Abstract

In this paper we investigate analytically the localized surface waves (SWs) based on the first integral of the nonlinear Helmholtz wave equation in a semi-infinite one-dimensional photonic crystal (1DPC) truncated with a self-defocusing nonlinear cap layer of a left-handed material (LHM). The intensity-dependent properties of nonlinear SWs at the interface between air background and 1DPC are of two types: conventional photonic crystal (PC) and left-handed PC (made of alternate LHM and right-handed material) ones. It is shown that there are unusual SWs with different peculiarities for these two types of PC. To confirm our results, we also compared the results obtained by this method and the alternative method based on δ-function, for the structure with a very thin nonlinear cap layer.

© 2010 Optical Society of America

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  1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef] [PubMed]
  2. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
    [CrossRef] [PubMed]
  3. V. G. Veselago, “Electromagnetics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  4. A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
    [CrossRef] [PubMed]
  5. J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37–43 (2004).
    [CrossRef]
  6. A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Excitation of backward Tamm states at an interface between a periodic photonic crystal and a left-handed metamaterial,” Phys. Rev. A 75, 053812 (2007).
    [CrossRef]
  7. X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103 (2003).
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  8. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).
  9. P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
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  12. F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
    [CrossRef]
  13. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
    [CrossRef] [PubMed]
  14. E. Moreno, L. Martìn-Moreno, and F. J. Garcìa-Vidal, “Efficient coupling of light into and out of a photonic crystal waveguide via surface modes,” Photonics Nanostruct. Fundam. Appl. 2, 97–102 (2004).
    [CrossRef]
  15. P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
    [CrossRef] [PubMed]
  16. R. R. Wei, X. Chen, J. W. Tao, and C. F. Li, “Giant and negative bistable shifts for one-dimensional photonic crystal containing a nonlinear metamaterial defect,” Phys. Lett. A 372, 6797–6800 (2008).
    [CrossRef]
  17. A. D. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A, Pure Appl. Opt. 11, 114302 (2009).
    [CrossRef]
  18. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
    [CrossRef]
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    [CrossRef]
  22. A. Namdar, “Tamm states in one dimensional photonic crystals containing left-handed materials,” Opt. Commun. 278, 194–198 (2007).
    [CrossRef]
  23. J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A, Pure Appl. Opt. 8, 630–638 (2006).
    [CrossRef]
  24. A. Yarive and P. Yeh, “Electromagnetic propagation in periodic media,” in Optical Waves in Crystals (Wiley, 1984), pp. 209–215.
  25. A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
    [CrossRef]
  26. M. Abramowitz and A. S. Stegun, Handbook of Mathematical Function (Dover, 1972).
  27. A. C. Dioxon, The Elementary Properties of the Elliptic Functions with Examples (Macmillan, 1894).
  28. Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
    [CrossRef]

2009

A. D. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A, Pure Appl. Opt. 11, 114302 (2009).
[CrossRef]

2008

R. R. Wei, X. Chen, J. W. Tao, and C. F. Li, “Giant and negative bistable shifts for one-dimensional photonic crystal containing a nonlinear metamaterial defect,” Phys. Lett. A 372, 6797–6800 (2008).
[CrossRef]

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials 2, 1–17 (2008).
[CrossRef]

2007

U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, “Dual-band negative index metamaterial: double negative at 813 nm and single negative at 772 nm,” Opt. Lett. 32, 1671–1673 (2007).
[CrossRef] [PubMed]

A. Namdar, “Tamm states in one dimensional photonic crystals containing left-handed materials,” Opt. Commun. 278, 194–198 (2007).
[CrossRef]

A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Excitation of backward Tamm states at an interface between a periodic photonic crystal and a left-handed metamaterial,” Phys. Rev. A 75, 053812 (2007).
[CrossRef]

2006

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A, Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Backward Tamm states in left-handed metamaterials,” Appl. Phys. Lett. 89, 114104 (2006).
[CrossRef]

2005

Y. Zeng, X. Chen, and W. Lu, “Electromagnetic modes in semi-infinite photonic crystals,” Physica E (Amsterdam) 30, 55–58 (2005).
[CrossRef]

2004

E. Moreno, L. Martìn-Moreno, and F. J. Garcìa-Vidal, “Efficient coupling of light into and out of a photonic crystal waveguide via surface modes,” Photonics Nanostruct. Fundam. Appl. 2, 97–102 (2004).
[CrossRef]

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
[CrossRef] [PubMed]

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37–43 (2004).
[CrossRef]

2003

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103 (2003).
[CrossRef]

2001

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

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

1999

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

1996

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

1986

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
[CrossRef]

1978

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[CrossRef]

1977

1968

V. G. Veselago, “Electromagnetics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Abramowitz, M.

M. Abramowitz and A. S. Stegun, Handbook of Mathematical Function (Dover, 1972).

Agio, M.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Birner, A.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Boardman, A. D.

A. D. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A, Pure Appl. Opt. 11, 114302 (2009).
[CrossRef]

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
[CrossRef]

Boltasseva, A.

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials 2, 1–17 (2008).
[CrossRef]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Cai, W.

Chan, C. T.

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

Chen, X.

R. R. Wei, X. Chen, J. W. Tao, and C. F. Li, “Giant and negative bistable shifts for one-dimensional photonic crystal containing a nonlinear metamaterial defect,” Phys. Lett. A 372, 6797–6800 (2008).
[CrossRef]

Y. Zeng, X. Chen, and W. Lu, “Electromagnetic modes in semi-infinite photonic crystals,” Physica E (Amsterdam) 30, 55–58 (2005).
[CrossRef]

Chettiar, U. K.

Chigrin, D. N.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Cho, A. Y.

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[CrossRef]

Dioxon, A. C.

A. C. Dioxon, The Elementary Properties of the Elliptic Functions with Examples (Macmillan, 1894).

Drachev, V. P.

Economou, E. N.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

Egan, P.

A. D. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A, Pure Appl. Opt. 11, 114302 (2009).
[CrossRef]

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
[CrossRef]

Erland, J.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Foteinopoulou, S.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

Gaponenko, S. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Garcìa-Vidal, F. J.

E. Moreno, L. Martìn-Moreno, and F. J. Garcìa-Vidal, “Efficient coupling of light into and out of a photonic crystal waveguide via surface modes,” Photonics Nanostruct. Fundam. Appl. 2, 97–102 (2004).
[CrossRef]

Gösele, U.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Halevi, P.

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

Ho, K. M.

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

Hong, C. S.

Hvam, J. M.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

Kildishev, A. V.

Kissel, V. N.

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
[CrossRef] [PubMed]

Kivshar, Y. S.

A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Excitation of backward Tamm states at an interface between a periodic photonic crystal and a left-handed metamaterial,” Phys. Rev. A 75, 053812 (2007).
[CrossRef]

A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Backward Tamm states in left-handed metamaterials,” Appl. Phys. Lett. 89, 114104 (2006).
[CrossRef]

Kramper, P.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Lagarkov, A. N.

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
[CrossRef] [PubMed]

Lavrinenko, A. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Leosson, K.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Li, C. F.

R. R. Wei, X. Chen, J. W. Tao, and C. F. Li, “Giant and negative bistable shifts for one-dimensional photonic crystal containing a nonlinear metamaterial defect,” Phys. Lett. A 372, 6797–6800 (2008).
[CrossRef]

Li, Q.

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

Lu, W.

Y. Zeng, X. Chen, and W. Lu, “Electromagnetic modes in semi-infinite photonic crystals,” Physica E (Amsterdam) 30, 55–58 (2005).
[CrossRef]

Martìn-Moreno, L.

E. Moreno, L. Martìn-Moreno, and F. J. Garcìa-Vidal, “Efficient coupling of light into and out of a photonic crystal waveguide via surface modes,” Photonics Nanostruct. Fundam. Appl. 2, 97–102 (2004).
[CrossRef]

Martorell, J.

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A, Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

Moreno, E.

E. Moreno, L. Martìn-Moreno, and F. J. Garcìa-Vidal, “Efficient coupling of light into and out of a photonic crystal waveguide via surface modes,” Photonics Nanostruct. Fundam. Appl. 2, 97–102 (2004).
[CrossRef]

Morozov, G. V.

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A, Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

Müller, F.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Namdar, A.

A. Namdar, “Tamm states in one dimensional photonic crystals containing left-handed materials,” Opt. Commun. 278, 194–198 (2007).
[CrossRef]

A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Excitation of backward Tamm states at an interface between a periodic photonic crystal and a left-handed metamaterial,” Phys. Rev. A 75, 053812 (2007).
[CrossRef]

A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Backward Tamm states in left-handed metamaterials,” Appl. Phys. Lett. 89, 114104 (2006).
[CrossRef]

Ong, C. K.

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103 (2003).
[CrossRef]

Pendry, J. B.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37–43 (2004).
[CrossRef]

Ramos-Mendieta, F.

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

Rao, X. S.

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103 (2003).
[CrossRef]

Sandoghdar, V.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

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] [PubMed]

Shadrivov, I. V.

A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Excitation of backward Tamm states at an interface between a periodic photonic crystal and a left-handed metamaterial,” Phys. Rev. A 75, 053812 (2007).
[CrossRef]

A. Namdar, I. V. Shadrivov, and Y. S. Kivshar, “Backward Tamm states in left-handed metamaterials,” Appl. Phys. Lett. 89, 114104 (2006).
[CrossRef]

Shalaev, V. M.

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] [PubMed]

Skovgaard, P. M. W.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37–43 (2004).
[CrossRef]

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

Soukoulis, C. M.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

Sprung, D. W. L.

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A, Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

Stegun, A. S.

M. Abramowitz and A. S. Stegun, Handbook of Mathematical Function (Dover, 1972).

Tao, J. W.

R. R. Wei, X. Chen, J. W. Tao, and C. F. Li, “Giant and negative bistable shifts for one-dimensional photonic crystal containing a nonlinear metamaterial defect,” Phys. Lett. A 372, 6797–6800 (2008).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “Electromagnetics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Wehrspohn, R. B.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Wei, R. R.

R. R. Wei, X. Chen, J. W. Tao, and C. F. Li, “Giant and negative bistable shifts for one-dimensional photonic crystal containing a nonlinear metamaterial defect,” Phys. Lett. A 372, 6797–6800 (2008).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

Xiao, S.

Yariv, A.

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of the problem. In our calculations we take the following values for conventional PC: d 1 = 1   cm , d 2 = 1.65   cm , ε 1 = 4 , μ 1 = 1 , ε 2 = 2.25 , μ 2 = 1 , and cap layer characterized by μ c = 1 , ε c = 1 , and thickness d c = 2 d 1 ; also the parameters used for LH PC are d 1 = 2.5   cm , d 2 = 1   cm , ε 1 = 4 , μ 1 = 1 , ε 2 = 2.25 , μ 2 = 1 , and cap layer characterized by μ c = 1 , ε c = 4 , and thickness d c = 0.6 d 1 .

Fig. 2
Fig. 2

Graphical representation of the solutions of dispersion relation [Eq. (9)] versus k in the first spectral gap of 1DPC for a typical value of β = 1.211 . Here the solid and dashed lines show the left-hand and right-hand sides of Eq. (9) for (a) conventional PC, γ 0 = 0.5 , and (b) LH PC, γ 0 = 0.5 .

Fig. 3
Fig. 3

Dispersion property of SWs for different values of γ 0 in the first spectral gap of (a) conventional PC and (b) LH PC. Shaded area: First bandgap of PC. Solid, dashed, and dotted lines: dispersion of the SWs for γ 0 = 0.5 , 0.2, and γ 0 0 .

Fig. 4
Fig. 4

(a),(b) The transverse profile of an order zero SWs versus coordinate z for β = 1.221 . Solid, dashed, and dotted lines are for γ 0 = + .5 , 0.2, and γ 0 0 , respectively. (c) The intensity distributions for γ 0 = + .2 in Fig. 3a. (d) The transverse profile of an order 1 SWs versus coordinate z for β = 1.17704 and γ 0 = + .2 . The other parameters are the same as those in Fig. 3.

Fig. 5
Fig. 5

Dispersion property of the SWs in the first spectral gap for d c = 0.01   cm , γ 0 = 0.2 . (a) Conventional PC and (b) LH PC. Here the solid and marked lines show dispersion properties of SWs based on Eqs. (9, 17), respectively.

Fig. 6
Fig. 6

Total energy flow of SWs in the first bandgap of LH PC versus β. Here the solid, dashed, and dotted curves correspond to γ 0 = 0.5 , 0.2, and γ 0 0 . The other parameters are the same as those in Fig. 3.

Fig. 7
Fig. 7

Total energy flow of Tamm states in the first bandgap of conventional PC versus β. Here the solid, dashed, and dotted curves correspond to γ 0 = 0.5 , 0.2, and γ 0 0 . The other parameters are the same as those in Fig. 3.

Equations (21)

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E = E y ( z ) exp ( i ( k β x ω t ) ) e ̂ y ,
H = ( H x ( z ) e ̂ x + H z ( z ) e ̂ z ) exp ( i ( k β x ω t ) ) ,
ε NL = ε c + α | E c ( z ) | 2 ,
[ 2 z 2 k 2 β 2 + ( ω 2 c 2 ) μ c ( ε c + α | E c ( z ) | 2 ) ] E c ( z ) = 0.
E d ( z ) = E 0 e q 0 ( z + d c ) ,
( E c ( z ) z ) 2 k c 2 E c 2 ( z ) + k 2 μ c α / 2 E c 4 ( z ) = C c ,
E d z = d c = E c z = d c = E 0 ;     | 1 μ 0 E d z | z = d c = | 1 μ c E c z | z = d c ,
E c z = 0 = E 1 z = 0 = E b ,     | 1 μ c E c z | z = 0 = | 1 μ 1 E 1 z | z = 0 .
C c = γ 0 ( μ c 2 μ 0 2 q 0 2 k c 2 + k 2 μ c γ 0 2 ) = γ b ( μ c 2 μ 1 2 k 1 z 2 R ̃ 2 k c 2 + k 2 μ c γ b 2 ) ,
E c z = | μ c | 1 / 2 k α 1 / 2 [ E c 2 + k c 2 + k c 4 2 C c | μ c | k 2 α | μ c | k 2 α ] 1 / 2 [ E c 2 + k c 2 k s 4 2 c c | μ c | k 2 α | μ c | k 2 α ] 1 / 2 .
E c 1 ( z ) = d s ( | μ c | 1 / 2 k ( α / 2 ) 1 / 2 ( a 1 2 + b 1 2 ) 1 / 2 ( z + d c ) + z 01 m 1 ) ( a 1 2 + b 1 2 ) 1 / 2 ,
a 1 2 = k c 2 + k c 4 + 2 | C c | | μ c | k 2 α | μ c | k 2 α ,
b 1 2 = k c 2 + k c 4 + 2 | C c | | μ c | k 2 α | μ c | k 2 α ,
E c 2 ( z ) = b 2 s n ( z 02 a 2 k ( | μ c | α 2 ) 1 / 2 ( z + d c ) m 2 ) ,
a 2 2 = | k c 2 | + | k c 2 | 2 2 C c k 2 | μ c | α k 2 | μ c | α ,
b 2 2 = | k c 2 | | k c 2 | 2 2 C c k 2 | μ c | α k 2 | μ c | α ,
E c 1 ( z ) = 2 k c E 0 k | μ c | γ 0 cosech ( cosech 1 ( E 0 a 1 ) + k c ( z + d c ) ) .
E c 2 ( z ) = a 2 n s ( n s 1 ( E 0 a 2 m 2 ) a 2 k | μ c | α / 2 ( z + d c ) m 2 ) .
E c 2 ( z ) = 2 | k c | E 0 k | μ c | γ 0 csc ( csc 1 ( E 0 a ) | k c | ( z + d c ) ) .
ε ( z + d c ) = ζ 0 ( 1 + λ | E c | 2 ) δ ( z + d c ) .
q 0 k 1 z = k 2 ζ 0 ( 1 + λ | E 0 | 2 ) μ 0 k 1 z + i μ 0 μ 1 B ̃ ( λ A ) B ̃ + ( λ A ) ,

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