Abstract

We report on the theoretical investigation of the plasmonic wave propagation in the coaxial cylindrical cables fabricated of both right-handed medium [with ϵ>0, μ>0] and left-handed medium [with ϵ(ω)<0, μ(ω)<0], using a Green’s-function (or a response function) theory in the absence of an applied magnetic field. The Green’s-function theory generalized to be applicable to such quasi-one-dimensional systems enables us to derive explicit expressions for the corresponding response functions (associated with the electromagnetic fields), which can in turn be used to derive various physical properties of the system. The confined plasmonic wave excitations in such multi-interface structures are characterized by the electromagnetic fields that are localized at and decay exponentially away from the interfaces. A rigorous analytical diagnosis of the general results in diverse situations leads us to reproduce exactly the previously well-known results in other geometries, obtained within the different theoretical frameworks. As an application, we present several illustrative examples on the dispersion characteristics of the confined (and extended) plasmonic waves in single- and double-interface structures made up of dispersive metamaterials interlaced with conventional dielectrics. These dispersive modes are also substantiated through the computation of local as well as total density of states. It is observed that the dispersive components enable the system to support the simultaneous existence of s- and p-polarization modes in the system. Such effects as this one are solely attributed to the negative-index metamaterials and are otherwise impossible. The readers will also notice the explicit μ-dependence of the dispersion relations for the s-polarization modes, obtained under special limits in some cases, for the single- and double-interface systems. The elegance of the theory lies in the fact that it does not require the matching of the boundary conditions and in its simplicity and the compact form of the desired (analytical) results.

© 2009 Optical Society of America

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  1. For an extensive review of electronic, optical, and transport properties of systems of reduced dimensionality, such as quantum wells, wires, dots, and electrically/magnetically modulated 2D systems, see M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1-416 (2001).
    [CrossRef]
  2. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  5. J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
    [CrossRef] [PubMed]
  6. 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]
  7. 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. B 69, 066613 (2004).
    [CrossRef]
  8. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Complete band gaps in one-dimensional left-handed periodic structures,” Phys. Rev. Lett. 95, 193903 (2005).
    [CrossRef] [PubMed]
  9. N. C. Panoiu, R. M. Osgood, S. Zhang, and S. R. J. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506-513 (2006).
    [CrossRef]
  10. B. Wood and J. B. Pendry, “Metamaterials at zero frequency,” J. Phys. Condens. Matter 19, 076208 (2007).
    [CrossRef] [PubMed]
  11. W. J. Hsueh, C. T. Chen, and C. H. Chen, “Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism,” Phys. Rev. A 78, 013836 (2008).
    [CrossRef]
  12. Y. Wu and Z. Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (2009).
    [CrossRef]
  13. R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61-64 (2000).
    [CrossRef]
  14. I. V. Shadrivov, A. A. Shukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
    [CrossRef]
  15. S. A. Darmanyan, M. Neviere, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233-240 (2003).
    [CrossRef]
  16. H. Cory and A. Barger, “Surface-wave propagation along a metamaterial slab,” Microwave Opt. Technol. Lett. 38, 392-395 (2003).
    [CrossRef]
  17. H. Cory and C. Zach, “Wave propagation in metamaterial multi-layered structures,” Microwave Opt. Technol. Lett. 40, 460-465 (2004).
    [CrossRef]
  18. Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245, 125-135 (2005).
    [CrossRef]
  19. L. G. Wang, H. Chen, and S. Y. Zhu, “Wave propagation inside one-dimensional photonic crystals with single-negative materials,” Phys. Lett. A 350, 410-415 (2006).
    [CrossRef]
  20. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15, 11133-11141 (2007).
    [CrossRef] [PubMed]
  21. Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
    [CrossRef]
  22. F. Tao, H. F. Zhang, X. H. Yang, and D. Cao, “Surface plasmon polaritons of the metamaterial four-layered structures,” J. Opt. Soc. Am. B 26, 50-59 (2009).
    [CrossRef]
  23. V. Kuzmiak and A. A. Maradudin, “Scattering properties of a cylinder fabricated from a left-handed material,” Phys. Rev. B 66, 045116 (2002).
    [CrossRef]
  24. N. C. Panoiu and R. M. Osgood, “Numerical investigation of negative refractive index metamaterials at infrared and optical frequencies,” Opt. Commun. 223, 331-337 (2003).
    [CrossRef]
  25. R. Ruppin, “Surface polaritons and extinction properties of a left-handed material cylinder,” J. Phys. Condens. Matter 16, 5991-5998 (2004).
    [CrossRef]
  26. S. Ancey, Y. Decanini, A. Folacci, and P. Gabrielli, “Surface polaritons on left-handed cylinders: a complex angular momentum analysis,” Phys. Rev. B 72, 085458 (2005).
    [CrossRef]
  27. H. Cory and T. Blum, “Surface-wave propagation along a metamaterial cylindrical guide,” Microwave Opt. Technol. Lett. 44, 31-35 (2005).
    [CrossRef]
  28. K. Y. Kim, J. H. Li, Y. K. Cho, and H. S. Tae, “Electromagnetic wave propagation through doubly dispersive subwavelength metamaterial hole,” Opt. Express 13, 3653-3665 (2005).
    [CrossRef] [PubMed]
  29. S. Arslanagic, R. W. Ziolkowski, and O. Breinbjerg, “Excitation of an electrically small metamaterial-coated cylinder by an arbitrarily located line source,” Microwave Opt. Technol. Lett. 48, 2598-2606 (2006).
    [CrossRef]
  30. E. Irci and V. K. Erturk, “Achieving transparency and maximizing scattering with metamaterial-coated conducting cylinders,” Phys. Rev. E 76, 056603 (2007).
    [CrossRef]
  31. K. Y. Kim, “Fundamental guided electromagnetic dispersion characteristics in lossless dispersive metamaterial clad circular air-hole waveguides,” J. Opt. A, Pure Appl. Opt. 9, 1062-1069 (2007).
    [CrossRef]
  32. S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from a perfect electromagnetic conductor circular cylinder coated with a metamaterial having negative permittivity and/or permeability,” Opt. Commun. 281, 5664-5670 (2008).
    [CrossRef]
  33. H. Y. She, L. W. Li, O. J. F. Martin, and J. R. Mosig, “Surface polaritons of small coated cylinders illuminated by normal incident TM and TE plane waves,” Opt. Express 16, 1007-1019 (2008).
    [CrossRef] [PubMed]
  34. C. Garcia-Meca, R. Ortuno, F. J. Rodriguez, J. Marti, and A. Martinez, “Negative refractive index metamaterials aided by extraordinary optical transmission,” Opt. Express 17, 6026-6031 (2009).
    [CrossRef] [PubMed]
  35. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow storage of light in metamaterials,” Nature 450, 397-401 (2007).
    [CrossRef] [PubMed]
  36. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
    [CrossRef] [PubMed]
  37. M. G. Silveirinha, “Anomalous refraction of light colors by a metamaterial prism,” Phys. Rev. Lett. 102, 193903 (2009).
    [CrossRef] [PubMed]
  38. J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37-44 (2004).
    [CrossRef]
  39. A. D. Boardman, N. King, and L. Velasco, “Negative refraction in perspective,” Electromagnetics 25, 365-389 (2005).
    [CrossRef]
  40. T. W. Ebbesen, H. J. Lezec, H. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
    [CrossRef]
  41. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845-2848 (1999).
    [CrossRef]
  42. L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
    [CrossRef] [PubMed]
  43. Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601-5603 (2001).
    [CrossRef] [PubMed]
  44. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847-848 (2004).
    [CrossRef] [PubMed]
  45. D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire medium for investigating plasmons at microwave frequencies,” Appl. Phys. Lett. 75, 1425-1427 (1999).
    [CrossRef]
  46. F. Yang and J. R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. 89, 063901 (2002).
    [CrossRef] [PubMed]
  47. J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
    [CrossRef] [PubMed]
  48. S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805 (2006).
    [CrossRef] [PubMed]
  49. Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008).
    [CrossRef]
  50. A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96, 073904 (2006).
    [CrossRef] [PubMed]
  51. A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “Coupled surface-plasmon-like modes between metamaterial,” Phys. Rev. B 76, 165431 (2007).
    [CrossRef]
  52. M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102, 073901 (2009).
    [CrossRef] [PubMed]
  53. L. Dobrzynski, “Interface response theory of discrete composite systems,” Surf. Sci. Rep. 6, 119-157 (1986).
    [CrossRef]
  54. L. Dobrzynski and H. Puszkarski, “Eigenvectors of composite systems. I. General theory,” J. Phys. Condens. Matter 1, 1239-1245 (1989).
    [CrossRef]
  55. M. S. Kushwaha and B. Djafari-Rouhani, “Theory of magnetoplasmons in semiconductor superlattices in the Voigt geometry: a Green-function approach,” Phys. Rev. B 43, 9021-9032 (1991).
    [CrossRef]
  56. B. Djafari-Rouhani and L. Dobrzynski, “Acoustic resonances of adsorbed wires and channels,” J. Phys. Condens. Matter 5, 8177-8194 (1993).
    [CrossRef]
  57. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I, Chap. 7.
  58. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  59. See, for example, R. Ruppin, in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), pp. 345-398.
  60. J. Wang and J. P. Leburton, “Plasmon dispersion relation of a quasi-one-dimensional electron gas,” Phys. Rev. B 41, 7846-7849 (1990).
    [CrossRef]
  61. Q. P. Li and S. Das Sarma, “Elementary excitation spectrum of one-dimensional electron systems in confined semiconductor structures: zero magnetic field,” Phys. Rev. B 43, 11768-11786 (1991).
    [CrossRef]

2009

Y. Wu and Z. Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (2009).
[CrossRef]

M. G. Silveirinha, “Anomalous refraction of light colors by a metamaterial prism,” Phys. Rev. Lett. 102, 193903 (2009).
[CrossRef] [PubMed]

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102, 073901 (2009).
[CrossRef] [PubMed]

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

C. Garcia-Meca, R. Ortuno, F. J. Rodriguez, J. Marti, and A. Martinez, “Negative refractive index metamaterials aided by extraordinary optical transmission,” Opt. Express 17, 6026-6031 (2009).
[CrossRef] [PubMed]

2008

H. Y. She, L. W. Li, O. J. F. Martin, and J. R. Mosig, “Surface polaritons of small coated cylinders illuminated by normal incident TM and TE plane waves,” Opt. Express 16, 1007-1019 (2008).
[CrossRef] [PubMed]

S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from a perfect electromagnetic conductor circular cylinder coated with a metamaterial having negative permittivity and/or permeability,” Opt. Commun. 281, 5664-5670 (2008).
[CrossRef]

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
[CrossRef]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008).
[CrossRef]

W. J. Hsueh, C. T. Chen, and C. H. Chen, “Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism,” Phys. Rev. A 78, 013836 (2008).
[CrossRef]

2007

B. Wood and J. B. Pendry, “Metamaterials at zero frequency,” J. Phys. Condens. Matter 19, 076208 (2007).
[CrossRef] [PubMed]

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

Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15, 11133-11141 (2007).
[CrossRef] [PubMed]

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “Coupled surface-plasmon-like modes between metamaterial,” Phys. Rev. B 76, 165431 (2007).
[CrossRef]

E. Irci and V. K. Erturk, “Achieving transparency and maximizing scattering with metamaterial-coated conducting cylinders,” Phys. Rev. E 76, 056603 (2007).
[CrossRef]

K. Y. Kim, “Fundamental guided electromagnetic dispersion characteristics in lossless dispersive metamaterial clad circular air-hole waveguides,” J. Opt. A, Pure Appl. Opt. 9, 1062-1069 (2007).
[CrossRef]

2006

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

S. Arslanagic, R. W. Ziolkowski, and O. Breinbjerg, “Excitation of an electrically small metamaterial-coated cylinder by an arbitrarily located line source,” Microwave Opt. Technol. Lett. 48, 2598-2606 (2006).
[CrossRef]

N. C. Panoiu, R. M. Osgood, S. Zhang, and S. R. J. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506-513 (2006).
[CrossRef]

L. G. Wang, H. Chen, and S. Y. Zhu, “Wave propagation inside one-dimensional photonic crystals with single-negative materials,” Phys. Lett. A 350, 410-415 (2006).
[CrossRef]

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96, 073904 (2006).
[CrossRef] [PubMed]

2005

A. D. Boardman, N. King, and L. Velasco, “Negative refraction in perspective,” Electromagnetics 25, 365-389 (2005).
[CrossRef]

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

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Complete band gaps in one-dimensional left-handed periodic structures,” Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

K. Y. Kim, J. H. Li, Y. K. Cho, and H. S. Tae, “Electromagnetic wave propagation through doubly dispersive subwavelength metamaterial hole,” Opt. Express 13, 3653-3665 (2005).
[CrossRef] [PubMed]

S. Ancey, Y. Decanini, A. Folacci, and P. Gabrielli, “Surface polaritons on left-handed cylinders: a complex angular momentum analysis,” Phys. Rev. B 72, 085458 (2005).
[CrossRef]

H. Cory and T. Blum, “Surface-wave propagation along a metamaterial cylindrical guide,” Microwave Opt. Technol. Lett. 44, 31-35 (2005).
[CrossRef]

2004

R. Ruppin, “Surface polaritons and extinction properties of a left-handed material cylinder,” J. Phys. Condens. Matter 16, 5991-5998 (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. B 69, 066613 (2004).
[CrossRef]

H. Cory and C. Zach, “Wave propagation in metamaterial multi-layered structures,” Microwave Opt. Technol. Lett. 40, 460-465 (2004).
[CrossRef]

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37-44 (2004).
[CrossRef]

J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847-848 (2004).
[CrossRef] [PubMed]

2003

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

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

H. Cory and A. Barger, “Surface-wave propagation along a metamaterial slab,” Microwave Opt. Technol. Lett. 38, 392-395 (2003).
[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] [PubMed]

N. C. Panoiu and R. M. Osgood, “Numerical investigation of negative refractive index metamaterials at infrared and optical frequencies,” Opt. Commun. 223, 331-337 (2003).
[CrossRef]

2002

V. Kuzmiak and A. A. Maradudin, “Scattering properties of a cylinder fabricated from a left-handed material,” Phys. Rev. B 66, 045116 (2002).
[CrossRef]

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

F. Yang and J. R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. 89, 063901 (2002).
[CrossRef] [PubMed]

2001

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601-5603 (2001).
[CrossRef] [PubMed]

For an extensive review of electronic, optical, and transport properties of systems of reduced dimensionality, such as quantum wells, wires, dots, and electrically/magnetically modulated 2D systems, see M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1-416 (2001).
[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]

2000

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

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

1999

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire medium for investigating plasmons at microwave frequencies,” Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

1998

T. W. Ebbesen, H. J. Lezec, H. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

1993

B. Djafari-Rouhani and L. Dobrzynski, “Acoustic resonances of adsorbed wires and channels,” J. Phys. Condens. Matter 5, 8177-8194 (1993).
[CrossRef]

1991

M. S. Kushwaha and B. Djafari-Rouhani, “Theory of magnetoplasmons in semiconductor superlattices in the Voigt geometry: a Green-function approach,” Phys. Rev. B 43, 9021-9032 (1991).
[CrossRef]

Q. P. Li and S. Das Sarma, “Elementary excitation spectrum of one-dimensional electron systems in confined semiconductor structures: zero magnetic field,” Phys. Rev. B 43, 11768-11786 (1991).
[CrossRef]

1990

J. Wang and J. P. Leburton, “Plasmon dispersion relation of a quasi-one-dimensional electron gas,” Phys. Rev. B 41, 7846-7849 (1990).
[CrossRef]

1989

L. Dobrzynski and H. Puszkarski, “Eigenvectors of composite systems. I. General theory,” J. Phys. Condens. Matter 1, 1239-1245 (1989).
[CrossRef]

1986

L. Dobrzynski, “Interface response theory of discrete composite systems,” Surf. Sci. Rep. 6, 119-157 (1986).
[CrossRef]

1968

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

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Ahmed, S.

S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from a perfect electromagnetic conductor circular cylinder coated with a metamaterial having negative permittivity and/or permeability,” Opt. Commun. 281, 5664-5670 (2008).
[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. B 69, 066613 (2004).
[CrossRef]

Ancey, S.

S. Ancey, Y. Decanini, A. Folacci, and P. Gabrielli, “Surface polaritons on left-handed cylinders: a complex angular momentum analysis,” Phys. Rev. B 72, 085458 (2005).
[CrossRef]

Andrews, S. R.

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

Arslanagic, S.

S. Arslanagic, R. W. Ziolkowski, and O. Breinbjerg, “Excitation of an electrically small metamaterial-coated cylinder by an arbitrarily located line source,” Microwave Opt. Technol. Lett. 48, 2598-2606 (2006).
[CrossRef]

Barger, A.

H. Cory and A. Barger, “Surface-wave propagation along a metamaterial slab,” Microwave Opt. Technol. Lett. 38, 392-395 (2003).
[CrossRef]

Blum, T.

H. Cory and T. Blum, “Surface-wave propagation along a metamaterial cylindrical guide,” Microwave Opt. Technol. Lett. 44, 31-35 (2005).
[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] [PubMed]

A. D. Boardman, N. King, and L. Velasco, “Negative refraction in perspective,” Electromagnetics 25, 365-389 (2005).
[CrossRef]

Breinbjerg, O.

S. Arslanagic, R. W. Ziolkowski, and O. Breinbjerg, “Excitation of an electrically small metamaterial-coated cylinder by an arbitrarily located line source,” Microwave Opt. Technol. Lett. 48, 2598-2606 (2006).
[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. B 69, 066613 (2004).
[CrossRef]

Brueck, S. R. J.

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]

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, C. H.

W. J. Hsueh, C. T. Chen, and C. H. Chen, “Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism,” Phys. Rev. A 78, 013836 (2008).
[CrossRef]

Chen, C. T.

W. J. Hsueh, C. T. Chen, and C. H. Chen, “Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism,” Phys. Rev. A 78, 013836 (2008).
[CrossRef]

Chen, H.

L. G. Wang, H. Chen, and S. Y. Zhu, “Wave propagation inside one-dimensional photonic crystals with single-negative materials,” Phys. Lett. A 350, 410-415 (2006).
[CrossRef]

Chen, Z.

Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008).
[CrossRef]

Cho, Y. K.

Cory, H.

H. Cory and T. Blum, “Surface-wave propagation along a metamaterial cylindrical guide,” Microwave Opt. Technol. Lett. 44, 31-35 (2005).
[CrossRef]

H. Cory and C. Zach, “Wave propagation in metamaterial multi-layered structures,” Microwave Opt. Technol. Lett. 40, 460-465 (2004).
[CrossRef]

H. Cory and A. Barger, “Surface-wave propagation along a metamaterial slab,” Microwave Opt. Technol. Lett. 38, 392-395 (2003).
[CrossRef]

Darmanyan, S. A.

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

Das Sarma, S.

Q. P. Li and S. Das Sarma, “Elementary excitation spectrum of one-dimensional electron systems in confined semiconductor structures: zero magnetic field,” Phys. Rev. B 43, 11768-11786 (1991).
[CrossRef]

Decanini, Y.

S. Ancey, Y. Decanini, A. Folacci, and P. Gabrielli, “Surface polaritons on left-handed cylinders: a complex angular momentum analysis,” Phys. Rev. B 72, 085458 (2005).
[CrossRef]

Djafari-Rouhani, B.

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. B 69, 066613 (2004).
[CrossRef]

B. Djafari-Rouhani and L. Dobrzynski, “Acoustic resonances of adsorbed wires and channels,” J. Phys. Condens. Matter 5, 8177-8194 (1993).
[CrossRef]

M. S. Kushwaha and B. Djafari-Rouhani, “Theory of magnetoplasmons in semiconductor superlattices in the Voigt geometry: a Green-function approach,” Phys. Rev. B 43, 9021-9032 (1991).
[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. B 69, 066613 (2004).
[CrossRef]

B. Djafari-Rouhani and L. Dobrzynski, “Acoustic resonances of adsorbed wires and channels,” J. Phys. Condens. Matter 5, 8177-8194 (1993).
[CrossRef]

L. Dobrzynski and H. Puszkarski, “Eigenvectors of composite systems. I. General theory,” J. Phys. Condens. Matter 1, 1239-1245 (1989).
[CrossRef]

L. Dobrzynski, “Interface response theory of discrete composite systems,” Surf. Sci. Rep. 6, 119-157 (1986).
[CrossRef]

Ebbesen, T. W.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

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. B 69, 066613 (2004).
[CrossRef]

Erturk, V. K.

E. Irci and V. K. Erturk, “Achieving transparency and maximizing scattering with metamaterial-coated conducting cylinders,” Phys. Rev. E 76, 056603 (2007).
[CrossRef]

Fang, Y.

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
[CrossRef]

Feng, Y.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I, Chap. 7.

Folacci, A.

S. Ancey, Y. Decanini, A. Folacci, and P. Gabrielli, “Surface polaritons on left-handed cylinders: a complex angular momentum analysis,” Phys. Rev. B 72, 085458 (2005).
[CrossRef]

Gabrielli, P.

S. Ancey, Y. Decanini, A. Folacci, and P. Gabrielli, “Surface polaritons on left-handed cylinders: a complex angular momentum analysis,” Phys. Rev. B 72, 085458 (2005).
[CrossRef]

Garcia-Meca, C.

Garcia-Vidal, F. J.

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847-848 (2004).
[CrossRef] [PubMed]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Ghaemi, H.

T. W. Ebbesen, H. J. Lezec, H. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Grzegorczyk, T. M.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

He, S.

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (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]

Hess, O.

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

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102, 073901 (2009).
[CrossRef] [PubMed]

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “Coupled surface-plasmon-like modes between metamaterial,” Phys. Rev. B 76, 165431 (2007).
[CrossRef]

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96, 073904 (2006).
[CrossRef] [PubMed]

J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
[CrossRef] [PubMed]

Hooper, I. R.

Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008).
[CrossRef]

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96, 073904 (2006).
[CrossRef] [PubMed]

Hsueh, W. J.

W. J. Hsueh, C. T. Chen, and C. H. Chen, “Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism,” Phys. Rev. A 78, 013836 (2008).
[CrossRef]

Huang, Y.

Irci, E.

E. Irci and V. K. Erturk, “Achieving transparency and maximizing scattering with metamaterial-coated conducting cylinders,” Phys. Rev. E 76, 056603 (2007).
[CrossRef]

Jiang, T.

Kim, K. Y.

K. Y. Kim, “Fundamental guided electromagnetic dispersion characteristics in lossless dispersive metamaterial clad circular air-hole waveguides,” J. Opt. A, Pure Appl. Opt. 9, 1062-1069 (2007).
[CrossRef]

K. Y. Kim, J. H. Li, Y. K. Cho, and H. S. Tae, “Electromagnetic wave propagation through doubly dispersive subwavelength metamaterial hole,” Opt. Express 13, 3653-3665 (2005).
[CrossRef] [PubMed]

King, N.

A. D. Boardman, N. King, and L. Velasco, “Negative refraction in perspective,” Electromagnetics 25, 365-389 (2005).
[CrossRef]

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Complete band gaps in one-dimensional left-handed periodic structures,” Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

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

Kong, J. A.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Kushwaha, M. S.

For an extensive review of electronic, optical, and transport properties of systems of reduced dimensionality, such as quantum wells, wires, dots, and electrically/magnetically modulated 2D systems, see M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1-416 (2001).
[CrossRef]

M. S. Kushwaha and B. Djafari-Rouhani, “Theory of magnetoplasmons in semiconductor superlattices in the Voigt geometry: a Green-function approach,” Phys. Rev. B 43, 9021-9032 (1991).
[CrossRef]

Kuzmiak, V.

V. Kuzmiak and A. A. Maradudin, “Scattering properties of a cylinder fabricated from a left-handed material,” Phys. Rev. B 66, 045116 (2002).
[CrossRef]

Lawrence, C. R.

J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
[CrossRef] [PubMed]

Leburton, J. P.

J. Wang and J. P. Leburton, “Plasmon dispersion relation of a quasi-one-dimensional electron gas,” Phys. Rev. B 41, 7846-7849 (1990).
[CrossRef]

Lezec, H. J.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Li, J.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

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, J. H.

Li, L. W.

Li, Q. P.

Q. P. Li and S. Das Sarma, “Elementary excitation spectrum of one-dimensional electron systems in confined semiconductor structures: zero magnetic field,” Phys. Rev. B 43, 11768-11786 (1991).
[CrossRef]

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102, 073901 (2009).
[CrossRef] [PubMed]

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “Coupled surface-plasmon-like modes between metamaterial,” Phys. Rev. B 76, 165431 (2007).
[CrossRef]

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96, 073904 (2006).
[CrossRef] [PubMed]

J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
[CrossRef] [PubMed]

Maier, S. A.

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

Maradudin, A. A.

V. Kuzmiak and A. A. Maradudin, “Scattering properties of a cylinder fabricated from a left-handed material,” Phys. Rev. B 66, 045116 (2002).
[CrossRef]

Marti, J.

Martin, O. J. F.

Martinez, A.

Martin-Moreno, L.

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Martín-Moreno, L.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I, Chap. 7.

Mosig, J. R.

Naqvi, Q. A.

S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from a perfect electromagnetic conductor circular cylinder coated with a metamaterial having negative permittivity and/or permeability,” Opt. Commun. 281, 5664-5670 (2008).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire medium for investigating plasmons at microwave frequencies,” Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Neviere, M.

S. A. Darmanyan, M. Neviere, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233-240 (2003).
[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. B 69, 066613 (2004).
[CrossRef]

Ortuno, R.

Osgood, R. M.

N. C. Panoiu, R. M. Osgood, S. Zhang, and S. R. J. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506-513 (2006).
[CrossRef]

N. C. Panoiu and R. M. Osgood, “Numerical investigation of negative refractive index metamaterials at infrared and optical frequencies,” Opt. Commun. 223, 331-337 (2003).
[CrossRef]

Pacheco, J.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Padilla, W.

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire medium for investigating plasmons at microwave frequencies,” Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Panoiu, N. C.

N. C. Panoiu, R. M. Osgood, S. Zhang, and S. R. J. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506-513 (2006).
[CrossRef]

N. C. Panoiu and R. M. Osgood, “Numerical investigation of negative refractive index metamaterials at infrared and optical frequencies,” Opt. Commun. 223, 331-337 (2003).
[CrossRef]

Pellerin, K. M.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

Pendry, J. B.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

B. Wood and J. B. Pendry, “Metamaterials at zero frequency,” J. Phys. Condens. Matter 19, 076208 (2007).
[CrossRef] [PubMed]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847-848 (2004).
[CrossRef] [PubMed]

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37-44 (2004).
[CrossRef]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

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

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Porto, J. A.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Preist, T. W.

J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
[CrossRef] [PubMed]

Puszkarski, H.

L. Dobrzynski and H. Puszkarski, “Eigenvectors of composite systems. I. General theory,” J. Phys. Condens. Matter 1, 1239-1245 (1989).
[CrossRef]

Rodriguez, F. J.

Ruppin, R.

R. Ruppin, “Surface polaritons and extinction properties of a left-handed material cylinder,” J. Phys. Condens. Matter 16, 5991-5998 (2004).
[CrossRef]

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

See, for example, R. Ruppin, in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), pp. 345-398.

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102, 073901 (2009).
[CrossRef] [PubMed]

Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008).
[CrossRef]

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “Coupled surface-plasmon-like modes between metamaterial,” Phys. Rev. B 76, 165431 (2007).
[CrossRef]

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96, 073904 (2006).
[CrossRef] [PubMed]

J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
[CrossRef] [PubMed]

F. Yang and J. R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. 89, 063901 (2002).
[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]

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire medium for investigating plasmons at microwave frequencies,” Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Complete band gaps in one-dimensional left-handed periodic structures,” Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

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

She, H. Y.

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]

Shen, Q.

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

Shukhorukov, A. A.

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

Silveirinha, M. G.

M. G. Silveirinha, “Anomalous refraction of light colors by a metamaterial prism,” Phys. Rev. Lett. 102, 193903 (2009).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37-44 (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]

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire medium for investigating plasmons at microwave frequencies,” Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Suckling, J. R.

J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
[CrossRef] [PubMed]

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Complete band gaps in one-dimensional left-handed periodic structures,” Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

Tae, H. S.

Takakura, Y.

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601-5603 (2001).
[CrossRef] [PubMed]

Tao, F.

Thio, T.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[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] [PubMed]

Velasco, L.

A. D. Boardman, N. King, and L. Velasco, “Negative refraction in perspective,” Electromagnetics 25, 365-389 (2005).
[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, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire medium for investigating plasmons at microwave frequencies,” Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Vigneron, J. P.

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. B 69, 066613 (2004).
[CrossRef]

Wang, J.

J. Wang and J. P. Leburton, “Plasmon dispersion relation of a quasi-one-dimensional electron gas,” Phys. Rev. B 41, 7846-7849 (1990).
[CrossRef]

Wang, L. G.

L. G. Wang, H. Chen, and S. Y. Zhu, “Wave propagation inside one-dimensional photonic crystals with single-negative materials,” Phys. Lett. A 350, 410-415 (2006).
[CrossRef]

Wolf, P. A.

T. W. Ebbesen, H. J. Lezec, H. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Wood, B.

B. Wood and J. B. Pendry, “Metamaterials at zero frequency,” J. Phys. Condens. Matter 19, 076208 (2007).
[CrossRef] [PubMed]

Wu, B. I.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Wu, Y.

Y. Wu and Z. Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (2009).
[CrossRef]

Yang, F.

F. Yang and J. R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. 89, 063901 (2002).
[CrossRef] [PubMed]

Yang, X. H.

Zach, C.

H. Cory and C. Zach, “Wave propagation in metamaterial multi-layered structures,” Microwave Opt. Technol. Lett. 40, 460-465 (2004).
[CrossRef]

Zakhidov, A. A.

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

Zhang, H. F.

Zhang, S.

Zhang, Y.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Zhang, Z. Q.

Y. Wu and Z. Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (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] [PubMed]

Zhu, S. Y.

L. G. Wang, H. Chen, and S. Y. Zhu, “Wave propagation inside one-dimensional photonic crystals with single-negative materials,” Phys. Lett. A 350, 410-415 (2006).
[CrossRef]

Ziolkowski, R. W.

S. Arslanagic, R. W. Ziolkowski, and O. Breinbjerg, “Excitation of an electrically small metamaterial-coated cylinder by an arbitrarily located line source,” Microwave Opt. Technol. Lett. 48, 2598-2606 (2006).
[CrossRef]

Appl. Phys. Lett.

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire medium for investigating plasmons at microwave frequencies,” Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Electromagnetics

A. D. Boardman, N. King, and L. Velasco, “Negative refraction in perspective,” Electromagnetics 25, 365-389 (2005).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

K. Y. Kim, “Fundamental guided electromagnetic dispersion characteristics in lossless dispersive metamaterial clad circular air-hole waveguides,” J. Opt. A, Pure Appl. Opt. 9, 1062-1069 (2007).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Condens. Matter

R. Ruppin, “Surface polaritons and extinction properties of a left-handed material cylinder,” J. Phys. Condens. Matter 16, 5991-5998 (2004).
[CrossRef]

B. Djafari-Rouhani and L. Dobrzynski, “Acoustic resonances of adsorbed wires and channels,” J. Phys. Condens. Matter 5, 8177-8194 (1993).
[CrossRef]

L. Dobrzynski and H. Puszkarski, “Eigenvectors of composite systems. I. General theory,” J. Phys. Condens. Matter 1, 1239-1245 (1989).
[CrossRef]

B. Wood and J. B. Pendry, “Metamaterials at zero frequency,” J. Phys. Condens. Matter 19, 076208 (2007).
[CrossRef] [PubMed]

Microwave Opt. Technol. Lett.

H. Cory and A. Barger, “Surface-wave propagation along a metamaterial slab,” Microwave Opt. Technol. Lett. 38, 392-395 (2003).
[CrossRef]

H. Cory and C. Zach, “Wave propagation in metamaterial multi-layered structures,” Microwave Opt. Technol. Lett. 40, 460-465 (2004).
[CrossRef]

H. Cory and T. Blum, “Surface-wave propagation along a metamaterial cylindrical guide,” Microwave Opt. Technol. Lett. 44, 31-35 (2005).
[CrossRef]

S. Arslanagic, R. W. Ziolkowski, and O. Breinbjerg, “Excitation of an electrically small metamaterial-coated cylinder by an arbitrarily located line source,” Microwave Opt. Technol. Lett. 48, 2598-2606 (2006).
[CrossRef]

Nature

T. W. Ebbesen, H. J. Lezec, H. Ghaemi, T. Thio, and P. A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

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

Opt. Commun.

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

S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from a perfect electromagnetic conductor circular cylinder coated with a metamaterial having negative permittivity and/or permeability,” Opt. Commun. 281, 5664-5670 (2008).
[CrossRef]

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

N. C. Panoiu and R. M. Osgood, “Numerical investigation of negative refractive index metamaterials at infrared and optical frequencies,” Opt. Commun. 223, 331-337 (2003).
[CrossRef]

Opt. Express

Phys. Lett. A

L. G. Wang, H. Chen, and S. Y. Zhu, “Wave propagation inside one-dimensional photonic crystals with single-negative materials,” Phys. Lett. A 350, 410-415 (2006).
[CrossRef]

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

Phys. Rev. A

W. J. Hsueh, C. T. Chen, and C. H. Chen, “Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism,” Phys. Rev. A 78, 013836 (2008).
[CrossRef]

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
[CrossRef]

Phys. Rev. B

S. Ancey, Y. Decanini, A. Folacci, and P. Gabrielli, “Surface polaritons on left-handed cylinders: a complex angular momentum analysis,” Phys. Rev. B 72, 085458 (2005).
[CrossRef]

M. S. Kushwaha and B. Djafari-Rouhani, “Theory of magnetoplasmons in semiconductor superlattices in the Voigt geometry: a Green-function approach,” Phys. Rev. B 43, 9021-9032 (1991).
[CrossRef]

J. Wang and J. P. Leburton, “Plasmon dispersion relation of a quasi-one-dimensional electron gas,” Phys. Rev. B 41, 7846-7849 (1990).
[CrossRef]

Q. P. Li and S. Das Sarma, “Elementary excitation spectrum of one-dimensional electron systems in confined semiconductor structures: zero magnetic field,” Phys. Rev. B 43, 11768-11786 (1991).
[CrossRef]

V. Kuzmiak and A. A. Maradudin, “Scattering properties of a cylinder fabricated from a left-handed material,” Phys. Rev. B 66, 045116 (2002).
[CrossRef]

Y. Wu and Z. Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (2009).
[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. B 69, 066613 (2004).
[CrossRef]

Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008).
[CrossRef]

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “Coupled surface-plasmon-like modes between metamaterial,” Phys. Rev. B 76, 165431 (2007).
[CrossRef]

Phys. Rev. E

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

E. Irci and V. K. Erturk, “Achieving transparency and maximizing scattering with metamaterial-coated conducting cylinders,” Phys. Rev. E 76, 056603 (2007).
[CrossRef]

Phys. Rev. Lett.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Complete band gaps in one-dimensional left-handed periodic structures,” Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

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]

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

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102, 073901 (2009).
[CrossRef] [PubMed]

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96, 073904 (2006).
[CrossRef] [PubMed]

F. Yang and J. R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. 89, 063901 (2002).
[CrossRef] [PubMed]

J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401 (2004).
[CrossRef] [PubMed]

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

M. G. Silveirinha, “Anomalous refraction of light colors by a metamaterial prism,” Phys. Rev. Lett. 102, 193903 (2009).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601-5603 (2001).
[CrossRef] [PubMed]

Phys. Today

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37-44 (2004).
[CrossRef]

Science

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847-848 (2004).
[CrossRef] [PubMed]

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

Sov. Phys. Usp.

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

Surf. Sci. Rep.

For an extensive review of electronic, optical, and transport properties of systems of reduced dimensionality, such as quantum wells, wires, dots, and electrically/magnetically modulated 2D systems, see M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1-416 (2001).
[CrossRef]

L. Dobrzynski, “Interface response theory of discrete composite systems,” Surf. Sci. Rep. 6, 119-157 (1986).
[CrossRef]

Other

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I, Chap. 7.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

See, for example, R. Ruppin, in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), pp. 345-398.

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Figures (13)

Fig. 1
Fig. 1

Schematics of the concept of three perturbations: [A], [B], and [C]. The blank (shaded) region refers to the material medium (black box) in the system. The sum of the first two perturbations defines a metamaterial (dielectric) cylinder embedded in a dielectric (metamaterial) and the sum of all three perturbations specifies, say, a metamaterial (dielectric) shell surrounded by two unidentical dielectrics (metamaterials). Here R j is the radius and X ϵ ( ω ) and/or μ ( ω ) for a specified medium.

Fig. 2
Fig. 2

Plasmonic wave dispersion for a dielectric (vacuum) cable embedded in a metamaterial background for different values of index m = 0 , 1 , 3 , 5 . The dimensionless plasma frequency used in the computation is specified by ω p R 1 / c = 3.5 . Dashed line and curve marked as LL1 and LL2 refer, respectively, to the light lines in the vacuum and the metamaterial. The horizontal dotted line stands for the characteristic resonance frequency ( ω 0 ) in the metamaterial. The shaded area represents the region within which both ϵ ( ω ) and μ ( ω ) are negative and prohibits the existence of the confined modes.

Fig. 3
Fig. 3

LDOS for the system discussed in Fig. 2 and for m = 0 and ζ = 1.0 . The rest of the parameters used are the same as in Fig. 2. The arrows in the panel indicate the peaks at ξ = 0.5119 and 0.7718.

Fig. 4
Fig. 4

TDOS for the system discussed in Fig. 2 and for m = 0 and ζ = 1.0 . The rest of the parameters used are the same as in Fig. 2. Both negative peaks are characteristic of the resonance frequency ω 0 and other characteristic frequency ω c in the system and bear no physical significance.

Fig. 5
Fig. 5

Plasmonic wave dispersion for a metamaterial cable in a dielectric (vacuum) background for different values of index m = 0 , 1 , 3 , 5 . The dimensionless plasma frequency used in the computation is specified by ω p R 1 / c = 3.5 . Dashed line and curve marked as LL1 and LL2 refer, respectively, to the light lines in the vacuum and the metamaterial. The horizontal dotted line stands for the characteristic resonance frequency ( ω 0 ) in the metamaterial. The shaded area represents the region within which both ϵ ( ω ) and μ ( ω ) are negative and disallows the existence of the confined modes.

Fig. 6
Fig. 6

LDOS for the system discussed in Fig. 5 and for m = 0 and ζ = 1.0 . The arrows in the panel indicate the peaks at ξ = 0.3947 , 0.4581, and 0.5577. The rest of the parameters used are the same as in Fig. 5.

Fig. 7
Fig. 7

TDOS for the system discussed in Fig. 5 and for m = 0 and ζ = 1.0 . The arrows in the panel indicate the peaks at ξ = 0.3947 , 0.4581, and 0.5577. The rest of the parameters used are the same as in Fig. 5.

Fig. 8
Fig. 8

Plasmonic wave dispersion for a metamaterial shell sandwiched between two identical dielectrics (vacuum) for different values of index m = 0 , 1 , 2 , 3 . The dimensionless plasma frequency used here is specified by ω p R 1 / c = 3.5 and the radii ratio by R 2 / R 1 = 1.2 . Dashed line and curve marked as LL1 and LL2 refer, respectively, to the light lines in the vacuum and the metamaterial. The horizontal dotted line stands for the characteristic resonance frequency ( ω 0 ) in the metamaterial. The shaded area represents the region within which both ϵ ( ω ) and μ ( ω ) are negative and forbids the existence of the confined modes. The parameters used in the computation are as listed in the picture.

Fig. 9
Fig. 9

LDOS at the interface R 1 ( R 2 ) in the lower (upper) panel for m = 0 and ζ = 1.0 for the system discussed in Fig. 8. We call attention to the DOS resonance peaks, indicated by the arrows, corresponding to the five modes in total at ζ = 1.0 in Fig. 8. Interface 1 (2) refers to the one specified by R 1 ( R 2 ) . The rest of the parameters used are the same as in Fig. 8.

Fig. 10
Fig. 10

TDOS for m = 0 and ζ = 1.0 for the system discussed in Fig. 8. We call attention to the DOS resonance peaks, indicated by the arrows, corresponding to the five modes in total at ζ = 1.0 in Fig. 8. The parameters used are the same as in Fig. 8.

Fig. 11
Fig. 11

Plasmonic wave dispersion for a dielectric (vacuum) shell sandwiched between two identical metamaterials for different values of index m = 0 , 1 , 2 , 3 . The dimensionless plasma frequency used here is specified by ω p R 1 / c = 3.5 and the radii ratio by R 2 / R 1 = 1.2 . Dashed line and curve marked as LL1 and LL2 refer, respectively, to the light lines in the vacuum and the metamaterial. The horizontal dotted line stands for the characteristic resonance frequency ( ω 0 ) in the metamaterial. The shaded area represents the region within which both ϵ ( ω ) and μ ( ω ) are negative and proscribes the existence of the confined modes. The parameters used in the computation are as listed in the picture.

Fig. 12
Fig. 12

LDOS at the interface R 1 ( R 2 ) in the lower (upper) panel for m = 0 and ζ = 1.5 for the system discussed in Fig. 11. We call attention to the DOS resonance peaks, indicated by the arrows, corresponding to the five modes in total at ζ = 1.5 in Fig. 11. Interface 1 (2) refers to the one specified by R 1 ( R 2 ) . The rest of the parameters used are the same as in Fig. 11.

Fig. 13
Fig. 13

TDOS for m = 0 and ζ = 1.5 for the system discussed in Fig. 11. We call attention to the DOS resonance peaks, indicated by the arrows, corresponding to the five modes in total at ζ = 1.5 in Fig. 11. The parameters used are the same as in Fig. 11. The DOSs are shown in arbitrary units throughout.

Equations (80)

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× ( × E ) q 0 2 ϵ μ E = 0.
[ 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 θ 2 k 2 ] E ρ 1 ρ 2 ( E ρ + 2 θ E θ ) + q 0 2 ϵ μ E ρ = 0 ,
[ 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 θ 2 k 2 ] E θ 1 ρ 2 ( E θ 2 θ E ρ ) + q 0 2 ϵ μ E θ = 0 ,
[ 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 θ 2 k 2 ] E z + q 0 2 ϵ μ E z = 0.
E ρ = 1 α 2 [ i q 0 μ 1 ρ θ H z i k ρ E z ] ,
E θ = 1 α 2 [ i q 0 μ ρ H z i k 1 ρ θ E z ] ,
H ρ = 1 α 2 [ i q 0 ϵ 1 ρ θ E z i k ρ H z ] ,
H θ = 1 α 2 [ i q 0 ϵ ρ E z i k 1 ρ θ H z ] .
i q 0 ϵ E z = 1 ρ ρ ( ρ H θ ) 1 ρ θ H ρ ,
i q 0 μ H z = 1 ρ ρ ( ρ E θ ) 1 ρ θ E ρ ,
2 ρ 2 A z + 1 ρ ρ A z + ( 1 ρ 2 2 θ 2 α 2 ) A z = 0 ,
ρ ( ρ E θ ) = 1 α 2 { [ i q 0 μ ρ H z + i q 0 μ ρ 2 ρ 2 H z i k θ ρ E z ] δ ( R ρ ) [ i q 0 μ ρ ρ H z i k θ E z ] } ,
ρ ( ρ H θ ) = 1 α 2 { [ i q 0 ϵ ρ E z + i q 0 ϵ ρ 2 ρ 2 E z i k θ ρ H z ] δ ( R ρ ) [ i q 0 ϵ ρ ρ E z i k θ H z ] } .
( i q 0 ϵ β 2 ) [ ( 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 θ 2 + β 2 ) E z δ ( R ρ ) ( ρ E z + k q 0 ϵ ρ θ H z ) ] = 0 ,
( i q 0 μ β 2 ) [ ( 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 θ 2 + β 2 ) H z δ ( R ρ ) ( ρ H z k q 0 μ ρ θ E z ) ] = 0 ,
G ( r , r ) G ( | r r | ) G ( ρ , θ ; ρ , θ ) ,
( 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 θ 2 + β 2 ) G ( r , r ) = 4 π δ ( r r ) = 4 π ρ δ ( ρ ρ ) δ ( θ θ ) .
G ( r , r ) = m = e i m ( θ θ ) G ( m ; ρ , ρ ) ,
G ( m ; ρ , ρ ) = i π × { J m ( β ρ ) H m ( β ρ ) , if   ρ ρ H m ( β ρ ) J m ( β ρ ) , if   ρ ρ , }
G ( m ; ρ , ρ ) = i π { [ 1 θ ( ρ ρ ) ] J m ( β ρ ) H m ( β ρ ) + θ ( ρ ρ ) H m ( β ρ ) J m ( β ρ ) } ,
[ ( q 0 2 ϵ β 2 ) [ 2 ρ 2 + 1 ρ ρ m 2 ρ 2 + β 2 ] 0 0 ( q 0 2 μ β 2 ) [ 2 ρ 2 + 1 ρ ρ m 2 ρ 2 + β 2 ] ] [ G E ( m ; ρ , ρ ) 0 0 G H ( m ; ρ , ρ ) ] = 2 ρ δ ( ρ ρ ) [ 1 0 0 1 ] ,
( q 0 2 ϵ β 2 ) G E ( m ; ρ , ρ ) = ( q 0 2 μ β 2 ) G H ( m ; ρ , ρ ) = i π × { J m ( β ρ ) H m ( β ρ ) , ρ ρ H m ( β ρ ) J m ( β ρ ) , ρ ρ . }
V ̃ 1 ( R 1 , ρ ) = R 1 2 q 0 2 β 1 2 [ ϵ 1 ρ i m k q 0 ρ i m k q 0 ρ μ 1 ρ ] ,
G ̃ 1 ( ρ , ρ ) = i π β 1 2 q 0 2 [ 1 ϵ 1 H m ( β 1 ρ ) J m ( β 1 ρ ) 0 0 1 μ 1 H m ( β 1 ρ ) J m ( β 1 ρ ) ] .
A ̃ 1 ( R 1 , R 1 ) = V ̃ 1 ( R 1 , ρ ) G ̃ 1 ( ρ , ρ ) ρ = R 1 = ρ = [ i π 2 β 1 R 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) π 2 m k q 0 μ 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) π 2 m k q 0 ϵ 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) i π 2 β 1 R 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) ] .
Δ ̃ 1 ( R 1 , R 1 ) = I ̃ + A ̃ 1 ( R 1 , R 1 ) = [ i π 2 β 1 R 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) π 2 m k q 0 μ 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) π 2 m k q 0 ϵ 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) i π 2 β 1 R 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) ] .
1 J ν ( z ) H ν ( z ) = π z 2 i [ H ν ( z ) H ν ( z ) J ν ( z ) J ν ( z ) ] .
G ̃ 1 1 ( R 1 , R 1 ) = q 0 2 i π β 1 2 1 H m ( β 1 R 1 ) J m ( β 1 R 1 ) [ ϵ 1 0 0 μ 1 ] .
g ̃ 1 1 ( R 1 , R 1 ) = Δ ̃ 1 ( R 1 , R 1 ) G ̃ 1 1 ( R 1 , R 1 ) .
g ̃ 1 1 ( R 1 , R 1 ) = q 0 2 2 β 1 2 [ β 1 R 1 ϵ 1 J m ( β 1 R 1 ) J m ( β 1 R 1 ) i m k q 0 i m k q 0 β 1 R 1 μ 1 J m ( β 1 R 1 ) J m ( β 1 R 1 ) ]
V ̃ 2 ( R 2 , ρ ) = R 2 2 q 0 2 β 2 2 [ ϵ 2 ρ i m k q 0 ρ i m k q 0 ρ μ 2 ρ ] ,
G ̃ 2 ( ρ , ρ ) = i π β 2 2 q 0 2 [ 1 ϵ 2 J m ( β 2 ρ ) H m ( β 2 ρ ) 0 0 1 μ 2 J m ( β 2 ρ ) H m ( β 2 ρ ) ] .
A ̃ 2 ( R 2 , R 2 ) = V ̃ 2 ( R 2 , ρ ) G ̃ 2 ( ρ , ρ ) ρ = R 2 = ρ = [ i π 2 β 2 R 2 J m ( β 2 R 2 ) H m ( β 2 R 2 ) + π 2 m k q 0 μ 2 J m ( β 2 R 2 ) H m ( β 2 R 2 ) π 2 m k q 0 ϵ 2 J m ( β 2 R 2 ) H m ( β 2 R 2 ) i π 2 β 2 R 2 J m ( β 2 R 2 ) H m ( β 2 R 2 ) ] .
Δ ̃ 2 ( R 2 , R 2 ) = I ̃ + A ̃ 2 ( R 2 , R 2 ) = [ i π 2 β 2 R 2 J m ( β 2 R 2 ) H m ( β 2 R 2 ) + π 2 m k q 0 μ 2 J m ( β 2 R 2 ) H m ( β 2 R 2 ) π 2 m k q 0 ϵ 2 J m ( β 2 R 2 ) H m ( β 2 R 2 ) i π 2 β 2 R 2 J m ( β 2 R 2 ) H m ( β 2 R 2 ) ] .
G ̃ 2 1 ( R 2 , R 2 ) = q 0 2 i π β 2 2 1 J m ( β 2 R 2 ) H m ( β 2 R 2 ) [ ϵ 2 0 0 μ 2 ] .
g ̃ 2 1 ( R 2 , R 2 ) = Δ ̃ 2 ( R 2 , R 2 ) G ̃ 2 1 ( R 2 , R 2 ) .
g ̃ 2 1 ( R 2 , R 2 ) = q 0 2 2 β 2 2 [ β 2 R 2 ϵ 2 H m ( β 2 R 2 ) H m ( β 2 R 2 ) i m k q 0 i m k q 0 β 2 R 2 μ 2 H m ( β 2 R 2 ) H m ( β 2 R 2 ) ]
V ̃ 3 ( R i , ρ ) = 1 2 q 0 2 β 3 2 [ ϵ 3 R 1 ρ i m k q 0 ρ R 1 0 0 i m k q 0 ρ R 1 μ 3 R 1 ρ 0 0 0 0 ϵ 3 R 2 ρ i m k q 0 ρ R 2 0 0 i m k q 0 ρ R 2 μ 3 R 2 ρ ] .
G ̃ 3 ( ρ , ρ ) = i π β 3 2 q 0 2 [ 1 ϵ 3 J m ( β 3 ρ ) H m ( β 3 ρ ) 0 1 ϵ 3 J m ( β 3 ρ ) H m ( β 3 ρ ) 0 0 1 μ 3 J m ( β 3 ρ ) H m ( β 3 ρ ) 0 1 μ 3 J m ( β 3 ρ ) H m ( β 3 ρ ) 1 ϵ 3 H m ( β 3 ρ ) J m ( β 3 ρ ) 0 1 ϵ 3 H m ( β 3 ρ ) J m ( β 3 ρ ) 0 0 1 μ 3 H m ( β 3 ρ ) J m ( β 3 ρ ) 0 1 μ 3 H m ( β 3 ρ ) J m ( β 3 ρ ) ] ,
A ̃ 3 ( M , M ) = V ̃ 3 ( M ) G ̃ 3 ( M , M ) = i π 2 [ β 3 R 1 J m ( β 3 R 1 ) H m ( β 3 R 1 ) i m k q 0 μ 3 J m ( β 3 R 1 ) H m ( β 3 R 1 ) β 3 R 1 J m ( β 3 R 1 ) H m ( β 3 R 2 ) i m k q 0 μ 3 J m ( β 3 R 1 ) H m ( β 3 R 2 ) i m k q 0 ϵ 3 J m ( β 3 R 1 ) H m ( β 3 R 1 ) β 3 R 1 J m ( β 3 R 1 ) H m ( β 3 R 1 ) i m k q 0 ϵ 3 J m ( β 3 R 1 ) H m ( β 3 R 2 ) β 3 R 1 J m ( β 3 R 1 ) H m ( β 3 R 2 ) β 3 R 2 H m ( β 3 R 2 ) J m ( β 3 R 1 ) i m k q 0 μ 3 H m ( β 3 R 2 ) J m ( β 3 R 1 ) β 3 R 2 H m ( β 3 R 2 ) J m ( β 3 R 2 ) i m k q 0 μ 3 H m ( β 3 R 2 ) J m ( β 3 R 2 ) i m k q 0 ϵ 3 H m ( β 3 R 2 ) J m ( β 3 R 1 ) β 3 R 2 H m ( β 3 R 2 ) J m ( β 3 R 1 ) i m k q 0 ϵ 3 H m ( β 3 R 2 ) J m ( β 3 R 2 ) β 3 R 2 H m ( β 3 R 2 ) J m ( β 3 R 2 ) ] .
Δ ̃ 3 ( M , M ) = I ̃ + A ̃ 3 ( M , M ) = i π 2 [ β 3 R 1 H m ( β 3 R 1 ) J m ( β 3 R 1 ) i m k q 0 μ 3 J m ( β 3 R 1 ) H m ( β 3 R 1 ) β 3 R 1 J m ( β 3 R 1 ) H m ( β 3 R 2 ) i m k q 0 μ 3 J m ( β 3 R 1 ) H m ( β 3 R 2 ) i m k q 0 ϵ 3 J m ( β 3 R 1 ) H m ( β 3 R 1 ) β 3 R 1 H m ( β 3 R 1 ) J m ( β 3 R 1 ) i m k q 0 ϵ 3 J m ( β 3 R 1 ) H m ( β 3 R 2 ) β 3 R 1 J m ( β 3 R 1 ) H m ( β 3 R 2 ) β 3 R 2 H m ( β 3 R 2 ) J m ( β 3 R 1 ) i m k q 0 μ 3 H m ( β 3 R 2 ) J m ( β 3 R 1 ) β 3 R 2 J m ( β 3 R 2 ) H m ( β 3 R 2 ) i m k q 0 μ 3 H m ( β 3 R 2 ) J m ( β 3 R 2 ) i m k q 0 ϵ 3 H m ( β 3 R 2 ) J m ( β 3 R 1 ) β 3 R 2 H m ( β 3 R 2 ) J m ( β 3 R 1 ) i m k q 0 ϵ 3 H m ( β 3 R 2 ) J m ( β 3 R 2 ) β 3 R 2 J m ( β 3 R 2 ) H m ( β 3 R 2 ) ] .
G ̃ 3 1 ( M , M ) = q 0 2 i π β 3 2 1 D [ ϵ 3 J m ( β 3 R 2 ) J m ( β 3 R 1 ) 0 ϵ 3 0 0 μ 3 J m ( β 3 R 2 ) J m ( β 3 R 1 ) 0 μ 3 ϵ 3 0 ϵ 3 H m ( β 3 R 1 ) H m ( β 3 R 2 ) 0 0 μ 3 0 μ 3 H m ( β 3 R 1 ) H m ( β 3 R 2 ) ] ,
D = H m ( β 3 R 1 ) J m ( β 3 R 2 ) J m ( β 3 R 1 ) H m ( β 3 R 2 ) .
g ̃ 3 1 ( M , M ) = Δ ̃ 3 ( M , M ) G ̃ 3 1 ( M , M ) ,
g ̃ 3 1 ( M , M ) = q 0 2 2 β 3 2 [ β 3 R 1 ϵ 3 Z 1 D i m k q 0 2 i ϵ 3 π D 0 i m k q 0 β 3 R 1 μ 3 Z 1 D 0 2 i μ 3 π D 2 i ϵ 3 π D 0 β 3 R 2 ϵ 3 Z 2 D i m k q 0 0 2 i μ 3 π D i m k q 0 β 3 R 2 μ 3 Z 2 D ] ,
Z 1 = H m ( β 3 R 1 ) J m ( β 3 R 2 ) J m ( β 3 R 1 ) H m ( β 3 R 2 ) ,
Z 2 = H m ( β 3 R 2 ) J m ( β 3 R 1 ) J m ( β 3 R 2 ) H m ( β 3 R 1 ) .
| g ̃ 1 ( M , M ) | = | g ̃ 1 1 ( M , M ) + g ̃ 2 1 ( M , M ) | = 0 ,
| [ ϵ 1 β 1 J m ( β 1 R ) J m ( β 1 R ) ϵ 2 β 2 H m ( β 2 R ) H m ( β 2 R ) ] i m k R q 0 ( 1 β 1 2 1 β 2 2 ) i m k R q 0 ( 1 β 1 2 1 β 2 2 ) [ μ 1 β 1 J m ( β 1 R ) J m ( β 1 R ) μ 2 β 2 H m ( β 2 R ) H m ( β 2 R ) ] | = 0
[ ϵ 1 β 1 J m ( β 1 R ) J m ( β 1 R ) ϵ 2 β 2 H m ( β 2 R ) H m ( β 2 R ) ] [ μ 1 β 1 J m ( β 1 R ) J m ( β 1 R ) μ 2 β 2 H m ( β 2 R ) H m ( β 2 R ) ] = ( m R ) 2 k 2 q 0 2 ( 1 β 1 2 1 β 2 2 ) 2 .
ϵ 1 β 1 J 1 ( β 1 R ) J 0 ( β 1 R ) ϵ 2 β 2 H 1 ( β 2 R ) H 0 ( β 2 R ) = 0 ,
ϵ 1 + 2 ϵ 2 α 2 R K 1 ( α 2 R ) K 0 ( α 2 R ) = 0 ,
ω = ω 0 β 2 R | ln ( β 2 R ) | 1 / 2 ,
| g ̃ 1 ( M , M ) | = | g ̃ 1 1 ( M , M ) + g ̃ 2 1 ( M , M ) + g ̃ 3 1 ( M , M ) | = 0.
| R 1 ( ϵ 1 β 1 A 1 ϵ 3 β 3 C 1 ) i m k q 0 ( 1 β 1 2 1 β 3 2 ) 2 i ϵ 3 π β 3 2 D 0 i m k q 0 ( 1 β 1 2 1 β 3 2 ) R 1 ( μ 1 β 1 A 1 μ 3 β 3 C 1 ) 0 2 i μ 3 π β 3 2 D 2 i ϵ 3 π β 3 2 D 0 R 2 ( ϵ 2 β 2 A 2 + ϵ 3 β 3 C 2 ) i m k q 0 ( 1 β 2 2 1 β 3 2 ) 0 2 i μ 3 π β 3 2 D i m k q 0 ( 1 β 2 2 1 β 3 2 ) R 2 ( μ 2 β 2 A 2 + μ 3 β 3 C 2 ) | = 0 ,
A 1 = J m ( β 1 R 1 ) / J m ( β 1 R 1 ) ,
A 2 = H m ( β 2 R 2 ) / H m ( β 2 R 2 ) ,
C 1 = Z 1 / D ,
C 2 = Z 2 / D .
| R ( ϵ 1 α 1 + ϵ 3 α 3 C ) i m k q 0 ( 1 α 1 2 1 α 3 2 ) R ϵ 3 α 3 S 0 i m k q 0 ( 1 α 1 2 1 α 3 2 ) R ( μ 1 α 1 + μ 3 α 3 C ) 0 R μ 3 α 3 S R ϵ 3 α 3 S 0 R ( ϵ 2 α 2 + ϵ 3 α 3 C ) i m k q 0 ( 1 α 2 2 1 α 3 2 ) 0 R μ 3 α 3 S i m k q 0 ( 1 α 2 2 1 α 3 2 ) R ( μ 2 α 2 + μ 3 α 3 C ) | = 0 ,
[ ϵ 1 ϵ 2 α 1 α 2 + ( ϵ 1 α 1 + ϵ 2 α 2 ) ϵ 3 α 3 coth   θ + ( ϵ 3 α 3 ) 2 ] [ μ 1 μ 2 α 1 α 2 + ( μ 1 α 1 + μ 2 α 2 ) μ 3 α 3 coth   θ + ( μ 3 α 3 ) 2 ] = 0.
coth   θ 1 θ + θ 3 θ 3 45 + 2 θ 5 945 ,
4 π χ + ϵ 1 α 1 + ϵ 2 α 2 = 0.
ω 2 = ( 2 π n s e 2 / m ϵ ) k ,
N L ( ω ) = 2 ω π Im { trace [ g ̃ ( M , M ) ] } ,
N T ( ω ) = 1 π d d ω ( Arg   det [ g ̃ i ( M , M ) g ̃ f ( M , M ) ] ) .
ϵ 1 β 1 + ϵ 2 β 2 = 0.
k 2 = q 0 2 μ 1 ϵ 1 μ 2 ϵ 2 1 ϵ 1 2 1 ϵ 2 2 .
μ 1 ϵ 1 > μ 2 ϵ 2 ,     1 ϵ 1 2 > 1 ϵ 2 2
μ 1 ϵ 1 < μ 2 ϵ 2 ,     1 ϵ 1 2 < 1 ϵ 2 2 .
μ 1 β 1 + μ 2 β 2 = 0.
k 2 = q 0 2 ϵ 1 μ 1 ϵ 2 μ 2 1 μ 1 2 1 μ 2 2 .
ϵ 1 μ 1 > ϵ 2 μ 2 ,     1 μ 1 2 > 1 μ 2 2
ϵ 1 μ 1 < ϵ 2 μ 2 ,     1 μ 1 2 < 1 μ 2 2 .
ω 2 c 2 k 2 = 1 ϵ 1 2 1 ϵ 2 2 μ 1 ϵ 1 μ 2 ϵ 2 ,
ω 2 c 2 k 2 = 1 μ 1 2 1 μ 2 2 ϵ 1 μ 1 ϵ 2 μ 2 ,
ϵ ( ω ) = 1 ω p 2 ω 2 ,
μ ( ω ) = 1 F ω 2 ω 2 ω 0 2 ,
W = ω ω p = 1 ϵ 1 + 1 ,
W = ω ω p = W 0 1 F μ 1 + 1 ,

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