Abstract

Starting from a general description of light scattering by a nanoparticle in homogeneous surroundings and situated near a substrate, we outline the connection to multipole expansion of scattered light and derive conditions and limits on achievable half-space scattering asymmetry, including the possibility of unidirectional scattering along the propagation direction of the incident light (i.e., generalized Kerker conditions). As a way of realizing strongly asymmetric scattering, we perform a parametric study of the optical properties of disk-shaped gap-surface plasmon (GSP) resonators, consisting of a glass spacer sandwiched between two gold disks, with numerical calculations that corroborate the conditions derived from the multipole expansion. Finally, we present proof-of-principle experiments of asymmetric scattering by GSP-resonators on a glass substrate.

© 2015 Optical Society of America

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2015 (2)

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

A. Pors and S. I. Bozhevolnyi, “Quantum emitters near layered plasmonic nanostructures: decay rate contributions,” ACS Photonics 2, 228–236 (2015).
[Crossref]

2014 (3)

E. Poutrina and A. Urbas, “Multipole analysis of unidirectional light scattering from plasmonic dimers,” J. Opt. 16, 114005 (2014).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref] [PubMed]

W. Liu, J. Zhang, B. Lei, H. Ma, W. Xie, and H. Hu, “Ultra-directional forward scattering by individual core-shell nanoparticles,” Opt. Express 22, 16178–16187 (2014).
[Crossref] [PubMed]

2013 (5)

E. Poutrina, A. Rose, D. Brown, A. Urbas, and D. R. Smith, “Forward and backward unidirectional scattering from plasmonic coupled wires,” Opt. Express 21, 31138–31154 (2013).
[Crossref]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (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, 7824–7832 (2013).
[Crossref] [PubMed]

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

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B. 30, 2589–2598 (2013).
[Crossref]

2012 (3)

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, 1171 (2012).
[Crossref] [PubMed]

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

P.-Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP304 (2012).
[PubMed]

2011 (7)

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13, 013001 (2011).
[Crossref]

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photon. 5, 523–530 (2011).

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

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys. 13, 023034 (2011).
[Crossref]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531–534 (2011).
[Crossref]

B. García-Cámara, R. A. de la Osa, J. M. Saiz, F. González, and F. Moreno, “Directionality in scattering by nanoparticles: Kerker’s null-scattering conditions revisited,” Opt. Lett. 36, 728–730 (2011).
[Crossref]

J. Muller, G. Parent, G. Jeandel, and D. Lacroix, “Finite-difference time-domain and near-field-to-far-field transformation in the spectral domain: application to scattering objects with complex shapes in the vicinity of a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 28, 868–878 (2011).
[Crossref]

2010 (6)

K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010).
[Crossref] [PubMed]

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, 707–715 (2010).
[Crossref]

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

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010).
[Crossref] [PubMed]

A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10, 4571–4577 (2010).
[Crossref] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

2009 (2)

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

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009).
[Crossref]

2008 (3)

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–21800 (2008).
[Crossref] [PubMed]

M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonances,” Phys. Rev. Lett. 100, 043903 (2008).
[Crossref] [PubMed]

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136–159 (2008).
[Crossref]

2007 (5)

2006 (1)

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96, 097401 (2006).
[Crossref] [PubMed]

1983 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Annalen der Physik 330, 377–445 (1908).
[Crossref]

Aizpurua, J.

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136–159 (2008).
[Crossref]

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, 1171 (2012).
[Crossref] [PubMed]

Albrektsen, O.

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys. 13, 023034 (2011).
[Crossref]

A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10, 4571–4577 (2010).
[Crossref] [PubMed]

Alù, A.

P.-Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP304 (2012).
[PubMed]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Bharadwaj, P.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH, 2004).

Boltasseva, A.

Borneman, J. D.

K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

A. Pors and S. I. Bozhevolnyi, “Quantum emitters near layered plasmonic nanostructures: decay rate contributions,” ACS Photonics 2, 228–236 (2015).
[Crossref]

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys. 13, 023034 (2011).
[Crossref]

A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10, 4571–4577 (2010).
[Crossref] [PubMed]

S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators,” Opt. Express 15, 10869–10877 (2007).
[Crossref] [PubMed]

Brener, I.

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, 7824–7832 (2013).
[Crossref] [PubMed]

Brown, D.

Bryant, G.

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136–159 (2008).
[Crossref]

Cai, W.

Capasso, F.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref] [PubMed]

Catchpole, K. R.

Chan, C. T.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531–534 (2011).
[Crossref]

Chen, J.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531–534 (2011).
[Crossref]

Chen, K.-P.

K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010).
[Crossref] [PubMed]

Chen, P.-Y.

P.-Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP304 (2012).
[PubMed]

Cheng, W.

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

Chettiar, U. K.

Chichkov, B. N.

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B. 30, 2589–2598 (2013).
[Crossref]

Chong, C. T.

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, 707–715 (2010).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

de la Osa, R. A.

de Silva, V. C.

Decker, M.

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, 7824–7832 (2013).
[Crossref] [PubMed]

Deutsch, B.

Dmitriev, A.

A. Dmitriev, T. Pakizeh, M. Kall, and D. S. Sutherland, “Gold-silica-gold nanosandwiches: tunable bimodal plasmonic resonators,” Small 3, 294–299 (2007).
[Crossref] [PubMed]

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

Drachev, V. P.

Evlyukhin, A. B.

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B. 30, 2589–2598 (2013).
[Crossref]

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys. 13, 023034 (2011).
[Crossref]

A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10, 4571–4577 (2010).
[Crossref] [PubMed]

Evlyukhin, E.

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B. 30, 2589–2598 (2013).
[Crossref]

Eyraud, C.

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M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonances,” Phys. Rev. Lett. 100, 043903 (2008).
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ACS Nano (2)

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, 7824–7832 (2013).
[Crossref] [PubMed]

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

ACS Photonics (1)

A. Pors and S. I. Bozhevolnyi, “Quantum emitters near layered plasmonic nanostructures: decay rate contributions,” ACS Photonics 2, 228–236 (2015).
[Crossref]

Adv. Mater. (1)

P.-Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP304 (2012).
[PubMed]

Adv. Opt. Photon. (1)

Annalen der Physik (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Annalen der Physik 330, 377–445 (1908).
[Crossref]

Annu. Rev. Phys. Chem. (1)

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58, 267–297 (2007).
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J. Appl. Phys. (1)

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

J. Opt. (2)

E. Poutrina and A. Urbas, “Multipole analysis of unidirectional light scattering from plasmonic dimers,” J. Opt. 16, 114005 (2014).
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C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13, 013001 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

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

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

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B. 30, 2589–2598 (2013).
[Crossref]

Laser Photonics Rev. (1)

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136–159 (2008).
[Crossref]

Nano Lett. (3)

A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10, 4571–4577 (2010).
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T. Pakizeh and M. Kall, “Unidirectional ultracompact optical nanoantennas,” Nano Lett. 9, 2343–2349 (2009).
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K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010).
[Crossref] [PubMed]

Nat. Commun. (2)

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Lukyanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (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, 1171 (2012).
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Nat. Mater. (3)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
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H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
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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, 707–715 (2010).
[Crossref]

Nat. Photon. (4)

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531–534 (2011).
[Crossref]

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

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photon. 5, 523–530 (2011).

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

New J. Phys. (1)

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys. 13, 023034 (2011).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. B (1)

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[Crossref]

Phys. Rev. Lett. (3)

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96, 097401 (2006).
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M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonances,” Phys. Rev. Lett. 100, 043903 (2008).
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Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010).
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Science (1)

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref] [PubMed]

Small (1)

A. Dmitriev, T. Pakizeh, M. Kall, and D. S. Sutherland, “Gold-silica-gold nanosandwiches: tunable bimodal plasmonic resonators,” Small 3, 294–299 (2007).
[Crossref] [PubMed]

Other (3)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH, 2004).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Optical properties of GSP-resonator in air. (a) Sketch of disk GSP-resonator consisting of a glass spacer of diameter d and thickness ts sandwiched between two gold disks of same diameter and thickness t. The direction and polarization of the incident plane wave are indicated in the figure. (b) Scattering and (c) absorption cross sections as a function of wavelength and spacer thickness ts for a GSP-resonator with d = 140 nm and t = 30 nm. The cross sections are normalized to the geometrical cross section π(d/2)2. (d, e) Color maps show the enhancement of the magnetic field in the xz-plane for ts = 30 nm at wavelengths λ = 615 nm and 765 nm, respectively. The cone plots correspond to the scattered electric near-field. (f–h) Decomposition of the scattering cross section into multipole contributions for spacer thicknesses ts = 30 nm, 65 nm, and 90 nm, respectively.
Fig. 2
Fig. 2 Asymmetric scattering by disk GSP-resonator in air. Full-wave numerical calculations of normalized (a) forward and (b) backward scattering cross sections of disk GSP-resonator (d = 140 nm and t = 30 nm) as a function of wavelength and spacer thickness ts. (c) Symmetric and (d) asymmetric part of the scattering cross section, calculated using Eq. (15)). (e) Color map of the ratio between forward and backward scattering, with the solid lines corresponding to the equality of the real and imaginary parts of my = αvdpx and Qxz = β6i/kdpx, when α = 12 / 17 and β = 5 51. Dashed lines indicate the positions of the ED and MD resonances.
Fig. 3
Fig. 3 Suppressing backward scattering using disk GSP-resonators in air. (a) Color map of the directivity D, with solid lines corresponding to equality of the real and imaginary parts of the Kerker condition for backward scattering suppression [i.e., Eq. (13))]. Dashed lines indicate the positions of the electric and magnetic dipole resonances. (b) Normalized radiation pattern on a linear scale at maximum directivity: (ts, λ) = (65nm, 720nm). The inset shows the three-dimensional radiation pattern.
Fig. 4
Fig. 4 Maximum scattering asymmetry between forward (z < 0) and backward (z > 0) half-space. (a) Theoretical maximum half-space scattering asymmetry as a function of the distance z0d, where z0 is the height of the multipole expansion above the interface and λd is the wavelength in the upper medium, and the relative refractive index N = ε s / ε d when considering ED, MD, and EQ contributions. The related proportionality constants α and β, as defined by my = αvdpx and Qxz = β6i/kdpx, are displayed in (b)–(e). The phase of α and β is presented in units of π. (f) Maximum achievable half-space scattering asymmetry when only considering ED and MD contributions, with the associated α-parameter shown in (g, h).
Fig. 5
Fig. 5 Optical properties of GSP-resonator situated on a glass substrate and surrounded by air. (a) Sketch of disk GSP-resonator consisting of a glass spacer of diameter d and thickness ts sandwiched between two gold disks of same diameter and thickness t. The direction and polarization of the incident plane wave are indicated in the figure. (b) Scattering and (c) absorption cross sections as a function of wavelength and spacer thickness ts for a GSP-resonator with d = 130 nm and t = 30 nm. The cross sections are normalized to the geometrical cross section π(d/2)2. (d, e) Color maps show the enhancement of the magnetic field in the xz-plane for ts = 30 nm at wavelengths λ = 640 nm and 750 nm, respectively. The cone plots correspond to the scattered electric near-field. (f–h) Decomposition of the scattering cross section into multipole contributions for spacer thicknesses ts = 20 nm, 30 nm, and 40 nm, respectively.
Fig. 6
Fig. 6 Asymmetric scattering by GSP-resonator on a glass substrate and surrounded by air. Numerical calculations of normalized (a–c) forward and (d–f) backward scattering cross sections of disk GSP-resonator (d = 130 nm and t = 30 nm) as a function of wavelength and spacer thickness ts. (c, f) Total half-space scattering cross sections, with (a, d) and (b, e) displaying the direct and interference part, respectively, as calculated from the multipole expansion in Eqs. (5))–(8)) and the formulas presented in the Appendix.
Fig. 7
Fig. 7 Half-space scattering asymmetry and directivity by GSP-resonator on a glass substrate and surrounded by air. Numerical calculations of (a) Forward-to-backward half-space scattering ratio and (b) directivity of disk GSP-resonator (d = 130 nm and t = 30 nm) as a function of wavelength and spacer thickness ts. (c) Normalized radiation pattern on a linear scale for (ts, λ) = (17.5 nm, 858 nm). The inset shows the three-dimensional radiation pattern. The radiation pattern is calculated following a near-field-to-far-field transformation of the electric field as outlined in [40, 41].
Fig. 8
Fig. 8 Experimental study of scattering asymmetry in GSP-resonators. (a) Sketch of dark-field configuration for detection of forward (red incident rays) or backward (blue incident rays) scattered light from a disk-shaped GSP-resonator. The black cone indicates the numerical aparture of the collection objective. (b) Forward (red) and backward (blue) scattered light from disk GSP-Resonator with diameter 120 nm (dashed) and 130 nm (solid). The spectra are normalized to a gold disk with diameter 700nm. Inset shows the ratio of forward to backward scattered light.

Tables (2)

Tables Icon

Table 1 Constants and functions for the calculation of scattered power in the upper medium, as defined in Eqs. (29)) and (30)). Here, τ = 2kdz0 cosθ.

Tables Icon

Table 2 Constants and functions for the calculation of scattered power in the lower medium (i.e., substrate), as defined in Eqs. (29)) and (31)).

Equations (33)

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E s c ( r ) = i ω μ 0 V p G ^ ( r , r ) J p ( r ) d 3 r ,
G ^ FF ( r , r ) = g ^ ( r ) e i k d N ( r ) r ,
E s c FF ( r ) = i ω μ 0 g ^ ( r ) e i k d ( N r 0 ) n = 0 ( i k d ) n n ! V p ( N Δ r ) n J p ( r ) d 3 r ,
E s c FF ( r ) ω 2 μ 0 g ^ ( r ) e i k d ( N r 0 ) [ p i k d 6 Q ^ N 1 v d ( N × m ) + i k d 2 v d ( N × M ^ N ) k d 2 6 Q ^ ( NN ) ] ,
p = i ω V p J p ( r ) d 3 r ,
m ( r 0 ) = 1 2 V p Δ r × J p d 3 r ,
Q ^ ( r 0 ) = i ω V p 3 [ Δ r J p ( r )+ J p ( r ) Δ r ] 2 ( J p ( r ) Δ r ) I ^ d 3 r ,
M ^ ( r 0 ) = 1 3 V p [ Δ r × J p ( r )] Δ r + Δ r [ Δ r × J p ( r )]d 3 r .
G ^ 0 FF ( r , r ) = e i k d r 4 π r ( I ^ nn ) e i k d n r ,
E s c 0 , FF ( r ) ω 2 μ 0 e i k d ( r n r 0 ) 4 π r [ n × ( p × n ) + i k d 6 [ n × ( n × Q ^ n ) ] 1 v d ( n × m ) + i k d 2 v d ( n × M ^ n ) ] .
E s c , x 0 , FF , + = ω 2 μ 0 e i k d ( z z 0 ) 4 π r [ p x + k d 6 i Q x z + 1 v d m y + k d 2 v d i M y z ] ,
E s c , x 0 , FF , = ω 2 μ 0 e i k d ( z + z 0 ) 4 π r [ p x k d 6 i Q x z 1 v d m y + k d 2 v d i M y z ] .
p x + k d 2 v d i M y z = 1 v d m y + k d 6 i Q x z ,
Re { p x + k d 2 v d i M y z } = Re { 1 v d m y + k d 6 i Q x z } .
P s c ± = ω 4 μ 0 2 32 π η d [ 4 3 | p x | 2 + 4 3 v d 2 | m y | 2 + k d 2 45 | Q x z | 2 + k d 2 5 v d 2 | M y z | 2 ± 2 v d Re { p x m y * } ± k d 6 Im { Q x z p x * } ± k d 2 v d 2 Im { M y z m y * } ] ,
p x + k d 2 v d i M y z = 1 + r p e i 2 k d z 0 1 r p e i 2 k d z 0 ( 1 v d m y + k d 6 i Q x z ) ,
p x + k d 2 v d i M y z = 1 v d m y + k d 6 i Q x z .
G ^ 0 FF ( r , r ) = e i k d r 4 π r e i k d ( n r ) ( I ^ nm ) ,
G ^ r FF ( r , r ) = e i k d r 4 π r e i k d ( n ˜ r ) R ^ ( r ) ,
G ^ r FF ( r , r ) = e i k s r 4 π r e i k d ( n ˜ ˜ r ) T ^ ( r ) ,
R ^ ( r ) = r p ( n x 2 n z 2 n ρ 2 n x n y n z 2 n ρ 2 n x n z n x n y n z 2 n ρ 2 n y 2 n z 2 n ρ 2 n y n z n x n z n y n z n ρ 2 ) + r s 1 n ρ 2 ( n y 2 n x n y 0 n x n y n x 2 0 0 0 0 ) ,
T ^ ( r ) = t p N n z ( n x 2 n z n ρ 2 n x n y n z n ρ 2 n x n z N 1 N 2 n ρ 2 n x n y n z n ρ 2 n y 2 n z n ρ 2 n y n z N 1 N 2 n ρ 2 n x n y n ρ 2 N 1 N 2 n ρ 2 ) + t s N n z n ρ 2 1 N 2 n ρ 2 ( n y 2 n x n y 0 n x n y n x 2 0 0 0 0 ) ,
r s ( θ ) = N 2 cos θ N 2 sin 2 θ N 2 cos θ + N 2 sin 2 θ , r s ( θ ) = cos θ N 2 sin 2 θ cos θ + N 2 sin 2 θ ,
t p ( θ t ) = 2 1 N 2 sin 2 θ t N 1 N 2 sin 2 θ t cos θ t , t s ( θ t ) = 2 1 N 2 sin 2 θ t 1 N 2 sin 2 θ t N cos θ t .
E s c FF , + ( r ) E s c 0 , FF + ω 2 μ 0 4 π r e i k d ( r n ˜ r 0 ) R ^ ( r ) [ p i k d 6 Q ^ n ˜ 1 v d ( n ˜ × m ) + i k d 2 v d ( n ˜ × M ^ n ˜ ) ] ,
E s c FF , ( r ) ω 2 μ 0 4 π r e i ( k s r k d n ˜ ˜ r 0 ) T ^ ( r ) [ p i k d 6 Q ^ n ˜ ˜ 1 v d ( n ˜ ˜ × m ) + i k d 2 v d ( n ˜ ˜ × M ^ n ˜ ˜ ) ] ,
E s c , x FF , + = ω 2 μ 0 e i k d ( z z 0 ) 4 π r [ ( 1 r p ( 0 ) e i 2 k d z 0 ) ( p x + k d 2 v d i M y x ) ( 1 + r p ( 0 ) e i 2 k d z 0 ) ( k d 6 i Q x z + 1 v d m y ) ] ,
E s c , x FF , = ω 2 μ 0 e i ( k s z k d z 0 ) 4 π r N t p ( π ) [ p x + k d 6 i Q x z + 1 v d m y k d 2 v d i M y z ] .
P s c ± = P ED ± + P MD ± + P EQ ± + P MQ ± + P ED MD ± + P EQ MQ ± + P ED EQ ± + P MD MQ ± + P ED MQ ± + P MD EQ ± ,
P j + = ω 4 μ 0 2 32 π η d [ D + C r 0 π / 2 ( f s ( θ ) | r s | 2 + f p ( θ ) | r p | 2 ) sin θ d θ + C i 0 π / 2 ( f s ( θ ) r s f p ( θ ) r p ) g ( θ ) sin θ d θ ] ,
P j = ω 4 μ 0 2 32 π η s C t [ π / 2 π ( f s ( θ t ) | t s | 2 + Γ f p ( θ t ) | t p | 2 ) cos 2 θ t sin θ t e 2 k d z 0 N 2 sin 2 θ t 1 Λ d θ t ,
C t = { C t f , π / 2 θ t < θ m C t a , θ m θ t π , Λ = { 1 , π / 2 θ t < θ m 0 , θ m θ t π ,
Γ = { 1 , θ m θ t π C t f C t a 1 , otherwise .

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