Abstract

The interference between electric and magnetic dipolar fields is known to lead to asymmetric angular distributions of the scattered intensity from small high refractive index (HRI) particles. Properly designed all-dielectric metasurfaces based on HRI spheres have been shown to exhibit zero reflectivity, a generalized Brewster’s effect, potentially for any angle, wavelength and polarization of choice. At normal incidence, the effect is related to the absence of backscattering from small dielectric spheres or disks at the, so-called, first Kerker condition. In contrast, homogeneous HRI cylinders do not fulfil the first Kerker condition due to the mismatch between the local electric and magnetic density of states. In this work, we show that although a zero back-scattering condition can never be achieved for individual cylinders, when they are arranged in a periodic array their mutual interaction leads to an anomalous Kerker condition, leading to a generalized Brewster’s effect in a nanorod-based metasurface. We derive a coupled electric and magnetic dipole (CEMD) analytical formulation to describe the properties of a periodic array of HRI nanorods in full agreement with exact numerical calculations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

W. Liu and A. E. Miroshnichenko, “Beam steering with dielectric metalattices,” ACS Photonics 5, 1733 (2018).
[Crossref]

Y. H. Ko and R. Magnusson, “Wideband dielectric metamaterial reflectors: Mie scattering or leaky bloch mode resonance?” Optica 5, 289–294 (2018).
[Crossref]

2017 (6)

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

P. Lalanne and P. Chavel, “Metalenses at visible wavelengths: past, present, perspectives,” Laser Photonics Rev. 111600295 (2017).
[Crossref]

W. Liu, “Generalized Magnetic Mirrors,” Phys. Rev. Lett. 119, 123902 (2017).
[Crossref]

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
[Crossref]

J. Cambiasso, G. Grinblat, Y. Li, A. Rakovich, E. Cortés, and S. A. Maier, “Bridging the Gap between Dielectric Nanophotonics and the Visible Regime with Effectively Lossless Gallium Phosphide Antennas,” Nano Lett. 17, 1219–1225 (2017).
[Crossref]

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95, 165426 (2017).
[Crossref]

2016 (6)

G. Grinblat, Y. Li, M. P. Nielsen, R. F. Oulton, and S. A. Maier, “Enhanced third harmonic generation in single germanium nanodisks excited at the anapole mode,” Nano Lett. 16, 4635–4640 (2016).
[Crossref]

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, 846 (2016).
[Crossref]

D. Sell, J. Yang, S. Doshay, K. Zhang, and J. A. Fan, “Visible Light Metasurfaces Based on Single-Crystal Silicon,” ACS Photonics 3, 1919–1925 (2016).
[Crossref]

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun.  7, 10362 (2016).
[Crossref] [PubMed]

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

2015 (4)

P. Moitra, B. A. Slovick, W. Li, I. I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-Scale All-Dielectric Metamaterial Perfect Reflectors,” ACS Photonics 2, 692–698 (2015).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

R. Paniagua-Domínguez, L. S. Froufe-Pérez, J. J. Sáenz, and J. A. Sánchez-Gil, “Localized magnetic plasmons in all-dielectric µ<0 metastructures,” Phys. Rev. B 91, 235120 (2015).
[Crossref]

M. Caldarola, P. Albella, E. Cortés, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun.  6, 7915 (2015).
[Crossref] [PubMed]

2014 (5)

M. L. Brongersma, Y. Cui, and S. Fan, “Light management for photovoltaics using high-index nanostructures,” Nat. Mater. 13, 451–460 (2014).
[Crossref]

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

P. Albella, R. Alcaraz de la Osa, F. Moreno, and S. A. Maier, “Electric and Magnetic Field Enhancement with Ultralow Heat Radiation Dielectric Nanoantennas: Considerations for Surface-Enhanced Spectroscopies,” ACS Photonics 1, 524–529 (2014).
[Crossref]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric Meta-Reflectarray for Broadband Linear Polarization Conversion and Optical Vortex Generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

H. Marinchio, R. Carminati, A. García-Martín, and J. J. Sáenz, “Magneto-optical Kerr effect in resonant subwavelength nanowire gratings,” New J. Phys.  16, 015007 (2014).
[Crossref]

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, 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, 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 Na 7, 7824–7832 (2013).
[Crossref]

2012 (8)

B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express 20, 1473–1478 (2012).
[Crossref]

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas - a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 930–933 (2012).

J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
[Crossref] [PubMed]

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep.  2, 492 (2012).
[Crossref] [PubMed]

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

C. J. Chang-Hasnain and W. Yang, “High–contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54, 10–35 (2012).
[Crossref]

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, and S. Collin, “Optical Extinction in a Single Layer of Nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
[Crossref] [PubMed]

2011 (5)

I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83, 245102 (2011).
[Crossref]

R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, and J. A. Sánchez-Gil, “Metallo-dielectric core-shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials,” New J. Phys.  13, 123017 (2011).
[Crossref]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815 (2011).
[Crossref] [PubMed]

M. Nieto-Vesperinas, R. Gomez-Medina, and J. Saenz, “Angle-suppressed scattering and optical forces on submi-crometer dielectric particles,” J. Opt. Soc. Am. A 28, 54–60 (2011).
[Crossref]

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

2010 (3)

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[Crossref]

S. Albaladejo, R. Gómez-Medina, L. Froufe-Pérez, H. Marinchio, R. Carminati, J. Torrado, G. Armelles, A. García-Martín, and J. J. Sáenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle,” Opt. Express 18, 3556–3567 (2010).
[Crossref] [PubMed]

G. Brönstrup, N. Jahr, C. Leiterer, A. Csáki, W. Fritzsche, and S. Christiansen, “Optical properties of individual silicon nanowires for photonic devices,” ACS Nano 4, 7113–7122 (2010).
[Crossref] [PubMed]

2009 (1)

K. Vynck, D. Felbacq, E. Centeno, A. Căbuz, D. Cassagne, and B. Guizal, “All-Dielectric Rod-Type Metamaterials at Optical Frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[Crossref]

2006 (2)

R. Gómez-Medina, M. Laroche, and J. J. Sáenz, “Extraordinary optical reflection from sub-wavelength cylinder arrays,” Opt. Express 14, 3730–3737 (2006).
[Crossref] [PubMed]

M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Sáenz, “Tuning the optical response of nanocylinder arrays: An analytical study,” Phys. Rev. B 74, 245422 (2006).
[Crossref]

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P. Belov, S. Maslovski, K. Simovski, and S. Tretyakov, “A condition imposed on the electromagnetic polarizability of a bianisotropic lossless scatterer,” Tech. Phys. Lett. 29, 718–720 (2003).
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1997 (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
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1967 (1)

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
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1965 (1)

1952 (1)

V. Twersky, “On a multiple scattering theory of the finite grating and the Wood anomalies,” J. Appl. Phys. 23, 1099–1118 (1952).
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P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
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F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
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Aizpurua, J.

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas - a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 930–933 (2012).

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815 (2011).
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Albella, P.

M. Caldarola, P. Albella, E. Cortés, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun.  6, 7915 (2015).
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P. Albella, R. Alcaraz de la Osa, F. Moreno, and S. A. Maier, “Electric and Magnetic Field Enhancement with Ultralow Heat Radiation Dielectric Nanoantennas: Considerations for Surface-Enhanced Spectroscopies,” ACS Photonics 1, 524–529 (2014).
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J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
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Alcaraz de la Osa, R.

P. Albella, R. Alcaraz de la Osa, F. Moreno, and S. A. Maier, “Electric and Magnetic Field Enhancement with Ultralow Heat Radiation Dielectric Nanoantennas: Considerations for Surface-Enhanced Spectroscopies,” ACS Photonics 1, 524–529 (2014).
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Armelles, G.

Bardou, N.

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, and S. Collin, “Optical Extinction in a Single Layer of Nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
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Belov, P.

P. Belov, S. Maslovski, K. Simovski, and S. Tretyakov, “A condition imposed on the electromagnetic polarizability of a bianisotropic lossless scatterer,” Tech. Phys. Lett. 29, 718–720 (2003).
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C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, N.Y., 1998).
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B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express 20, 1473–1478 (2012).
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C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54, 10–35 (2012).
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Bozhevolnyi, S. I.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region,” Nano Lett. 12, 3749–3755 (2012).
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Bragas, A. V.

M. Caldarola, P. Albella, E. Cortés, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun.  6, 7915 (2015).
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Brener, I.

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
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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 Na 7, 7824–7832 (2013).
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P. Moitra, B. A. Slovick, W. Li, I. I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-Scale All-Dielectric Metamaterial Perfect Reflectors,” ACS Photonics 2, 692–698 (2015).
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Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric Meta-Reflectarray for Broadband Linear Polarization Conversion and Optical Vortex Generation,” Nano Lett. 14, 1394–1399 (2014).
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A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, 846 (2016).
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M. L. Brongersma, Y. Cui, and S. Fan, “Light management for photovoltaics using high-index nanostructures,” Nat. Mater. 13, 451–460 (2014).
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Brönstrup, G.

G. Brönstrup, N. Jahr, C. Leiterer, A. Csáki, W. Fritzsche, and S. Christiansen, “Optical properties of individual silicon nanowires for photonic devices,” ACS Nano 4, 7113–7122 (2010).
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Cabuz, A.

K. Vynck, D. Felbacq, E. Centeno, A. Căbuz, D. Cassagne, and B. Guizal, “All-Dielectric Rod-Type Metamaterials at Optical Frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
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Caldarola, M.

M. Caldarola, P. Albella, E. Cortés, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun.  6, 7915 (2015).
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Cambiasso, J.

J. Cambiasso, G. Grinblat, Y. Li, A. Rakovich, E. Cortés, and S. A. Maier, “Bridging the Gap between Dielectric Nanophotonics and the Visible Regime with Effectively Lossless Gallium Phosphide Antennas,” Nano Lett. 17, 1219–1225 (2017).
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Capasso, F.

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
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M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
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F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
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Carminati, R.

Cassagne, D.

K. Vynck, D. Felbacq, E. Centeno, A. Căbuz, D. Cassagne, and B. Guizal, “All-Dielectric Rod-Type Metamaterials at Optical Frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
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Centeno, E.

K. Vynck, D. Felbacq, E. Centeno, A. Căbuz, D. Cassagne, and B. Guizal, “All-Dielectric Rod-Type Metamaterials at Optical Frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
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Chang-Hasnain, C. J.

C. J. Chang-Hasnain and W. Yang, “High–contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
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Chantada, L.

Chavel, P.

P. Lalanne and P. Chavel, “Metalenses at visible wavelengths: past, present, perspectives,” Laser Photonics Rev. 111600295 (2017).
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M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
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Chichkov, B. N.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region,” Nano Lett. 12, 3749–3755 (2012).
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A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
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Christiansen, S.

G. Brönstrup, N. Jahr, C. Leiterer, A. Csáki, W. Fritzsche, and S. Christiansen, “Optical properties of individual silicon nanowires for photonic devices,” ACS Nano 4, 7113–7122 (2010).
[Crossref] [PubMed]

Collin, S.

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, and S. Collin, “Optical Extinction in a Single Layer of Nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
[Crossref] [PubMed]

Cortés, E.

J. Cambiasso, G. Grinblat, Y. Li, A. Rakovich, E. Cortés, and S. A. Maier, “Bridging the Gap between Dielectric Nanophotonics and the Visible Regime with Effectively Lossless Gallium Phosphide Antennas,” Nano Lett. 17, 1219–1225 (2017).
[Crossref]

M. Caldarola, P. Albella, E. Cortés, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun.  6, 7915 (2015).
[Crossref] [PubMed]

Csáki, A.

G. Brönstrup, N. Jahr, C. Leiterer, A. Csáki, W. Fritzsche, and S. Christiansen, “Optical properties of individual silicon nanowires for photonic devices,” ACS Nano 4, 7113–7122 (2010).
[Crossref] [PubMed]

Cui, Y.

M. L. Brongersma, Y. Cui, and S. Fan, “Light management for photovoltaics using high-index nanostructures,” Nat. Mater. 13, 451–460 (2014).
[Crossref]

Decker, M.

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

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 Na 7, 7824–7832 (2013).
[Crossref]

Devlin, R.

Devlin, R. C.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Dominguez, J.

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 Na 7, 7824–7832 (2013).
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Doshay, S.

D. Sell, J. Yang, S. Doshay, K. Zhang, and J. A. Fan, “Visible Light Metasurfaces Based on Single-Crystal Silicon,” ACS Photonics 3, 1919–1925 (2016).
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Eriksen, R. L.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

Esteban, R.

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas - a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 930–933 (2012).

Evlyukhin, A. B.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[Crossref]

Eyraud, C.

J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
[Crossref] [PubMed]

Ezhov, A. A.

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

Fan, J. A.

D. Sell, J. Yang, S. Doshay, K. Zhang, and J. A. Fan, “Visible Light Metasurfaces Based on Single-Crystal Silicon,” ACS Photonics 3, 1919–1925 (2016).
[Crossref]

Fan, S.

M. L. Brongersma, Y. Cui, and S. Fan, “Light management for photovoltaics using high-index nanostructures,” Nat. Mater. 13, 451–460 (2014).
[Crossref]

Fang, H.

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
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Fedyanin, A. A.

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

Felbacq, D.

K. Vynck, D. Felbacq, E. Centeno, A. Căbuz, D. Cassagne, and B. Guizal, “All-Dielectric Rod-Type Metamaterials at Optical Frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[Crossref]

Fofang, N. T.

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 Na 7, 7824–7832 (2013).
[Crossref]

Friesem, A. A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

Fritzsche, W.

G. Brönstrup, N. Jahr, C. Leiterer, A. Csáki, W. Fritzsche, and S. Christiansen, “Optical properties of individual silicon nanowires for photonic devices,” ACS Nano 4, 7113–7122 (2010).
[Crossref] [PubMed]

Froufe-Pérez, L.

J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
[Crossref] [PubMed]

S. Albaladejo, R. Gómez-Medina, L. Froufe-Pérez, H. Marinchio, R. Carminati, J. Torrado, G. Armelles, A. García-Martín, and J. J. Sáenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle,” Opt. Express 18, 3556–3567 (2010).
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Froufe-Pérez, L. S.

Fu, Y. H.

R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun.  7, 10362 (2016).
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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, 1527 (2013).
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A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep.  2, 492 (2012).
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Garcia-Camara, B.

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

García-Cámara, B.

J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
[Crossref] [PubMed]

García-Etxarri, A.

García-Martín, A.

Geffrin, J.

J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
[Crossref] [PubMed]

Genevet, P.

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

Ghenuche, P.

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, and S. Collin, “Optical Extinction in a Single Layer of Nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
[Crossref] [PubMed]

Gomez-Medina, R.

M. Nieto-Vesperinas, R. Gomez-Medina, and J. Saenz, “Angle-suppressed scattering and optical forces on submi-crometer dielectric particles,” J. Opt. Soc. Am. A 28, 54–60 (2011).
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R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

Gómez-Medina, R.

Gonzaga, L.

R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun.  7, 10362 (2016).
[Crossref] [PubMed]

Gonzales, E.

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 Na 7, 7824–7832 (2013).
[Crossref]

González, F.

J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
[Crossref] [PubMed]

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

Gordon, J. A.

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54, 10–35 (2012).
[Crossref]

Grinblat, G.

J. Cambiasso, G. Grinblat, Y. Li, A. Rakovich, E. Cortés, and S. A. Maier, “Bridging the Gap between Dielectric Nanophotonics and the Visible Regime with Effectively Lossless Gallium Phosphide Antennas,” Nano Lett. 17, 1219–1225 (2017).
[Crossref]

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[Crossref]

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, 1527 (2013).
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A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep.  2, 492 (2012).
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[Crossref]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric Meta-Reflectarray for Broadband Linear Polarization Conversion and Optical Vortex Generation,” Nano Lett. 14, 1394–1399 (2014).
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[Crossref]

Odit, M.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95, 165426 (2017).
[Crossref]

Oh, J.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Oliner, A. A.

Oulton, R. F.

G. Grinblat, Y. Li, M. P. Nielsen, R. F. Oulton, and S. A. Maier, “Enhanced third harmonic generation in single germanium nanodisks excited at the anapole mode,” Nano Lett. 16, 4635–4640 (2016).
[Crossref]

M. Caldarola, P. Albella, E. Cortés, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun.  6, 7915 (2015).
[Crossref] [PubMed]

Paniagua-Domínguez, R.

R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun.  7, 10362 (2016).
[Crossref] [PubMed]

R. Paniagua-Domínguez, L. S. Froufe-Pérez, J. J. Sáenz, and J. A. Sánchez-Gil, “Localized magnetic plasmons in all-dielectric µ<0 metastructures,” Phys. Rev. B 91, 235120 (2015).
[Crossref]

R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, and J. A. Sánchez-Gil, “Metallo-dielectric core-shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials,” New J. Phys.  13, 123017 (2011).
[Crossref]

Pelouard, J.-L.

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, and S. Collin, “Optical Extinction in a Single Layer of Nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
[Crossref] [PubMed]

Person, S.

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, 1806–1809 (2013).
[Crossref] [PubMed]

Qiu, M.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95, 165426 (2017).
[Crossref]

Rahmani, M.

M. Caldarola, P. Albella, E. Cortés, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun.  6, 7915 (2015).
[Crossref] [PubMed]

Rakovich, A.

J. Cambiasso, G. Grinblat, Y. Li, A. Rakovich, E. Cortés, and S. A. Maier, “Bridging the Gap between Dielectric Nanophotonics and the Visible Regime with Effectively Lossless Gallium Phosphide Antennas,” Nano Lett. 17, 1219–1225 (2017).
[Crossref]

Reardon, C. P.

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

Reinhardt, C.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[Crossref]

Rolly, B.

B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express 20, 1473–1478 (2012).
[Crossref]

Roschuk, T.

M. Caldarola, P. Albella, E. Cortés, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun.  6, 7915 (2015).
[Crossref] [PubMed]

Rosenblatt, D.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

Saenz, J.

Sáenz, J.

J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
[Crossref] [PubMed]

Sáenz, J. J.

R. Paniagua-Domínguez, L. S. Froufe-Pérez, J. J. Sáenz, and J. A. Sánchez-Gil, “Localized magnetic plasmons in all-dielectric µ<0 metastructures,” Phys. Rev. B 91, 235120 (2015).
[Crossref]

H. Marinchio, R. Carminati, A. García-Martín, and J. J. Sáenz, “Magneto-optical Kerr effect in resonant subwavelength nanowire gratings,” New J. Phys.  16, 015007 (2014).
[Crossref]

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, 1806–1809 (2013).
[Crossref] [PubMed]

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas - a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 930–933 (2012).

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815 (2011).
[Crossref] [PubMed]

S. Albaladejo, R. Gómez-Medina, L. Froufe-Pérez, H. Marinchio, R. Carminati, J. Torrado, G. Armelles, A. García-Martín, and J. J. Sáenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle,” Opt. Express 18, 3556–3567 (2010).
[Crossref] [PubMed]

R. Gómez-Medina, M. Laroche, and J. J. Sáenz, “Extraordinary optical reflection from sub-wavelength cylinder arrays,” Opt. Express 14, 3730–3737 (2006).
[Crossref] [PubMed]

M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Sáenz, “Tuning the optical response of nanocylinder arrays: An analytical study,” Phys. Rev. B 74, 245422 (2006).
[Crossref]

Sánchez-Gil, J. A.

R. Paniagua-Domínguez, L. S. Froufe-Pérez, J. J. Sáenz, and J. A. Sánchez-Gil, “Localized magnetic plasmons in all-dielectric µ<0 metastructures,” Phys. Rev. B 91, 235120 (2015).
[Crossref]

R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, and J. A. Sánchez-Gil, “Metallo-dielectric core-shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials,” New J. Phys.  13, 123017 (2011).
[Crossref]

Scheffold, F.

Schmidt, M. K.

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas - a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 930–933 (2012).

Seidel, A.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[Crossref]

Sell, D.

D. Sell, J. Yang, S. Doshay, K. Zhang, and J. A. Fan, “Visible Light Metasurfaces Based on Single-Crystal Silicon,” ACS Photonics 3, 1919–1925 (2016).
[Crossref]

Sersic, I.

I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83, 245102 (2011).
[Crossref]

Sharon, A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

Shcherbakov, M. R.

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

Shorokhov, A. S.

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

Simovski, K.

P. Belov, S. Maslovski, K. Simovski, and S. Tretyakov, “A condition imposed on the electromagnetic polarizability of a bianisotropic lossless scatterer,” Tech. Phys. Lett. 29, 718–720 (2003).
[Crossref]

Slovick, B. A.

P. Moitra, B. A. Slovick, W. Li, I. I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-Scale All-Dielectric Metamaterial Perfect Reflectors,” ACS Photonics 2, 692–698 (2015).
[Crossref]

Smith, D. R.

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54, 10–35 (2012).
[Crossref]

Staude, I.

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

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 Na 7, 7824–7832 (2013).
[Crossref]

Stellinga, D.

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

Stout, B.

B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express 20, 1473–1478 (2012).
[Crossref]

Su, R.

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

Suárez-Lacalle, I.

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas - a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 930–933 (2012).

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

Toh, Y. T.

R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun.  7, 10362 (2016).
[Crossref] [PubMed]

Torrado, J.

Tretyakov, S.

P. Belov, S. Maslovski, K. Simovski, and S. Tretyakov, “A condition imposed on the electromagnetic polarizability of a bianisotropic lossless scatterer,” Tech. Phys. Lett. 29, 718–720 (2003).
[Crossref]

Tuambilangana, C.

I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83, 245102 (2011).
[Crossref]

Twersky, V.

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
[Crossref]

V. Twersky, “On a multiple scattering theory of the finite grating and the Wood anomalies,” J. Appl. Phys. 23, 1099–1118 (1952).
[Crossref]

Vaillon, R.

J. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, 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, 1171 (2012).
[Crossref] [PubMed]

Valentine, J.

P. Moitra, B. A. Slovick, W. Li, I. I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-Scale All-Dielectric Metamaterial Perfect Reflectors,” ACS Photonics 2, 692–698 (2015).
[Crossref]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric Meta-Reflectarray for Broadband Linear Polarization Conversion and Optical Vortex Generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Valuckas, V.

R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun.  7, 10362 (2016).
[Crossref] [PubMed]

Vincent, G.

P. Ghenuche, G. Vincent, M. Laroche, N. Bardou, R. Haïdar, J.-L. Pelouard, and S. Collin, “Optical Extinction in a Single Layer of Nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
[Crossref] [PubMed]

Vynck, K.

K. Vynck, D. Felbacq, E. Centeno, A. Căbuz, D. Cassagne, and B. Guizal, “All-Dielectric Rod-Type Metamaterials at Optical Frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[Crossref]

Wang, W.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric Meta-Reflectarray for Broadband Linear Polarization Conversion and Optical Vortex Generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Wang, X.

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

Wicks, G.

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, 1806–1809 (2013).
[Crossref] [PubMed]

Yang, J.

D. Sell, J. Yang, S. Doshay, K. Zhang, and J. A. Fan, “Visible Light Metasurfaces Based on Single-Crystal Silicon,” ACS Photonics 3, 1919–1925 (2016).
[Crossref]

Yang, W.

C. J. Chang-Hasnain and W. Yang, “High–contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

Yang, Y.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95, 165426 (2017).
[Crossref]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric Meta-Reflectarray for Broadband Linear Polarization Conversion and Optical Vortex Generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Yao, B.

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

Yu, Y. F.

R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun.  7, 10362 (2016).
[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, 1527 (2013).
[Crossref] [PubMed]

Zhang, J.

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep.  2, 492 (2012).
[Crossref] [PubMed]

Zhang, K.

D. Sell, J. Yang, S. Doshay, K. Zhang, and J. A. Fan, “Visible Light Metasurfaces Based on Single-Crystal Silicon,” ACS Photonics 3, 1919–1925 (2016).
[Crossref]

Zhou, L.

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

Zhou, Z.

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

Zhu, A. Y.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Zywietz, U.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

ACS Na (1)

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 Na 7, 7824–7832 (2013).
[Crossref]

ACS Nano (1)

G. Brönstrup, N. Jahr, C. Leiterer, A. Csáki, W. Fritzsche, and S. Christiansen, “Optical properties of individual silicon nanowires for photonic devices,” ACS Nano 4, 7113–7122 (2010).
[Crossref] [PubMed]

ACS Photonics (5)

W. Liu and A. E. Miroshnichenko, “Beam steering with dielectric metalattices,” ACS Photonics 5, 1733 (2018).
[Crossref]

P. Albella, R. Alcaraz de la Osa, F. Moreno, and S. A. Maier, “Electric and Magnetic Field Enhancement with Ultralow Heat Radiation Dielectric Nanoantennas: Considerations for Surface-Enhanced Spectroscopies,” ACS Photonics 1, 524–529 (2014).
[Crossref]

P. Moitra, B. A. Slovick, W. Li, I. I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-Scale All-Dielectric Metamaterial Perfect Reflectors,” ACS Photonics 2, 692–698 (2015).
[Crossref]

D. Sell, J. Yang, S. Doshay, K. Zhang, and J. A. Fan, “Visible Light Metasurfaces Based on Single-Crystal Silicon,” ACS Photonics 3, 1919–1925 (2016).
[Crossref]

Z. Zhou, J. Li, R. Su, B. Yao, H. Fang, K. Li, L. Zhou, J. Liu, D. Stellinga, C. P. Reardon, T. F. Krauss, and X. Wang, “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4, 544–551 (2017).
[Crossref]

Adv. Opt. Photonics (1)

C. J. Chang-Hasnain and W. Yang, “High–contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

Appl. Opt. (1)

IEEE Antennas Propag. Mag. (1)

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54, 10–35 (2012).
[Crossref]

IEEE J. Quantum Electron. (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

J. Appl. Phys. (1)

V. Twersky, “On a multiple scattering theory of the finite grating and the Wood anomalies,” J. Appl. Phys. 23, 1099–1118 (1952).
[Crossref]

J. Math. Phys. (1)

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
[Crossref]

J. Nanophotonics (1)

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

J. Opt. Soc. Am. A (1)

Laser Photonics Rev. (1)

P. Lalanne and P. Chavel, “Metalenses at visible wavelengths: past, present, perspectives,” Laser Photonics Rev. 111600295 (2017).
[Crossref]

Nano Lett. (6)

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

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Nat. Commun (4)

R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun.  7, 10362 (2016).
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Figures (7)

Fig. 1
Fig. 1 Contributions of the first terms in Mie theory to the scattering efficiency, Qsca, for (a) p-polarized TE and (b) s-polarized TM waves of a dielectric cylinder of radius R = 50 nm and relative permittivity ε = 12.25 as a function of the energy ω. The second order terms a2, b2 are also shown to indicate the limit of validity of the dipolar approximation.
Fig. 2
Fig. 2 Sketch of the metafilm based on an array of parallel nanorods and scattering geometry for both s and p polarizations of the incoming light. Continuous blue lines correspond to radiation intensity diagrams of a single cylinder HRI cylinder for different frequencies and polarizations (same parameters as in Fig. 1).
Fig. 3
Fig. 3 The two-dimensional maps correspond to the metafilm normalized specular reflection versus angle of incidence θ0 and frequency ω (or a/λ = ω/(2πc)). (a,c) Maps calculated from single scattering and neglecting interference effects corresponding to the asymmetric and resonant scattering from a single cylinder. (b) Schematic of the single scattering geometry. (d,f) Results of a (COMSOL) full wave numerical solution. Black solid line delimit the onset of diffraction. Notice that we use two different color palettes to emphasise both transparent and total reflection regions. (e) Illustration of the metafilm and the scattering configuration.
Fig. 4
Fig. 4 (a,b) Total reflectance calculated only considering α ˜ x ( E ) and α ˜ x ( M ) on Eq. (31) for a/R = 4 and n = 3.5 for (a) p and (b) s waves as a function of the energy ω of the incident wave and the angle of incidence θ0. The black, vertical dashed lines marks the position of the resonance of the polarizabilities, while the black solid lines delimit the diffractive region. Middle panel: Configuration of the equivalent longitudinal dipole multipoles (bear in mind that, in infinitely long cylinders, a dipole along the axis is not strictly speaking meaningful, but rather a current line).
Fig. 5
Fig. 5 (a,b) Reflectance calculated with Eq. (31) setting a ˜ x ( E ) = a ˜ x ( M ) = 0 for a/R = 4 and n = 3.5: (a) p and (b) s waves as a function of the energy ω of the incident wave and the angle of incidence θ0. The black, vertical dashed lines marks the position of the resonance of the polarizabilities, while the blue dashed lines show the zero reflectance (total transmission) bands. The black solid lines delimit the diffractive region. Middle panel: Configuration of the equivalent transverse dipole multipoles.
Fig. 6
Fig. 6 (a,b) Contribution to the p total reflectance in Fig. 5(a) from the projection of the transverse dipole multipoles along the y (a) and z axis (b). (c,d) Contribution to the s total reflectance in Fig. 5(b) from the projection of the transverse dipole multipoles along the y (c) and z axis (d). Top panel: Configuration of the equivalent transverse dipole multipoles.
Fig. 7
Fig. 7 (a,b) Reflectance calculated considering all terms on Eq. (31) for a/R = 4 and n = 3.5: (a) p and (b) s waves as a function of the energy ω of the incident wave and the angle of incidence θ0. The black, vertical dashed lines mark the position of the resonance of the polarizabilities, labeled for a better identification. The blue dashed lines show the zero reflectance (total transmission) bands, also labeled. The black solid line delimits the diffractive region. Middle panel: Configuration of the equivalent dipole multipoles (bear in mind that, in infinitely long cylinders, a dipole along the axis is not strictly speaking meaningful).

Equations (59)

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Ψ ( r ) = [ E ( r ) Z H ( r ) ] , E ( r ) [ E x ( r ) E y ( r ) E z ( r ) ] , H ( r ) [ H x ( r ) H y ( r ) H z ( r ) ] ,
α = ( α ( E ) 0 0 α ( M ) ) , k 2 α ( E ) = ( 4 i b 0 0 0 0 8 i a 1 0 0 0 8 i a 1 ) , k 2 α ( M ) = ( 4 i a 0 0 0 0 8 i b 1 0 0 0 8 i b 1 ) ,
α x α x x , α y = α z = α y , z α y y = α z z
Ψ sca ( r ) = k 2 ( G ( r , r ) G E M ( r , r ) G E M ( r , r ) G ( r , r ) ) α Ψ inc ( r ) = k 2 G ( r , r ) α Ψ inc ( r )
G ( r , r ) = { I k 2 } g ( r r )
G E M ( r , r ) = i k × ( g ( r r ) I )
I ff ( T E ) ( ω , ϕ ) | a 0 + 2 a 1 cos ϕ | 2 | α x ( M ) + α y , z ( E ) cos ϕ | 2 , I ff ( T E ) ( ω , ϕ ) | b 0 + 2 b 1 cos ϕ | 2 | α x ( E ) + α y , z ( M ) cos ϕ | 2 ,
Re [ 1 / a n ] = Re [ 1 / b n ] = 1 { Im [ 1 / ( k 2 α x ) ] = 1 / 4 , Im [ 1 / ( k 2 α y , z ) ] = 1 / 8 }
Im { α 1 } = k 2 Im { G ( r 0 ) } ,
K 0 = k sin θ 0 , q 0 = k cos θ 0 , k 2 = K 0 2 + q 0 2 ,
0 | sin θ 0 | λ a 1
E ( 0 ) ( r ) = ( E s 0 x ^ + E p 0 [ ( K 0 / k ) z ^ + ( q 0 / k ) y ^ ] ) e i k 0 r
Z H ( 0 ) ( r ) = ( E s 0 [ ( K 0 / k ) z ^ + ( q 0 / k ) y ^ ] + E p 0 x ^ ) e i k 0 r
Ψ inc = ( r 0 ) = Ψ ( 0 ) ( r 0 ) + n 0 k 2 G ( r n ) α Ψ inc ( r n ) .
Ψ inc ( r 0 ) = Ψ ( 0 ) ( r 0 ) + k 2 { n 0 G ( r n ) e i K 0 n a } α Ψ inc ( r 0 ) Ψ ( 0 ) ( r 0 ) + k 2 G b α Ψ inc ( r 0 )
G b n G ( r n ) e i K 0 n a .
Ψ inc ( r 0 ) = [ I k 2 G b α ] 1 Ψ ( 0 ) ( r 0 ) ,
Ψ scat ( r ) = k 2 G ( r r n ) α Ψ inc ( r n ) ,
Ψ scat-tot ( r ) = k 2 { n = G ( r r n ) e i K 0 a n } α Ψ inc ( r 0 ) k 2 G ± ( r ) α Ψ inc ( r 0 ) ,
G ± ( r ) n = G ( r r n ) e i K 0 a n = m = G m ± e i K m y e ± i q m z ,
Ψ r ( r ) = k 2 G + ( r ) α ˜ Ψ ( 0 ) ( r 0 ) ,
Ψ t ( r ) = Ψ ( 0 ) ( r ) k 2 G ( r ) α ˜ Ψ ( 0 ) ( r 0 ) ,
α ˜ = α [ I k 2 G b α ] 1 = [ α 1 k 2 G b ] 1 .
S ( 0 ) = 1 2 Re { E ( 0 ) × H ( 0 ) * } z ^ = q 0 4 k Z | Ψ ( 0 ) | 2
S R = k 4 4 k Z Ψ ( 0 ) ( r 0 ) α ˜ ( m = 0 N p q m G m +   G m + ) α ˜ Ψ ( 0 ) ( r 0 ) ,
S T = k 4 4 k Z Ψ ( 0 ) ( r 0 ) α ˜ ( m = 0 N p q m G m   G m ) α ˜ Ψ ( 0 ) ( r 0 ) + q 0 4 k Z Ψ ( 0 ) ( r 0 ) ( I + i k 2 a q 0 { α ˜ α ˜ } ) Ψ ( 0 ) ( r 0 ) ,
m N p q m [ G m +   G m + G m   G m ] = 2 a Im { G ( r 0 ) } ,
2 i { α ˜ α ˜ } = k 2 α ˜ Im { G ( r 0 ) } α ˜ .
α ˜ 1 = α ˜ 0 1 i k 2 Im { G ( r 0 ) } ,
α ˜ 1 = Re { α 1 k 2 G b } i k 2 Im { G ( r 0 ) } ,
R 0 ( p ) = ( k 2 2 k a cos θ 0 ) 2 | γ ( p ) ( α ˜ x ( M ) + sin 2 θ 0 α ˜ z ( E ) + 2 k 2 sin θ 0 G b z x α ˜ x ( M ) α ˜ z ( E ) ) cos 2 θ 0 α ˜ y ( E ) | 2 R 0 ( s ) = ( k 2 2 k a cos θ 0 ) 2 | γ ( s ) ( α ˜ x ( E ) + sin 2 θ 0 α ˜ z ( M ) + 2 k 2 sin θ 0 G b z x α ˜ x ( E ) α ˜ z ( M ) ) cos 2 θ 0 α ˜ y ( M ) | 2 ,
k 2 α ˜ i ( E ) = ( 1 k 2 α i ( E ) G b i i ) 1 , k 2 α ˜ i ( M ) = ( 1 k 2 α i ( M ) G b i i ) 1
γ ( p ) = 1 1 k 4 G b z x 2 α ˜ x ( M ) α ˜ x ( E ) , γ ( s ) = 1 1 k 4 G b z x 2 α ˜ x ( E ) α ˜ z ( M ) ,
R 0 ( p ) = ( k 2 2 k a ) 2 | α ˜ x ( M ) α ˜ y ( E ) | 2 = 0 , R 0 ( s ) = ( k 2 2 k a ) 2 | α ˜ x ( E ) α ˜ y ( M ) | 2 = 0 ,
[ 1 / α x ( M ) ] [ 1 / α y , z ( E ) ] k 2 ( G b x x G b y y ) = 0 for p waves , [ 1 / α x ( E ) ] [ 1 / α y , z ( M ) ] k 2 ( G b x x G b y y ) = 0 for s waves .
R 0 ( p , s ) = ( 1 2 k a cos θ 0 ) 2 | ( { 1 k 2 α x ( M , E ) G b x x } i 1 2 k a cos θ 0 ) 1 | 2 .
R 0 ( p , s ) = ( 1 2 k a cos θ 0 ) 2 | sin 2 θ 0 ( { 1 k 2 α y , z ( E , M ) G b z z } i sin 2 θ 0 2 k a cos θ 0 ) 1 cos 2 θ 0 ( { 1 k 2 α y , z ( E , M ) G b y y } i cos 2 θ 0 2 k a cos θ 0 ) 1 | 2 ,
R 0 ( p , s ) | [ 1 k 2 α y , z ( E , M ) ] ( cos 2 θ 0 sin 2 θ 0 ) ( [ G b z z ] cos 2 θ 0 [ G b y y ] sin 2 θ 0 ) | 2 .
J [ 1 / α ˜ y ( E , M ) ] k 2 cos 2 θ 0 , J [ 1 / α ˜ z ( E , M ) ] k 2 sin 2 θ 0 ,
g ( r r n ) = i 4 H 0 ( k | r r n | ) = + d K 2 π e i K ( y y n ) { i 2 q e i q | z z n | } , q k 2 K 2 .
G ( r , r n ) = { I + k 2 } g ( r r n ) = ( G x x 0 0 0 G y y G y z 0 G y z G z z ) = 1 k 2 ( k 2 0 0 0 k 2 + 2 / y 2 2 / y z 0 2 / y z k 2 + 2 / z 2 ) g ( r r n ) ,
G E M ( r , r n ) = i k × ( g ( r r n ) I ) = ( 0 G E M x y G E M x z G E M x y 0 0 G E M x z 0 0 ) = i k ( 0 / z / y / z 0 0 / y 0 0 ) g ( r r n ) .
G b n 0 G ( r n ) e i K 0 n a = lim r r 0 { G ( r ) G ( r ) } = ( G b G b ( E M ) G b ( E M ) G b )
G b x x ( G b ) x x = i [ 1 2 q 0 a 1 4 ] + 1 2 a m = 1 ( i q m + i q m i k m ) + 1 2 π [ ln k a 4 π + γ e ] ,
G b y y ( G b ) y y = i 4 [ 2 q 0 k 2 a = 1 2 ] + 1 2 k 2 a m = 1 ( i q m + i q m + 2 k m k 2 k m ) + 1 4 π [ ln k a 4 π + γ e 1 2 ] q 0 2 4 k 2 π + 1 6 π k 2 a 2 ,
G b z z ( G b ) z z = i 4 [ 2 K 0 2 k 2 q 0 a 1 2 ] + 1 2 k 2 a m = 1 ( i K m 2 q m + K m 2 q m 2 k m k 2 k m ) + 1 4 π [ ln k a 4 π + γ e 1 2 ] K 0 2 4 k 2 π 1 6 π k 2 a 2 ,
k m = 2 m π a , K m K 0 k m , q m = k 2 K m 2 .
G b z x ( G b ( E M ) ) z x = ( G b ( E M ) ) x z = i K 0 2 k q 0 a + i 2 k a m = 1 ( K m q m + K m q m ) 1 2 π K 0 k
2 π a m = + δ ( K 2 π m a ) = n = + e i K a n
n = e i K 0 a n g ( r r n ) = + d K 2 π { n = e i ( K 0 K ) a n } e i K y { 1 2 q e ± i q z } = { m = e i K m y e ± i q m z i 2 a q m } .
G ( r ) n = e i K 0 a n G ( r r n ) = n = e i K 0 a n ( G ( r r n ) G E M ( r r n ) G E M ( r r n ) G ( r r n ) ) = m = e i K m y e ± i q m z G m ± = m = e i K m y e ± i q m z ( G m ± G E M m ± G E M m ± G m ± )
G m ± = i 2 q m a 1 k 2 ( k 2 0 0 0 k 2 K m 2 q m K m 0 q m K m k 2 q m 2 ) , G E M m ± = i 2 q m a 1 k ( 0 ± q m K m q m 0 0 K m 0 0 ) ,
G ( r ) ~ e i K 0 y e ± i q 0 z ( G 0 ± G E M 0 ± G E M 0 ± G 0 ± ) .
G 0 ± = i 2 q 0 a 1 k 2 ( k 2 0 0 0 q 0 2 q 0 K 0 0 q 0 K 0 K 0 2 ) , G E M m ± = i 2 q 0 a 1 k ( 0 ± q 0 K 0 q 0 0 0 K 0 0 0 ) ,
Im { G ( 0 ) } 1 2 Im { G + ( 0 ) + G ( 0 ) } = Im { ( G 0 G E M 0 G E M 0 G 0 ) . } + m 0 diffracted Im { ( G m G E M 0 G E M 0 G m ) . }
Im { G 0 } = 1 2 q 0 a 1 k 2 ( k 2 0 0 0 q 0 2 0 0 0 K 0 2 ) , Im { G E M 0 } = 1 2 q 0 a K 0 k ( 0 0 1 0 0 0 1 0 0 ) ,
T 0 ( p ) = | 1 + i 2 k a cos θ Z H x 0 ( T ) / E p 0 | 2 , T 0 ( s ) = | 1 + i 2 k a cos θ E x 0 ( T ) / E s 0 | 2 ,
Z H x 0 ( T ) / E p 0 = k 2 γ ( p ) ( α ˜ x ( M ) + sin 2 θ 0 α ˜ z ( E ) + 2 k 2 sin θ 0 G b z x α ˜ x ( M ) α ˜ z ( E ) ) + k 2 cos 2 θ 0 α ˜ y ( E ) , E x 0 ( T ) / E s 0 = k 2 γ ( s ) ( α ˜ x ( E ) + sin 2 θ 0 α ˜ z ( M ) + 2 k 2 sin θ 0 G b z x α ˜ x ( E ) α ˜ z ( M ) ) + k 2 cos 2 θ 0 α ˜ y ( M ) .
R m ( p ) = ( 1 2 k a cos θ 0 ) 2 | γ ( p ) ( k 2 α ˜ x ( M ) + K 0 K m α ˜ z ( E ) + k 3 ( K 0 + K m ) G b z x α ˜ x ( M ) α ˜ z ( E ) ) q 0 q m α ˜ y ( E ) | 2 , R m ( s ) = ( 1 2 k a cos θ 0 ) 2 | γ ( s ) ( k 2 α ˜ x ( E ) + K 0 K m α ˜ z ( M ) + k 3 ( K 0 + K m ) G b z x α ˜ x ( E ) α ˜ z ( M ) ) q 0 q m α ˜ y ( M ) | 2 .

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