Abstract

We verify numerically and experimentally the accuracy of an analytical model used to derive the effective nonlinear susceptibilities of a varactor-loaded split ring resonator (VLSRR) magnetic medium. For the numerical validation, a nonlinear oscillator model for the effective magnetization of the metamaterial is applied in conjunction with Maxwell equations and the two sets of equations solved numerically in the time-domain. The computed second harmonic generation (SHG) from a slab of a nonlinear material is then compared with the analytical model. The computed SHG is in excellent agreement with that predicted by the analytical model, both in terms of magnitude and spectral characteristics. Moreover, experimental measurements of the power transmitted through a fabricated VLSRR metamaterial at several power levels are also in agreement with the model, illustrating that the effective medium techniques associated with metamaterials can accurately be transitioned to nonlinear systems.

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  1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  2. M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
    [CrossRef]
  3. I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express 14, 9344–9349 (2006).
    [CrossRef] [PubMed]
  4. B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Express 16, 16058–16063 (2008).
    [CrossRef] [PubMed]
  5. I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar, “Tunable transmission and harmonic generation in nonlinear metamaterials,” Appl. Phys. Lett. 93, 161903 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  7. A. B. Kozyrev and D. W. van derWeide, “Nonlinear left-handed transmission line metamaterials,” J. Phys. D: Appl. Phys. 41, 173001 (10pp) (2008).
  8. E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” N. J. Phys. 12, 093010 (2010).
    [CrossRef]
  9. R. W. Boyd, Nonlinear Optics , 3rd ed. (Academic Press, 2008).
  10. S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial,” Appl. Phys. Lett. 97, 011109 (2010).
    [CrossRef]
  11. S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
    [CrossRef]
  12. D. Huang, E. Poutrina, and D. R. Smith, “Analysis of the power dependent tuning of a varactor-loaded metamaterial at microwave frequencies,” Appl. Phys. Lett. 96, 104104 (3pp) (2010).
  13. J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelinek , “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. 98, 033103 (2005).
    [CrossRef]
  14. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
  15. E. Poutrina, S. Larouche, and D. R. Smith, “Parameteric oscillator based on a single-layer resonant metamaterial,” Opt. Commun. 283, 1640–1646 (2010).
    [CrossRef]
  16. V.V. Migulin, V.I. Medvedev, E. R. Mustel, and V.N. Parygin, Theory of Oscillations . Mir Publishers, Moscow, 1983.
  17. Skyworks, “Skyworks smv123x series hyperabrupt junctiontuning varactors,” Technical report, Skyworks, September 2009. Data sheet. http://pdf1.alldatasheet.com/datasheet-pdf/view/155290/SKYWORKS/SMV1231-079.html ).
  18. I. Shadrivov, N. Zharova, A. Zharov, and Y. Kivshar, “Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction,” Opt. Express 13, 1291–1298 (2005).
    [CrossRef] [PubMed]
  19. B. Popa and S. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100, 207401 (2008).
    [CrossRef] [PubMed]

2010 (4)

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” N. J. Phys. 12, 093010 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
[CrossRef]

E. Poutrina, S. Larouche, and D. R. Smith, “Parameteric oscillator based on a single-layer resonant metamaterial,” Opt. Commun. 283, 1640–1646 (2010).
[CrossRef]

2008 (3)

B. Popa and S. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100, 207401 (2008).
[CrossRef] [PubMed]

B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Express 16, 16058–16063 (2008).
[CrossRef] [PubMed]

I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar, “Tunable transmission and harmonic generation in nonlinear metamaterials,” Appl. Phys. Lett. 93, 161903 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (2)

I. Shadrivov, N. Zharova, A. Zharov, and Y. Kivshar, “Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction,” Opt. Express 13, 1291–1298 (2005).
[CrossRef] [PubMed]

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelinek , “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

2003 (1)

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

1999 (1)

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

1951 (1)

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).

Baena, J. D.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelinek , “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics , 3rd ed. (Academic Press, 2008).

Cummer, S.

B. Popa and S. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100, 207401 (2008).
[CrossRef] [PubMed]

Feth, N.

Garcia-Garcia, J.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelinek , “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Gorkunov, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[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. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Huang, D.

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” N. J. Phys. 12, 093010 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

D. Huang, E. Poutrina, and D. R. Smith, “Analysis of the power dependent tuning of a varactor-loaded metamaterial at microwave frequencies,” Appl. Phys. Lett. 96, 104104 (3pp) (2010).

Jelinek, L.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelinek , “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Kivshar, Y.

Kivshar, Y. S.

I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar, “Tunable transmission and harmonic generation in nonlinear metamaterials,” Appl. Phys. Lett. 93, 161903 (2008).
[CrossRef]

I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express 14, 9344–9349 (2006).
[CrossRef] [PubMed]

Klein, M. W.

Koschny, T.

Kozyrev, A. B.

I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar, “Tunable transmission and harmonic generation in nonlinear metamaterials,” Appl. Phys. Lett. 93, 161903 (2008).
[CrossRef]

A. B. Kozyrev and D. W. van derWeide, “Nonlinear left-handed transmission line metamaterials,” J. Phys. D: Appl. Phys. 41, 173001 (10pp) (2008).

Lapine, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Larouche, S.

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
[CrossRef]

E. Poutrina, S. Larouche, and D. R. Smith, “Parameteric oscillator based on a single-layer resonant metamaterial,” Opt. Commun. 283, 1640–1646 (2010).
[CrossRef]

Linden, S.

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).

Marques, R.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelinek , “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Martin, F.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelinek , “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Medvedev, V.I.

V.V. Migulin, V.I. Medvedev, E. R. Mustel, and V.N. Parygin, Theory of Oscillations . Mir Publishers, Moscow, 1983.

Migulin, V.V.

V.V. Migulin, V.I. Medvedev, E. R. Mustel, and V.N. Parygin, Theory of Oscillations . Mir Publishers, Moscow, 1983.

Morrison, S. K.

Mustel, E. R.

V.V. Migulin, V.I. Medvedev, E. R. Mustel, and V.N. Parygin, Theory of Oscillations . Mir Publishers, Moscow, 1983.

Parygin, V.N.

V.V. Migulin, V.I. Medvedev, E. R. Mustel, and V.N. Parygin, Theory of Oscillations . Mir Publishers, Moscow, 1983.

Pendry, J. B.

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

Popa, B.

B. Popa and S. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100, 207401 (2008).
[CrossRef] [PubMed]

Poutrina, E.

E. Poutrina, S. Larouche, and D. R. Smith, “Parameteric oscillator based on a single-layer resonant metamaterial,” Opt. Commun. 283, 1640–1646 (2010).
[CrossRef]

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” N. J. Phys. 12, 093010 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

D. Huang, E. Poutrina, and D. R. Smith, “Analysis of the power dependent tuning of a varactor-loaded metamaterial at microwave frequencies,” Appl. Phys. Lett. 96, 104104 (3pp) (2010).

Ringhofer, K. H.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[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. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Rose, A.

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).

Shadrivov, I.

Shadrivov, I. V.

I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar, “Tunable transmission and harmonic generation in nonlinear metamaterials,” Appl. Phys. Lett. 93, 161903 (2008).
[CrossRef]

I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express 14, 9344–9349 (2006).
[CrossRef] [PubMed]

Smith, D. R.

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” N. J. Phys. 12, 093010 (2010).
[CrossRef]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

E. Poutrina, S. Larouche, and D. R. Smith, “Parameteric oscillator based on a single-layer resonant metamaterial,” Opt. Commun. 283, 1640–1646 (2010).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).

D. Huang, E. Poutrina, and D. R. Smith, “Analysis of the power dependent tuning of a varactor-loaded metamaterial at microwave frequencies,” Appl. Phys. Lett. 96, 104104 (3pp) (2010).

Soukoulis, C. M.

B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Express 16, 16058–16063 (2008).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).

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. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

van der Weide, D. W.

I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar, “Tunable transmission and harmonic generation in nonlinear metamaterials,” Appl. Phys. Lett. 93, 161903 (2008).
[CrossRef]

van derWeide, D. W.

A. B. Kozyrev and D. W. van derWeide, “Nonlinear left-handed transmission line metamaterials,” J. Phys. D: Appl. Phys. 41, 173001 (10pp) (2008).

Wang, B.

Wegener, M.

Zharov, A.

Zharova, N.

Zhou, J.

Appl. Phys. Lett. (3)

I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar, “Tunable transmission and harmonic generation in nonlinear metamaterials,” Appl. Phys. Lett. 93, 161903 (2008).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

D. Huang, E. Poutrina, and D. R. Smith, “Analysis of the power dependent tuning of a varactor-loaded metamaterial at microwave frequencies,” Appl. Phys. Lett. 96, 104104 (3pp) (2010).

IEEE Trans. Microwave Theory Tech. (1)

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

J. Appl. Phys. (1)

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelinek , “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

J. Phys. D: Appl. Phys. (1)

A. B. Kozyrev and D. W. van derWeide, “Nonlinear left-handed transmission line metamaterials,” J. Phys. D: Appl. Phys. 41, 173001 (10pp) (2008).

N. J. Phys. (1)

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” N. J. Phys. 12, 093010 (2010).
[CrossRef]

Opt. Commun. (2)

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
[CrossRef]

E. Poutrina, S. Larouche, and D. R. Smith, “Parameteric oscillator based on a single-layer resonant metamaterial,” Opt. Commun. 283, 1640–1646 (2010).
[CrossRef]

Opt. Express (4)

Phys. Rev. B (1)

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).

Phys. Rev. E (1)

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

B. Popa and S. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100, 207401 (2008).
[CrossRef] [PubMed]

Other (3)

V.V. Migulin, V.I. Medvedev, E. R. Mustel, and V.N. Parygin, Theory of Oscillations . Mir Publishers, Moscow, 1983.

Skyworks, “Skyworks smv123x series hyperabrupt junctiontuning varactors,” Technical report, Skyworks, September 2009. Data sheet. http://pdf1.alldatasheet.com/datasheet-pdf/view/155290/SKYWORKS/SMV1231-079.html ).

R. W. Boyd, Nonlinear Optics , 3rd ed. (Academic Press, 2008).

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Figures (3)

Fig. 1
Fig. 1

(a) Orientation of the unit cell; (b) Equivalent effective circuit model; (c) Simulations geometry; (d) Second order effective nonlinear susceptibility χ (2)(2ω) calculated from Eq. (3).

Fig. 2
Fig. 2

(a) Field distribution inside the transmission line at the crossection before the sample; the inset shows the transmission line structure with the material inside. (b) Field distribution after the sample inside the transmission line.

Fig. 3
Fig. 3

(a) Experimental amplitude of the Hy field component at the second harmonic, versus the simulation results; (b) Magnetization at the second harmonic obtained by the numerical solution to NL oscillator model (black lines) and analytically (red lines), for several values of the driving field (c) Same as in (b), with the numerical solution obtained from Comsol simulations, for the lower three values of the excitation field shown in (b). The grey gradient color areas indicate the range of the second-order magnetization amplitude that could be produced with a variation of the fundamental driving magnetic field shown on the inset.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

q ˜ ¨ + γ q ˜ ˙ + ω 0 2 V D ( q ˜ ) = ω 0 2 A μ 0 H ˜ ˙ y
χ y ( 1 ) ( ω ) = F ω 2 D ( ω )
χ y y y ( 2 ) ( ω r ; ω n , ω m ) = i a ω 0 4 ( ω n + ω m ) ω n ω m μ 0 A F D ( ω n ) D ( ω m ) D ( ω n + ω m ) ,
M ˜ ¨ y + γ M ˜ ˙ y + ω 0 2 M ˜ y = F H ˜ ¨ y α M ˜ y M ˜ y d t

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