Abstract

Slabs of materials with near-zero permittivity display enhanced nonlinear processes. We show that field enhancement due to the continuity of the longitudinal component of the displacement field drastically enhances harmonic generation. We investigate the impact of losses with and without bulk nonlinearities and demonstrate that in the latter scenario surface, magnetic and quadrupolar nonlinear sources cannot always be ignored.

© 2013 Optical Society of America

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  1. J. Brown, “Artificial dielectrics having refractive indices less than unity,” IEE Proc. Part IV Inst. Monogr.100, 51–62 (1953).
  2. W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” IRE Trans. Antennas Propag.10(1), 82–95 (1962).
    [CrossRef]
  3. S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett.89(21), 213902 (2002).
    [CrossRef] [PubMed]
  4. G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag.54(3), 1017–1030 (2006).
    [CrossRef]
  5. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97(15), 157403 (2006).
    [CrossRef] [PubMed]
  6. S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ɛ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B87(3), 035120 (2013).
    [CrossRef]
  7. D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
    [CrossRef]
  8. M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A84(6), 063826 (2011).
    [CrossRef]
  9. A. Ciattoni and E. Spinozzi, “Efficient second-harmonic generation in micrometer-thick slabs with indefinite permittivity,” Phys. Rev. A85(4), 043806 (2012).
    [CrossRef]
  10. M. A. Vincenti, S. Campione, D. de Ceglia, F. Capolino, and M. Scalora, “Gain-assisted harmonic generation in near-zero permittivity metamaterials made of plasmonic nanoshells,” New J. Phys.14(10), 103016 (2012).
    [CrossRef]
  11. A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. A81(4), 043839 (2010).
    [CrossRef]
  12. D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: Nonlinear and nonlocal optical properties,” Phys. Rev. B87(15), 155140 (2013).
    [CrossRef]
  13. C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A83(5), 053805 (2011).
    [CrossRef]
  14. E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998).
  15. S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express1(6), 1077–1089 (2011).
    [CrossRef]
  16. S. Campione and F. Capolino, “Composite material made of plasmonic nanoshells with quantum dot cores: loss-compensation and ε-near-zero physical properties,” Nanotechnology23(23), 235703 (2012).
    [CrossRef] [PubMed]
  17. A. Ciattoni, R. Marinelli, C. Rizza, and E. Palange, “|ε|-Near-zero materials in the near-infrared,” Appl. Phys. B110(1), 23–26 (2013).
    [CrossRef]
  18. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
    [CrossRef] [PubMed]
  19. M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
    [CrossRef]
  20. D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys.107(12), 123114 (2010).
    [CrossRef]
  21. M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
    [CrossRef] [PubMed]
  22. N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced Nonlinear Optical Response of One-Dimensional Metal-Dielectric Photonic Crystals,” Phys. Rev. Lett.93(12), 123902 (2004).
    [CrossRef] [PubMed]
  23. M. Airola, Y. Liu, and S. Blair, “Second-harmonic generation from an array of sub-wavelength metal apertures,” J. Opt. A7(2), S118–S123 (2005).
    [CrossRef]
  24. A. Lesuffleur, L. K. S. Kumar, and R. Gordon, “Enhanced second harmonic generation from nanoscale double-hole arrays in a gold film,” Appl. Phys. Lett.88(26), 261104 (2006).
    [CrossRef]
  25. N. Rakov, F. E. Ramos, and M. Xiao, “Strong second-harmonic radiation from a thin silver film with randomly distributed small holes,” J. Phys. Condens. Matter15(23), L349–L352 (2003).
    [CrossRef]
  26. T. Xu, X. Jiao, and S. Blair, “Third-harmonic generation from arrays of sub-wavelength metal apertures,” Opt. Express17(26), 23582–23588 (2009).
    [CrossRef] [PubMed]
  27. M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
    [CrossRef]
  28. M. A. Vincenti, D. de Ceglia, J. W. Haus, and M. Scalora, “Harmonic generation in multiresonant plasma films,” Phys. Rev. A88(4), 043812 (2013).
    [CrossRef]

2013 (4)

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ɛ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B87(3), 035120 (2013).
[CrossRef]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: Nonlinear and nonlocal optical properties,” Phys. Rev. B87(15), 155140 (2013).
[CrossRef]

A. Ciattoni, R. Marinelli, C. Rizza, and E. Palange, “|ε|-Near-zero materials in the near-infrared,” Appl. Phys. B110(1), 23–26 (2013).
[CrossRef]

M. A. Vincenti, D. de Ceglia, J. W. Haus, and M. Scalora, “Harmonic generation in multiresonant plasma films,” Phys. Rev. A88(4), 043812 (2013).
[CrossRef]

2012 (4)

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

S. Campione and F. Capolino, “Composite material made of plasmonic nanoshells with quantum dot cores: loss-compensation and ε-near-zero physical properties,” Nanotechnology23(23), 235703 (2012).
[CrossRef] [PubMed]

A. Ciattoni and E. Spinozzi, “Efficient second-harmonic generation in micrometer-thick slabs with indefinite permittivity,” Phys. Rev. A85(4), 043806 (2012).
[CrossRef]

M. A. Vincenti, S. Campione, D. de Ceglia, F. Capolino, and M. Scalora, “Gain-assisted harmonic generation in near-zero permittivity metamaterials made of plasmonic nanoshells,” New J. Phys.14(10), 103016 (2012).
[CrossRef]

2011 (4)

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A84(6), 063826 (2011).
[CrossRef]

C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A83(5), 053805 (2011).
[CrossRef]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express1(6), 1077–1089 (2011).
[CrossRef]

2010 (4)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys.107(12), 123114 (2010).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. A81(4), 043839 (2010).
[CrossRef]

2009 (2)

D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
[CrossRef]

T. Xu, X. Jiao, and S. Blair, “Third-harmonic generation from arrays of sub-wavelength metal apertures,” Opt. Express17(26), 23582–23588 (2009).
[CrossRef] [PubMed]

2006 (3)

A. Lesuffleur, L. K. S. Kumar, and R. Gordon, “Enhanced second harmonic generation from nanoscale double-hole arrays in a gold film,” Appl. Phys. Lett.88(26), 261104 (2006).
[CrossRef]

G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag.54(3), 1017–1030 (2006).
[CrossRef]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97(15), 157403 (2006).
[CrossRef] [PubMed]

2005 (1)

M. Airola, Y. Liu, and S. Blair, “Second-harmonic generation from an array of sub-wavelength metal apertures,” J. Opt. A7(2), S118–S123 (2005).
[CrossRef]

2004 (1)

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced Nonlinear Optical Response of One-Dimensional Metal-Dielectric Photonic Crystals,” Phys. Rev. Lett.93(12), 123902 (2004).
[CrossRef] [PubMed]

2003 (1)

N. Rakov, F. E. Ramos, and M. Xiao, “Strong second-harmonic radiation from a thin silver film with randomly distributed small holes,” J. Phys. Condens. Matter15(23), L349–L352 (2003).
[CrossRef]

2002 (1)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett.89(21), 213902 (2002).
[CrossRef] [PubMed]

1962 (1)

W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” IRE Trans. Antennas Propag.10(1), 82–95 (1962).
[CrossRef]

Airola, M.

M. Airola, Y. Liu, and S. Blair, “Second-harmonic generation from an array of sub-wavelength metal apertures,” J. Opt. A7(2), S118–S123 (2005).
[CrossRef]

Akozbek, N.

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

Albani, M.

Alù, A.

D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
[CrossRef]

Bennink, R. S.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced Nonlinear Optical Response of One-Dimensional Metal-Dielectric Photonic Crystals,” Phys. Rev. Lett.93(12), 123902 (2004).
[CrossRef] [PubMed]

Blair, S.

T. Xu, X. Jiao, and S. Blair, “Third-harmonic generation from arrays of sub-wavelength metal apertures,” Opt. Express17(26), 23582–23588 (2009).
[CrossRef] [PubMed]

M. Airola, Y. Liu, and S. Blair, “Second-harmonic generation from an array of sub-wavelength metal apertures,” J. Opt. A7(2), S118–S123 (2005).
[CrossRef]

Bloemer, M.

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

Boyd, R. W.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced Nonlinear Optical Response of One-Dimensional Metal-Dielectric Photonic Crystals,” Phys. Rev. Lett.93(12), 123902 (2004).
[CrossRef] [PubMed]

Brown, J.

J. Brown, “Artificial dielectrics having refractive indices less than unity,” IEE Proc. Part IV Inst. Monogr.100, 51–62 (1953).

Burghignoli, P.

G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag.54(3), 1017–1030 (2006).
[CrossRef]

Campione, S.

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: Nonlinear and nonlocal optical properties,” Phys. Rev. B87(15), 155140 (2013).
[CrossRef]

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ɛ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B87(3), 035120 (2013).
[CrossRef]

M. A. Vincenti, S. Campione, D. de Ceglia, F. Capolino, and M. Scalora, “Gain-assisted harmonic generation in near-zero permittivity metamaterials made of plasmonic nanoshells,” New J. Phys.14(10), 103016 (2012).
[CrossRef]

S. Campione and F. Capolino, “Composite material made of plasmonic nanoshells with quantum dot cores: loss-compensation and ε-near-zero physical properties,” Nanotechnology23(23), 235703 (2012).
[CrossRef] [PubMed]

S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express1(6), 1077–1089 (2011).
[CrossRef]

Capolino, F.

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ɛ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B87(3), 035120 (2013).
[CrossRef]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: Nonlinear and nonlocal optical properties,” Phys. Rev. B87(15), 155140 (2013).
[CrossRef]

S. Campione and F. Capolino, “Composite material made of plasmonic nanoshells with quantum dot cores: loss-compensation and ε-near-zero physical properties,” Nanotechnology23(23), 235703 (2012).
[CrossRef] [PubMed]

M. A. Vincenti, S. Campione, D. de Ceglia, F. Capolino, and M. Scalora, “Gain-assisted harmonic generation in near-zero permittivity metamaterials made of plasmonic nanoshells,” New J. Phys.14(10), 103016 (2012).
[CrossRef]

S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express1(6), 1077–1089 (2011).
[CrossRef]

G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag.54(3), 1017–1030 (2006).
[CrossRef]

Centini, M.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Ciattoni, A.

A. Ciattoni, R. Marinelli, C. Rizza, and E. Palange, “|ε|-Near-zero materials in the near-infrared,” Appl. Phys. B110(1), 23–26 (2013).
[CrossRef]

A. Ciattoni and E. Spinozzi, “Efficient second-harmonic generation in micrometer-thick slabs with indefinite permittivity,” Phys. Rev. A85(4), 043806 (2012).
[CrossRef]

C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A83(5), 053805 (2011).
[CrossRef]

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A84(6), 063826 (2011).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. A81(4), 043839 (2010).
[CrossRef]

de Ceglia, D.

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: Nonlinear and nonlocal optical properties,” Phys. Rev. B87(15), 155140 (2013).
[CrossRef]

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ɛ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B87(3), 035120 (2013).
[CrossRef]

M. A. Vincenti, D. de Ceglia, J. W. Haus, and M. Scalora, “Harmonic generation in multiresonant plasma films,” Phys. Rev. A88(4), 043812 (2013).
[CrossRef]

M. A. Vincenti, S. Campione, D. de Ceglia, F. Capolino, and M. Scalora, “Gain-assisted harmonic generation in near-zero permittivity metamaterials made of plasmonic nanoshells,” New J. Phys.14(10), 103016 (2012).
[CrossRef]

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A84(6), 063826 (2011).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Edwards, B.

D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
[CrossRef]

Engheta, N.

D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
[CrossRef]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97(15), 157403 (2006).
[CrossRef] [PubMed]

Enoch, S.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett.89(21), 213902 (2002).
[CrossRef] [PubMed]

Fuentes-Hernandez, C.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys.107(12), 123114 (2010).
[CrossRef]

Gordon, R.

A. Lesuffleur, L. K. S. Kumar, and R. Gordon, “Enhanced second harmonic generation from nanoscale double-hole arrays in a gold film,” Appl. Phys. Lett.88(26), 261104 (2006).
[CrossRef]

Guérin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett.89(21), 213902 (2002).
[CrossRef] [PubMed]

Hales, J. M.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys.107(12), 123114 (2010).
[CrossRef]

Haus, J.

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

Haus, J. W.

M. A. Vincenti, D. de Ceglia, J. W. Haus, and M. Scalora, “Harmonic generation in multiresonant plasma films,” Phys. Rev. A88(4), 043812 (2013).
[CrossRef]

Jackson, D. R.

G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag.54(3), 1017–1030 (2006).
[CrossRef]

Jiao, X.

Kildishev, A. V.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Kippelen, B.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys.107(12), 123114 (2010).
[CrossRef]

Kivshar, Y. S.

D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
[CrossRef]

Kumar, L. K. S.

A. Lesuffleur, L. K. S. Kumar, and R. Gordon, “Enhanced second harmonic generation from nanoscale double-hole arrays in a gold film,” Appl. Phys. Lett.88(26), 261104 (2006).
[CrossRef]

Lepeshkin, N. N.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced Nonlinear Optical Response of One-Dimensional Metal-Dielectric Photonic Crystals,” Phys. Rev. Lett.93(12), 123902 (2004).
[CrossRef] [PubMed]

Lesuffleur, A.

A. Lesuffleur, L. K. S. Kumar, and R. Gordon, “Enhanced second harmonic generation from nanoscale double-hole arrays in a gold film,” Appl. Phys. Lett.88(26), 261104 (2006).
[CrossRef]

Liu, Y.

M. Airola, Y. Liu, and S. Blair, “Second-harmonic generation from an array of sub-wavelength metal apertures,” J. Opt. A7(2), S118–S123 (2005).
[CrossRef]

Lovat, G.

G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag.54(3), 1017–1030 (2006).
[CrossRef]

Marinelli, R.

A. Ciattoni, R. Marinelli, C. Rizza, and E. Palange, “|ε|-Near-zero materials in the near-infrared,” Appl. Phys. B110(1), 23–26 (2013).
[CrossRef]

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Owens, D. T.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys.107(12), 123114 (2010).
[CrossRef]

Palange, E.

A. Ciattoni, R. Marinelli, C. Rizza, and E. Palange, “|ε|-Near-zero materials in the near-infrared,” Appl. Phys. B110(1), 23–26 (2013).
[CrossRef]

C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A83(5), 053805 (2011).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. A81(4), 043839 (2010).
[CrossRef]

Perry, J. W.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys.107(12), 123114 (2010).
[CrossRef]

Piredda, G.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced Nonlinear Optical Response of One-Dimensional Metal-Dielectric Photonic Crystals,” Phys. Rev. Lett.93(12), 123902 (2004).
[CrossRef] [PubMed]

Powell, D. A.

D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
[CrossRef]

Rakov, N.

N. Rakov, F. E. Ramos, and M. Xiao, “Strong second-harmonic radiation from a thin silver film with randomly distributed small holes,” J. Phys. Condens. Matter15(23), L349–L352 (2003).
[CrossRef]

Ramos, F. E.

N. Rakov, F. E. Ramos, and M. Xiao, “Strong second-harmonic radiation from a thin silver film with randomly distributed small holes,” J. Phys. Condens. Matter15(23), L349–L352 (2003).
[CrossRef]

Rizza, C.

A. Ciattoni, R. Marinelli, C. Rizza, and E. Palange, “|ε|-Near-zero materials in the near-infrared,” Appl. Phys. B110(1), 23–26 (2013).
[CrossRef]

C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A83(5), 053805 (2011).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. A81(4), 043839 (2010).
[CrossRef]

Roppo, V.

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

Rotman, W.

W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” IRE Trans. Antennas Propag.10(1), 82–95 (1962).
[CrossRef]

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett.89(21), 213902 (2002).
[CrossRef] [PubMed]

Scalora, M.

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: Nonlinear and nonlocal optical properties,” Phys. Rev. B87(15), 155140 (2013).
[CrossRef]

M. A. Vincenti, D. de Ceglia, J. W. Haus, and M. Scalora, “Harmonic generation in multiresonant plasma films,” Phys. Rev. A88(4), 043812 (2013).
[CrossRef]

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ɛ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B87(3), 035120 (2013).
[CrossRef]

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

M. A. Vincenti, S. Campione, D. de Ceglia, F. Capolino, and M. Scalora, “Gain-assisted harmonic generation in near-zero permittivity metamaterials made of plasmonic nanoshells,” New J. Phys.14(10), 103016 (2012).
[CrossRef]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A84(6), 063826 (2011).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

Schweinsberg, A.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced Nonlinear Optical Response of One-Dimensional Metal-Dielectric Photonic Crystals,” Phys. Rev. Lett.93(12), 123902 (2004).
[CrossRef] [PubMed]

Shalaev, V. M.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Silveirinha, M.

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97(15), 157403 (2006).
[CrossRef] [PubMed]

Spinozzi, E.

A. Ciattoni and E. Spinozzi, “Efficient second-harmonic generation in micrometer-thick slabs with indefinite permittivity,” Phys. Rev. A85(4), 043806 (2012).
[CrossRef]

Tayeb, G.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett.89(21), 213902 (2002).
[CrossRef] [PubMed]

Vakil, A.

D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
[CrossRef]

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett.89(21), 213902 (2002).
[CrossRef] [PubMed]

Vincenti, M.

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

Vincenti, M. A.

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ɛ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B87(3), 035120 (2013).
[CrossRef]

M. A. Vincenti, D. de Ceglia, J. W. Haus, and M. Scalora, “Harmonic generation in multiresonant plasma films,” Phys. Rev. A88(4), 043812 (2013).
[CrossRef]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: Nonlinear and nonlocal optical properties,” Phys. Rev. B87(15), 155140 (2013).
[CrossRef]

M. A. Vincenti, S. Campione, D. de Ceglia, F. Capolino, and M. Scalora, “Gain-assisted harmonic generation in near-zero permittivity metamaterials made of plasmonic nanoshells,” New J. Phys.14(10), 103016 (2012).
[CrossRef]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A84(6), 063826 (2011).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

Wilton, D. R.

G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag.54(3), 1017–1030 (2006).
[CrossRef]

Xiao, M.

N. Rakov, F. E. Ramos, and M. Xiao, “Strong second-harmonic radiation from a thin silver film with randomly distributed small holes,” J. Phys. Condens. Matter15(23), L349–L352 (2003).
[CrossRef]

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Xu, T.

Yuan, H.-K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Appl. Phys. B (1)

A. Ciattoni, R. Marinelli, C. Rizza, and E. Palange, “|ε|-Near-zero materials in the near-infrared,” Appl. Phys. B110(1), 23–26 (2013).
[CrossRef]

Appl. Phys. Lett. (1)

A. Lesuffleur, L. K. S. Kumar, and R. Gordon, “Enhanced second harmonic generation from nanoscale double-hole arrays in a gold film,” Appl. Phys. Lett.88(26), 261104 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag.54(3), 1017–1030 (2006).
[CrossRef]

IRE Trans. Antennas Propag. (1)

W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” IRE Trans. Antennas Propag.10(1), 82–95 (1962).
[CrossRef]

J. Appl. Phys. (1)

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys.107(12), 123114 (2010).
[CrossRef]

J. Opt. A (1)

M. Airola, Y. Liu, and S. Blair, “Second-harmonic generation from an array of sub-wavelength metal apertures,” J. Opt. A7(2), S118–S123 (2005).
[CrossRef]

J. Phys. Condens. Matter (1)

N. Rakov, F. E. Ramos, and M. Xiao, “Strong second-harmonic radiation from a thin silver film with randomly distributed small holes,” J. Phys. Condens. Matter15(23), L349–L352 (2003).
[CrossRef]

Nanotechnology (1)

S. Campione and F. Capolino, “Composite material made of plasmonic nanoshells with quantum dot cores: loss-compensation and ε-near-zero physical properties,” Nanotechnology23(23), 235703 (2012).
[CrossRef] [PubMed]

Nature (1)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010).
[CrossRef] [PubMed]

New J. Phys. (1)

M. A. Vincenti, S. Campione, D. de Ceglia, F. Capolino, and M. Scalora, “Gain-assisted harmonic generation in near-zero permittivity metamaterials made of plasmonic nanoshells,” New J. Phys.14(10), 103016 (2012).
[CrossRef]

Opt. Express (2)

Opt. Mater. Express (1)

Phys. Rev. A (7)

M. Scalora, M. Vincenti, D. de Ceglia, N. Akozbek, V. Roppo, M. Bloemer, and J. Haus, “Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths,” Phys. Rev. A85(5), 053809 (2012).
[CrossRef]

M. A. Vincenti, D. de Ceglia, J. W. Haus, and M. Scalora, “Harmonic generation in multiresonant plasma films,” Phys. Rev. A88(4), 043812 (2013).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. A81(4), 043839 (2010).
[CrossRef]

C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A83(5), 053805 (2011).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A82(4), 043828 (2010).
[CrossRef]

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A84(6), 063826 (2011).
[CrossRef]

A. Ciattoni and E. Spinozzi, “Efficient second-harmonic generation in micrometer-thick slabs with indefinite permittivity,” Phys. Rev. A85(4), 043806 (2012).
[CrossRef]

Phys. Rev. B (3)

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ɛ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B87(3), 035120 (2013).
[CrossRef]

D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an epsilon-near-zero channel,” Phys. Rev. B79(24), 245135 (2009).
[CrossRef]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: Nonlinear and nonlocal optical properties,” Phys. Rev. B87(15), 155140 (2013).
[CrossRef]

Phys. Rev. Lett. (3)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett.89(21), 213902 (2002).
[CrossRef] [PubMed]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97(15), 157403 (2006).
[CrossRef] [PubMed]

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced Nonlinear Optical Response of One-Dimensional Metal-Dielectric Photonic Crystals,” Phys. Rev. Lett.93(12), 123902 (2004).
[CrossRef] [PubMed]

Other (2)

E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998).

J. Brown, “Artificial dielectrics having refractive indices less than unity,” IEE Proc. Part IV Inst. Monogr.100, 51–62 (1953).

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Figures (5)

Fig. 1
Fig. 1

(a) A TM-polarized pump impinges on a slab with thickness d and angle of incidence ϑ i ; (b) Transmission, (c) Reflection and (d) Absorption when d = 200nm. Fundamental (red, dashed line marked λω), second (green, dashed line marked λ) and third (blue, dashed line marked λ) harmonic wavelengths are shown on the absorption map.

Fig. 2
Fig. 2

Electric field enhancement [ max ( | E z | / | E z,0 | ) ] vs. wavelength and angle of incidence for a d = 200nm thick, modeled with ω p1 = 0.906049 ω r , ω 01 = 0.25 ω r and (a) γ 1 = 0.01 ω r , (b) γ 1 = 0.001 ω r and (c) γ 1 = 0.0001 ω r .

Fig. 3
Fig. 3

Total (a) SH and (b) TH conversion efficiencies for a slab d = 200 nm thick as a function of incident angle and pump wavelength. The slab is modeled assuming ω p1 = 0.906049 ω r , ω 01 = 0.25 ω r , and γ 1 = 0.01 ω r . (c) and (d) same as in (a) and (b) for γ 1 = 0.001 ω r .

Fig. 4
Fig. 4

(a) SH efficiency vs. angle of incidence for a slab d = 200 nm thick, modeled with ω p1 = 0.906049 ω r , ω 01 = 0.25 ω r and γ 1 = 0.01 ω r (blue, solid line) and γ 1 = 0.001 ω r (red, dashed line). (b) Same as in (a) for TH.

Fig. 5
Fig. 5

(a) SH efficiency vs. angle of incidence for a slab d = 200 nm thick, modeled with ω p1 = ω p2 = ω p3 = 0.906049 ω r , ω 01 = 0.25 ω r , ω 02 = 1.652 ω r , ω 03 = 2.678 ω r γ 1 = γ 2 = γ 3 = 0.01 ω r (blue, solid line) and γ 1 = γ 2 = γ 3 = 0.001 ω r (red, dashed line); (b) Same as in (a) for TH.

Equations (1)

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ε M ( ω ) = 1 j ω p j 2 ω 2 ω 0j 2 + i ω γ j .

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