Abstract

Characteristics of surface modes at the interface between an isotropic medium and an indefinite medium that has a dispersion relation of hyperbolic form are studied. Four cases for the isotropic medium, including normal, left-handed, magnetic (with negative permeability), and metallic media, are considered. The conditions for the existence of surface modes in each case are analyzed in detail, and the results are expressed explicitly, indicating that the existence of surface modes is determined by the nature of the indefinite medium as well as the orientation of the boundary surface of this anisotropic medium. The energy flows associated with the surface modes are also discussed.

© 2007 Optical Society of America

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References

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  1. D. R. Smith, P. Kolinko, and D. Schurig, "Negative refraction in indefinite media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
    [CrossRef]
  2. P. A. Belov, "Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis," Microwave Opt. Technol. Lett. 37, 259-263 (2003).
    [CrossRef]
  3. L. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
    [CrossRef]
  4. D. R. Smith and S. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
    [CrossRef] [PubMed]
  5. I. V. Lindell and S. Ilvonen, "Waves in a slab of uniaxial bw medium," J. Electromagn. Waves Appl. 16, 303-318 (2002).
    [CrossRef]
  6. S. Schurig and D. R. Smith, "Spatial filtering using media with indefinite permittivity and permeability tensors," Appl. Phys. Lett. 82, 2215-2217 (2003).
    [CrossRef]
  7. D. R. Smith, S. Schurig, J. J. Mock, P. Kolinko, and P. Rye, "Partial focusing of radiation by a slab of indefinite media," Appl. Phys. Lett. 84, 2244-2246 (2004).
    [CrossRef]
  8. R. A. Depine and A. Lakhtakia, "Diffraction by a grating made of a uniaxial dielectric magnetic medium exhibiting negative refraction," New J. Phys. 7, 158 (2005).
    [CrossRef]
  9. Y. Urzhumov and G. Shvets, "Extreme anisotropy of wave propagation in two-dimensional photonic crystals," Phys. Rev. E 72, 026608 (2005).
    [CrossRef]
  10. 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]
  11. D. B. Walker, E. N. Glytsis, and T. K. Gaylord, "Surface mode at isotropic-uniaxial and isotropic-biaxial interfaces," J. Opt. Soc. Am. A 15, 248-260 (1998).
    [CrossRef]
  12. V.M.Agranovich and D.L.Mills, eds., Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).
  13. J. F. McGilp, D. Weaire, and C. H. Patterson, Epioptics: Linear and Nonlinear Optical Spectroscopy of Surfaces and Interfaces (Springer-Verlag, 1995).
  14. H. Raether, Surface Plasmons (Springer-Verlag, l988).
  15. T. J. Cui and J. A. Kong, "Time-domain electromagnetic energy in a frequency-dispersive left-handed medium," Phys. Rev. B 70, 205106 (2004).
    [CrossRef]

2005 (2)

R. A. Depine and A. Lakhtakia, "Diffraction by a grating made of a uniaxial dielectric magnetic medium exhibiting negative refraction," New J. Phys. 7, 158 (2005).
[CrossRef]

Y. Urzhumov and G. Shvets, "Extreme anisotropy of wave propagation in two-dimensional photonic crystals," Phys. Rev. E 72, 026608 (2005).
[CrossRef]

2004 (3)

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

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

T. J. Cui and J. A. Kong, "Time-domain electromagnetic energy in a frequency-dispersive left-handed medium," Phys. Rev. B 70, 205106 (2004).
[CrossRef]

2003 (4)

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

P. A. Belov, "Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis," Microwave Opt. Technol. Lett. 37, 259-263 (2003).
[CrossRef]

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

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]

2002 (2)

I. V. Lindell and S. Ilvonen, "Waves in a slab of uniaxial bw medium," J. Electromagn. Waves Appl. 16, 303-318 (2002).
[CrossRef]

L. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

1998 (1)

Belov, P. A.

P. A. Belov, "Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis," Microwave Opt. Technol. Lett. 37, 259-263 (2003).
[CrossRef]

Chui, S. T.

L. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

Cui, T. J.

T. J. Cui and J. A. Kong, "Time-domain electromagnetic energy in a frequency-dispersive left-handed medium," Phys. Rev. B 70, 205106 (2004).
[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]

Depine, R. A.

R. A. Depine and A. Lakhtakia, "Diffraction by a grating made of a uniaxial dielectric magnetic medium exhibiting negative refraction," New J. Phys. 7, 158 (2005).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Hu, L.

L. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

Ilvonen, S.

I. V. Lindell and S. Ilvonen, "Waves in a slab of uniaxial bw medium," J. Electromagn. Waves Appl. 16, 303-318 (2002).
[CrossRef]

Kolinko, P.

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

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

Kong, J. A.

T. J. Cui and J. A. Kong, "Time-domain electromagnetic energy in a frequency-dispersive left-handed medium," Phys. Rev. B 70, 205106 (2004).
[CrossRef]

Lakhtakia, A.

R. A. Depine and A. Lakhtakia, "Diffraction by a grating made of a uniaxial dielectric magnetic medium exhibiting negative refraction," New J. Phys. 7, 158 (2005).
[CrossRef]

Lindell, I. V.

I. V. Lindell and S. Ilvonen, "Waves in a slab of uniaxial bw medium," J. Electromagn. Waves Appl. 16, 303-318 (2002).
[CrossRef]

McGilp, J. F.

J. F. McGilp, D. Weaire, and C. H. Patterson, Epioptics: Linear and Nonlinear Optical Spectroscopy of Surfaces and Interfaces (Springer-Verlag, 1995).

Mock, J. J.

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

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]

Patterson, C. H.

J. F. McGilp, D. Weaire, and C. H. Patterson, Epioptics: Linear and Nonlinear Optical Spectroscopy of Surfaces and Interfaces (Springer-Verlag, 1995).

Raether, H.

H. Raether, Surface Plasmons (Springer-Verlag, l988).

Rye, P.

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

Schurig, D.

Schurig, S.

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

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

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

Shvets, G.

Y. Urzhumov and G. Shvets, "Extreme anisotropy of wave propagation in two-dimensional photonic crystals," Phys. Rev. E 72, 026608 (2005).
[CrossRef]

Smith, D. R.

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

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

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

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

Urzhumov, Y.

Y. Urzhumov and G. Shvets, "Extreme anisotropy of wave propagation in two-dimensional photonic crystals," Phys. Rev. E 72, 026608 (2005).
[CrossRef]

Walker, D. B.

Weaire, D.

J. F. McGilp, D. Weaire, and C. H. Patterson, Epioptics: Linear and Nonlinear Optical Spectroscopy of Surfaces and Interfaces (Springer-Verlag, 1995).

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]

Appl. Phys. Lett. (2)

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

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

J. Electromagn. Waves Appl. (1)

I. V. Lindell and S. Ilvonen, "Waves in a slab of uniaxial bw medium," J. Electromagn. Waves Appl. 16, 303-318 (2002).
[CrossRef]

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

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

Microwave Opt. Technol. Lett. (1)

P. A. Belov, "Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis," Microwave Opt. Technol. Lett. 37, 259-263 (2003).
[CrossRef]

New J. Phys. (1)

R. A. Depine and A. Lakhtakia, "Diffraction by a grating made of a uniaxial dielectric magnetic medium exhibiting negative refraction," New J. Phys. 7, 158 (2005).
[CrossRef]

Opt. Commun. (1)

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]

Phys. Rev. B (2)

T. J. Cui and J. A. Kong, "Time-domain electromagnetic energy in a frequency-dispersive left-handed medium," Phys. Rev. B 70, 205106 (2004).
[CrossRef]

L. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

Phys. Rev. E (1)

Y. Urzhumov and G. Shvets, "Extreme anisotropy of wave propagation in two-dimensional photonic crystals," Phys. Rev. E 72, 026608 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (3)

V.M.Agranovich and D.L.Mills, eds., Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).

J. F. McGilp, D. Weaire, and C. H. Patterson, Epioptics: Linear and Nonlinear Optical Spectroscopy of Surfaces and Interfaces (Springer-Verlag, 1995).

H. Raether, Surface Plasmons (Springer-Verlag, l988).

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Figures (4)

Fig. 1
Fig. 1

Diagram of the physical model. The ξ and η axes indicate the principal axes of the indefinite medium.

Fig. 2
Fig. 2

Contour plot of critical angle θ c T M for the existence of the surface TM mode. The surface wave can exist when μ y > 0 and θ < θ c T M for (a) case (i) and (c) case (iii) and when μ y < 0 and θ c T M < θ π 2 for (b) case (ii) and (d) case (iv). Shaded area in each figure indicates the parameter domain where surface TM waves are forbidden.

Fig. 3
Fig. 3

Dispersion curves for the isotropic medium and indefinite medium. (a) Case (i) and μ y > 0 ; (b) case (ii) and μ y < 0 .

Fig. 4
Fig. 4

Critical angle θ c 1 T M versus frequency. Shaded areas indicate the parameter domains of existence of the surface TM mode. (a) Case (i) where ϵ 1 = ϵ 0 , μ 1 = μ 0 , ϵ ξ = 2 ϵ 0 , ϵ η = ϵ 0 [ 1 0.5 f 0 2 ( f 2 f 0 2 ) ] , and μ y = μ 0 ; (b) case (ii) where ϵ 1 = ϵ 0 [ 1 0.375 f 0 2 ( f 2 1.1 f 0 2 ) ] , μ 1 = μ 0 [ 1 0.375 f 0 2 ( f 2 1.1 f 0 2 ) ] , ϵ ξ = ϵ 0 , ϵ η = ϵ 0 [ 1 0.5 f 0 2 ( f 2 f 0 2 ) ] , and μ y = μ 0 [ 1 0.5 f 0 2 ( f 2 f 0 2 ) ] . In both cases f 0 = 10 GHz .

Tables (2)

Tables Icon

Table 1 Conditions for the Existence of the Surface TM Mode

Tables Icon

Table 2 Conditions for the Existence of the Surface TE Mode (When μ ξ > 0 and μ η < 0 )

Equations (27)

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ϵ ̿ 2 = [ ϵ ξ 0 0 0 ϵ ς 0 0 0 ϵ η ] , μ ̿ 2 = [ μ ξ 0 0 0 μ ς 0 0 0 μ η ] ,
ϵ ̿ 2 = [ ϵ x x 0 ϵ x z 0 ϵ y 0 ϵ z x 0 ϵ z z ] , μ ̿ 2 = [ μ x x 0 μ x z 0 μ y 0 μ z x 0 μ z z ] ,
ϵ x x = ϵ ξ cos 2 θ + ϵ η sin 2 θ ,
ϵ x z = ( ϵ ξ ϵ η ) sin θ cos θ ,
ϵ z x = ϵ x z ,
ϵ z z = ϵ ξ sin 2 θ + ϵ η cos 2 θ ,
ϵ y = ϵ ζ ,
H 1 = y ̂ A exp ( α 1 x + i β z ) exp ( i ω t ) ( x < 0 ) ,
H 2 = y ̂ A exp ( α 2 x + i β z ) exp ( i ω t ) ( x > 0 ) ,
ω 2 ϵ 1 μ 1 = β 2 α 1 2 ,
ω 2 μ y = ( i α 2 cos θ + β sin θ ) 2 ϵ η + ( i α 2 sin θ + β cos θ ) 2 ϵ ξ .
β 2 = ω 2 ϵ 1 ( ϵ 1 ϵ x x μ y ϵ ξ ϵ η μ 1 ) ϵ 1 2 ϵ ξ ϵ η .
α 1 = ω ϵ 1 ( ϵ x x μ y ϵ 1 μ 1 ) ϵ 1 2 ϵ ξ ϵ η ,
α 2 = α 2 r + i α 2 i ,
α 2 r = α 1 ϵ ξ ϵ η ϵ 1 ϵ x x ,
α 2 i = β ϵ x z ϵ x x ,
cos 2 θ > 1 ϵ η 1 μ y 1 ϵ ξ 1 μ y 1 ϵ η 1 μ y 1 ,
cos 2 θ < 1 ϵ η 1 μ y 1 ϵ ξ 1 μ y 1 ϵ η 1 μ y 1 ,
cos 2 θ > ( ϵ η 1 μ y 1 ) ( ϵ ξ 1 μ y 1 ) ϵ η 1 μ y 1 ϵ ξ 1 μ y 1 ϵ η 1 μ y 1 ,
cos 2 θ < ( ϵ η 1 μ y 1 ) ( ϵ ξ 1 μ y 1 ) ϵ η 1 μ y 1 ϵ ξ 1 μ y 1 ϵ η 1 μ y 1 ,
cos 2 θ c 1 T M = 1 ϵ η 1 μ y 1 ϵ ξ 1 μ y 1 ϵ η 1 μ y 1 ,
cos 2 θ c 2 T M = ( ϵ η 1 μ y 1 ) ( ϵ ξ 1 μ y 1 ) ϵ η 1 μ y 1 ϵ ξ 1 μ y 1 ϵ η 1 μ y 1 .
cos 2 θ c 1 T E = 1 μ η 1 ϵ y 1 μ ξ 1 ϵ y 1 μ η 1 ϵ y 1 ,
cos 2 θ c 2 T E = ( μ η 1 ϵ y 1 ) ( μ ξ 1 ϵ y 1 ) μ η 1 ϵ y 1 μ ξ 1 ϵ y 1 μ η 1 ϵ y 1 ,
S s 1 = z ̂ β 2 ϵ 1 ω A 2 e 2 α 1 x ,
S s 2 = z ̂ β 2 ϵ x x ω A 2 e 2 α 2 r x ,
S s = 0 S s 2 d x + 0 S s 1 d x = z ̂ A 2 β 4 ω ( 1 ϵ x x α 2 r + 1 ϵ 1 α 1 ) .

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