Abstract

The contradirectional coupling between a pair of planar dielectric waveguides, whose guiding layers are respectively a negative index medium and a positive index medium (PIM, i.e., the conventional medium), is theoretically investigated using the coupled mode theory. It is shown that, by tapering the middle segment of the PIM guiding layer to introduce a certain phase shift, the energy flow circulation can be established based on the contradirectional coupling effect, and, thus, the electromagnetic energy is localized in the waveguide system. This phenomenon is further verified numerically with the finite-difference time domain method. The quality factor of this open waveguide cavity is also discussed.

© 2011 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. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [CrossRef] [PubMed]
  3. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79(2001).
    [CrossRef] [PubMed]
  4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  5. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
    [CrossRef]
  6. B.-L. 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. Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
    [CrossRef]
  8. W. Yan and L. F. Shen, “Open waveguide cavity using a negative index medium,” Opt. Lett. 33, 2806–2808 (2008).
    [CrossRef] [PubMed]
  9. H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
    [CrossRef]
  10. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).
  11. W. Yan, L. F. Shen, and T. J. Yang, “Interaction between negative and positive index medium waveguides,” J. Lightwave Technol. 26, 3560–3566 (2008).
    [CrossRef]
  12. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  13. A. Taflove, Computational Electrodynamics, The Finite- Difference Time-Domain Method (Artech House, 1995).

2008

2006

Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

2003

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

B.-L. 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

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

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

2000

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

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

1987

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[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]

Chen, H. S.

Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Grzegorczyk, T. M.

Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

B.-L. 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]

Haus, H. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Heyman, E.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Huang, W. P.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Huangfu, J. T.

Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Kivshar, Y. S.

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

Kong, J. A.

Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

B.-L. 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]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

Nemat-Nasser, S. C.

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

Padilla, W. J.

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

Pendry, J. B.

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

Ran, L. X.

Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

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

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-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. F.

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]

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

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

Sukhorukov, A. A.

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

Taflove, A.

A. Taflove, Computational Electrodynamics, The Finite- Difference Time-Domain Method (Artech House, 1995).

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

Whitaker, N. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Wu, B.-L.

B.-L. 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]

Yan, W.

Yang, T. J.

Yuan, Y.

Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Zhang, Y.

B.-L. 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]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Appl. Phys. Lett.

Y. Yuan, L. X. Ran, H. S. Chen, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

J. Appl. Phys.

B.-L. 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. Lightwave Technol.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

W. Yan, L. F. Shen, and T. J. Yang, “Interaction between negative and positive index medium waveguides,” J. Lightwave Technol. 26, 3560–3566 (2008).
[CrossRef]

Opt. Lett.

Phys. Rev. E

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

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

Phys. Rev. Lett.

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

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

Science

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]

Other

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

A. Taflove, Computational Electrodynamics, The Finite- Difference Time-Domain Method (Artech House, 1995).

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Figures (5)

Fig. 1
Fig. 1

Schematic of the open waveguide cavity. Arrows indicate the energy and phase flows in the NIM and PIM waveguides with contradirectional coupling.

Fig. 2
Fig. 2

(a) Calculated real and imaginary parts of σ versus L c . (b) Dependence of L c on d and w.

Fig. 3
Fig. 3

Spectrum for the FDTD simulated electric field.

Fig. 4
Fig. 4

(a) Spatial variation of the amplitude of E y at the resonant frequency ω = 0.4 ( 2 π c / a ) . (b) Envelope of the distribution of E y amplitude along the PIM waveguide axis.

Fig. 5
Fig. 5

Q factor versus the cavity length.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

d U ( z ) d z = i β a U ( z ) + i C a b V ( z ) ,
d V ( z ) d z = i β b V ( z ) + i C b a U ( z ) ,
C p q = ω S p [ Δ ε q e q · e p * + Δ μ q h q · h p * ] d x ,
U 1 ( z ) = A 1 e i β ¯ z + α z , V 1 ( z ) = A 1 Δ β i α C a b e i β ¯ z + α z ,
U 3 ( z ) = A 3 e i β ¯ z α z , V 3 ( z ) = A 3 Δ β + i α C a b e i β ¯ z α z ,
U 2 ( L c / 2 ) = U 1 ( L c / 2 ) , U 2 ( L c / 2 ) = U 3 ( L c / 2 ) ,
V 2 ( L c / 2 ) = V 1 ( L c / 2 ) , V 2 ( L c / 2 ) = V 3 ( L c / 2 ) .
U 3 ( L c / 2 ) U 1 ( L c / 2 ) = V 3 ( L c / 2 ) V 1 ( L c / 2 ) ,
U 2 ( L c / 2 ) U 2 ( L c / 2 ) = V 2 ( L c / 2 ) V 2 ( L c / 2 ) .
σ = V 2 ( L c / 2 ) U 2 ( L c / 2 ) + V 2 ( L c / 2 ) U 2 ( L c / 2 ) ,
E y z = μ 0 H x t K x ,
E y x = = μ 0 H z t K z ,
H x z H z x = ε 0 E y t + J y ,
K x t = μ 0 ω p m 2 H x ,
K z t = μ 0 ω p m 2 H z ,
J y t = ε 0 ω p e 2 E y .
F ˜ ( ω m ) = 1 N ¯ t n = 2001 N t F ( t n ) e i ω m t n ,

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