Abstract

We make a theoretical investigation of nonlinear surface polaritons localized at a single interface separating a nonlinear conventional dielectric medium and a linear indefinite medium. The propagation properties of such nonlinear surface polaritons are demonstrated in several distinct frequency intervals. The surface effect of optical beams suffers different dynamics from the propagation (transmission) effect in indefinite media. The existence conditions, frequency thresholds, group velocity, and the power fluxes of the nonlinear surface polaritons are also discussed in detail.

© 2012 Optical Society of America

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  1. D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Shultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [CrossRef]
  2. D. R. Smith and N. Kroll, “Negative refractive index in left-handed material,” Phys. Rev. Lett. 85, 2933–2936 (2000).
    [CrossRef]
  3. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef]
  4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef]
  5. R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
    [CrossRef]
  6. S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
    [CrossRef]
  7. H. Zhang, Q. Wang, N. Shen, R. Li, J. Chen, J. Ding, and H. Wang, “Surface plasmon polaritons at interfaces associated with artificial composite materials,” J. Opt. Soc. Am. B 22, 2686–2696 (2005).
    [CrossRef]
  8. F. Tao, H. Zhang, X. Yang, and D. Cao, “Surface plasmon polaritons of the metamaterial four-layered structures,” J. Opt. Soc. Am. B 26, 50–59 (2009).
    [CrossRef]
  9. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refraction-index waveguides,” Phys. Rev. E 67, 057602 (2003).
    [CrossRef]
  10. G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
    [CrossRef]
  11. Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245, 125–135 (2005).
    [CrossRef]
  12. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterial,” Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef]
  13. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
    [CrossRef]
  14. S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Nonlinear surface waves at the interfaces of left-handed electromagnetic media,” Phys. Rev. E 72, 036615 (2005).
    [CrossRef]
  15. I. V. Shadrivov, “Nonlinear guided waves and symmetry breaking in left-handed waveguides,” Photon. Nanostr. Fundam. Appl. 2, 175–180 (2004).
    [CrossRef]
  16. S. A. Darmanyan, A. Kobyakov, and D. Q. Chowdhury, “Nonlinear guided waves in a negative-index slab waveguide,” Phys. Lett. A 363, 159–163 (2007).
    [CrossRef]
  17. R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
    [CrossRef]
  18. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
    [CrossRef]
  19. D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003).
    [CrossRef]
  20. D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21, 1032–1043 (2004).
    [CrossRef]
  21. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84, 2244–2246 (2004).
    [CrossRef]
  22. T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
    [CrossRef]
  23. W. Yan, L. Shen, L. Ran, and J. A. Kong, “Surface modes at the interfaces between isotropic media and indefinite media,” J. Opt. Soc. Am. A 24, 530–535 (2007).
    [CrossRef]
  24. G. Xu, T. Pan, T. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
    [CrossRef]
  25. G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Nonlinear surface polaritons in anisotropic Kerr-type metamaterials,” J. Phys. D 42, 045303 (2009).
    [CrossRef]
  26. W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Modulation of bistable lateral shifts in a one-dimensional nonlinear photonic crystal consisting of indefinite metamaterials,” Phys. Rev. E 81, 046603 (2010).
    [CrossRef]
  27. W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Transformation of nonlinear behaviors: from bright- to dark-gap soliton in a one-dimensional photonic crystal containing a nonlinear indefinite metamaterial defect,” Phys. Rev. E 82, 066601(2010).
    [CrossRef]
  28. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  29. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
    [CrossRef]
  30. R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
    [CrossRef]
  31. M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Bistable diode action in left-handed periodic structures,” Phys. Rev. E 71, 037602 (2005).
    [CrossRef]
  32. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
    [CrossRef]
  33. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
    [CrossRef]

2010 (2)

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Modulation of bistable lateral shifts in a one-dimensional nonlinear photonic crystal consisting of indefinite metamaterials,” Phys. Rev. E 81, 046603 (2010).
[CrossRef]

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Transformation of nonlinear behaviors: from bright- to dark-gap soliton in a one-dimensional photonic crystal containing a nonlinear indefinite metamaterial defect,” Phys. Rev. E 82, 066601(2010).
[CrossRef]

2009 (2)

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Nonlinear surface polaritons in anisotropic Kerr-type metamaterials,” J. Phys. D 42, 045303 (2009).
[CrossRef]

F. Tao, H. Zhang, X. Yang, and D. Cao, “Surface plasmon polaritons of the metamaterial four-layered structures,” J. Opt. Soc. Am. B 26, 50–59 (2009).
[CrossRef]

2008 (2)

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

G. Xu, T. Pan, T. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

2007 (2)

W. Yan, L. Shen, L. Ran, and J. A. Kong, “Surface modes at the interfaces between isotropic media and indefinite media,” J. Opt. Soc. Am. A 24, 530–535 (2007).
[CrossRef]

S. A. Darmanyan, A. Kobyakov, and D. Q. Chowdhury, “Nonlinear guided waves in a negative-index slab waveguide,” Phys. Lett. A 363, 159–163 (2007).
[CrossRef]

2006 (2)

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef]

2005 (6)

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Bistable diode action in left-handed periodic structures,” Phys. Rev. E 71, 037602 (2005).
[CrossRef]

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245, 125–135 (2005).
[CrossRef]

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Nonlinear surface waves at the interfaces of left-handed electromagnetic media,” Phys. Rev. E 72, 036615 (2005).
[CrossRef]

H. Zhang, Q. Wang, N. Shen, R. Li, J. Chen, J. Ding, and H. Wang, “Surface plasmon polaritons at interfaces associated with artificial composite materials,” J. Opt. Soc. Am. B 22, 2686–2696 (2005).
[CrossRef]

2004 (4)

I. V. Shadrivov, “Nonlinear guided waves and symmetry breaking in left-handed waveguides,” Photon. Nanostr. Fundam. Appl. 2, 175–180 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21, 1032–1043 (2004).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84, 2244–2246 (2004).
[CrossRef]

2003 (6)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterial,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003).
[CrossRef]

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refraction-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

2002 (1)

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

2001 (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

2000 (4)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[CrossRef]

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Shultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed material,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

Bloemer, M. J.

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

Boardman, A. D.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

Cao, D.

Cao, Z.

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245, 125–135 (2005).
[CrossRef]

Chen, J.

Chen, Y.

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Modulation of bistable lateral shifts in a one-dimensional nonlinear photonic crystal consisting of indefinite metamaterials,” Phys. Rev. E 81, 046603 (2010).
[CrossRef]

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Transformation of nonlinear behaviors: from bright- to dark-gap soliton in a one-dimensional photonic crystal containing a nonlinear indefinite metamaterial defect,” Phys. Rev. E 82, 066601(2010).
[CrossRef]

Cheng, Q.

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

Chowdhury, D. Q.

S. A. Darmanyan, A. Kobyakov, and D. Q. Chowdhury, “Nonlinear guided waves in a negative-index slab waveguide,” Phys. Lett. A 363, 159–163 (2007).
[CrossRef]

Cui, T. J.

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

Cummer, S. A.

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

D’Aguanno, G.

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

Darmanyan, S. A.

S. A. Darmanyan, A. Kobyakov, and D. Q. Chowdhury, “Nonlinear guided waves in a negative-index slab waveguide,” Phys. Lett. A 363, 159–163 (2007).
[CrossRef]

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Nonlinear surface waves at the interfaces of left-handed electromagnetic media,” Phys. Rev. E 72, 036615 (2005).
[CrossRef]

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

Ding, J.

Dolling, G.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef]

Egan, P.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef]

Feise, M. W.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Bistable diode action in left-handed periodic structures,” Phys. Rev. E 71, 037602 (2005).
[CrossRef]

Grzegorczyk, T. M.

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[CrossRef]

Hand, T.

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

He, Y.

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245, 125–135 (2005).
[CrossRef]

Heyman, E.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Kivshar, Y. S.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Bistable diode action in left-handed periodic structures,” Phys. Rev. E 71, 037602 (2005).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refraction-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterial,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Kobyakov, A.

S. A. Darmanyan, A. Kobyakov, and D. Q. Chowdhury, “Nonlinear guided waves in a negative-index slab waveguide,” Phys. Lett. A 363, 159–163 (2007).
[CrossRef]

Kolinko, P.

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21, 1032–1043 (2004).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84, 2244–2246 (2004).
[CrossRef]

Kong, J. A.

W. Yan, L. Shen, L. Ran, and J. A. Kong, “Surface modes at the interfaces between isotropic media and indefinite media,” J. Opt. Soc. Am. A 24, 530–535 (2007).
[CrossRef]

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[CrossRef]

Kroll, N.

D. R. Smith and N. Kroll, “Negative refractive index in left-handed material,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

Li, R.

Linden, S.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef]

Liu, R. P.

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

Marqus, R.

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Mattiucci, N.

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

Medina, F.

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Mock, J. J.

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84, 2244–2246 (2004).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Shultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Nevière, M.

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Nonlinear surface waves at the interfaces of left-handed electromagnetic media,” Phys. Rev. E 72, 036615 (2005).
[CrossRef]

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

Padilla, W.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Shultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Pan, T.

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Nonlinear surface polaritons in anisotropic Kerr-type metamaterials,” J. Phys. D 42, 045303 (2009).
[CrossRef]

G. Xu, T. Pan, T. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

Rafii-El-Idrissi, R.

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Ran, L.

Ruppin, R.

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[CrossRef]

Rye, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84, 2244–2246 (2004).
[CrossRef]

Scalora, M.

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

Schurig, D.

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84, 2244–2246 (2004).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21, 1032–1043 (2004).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003).
[CrossRef]

Shadrivov, I. V.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Bistable diode action in left-handed periodic structures,” Phys. Rev. E 71, 037602 (2005).
[CrossRef]

I. V. Shadrivov, “Nonlinear guided waves and symmetry breaking in left-handed waveguides,” Photon. Nanostr. Fundam. Appl. 2, 175–180 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterial,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refraction-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

Shen, L.

Shen, N.

Shen, Q.

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245, 125–135 (2005).
[CrossRef]

Shi, J.

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Transformation of nonlinear behaviors: from bright- to dark-gap soliton in a one-dimensional photonic crystal containing a nonlinear indefinite metamaterial defect,” Phys. Rev. E 82, 066601(2010).
[CrossRef]

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Modulation of bistable lateral shifts in a one-dimensional nonlinear photonic crystal consisting of indefinite metamaterials,” Phys. Rev. E 81, 046603 (2010).
[CrossRef]

Shultz, S.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Shultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Smith, D. R.

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84, 2244–2246 (2004).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21, 1032–1043 (2004).
[CrossRef]

D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed material,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef]

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Shultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Soukoulis, C. M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef]

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refraction-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

Sun, J.

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Nonlinear surface polaritons in anisotropic Kerr-type metamaterials,” J. Phys. D 42, 045303 (2009).
[CrossRef]

G. Xu, T. Pan, T. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

Tao, F.

Thomas, Z. M.

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[CrossRef]

Vier, D. C.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Shultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Wang, H.

Wang, Q.

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Modulation of bistable lateral shifts in a one-dimensional nonlinear photonic crystal consisting of indefinite metamaterials,” Phys. Rev. E 81, 046603 (2010).
[CrossRef]

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Transformation of nonlinear behaviors: from bright- to dark-gap soliton in a one-dimensional photonic crystal containing a nonlinear indefinite metamaterial defect,” Phys. Rev. E 82, 066601(2010).
[CrossRef]

H. Zhang, Q. Wang, N. Shen, R. Li, J. Chen, J. Ding, and H. Wang, “Surface plasmon polaritons at interfaces associated with artificial composite materials,” J. Opt. Soc. Am. B 22, 2686–2696 (2005).
[CrossRef]

Wegener, M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef]

Xu, G.

G. Xu, T. Pan, T. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

Xu, G.-D.

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Nonlinear surface polaritons in anisotropic Kerr-type metamaterials,” J. Phys. D 42, 045303 (2009).
[CrossRef]

Yan, W.

Yang, X.

Zakhidov, A. A.

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Nonlinear surface waves at the interfaces of left-handed electromagnetic media,” Phys. Rev. E 72, 036615 (2005).
[CrossRef]

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

Zang, T.

G. Xu, T. Pan, T. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

Zang, T.-C.

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Nonlinear surface polaritons in anisotropic Kerr-type metamaterials,” J. Phys. D 42, 045303 (2009).
[CrossRef]

Zhang, H.

Zhang, W.

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Modulation of bistable lateral shifts in a one-dimensional nonlinear photonic crystal consisting of indefinite metamaterials,” Phys. Rev. E 81, 046603 (2010).
[CrossRef]

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Transformation of nonlinear behaviors: from bright- to dark-gap soliton in a one-dimensional photonic crystal containing a nonlinear indefinite metamaterial defect,” Phys. Rev. E 82, 066601(2010).
[CrossRef]

Zharov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterial,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Appl. Phys. Lett. (4)

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84, 2244–2246 (2004).
[CrossRef]

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[CrossRef]

D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003).
[CrossRef]

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

J. Phys. D (1)

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Nonlinear surface polaritons in anisotropic Kerr-type metamaterials,” J. Phys. D 42, 045303 (2009).
[CrossRef]

Opt. Commun. (3)

G. Xu, T. Pan, T. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245, 125–135 (2005).
[CrossRef]

Photon. Nanostr. Fundam. Appl. (1)

I. V. Shadrivov, “Nonlinear guided waves and symmetry breaking in left-handed waveguides,” Photon. Nanostr. Fundam. Appl. 2, 175–180 (2004).
[CrossRef]

Phys. Lett. A (2)

S. A. Darmanyan, A. Kobyakov, and D. Q. Chowdhury, “Nonlinear guided waves in a negative-index slab waveguide,” Phys. Lett. A 363, 159–163 (2007).
[CrossRef]

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[CrossRef]

Phys. Rev. B (1)

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Phys. Rev. E (8)

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Nonlinear surface waves at the interfaces of left-handed electromagnetic media,” Phys. Rev. E 72, 036615 (2005).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refraction-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Bistable diode action in left-handed periodic structures,” Phys. Rev. E 71, 037602 (2005).
[CrossRef]

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Modulation of bistable lateral shifts in a one-dimensional nonlinear photonic crystal consisting of indefinite metamaterials,” Phys. Rev. E 81, 046603 (2010).
[CrossRef]

W. Zhang, Y. Chen, J. Shi, and Q. Wang, “Transformation of nonlinear behaviors: from bright- to dark-gap soliton in a one-dimensional photonic crystal containing a nonlinear indefinite metamaterial defect,” Phys. Rev. E 82, 066601(2010).
[CrossRef]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Phys. Rev. Lett. (6)

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903(2008).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Shultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed material,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterial,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Science (3)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Structure of the interface between two semi-infinite media of nonlinear conventional media and linear indefinite media. The permittivity and permeability of the nonlinear conventional media are ϵ1NL=ϵ1+αE2 and μ1, and ϵ^ and μ^ of the linear indefinite media.

Fig. 2.
Fig. 2.

The nonlinear surface polaritons when 0<Ω<0.8315. (a) Indexes of ϵx (dot), μy (solid), and μz (dash) versus the normalized frequency Ω; the dash-dotted line represents constant value 0. (b) Spatial profile of the nonlinear surface polaritons when Ω=0.75 and αE22/2=0.1. (c) The existence conditions of the nonlinear surface polaritons. (d), (e), and (f) represent the dispersion, the power fluxes, and the group velocity of the nonlinear surface polaritons with different nonlinearity, respectively.

Fig. 3.
Fig. 3.

The propagation constant kz [(a), (c), and (e)] of nonlinear surface polaritons and their power versus the propagation constant [(b), (d), and (f)] in different frequency intervals. (a) and (b) 0<Ω<0.8315, ϵx<0, μy>0, and μz>0. (c) and (d) 0.8315<Ω<1, ϵx<0, μy<0, and μz<0. (e) and (f) 1<Ω<1.2536, ϵx>0, μy<0, and μz<0.

Fig. 4.
Fig. 4.

The nonlinear surface polaritons when 0.8315<Ω<1. (a) Indexes of ϵx, μy, and μz versus the normalized frequency Ω. (b) Spatial profile of the nonlinear surface polaritons when Ω=0.95 and αE22/2=0.1. (c) The existence conditions of the nonlinear surface polaritons. (d), (e), and (f) represent the dispersion, the power fluxes, and the group velocity of the nonlinear surface polaritons with different nonlinearity, respectively.

Fig. 5.
Fig. 5.

The nonlinear surface polaritons when 1<Ω<1.2536. (a) Indexes of ϵx (dot), μy (solid), and μz (dash) versus the normalized frequency Ω; the dash-dotted line represents constant value 0. (b) Spatial profile of the nonlinear surface polaritons when Ω=1.002 and αE22/2=0.1. (c) The existence conditions of the nonlinear surface polaritons. (d), (e), and (f) represent the dispersion, the power fluxes, and the group velocity of the nonlinear surface polaritons with different nonlinearity, respectively.

Fig. 6.
Fig. 6.

The nonlinear surface polaritons when 1.2536<Ω<1.7728. (a) Indexes of ϵx (dot), μy (solid), and μz (dash) versus the normalized frequency Ω; the dash-dotted line represents constant value 0. (b) The existence conditions of the nonlinear surface polaritons.

Fig. 7.
Fig. 7.

The nonlinear surface polaritons when 1.7728<Ω. (a) Indexes of ϵx (dot), μy (solid), and μz (dash) versus the normalized frequency Ω; the dash-dotted line represents constant value 0. (b) The existence conditions of the nonlinear surface polaritons.

Equations (19)

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

ϵ^=(ϵx00010001),μ^=(1000μy000μz)
ϵx=11Ω2,μy=1FyΩ2Ω2Ωry2,μz=1FzΩ2Ω2Ωrz2,
kz2μy+ky2μz=ϵxω2c2,
2Ex(y)y2(kz2k02ϵ1NLμ1)Ex(y)=0,y>0,
2Ex(y)y2+ky2Ex(y)=0,y<0,
E1x(y)=k1k02αμ1sech[k1(yy0)],y>0,
E2x(y)=E2exp(ikyy),y<0,
ky2=ϵxμzω2c2μzμykz2<0.
E2x(y)=E2exp(kyy),y<0,
E2=k1k02αμ1sech(k1y0),
kyμzE2=k12μ1k02αμ1tanh(k1y0)cosh(k1y0),
tanh(k1y0)=μ1kyμzk1,
kz2k02=μyμ12μyμzμ12[μzμ1(ϵ1+αE222)ϵx].
PNL=k1kzμ12k02ωα(1+μ1kyμzk1),
PL=kzk122μ1μykyk02ωα[1(μ1kyμzk1)2]
kz=k0{μyμ12μyμzμ12[μzμ1(ϵ1+αE222)ϵx]}1/2,
dkzdω=kzω+k022kz{(μyμ12μyμzμ12)[μzμ1(ϵ1+αE222)ϵx]+μyμ12μyμzμ12[μzμ1(ϵ1+αE222)ϵx]},
vg=dω/dκ=(dκ/dω)1.
kz2>k02ϵ1μ1,ky2=μzμykz2ϵxμzω2c2>0.

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