Abstract

Parametric optics and second harmonic generation in pure plasmonic particle chains are studied. By a proper design of the plasmonic particle geometry, the modes supported by the chain can achieve phase-matching conditions. Then the magnetic-field dependence of the plasmon electric susceptibility can provide the nonlinearity and the coupling mechanism leading to parametric processes, sum frequency and second harmonic generation. Hence, chains of plasmonic particles can support parametric optics and higher harmonic generation by using its own modes only. Since the second order nonlinearity involves both electric and magnetic fields, the SHG reported here is supported also by centrosymmetric particle chains.

© 2011 OSA

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  4. Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (2006).
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  7. Y. Hadad and Ben Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105, 233904 (2010).
    [CrossRef]
  8. Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
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    [CrossRef] [PubMed]
  13. J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

2010

H. Husu, J. Makitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonance in L-shaped gold nano-particles,” Opt. Express 18(16), 16601–16606 (2010).
[CrossRef] [PubMed]

D. V. Orden, Y. Fainman, and V. Lomakin, “Twisted chains of resonant particles: optical polarization control, waveguidance, and radiation,” Opt. Lett. 35(15), 2579–2581 (2010).
[CrossRef] [PubMed]

Y. Hadad and Ben Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105, 233904 (2010).
[CrossRef]

Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
[CrossRef]

2009

2008

2007

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, and M. Kauranen, “Multipole interference of second harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

2006

Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73, 023819 (2006).
[CrossRef]

2005

J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[CrossRef]

2003

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

2000

S. A. Tretyakov and A. J. Viitanen, “Line of periodically arranged passive dipole scatterers,” Electrical Engineering 82, 353–361 (2000).
[CrossRef]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356 (2000).
[CrossRef]

1998

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover1972).

Alu, Andrea

Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

Antoine, I. R.

G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25(6) 955–960 (2008).
[CrossRef]

J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356 (2000).
[CrossRef]

Aussenegg, F. R.

Bachelier, G.

Bai, B.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Benichou, E.

G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25(6) 955–960 (2008).
[CrossRef]

J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[CrossRef]

Brevet, P. F.

G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25(6) 955–960 (2008).
[CrossRef]

J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[CrossRef]

Brongersma, M. L.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356 (2000).
[CrossRef]

Canfield, B. K.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, and M. Kauranen, “Multipole interference of second harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Dadap, J. I.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73, 023819 (2006).
[CrossRef]

Davoyan, A. R.

Engheta, Nader

Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

Fainman, Y.

Giessen, H.

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

Hadad, Y.

Y. Hadad and Ben Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105, 233904 (2010).
[CrossRef]

Hartman, J. W.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356 (2000).
[CrossRef]

Heinz, T. F.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73, 023819 (2006).
[CrossRef]

Hoyer, W.

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

Hu, Wei

Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
[CrossRef]

Hu, Xi-kui

Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
[CrossRef]

Husu, H.

H. Husu, J. Makitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonance in L-shaped gold nano-particles,” Opt. Express 18(16), 16601–16606 (2010).
[CrossRef] [PubMed]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Whiley1999).

Jonin, C.

G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25(6) 955–960 (2008).
[CrossRef]

J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[CrossRef]

Kauranen, M.

H. Husu, J. Makitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonance in L-shaped gold nano-particles,” Opt. Express 18(16), 16601–16606 (2010).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, and M. Kauranen, “Multipole interference of second harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Kivshar, Y. S.

Koch, S. W.

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

Krenn, J. R.

Kuittinen, M.

H. Husu, J. Makitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonance in L-shaped gold nano-particles,” Opt. Express 18(16), 16601–16606 (2010).
[CrossRef] [PubMed]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Kujala, S.

S. Kujala, B. K. Canfield, and M. Kauranen, “Multipole interference of second harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Laukkanen, J.

H. Husu, J. Makitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonance in L-shaped gold nano-particles,” Opt. Express 18(16), 16601–16606 (2010).
[CrossRef] [PubMed]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Lederer, F.

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

Leitner, A.

Lewin, Leonard

Leonard Lewin, Polylogarithms and Associated Functions (Elsevier1981).

Lippitz, M.

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

Liu, J.

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

Lomakin, V.

Lu, Yan-qing

Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Makitalo, J.

Moloney, J.

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

Nappa, J.

J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[CrossRef]

Orden, D. V.

Paul, T.

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

Quinten, M.

Reider, G. A.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73, 023819 (2006).
[CrossRef]

Revillod, G.

J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[CrossRef]

Rockstuhl, C.

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

Shadrivov, I. V.

Shan, J.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73, 023819 (2006).
[CrossRef]

Sihvola, A. H.

A. H. Sihvola, Electromagnetic Mixing Formulas and Applications, Electromagnetic Waves Series (IEE1999).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover1972).

Steinberg, Ben Z.

Y. Hadad and Ben Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105, 233904 (2010).
[CrossRef]

Stiopkin, I.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73, 023819 (2006).
[CrossRef]

Tretyakov, S. A.

S. A. Tretyakov and A. J. Viitanen, “Line of periodically arranged passive dipole scatterers,” Electrical Engineering 82, 353–361 (2000).
[CrossRef]

Turunen, J.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Utikel, T.

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

Viitanen, A. J.

S. A. Tretyakov and A. J. Viitanen, “Line of periodically arranged passive dipole scatterers,” Electrical Engineering 82, 353–361 (2000).
[CrossRef]

Wu, Zi-jian

Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
[CrossRef]

Xu, Fei

Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
[CrossRef]

Yu, Zi-yan

Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
[CrossRef]

Zeng, Y.

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

Zentgraf, T.

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

Electrical Engineering

S. A. Tretyakov and A. J. Viitanen, “Line of periodically arranged passive dipole scatterers,” Electrical Engineering 82, 353–361 (2000).
[CrossRef]

J. Opt. Soc. Am. B

Nano Letters

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. A

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73, 023819 (2006).
[CrossRef]

Phys. Rev. B

J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[CrossRef]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356 (2000).
[CrossRef]

Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82, 155107 (2010).
[CrossRef]

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

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Phys. Rev. Lett.

Y. Hadad and Ben Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105, 233904 (2010).
[CrossRef]

S. Kujala, B. K. Canfield, and M. Kauranen, “Multipole interference of second harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Whiley1999).

A. H. Sihvola, Electromagnetic Mixing Formulas and Applications, Electromagnetic Waves Series (IEE1999).
[CrossRef]

Leonard Lewin, Polylogarithms and Associated Functions (Elsevier1981).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover1972).

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Figures (4)

Fig. 1
Fig. 1

A subdiffraction chain of ellipsoidal particles (dλ). Resonances, phase-matching conditions, and gain can be achieved by designing d and the ellipsoids semi-axis ax, ay, az. (a) Chain Geometry for prolate ellipsoid particles. (b) Source of non-linearity. The chain supports a modal dipole response of the form pm = x̂p0eimβ. We examine the specific case of βπ/2; for any n, there is a phase difference of π between the dipole responses of the two nearest neighbors of the n-th particle. As a result their (ŷ-directed) magnetic fields add constructively at the n-th site, creating a modulation of the n-th particle (in blue) electric susceptibility component relevant for a -directed dipole. Hence a parametric coupling between the and the polarized dipole excitations is created, both can be supported by the chain dispersion. Generally, at the n-th site, the magnetic fields of the n ± neighbors add constructively (destructively) for odd (even) . The strongest contributions are from = 1. This general picture holds also if β deviates from π/2. In fact, the maximal value of H is obtained for β ≈ 0.4π.

Fig. 2
Fig. 2

Geometrical parameters of prolate or oblate ellipsoid SDC, supporting (a) SHG of Eq. (8) for various values of β1, and (b) SFG of Eq. (9) for various values of β2.

Fig. 3
Fig. 3

The n-th particle magnetization due to the chain’s own electric-dipole mode. (a) The exact Hn of Eq. (15) shown in the kd,β/π plane. The field is in units of ×pnck3/(4π) on logarithmic scale. (b) The NNA error for values of β, near π/2.

Fig. 4
Fig. 4

SHG example in a lossy plasmonic chain, for pump of 1V/m at the chain input.

Equations (39)

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α _ = ɛ 0 V ( I _ 3 + χ _ L _ ) 1 χ _
χ _ = χ e e [ I _ 3 + i B _ H ] , χ e e = ω p 2 ω ( ω + i / τ )
B _ H = e μ 0 ω m e ( 0 H z H y H z 0 H x H y H x 0 ) e μ 0 ω m e H ×
p n = 1 4 π ɛ 0 d 3 α _ [ 3 z ^ ( z ^ S n ) S n ]
S n p n + 1 + p n 1 .
p n = p 0 e i β n
( ω / ω p ) 2 = N u + σ u cos ( β ) , u = x , y , z
SHG : ( ω 3 , β 3 ) = ( 2 ω 1 , 2 β 1 )
SFG : ( ω 3 , β 3 ) = ( ω 1 + ω 2 , β 1 + β 2 ) .
N z 4 N x = 2 σ [ cos ( 2 β 1 ) + 2 cos ( β 1 ) ]
σ cos ( β 2 ) + 2 ( N x N y + N x σ cos β 2 ) 1 / 2 = 2 N z 1 ± 2 σ sin β 2 .
H n m = z ^ × p m sgn ( m n ) c k 2 g ( d m n ) [ 1 + i / ( k d m n ) ]
H n = m , m n H n m = z ^ × p n c k 3 4 π F ( k d , β )
F ( k d , β ) = 1 k d [ L i 1 ( e + ) L i 1 ( e ) + i k d L i 2 ( e + ) i k d L i 2 ( e ) ] , e ± e i k d ± i β
L i s ( z ) = n = 1 z n n s L i 0 ( z ) = z 1 z , L i 1 ( z ) = ln ( 1 z ) .
F ( k d , β ) 2 ( k d ) 2 sin β .
[ ( ω / ω p ) 2 + i ω / ( ω p 2 τ ) ] p n = L _ p n ( σ / 2 ) [ 3 z ^ ( z ^ S n ) S n ] + Ψ
Ψ = i A [ p n ( z ^ L _ p n ) z ^ ( p n L _ p n ) ] i A ( σ / 2 ) [ 2 p n ( z ^ S n ) + z ^ ( S n p n ) 3 z ^ ( z ^ p n ) ( z ^ S n ) ]
A = A ( k d , β ) = k 2 e μ 0 4 π m e F ( k d , β ) .
p n ( 1 ) = x ^ p ˜ n ( 1 ) e i β 1 n , p n ( 3 ) = z ^ p ˜ n ( 3 ) e i β 3 n
S ˜ n ( 1 , 3 ) = p ˜ n + 1 ( 1 , 3 ) e i β 1 , 3 + p ˜ n 1 ( 1 , 3 ) e i β 1 , 3
[ ( ω 3 / ω p ) 2 + i ω 3 / ( ω p 2 τ ) ] p ˜ n ( 3 ) = N z p ˜ n ( 3 ) σ S ˜ n ( 3 ) i ( A 1 / 4 ) p ˜ n ( 1 ) [ σ S ˜ n ( 1 ) + 2 N x p ˜ n ( 1 ) ] .
[ ( ω 1 / ω p ) 2 + i ω 1 / ( ω p 2 τ ) ] p ˜ n ( 1 ) = N x p ˜ n ( 1 ) + ( σ / 2 ) S ˜ n ( 1 ) i ( A 1 / 2 ) [ p ˜ n ( 1 ) ] * [ σ S ˜ n ( 3 ) N z p ˜ n ( 3 ) ] .
p ˜ n ( 1 ) = p ˜ 0 ( 1 ) n , p ˜ 0 ( 3 ) = 0 .
( ω 3 / ω p ) 2 p ˜ n ( 3 ) = N z p ˜ n ( 3 ) σ S ˜ n ( 3 ) i N x ( A 1 / 2 ) [ p ˜ 0 ( 1 ) ] 2 .
p ˜ n ( 3 ) = i N x ( A 1 / 4 σ ) [ p ˜ 0 ( 1 ) ] 2 n 2 .
p ˜ n ( 1 ) = p ˜ 0 ( 1 ) e γ n
p ˜ 0 ( 3 ) = 0 , lim n p ˜ n ( 3 ) = 0 .
B p ˜ n ( 3 ) + e i β 3 p ˜ n + 1 ( 3 ) + e i β 3 p ˜ n 1 ( 3 ) = [ p ˜ 0 ( 1 ) ] 2 D e 2 γ n
B = 2 cos β 3 + i ω 3 ω p 2 τ σ , D = i A 1 2 σ [ N x + σ cosh ( γ i β 1 ) ] .
p ˜ n ( 3 ) , p = [ p ˜ 0 ( 1 ) ] 2 a p e 2 γ m , a p = D B + 2 cosh ( 2 γ i β 3 ) .
r 2 + B e i β 3 r + e 2 i β 3 = 0 r 1 , 2 = e i β 3 2 ( B ± B 2 4 ) .
p ˜ n ( 3 ) = [ p ˜ 0 ( 1 ) ] 2 a p ( e 2 γ n r 1 n ) .
E ˜ n ( 3 ) = ( α _ 1 x ^ ) 2 α _ 3 z ^ [ E ˜ 0 ( 1 ) ] 2 a p ( e 2 γ n r 1 n ) .
E ˜ n ( 3 ) E ˜ n ( 1 ) = ( α _ 1 x ^ ) 2 α _ 3 z ^ E ˜ 0 ( 1 ) a p ( e 2 γ n r 1 n ) e γ n .
m e r ¨ 2 = e μ 0 r ˙ 1 × H
r 2 = i ω 1 e ω 3 2 m e r 1 × B
P 2 = i ω 1 e ω 3 2 m e P 1 × B P 2 = i ω 1 e ω 3 2 m e χ e e ( ω 1 ) E × B
P 2 = i ω 1 e ω 3 2 m e χ e e B × E L P = P 1 + P 2 = χ e e ( ω 1 ) ( I _ 3 + i e ω 1 m e * B × ) E

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