Abstract

We introduce a novel bottom-up approach to fabricate by self assembly a metamaterial from metallic nanoparticles in a two-step process. In the first step, a metamaterial made of densely packed silver nanoparticles is required. The material dispersion with increasing nanoparticle densities, from dispersed to randomly packed nanoparticles, was measured by spectroscopic ellipsometry, demonstrating high permittivity values in the visible. In the second step, this material was used to prepare spherical clusters by a method based on oil-in-water emulsion. The optical properties of these clusters were equally investigated by spectroscopic means. Comparisons with rigorous numerical simulations clearly indicate that, depending on the cluster size, their spectral response can be unambiguously associated with the excitation of a magnetic dipole resonance. As a consequence, such spherical clusters are promising building blocks for future metamaterials possessing a magnetic response in the visible range.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2011

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

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(12), 123017 (2011).
[CrossRef]

S. Mühlig, C. Rockstuhl, V. Yannopapas, T. Bürgi, N. Shalkevich, and F. Lederer, “Optical properties of a fabricated self-assembled bottom-up bulk metamaterial,” Opt. Express19(10), 9607–9616 (2011).
[CrossRef] [PubMed]

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express19(3), 2754–2772 (2011).
[CrossRef] [PubMed]

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials (Amst.)5(2-3), 64–73 (2011).
[CrossRef]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83(24), 245119 (2011).
[CrossRef]

2010

J. Petschulat, J. Yang, C. Menzel, C. Rockstuhl, A. Chipouline, P. Lalanne, A. Tüennermann, F. Lederer, and T. Pertsch, “Understanding the electric and magnetic response of isolated metaatoms by means of a multipolar field decomposition,” Opt. Express18(14), 14454–14466 (2010).
[CrossRef] [PubMed]

I. Hussain, H. Zhang, M. Brust, J. Barauskas, and A. I. Cooper, “Emulsions-directed assembly of gold nanoparticles to molecularly-linked and size-controlled spherical aggregates,” J. Colloid Interface Sci.350(1), 368–372 (2010).
[CrossRef] [PubMed]

P. Qiu, C. Jensen, N. Charity, R. Towner, and C. Mao, “Oil phase evaporation-induced self-assembly of hydrophobic nanoparticles into spherical clusters with controlled surface chemistry in an oil-in-water dispersion and comparison of behaviors of individual and clustered iron oxide nanoparticles,” J. Am. Chem. Soc.132(50), 17724–17732 (2010).
[CrossRef] [PubMed]

S. Mühlig, C. Rockstuhl, J. Pniewski, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Three-dimensional metamaterial nanotips,” Phys. Rev. B81(7), 075317 (2010).
[CrossRef]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk'yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82(4), 045404 (2010).
[CrossRef]

N. I. Zheludev, “Applied physics. The road ahead for metamaterials,” Science328(5978), 582–583 (2010).
[CrossRef] [PubMed]

2009

C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B79(4), 045111 (2009).
[CrossRef]

2007

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science315(5808), 47–49 (2007).
[CrossRef] [PubMed]

V. Yannopapas, “Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies,” Appl. Phys., A Mater. Sci. Process.87(2), 259–264 (2007).
[CrossRef]

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett.99(1), 017401 (2007).
[CrossRef] [PubMed]

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett.98(15), 157403 (2007).
[CrossRef] [PubMed]

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99(10), 107401 (2007).
[CrossRef] [PubMed]

2006

V. Yannopapas and N. V. Vitanov, “Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength,” Phys. Rev. B74(19), 193304 (2006).
[CrossRef]

2005

2004

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

2002

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter14(15), 4035–4044 (2002).
[CrossRef]

2000

R. Ruppin, “Evaluation of extended Maxwell-Garnett theories,” Opt. Commun.182(4-6), 273–279 (2000).
[CrossRef]

1999

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999).
[CrossRef]

1995

1989

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B Condens. Matter39(14), 9852–9858 (1989).
[CrossRef] [PubMed]

1972

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

1947

L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng.94, 65–68 (1947).

Albani, M.

Bai, F.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Barauskas, J.

I. Hussain, H. Zhang, M. Brust, J. Barauskas, and A. I. Cooper, “Emulsions-directed assembly of gold nanoparticles to molecularly-linked and size-controlled spherical aggregates,” J. Colloid Interface Sci.350(1), 368–372 (2010).
[CrossRef] [PubMed]

Brongersma, M. L.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99(10), 107401 (2007).
[CrossRef] [PubMed]

Brust, M.

I. Hussain, H. Zhang, M. Brust, J. Barauskas, and A. I. Cooper, “Emulsions-directed assembly of gold nanoparticles to molecularly-linked and size-controlled spherical aggregates,” J. Colloid Interface Sci.350(1), 368–372 (2010).
[CrossRef] [PubMed]

Bürgi, T.

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

S. Mühlig, C. Rockstuhl, V. Yannopapas, T. Bürgi, N. Shalkevich, and F. Lederer, “Optical properties of a fabricated self-assembled bottom-up bulk metamaterial,” Opt. Express19(10), 9607–9616 (2011).
[CrossRef] [PubMed]

Burnett, J. H.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Cai, W. S.

Capolino, F.

Charity, N.

P. Qiu, C. Jensen, N. Charity, R. Towner, and C. Mao, “Oil phase evaporation-induced self-assembly of hydrophobic nanoparticles into spherical clusters with controlled surface chemistry in an oil-in-water dispersion and comparison of behaviors of individual and clustered iron oxide nanoparticles,” J. Am. Chem. Soc.132(50), 17724–17732 (2010).
[CrossRef] [PubMed]

Chen, C.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Chen, H.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett.98(15), 157403 (2007).
[CrossRef] [PubMed]

Chen, W.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Chettiar, U. K.

Chichkov, B. N.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk'yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82(4), 045404 (2010).
[CrossRef]

Chipouline, A.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Cooney, G.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Cooper, A. I.

I. Hussain, H. Zhang, M. Brust, J. Barauskas, and A. I. Cooper, “Emulsions-directed assembly of gold nanoparticles to molecularly-linked and size-controlled spherical aggregates,” J. Colloid Interface Sci.350(1), 368–372 (2010).
[CrossRef] [PubMed]

Cunningham, A.

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

Dolling, G.

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B Condens. Matter39(14), 9852–9858 (1989).
[CrossRef] [PubMed]

Drachev, V. P.

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett.30(23), 3198–3200 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Etrich, C.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83(24), 245119 (2011).
[CrossRef]

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett.99(1), 017401 (2007).
[CrossRef] [PubMed]

Evlyukhin, A. B.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk'yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82(4), 045404 (2010).
[CrossRef]

French, R. H.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Green, S. E.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Grzegorczyk, T. M.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett.98(15), 157403 (2007).
[CrossRef] [PubMed]

Helgert, C.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83(24), 245119 (2011).
[CrossRef]

Herzinger, C. M.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999).
[CrossRef]

Huo, Z.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Hussain, I.

I. Hussain, H. Zhang, M. Brust, J. Barauskas, and A. I. Cooper, “Emulsions-directed assembly of gold nanoparticles to molecularly-linked and size-controlled spherical aggregates,” J. Colloid Interface Sci.350(1), 368–372 (2010).
[CrossRef] [PubMed]

Jensen, C.

P. Qiu, C. Jensen, N. Charity, R. Towner, and C. Mao, “Oil phase evaporation-induced self-assembly of hydrophobic nanoparticles into spherical clusters with controlled surface chemistry in an oil-in-water dispersion and comparison of behaviors of individual and clustered iron oxide nanoparticles,” J. Am. Chem. Soc.132(50), 17724–17732 (2010).
[CrossRef] [PubMed]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Kaplan, S.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Kildishev, A. V.

Kong, J. A.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett.98(15), 157403 (2007).
[CrossRef] [PubMed]

Koschny, T.

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Lalanne, P.

Lederer, F.

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials (Amst.)5(2-3), 64–73 (2011).
[CrossRef]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83(24), 245119 (2011).
[CrossRef]

S. Mühlig, C. Rockstuhl, V. Yannopapas, T. Bürgi, N. Shalkevich, and F. Lederer, “Optical properties of a fabricated self-assembled bottom-up bulk metamaterial,” Opt. Express19(10), 9607–9616 (2011).
[CrossRef] [PubMed]

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

J. Petschulat, J. Yang, C. Menzel, C. Rockstuhl, A. Chipouline, P. Lalanne, A. Tüennermann, F. Lederer, and T. Pertsch, “Understanding the electric and magnetic response of isolated metaatoms by means of a multipolar field decomposition,” Opt. Express18(14), 14454–14466 (2010).
[CrossRef] [PubMed]

S. Mühlig, C. Rockstuhl, J. Pniewski, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Three-dimensional metamaterial nanotips,” Phys. Rev. B81(7), 075317 (2010).
[CrossRef]

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett.99(1), 017401 (2007).
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L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng.94, 65–68 (1947).

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F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Liang, X.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science315(5808), 47–49 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett.30(23), 3198–3200 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Liu, L.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

López-Tejeira, F.

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(12), 123017 (2011).
[CrossRef]

Luk'yanchuk, B. S.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk'yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82(4), 045404 (2010).
[CrossRef]

Mao, C.

P. Qiu, C. Jensen, N. Charity, R. Towner, and C. Mao, “Oil phase evaporation-induced self-assembly of hydrophobic nanoparticles into spherical clusters with controlled surface chemistry in an oil-in-water dispersion and comparison of behaviors of individual and clustered iron oxide nanoparticles,” J. Am. Chem. Soc.132(50), 17724–17732 (2010).
[CrossRef] [PubMed]

Marqués, R.

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(12), 123017 (2011).
[CrossRef]

Menzel, C.

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials (Amst.)5(2-3), 64–73 (2011).
[CrossRef]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83(24), 245119 (2011).
[CrossRef]

J. Petschulat, J. Yang, C. Menzel, C. Rockstuhl, A. Chipouline, P. Lalanne, A. Tüennermann, F. Lederer, and T. Pertsch, “Understanding the electric and magnetic response of isolated metaatoms by means of a multipolar field decomposition,” Opt. Express18(14), 14454–14466 (2010).
[CrossRef] [PubMed]

Moroz, A.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter17(25), 3717–3734 (2005).
[CrossRef] [PubMed]

Mühlig, S.

S. Mühlig, C. Rockstuhl, V. Yannopapas, T. Bürgi, N. Shalkevich, and F. Lederer, “Optical properties of a fabricated self-assembled bottom-up bulk metamaterial,” Opt. Express19(10), 9607–9616 (2011).
[CrossRef] [PubMed]

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83(24), 245119 (2011).
[CrossRef]

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials (Amst.)5(2-3), 64–73 (2011).
[CrossRef]

S. Mühlig, C. Rockstuhl, J. Pniewski, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Three-dimensional metamaterial nanotips,” Phys. Rev. B81(7), 075317 (2010).
[CrossRef]

O'Brien, S.

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter14(15), 4035–4044 (2002).
[CrossRef]

Pacholski, C.

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

Paniagua-Domínguez, R.

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(12), 123017 (2011).
[CrossRef]

Pendry, J. B.

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter14(15), 4035–4044 (2002).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999).
[CrossRef]

Peng, L.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett.98(15), 157403 (2007).
[CrossRef] [PubMed]

Peng, Q.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Pertsch, T.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83(24), 245119 (2011).
[CrossRef]

J. Petschulat, J. Yang, C. Menzel, C. Rockstuhl, A. Chipouline, P. Lalanne, A. Tüennermann, F. Lederer, and T. Pertsch, “Understanding the electric and magnetic response of isolated metaatoms by means of a multipolar field decomposition,” Opt. Express18(14), 14454–14466 (2010).
[CrossRef] [PubMed]

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett.99(1), 017401 (2007).
[CrossRef] [PubMed]

Petschulat, J.

Pniewski, J.

S. Mühlig, C. Rockstuhl, J. Pniewski, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Three-dimensional metamaterial nanotips,” Phys. Rev. B81(7), 075317 (2010).
[CrossRef]

Pribil, G. K.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Qiu, P.

P. Qiu, C. Jensen, N. Charity, R. Towner, and C. Mao, “Oil phase evaporation-induced self-assembly of hydrophobic nanoparticles into spherical clusters with controlled surface chemistry in an oil-in-water dispersion and comparison of behaviors of individual and clustered iron oxide nanoparticles,” J. Am. Chem. Soc.132(50), 17724–17732 (2010).
[CrossRef] [PubMed]

Ran, L.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett.98(15), 157403 (2007).
[CrossRef] [PubMed]

Reinhardt, C.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk'yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82(4), 045404 (2010).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999).
[CrossRef]

Rockstuhl, C.

S. Mühlig, C. Rockstuhl, V. Yannopapas, T. Bürgi, N. Shalkevich, and F. Lederer, “Optical properties of a fabricated self-assembled bottom-up bulk metamaterial,” Opt. Express19(10), 9607–9616 (2011).
[CrossRef] [PubMed]

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials (Amst.)5(2-3), 64–73 (2011).
[CrossRef]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B83(24), 245119 (2011).
[CrossRef]

J. Petschulat, J. Yang, C. Menzel, C. Rockstuhl, A. Chipouline, P. Lalanne, A. Tüennermann, F. Lederer, and T. Pertsch, “Understanding the electric and magnetic response of isolated metaatoms by means of a multipolar field decomposition,” Opt. Express18(14), 14454–14466 (2010).
[CrossRef] [PubMed]

S. Mühlig, C. Rockstuhl, J. Pniewski, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Three-dimensional metamaterial nanotips,” Phys. Rev. B81(7), 075317 (2010).
[CrossRef]

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett.99(1), 017401 (2007).
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R. Ruppin, “Evaluation of extended Maxwell-Garnett theories,” Opt. Commun.182(4-6), 273–279 (2000).
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Sánchez-Gil, J. A.

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(12), 123017 (2011).
[CrossRef]

Sarychev, A. K.

Scharf, T.

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett.99(1), 017401 (2007).
[CrossRef] [PubMed]

Scheeler, S.

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

Schuller, J. A.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99(10), 107401 (2007).
[CrossRef] [PubMed]

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. B82(4), 045404 (2010).
[CrossRef]

Shalaev, V. M.

Shalkevich, N.

Simovski, C. R.

S. Mühlig, C. Rockstuhl, J. Pniewski, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Three-dimensional metamaterial nanotips,” Phys. Rev. B81(7), 075317 (2010).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B79(4), 045111 (2009).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science315(5808), 47–49 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett.30(23), 3198–3200 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999).
[CrossRef]

Synowicki, R. A.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Taubner, T.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99(10), 107401 (2007).
[CrossRef] [PubMed]

Towner, R.

P. Qiu, C. Jensen, N. Charity, R. Towner, and C. Mao, “Oil phase evaporation-induced self-assembly of hydrophobic nanoparticles into spherical clusters with controlled surface chemistry in an oil-in-water dispersion and comparison of behaviors of individual and clustered iron oxide nanoparticles,” J. Am. Chem. Soc.132(50), 17724–17732 (2010).
[CrossRef] [PubMed]

Tretyakov, S. A.

S. Mühlig, C. Rockstuhl, J. Pniewski, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Three-dimensional metamaterial nanotips,” Phys. Rev. B81(7), 075317 (2010).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B79(4), 045111 (2009).
[CrossRef]

Tüennermann, A.

Vallecchi, A.

Vitanov, N. V.

V. Yannopapas and N. V. Vitanov, “Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength,” Phys. Rev. B74(19), 193304 (2006).
[CrossRef]

Wang, D.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Wang, X.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Wegener, M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science315(5808), 47–49 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett.30(23), 3198–3200 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Xu, Y. L.

Yang, J.

Yang, M. K.

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Yannopapas, V.

S. Mühlig, C. Rockstuhl, V. Yannopapas, T. Bürgi, N. Shalkevich, and F. Lederer, “Optical properties of a fabricated self-assembled bottom-up bulk metamaterial,” Opt. Express19(10), 9607–9616 (2011).
[CrossRef] [PubMed]

V. Yannopapas, “Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies,” Appl. Phys., A Mater. Sci. Process.87(2), 259–264 (2007).
[CrossRef]

V. Yannopapas and N. V. Vitanov, “Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength,” Phys. Rev. B74(19), 193304 (2006).
[CrossRef]

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter17(25), 3717–3734 (2005).
[CrossRef] [PubMed]

Yuan, H. K.

Zhang, H.

I. Hussain, H. Zhang, M. Brust, J. Barauskas, and A. I. Cooper, “Emulsions-directed assembly of gold nanoparticles to molecularly-linked and size-controlled spherical aggregates,” J. Colloid Interface Sci.350(1), 368–372 (2010).
[CrossRef] [PubMed]

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett.98(15), 157403 (2007).
[CrossRef] [PubMed]

Zheludev, N. I.

N. I. Zheludev, “Applied physics. The road ahead for metamaterials,” Science328(5978), 582–583 (2010).
[CrossRef] [PubMed]

Zhou, J. F.

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett.30(23), 3198–3200 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Zia, R.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99(10), 107401 (2007).
[CrossRef] [PubMed]

ACS Nano

S. Mühlig, A. Cunningham, S. Scheeler, C. Pacholski, T. Bürgi, C. Rockstuhl, and F. Lederer, “Self-assembled plasmonic core-shell clusters with an isotropic magnetic dipole response in the visible range,” ACS Nano5(8), 6586–6592 (2011).
[CrossRef] [PubMed]

Angew. Chem. Int. Ed. Engl.

F. Bai, D. Wang, Z. Huo, W. Chen, L. Liu, X. Liang, C. Chen, X. Wang, Q. Peng, and Y. Li, “A versatile bottom-up assembly approach to colloidal spheres from nanocrystals,” Angew. Chem. Int. Ed. Engl.46(35), 6650–6653 (2007).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys., A Mater. Sci. Process.

V. Yannopapas, “Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies,” Appl. Phys., A Mater. Sci. Process.87(2), 259–264 (2007).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999).
[CrossRef]

J. Am. Chem. Soc.

P. Qiu, C. Jensen, N. Charity, R. Towner, and C. Mao, “Oil phase evaporation-induced self-assembly of hydrophobic nanoparticles into spherical clusters with controlled surface chemistry in an oil-in-water dispersion and comparison of behaviors of individual and clustered iron oxide nanoparticles,” J. Am. Chem. Soc.132(50), 17724–17732 (2010).
[CrossRef] [PubMed]

J. Colloid Interface Sci.

I. Hussain, H. Zhang, M. Brust, J. Barauskas, and A. I. Cooper, “Emulsions-directed assembly of gold nanoparticles to molecularly-linked and size-controlled spherical aggregates,” J. Colloid Interface Sci.350(1), 368–372 (2010).
[CrossRef] [PubMed]

J. Phys. Condens. Matter

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter17(25), 3717–3734 (2005).
[CrossRef] [PubMed]

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter14(15), 4035–4044 (2002).
[CrossRef]

J. Vac. Sci. Technol. B

R. A. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B22(6), 3450–3453 (2004).
[CrossRef]

Metamaterials (Amst.)

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials (Amst.)5(2-3), 64–73 (2011).
[CrossRef]

Nat. Photonics

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Figures (6)

Fig. 1
Fig. 1

A silver NP metasphere behaves as high permittivity dielectric sphere. When the first Mie resonance is excited, it acts as magnetic dipole m , induced by displacement current j circulating in a plane perpendicular to the polarization of the incident magnetic field.

Fig. 2
Fig. 2

(a) Experimental (plain) and calculated (dashed) absorption spectrum of the silver NP ink. Inset: TEM image of the silver NPs. (b) SEM image of the cross-section of a spin coated film of silver NPs. Inset: top view of the film.

Fig. 3
Fig. 3

(a) Real (left) and imaginary (right) part of the permittivity of silver NPs dispersed at different NP volume filling fractions f. For more clarity, the data obtained for the highest f (e.g. f = 0.3-0.4) have been divided by a factor 4. (b) Maximum and minimum values of the real permittivity as a function of the filling fraction as measured by ellipsometry (circle) and calculated from the Maxwell-Garnett formula (line).

Fig. 4
Fig. 4

(a) Sketch of the formation of spherical silver NP clusters in oil-in-water emulsions following the addition of a molecular dithiol linker. (b) SEM images of the raw emulsion showing the polymeric nanocapsules. (c) TEM images of the NP clusters.

Fig. 5
Fig. 5

(a) Experimental extinction spectra of the silver NP cluster dispersions for increasing cluster size. The black curve is the spectrum of isolated NPs. Inset: photo of the vials containing the NP clusters. (b) Simulated extinction spectra spherical silver NP clusters with different radiuses.

Fig. 6
Fig. 6

(a) Contribution of the magnetic dipole moment to the scattering cross section of the NP cluster as a function of its radius. (b) Contribution of the various multipole moments (p, electric dipole; m, magnetic dipole; Q,electric quadrupole) to the scattering cross section for the larger cluster simulated (r = 75nm). The cluster was illuminated by a plane wave propagating along z direction with a polarization parallel to x direction. (c) Simulated effective permeability of a fcc arrangement of silver NP clusters (f = 0.68).

Equations (1)

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ε 1 +i ε 2 = sin 2 θ{ 1+ ( 1ρ 1+ρ ) 2 tan 2 θ }withρ= r p r s =tanΨ e iΔ

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