Abstract

We present a comprehensive theoretical and numerical analysis of the physical mechanisms pertaining to the nonlinear pulsed excitation of optical modes in plasmonic cavities made of metallic nanowires. Our analysis is based on extensive numerical simulations carried out both in the frequency and time domains. The numerical algorithm used in our study is based on the multiple scattering method and allows us to include in our analysis the effects of both the surface and bulk nonlinear polarizations generated at the second harmonic (SH). In particular, we investigate the physical roperties of plasmonic modes excited at the SH as the result of the interaction of femtosecond optical pulses with plasmonic nanocavities. We show that such cavities have two distinct types of modes, namely, plasmonic cavity modes and multipole plasmon modes generated via the hybridization of modes of single nanowires. Our analysis reveals that the properties of the latter modes depend only weakly on the cavity geometry, whereas the lifetime and quality factor of plasmonic cavity modes vary considerably with the system parameters.

© 2010 Optical Society of America

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  36. C. G. Biris, and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogenous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
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  37. J. A. Sanchez-Gil, and A. A. Maradudin, “Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects,” Opt. Express 12, 883–894 (2004).
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  40. C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21, 36–44 (2004).
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  41. C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150–2161 (2003).
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  44. J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-Harmonic Rayleigh Scattering from a Sphere of Centrosymmetric Material,” Phys. Rev. Lett. 83, 4045 (1999).
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  48. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813–822 (1968).
    [CrossRef]
  49. M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of Dark Plasmons in Metal Nanoparticles by a Localized Emitter,” Phys. Rev. Lett. 102, 107401 (2009).
    [CrossRef] [PubMed]

2010

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength Plasmonic Lattice Solitons in Arrays of Metallic Nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef] [PubMed]

M. P. Nezhand, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[CrossRef] [PubMed]

C. G. Biris, and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogenous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
[CrossRef]

2009

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–459 (2009).
[CrossRef] [PubMed]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

S. Kocaman, R. Chatterjee, N. C. Panoiu, J. F. McMillan, M. B. Yu, R. M. Osgood, D. L. Kwong, and C. W. Wong, “Observation of Zeroth-Order Band Gaps in Negative-Refraction Photonic Crystal Superlattices at Near-Infrared Frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

Y. Zeng, W. Hoyer, J. J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of Dark Plasmons in Metal Nanoparticles by a Localized Emitter,” Phys. Rev. Lett. 102, 107401 (2009).
[CrossRef] [PubMed]

2008

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78, 205322 (2008).
[CrossRef]

G. Kozyreff, J. L. D. Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[CrossRef]

Y. Xu, M. Han, A. B. Wang, Z. Liu, and J. R. Heflin, “Second order parametric processes in nonlinear silica microspheres,” Phys. Rev. Lett. 100, 163905 (2008).
[CrossRef] [PubMed]

R. D. R. Bhat, N. C. Panoiu, S. R. J. Brueck, and R. M. Osgood, “Enhancing the signal-to-noise ratio of an infrared photodetector with a circular metal grating,” Opt. Express 16, 4588–4596 (2008).
[CrossRef] [PubMed]

2007

N. C. Panoiu, and R. M. Osgood, “Enhanced optical absorption for photovoltaics via excitation of waveguide and plasmon-polariton modes,” Opt. Lett. 32, 2825–2827 (2007).
[CrossRef] [PubMed]

L. Cao, N. C. Panoiu, and R. M. Osgood, “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007).
[CrossRef]

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ /4 Resonance of an Optical Monopole Antenna Probed by Single Molecule Fluorescence,” Nano Lett. 7, 28 (2007).
[CrossRef] [PubMed]

2006

X. W. Wang, G. C. Schatz, and S. K. Gray, “Ultrafast pulse excitation of a metallic nanosystem containing a Kerr nonlinear material,” Phys. Rev. B 74, 195439 (2006).
[CrossRef]

N. C. Panoiu, R. M. Osgood, S. Zhang, and S. R. J. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[CrossRef]

W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second Harmonic Generation from a Nano-patterned Isotropic Nonlinear Material,” Nano Lett. 6, 1027–1030 (2006).
[CrossRef]

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong Modification of the Nonlinear Optical Response of Metallic Subwavelength Hole Arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-plasmon field enhancement from asymmetrically illuminated conical metallic-probe tips,” Opt. Express 14, 2921–2913 (2006).
[CrossRef] [PubMed]

Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmons on grating structures,” Appl. Phys. Lett. 89, 151116 (2006).
[CrossRef]

E. Centeno, and D. Felbacq, “Second-harmonic emission in two-dimensional photonic crystals,” J. Opt. Soc. Am. B 23, 2257–2264 (2006).
[CrossRef]

2005

T. Ishi, J. Fujikata, K. Makita, T. Baba, and K. Ohashi, “Si nano-photodiode with a surface plasmon antenna,” Jpn. J. Appl. Phys. 44, L364–L366 (2005).
[CrossRef]

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

2004

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E 70, 046608 (2004).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004).
[CrossRef] [PubMed]

N. C. Panoiu, and R. M. Osgood, “Subwavelength Nonlinear Plasmonic Nanowire,” Nano Lett. 4, 2427–2430 (2004).
[CrossRef]

J. A. Sanchez-Gil, and A. A. Maradudin, “Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects,” Opt. Express 12, 883–894 (2004).
[CrossRef] [PubMed]

C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21, 36–44 (2004).
[CrossRef]

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96, 3626–3635 (2004).
[CrossRef]

2003

C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150–2161 (2003).
[CrossRef]

D. G. Gusev, I. V. Soboleva, M. G. Martemyanov, T. V. Dolgova, A. A. Fedyanin, and O. A. Aktsipetrov, “Enhanced second-harmonic generation in coupled microcavities based on all-silicon photonic crystals,” Phys. Rev. B 68, 233303 (2003).
[CrossRef]

S. I. Bozhevolnyi, J. Beermann, and V. Coello, “Direct Observation of Localized Second-Harmonic Enhancement in Random Metal Nanostructures,” Phys. Rev. Lett. 90, 197403 (2003).
[CrossRef] [PubMed]

2002

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-Photon Tunneling via Localized Surface Plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[CrossRef] [PubMed]

2001

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

2000

1999

B. Knoll, and F. Keilmann, “Near-field probing of vibrational absorption for chemical microscopy,” Nature 399, 134–137 (1999).
[CrossRef]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-Harmonic Rayleigh Scattering from a Sphere of Centrosymmetric Material,” Phys. Rev. Lett. 83, 4045 (1999).
[CrossRef]

1997

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single Molecule Detection Using Surface-Enhanced Raman Scattering (SERS),” Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

V. Berger, “Second-harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B 14, 1351–1360 (1997).
[CrossRef]

1985

1971

J. Rudnick, and E. A. Stern, “Second-harmonic Radiation from Metal Surfaces,” Phys. Rev. B 4, 4274 (1971).
[CrossRef]

1968

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Abdenour, A.

W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second Harmonic Generation from a Nano-patterned Isotropic Nonlinear Material,” Nano Lett. 6, 1027–1030 (2006).
[CrossRef]

Adams, M. M.

Aktsipetrov, O. A.

D. G. Gusev, I. V. Soboleva, M. G. Martemyanov, T. V. Dolgova, A. A. Fedyanin, and O. A. Aktsipetrov, “Enhanced second-harmonic generation in coupled microcavities based on all-silicon photonic crystals,” Phys. Rev. B 68, 233303 (2003).
[CrossRef]

Alexander, R. W.

Baba, T.

T. Ishi, J. Fujikata, K. Makita, T. Baba, and K. Ohashi, “Si nano-photodiode with a surface plasmon antenna,” Jpn. J. Appl. Phys. 44, L364–L366 (2005).
[CrossRef]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Beermann, J.

S. I. Bozhevolnyi, J. Beermann, and V. Coello, “Direct Observation of Localized Second-Harmonic Enhancement in Random Metal Nanostructures,” Phys. Rev. Lett. 90, 197403 (2003).
[CrossRef] [PubMed]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Bell, R. J.

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S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004).
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M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of Dark Plasmons in Metal Nanoparticles by a Localized Emitter,” Phys. Rev. Lett. 102, 107401 (2009).
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S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004).
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Y. Zeng, W. Hoyer, J. J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
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M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of Dark Plasmons in Metal Nanoparticles by a Localized Emitter,” Phys. Rev. Lett. 102, 107401 (2009).
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Y. Xu, M. Han, A. B. Wang, Z. Liu, and J. R. Heflin, “Second order parametric processes in nonlinear silica microspheres,” Phys. Rev. Lett. 100, 163905 (2008).
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M. P. Nezhand, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second Harmonic Generation from a Nano-patterned Isotropic Nonlinear Material,” Nano Lett. 6, 1027–1030 (2006).
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G. Kozyreff, J. L. D. Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
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T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ /4 Resonance of an Optical Monopole Antenna Probed by Single Molecule Fluorescence,” Nano Lett. 7, 28 (2007).
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Y. Zeng, W. Hoyer, J. J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
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R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-plasmon field enhancement from asymmetrically illuminated conical metallic-probe tips,” Opt. Express 14, 2921–2913 (2006).
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S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
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N. C. Panoiu, and R. M. Osgood, “Subwavelength Nonlinear Plasmonic Nanowire,” Nano Lett. 4, 2427–2430 (2004).
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B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–459 (2009).
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F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength Plasmonic Lattice Solitons in Arrays of Metallic Nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
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C. G. Biris, and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogenous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
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S. Kocaman, R. Chatterjee, N. C. Panoiu, J. F. McMillan, M. B. Yu, R. M. Osgood, D. L. Kwong, and C. W. Wong, “Observation of Zeroth-Order Band Gaps in Negative-Refraction Photonic Crystal Superlattices at Near-Infrared Frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
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R. D. R. Bhat, N. C. Panoiu, S. R. J. Brueck, and R. M. Osgood, “Enhancing the signal-to-noise ratio of an infrared photodetector with a circular metal grating,” Opt. Express 16, 4588–4596 (2008).
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N. C. Panoiu, and R. M. Osgood, “Enhanced optical absorption for photovoltaics via excitation of waveguide and plasmon-polariton modes,” Opt. Lett. 32, 2825–2827 (2007).
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W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second Harmonic Generation from a Nano-patterned Isotropic Nonlinear Material,” Nano Lett. 6, 1027–1030 (2006).
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N. C. Panoiu, R. M. Osgood, S. Zhang, and S. R. J. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
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R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-plasmon field enhancement from asymmetrically illuminated conical metallic-probe tips,” Opt. Express 14, 2921–2913 (2006).
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S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
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N. C. Panoiu, and R. M. Osgood, “Subwavelength Nonlinear Plasmonic Nanowire,” Nano Lett. 4, 2427–2430 (2004).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
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M. P. Nezhand, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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M. P. Nezhand, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004).
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D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96, 3626–3635 (2004).
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B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–459 (2009).
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S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
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Supplementary Material (2)

» Media 1: AVI (2045 KB)     
» Media 2: AVI (2009 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Schematic of the system geometry.

Fig. 2.
Fig. 2.

Top panels show logarithmic plots of the absorption cross sections, calculated for a plasmonic cavity containing 6 nanowires. The legend indicates the separation distance, in nanometers. Bottom panels show a snapshot of the temporal evolution of the intensity of the electric field. The left and right panels correspond to the FF (Media 1) and SH (Media 2), respectively. The plasmonic cavity consists of Ag cylinders with R = 200 nm and d = 60 nm. The wavelength at the FF is λFF = 858 nm and the angle of incidence is ϕ 0 = 90°.

Fig. 3.
Fig. 3.

Top panels show logarithmic plots of the absorption cross sections. The legends indicate the number of cylinders. Bottom panels present the amplitude of the electric field at the SH, for Ag cylinders with R = 200 nm and d = 20 nm, at λSH = 578 nm and ϕ 0 = 0 (for better visualization, we plot the fourth-order square root of the field amplitude).

Fig. 4.
Fig. 4.

Distribution of the amplitude of the electric field (top panels) and the real part of the magnetic field (bottom panels) at the SH, calculated for three different plasmonic cavities made of Ag cylinders with R = 200 nm and separation distance d = 20 nm. The wavelength at the SH is λSH = 336 nm and ϕ 0 = 0 (for better visualization, in the case of the electric field, we plot the fourth-order square root of the field amplitude).

Fig. 5.
Fig. 5.

Logarithmic plots of absorption cross section spectra at the SH vs. the angle of incidence ϕ 0 (top) and separation distance d (bottom), determined for a three-cylinder geometry. The top and bottom panels correspond to d = 60 nm and ϕ 0 = 0, respectively.

Fig. 6.
Fig. 6.

Top panels show logarithmic plots of the absorption cross section at the FF and the SH. The legend indicates the number of cylinders forming the plasmonic cavity. Bottom panels present the distribution of the amplitude of the electric field at the SH, for Ag cylinders with R=200 nm and separation distance d=60 nm. From left to right, the wavelength at the SH is λSH = 321 nm, λSH = 429 nm, and λSH = 333 nm.

Fig. 7.
Fig. 7.

The same as in Fig. 5 but for a hexagonal plasmonic cavity.

Fig. 8.
Fig. 8.

a) Normalized electric field inside the cavity vs. time. The numbers in the legend refer to the separation distance in nanometers. b) The dependence of the Q factor on the separation distance d: circles represent simulation results whereas the dotted line is provided as a guide to the eye. The cavity consists of Ag cylinders with R = 200 nm and the incoming pulse has T 0 = 283 fs.

Equations (18)

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H z inc ( r , t ) = e i ( k 0 · r ω 0 t ) 1 2 π H ˜ z inc ( ω ¯ ) e i ω ¯ t d ω ¯ ,
H z inc ( t ) = H 0 e t 2 2 T 0 2 e i ω 0 t ,
H ˜ z inc ( ω ω 0 ) = H 0 π T 0 e T 0 2 ( ω ω 0 ) 2 4 ,
H z inc ( r , φ ) = Σ m = a m J m ( k 0 r ) e im φ ,
H z , j sc ( r , φ ) = Σ m = b mj H m ( 2 ) ( k 0 r P j ) e im φ P j ,
H z tot ( P ) = Σ m = a m J m ( k 0 r P ) e im φ P + Σ j = 1 N Σ m = b mj H m ( 2 ) ( k 0 r P j ) e im φ P j .
Σ k = 1 N [ δ kj I ( 1 δ kj ) S j T jk ] b k = S j a j , j = 1 , 2 , . . . , N ,
S ω B = G ,
S ω = ( I S 1 T 12 S 1 T 13 S 2 T 21 I S 2 T 23 S 3 T 31 S 3 T 32 I ) .
P s ( r ; 2 ω ) = ε 0 χ ̂ s ( 2 ) ( r ; 2 ω ) : E ( r ; ω ) E ( r ; ω ) δ ( r r s ) ,
P b ( 2 ω ) = α [ E ( ω ) · ] E ( ω ) + β E ( ω ) [ · E ( ω ) ] + γ [ E ( ω ) · E ( ω ) ] ,
α = 0 ; β = ε 0 e 2 m 0 ω 2 ; γ = β 4 [ 1 ε r ( ω ) ] ,
H z , j tot ( P , Ω ) = Σ m = ( a Ω , mj + b Ω , mj ) H m ( 2 ) ( kr P j ) e im φ P j
+ Σ k = 1 k j N Σ m , q = T jk , mq ( a Ω , qk + b Ω , qk ) J m ( kr P j ) e im φ P j .
Σ k = 1 k j N S Ω , j T jk ( a Ω , k + b Ω , k ) = ( a Ω , j + b Ω , j ) g Ω , j self , j = 1 , 2 , . . . , N ,
S Ω B Ω = S Ω A Ω + G Ω self .
E ( t ) = E 0 e ω r t Q ,
1 Q = 1 Q abs + 1 Q rad ,

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