlyanov, and E. K. Panina, “Photonic Nanonanojets from Nonspherical Dielectric Microparticles,” Russ. Phys. J. 58(7), 904–910 (2015).
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Yu. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Characteristics of photonic jets from microcones,” Opt. Spectrosc. 119(5), 849–854 (2015).
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P. Wu, J. Li, K. Wei, and W. Yue, “Tunable and ultra-elongated photonic nanojet generated by a liquid-immersed core-shell dielectric microsphere,” Appl. Phys. Express 8(11), 112001 (2015).
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B. Luk’yanchuk, N. Voshchinnikov, R. Paniagua-Dominguez, and A. Kuznetsov, “Optimum Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Index,” ACS Photonics 2(7), 993–999 (2015).
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Y. Zhang, M. Nieto-Vesperinas, and J. J. Sáenz, “Dielectric spheres with maximum forward scattering and zero backscattering: a search for their material composition,” J. Opt. 17(10), 105612 (2015).
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B. Ounnas, B. Sauviac, Y. Takakura, S. Lecler, B. Bayard, and S. Robert, “Single and Dual Photonic Jets and Corresponding Backscattering Enhancement With Tipped Waveguides: Direct Observation at Microwave Frequencies,” IEEE Trans. Antenn. Propag. 63(12), 5612–5618 (2015).
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M. Guo, Y. H. Ye, J. Hou, and B. Du, “Experimental far-field imaging properties of high refractive index microsphere lens,” Photonics Research 3(6), 339–342 (2015).
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C.-Y. Liu and K.-L. Hsiao, “Direct imaging of optimal photonic nanojets from core-shell microcylinders,” Opt. Lett. 40(22), 5303–5306 (2015).
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I. V. Minin, O. V. Minin, V. Pacheco-Peña, and M. Beruete, “Localized photonic jets from flat, three-dimensional dielectric cuboids in the reflection mode,” Opt. Lett. 40(10), 2329–2332 (2015).
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M. X. Wu, B. J. Huang, R. Chen, Y. Yang, J. F. Wu, R. Ji, X. D. Chen, and M. H. Hong, “Modulation of photonic nanojets generated by microspheres decorated with concentric rings,” Opt. Express 23(15), 20096–20103 (2015).
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A. Darafsheh, C. Guardiola, A. Palovcak, J. C. Finlay, and A. Cárabe, “Optical super-resolution imaging by high-index microspheres embedded in elastomers,” Opt. Lett. 40(1), 5–8 (2015).
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M. Guo, Y.-H. Ye, J. Hou, and B. Du, “Experimental far-field imaging properties of high refractive index microsphere lens,” Photonics Research 3(6), 339–342 (2015).
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H. Pang, A. Cao, C. Du, Q. Qiu, Q. Deng, and S. Yin, “Spectrum analysis of liquid immersion to transparent microsphere based optical nanoscopy,” Optik (Stuttg.) 126(21), 3079–3083 (2015).
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H. Zhu, B. Yan, S. Zhou, Z. Wang, and L. Wu, “Synthesis and superresolution imaging performance of a refractive-index-controllable microsphere superlens,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(41), 10907–10915 (2015).
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S. Lee and L. Li, “Rapid super-resolution imaging of sub-surface nanostructures beyond diffraction limit by high refractive index optical nanoscopy,” Opt. Commun. 334, 253–257 (2015).
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K. W. Allen, N. Farahi, Y. Li, N. I. Limberopoulos, D. E. Walker, A. M. Urbas, and V. N. Astratov, “Overcoming the diffraction limit of imaging nanoplasmonic arrays by microspheres and microfibers,” Opt. Express 23(19), 24484–24496 (2015).
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T. X. Hoang, Y. Duan, X. Chen, and G. Barbastathis, “Focusing and imaging in microsphere-based microscopy,” Opt. Express 23(9), 12337–12353 (2015).
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S. Lee and L. Li, “Rapid super-resolution imaging of sub-surface nanostructures beyond diffraction limit by high refractive index microsphere optical nanoscopy,” Opt. Commun. 334, 253–257 (2015).
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K. W. Allen, N. Farahi, Y. Li, N. I. Limberopoulos, D. E. Walker, A. M. Urbas, V. Liberman, and V. N. Astratov, “Super-resolution microscopy by movable thin-films with embedded microspheres: Resolution analysis,” Ann. Phys. 527(7-8), 513–522 (2015).
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C. Zheng, A. Hu, K. D. Kihm, Q. Ma, R. Li, T. Chen, and W. W. Duley, “Femtosecond Laser Fabrication of Cavity Microball Lens (CMBL) inside a PMMA Substrate for Super-Wide Angle Imaging,” Small 11(25), 3007–3016 (2015).
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L. Yao, Y. H. Ye, H. F. Ma, L. Cao, and J. Hou, “Role of the immersion medium in the microscale spherical lens imaging,” Opt. Commun. 335, 23–27 (2015).
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H. Cang, A. Salandrino, Y. Wang, and X. Zhang, “Adiabatic far-field sub-diffraction imaging,” Nat. Commun. 6, 7942 (2015).
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L. Cao, Y. H. Ye, L. Yao, and M. Guo, “Dependence of focal position on the microscale spherical lens imaging,” Opt. Commun. 353, 184–188 (2015).
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H. Yang and M. A. M. Gijs, “Optical microscopy using a glass microsphere for metrology of sub-wavelength nanostructures,” Microelectron. Eng. 143, 86–90 (2015).
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J. Hou, L. Yao, J. Ren, M. Guo, and Y.-H. Ye, “Magnification and resolution of microlenses with different shapes,” IET Micro & Nano Letters 10(7), 351–354 (2015).
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Y. Chen, X. Xie, L. Li, G. Chen, L. Guo, and X. Lin, “Improving field enhancement of 2D hollow tapered waveguides via dielectric microcylinder coupling,” J. Phys. D Appl. Phys. 48(6), 065103 (2015).
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Y. Li, Z. Shi, S. Shuai, and L. Wang, “Widefield scanning imaging with optical super-resolution,” J. Mod. Opt. 62(14), 1033–1036 (2015).
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H. Pang, A. Cao, C. Du, Q. Qiu, Q. Deng, and S. Yin, “Spectrum analysis of liquid immersion to transparent microsphere based optical nanoscopy,” Optik (Stuttg.) 126(21), 3079–3083 (2015).
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M. Guo, Y. H. Ye, J. Hou, and B. Du, “Experimental far-field imaging properties of high refractive index microsphere lens,” Photonics Research 3(6), 339–342 (2015).
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R. Ye, Y. H. Ye, H. F. Ma, L. Cao, J. Ma, F. Wyrowski, R. Shi, and J. Y. Zhang, “Experimental imaging properties of immersion microscale spherical lenses,” Sci. Rep. 4(1), 3769 (2015).
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R. Rezvani Naraghi, S. Sukhov, J. J. Sáenz, and A. Dogariu, “Near-Field Effects in Mesoscopic Light Transport,” Phys. Rev. Lett. 115(20), 203903 (2015).
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2014 (18)

H. Yang, N. Moullan, J. Auwerx, and M. A. M. Gijs, “Super-resolution biological microscopy using virtual imaging by a microsphere nanoscope,” Small 10(9), 1712–1718 (2014).
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X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113(26), 263901 (2014).
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X. Huang, X. N. He, W. Xiong, Y. Gao, L. J. Jiang, L. Liu, Y. S. Zhou, L. Jiang, J. F. Silvain, and Y. F. Lu, “Contrast enhancement using silica microspheres in coherent anti-Stokes Raman spectroscopic imaging,” Opt. Express 22(3), 2889–2896 (2014).
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E. McLeod and A. Ozcan, “Nano-imaging enabled via self-assembly,” Nano Today 9(5), 560–573 (2014).
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S. Peng, C. Xu, T. C. Hughes, and X. Zhang, “From nanodroplets by the ouzo effect to interfacial nanolenses,” Langmuir 30(41), 12270–12277 (2014).
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Y. Yan, Y. Zeng, Y. Wu, Y. Zhao, L. Ji, Y. Jiang, and L. Li, “Ten-fold enhancement of ZnO thin film ultraviolet-luminescence by dielectric microsphere arrays,” Opt. Express 22(19), 23552–23564 (2014).
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P. K. Upputuri, Z. Wu, L. Gong, C. K. Ong, and H. Wang, “Super-resolution coherent anti-Stokes Raman scattering microscopy with photonic nanojets,” Opt. Express 22(11), 12890–12899 (2014).
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V. M. Sundaram and S. B. Wen, “Analysis of deep sub-micron resolution in microsphere based imaging,” Appl. Phys. Lett. 105(20), 204102 (2014).
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Y. V. Miklyaev, S. A. Asselborn, K. A. Zaytsev, and M. Y. Darscht, “Superresolution microscopy in far-field by near-field optical random mapping nanoscopy,” Appl. Phys. Lett. 105(11), 113103 (2014).
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M. Michihata, K. Takami, T. Hayashi, and Y. Takaya, “Fundamental validation for surface texture imaging using a microsphere as a laser-trapping-based microprobe,” Adv. Opt. Technol. 3, 417–423 (2014).

S. Mandal, “Superlens-Based Nanoscale Imaging,” IEEE Potentials 33(2), 17–20 (2014).
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Y. Yan, L. Li, C. Feng, W. Guo, S. Lee, and M. Hong, “Microsphere-coupled scanning laser confocal nanoscope for sub-diffraction-limited imaging at 25 nm lateral resolution in the visible spectrum,” ACS Nano 8(2), 1809–1816 (2014).
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A. Darafsheh, N. I. Limberopoulos, J. S. Derov, D. E. Walker, and V. N. Astratov, “Advantages of microsphere-assisted super-resolution imaging technique over solid immersion lens and confocal microscopies,” Appl. Phys. Lett. 104(6), 061117 (2014).
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K. W. Allen, A. Darafsheh, F. Abolmaali, N. Mojaverian, N. I. Limberopoulos, A. Lupu, and V. N. Astratov, “Microsphere-chain waveguides: Focusing and transport properties,” Appl. Phys. Lett. 105(2), 021112 (2014).
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V. M. Sundaram and S.-B. Wen, “Analysis of deep sub-micron resolution in micro-sphere based imaging,” Appl. Phys. Lett. 105(20), 204102 (2014).
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C.-Y. Liu and Y. Wang, “Real-space observation of photonic nanojet in microspheres,” Physica E 61, 141–147 (2014).
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L. Han, Y. Han, J. Wang, G. Gouesbet, and G. Gréhan, “Controllable and enhanced photonic jet generated by fiber combined with spheroid,” Opt. Lett. 39(6), 1585–1588 (2014).
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V. Pacheco-Peña, M. Beruete, I. V. Minin, and O. V. Minin, “Terajets produced by 3D dielectric cuboids,” Appl. Phys. Lett. 105(8), 084102 (2014).
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2013 (14)

P. Albella, M. A. Poyli, M. K. Schmidt, S. A. Maier, F. Moreno, J. J. Saénz, and J. Aizpurua, “Low-Loss Electric and Magnetic Field-Enhanced Spectroscopy with Subwavelength Silicon Dimers,” J. Phys. Chem. C 117(26), 13573–13584 (2013).
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H. Guo, Y. Han, X. Weng, Y. Zhao, G. Sui, Y. Wang, and S. Zhuang, “Near-field focusing of the dielectric microsphere with wavelength scale radius,” Opt. Express 21(2), 2434–2443 (2013).
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C.-Y. Liu, “Tunable photonic nanojet achieved by core-shell microcylinder with nematic liquid crystal,” J. Mod. Opt. 60(7), 538–543 (2013).
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L. A. Krivitsky, J. J. Wang, Z. Wang, and B. Luk’yanchuk, “Locomotion of microspheres for super-resolution imaging,” Sci. Rep. 3(1), 3501 (2013).
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L. Li, W. Guo, Y. Yan, S. Lee, and T. Wang, “Label-free super-resolution imaging of adenoviruses by submerged microsphere optical nanoscopy,” Light Sci. Appl. 2(9), e104 (2013).
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X. Hao, C. Kuang, Z. Gu, Y. Wang, S. Li, Y. Ku, Y. Li, J. Ge, and X. Liu, “From microscopy to nanoscopy via visible light,” Light Sci. Appl. 2(10), e108 (2013).
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H. Guo, Y. Han, X. Weng, Y. Zhao, G. Sui, Y. Wang, and S. Zhuang, “Near-field focusing of the dielectric microsphere with wavelength scale radius,” Opt. Express 21(2), 2434–2443 (2013).
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S. Lee, L. Li, Y. Ben-Aryeh, Z. Wang, and W. Guo, “Overcoming the diffraction limit induced by microsphere optical nanoscopy,” J. Opt. 15(12), 125710 (2013).
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X. Hao, C. Kuang, Y. Li, and X. Liu, “Evanescent-wave-induced frequency shift for optical superresolution imaging,” Opt. Lett. 38(14), 2455–2458 (2013).
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R. Ye, Y. H. Ye, H. F. Ma, J. Ma, B. Wang, J. Yao, S. Liu, L. Cao, H. Xu, and J. Y. Zhang, “Experimental far-field imaging properties of a ~5-μm diameter spherical lens,” Opt. Lett. 38(11), 1829–1831 (2013).
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S. Lee, L. Li, Z. Wang, W. Guo, Y. Yan, and T. Wang, “Immersed transparent microsphere magnifying sub-diffraction-limited objects,” Appl. Opt. 52(30), 7265–7270 (2013).
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C. Pérez-Rodríguez, S. Ríos, I. R. Martín, L. L. Martín, P. Haro-González, and D. Jaque, “Upconversion emission obtained in Yb3+-Er3+ doped fluoroindate glasses using silica microspheres as focusing lens,” Opt. Express 21(9), 10667–10675 (2013).
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Y. Duan, G. Barbastathis, and B. Zhang, “Classical imaging theory of a microlens with super-resolution,” Opt. Lett. 38(16), 2988–2990 (2013).
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L. Li, W. Guo, Y. Yan, S. Lee, and T. Wang, “Label-free super-resolution imaging of adenoviruses by submerged microsphere optical nanoscopy,” Light Sci. Appl. 2(9), e104 (2013).
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2012 (7)

G. F. W. Darafsheh, L. D. Negro, and V. N. Astratov, “Optical super-resolution by high-index liquid-immersed microspheres,” Appl. Phys. Lett. 101(14), 141128 (2012).
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A. Vlad, I. Huynen, and S. Melinte, “Wavelength-scale lens microscopy via thermal reshaping of colloidal particles,” Nanotechnology 23(28), 285708 (2012).
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A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
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A. Darafsheh and V. N. Astratov, “Periodically focused modes in chains of dielectric spheres,” Appl. Phys. Lett. 100(6), 061123 (2012).
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C.-Y. Liu, “Ultra-high transmission of photonic nanojet induced modes in chains of core-shell microcylinders,” Phys. Lett. A 376(45), 3261–3266 (2012).
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C.-Y. Liu, “Superenhanced photonic nanojet by core-shell microcylinders,” Phys. Lett. A 376(23), 1856–1860 (2012).
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Y. Takakura, H. Halaq, S. Lecler, S. Robert, and B. Sauviac, “Single and dual photonic jets with tipped waveguides: An integral approach,” IEEE Photonics Technol. Lett. 24(17), 1516–1518 (2012).
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T. Wang, C. Kuang, X. Hao, and X. Liu, “Subwavelength focusing by a microsphere array,” J. Opt. 13(3), 035702 (2011).
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S. Yang, A. Taflove, and V. Backman, “Experimental confirmation at visible light wavelengths of the backscattering enhancement phenomenon of the photonic nanojet,” Opt. Express 19(8), 7084–7093 (2011).
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M. Nieto-Vesperinas, R. Goḿez-Medina, and J. J. Sáenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A 28(1), 54–60 (2011).
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Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 218 (2011).
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S. Yang, A. Taflove, and V. Backman, “Experimental confirmation at visible light wavelengths of the backscattering enhancement phenomenon of the photonic nanojet,” Opt. Express 19(8), 7084–7093 (2011).
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X. Hao, C. Kuang, X. Liu, H. Zhang, and Y. Li, “Microsphere based microscope with optical super-resolution capability,” Appl. Phys. Lett. 99(20), 203102 (2011).
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2010 (2)

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
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C. M. Ruiz and J. J. Simpson, “Detection of embedded ultra-subwavelength-thin dielectric features using elongated photonic nanojets,” Opt. Express 18(16), 16805–16812 (2010).
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2009 (5)

L. Zhao and C. K. Ong, “Direct observation of photonic jets and corresponding backscattering enhancement at microwave frequencies,” J. Appl. Phys. 105(12), 123512 (2009).
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K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A, Pure Appl. Opt. 11(5), 054009 (2009).
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A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6(9), 1979–1992 (2009).
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J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature 460(7254), 498–501 (2009).
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S. A. Claridge, A. W. Castleman, S. N. Khanna, C. B. Murray, A. Sen, and P. S. Weiss, “Cluster-assembled materials,” ACS Nano 3(2), 244–255 (2009).
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2008 (7)

Z. B. Wang, W. Guo, A. Pena, D. J. Whitehead, B. S. Luk’yanchuk, L. Li, Z. Liu, Y. Zhou, and M. H. Hong, “Laser micro/nano fabrication in glass with tunable-focus particle lens array,” Opt. Express 16(24), 19706–19711 (2008).
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S.-C. Kong, A. Sahakian, A. Taflove, and V. Backman, “Photonic nanojet-enabled optical data storage,” Opt. Express 16(18), 13713–13719 (2008).
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E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
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S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92(26), 261111 (2008).
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A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008).
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P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008).
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S.-C. Kong, A. Sahakian, A. Heifetz, A. Taflove, and V. Backman, “Robust detection of deeply subwavelength pits in simulated optical data-storage disks using photonic jets,” Appl. Phys. Lett. 92(21), 211102 (2008).
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A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88(11), 111111 (2006).
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Z. Chen, A. Taflove, X. Li, and V. Backman, “Superenhanced backscattering of light by nanoparticles,” Opt. Lett. 31(2), 196–198 (2006).
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Z. Chen, X. Li, A. Taflove, and V. Backman, “Backscattering enhancement of light by nanoparticles positioned in localized optical intensity peaks,” Appl. Opt. 45(4), 633–638 (2006).
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A. Heifetz, K. Huang, A. V. Sahakian, X. Li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. 89(22), 221118 (2006).
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J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006).
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V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85(23), 5508–5510 (2004).
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Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004).
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P. Leiderer, C. Bartels, J. König-Birk, M. Mosbacher, and J. Boneberg, “Imaging optical near-fields of nanostructures,” Appl. Phys. Lett. 85(22), 5370–5372 (2004).
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B. Luk’yanchuk, N. Arnold, S. M. Huang, Z. B. Wang, and M. H. Hong, “Three-dimensional effects in dry laser cleaning,” Appl. Phys., A Mater. Sci. Process. 77, 209–215 (2003).

N. Chaoui, J. Solis, C. N. Afonso, T. Fourrier, T. Muehlberger, G. Schrems, M. Mosbacher, D. Bäuerle, M. Bertsch, and P. Leiderer, “A high-sensitivity in situ optical diagnostic technique for laser cleaning of transparent substrates,” Appl. Phys., A Mater. Sci. Process. 76(5), 767–771 (2003).
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J. F. Nye, “Evolution from a Fraunhofer to a Pearcey diffraction pattern,” J. Opt. A, Pure Appl. Opt. 5(5), 495–502 (2003).
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K. Piglmayer, R. Denk, and D. Bäuerle, “Laser-induced surface patterning by means of microspheres,” Appl. Phys. Lett. 80(25), 4693–4695 (2002).
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H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(Pt 1), 129–135 (2001).
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2000 (2)

Y. F. Lu, L. Zhang, W. D. Song, Y. W. Zheng, and B. S. Luk’yanchuk, “Laser writing of a subwavelength structure on silicon (100) surfaces with particle enhanced optical irradiation,” JETP Lett. 72(9), 457–459 (2000).
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K. W. Allen, A. Darafsheh, F. Abolmaali, N. Mojaverian, N. I. Limberopoulos, A. Lupu, and V. N. Astratov, “Microsphere-chain waveguides: Focusing and transport properties,” Appl. Phys. Lett. 105(2), 021112 (2014).
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Zhao, Y.

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C. Zheng, A. Hu, K. D. Kihm, Q. Ma, R. Li, T. Chen, and W. W. Duley, “Femtosecond Laser Fabrication of Cavity Microball Lens (CMBL) inside a PMMA Substrate for Super-Wide Angle Imaging,” Small 11(25), 3007–3016 (2015).
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Y. W. Zheng, B. S. Luk’yanchuk, Y. F. Lu, W. D. Song, and Z. H. Mai, “Dry laser cleaning of particles from solid substrates: experiments and theory,” J. Appl. Phys. 90(5), 2135–2142 (2001).
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Y. F. Lu, L. Zhang, W. D. Song, Y. W. Zheng, and B. S. Luk’yanchuk, “Laser writing of a subwavelength structure on silicon (100) surfaces with particle enhanced optical irradiation,” JETP Lett. 72(9), 457–459 (2000).
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H. Zhu, B. Yan, S. Zhou, Z. Wang, and L. Wu, “Synthesis and superresolution imaging performance of a refractive-index-controllable microsphere superlens,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(41), 10907–10915 (2015).
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H. Zhu, W. Fan, S. Zhou, M. Chen, and L. Wu, “Polymer Colloidal Sphere-Based Hybrid Solid Immersion Lens for Optical Super-Resolution Imaging,” ACS Nano 10(10), 9755–9761 (2016).
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ACS Nano (4)

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ACS Photonics (1)

B. Luk’yanchuk, N. Voshchinnikov, R. Paniagua-Dominguez, and A. Kuznetsov, “Optimum Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Index,” ACS Photonics 2(7), 993–999 (2015).
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M. Michihata, K. Takami, T. Hayashi, and Y. Takaya, “Fundamental validation for surface texture imaging using a microsphere as a laser-trapping-based microprobe,” Adv. Opt. Technol. 3, 417–423 (2014).

Ann. Phys. (2)

K. W. Allen, N. Farahi, Y. Li, N. I. Limberopoulos, D. E. Walker, A. M. Urbas, V. Liberman, and V. N. Astratov, “Super-resolution microscopy by movable thin-films with embedded microspheres: Resolution analysis,” Ann. Phys. 527(7-8), 513–522 (2015).
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Appl. Opt. (3)

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Appl. Phys. Express (1)

P. Wu, J. Li, K. Wei, and W. Yue, “Tunable and ultra-elongated photonic nanojet generated by a liquid-immersed core-shell dielectric microsphere,” Appl. Phys. Express 8(11), 112001 (2015).
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B. Du, Y. H. Ye, J. Hou, M. Guo, and T. Wang, “Sub-wavelength image stitching with removable microsphere-embedded thin film,” Appl. Phys., A Mater. Sci. Process. 122(1), 1–6 (2016).
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M. Mosbacher, H.-J. Münzer, J. Zimmermann, J. Solis, J. Boneberg, and P. Leiderer, “Optical field enhancement effects in laser-assisted particle removal,” Appl. Phys., A Mater. Sci. Process. 72(1), 41–44 (2001).
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Y. W. Zheng, B. S. Luk’yanchuk, Y. F. Lu, W. D. Song, and Z. H. Mai, “Dry laser cleaning of particles from solid substrates: experiments and theory,” J. Appl. Phys. 90(5), 2135–2142 (2001).
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S. M. Huang, M. H. Hong, Y. F. Lu, B. S. Luk’yanchuk, W. D. Song, and T. C. Chong, “Pulsed laser-assisted surface structuring with optical near-field enhanced effects,” J. Appl. Phys. 92(5), 2495–2500 (2002).
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H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(Pt 1), 129–135 (2001).
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C.-Y. Liu, “Tunable photonic nanojet achieved by core-shell microcylinder with nematic liquid crystal,” J. Mod. Opt. 60(7), 538–543 (2013).
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JETP Lett. (1)

Y. F. Lu, L. Zhang, W. D. Song, Y. W. Zheng, and B. S. Luk’yanchuk, “Laser writing of a subwavelength structure on silicon (100) surfaces with particle enhanced optical irradiation,” JETP Lett. 72(9), 457–459 (2000).
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Langmuir (1)

S. Peng, C. Xu, T. C. Hughes, and X. Zhang, “From nanodroplets by the ouzo effect to interfacial nanolenses,” Langmuir 30(41), 12270–12277 (2014).
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Laser Photonics Rev. (1)

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L. Li, W. Guo, Y. Yan, S. Lee, and T. Wang, “Label-free super-resolution imaging of adenoviruses by submerged microsphere optical nanoscopy,” Light Sci. Appl. 2(9), e104 (2013).
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Macromolecules (1)

H. Zhu, M. Chen, S. Zhou, and L. Wu, “Synthesis of High Refractive Index and Shape Controllable Colloidal Polymer Microspheres for Super-Resolution Imaging,” Macromolecules 50(2), 660–665 (2017).
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H. Yang and M. A. M. Gijs, “Optical microscopy using a glass microsphere for metrology of sub-wavelength nanostructures,” Microelectron. Eng. 143, 86–90 (2015).
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Nat. Mater. (2)

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
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Supplementary Material (2)

NameDescription
» Visualization 1: MP4 (7100 KB)      Supplementary Video showing nanojet formation in spheres
» Visualization 2: MP4 (6607 KB)      Supplementary Video showing nanojet formation in cylinders

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Figures (20)

Fig. 1
Fig. 1 The image of photonic jet for a particle with n=1.6 from the front page of the book [10]. Original publications related to similar jets one can find in the papers [3, 6–17].
Fig. 2
Fig. 2 Intensity E 2 distribution within the {xz} plane through the particle center calculated from the Mie theory for the particle with n=1.5 and different values of the size parameter q.
Fig. 3
Fig. 3 Left panels (a) show first three electric amplitudes: electric dipole | a 1 | 2 (ed), electric quadrupole | a 2 | 2 (eq), electric octupole | a 3 | 2 (eo) and corresponding first three magnetic amplitudes | b 1 | 2 (md), | b 2 | 2 (mq) and | b 3 | 2 (mo) for a spherical particle as a function of the size parameter for three different values of the index of refraction (indicated in the inset). The color panel on the right shows the contour plot of scattering efficiency Q sca on the {q,n} parameters plane, with the trajectories corresponding to the excitation of the different resonances indicated. The panel (b) shows the same as (a) but for a cylindrical particle under TE-polarized light illumination.
Fig. 4
Fig. 4 (a, e) Total scattering efficiency, Q sca , vs size parameter for refractive index n=2 sphere and cylinder. Six partial scattering efficiencies, corresponding to the electric dipole (ed), magnetic dipole (md), electric quadrupole (eq), magnetic quadrupole (md), electric octupole (eo) and magnetic octupole (mo) contributions. (b, f) Backscattering efficiency Q BS vs size parameter for three different values of refractive index n exhibiting pronounced minima at particular values of size parameter. (c, g) The trajectory of the first Kerker condition [60–62] corresponding to a 1 = b 1 on the plane of parameters ( n,q ) . For a spherical particle it corresponds to a single “canyon” on the Q BS ( n,q ) surface, see inset. For a cylindrical particle these minima are presented by multiple canyon surfaces. Along these trajectories on the bottom of canyons the polar scattering diagram shows preferentially forward scattering. (d, h) Forward scattering efficiency Q FS vs size parameter for three different values of refractive index n. Interference of electric dipole and magnetic dipole produces the asymmetrical shape in forward scattering.
Fig. 5
Fig. 5 (a) Intensity distribution along the line through the center of a particle. The particle has a refractive index n=1.5 and size parameter q=20 , corresponding to Fig. 4.32 in Ref. [65]. The inset shows the formation of a caustic under the approximation of geometrical optics. (b) Field enhancement vs refractive index for different size parameters. (c) Field distribution in the { z,y } cross section of the particle with n=1.515 and q=11 . One can see the formation of a whispering gallery mode for the spherical particle. (d) Field enhancement vs size parameter for particle with refractive index n=1.7 .
Fig. 6
Fig. 6 (a) Color panel representing (in logarithmic scale) the intensity E 2 under a spherical particle on the plane of {n,q} parameters. The distribution of intensity vs size parameter for a particle with refractive index n=2 is shown along the right axes of the picture. It clearly shows the excitation of three resonances q = 1.47, 2.09 and 2.71. The latest resonance yields intensity E 2 16 . (b) The Poynting vector field distribution for a particle with n=1.5 and size parameter q=10 . The colour contour plot represents the modulus | S | in logarithmic scale. There are a number of singular points (52 in total) where | S |=0 . An enlarged picture (zoomed) of the Poynting vector field is shown in plot (c) which represents the field lines and singularities of the Poynting vector distribution in the region containing the singular points marked as 2-15. These optical vortices are important in the process of formation of the whispering gallery mode.
Fig. 7
Fig. 7 Cross-sectional view of the normalized local field distribution | E | 2 underneath a single particle with 2R=5 μm immersed in (a) air and (b) water [38]. The incident laser beam has a wavelength λ=800 nm, is linearly polarized along x-axis and propagates along z-axis. The sphere is assumed to be made of quartz n p =1.45332 and an index n m =1.326 is assumed for water. The lower panel shows the corresponding distributions along the cut-line passing through the center of the sphere, together with the magnetic field intensity and the z-component S z of the Poynting vector.
Fig. 8
Fig. 8 (a) Position of focal point vs size parameter for different values of refractive index. (b) Distribution of intensity vs z-coordinate (as measured from the center of the particle) for a particle with refractive index n=1.5 and different size parameters. The dashed curve follows the intensity maxima (indicated by solid black points).
Fig. 9
Fig. 9 Field distribution of the Poynting vector for a TiO2 particle immersed in water. The right panel represents a zoomed view of the intensity distribution near the focal point.
Fig. 10
Fig. 10 Distributions of intensity in the {x-z} plane of (a) the electric field E 2 and (b) the magnetic field H 2 and (c) the modulus of the Poynting vector | S | for a spherical particle with n=1.5 and size parameter q=10 .
Fig. 11
Fig. 11 Electric field intensity patterns for a sphere (a) and cylinder (b) from the same material with refractive index n= 2 and size parameter q=30 , as seen in Ref. 17.
Fig. 12
Fig. 12 Electric field intensity distributions calculated for (left panel) a Solid Immersion Lens (SIL): height h=R( 1+1/n )<2R , (central panel) a sphere in a homogeneous environment and (right panel) a particle on top of a 40-nm-thick gold film [75]. In all cases the sphere has a radius R=2.37 μm and a refractive index n=1.46. The operating wavelength is λ=600 nm.
Fig. 13
Fig. 13 Dielectric structures of different shapes (top) and intensity distribution of the corresponding photonic nanojets (bottom). (a) A conical structure with n=1.6 , height =λ and radius of axicon base =0.5λ . (b) A trihedral prism with n=1.6 , height =0.5λ and basis =λ . (c) A hexagonal prism with n=1.6 . (d) A deformed cube with chirality, n=1.6 . (e) A cube, n=1.6 . Adapted from the papers [78, 79, 84, 85].
Fig. 14
Fig. 14 (a) The intensity I= S z distribution within the xz plane for radiation with λ = 248 nm, scattered by a polystyrene particle (n = 1.6, R = 0.5 μm) on a silicon surface (calculated using the full analytical solution [17]). The color scale of the intensity varies from negative (dark) to positive (light) values. The dark area on top of the particle corresponds to energy flux directed upwards, while the white area under the particle corresponds to the energy flux directed towards the substrate. (b) Photonic jet formed upon reflection of a plane (unfocused) wave from a flat screen with a square dielectric plate embedded in air (the photonic jet length in this example at the level of half the power is 15 λ of the incident light). (c) Same as in (b) but when the dielectric plate is embedded in water. The inset shows the corresponding SiO2 parallelepiped ( n=1.46 ) placed on the metal (gold) substrate. The thickness of the dielectric is h=1 mm and the side of the square is L=3.17 mm, the incident radiation has a wavelength of λ = 671 nm. Taken from [86]. (d) and (e) Angular reflected photonic jets. The incident wavelength is 532 nm and the supporting material is a silicon wafer. The focal spot of the laser beam is about 1 mm and the parallelepiped is made of SiO2 with lateral dimensions of 5x5 μ m 2 and height 1 μm (e).
Fig. 15
Fig. 15 (a) Ray tracing of virtual image formation in a solid immersion lens (SIL) with thickness h=R( 1+1/n ) . (b) Ray tracing of virtual image formation in a spherical particle. (c) Magnification of virtual image versus refractive index in SIL and spherical particle [75].
Fig. 16
Fig. 16 Schematic showing a white-light microsphere nanoscope (a microsphere superlens integrated with a classical widefield optical microscope) with λ/8 imaging resolution. The spheres collect the near-field information of the object and form virtual images that are then captured by the conventional lens. (a) to (c) show two examples of microsphere superlenses imaging in transmission mode. In (a)-(b) the object is a diffraction grating with 360-nm-wide lines spaced 130 nm apart. (a) shows an image taken by scanning electron microscope (SEM), while (b) shows the optical nanoscope (ON) image in which the lines are clearly resolved. The magnified image corresponds to a 4.17 X magnification factor. In (c)-(d) the object is gold-coated fishnet membrane sample imaged with a microsphere (size 4.7 μm); (c) shows an SEM image of the object and the microsphere while (d) shows the ON image. The size of the optical image between the pores within the image plane is 400 nm and corresponds to a magnification factor of approximately 8X. The borders of the sphere are shown by white lines. The ON image clearly resolves the pores that are 50 nm in diameter and spaced 50 nm apart.
Fig. 17
Fig. 17 (a) Scanning superlens attached to a AFM cantilever [168]. (b) Virtual image observed using the microsphere superlens [168]. The inset shows an SEM image of the imaged object [168]. (c, d) Backside and frontside images, respectively, of the AFM cantilever with the attached microsphere superlens. Scale bars correspond to 2 mm in (b) and to 50 mm in (c, d). (e) Swimming micro robot optical nanoscopy [153] (SMON) - schematics and particle scanning mechanism. (f-h) Superlensing objective lenses [169]. In (f) the BaTiO3 superlens was fabricated by encapsulating a monolayer of BaTiO3 microspheres (3-80 µm diameter) inside Polydimethylsiloxane (PDMS). In (g) the superlensing objective was made by integrating a conventional microscope objective lens (e.g. 50×, NA = 0.7, or 100×, NA = 0.95) with a BaTiO3 microsphere superlens using a 3D printed adapter. (c) Experimental configuration for super-resolution imaging using the developed objective fitted onto a standard white light optical microscope.
Fig. 18
Fig. 18 (a) Schematic description of the decoupling mechanism of high spatial frequency evanescent waves, with wavenumber k e , by the particle-on-substrate system in a microsphere nanoscope. (b) Calculated distribution of light intensity for a SiO 2 particle on a glass substrate. The particle size is 4.7 μm and the incident radiation wavelength λ = 600 nm. (c) Distribution of the Poynting vector in the same situation as (b). A number of vortices can be seen, similar to those shown in Fig. 6, associated to the whispering gallery mode excitation.
Fig. 19
Fig. 19 Typical |(E)|2 field distributions of (a-c) spheres and cylinders (d-f) with refractive index n = 1.5 at varying size parameter q with calculation step size q = 0.1. (a) super-resonance mode of sphere, (b) usual jet mode of sphere, (c) whispery gallery mode of sphere (d) usual jet mode of cylinder, (e) strong whispery gallery mode of cylinder (f) weakly excited whispery gallery mode of cylinder. See Visualization 1 and Visualization 2 for details.
Fig. 20
Fig. 20 (a) Schematic depiction of a metamaterial solid immersion lens fabricated from an assembly of TiO2 particles. (b) SEM image of the test structure, consisting of different shapes and 60 nm pitches. (c) Optical image of the test structure imaged through the metamaterial solid immersion lens using white light [142].

Equations (16)

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I max I 0 R 2 w 2 = 27 n 4 ( 4 n 2 ) 3 .
I max I 0 R w = 27 n 4 ( 4 n 2 ) 3 .
Q sca = 2 q 2 =1 ( 2+1 ) [ | a | 2 + | b | 2 ],
Q FS = 1 q 2 | =1 ( 2+1 ) ( a + b ) | 2 , Q BS = 1 q 2 | =1 ( 2+1 ) ( 1 ) ( a b ) | 2 ,
a = F ( a ) F ( a ) +i G ( a ) , b = F ( b ) F ( b ) +i G ( b ) ,
F ( a ) =n ψ ( q ) ψ ( nq ) ψ ( q ) ψ ( nq ), G ( a ) =n χ ( q ) ψ ( nq ) ψ ( nq ) χ ( q ),
F ( b ) =n ψ ( nq ) ψ ( q ) ψ ( nq ) ψ ( q ), G ( b ) =n χ ( q ) ψ ( nq ) ψ ( nq ) χ ( q ).
d = in F ( a ) +i G ( a ) , c = in F ( b ) +i G ( b ) .
Q ¯ sca = 2 q = | a ¯ | 2 , Q ¯ FS = 2 πq | f 0 | 2 , Q ¯ BS = 2 πq | f π | 2 ,
f 0 = = ( i ) e iπ /2 a ¯ , f π = = ( i ) ( 1 ) e iπ /2 a ¯ , a ¯ = +i ,
=n J ( nq ) J ( q ) J ( nq ) J ( q ), =n J ( nq ) N ( q ) J ( nq ) N ( q ).
d ¯ = 2 πq i +i .
Ψ( x,z )= 1 2π dsexp[ i( s 4 4 + s 2 2 z+sx ) ]
M SIL = n 2 M sphere =n/ ( 2n )
I( x,y )= O( u,v ) PSF( x M u, y M v )dudv,
d=R R 2 x 2 4 x= x p M ,

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