Abstract

We study the electromagnetic surface modes supported by the interface between semi-infinite one-dimensional photonic crystals composed of dispersive metamaterials and a homogeneous medium in the presence of the cap layer. A new type of surface mode is found in such structure, and these surface modes can exhibit strong resonance at the first period of the photonic crystal. These resonant surface modes originate from the material dispersion. Another type of surface mode with structural resonance can appear in the thick cap layer. The influences of physical parameters of the structure on the surface modes have also been investigated.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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  26. C. Yeh and F. Shimabukuro, The Essence of Dielectric Waveguides (Springer, 2008).
    [CrossRef]
  27. J. W. Dong and H. Z. Wang, “Slow electromagnetic propagation with low group velocity dispersion in an all-metamaterial-based waveguide,” Appl. Phys. Lett. 91, 111909 (2007).
    [CrossRef]
  28. A. Alú and N. Engheta, “Pairing an epsilon-negative slab with a μ-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
    [CrossRef]

2008 (3)

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

J. Barvestani, M. Kalafi, A. Soltani-Vala, and A. Namdar, “Backward surface electromagnetic waves in semi-infinite one-dimensional photonic crystals containing left-handed materials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

C. Vandenbem, “Electromagnetic surface waves of multilayer stacks: coupling between guided modes and Bloch modes,” Opt. Lett. 33, 2260-2262 (2008).
[CrossRef] [PubMed]

2007 (5)

J. W. Dong and H. Z. Wang, “Slow electromagnetic propagation with low group velocity dispersion in an all-metamaterial-based waveguide,” Appl. Phys. Lett. 91, 111909 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53-55 (2007).
[CrossRef]

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]

Y. El Hassouani, E. H. El Boudouti, H. Aynaou, B. Djafari-Rouhani, and V. R. Velasco, “Comment on 'Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals',” J. Opt. A, Pure Appl. Opt. 9, 308-313 (2007).
[CrossRef]

M. Kalafi, A. Soltani-Vala, and J. Barvestani, “Surface optical waves in semi-infinite one-dimensional photonic crystals with a thin nonlinear cap layer,” Opt. Commun. 272, 403-406 (2007).
[CrossRef]

2006 (1)

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

2005 (3)

S. Feng, H. Sang, Z. Li, B. Cheng, and D. Zhang, “Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals,” J. Opt. A, Pure Appl. Opt. 7, 374-381 (2005).
[CrossRef]

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[CrossRef] [PubMed]

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

2004 (1)

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

2003 (3)

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

A. Alú and N. Engheta, “Pairing an epsilon-negative slab with a μ-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

2002 (1)

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

2001 (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]

2000 (1)

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

1999 (1)

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800-1802 (1999).
[CrossRef]

1996 (2)

E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, and L. Dobrzynski, “Theory of surface and interface transverse elastic waves in N-layer superlattices,” Phys. Rev. B 54, 14728-14741 (1996).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

1988 (1)

M. I. D'yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714-716 (1988).

1978 (1)

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

1977 (2)

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Akjouj, A.

E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, and L. Dobrzynski, “Theory of surface and interface transverse elastic waves in N-layer superlattices,” Phys. Rev. B 54, 14728-14741 (1996).
[CrossRef]

Alú, A.

A. Alú and N. Engheta, “Pairing an epsilon-negative slab with a μ-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Artigas, D.

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[CrossRef] [PubMed]

Aynaou, H.

Y. El Hassouani, E. H. El Boudouti, H. Aynaou, B. Djafari-Rouhani, and V. R. Velasco, “Comment on 'Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals',” J. Opt. A, Pure Appl. Opt. 9, 308-313 (2007).
[CrossRef]

Barvestani, J.

J. Barvestani, M. Kalafi, A. Soltani-Vala, and A. Namdar, “Backward surface electromagnetic waves in semi-infinite one-dimensional photonic crystals containing left-handed materials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

M. Kalafi, A. Soltani-Vala, and J. Barvestani, “Surface optical waves in semi-infinite one-dimensional photonic crystals with a thin nonlinear cap layer,” Opt. Commun. 272, 403-406 (2007).
[CrossRef]

Cai, W.

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Chen, H.

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Cheng, B.

S. Feng, H. Sang, Z. Li, B. Cheng, and D. Zhang, “Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals,” J. Opt. A, Pure Appl. Opt. 7, 374-381 (2005).
[CrossRef]

Chettiar, U. K.

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]

Djafari-Rouhani, B.

Y. El Hassouani, E. H. El Boudouti, H. Aynaou, B. Djafari-Rouhani, and V. R. Velasco, “Comment on 'Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals',” J. Opt. A, Pure Appl. Opt. 9, 308-313 (2007).
[CrossRef]

E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, and L. Dobrzynski, “Theory of surface and interface transverse elastic waves in N-layer superlattices,” Phys. Rev. B 54, 14728-14741 (1996).
[CrossRef]

Dobrzynski, L.

E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, and L. Dobrzynski, “Theory of surface and interface transverse elastic waves in N-layer superlattices,” Phys. Rev. B 54, 14728-14741 (1996).
[CrossRef]

Dolling, G.

Dong, J. W.

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

J. W. Dong and H. Z. Wang, “Slow electromagnetic propagation with low group velocity dispersion in an all-metamaterial-based waveguide,” Appl. Phys. Lett. 91, 111909 (2007).
[CrossRef]

Drachev, V. P.

D'yakonov, M. I.

M. I. D'yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714-716 (1988).

El Boudouti, E. H.

Y. El Hassouani, E. H. El Boudouti, H. Aynaou, B. Djafari-Rouhani, and V. R. Velasco, “Comment on 'Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals',” J. Opt. A, Pure Appl. Opt. 9, 308-313 (2007).
[CrossRef]

E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, and L. Dobrzynski, “Theory of surface and interface transverse elastic waves in N-layer superlattices,” Phys. Rev. B 54, 14728-14741 (1996).
[CrossRef]

El Hassouani, Y.

Y. El Hassouani, E. H. El Boudouti, H. Aynaou, B. Djafari-Rouhani, and V. R. Velasco, “Comment on 'Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals',” J. Opt. A, Pure Appl. Opt. 9, 308-313 (2007).
[CrossRef]

Engheta, N.

A. Alú and N. Engheta, “Pairing an epsilon-negative slab with a μ-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Enkrich, C.

Feng, S.

S. Feng, H. Sang, Z. Li, B. Cheng, and D. Zhang, “Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals,” J. Opt. A, Pure Appl. Opt. 7, 374-381 (2005).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Hong, C. S.

Jiang, H. T.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

Kalafi, M.

J. Barvestani, M. Kalafi, A. Soltani-Vala, and A. Namdar, “Backward surface electromagnetic waves in semi-infinite one-dimensional photonic crystals containing left-handed materials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

M. Kalafi, A. Soltani-Vala, and J. Barvestani, “Surface optical waves in semi-infinite one-dimensional photonic crystals with a thin nonlinear cap layer,” Opt. Commun. 272, 403-406 (2007).
[CrossRef]

Kavokin, A. V.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

Kildishev, A. V.

Kivshar, Y.

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

Li, H. Q.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Li, Z.

S. Feng, H. Sang, Z. Li, B. Cheng, and D. Zhang, “Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals,” J. Opt. A, Pure Appl. Opt. 7, 374-381 (2005).
[CrossRef]

Linden, S.

Lindgren, G.

Liu, N. H.

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Malpuech, G.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

May, M. S.

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800-1802 (1999).
[CrossRef]

Namdar, A.

J. Barvestani, M. Kalafi, A. Soltani-Vala, and A. Namdar, “Backward surface electromagnetic waves in semi-infinite one-dimensional photonic crystals containing left-handed materials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

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

Nemat-Nasser, S. C.

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

Padilla, W. J.

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

Pendry, J. B.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Robertson, W. M.

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800-1802 (1999).
[CrossRef]

Sang, H.

S. Feng, H. Sang, Z. Li, B. Cheng, and D. Zhang, “Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals,” J. Opt. A, Pure Appl. Opt. 7, 374-381 (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] [PubMed]

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

Shadrivov, I.

A. Namdar, I. Shadrivov, and Y. 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]

Shelykh, I. A.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Shimabukuro, F.

C. Yeh and F. Shimabukuro, The Essence of Dielectric Waveguides (Springer, 2008).
[CrossRef]

Smith, D. R.

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

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

Soltani-Vala, A.

J. Barvestani, M. Kalafi, A. Soltani-Vala, and A. Namdar, “Backward surface electromagnetic waves in semi-infinite one-dimensional photonic crystals containing left-handed materials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

M. Kalafi, A. Soltani-Vala, and J. Barvestani, “Surface optical waves in semi-infinite one-dimensional photonic crystals with a thin nonlinear cap layer,” Opt. Commun. 272, 403-406 (2007).
[CrossRef]

Soukoulis, C. M.

Stewart, W. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Torner, L.

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[CrossRef] [PubMed]

Vandenbem, C.

Velasco, V. R.

Y. El Hassouani, E. H. El Boudouti, H. Aynaou, B. Djafari-Rouhani, and V. R. Velasco, “Comment on 'Sensitivity of surface states to the stack sequence of one-dimensional photonic crystals',” J. Opt. A, Pure Appl. Opt. 9, 308-313 (2007).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vier, D. C.

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

Wang, H. Z.

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

J. W. Dong and H. Z. Wang, “Slow electromagnetic propagation with low group velocity dispersion in an all-metamaterial-based waveguide,” Appl. Phys. Lett. 91, 111909 (2007).
[CrossRef]

Wang, L. G.

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Wang, T. B.

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

Wegener, M.

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Figures (10)

Fig. 1
Fig. 1

Schematic representation of the problem. The periodic structure is a semi-infinite 1D PC stacked with materials A and B; the left is a homogeneous medium, and C is the cap layer.

Fig. 2
Fig. 2

Calculated surface modes for TE polarization in a semi-infinite 1D PC composed of metamaterials without the cap layer. The parameters of the structure are d A = d B = 0.5 L . The dashed lines and dotted lines are the surface modes of samples I and II, respectively. The solid line is the light line of air.

Fig. 3
Fig. 3

The electric field distributions of surface modes of the structure without cap layer. (a) Ω = 2.41 , k = 2.9988 ; (b) Ω = 1.632 , k = 3.0758 ; (c) Ω = 2.6570 , k = 3.0142 ; (d) Ω = 0.3765 , k = 2.8896 .

Fig. 4
Fig. 4

The surface modes as a function of d A L at k = 3.0 without the cap layer. The dashed lines and dotted lines are the surface modes of samples I and II, respectively.

Fig. 5
Fig. 5

Surface modes of 1D PC composed of metamaterials with a PIM cap layer. The parameters of the structure are d A = d B = 0.5 L , d C = 0.5 L , ϵ C = 4.0 , μ C = 1.0 . The dashed lines and dotted lines are the surface modes of samples I and II, respectively. The solid line is the light line of air.

Fig. 6
Fig. 6

The electric field distributions of surface modes of the structure with a PIM cap layer. The parameters of the cap layer are the same as that in Fig. 5. (a) Ω = 2.447 , k = 2.613 ; (b) Ω = 1.584 , k = 4.536 ; (c) Ω = 2.8696 , k = 5.0490 ; (d) Ω = 0.3732 , k = 1.7040 .

Fig. 7
Fig. 7

Surface modes of semi-infinite 1D PC composed of metamaterials as a function of the thickness of the cap layer at k = 3.0 , ϵ C = 4.0 . The dashed lines and dotted lines are the surface modes of samples I and II, respectively.

Fig. 8
Fig. 8

The electric field distributions of surface modes of the structure with a PIM cap layer. The parameters are the same as that in Fig. 7. (a) d C = 0.6 L , Ω = 2.4924 ; (b) d C = 0.5 L , Ω = 1.3497 ; (c) d C = 0.1 L , Ω = 0.3761 ; (d) d C = 0.8 L , Ω = 2.5958 .

Fig. 9
Fig. 9

Surface modes of 1D PC composed of metamaterials as a function of the permittivity of the cap layer at k = 3.0 , d C = 1.0 L . The dashed lines and dotted lines are the surface modes of samples I and II, respectively.

Fig. 10
Fig. 10

The electric field distributions of surface modes of the structure with a PIM cap layer. The parameters are the same as that in Fig. 9. (a) ϵ C = 4.0 , Ω = 2.6295 ; (b) ϵ C = 5.5 , Ω = 1.7543 ; (c) ϵ C = 2.04 , Ω = 0.3761 ; (d) ϵ C = 3.51 , Ω = 2.5114 .

Equations (6)

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M i ( d i ) = ( cos ( k i d i ) 1 σ i sin ( k i d i ) σ i sin ( k i d i ) cos ( k i d i ) ) , ( i = A , B , C ) ,
| ( E y ( z ) j c H x ( z ) ) | z = 0 = ( M 12 e j K L M 11 ) ,
| ( E y ( z ) j c H x ( z ) ) | z = d c = | ( C e k 0 z F 0 C e k 0 z ) | z = d c ,
( M 12 e j K L M 11 ) = M C | ( C e k 0 z F 0 C e k 0 z ) | z = d c .
M 11 e j K L M 12 = σ C ( σ C sin ( k C d C ) F 0 cos ( k C d C ) ) σ C cos ( k C d C ) + F 0 sin ( k C d C ) .
ϵ i = ϵ 0 i ω ep , i 2 ω 2 , μ i = μ 0 i ω mp , i 2 ω 2 , ( i = A , B ) ,

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