Abstract

The THz response of slit structures and split-ring resonators (SRRs) featuring extremely small gaps on the micro- or nanoscale is investigated numerically. Both structures exhibit strong field enhancement in the gap region due to light-induced current flows and capacitive charging across the gap. Whereas nanoslits allow for broadband enhancement the resonant behavior of the SRRs leads to narrowband amplification and results in significantly higher field enhancement factors reaching several 10,000. This property is particularly beneficial for the realization of nonlinear THz experiments which is exemplarily demonstrated by a second harmonic generation process in a nonlinear substrate material. Positioning nanostructures on top of the substrate is found to result in a significant increase of the generation efficiency for the frequency doubled component.

© 2011 OSA

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2010 (3)

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010).
[CrossRef] [PubMed]

J. Kyoung, M. Seo, H. Park, S. Koo, H. sun Kim, Y. Park, B.-J. Kim, K. Ahn, N. Park, H.-T. Kim, and D.-S. Kim, “Giant nonlinear response of terahertz nanoresonators on VO2 thin film,” Opt. Express 18, 16452–16459 (2010).
[CrossRef] [PubMed]

2009 (3)

M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. 95, 231105 (2009).
[CrossRef]

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

A. Bitzer, J. Wallauer, H. Helm, H. Merbold, T. Feurer, and M. Walther, “Lattice modes mediate radiative coupling in metamaterial arrays,” Opt. Express 17, 22108–22113 (2009).
[CrossRef] [PubMed]

2008 (2)

N. Laman and D. Grischkowsky, “Terahertz conductivity of thin metal films,” Appl. Phys. Lett. 93, 051105 (2008).
[CrossRef]

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008).
[CrossRef] [PubMed]

2007 (3)

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007).
[CrossRef]

2006 (2)

G. P. Williams, “Filling the THz gap-high power sources and applications,” Rep. Prog. Phys. 69, 301 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

2005 (3)

T. Bartel, P. Gaal, K. Reimann, M. Woerner, and T. Elsaesser, “Generation of single-cycle THz transients with high electric-field amplitudes,” Opt. Lett. 30, 2805–2807 (2005).
[CrossRef] [PubMed]

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. 7, S12–S22 (2005).
[CrossRef]

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

2004 (2)

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (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]

2002 (2)

J. Jin, The Finite Element Method in Electromagnetics , 2nd ed. (Wiley-IEEE Press, 2002).

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

1999 (2)

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]

V. Romero-Rochin, R. M. Koehl, C. J. Brennan, and K. A. Nelson, “Anharmonic phonon-polariton excitation through impulsive stimulated Raman scattering and detection through wave vector overtone spectroscopy: theory and comparison to experiments on lithium tantalate,” J. Chem. Phys. 11, 3559 (1999).
[CrossRef]

1997 (1)

R. Hsu, E. N. Maslen, D. du Boulay, and N. Ishizawa, “Synchrotron X-ray Studies of LiNbO3 and LiTaO3,” Acta Crystallogr., Sect. B 53, 420 (1997).
[CrossRef]

1996 (1)

I. Inbar and R. E. Cohen, “Comparison of the electronic structures and energetics of ferroelectric LiNbO3 and LiTaO3,” Phys. Rev. B 53, 1193 (1996).
[CrossRef]

1994 (1)

H. Boysen and F. Altorfer, “A neutron powder investigation of the high-temperature structure and phase transition in LiNbO3,” Acta Crystallogr., Sect. B 50, 405 (1994).
[CrossRef]

1992 (1)

R. W. Boyd, Nonlinear Optics , 1st ed. (Academic Press, 1992).

1985 (1)

1839 (1)

E. Budiarto, J. Margolies, S. Jeong, and J. Song, “High-intensity terahertz pulses at 1-kHz repetition rate,” IEEE J. Quantum Electron. 32, 1839 (1996).

Ahn, K.

Ahn, K. J.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

Ahn, Y. H.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

Alexander, R. W.

Altorfer, F.

H. Boysen and F. Altorfer, “A neutron powder investigation of the high-temperature structure and phase transition in LiNbO3,” Acta Crystallogr., Sect. B 50, 405 (1994).
[CrossRef]

Baena, J. D.

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

Bartel, T.

Bell, R. J.

Bitzer, A.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics , 1st ed. (Academic Press, 1992).

Boysen, H.

H. Boysen and F. Altorfer, “A neutron powder investigation of the high-temperature structure and phase transition in LiNbO3,” Acta Crystallogr., Sect. B 50, 405 (1994).
[CrossRef]

Brandt, N. C.

M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. 95, 231105 (2009).
[CrossRef]

Brennan, C. J.

V. Romero-Rochin, R. M. Koehl, C. J. Brennan, and K. A. Nelson, “Anharmonic phonon-polariton excitation through impulsive stimulated Raman scattering and detection through wave vector overtone spectroscopy: theory and comparison to experiments on lithium tantalate,” J. Chem. Phys. 11, 3559 (1999).
[CrossRef]

Budiarto, E.

E. Budiarto, J. Margolies, S. Jeong, and J. Song, “High-intensity terahertz pulses at 1-kHz repetition rate,” IEEE J. Quantum Electron. 32, 1839 (1996).

Choi, S. S.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Cohen, R. E.

I. Inbar and R. E. Cohen, “Comparison of the electronic structures and energetics of ferroelectric LiNbO3 and LiTaO3,” Phys. Rev. B 53, 1193 (1996).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

David, C.

S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010).
[CrossRef] [PubMed]

du Boulay, D.

R. Hsu, E. N. Maslen, D. du Boulay, and N. Ishizawa, “Synchrotron X-ray Studies of LiNbO3 and LiTaO3,” Acta Crystallogr., Sect. B 53, 420 (1997).
[CrossRef]

Economou, E. N.

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. 7, S12–S22 (2005).
[CrossRef]

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

Elsaesser, T.

Enkrich, C.

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]

Feurer, T.

A. Bitzer, J. Wallauer, H. Helm, H. Merbold, T. Feurer, and M. Walther, “Lattice modes mediate radiative coupling in metamaterial arrays,” Opt. Express 17, 22108–22113 (2009).
[CrossRef] [PubMed]

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007).
[CrossRef]

Gaal, P.

García-García, J.

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

Gay-Balmaz, P.

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

Genov, D. A.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Gorelick, S.

S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010).
[CrossRef] [PubMed]

Grischkowsky, D.

N. Laman and D. Grischkowsky, “Terahertz conductivity of thin metal films,” Appl. Phys. Lett. 93, 051105 (2008).
[CrossRef]

Gundogdu, T. F.

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. 7, S12–S22 (2005).
[CrossRef]

Guzenko, V. A.

S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010).
[CrossRef] [PubMed]

Hebling, J.

K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007).
[CrossRef]

Helm, H.

Hoffmann, M. C.

M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. 95, 231105 (2009).
[CrossRef]

K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007).
[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]

Hsu, R.

R. Hsu, E. N. Maslen, D. du Boulay, and N. Ishizawa, “Synchrotron X-ray Studies of LiNbO3 and LiTaO3,” Acta Crystallogr., Sect. B 53, 420 (1997).
[CrossRef]

Hwang, H. Y.

M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. 95, 231105 (2009).
[CrossRef]

Inbar, I.

I. Inbar and R. E. Cohen, “Comparison of the electronic structures and energetics of ferroelectric LiNbO3 and LiTaO3,” Phys. Rev. B 53, 1193 (1996).
[CrossRef]

Ishizawa, N.

R. Hsu, E. N. Maslen, D. du Boulay, and N. Ishizawa, “Synchrotron X-ray Studies of LiNbO3 and LiTaO3,” Acta Crystallogr., Sect. B 53, 420 (1997).
[CrossRef]

Jelinek, L.

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

Jeong, S.

E. Budiarto, J. Margolies, S. Jeong, and J. Song, “High-intensity terahertz pulses at 1-kHz repetition rate,” IEEE J. Quantum Electron. 32, 1839 (1996).

Jin, J.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008).
[CrossRef] [PubMed]

J. Jin, The Finite Element Method in Electromagnetics , 2nd ed. (Wiley-IEEE Press, 2002).

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Kafesaki, M.

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. 7, S12–S22 (2005).
[CrossRef]

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

Kang, J. H.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Katsarakis, N.

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

Kim, B.-J.

Kim, D. S.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Kim, D.-S.

Kim, H. S.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

Kim, H.-T.

Kim, S.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008).
[CrossRef] [PubMed]

Kim, S.-W.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008).
[CrossRef] [PubMed]

Kim, Y.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008).
[CrossRef] [PubMed]

Kim, Y.-J.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008).
[CrossRef] [PubMed]

Koehl, R. M.

V. Romero-Rochin, R. M. Koehl, C. J. Brennan, and K. A. Nelson, “Anharmonic phonon-polariton excitation through impulsive stimulated Raman scattering and detection through wave vector overtone spectroscopy: theory and comparison to experiments on lithium tantalate,” J. Chem. Phys. 11, 3559 (1999).
[CrossRef]

Koo, S.

Koo, S. M.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Koschny, T.

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. 7, S12–S22 (2005).
[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. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

Kyoung, J.

Kyoung, J. S.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

Laman, N.

N. Laman and D. Grischkowsky, “Terahertz conductivity of thin metal films,” Appl. Phys. Lett. 93, 051105 (2008).
[CrossRef]

Linden, S.

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]

Liu, H.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Liu, Y. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Liu, Z. W.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Long, L. L.

Margolies, J.

E. Budiarto, J. Margolies, S. Jeong, and J. Song, “High-intensity terahertz pulses at 1-kHz repetition rate,” IEEE J. Quantum Electron. 32, 1839 (1996).

Marqués, R.

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

Martin, O. J. F.

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

Martín, F.

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

Maslen, E. N.

R. Hsu, E. N. Maslen, D. du Boulay, and N. Ishizawa, “Synchrotron X-ray Studies of LiNbO3 and LiTaO3,” Acta Crystallogr., Sect. B 53, 420 (1997).
[CrossRef]

Merbold, H.

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Nelson, K. A.

M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. 95, 231105 (2009).
[CrossRef]

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007).
[CrossRef]

K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007).
[CrossRef]

V. Romero-Rochin, R. M. Koehl, C. J. Brennan, and K. A. Nelson, “Anharmonic phonon-polariton excitation through impulsive stimulated Raman scattering and detection through wave vector overtone spectroscopy: theory and comparison to experiments on lithium tantalate,” J. Chem. Phys. 11, 3559 (1999).
[CrossRef]

Ordal, M. A.

Park, D. J.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Park, G. S.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Park, H.

Park, H. R.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Park, I.-Y.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008).
[CrossRef] [PubMed]

Park, N.

Park, N. K.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Park, Q. H.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Park, Y.

Park, Y. M.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

Penciu, R. S.

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. 7, S12–S22 (2005).
[CrossRef]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

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]

Planken, P. C. M.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Querry, M. R.

Reimann, K.

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]

Romero-Rochin, V.

V. Romero-Rochin, R. M. Koehl, C. J. Brennan, and K. A. Nelson, “Anharmonic phonon-polariton excitation through impulsive stimulated Raman scattering and detection through wave vector overtone spectroscopy: theory and comparison to experiments on lithium tantalate,” J. Chem. Phys. 11, 3559 (1999).
[CrossRef]

Satz, E. R.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007).
[CrossRef]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Seo, M.

Seo, M. A.

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Song, J.

E. Budiarto, J. Margolies, S. Jeong, and J. Song, “High-intensity terahertz pulses at 1-kHz repetition rate,” IEEE J. Quantum Electron. 32, 1839 (1996).

Soukoulis, C. M.

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. 7, S12–S22 (2005).
[CrossRef]

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (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]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[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. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Stoyanov, N. S.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007).
[CrossRef]

Sun, C.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

sun Kim, H.

Suwal, O. K.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Vaughan, J. C.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007).
[CrossRef]

Vila-Comamala, J.

S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010).
[CrossRef] [PubMed]

Wallauer, J.

Walther, M.

Ward, D. W.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007).
[CrossRef]

Wegener, M.

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]

Williams, G. P.

G. P. Williams, “Filling the THz gap-high power sources and applications,” Rep. Prog. Phys. 69, 301 (2006).
[CrossRef]

Woerner, M.

Wu, D. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Yeh, K.

M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. 95, 231105 (2009).
[CrossRef]

K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007).
[CrossRef]

Zhang, X.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Zhou, J.

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]

Zhu, S. N.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Acta Crystallogr., Sect. B (2)

H. Boysen and F. Altorfer, “A neutron powder investigation of the high-temperature structure and phase transition in LiNbO3,” Acta Crystallogr., Sect. B 50, 405 (1994).
[CrossRef]

R. Hsu, E. N. Maslen, D. du Boulay, and N. Ishizawa, “Synchrotron X-ray Studies of LiNbO3 and LiTaO3,” Acta Crystallogr., Sect. B 53, 420 (1997).
[CrossRef]

Annu. Rev. Mater. Res. (1)

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37, 317–350 (2007).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (5)

N. Laman and D. Grischkowsky, “Terahertz conductivity of thin metal films,” Appl. Phys. Lett. 93, 051105 (2008).
[CrossRef]

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007).
[CrossRef]

H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. 96, 121106 (2010).
[CrossRef]

M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. 95, 231105 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Budiarto, J. Margolies, S. Jeong, and J. Song, “High-intensity terahertz pulses at 1-kHz repetition rate,” IEEE J. Quantum Electron. 32, 1839 (1996).

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. (2)

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

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

J. Chem. Phys. (1)

V. Romero-Rochin, R. M. Koehl, C. J. Brennan, and K. A. Nelson, “Anharmonic phonon-polariton excitation through impulsive stimulated Raman scattering and detection through wave vector overtone spectroscopy: theory and comparison to experiments on lithium tantalate,” J. Chem. Phys. 11, 3559 (1999).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. 7, S12–S22 (2005).
[CrossRef]

Nanotechnology (1)

S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Nature (1)

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (2)

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

I. Inbar and R. E. Cohen, “Comparison of the electronic structures and energetics of ferroelectric LiNbO3 and LiTaO3,” Phys. Rev. B 53, 1193 (1996).
[CrossRef]

Rep. Prog. Phys. (1)

G. P. Williams, “Filling the THz gap-high power sources and applications,” Rep. Prog. Phys. 69, 301 (2006).
[CrossRef]

Science (2)

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]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Other (5)

M. Shalaby, H. Merbold, M. Peccianti, L. Razzari, G. Sharma, R. Morandotti, T. Ozaki, T. Feurer, A. Weber, L. Heyderman, H. Sigg, and B. Patterson, “Concurrent field enhancement and high transmission of THz radiation in nanoslit arrays,” in preparation (2011).

R. W. Boyd, Nonlinear Optics , 1st ed. (Academic Press, 1992).

D. W. Ward, “Polaritonics: An Intermediate Regime Between Electronics and Photonics,” Ph.D. thesis, Department of Chemistry – Massachusetts Institute of Technology (2005).

J. Jin, The Finite Element Method in Electromagnetics , 2nd ed. (Wiley-IEEE Press, 2002).

COMSOL Multiphysics 3.5.

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Figures (7)

Fig. 1
Fig. 1

Schematic drawing of the micro- and nanogap structures under study. (a) Single slit, (b) split-ring resonator, and (c) slit array.

Fig. 2
Fig. 2

(a) Mid-gap field enhancement F in single slits as a function of slit width D and frequency ν. (b) For sufficiently subwavelength slits (Dλ) the field enhancement scales approximately linear with the ratio λ /D.

Fig. 3
Fig. 3

Field enhancement in mid-gap versus frequency for SRRs with (a) 100 μm, (b) 10 μm, and (c) 1 μm gap width. (d) Mid-gap field enhancement at resonance as a function of resonance wavelength divided by gap width. (e) Mid-gap field enhancement of d =1μm SRR under combined E- and H-field excitation and only E-field excitation.

Fig. 4
Fig. 4

(a) Study of the ideal periodicity p of slit arrays for the realization of nonlinear χ (2) (scaling with F (2) / p) and χ (3) (scaling with F (3) / p) effects for d = 40 nm wide slits at ν = 0.5 THz. (b) F (2)(p)/p obtained for slit arrays as a function of periodicity p and frequency ν.

Fig. 5
Fig. 5

(a) Field enhancement at resonance in mid gap for SRRs with d =1μm and varying split length s. (b) Second and third power of absolute field amplitude integrated over the gap volume for varying s.

Fig. 6
Fig. 6

(a) Transmission spectra of only the LiTaO3 substrate, slit arrays with 40 nm slit width and 30 μm periodicity deposited on the substrate, and slit arrays where additionally the gap volume was filled with LiTaO3. (b) Absolute field amplitude in a xz-slice cutting through the slit. (c) As in (a) but for SRR arrays with d = 1 μm, L = 70 μm, and 100 μm periodicity.

Fig. 7
Fig. 7

(a) Relative peak amplitude of the SHG spectra versus the incident field strength obtained with unfilled slit and SRR structures. (b) Normalized SHG spectra for simulated slits and SRRs on LiTaO3 and calculated from the incident waveform. (c) Normalized SHG spectra for SRRs on substrate, SRRs where additionally the gap volume was filled with LiTaO3, and SRRs on substrate with a 10% smaller side length.

Tables (1)

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Table 1 Constants Used for the Calculation of the Nonlinear Coefficients

Equations (21)

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F ( ν ) = E gap ( ν ) E inc ( ν ) ,
U ( 2 ) = V gap | E | 2 d V and U ( 3 ) = V gap | E | 3 d V ,
V ( Q ) = 1 4 M ω T 2 Q 2 + 1 64 V 0 M 2 ω T 4 Q 4 ,
P = ε 0 ( χ ( 1 ) E + χ ( 2 ) E 2 + χ ( 3 ) E 3 + ) ,
M 2 Q t 2 + M Γ Q t + V Q = q E ( t ) ,
Q ¨ + Γ Q ˙ + ω T 2 Q + a Q 2 = q M E ( t ) ,
a = 3 ω T 3 M 4 2 V 0 .
Q ¨ ( 1 ) + Γ Q ˙ ( 1 ) + ω T 2 Q ( 1 ) = q M E ( t ) ,
Q ¨ ( 2 ) + Γ Q ˙ ( 2 ) + ω T 2 Q ( 2 ) + a [ Q ( 1 ) ] 2 = 0 ,
Q ¨ ( 3 ) + Γ Q ˙ ( 3 ) + ω T 2 Q ( 3 ) + 2 a Q ( 1 ) Q ( 2 ) = 0.
P ( n ) = N q Q ( n ) = ε 0 χ ( n ) E n .
Q ( n ) ( t ) = Q ( n ) ( n ω ) e ι n ω t ,
χ ( 1 ) = N q 2 M ε 0 1 D ( ω ) ,
χ ( 2 ) = N q 3 M 2 ε 0 a D 2 ( ω ) D ( 2 ω ) ,
χ ( 3 ) = 2 N q 4 M 3 ε 0 a 2 D 3 ( ω ) D ( 2 ω ) D ( 3 ω ) ,
D ( ω ) = ω T 2 ( ω ) 2 + ι Γ ω .
N = 4 3 a 2 c ,
1 M = j 1 m j ,
ε ( ω ) = ε + f ω T 2 ω 2 + ι Γ ω .
f = N q 2 M ε 0 ,
q = ω T ε 0 ( ε s ε ) M N ,

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