Abstract

We investigate in detail the guided modes in a two-layered planar waveguide where one layer is filled with an ordinary right-handed material (RHM) and the other is filled with a biaxially anisotropic metamaterial. We show that the mode properties are closely dependent on the spatial dispersion relation of the anisotropic medium. When the dispersion equation for the anisotropic medium becomes a two-sheet or a one-sheet hyperbola type, an infinite number of guided modes can be supported simultaneously in the waveguide, which is completely different from the cases of RHM and isotropic metamaterial. We also investigate the mode distributions of the planar waveguide in the lossy case, where we discover that the dominant mode in the waveguide is a forward wave while the higher-order modes are backward waves under the two-sheet hyperbolic dispersion. Numerical results validate our theoretical analysis.

© 2006 Optical Society of America

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References

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  1. B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003).
    [Crossref]
  2. H. Cory and A. Barger, "Surface-wave propagation along a metamaterial slab," Microwave Opt. Technol. Lett. 38, 392-395 (2003).
    [Crossref]
  3. N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Wireless Propag. Lett. 1, 10-13 (2002).
    [Crossref]
  4. A. Alú and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers," IEEE Trans. Microwave Theory Tech. 52, 199-210 (2004).
    [Crossref]
  5. I. S. Nefedov and S. A. Tretyakov, "Waveguide containing a backward-wave slab," Radio Sci. 38, 1101-1109 (2003).
    [Crossref]
  6. Q. Cheng and T. J. Cui, "High-power generation and transmission through a left-handed material," Phys. Rev. B 72, 113112 (2005).
    [Crossref]
  7. Q. Cheng and T. J. Cui, "High-power generation and transmission in a left-handed planar waveguide excited by an electric dipole," Opt. Express 13, 10230-10237 (2005).
    [Crossref] [PubMed]
  8. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [Crossref] [PubMed]
  9. D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 0774051 (2003).
    [Crossref]

2005 (2)

2004 (1)

A. Alú and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers," IEEE Trans. Microwave Theory Tech. 52, 199-210 (2004).
[Crossref]

2003 (4)

I. S. Nefedov and S. A. Tretyakov, "Waveguide containing a backward-wave slab," Radio Sci. 38, 1101-1109 (2003).
[Crossref]

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003).
[Crossref]

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

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 0774051 (2003).
[Crossref]

2002 (1)

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Wireless Propag. Lett. 1, 10-13 (2002).
[Crossref]

2000 (1)

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

Alú, A.

A. Alú and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers," IEEE Trans. Microwave Theory Tech. 52, 199-210 (2004).
[Crossref]

Barger, A.

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

Cheng, Q.

Cory, H.

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

Cui, T. J.

Engheta, N.

A. Alú and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers," IEEE Trans. Microwave Theory Tech. 52, 199-210 (2004).
[Crossref]

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Wireless Propag. Lett. 1, 10-13 (2002).
[Crossref]

Grzegorczyk, T. M.

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003).
[Crossref]

Kong, J. A.

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003).
[Crossref]

Nefedov, I. S.

I. S. Nefedov and S. A. Tretyakov, "Waveguide containing a backward-wave slab," Radio Sci. 38, 1101-1109 (2003).
[Crossref]

Pendry, J. B.

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

Schurig, D.

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 0774051 (2003).
[Crossref]

Smith, D. R.

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 0774051 (2003).
[Crossref]

Tretyakov, S. A.

I. S. Nefedov and S. A. Tretyakov, "Waveguide containing a backward-wave slab," Radio Sci. 38, 1101-1109 (2003).
[Crossref]

Wu, B.-I.

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003).
[Crossref]

Zhang, Y.

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003).
[Crossref]

IEEE Antennas Wireless Propag. Lett. (1)

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Wireless Propag. Lett. 1, 10-13 (2002).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

A. Alú and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers," IEEE Trans. Microwave Theory Tech. 52, 199-210 (2004).
[Crossref]

J. Appl. Phys. (1)

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003).
[Crossref]

Microwave Opt. Technol. Lett. (1)

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

Opt. Express (1)

Phys. Rev. B (1)

Q. Cheng and T. J. Cui, "High-power generation and transmission through a left-handed material," Phys. Rev. B 72, 113112 (2005).
[Crossref]

Phys. Rev. Lett. (2)

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

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 0774051 (2003).
[Crossref]

Radio Sci. (1)

I. S. Nefedov and S. A. Tretyakov, "Waveguide containing a backward-wave slab," Radio Sci. 38, 1101-1109 (2003).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Electric line source located within a two-layered composite waveguide partially filled with the biaxially anisotropic metamaterial. PEC, perfectly electrically conducting boundary.

Fig. 2
Fig. 2

Graphical method to determine the cross points of the curve defined by Eq. (9) (circles) and the real k y axis (dotted line), where h 0 = h 1 = 2 cm and f = 10 GHz . (a) ϵ 1 x = 4 and μ 1 y = μ 1 z = 1 for the elliptic dispersion, where two bulk modes and two partial modes are excited. (b) ϵ 1 x = 4 , μ 1 y = 1 , and μ 1 z = 1 for the two-sheet hyperbolic dispersion, where an infinite number of guided modes can be excited. (c) ϵ 1 x = 4 , μ 1 y = 1 , and μ 1 z = 1 for the one-sheet hyperbolic dispersion, where there are also an infinite number of guided modes excited in the waveguide. (d) ϵ 1 x = 4 , μ 1 y = 1 , and μ 1 z = 1 , where Eq. (6) has no real solution and only one partial mode can be excited.

Tables (4)

Tables Icon

Table 1 Case I: Comparison of the Locations of Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases a

Tables Icon

Table 2 Case II: Comparison of the Locations of the First Ten Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases

Tables Icon

Table 3 Case III: Comparison of the Locations of the First Ten Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases a

Tables Icon

Table 4 Case IV: Comparison of the Locations of Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases a

Equations (9)

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E 0 x ( y , z ) = ω μ 0 I 4 π d k y 1 k 0 z e i k y y ( e i k 0 z z + R + e i k 0 z z + R e i k 0 z z ) ,
E 1 x ( y , z ) = ω μ 0 I 4 π d k y 1 k 0 z e i k y y ( C + e i k 1 z z + C e i k 1 z z ) ,
μ r 0 k 0 z tan k 0 z ( d 1 d 0 ) = μ 1 y k 1 z tan k 1 z ( d 2 d 1 ) ,
k 0 z ϵ 0 tan k 0 z ( d 1 d 0 ) = k 1 z ϵ 1 y tan k 1 z ( d 2 d 1 ) .
k y 2 + k 0 z 2 = ( ω c ) 2 ϵ r 0 μ r 0 .
k y 2 μ 1 z + k 1 z 2 μ 1 y = ( ω c ) 2 ϵ 1 x ,
k y 2 ϵ 1 z + k 1 z 2 ϵ 1 y = ( ω c ) 2 μ 1 x .
μ r 0 α 0 tanh α 0 ( d 1 d 0 ) = μ 1 y α 1 tanh α 1 ( d 2 d 1 ) .
s ( k y ) = μ r 0 k 0 z tan k 0 z h 0 + μ 1 y k 1 z tan k 1 z h 1 .

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