Abstract

We propose a scheme to extend the measuring range of a transverse displacement sensor by exploiting the interaction of an azimuthally polarized beam (APB) with a single metal-dielectric core-shell nanoparticle. The focused APB illumination induces a longitudinal magnetic dipole (MD) in the core-shell nanoparticle, which interferes with the induced transverse electric dipole (ED) to bring forth a transverse unidirectional scattering at a specific position within the focal plane. Emphatically, the rapidly varying electromagnetic field within the focal plane of an APB leads to a remarkable sensitivity of the far-field scattering directivity to nanoscale displacements as the nanoparticle moves away from the optical axis. Moreover, the scattering directivity of the APB illuminated core-shell nanoparticle is also a function of structure-dependent Mie scattering coefficients, rendering the measuring range of the transverse displacement sensor widely tunable. The culmination of all these features enables the continuous tuning of the displacement measuring range from several nanometers to a few micrometers. Thus, we envision the proposed scheme is of high value for modern optical nanometrology.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (6)

Y. Wang, Y. Lu, and P. Wang, “Nanoscale displacement sensing based on the interaction of a Gaussian beam with dielectric nano-dimer antennas,” Opt. Express 26(2), 1000–1011 (2018).
[Crossref] [PubMed]

W. Shang, F. Xiao, L. Han, M. Premaratne, T. Mei, and J. Zhao, “Enhanced second harmonic generation from a plasmonic Fano structure subjected to an azimuthally polarized light beam,” J. Phys. Condens. Matter 30(6), 064004 (2018).
[Crossref] [PubMed]

L. Han, S. Liu, P. Li, Y. Zhang, H. C. Cheng, and J. L. Zhao, “Catalystlike effect of orbital angular momentum on the conversion of transverse to three-dimensional spin states within tightly focused radially polarized beams,” Phys. Rev. A (Coll. Park) 97(5), 053802 (2018).
[Crossref]

F. Xiao, Y. Ren, W. Shang, W. Zhu, L. Han, H. Lu, T. Mei, M. Premaratne, and J. Zhao, “Sub-10 nm particle trapping enabled by a plasmonic dark mode,” Opt. Lett. 43(14), 3413–3416 (2018).
[Crossref] [PubMed]

G. Bautista, C. Dreser, X. Zang, D. P. Kern, M. Kauranen, and M. Fleischer, “Collective Effects in Second-Harmonic Generation from Plasmonic Oligomers,” Nano Lett. 18(4), 2571–2580 (2018).
[Crossref] [PubMed]

F. J. Xiao, W. Y. Shang, W. R. Zhu, L. Han, M. Premaratne, T. Mei, and J. L. Zhao, “Cylindrical vector beam-excited frequency-tunable second harmonic generation in a plasmonic octamer,” Photon. Res. 6(3), 157–161 (2018).
[Crossref]

2017 (6)

L. Wei, N. Bhattacharya, and H. Paul Urbach, “Adding a spin to Kerker’s condition: angular tuning of directional scattering with designed excitation,” Opt. Lett. 42(9), 1776–1779 (2017).
[Crossref] [PubMed]

J. Y. Lee, A. E. Miroshnichenko, and R.-K. Lee, “Reexamination of Kerker’s conditions by means of the phase diagram,” Phys. Rev. A (Coll. Park) 96(4), 043846 (2017).
[Crossref]

F. Balzarotti, Y. Eilers, K. C. Gwosch, A. H. Gynnå, V. Westphal, F. D. Stefani, J. Elf, and S. W. Hell, “Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes,” Science 355(6325), 606–612 (2017).
[Crossref] [PubMed]

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

U. Manna, J. H. Lee, T. S. Deng, J. Parker, N. Shepherd, Y. Weizmann, and N. F. Scherer, “Selective Induction of Optical Magnetism,” Nano Lett. 17(12), 7196–7206 (2017).
[Crossref] [PubMed]

E. V. Melik-Gaykazyan, S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, K. Zangeneh Kamali, A. Lamprianidis, A. E. Miroshnichenko, A. A. Fedyanin, D. N. Neshev, and Y. S. Kivshar, “Selective Third-Harmonic Generation by Structured Light in Mie-Resonant Nanoparticles,” ACS Photonics 5(3), 728–733 (2017).
[Crossref]

2016 (8)

Z. Xi, L. Wei, A. J. L. Adam, H. P. Urbach, and L. Du, “Accurate Feeding of Nanoantenna by Singular Optics for Nanoscale Translational and Rotational Displacement Sensing,” Phys. Rev. Lett. 117(11), 113903 (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]

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11(1), 23–36 (2016).
[Crossref] [PubMed]

Z. Xi, L. Wei, A. J. L. Adam, and H. P. Urbach, “Broadband active tuning of unidirectional scattering from nanoantenna using combined radially and azimuthally polarized beams,” Opt. Lett. 41(1), 33–36 (2016).
[Crossref] [PubMed]

M. Veysi, C. Guclu, and F. Capolino, “Focused azimuthally polarized vector beam and spatial magnetic resolution below the diffraction limit,” J. Opt. Soc. Am. B 33(11), 2265–2277 (2016).
[Crossref]

M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7(1), 11286 (2016).
[Crossref] [PubMed]

P. Li, Y. Zhang, S. Liu, C. Ma, L. Han, H. Cheng, and J. Zhao, “Generation of perfect vectorial vortex beams,” Opt. Lett. 41(10), 2205–2208 (2016).
[Crossref] [PubMed]

G. Bautista and M. Kauranen, “Vector-Field Nonlinear Microscopy of Nanostructures,” ACS Photonics 3(8), 1351–1370 (2016).
[Crossref]

2015 (7)

S. Roy, K. Ushakova, Q. van den Berg, S. F. Pereira, and H. P. Urbach, “Radially polarized light for detection and nanolocalization of dielectric particles on a planar substrate,” Phys. Rev. Lett. 114(10), 103903 (2015).
[Crossref] [PubMed]

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015).
[Crossref]

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 241110 (2015).
[Crossref]

P. Bon, N. Bourg, S. Lécart, S. Monneret, E. Fort, J. Wenger, and S. Lévêque-Fort, “Three-dimensional nanometre localization of nanoparticles to enhance super-resolution microscopy,” Nat. Commun. 6(1), 7764 (2015).
[Crossref] [PubMed]

D. Sikdar, W. Cheng, and M. Premaratne, “Optically resonant magneto-electric cubic nanoantennas for ultra-directional light scattering,” J. Appl. Phys. 117(8), 083101 (2015).
[Crossref]

S. Babar and J. H. Weaver, “Optical constants of Cu, Ag, and Au revisited,” Appl. Opt. 54(3), 477–481 (2015).
[Crossref]

R. Alaee, R. Filter, D. Lehr, F. Lederer, and C. Rockstuhl, “A generalized Kerker condition for highly directive nanoantennas,” Opt. Lett. 40(11), 2645–2648 (2015).
[Crossref] [PubMed]

2014 (2)

H. Deschout, F. Cella Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. T. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods 11(3), 253–266 (2014).
[Crossref] [PubMed]

M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs, “Polarization tailored light driven directional optical nanobeacon,” Nano Lett. 14(5), 2546–2551 (2014).
[Crossref] [PubMed]

2013 (3)

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
[Crossref] [PubMed]

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4(1), 1527 (2013).
[Crossref] [PubMed]

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7(9), 7824–7832 (2013).
[Crossref] [PubMed]

2012 (5)

A. E. Krasnok, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “All-dielectric optical nanoantennas,” Opt. Express 20(18), 20599–20604 (2012).
[Crossref] [PubMed]

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core-shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012).
[Crossref] [PubMed]

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3(1), 1171 (2012).
[Crossref] [PubMed]

J. Shamir, “Singular beams in metrology and nanotechnology,” Opt. Eng. 51(7), 073605 (2012).
[Crossref]

T. Shegai, P. Johansson, C. Langhammer, and M. Käll, “Directional scattering and hydrogen sensing by bimetallic Pd-Au nanoantennas,” Nano Lett. 12(5), 2464–2469 (2012).
[Crossref] [PubMed]

2011 (1)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[Crossref]

2010 (1)

T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi–Uda antenna,” Nat. Photonics 4(5), 312–315 (2010).
[Crossref]

2009 (3)

T. Pakizeh and M. Käll, “Unidirectional ultracompact optical nanoantennas,” Nano Lett. 9(6), 2343–2349 (2009).
[Crossref] [PubMed]

Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

O. Peña and U. Pal, “Scattering of electromagnetic radiation by a multilayered sphere,” Comput. Phys. Commun. 180(11), 2348–2354 (2009).
[Crossref]

2005 (2)

C. Sönnichsen, B. M. Reinhard, J. Liphardt, and A. P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23(6), 741–745 (2005).
[Crossref] [PubMed]

P. Mühlschlegel, H.-J. Eisler, O. J. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308(5728), 1607–1609 (2005).
[Crossref] [PubMed]

2000 (1)

1983 (1)

Adam, A. J. L.

Z. Xi, L. Wei, A. J. L. Adam, H. P. Urbach, and L. Du, “Accurate Feeding of Nanoantenna by Singular Optics for Nanoscale Translational and Rotational Displacement Sensing,” Phys. Rev. Lett. 117(11), 113903 (2016).
[Crossref] [PubMed]

Z. Xi, L. Wei, A. J. L. Adam, and H. P. Urbach, “Broadband active tuning of unidirectional scattering from nanoantenna using combined radially and azimuthally polarized beams,” Opt. Lett. 41(1), 33–36 (2016).
[Crossref] [PubMed]

Alaee, R.

Albella, P.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3(1), 1171 (2012).
[Crossref] [PubMed]

Alivisatos, A. P.

C. Sönnichsen, B. M. Reinhard, J. Liphardt, and A. P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23(6), 741–745 (2005).
[Crossref] [PubMed]

Babar, S.

Bag, A.

M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7(1), 11286 (2016).
[Crossref] [PubMed]

Balzarotti, F.

F. Balzarotti, Y. Eilers, K. C. Gwosch, A. H. Gynnå, V. Westphal, F. D. Stefani, J. Elf, and S. W. Hell, “Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes,” Science 355(6325), 606–612 (2017).
[Crossref] [PubMed]

Banzer, P.

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G. Bautista, C. Dreser, X. Zang, D. P. Kern, M. Kauranen, and M. Fleischer, “Collective Effects in Second-Harmonic Generation from Plasmonic Oligomers,” Nano Lett. 18(4), 2571–2580 (2018).
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P. Mühlschlegel, H.-J. Eisler, O. J. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308(5728), 1607–1609 (2005).
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E. V. Melik-Gaykazyan, S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, K. Zangeneh Kamali, A. Lamprianidis, A. E. Miroshnichenko, A. A. Fedyanin, D. N. Neshev, and Y. S. Kivshar, “Selective Third-Harmonic Generation by Structured Light in Mie-Resonant Nanoparticles,” ACS Photonics 5(3), 728–733 (2017).
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Fleischer, M.

G. Bautista, C. Dreser, X. Zang, D. P. Kern, M. Kauranen, and M. Fleischer, “Collective Effects in Second-Harmonic Generation from Plasmonic Oligomers,” Nano Lett. 18(4), 2571–2580 (2018).
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I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7(9), 7824–7832 (2013).
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P. Bon, N. Bourg, S. Lécart, S. Monneret, E. Fort, J. Wenger, and S. Lévêque-Fort, “Three-dimensional nanometre localization of nanoparticles to enhance super-resolution microscopy,” Nat. Commun. 6(1), 7764 (2015).
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I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7(9), 7824–7832 (2013).
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J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3(1), 1171 (2012).
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Gwosch, K. C.

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F. Balzarotti, Y. Eilers, K. C. Gwosch, A. H. Gynnå, V. Westphal, F. D. Stefani, J. Elf, and S. W. Hell, “Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes,” Science 355(6325), 606–612 (2017).
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F. Balzarotti, Y. Eilers, K. C. Gwosch, A. H. Gynnå, V. Westphal, F. D. Stefani, J. Elf, and S. W. Hell, “Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes,” Science 355(6325), 606–612 (2017).
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H. Deschout, F. Cella Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. T. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods 11(3), 253–266 (2014).
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T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi–Uda antenna,” Nat. Photonics 4(5), 312–315 (2010).
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T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 241110 (2015).
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T. Shegai, P. Johansson, C. Langhammer, and M. Käll, “Directional scattering and hydrogen sensing by bimetallic Pd-Au nanoantennas,” Nano Lett. 12(5), 2464–2469 (2012).
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G. Bautista, C. Dreser, X. Zang, D. P. Kern, M. Kauranen, and M. Fleischer, “Collective Effects in Second-Harmonic Generation from Plasmonic Oligomers,” Nano Lett. 18(4), 2571–2580 (2018).
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E. V. Melik-Gaykazyan, S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, K. Zangeneh Kamali, A. Lamprianidis, A. E. Miroshnichenko, A. A. Fedyanin, D. N. Neshev, and Y. S. Kivshar, “Selective Third-Harmonic Generation by Structured Light in Mie-Resonant Nanoparticles,” ACS Photonics 5(3), 728–733 (2017).
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E. V. Melik-Gaykazyan, S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, K. Zangeneh Kamali, A. Lamprianidis, A. E. Miroshnichenko, A. A. Fedyanin, D. N. Neshev, and Y. S. Kivshar, “Selective Third-Harmonic Generation by Structured Light in Mie-Resonant Nanoparticles,” ACS Photonics 5(3), 728–733 (2017).
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T. Shegai, P. Johansson, C. Langhammer, and M. Käll, “Directional scattering and hydrogen sensing by bimetallic Pd-Au nanoantennas,” Nano Lett. 12(5), 2464–2469 (2012).
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S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
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P. Bon, N. Bourg, S. Lécart, S. Monneret, E. Fort, J. Wenger, and S. Lévêque-Fort, “Three-dimensional nanometre localization of nanoparticles to enhance super-resolution microscopy,” Nat. Commun. 6(1), 7764 (2015).
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U. Manna, J. H. Lee, T. S. Deng, J. Parker, N. Shepherd, Y. Weizmann, and N. F. Scherer, “Selective Induction of Optical Magnetism,” Nano Lett. 17(12), 7196–7206 (2017).
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Leuchs, G.

M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7(1), 11286 (2016).
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M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs, “Polarization tailored light driven directional optical nanobeacon,” Nano Lett. 14(5), 2546–2551 (2014).
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P. Bon, N. Bourg, S. Lécart, S. Monneret, E. Fort, J. Wenger, and S. Lévêque-Fort, “Three-dimensional nanometre localization of nanoparticles to enhance super-resolution microscopy,” Nat. Commun. 6(1), 7764 (2015).
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L. Han, S. Liu, P. Li, Y. Zhang, H. C. Cheng, and J. L. Zhao, “Catalystlike effect of orbital angular momentum on the conversion of transverse to three-dimensional spin states within tightly focused radially polarized beams,” Phys. Rev. A (Coll. Park) 97(5), 053802 (2018).
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P. Li, Y. Zhang, S. Liu, C. Ma, L. Han, H. Cheng, and J. Zhao, “Generation of perfect vectorial vortex beams,” Opt. Lett. 41(10), 2205–2208 (2016).
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Figures (6)

Fig. 1
Fig. 1 (a) Schematic diagram of light scattered by a spherical nanoparticle. (b) Cross-sectional view of the studied spherical core-shell nanoparticle with inner radius of Rin and outer radius of Rout.
Fig. 2
Fig. 2 (a) Transverse electric field intensity |E|2 = |Ex|2 + |Ey|2, (b) transverse magnetic field intensity |ZH|2 = |ZHx|2 + |ZHy|2 and (c) longitudinal magnetic field intensity |ZHz|2 distributions at the focal plane of a focused APB (NA = 0.3, λ = 1.55 μm), which are all normalized to the maximum value of total field intensity Itotal = |(E)|2 + |Z(H)|2. The insets are their corresponding phase distributions. (d) Normalized amplitudes of Ey, ZHz and ZHx, as well as |Ey|/|ZHz|, along the x axis.
Fig. 3
Fig. 3 (a) Amplitudes of a1 and b1, the ratio (|b1|/|a1|) and (b) phase difference ( ϕ b 1 ϕ a 1 ) between b1 and a1 of a core-shell nanoparticle with Rin = 106 nm and Rout = 206 nm.
Fig. 4
Fig. 4 (a) Scattering efficiency spectra (total and the contributions from py, mx and mz), (b) electric near-field intensity distribution at the resonance wavelength of 1.263 μm in xy cut plane and corresponding (c) far-field radiation pattern (3D radiation pattern and radiation pattern in xz plane) of a core-shell nanoparticle (Rin = 106 nm, Rout = 206 nm) under an APB illumination when the particle located at the optical axis (x = 0 nm). (d) Scattering efficiency spectra, as well as (e) electric near-field intensity distribution and (f) far-field radiation pattern at the wavelength of 1.55 μm for the nanoparticle located at the position of x = 73 nm on the x axis. The insets in (a) and (d) are the schematic views of the induced ED (p) and MD (m) modes. The values of σd for outer circle of polar plots in (c) and (f) are shown in black on the side, as is the case for other figures below. The numerically calculated results based on FDTD method are given in (a), (c), (d) and (f) as red dots.
Fig. 5
Fig. 5 (a) Amplitudes of a1 and b1, the ratio (|b1|/|a1|) and (b) phase difference ( ϕ b 1 ϕ a 1 ) between b1 and a1 of a core-shell nanoparticle with Rin = 113 nm and Rout = 206 nm. Changes of APB excited (c) far-field radiation patterns and corresponding (d) LRR as the nanoparticle moved from the optical axis to a displacement of 50 nm along the x axis at the wavelength of 1.55 μm. (e) LRR map of the focused APB illuminated nanoparticle as the function of wavelength and lateral displacement (Δx). (f) LRRs change with respect to lateral displacement for the wavelength of 1.55 μm, 1.523 μm, 1.518 μm and 1.514 μm. The corresponding measuring ranges are 50 nm, 55 nm, 60 nm and 65 nm. The numerically calculated results based on FDTD method are given in (c), (d) and (f) as hollow and solid dots.
Fig. 6
Fig. 6 (a) Amplitude ratio (|b1|/|a1|) and phase difference ( ϕ b 1 ϕ a 1 ) between b1 and a1 for a core-shell nanoparticle with Rin = 78.5 nm and Rout = 220.5 nm. (b) Changes of far-field radiation pattern as the nanoparticle moved from the optical axis to a displacement of 500 nm along the x axis within the APB focal plane, where the radiation for the displacements of 200 nm, 300 nm, 400 nm and 500 nm are multiplied by 1.1, 1.3, 1.6 and 1.9 for clear comparison. (c) LRR map of an APB illuminated (NA = 0.3, λ = 1.55 μm) core-shell nanoparticle as a function of |b1|/|a1| and lateral displacement when ϕ b 1 ϕ a 1 = −π/2. (d) LRRs and (e) leftward differential scattering cross sections (σd, l ) change with respect to lateral displacement for |b1|/|a1| (as well as ϕ b 1 ϕ a 1 ) with the values of 0.022 (−1.384 rad), 0.114 (−1.403 rad), 0.631 (−1.401 rad), 1.145 (−1.388 rad), 1.941 (−1.4 rad) and 2.776 (−1.385 rad). The corresponding inner and outer radii (Rin, Rout) of the nanoparticles are (176 nm, 244 nm), (113 nm, 206 nm), (79 nm, 215 nm), (78.5 nm, 220.5 nm), (80.5 nm, 224.5 nm) and (82.5 nm, 227 nm), with the measuring ranges tuned from 10 nm, 50 nm, 280 nm, 500 nm, 800 nm to 1.1 μm. The numerically calculated results based on FDTD method are given in (b) and (d) as hollow dots.

Equations (9)

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E s c a ( r ) = 1 4 π ε 0 k 2 e i k r r [ ( i r × p ) × i r i r × m / c ] ,
E s c a ( r ) = 1 4 π ε 0 k 2 e i k r r [ p y ( cos θ sin φ i θ + cos φ i φ ) m x c ( sin φ i θ + cos θ cos φ i φ ) + m z c sin θ i φ ] ,
σ d ( θ , φ ) = r 2 | E s c a | 2 I t o t a l = 9 4 k 2 I t o t a l ( | a 1 E y cos θ sin φ Z b 1 H x sin φ | 2 + | a 1 E y cos φ Z b 1 H x cos θ cos φ + Z b 1 H z sin θ | 2 ) .
σ d , l = σ d ( π 2 , π ) = 9 4 k 2 I t o t a l | a 1 E y Z b 1 H z | 2 , σ d , r = σ d ( π 2 , 0 ) = 9 4 k 2 I t o t a l | a 1 E y + Z b 1 H z | 2 .
| E y | | Z H z | = | b 1 | | a 1 | , ϕ E y ϕ Z H z = ϕ b 1 ϕ a 1 ,
| E y | | Z H z | = | b 1 | | a 1 | , ( ϕ E y ϕ Z H z ) ( ϕ b 1 ϕ a 1 ) = π .
Q s c a = σ s c a π R o u t 2 ,
P s c a = 3 π k ω μ [ | a 1 | 2 | E i n c ( r p ) | 2 + | Z b 1 | 2 | H i n c ( r p ) | 2 ] ,
LRR = 10 log 10 ( | E y Z H z b 1 a 1 | 2 / | E y Z H z + b 1 a 1 | 2 ) ,

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