Abstract

A ray model for an evanescent plane wave has been introduced to describe the property of propagation. It is found that an evanescent wave undergoes an amplitude jump during refraction and reflection on a plane interface. TE surface modes guided by a single interface and a slab waveguide containing left-handed materials are then analyzed by using a ray picture. In a slab waveguide with left-handed material cover or substrate, the odd surface mode does not exist, but there exist two even surface modes. We derive dispersion relations of TE surface guided modes according to physical principles, thus describing the surface modes purely on an intuitive and simple basis. Besides, guiding mechanics and conditions are compared between oscillating and surface guided modes.

© 2008 Optical Society of America

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References

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  4. S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171-173 (2004).
    [CrossRef]
  5. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refraction-index waveguides,” Phys. Rev. E 67, 057602 (2003).
    [CrossRef]
  6. 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]
  7. Z. Y. Xiao and Z. H. Wang, “Dispersion characteristics of asymmetric double negative material slab waveguides,” J. Opt. Soc. Am. B 23, 1757-1760 (2006).
    [CrossRef]
  8. K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  10. Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left handed material cover or substrate,” Opt. Commun. 281, 607-613 (2008).
    [CrossRef]

2008 (1)

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left handed material cover or substrate,” Opt. Commun. 281, 607-613 (2008).
[CrossRef]

2006 (2)

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Z. Y. Xiao and Z. H. Wang, “Dispersion characteristics of asymmetric double negative material slab waveguides,” J. Opt. Soc. Am. B 23, 1757-1760 (2006).
[CrossRef]

2004 (1)

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171-173 (2004).
[CrossRef]

2003 (2)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refraction-index waveguides,” Phys. Rev. E 67, 057602 (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]

2001 (1)

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

2000 (1)

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

1971 (1)

1968 (1)

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

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]

He, S.

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171-173 (2004).
[CrossRef]

Hermann, C.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Hess, O.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Jamois, C.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Kivshar, Y. S.

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

Klaedtke, A.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
[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]

Li, S. P.

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left handed material cover or substrate,” Opt. Commun. 281, 607-613 (2008).
[CrossRef]

Pendry, J. B.

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

Shadrivov, I. V.

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

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, L.

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171-173 (2004).
[CrossRef]

Smith, D. R.

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

Sukhorukov, A. A.

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

Tien, P. K.

Tsakmakidis, K. L.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
[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]

Wang, Z. H.

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left handed material cover or substrate,” Opt. Commun. 281, 607-613 (2008).
[CrossRef]

Z. Y. Xiao and Z. H. Wang, “Dispersion characteristics of asymmetric double negative material slab waveguides,” J. Opt. Soc. Am. B 23, 1757-1760 (2006).
[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]

Xiao, S.

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171-173 (2004).
[CrossRef]

Xiao, Z. Y.

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left handed material cover or substrate,” Opt. Commun. 281, 607-613 (2008).
[CrossRef]

Z. Y. Xiao and Z. H. Wang, “Dispersion characteristics of asymmetric double negative material slab waveguides,” J. Opt. Soc. Am. B 23, 1757-1760 (2006).
[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]

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (1)

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171-173 (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]

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

Opt. Commun. (1)

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left handed material cover or substrate,” Opt. Commun. 281, 607-613 (2008).
[CrossRef]

Phys. Rev. B (1)

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (1)

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

Science (1)

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

Sov. Phys. Usp. (1)

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

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Figures (5)

Fig. 1
Fig. 1

Geometry of the interface between a LHM and a RHM.

Fig. 2
Fig. 2

Field amplitudes of surface waves at the interface between a LHM and a RHM: (a) surface wave, (b) antisurface wave.

Fig. 3
Fig. 3

Geometry for an asymmetric LHM slab waveguide

Fig. 4
Fig. 4

Field amplitudes of forward and backward evanescent waves (dashed curves). They are added to create the surface mode field (solid line). (a) Even surface modes, (b) odd surface mode.

Fig. 5
Fig. 5

Filed amplitudes of forward and backward evanescent waves (dashed curves). They are added to create the surface mode field (solid line), both are even modes. (a) TE 0 surface mode, (b) TE 1 surface mode.

Equations (9)

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r = α 1 μ 1 α 2 μ 2 α 1 μ 1 + α 2 μ 2 ,
t = 2 α 1 μ 1 α 1 μ 1 + α 2 μ 2 .
α 1 μ 1 + α 2 μ 2 = 0 ,
α 1 μ 1 α 2 μ 2 = 0 .
e 2 α f h r s r c = 1 ,
r s = ( α f μ f α s μ s ) ( α f μ f + α s μ s ) ,
r c = ( α f μ f α c μ c ) ( α f μ f + α c μ c ) .
α f h = ( tanh 1 μ f μ s α s α f + tanh 1 μ f μ c α c α f ) , if μ f μ s α s α f < 1 , μ f μ c α c α f < 1
α f h = ( coth 1 μ f μ s α s α f + coth 1 μ f μ c α c α f ) , if μ f μ s α s α f > 1 , μ f μ c α c α f > 1

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