Abstract

In this paper, we theoretically study the electromagnetic surface waves localized at the interface between a homogeneous dielectric medium and a semi-infinite, one-dimensional photonic crystal (1D PC). The semi-infinite 1D PC is made of alternative layers of right-handed (RH) and dispersive left-handed (LH) materials in the presence of a liquid crystal (LC) cap layer. In this structure, we derive the surface waves dispersion relation with tunable switching and localization by using an analytical direct matching procedure within the Kronig–Penny model. It is shown that, for both of layer arrangements, the variation of molecules orientation of the LC cap layer acts as an effective tool to tune the type (tuned switching) and localization of the surface waves and it also can create a surface mode with maximum localization in the first frequency bandgap.

© 2012 Optical Society of America

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    [CrossRef]
  36. M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S. G. Davison, “Localized electronic states in semiconductor superlattices,” Surf. Sci. Rep. 47, 93–196 (2002).
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2010 (4)

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Tunable fishnet metamaterials infiltrated by liquid crystals,” Appl. Phys. Lett. 96, 193103 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, S. I. Bozhevolnyi, and Y. S. Kivshar, “Backward and forward modes guided by metal-dielectric-metal plasmonic waveguides,” J. Nanophoton. 4, 043509 (2010).
[CrossRef]

H. Hajian, A. Soltani-Vala, and M. Kalafi, “Controlled switching of surface waves in 1D photonic crystals by a thin nonlinear cap layer,” Opt. Commun. 283, 4847–4854 (2010).
[CrossRef]

2009 (8)

X. Qi, I. L. Garanovich, Z. Xu, A. A. Sukhorukov, W. Krolikowski, A. Mitchell, G. Zhang, D. N. Neshev, and Y. S. Kivshar, “Observation of nonlinear surface waves in modulated waveguide arrays,” Opt. Lett. 34, 2751–2753 (2009).
[CrossRef]

B. Rezaei and M. Kalafi, “Tunable full band gap in two-dimensional anisotropic photonic crystals infiltrated with liquid crystals,” Opt. Commun. 282, 1584–1588 (2009).
[CrossRef]

T. B. Wang, C. P. Yin, W. Y. Liang, J. W. Dong, and H. Z. Wang, “Electromagnetic surface modes in one-dimensional photonic crystals with dispersive metamaterials,” J. Opt. Soc. Am. B 26, 1635–1640 (2009).
[CrossRef]

D. V. Kulagin, A. S. Savchenko, and S. V. Tarasenko, “Polariton dynamics of a one-dimensional gyrotropic magnetic photonic crystal in a dc electric field: II. Surface waves,” Opt. Spectrosc. 107, 803–810 (2009).
[CrossRef]

S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterials and metal-dielectric superlatices,” Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

D. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

S. Zhang, Y. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef]

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. S. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95, 084105 (2009).
[CrossRef]

2008 (3)

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

M. de Dios-Leyva and J. A. Leyva-Galano, “Influence of absorption on the zero-n¯ gap in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 78, 115106 (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 metamaterials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

2007 (5)

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]

A. Mendoza-Suarez, F. Villa-Villa, and G. A. Gasper-Armenta, “Band structure of two dimensional photonic crystals that include dispersive left-handed materials and dielectrics in the unit cells,” J. Opt. Soc. Am. B 24, 3091–3098 (2007).
[CrossRef]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow storage of light in metamaterials,” Nature 450, 397–401 (2007).
[CrossRef]

R. A. Depine, M. L. Martinez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andres, “Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystal,” Phys. Lett. A 364, 352–355 (2007).
[CrossRef]

G. Bordin, M. Marklund, L. Stenflo, and P. K. Shukla, “Anomalous reflection and excitation of surface waves in metamaterials,” Phys. Lett. A 367, 233–236 (2007).
[CrossRef]

2006 (4)

F. Villa-Villa and G. A. Gasper-Armenta, “Brewster angle and optical tunneling in one-dimensional photonic crystals composed of left- and right-handed materials,” J. Opt. Soc. Am. B 23, 375–380 (2006).
[CrossRef]

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

J. Martoller, 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. I. Rahachou and I. V. Zozoulenko, “Waveguiding properties of surface states in photonic crystals,” J. Opt. Soc. Am. B 23, 1679–1683 (2006).
[CrossRef]

2005 (3)

S. Feng, H.-Y. Sang, Z.-Y. Li, B.-Y. Cheng, and D.-Z. 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]

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Mid-infrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

2004 (3)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[CrossRef]

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Nonlinear surface waves in left-handed metamaterials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

2003 (1)

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]

2002 (2)

N. Engheta, “An idea for thin sub-wavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propagat. Lett. 1, 10–13 (2002).

M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S. G. Davison, “Localized electronic states in semiconductor superlattices,” Surf. Sci. Rep. 47, 93–196 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, D. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

2000 (1)

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

1968 (1)

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

Akjouj, A.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[CrossRef]

M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S. G. Davison, “Localized electronic states in semiconductor superlattices,” Surf. Sci. Rep. 47, 93–196 (2002).
[CrossRef]

Andres, P.

R. A. Depine, M. L. Martinez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andres, “Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystal,” Phys. Lett. A 364, 352–355 (2007).
[CrossRef]

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 metamaterials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow storage of light in metamaterials,” Nature 450, 397–401 (2007).
[CrossRef]

Bordin, G.

G. Bordin, M. Marklund, L. Stenflo, and P. K. Shukla, “Anomalous reflection and excitation of surface waves in metamaterials,” Phys. Lett. A 367, 233–236 (2007).
[CrossRef]

Boudouti, E. H. El

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]

Bozhevolnyi, S. I.

A. R. Davoyan, I. V. Shadrivov, S. I. Bozhevolnyi, and Y. S. Kivshar, “Backward and forward modes guided by metal-dielectric-metal plasmonic waveguides,” J. Nanophoton. 4, 043509 (2010).
[CrossRef]

Bria, D.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Brueck, S. R. J.

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Mid-infrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef]

Cai, W.

Caloz, C.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).

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]

Cheng, B.-Y.

S. Feng, H.-Y. Sang, Z.-Y. Li, B.-Y. Cheng, and D.-Z. 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.

Davison, S. G.

M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S. G. Davison, “Localized electronic states in semiconductor superlattices,” Surf. Sci. Rep. 47, 93–196 (2002).
[CrossRef]

Davoyan, A. R.

A. R. Davoyan, I. V. Shadrivov, S. I. Bozhevolnyi, and Y. S. Kivshar, “Backward and forward modes guided by metal-dielectric-metal plasmonic waveguides,” J. Nanophoton. 4, 043509 (2010).
[CrossRef]

de Dios-Leyva, M.

M. de Dios-Leyva and J. A. Leyva-Galano, “Influence of absorption on the zero-n¯ gap in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 78, 115106 (2008).
[CrossRef]

Depine, R. A.

R. A. Depine, M. L. Martinez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andres, “Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystal,” Phys. Lett. A 364, 352–355 (2007).
[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]

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[CrossRef]

M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S. G. Davison, “Localized electronic states in semiconductor superlattices,” Surf. Sci. Rep. 47, 93–196 (2002).
[CrossRef]

Dobrzynski, L.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[CrossRef]

M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S. G. Davison, “Localized electronic states in semiconductor superlattices,” Surf. Sci. Rep. 47, 93–196 (2002).
[CrossRef]

Dong, J. W.

Drachev, V. P.

El Boudouti, E. H.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Engheta, N.

N. Engheta, “An idea for thin sub-wavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propagat. Lett. 1, 10–13 (2002).

Fan, W.

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Mid-infrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef]

Feng, S.

S. Feng, H.-Y. Sang, Z.-Y. Li, B.-Y. Cheng, and D.-Z. 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]

Frauenglass, A.

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Mid-infrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef]

Garanovich, I. L.

Gasper-Armenta, G. A.

Gorkunov, M.

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. S. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95, 084105 (2009).
[CrossRef]

Hajian, H.

H. Hajian, A. Soltani-Vala, and M. Kalafi, “Controlled switching of surface waves in 1D photonic crystals by a thin nonlinear cap layer,” Opt. Commun. 283, 4847–4854 (2010).
[CrossRef]

Hassouani, Y. El

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]

Hattori, H. T.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow storage of light in metamaterials,” Nature 450, 397–401 (2007).
[CrossRef]

Itoh, T.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).

Jagadish, C.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

Kalafi, M.

H. Hajian, A. Soltani-Vala, and M. Kalafi, “Controlled switching of surface waves in 1D photonic crystals by a thin nonlinear cap layer,” Opt. Commun. 283, 4847–4854 (2010).
[CrossRef]

B. Rezaei and M. Kalafi, “Tunable full band gap in two-dimensional anisotropic photonic crystals infiltrated with liquid crystals,” Opt. Commun. 282, 1584–1588 (2009).
[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 metamaterials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

Khoo, I. C.

I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993).

I. C. Khoo, Liquid Crystals (Wiley, 2007).

Kildishev, A. V.

Kivshar, Y.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

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

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Nonlinear surface waves in left-handed metamaterials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Kivshar, Y. S.

A. R. Davoyan, I. V. Shadrivov, S. I. Bozhevolnyi, and Y. S. Kivshar, “Backward and forward modes guided by metal-dielectric-metal plasmonic waveguides,” J. Nanophoton. 4, 043509 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Tunable fishnet metamaterials infiltrated by liquid crystals,” Appl. Phys. Lett. 96, 193103 (2010).
[CrossRef]

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. S. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95, 084105 (2009).
[CrossRef]

D. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterials and metal-dielectric superlatices,” Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

X. Qi, I. L. Garanovich, Z. Xu, A. A. Sukhorukov, W. Krolikowski, A. Mitchell, G. Zhang, D. N. Neshev, and Y. S. Kivshar, “Observation of nonlinear surface waves in modulated waveguide arrays,” Opt. Lett. 34, 2751–2753 (2009).
[CrossRef]

Krolikowski, W.

Kucharczyk, R.

M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S. G. Davison, “Localized electronic states in semiconductor superlattices,” Surf. Sci. Rep. 47, 93–196 (2002).
[CrossRef]

Kulagin, D. V.

D. V. Kulagin, A. S. Savchenko, and S. V. Tarasenko, “Polariton dynamics of a one-dimensional gyrotropic magnetic photonic crystal in a dc electric field: II. Surface waves,” Opt. Spectrosc. 107, 803–810 (2009).
[CrossRef]

Lapine, M.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. S. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95, 084105 (2009).
[CrossRef]

Leyva-Galano, J. A.

M. de Dios-Leyva and J. A. Leyva-Galano, “Influence of absorption on the zero-n¯ gap in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 78, 115106 (2008).
[CrossRef]

Li, J.

S. Zhang, Y. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef]

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]

Li, Z.-Y.

S. Feng, H.-Y. Sang, Z.-Y. Li, B.-Y. Cheng, and D.-Z. 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]

Liang, W. Y.

Lu, X.

S. Zhang, Y. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef]

Malloy, K. J.

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Mid-infrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef]

Marklund, M.

G. Bordin, M. Marklund, L. Stenflo, and P. K. Shukla, “Anomalous reflection and excitation of surface waves in metamaterials,” Phys. Lett. A 367, 233–236 (2007).
[CrossRef]

Marques, R.

R. Marques, F. Martin, and Mosorolla, Metamaterials with Negative Parameters (Wiley, 2008).

Marqués, R.

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. S. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95, 084105 (2009).
[CrossRef]

Martin, F.

R. Marques, F. Martin, and Mosorolla, Metamaterials with Negative Parameters (Wiley, 2008).

Martinez-Ricci, M. L.

R. A. Depine, M. L. Martinez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andres, “Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystal,” Phys. Lett. A 364, 352–355 (2007).
[CrossRef]

Martoller, J.

J. Martoller, 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]

McKerracher, I.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

Mendoza-Suarez, A.

Minhas, B. K.

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Mid-infrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef]

Minovich, A.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Tunable fishnet metamaterials infiltrated by liquid crystals,” Appl. Phys. Lett. 96, 193103 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

Mitchell, A.

Monsoriu, J. A.

R. A. Depine, M. L. Martinez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andres, “Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystal,” Phys. Lett. A 364, 352–355 (2007).
[CrossRef]

Morozov, G. V.

J. Martoller, 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]

Mosorolla,

R. Marques, F. Martin, and Mosorolla, Metamaterials with Negative Parameters (Wiley, 2008).

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 metamaterials,” 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, D. C.

R. A. Shelby, D. R. Smith, D. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

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

Neshev, D. N.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Tunable fishnet metamaterials infiltrated by liquid crystals,” Appl. Phys. Lett. 96, 193103 (2010).
[CrossRef]

X. Qi, I. L. Garanovich, Z. Xu, A. A. Sukhorukov, W. Krolikowski, A. Mitchell, G. Zhang, D. N. Neshev, and Y. S. Kivshar, “Observation of nonlinear surface waves in modulated waveguide arrays,” Opt. Lett. 34, 2751–2753 (2009).
[CrossRef]

Nougaoui, A.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Padilla, W. J.

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

Park, Y.

S. Zhang, Y. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

Powell, D.

D. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. S. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95, 084105 (2009).
[CrossRef]

Powell, D. A.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Tunable fishnet metamaterials infiltrated by liquid crystals,” Appl. Phys. Lett. 96, 193103 (2010).
[CrossRef]

Qi, X.

Rahachou, A. I.

Rezaei, B.

B. Rezaei and M. Kalafi, “Tunable full band gap in two-dimensional anisotropic photonic crystals infiltrated with liquid crystals,” Opt. Commun. 282, 1584–1588 (2009).
[CrossRef]

Sang, H.-Y.

S. Feng, H.-Y. Sang, Z.-Y. Li, B.-Y. Cheng, and D.-Z. 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]

Sarychev, A. K.

Savchenko, A. S.

D. V. Kulagin, A. S. Savchenko, and S. V. Tarasenko, “Polariton dynamics of a one-dimensional gyrotropic magnetic photonic crystal in a dc electric field: II. Surface waves,” Opt. Spectrosc. 107, 803–810 (2009).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, D. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

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

Shadrivov, I.

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. S. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95, 084105 (2009).
[CrossRef]

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

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Nonlinear surface waves in left-handed metamaterials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Shadrivov, I. V.

A. R. Davoyan, I. V. Shadrivov, S. I. Bozhevolnyi, and Y. S. Kivshar, “Backward and forward modes guided by metal-dielectric-metal plasmonic waveguides,” J. Nanophoton. 4, 043509 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Tunable fishnet metamaterials infiltrated by liquid crystals,” Appl. Phys. Lett. 96, 193103 (2010).
[CrossRef]

D. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterials and metal-dielectric superlatices,” Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, D. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[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]

Shukla, P. K.

G. Bordin, M. Marklund, L. Stenflo, and P. K. Shukla, “Anomalous reflection and excitation of surface waves in metamaterials,” Phys. Lett. A 367, 233–236 (2007).
[CrossRef]

Silvestre, E.

R. A. Depine, M. L. Martinez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andres, “Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystal,” Phys. Lett. A 364, 352–355 (2007).
[CrossRef]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

R. A. Shelby, D. R. Smith, D. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

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

Soltani-Vala, A.

H. Hajian, A. Soltani-Vala, and M. Kalafi, “Controlled switching of surface waves in 1D photonic crystals by a thin nonlinear cap layer,” Opt. Commun. 283, 4847–4854 (2010).
[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 metamaterials,” Phys. Rev. A 77, 013805 (2008).
[CrossRef]

Sprung, D. W. L.

J. Martoller, 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]

Stenflo, L.

G. Bordin, M. Marklund, L. Stenflo, and P. K. Shukla, “Anomalous reflection and excitation of surface waves in metamaterials,” Phys. Lett. A 367, 233–236 (2007).
[CrossRef]

Steslicka, M.

M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S. G. Davison, “Localized electronic states in semiconductor superlattices,” Surf. Sci. Rep. 47, 93–196 (2002).
[CrossRef]

Sukhorukov, A.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Nonlinear surface waves in left-handed metamaterials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Sukhorukov, A. A.

Tan, H. H.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Y. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010).
[CrossRef]

Tarasenko, S. V.

D. V. Kulagin, A. S. Savchenko, and S. V. Tarasenko, “Polariton dynamics of a one-dimensional gyrotropic magnetic photonic crystal in a dc electric field: II. Surface waves,” Opt. Spectrosc. 107, 803–810 (2009).
[CrossRef]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow storage of light in metamaterials,” Nature 450, 397–401 (2007).
[CrossRef]

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).
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V. G. Veselago, “The electrodynamics of substances with simultaneous negative value of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
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D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Villa-Villa, F.

Vukovic, S. M.

S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterials and metal-dielectric superlatices,” Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

Wang, H. Z.

Wang, T. B.

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

Wu, S. T.

I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993).

Xu, Z.

Yin, C. P.

Yuan, H.-K.

Zhang, D.-Z.

S. Feng, H.-Y. Sang, Z.-Y. Li, B.-Y. Cheng, and D.-Z. 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]

Zhang, G.

Zhang, S.

S. Zhang, Y. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef]

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Mid-infrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef]

Zhang, W.

S. Zhang, Y. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef]

Zhang, X.

S. Zhang, Y. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef]

Zhou, L.

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]

Zozoulenko, I. V.

Appl. Phys. Lett. (6)

R. A. Shelby, D. R. Smith, D. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

D. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. S. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95, 084105 (2009).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Tunable fishnet metamaterials infiltrated by liquid crystals,” Appl. Phys. Lett. 96, 193103 (2010).
[CrossRef]

S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterials and metal-dielectric superlatices,” Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

A. Namdar, I. Shadrivov, and Y. Kivshar, “Backward Tamm states in left-handed metamaterials,” Appl. Phys. Lett. 89, 114104 (2006).
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J. Nanophoton. (1)

A. R. Davoyan, I. V. Shadrivov, S. I. Bozhevolnyi, and Y. S. Kivshar, “Backward and forward modes guided by metal-dielectric-metal plasmonic waveguides,” J. Nanophoton. 4, 043509 (2010).
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J. Opt. A: Pure Appl. Opt. (3)

S. Feng, H.-Y. Sang, Z.-Y. Li, B.-Y. Cheng, and D.-Z. 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]

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]

J. Martoller, 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]

J. Opt. Soc. Am. B (4)

Nature (1)

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Figures (9)

Fig. 1.
Fig. 1.

Photonic crystal structures under consideration. (a) An LH layer is adjacent to LC cap layer; (b) an RH layer is adjacent to an LC cap layer. The orange region (dLCz0) shows that the LC cap layer. n⃗ is the director of LC molecules, and ϕ is its angle with the Z axis. The left is a homogeneous medium.

Fig. 2.
Fig. 2.

Variation of rotation angle ϕ of LC directors versus applied normalized voltage.

Fig. 3.
Fig. 3.

Calculated band structure and surface waves for TM polarization in a semi-infinite one-dimensional dispersive L-R PC. The aqua and white regions show allowed frequency regions and frequency bandgaps, respectively. The blue curves with an asterisk show backward modes that are related to the case of LC cap layer absence. The dotted curves (backward modes) and solid curves (forward modes) are related to the presence of the LC cap layer with ϕ=86°. The solid straight line shows the light line of homogeneous medium with εs=μs=1.

Fig. 4.
Fig. 4.

Profiles of four surface waves with kd=6.05, which was shown in Fig. 3. (a) The mode in the first gap, (b) the mode in the third gap; both correspond to ϕ=86°. Panels (c) and (d) are similar to panels (a) and (b), except that they correspond to the absence of an LC cap layer.

Fig. 5.
Fig. 5.

(a) Variations of the surface waves with kd=6.05 versus the orientation of LC molecules (ϕ). Dotted curves are backward and solid curves are forward, and the red arrow shows the best switching of modes. (b) Tthe localization lengths of the surface waves presented in (a); and the red arrow shows the strongest localization. The horizontal dashed lines in (a) and (b) show the midgap.

Fig. 6.
Fig. 6.

Same as Fig. 3, but the arrangement of layers corresponds to Fig. 1(b). Solid red and green curves represent forward modes at ϕ=89.90°.

Fig. 7.
Fig. 7.

Profiles of four surface waves with k=6.05, which was shown in Fig. 6. Panel (a) is related to the mode in the first gap, and panel (b) is related to the mode in the fourth one. Both correspond to ϕ=89.90°. Panels (c) and (d) are similar to panels (a) and (b), but they correspond to the absence of LC cap layer.

Fig. 8.
Fig. 8.

Same as Fig. 5, and the horizontal dashed line in panels (a) and (c) shows the midgap.

Fig. 9.
Fig. 9.

(a), (b) Variations of the surface waves with kd=6.05 versus the width of LC cap layer (dLC). The red arrows show the switching of modes, which corresponds to dLC=0.33d. Panel (a) is achieved when a dispersive LH layer is adjacent to the LC cap layer, and panel (b) is achieved when an RH layer is adjacent to it.

Equations (15)

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cos(KBd)=cosh(k1zd1)cosh(k2zd2)+12(F1F2+F2F1)sinh(k1zd1)sinh(k2zd2)=B(ω).
KB=iη+mπ/d,η>0,m=0,±1,±2.
HyS(x,z)=ASekSzeikx,<zdLCHyPC(x,z)=C1(sinh(k1zz)+γcosh(k1zz))eikx,0zd1,
γ=sinh(k1zd1)+F1F2eiKBdsinh(k2zd2)eiKBdcosh(k2zd2)cosh(k1zd1).
εxx(r⃗)=εo+(εeεo)sin2(ϕ),εxy(r⃗)=0,εxz(r⃗)=(εeεo)sin(ϕ)cos(ϕ),
εyx(r⃗)=0,εyy(r⃗)=εo,εyz(r⃗)=0,
εzx(r⃗)=εxz(r⃗),εzy(r⃗)=0,εzz(r⃗)=εo+(εeεo)cos2(ϕ).
∇⃗×[εLC(r⃗)1∇⃗×H⃗LC(r⃗)]=ω2c2H⃗LC(r⃗),
HyLC(x,z)=eip/2(ALCsinh(kLCz)+BLCcosh(kLCz))eikx,dLCz0,
tanh(kLCdLC)=Q/P.
P=FLC/(F1FSγ)+ip/(2kLCFS)+ip/(2F1kLCγ)p2/(4FLCkLC2)+1/FLC,
Q=1/FS1/(F1γ),
eiKBd=(1)meηd=S(ω)=B(ω)[B(ω)21]1/2,
εL(ω)=1ωpe2/ω2,
μL(ω)=1Fω2/(ω2ω0m2),

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