Abstract

The guided modes of a negative refractive index channel waveguide have been numerically investigated. It has been found that the modes exhibit a number of unusual properties that differ considerably from those of a conventional waveguide. In particular, it has been shown that these waveguides can exhibit low or negative group velocity as well as extraordinarily large group velocity dispersion. Calculation of the Poynting vector reveals that it is possible to support a mode with a zero energy flux motivating a simple design for an optical trap.

© 2003 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. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, �??Extremely low frequency plasmons in metallic mesostructures,�?? Phys. Rev. Lett. 76, 4773�??4776, (1996).
    [CrossRef] [PubMed]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. Microwave Theory Tech. 47, 2075�??2084, (1999).
    [CrossRef]
  4. 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]
  5. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two-dimensional, isotropic left-handed metamaterial,�?? Appl. Phys. Lett. 78, 489�??491, (2001).
    [CrossRef]
  6. R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of a negative index of refraction,�?? Science 292, 77�??79, (2001).
    [CrossRef] [PubMed]
  7. M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696 (2000).
    [CrossRef]
  8. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??All-angle negative refraction without negative effective index,�?? Phys. Rev. B 65, 201104 (2002).
    [CrossRef]
  9. E. Cubukcu, K. Aydin, E. Ozbay, S. Forteinopoulou, and C. M. Soukoulis, �??Electromagnetic waves: Negative refraction by photonic crystals,�?? Nature 423, 604�??605 (2003).
    [CrossRef] [PubMed]
  10. M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, �??First order quasi-phase-matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,�?? Appl. Phys. Lett. 62, 435�??436 (1993).
    [CrossRef]
  11. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, �??Guided modes in negative-refractive-index waveguides,�?? Phys. Rev. E 67, 057602 (2003).
    [CrossRef]
  12. E. A. Marcatili, �??Dielectric rectangular waveguide and directional coupler for integrated optics,�?? Bell Syst. Tech. J. 48, 2071�??2102 (1969).
  13. K. Okamoto, �??Fundamentals of optical waveguides,�?? Academic Press, 2000.
  14. G. I. Stegeman, J. J. Burke, and T. Tamir, �??Surface-polaritonlike waves guided by thin, lossy metal films,�?? Opt. Lett. 8, 383�??385 (1983).
    [CrossRef] [PubMed]
  15. S. Ramo, J. R. Whinnery, and T. Van Duzer, �??Fields and waves in communication electronics,�?? John Wiley & Sons, Inc., 1965.
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    [CrossRef]
  17. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, �??Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,�?? Phys. Rev. Lett. 87, 253902 (2001).
    [CrossRef] [PubMed]
  18. T. M. Monro and D. J. Richardson, �??Holey optical fibres: Fundamental properties and device applications,�?? C. R. Physique 4, 175�??186 (2003).
    [CrossRef]
  19. N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, �??Nonlinearity in holey optical fibers: Measurement and future opportunities,�?? Opt. Lett. 15, 1395�??1397 (1999).
    [CrossRef]

Appl. Phys. Lett. (2)

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two-dimensional, isotropic left-handed metamaterial,�?? Appl. Phys. Lett. 78, 489�??491, (2001).
[CrossRef]

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, �??First order quasi-phase-matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,�?? Appl. Phys. Lett. 62, 435�??436 (1993).
[CrossRef]

Bell Syst. Tech. J. (1)

E. A. Marcatili, �??Dielectric rectangular waveguide and directional coupler for integrated optics,�?? Bell Syst. Tech. J. 48, 2071�??2102 (1969).

C. R. Physique (1)

T. M. Monro and D. J. Richardson, �??Holey optical fibres: Fundamental properties and device applications,�?? C. R. Physique 4, 175�??186 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. Microwave Theory Tech. 47, 2075�??2084, (1999).
[CrossRef]

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

Nature (1)

E. Cubukcu, K. Aydin, E. Ozbay, S. Forteinopoulou, and C. M. Soukoulis, �??Electromagnetic waves: Negative refraction by photonic crystals,�?? Nature 423, 604�??605 (2003).
[CrossRef] [PubMed]

Opt. Lett. (2)

N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, �??Nonlinearity in holey optical fibers: Measurement and future opportunities,�?? Opt. Lett. 15, 1395�??1397 (1999).
[CrossRef]

G. I. Stegeman, J. J. Burke, and T. Tamir, �??Surface-polaritonlike waves guided by thin, lossy metal films,�?? Opt. Lett. 8, 383�??385 (1983).
[CrossRef] [PubMed]

Phys. Rev. B (2)

M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696 (2000).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??All-angle negative refraction without negative effective index,�?? Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Phys. Rev. E (1)

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. (3)

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, �??Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,�?? Phys. Rev. Lett. 87, 253902 (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]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, �??Extremely low frequency plasmons in metallic mesostructures,�?? Phys. Rev. Lett. 76, 4773�??4776, (1996).
[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]

Other (2)

K. Okamoto, �??Fundamentals of optical waveguides,�?? Academic Press, 2000.

S. Ramo, J. R. Whinnery, and T. Van Duzer, �??Fields and waves in communication electronics,�?? John Wiley & Sons, Inc., 1965.

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Figures (6)

Fig. 1.
Fig. 1.

Waveguide geometry and parameters.

Fig. 2.
Fig. 2.

Typical solutions for the x (top) and y (bottom) components of the guided modes of a negative index channel waveguide. The solid lines are the right-hand-sides of Eqs. (5) and (6) and the dashed lines are obtained from the right-hand-sides of Eqs. (7) and (8).

Fig. 3.
Fig. 3.

Examples of the mode profiles for the solutions in Fig. 2. Top row : H2,1y solutions with L=0.1cm corresponding to intersections (a,α) and (b,β). Middle row : H3,1y solutions with L=1cm corresponding to intersections (c,δ) and (d,δ). Bottom row : H3,2y solutions with L=2cm corresponding to intersections (c,η) and (d,η).

Fig. 4.
Fig. 4.

(a) Propagation constant, (b) group velocity, and (c) group velocity dispersion parameter of the H3,1y solutions from the middle row of Fig. 3. The solid and dashed lines correspond to the strongly (c,δ) and weakly (d,δ) localized modes, respectively.

Fig. 5.
Fig. 5.

Normalized energy flux as calculated for the H3,1y solutions of Figs. 3 and 4.

Fig. 6.
Fig. 6.

(a) Propagation constant and (b) normalized energy flux of the H3,1y solutions from the middle row of Fig. 3, as functions of the waveguide width L. The solid and dashed lines correspond to the strongly (c,δ) and weakly (d,δ) localized modes, respectively.

Equations (11)

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E ˜ ( x , y , z , t ) = E ( x , y ) e i ( ω t β z ) ,
H ˜ ( x , y , z , t ) = H ( x , y ) e i ( ω t β z ) ,
2 H y x 2 + 2 H y y 2 + ( ω 2 c 2 ε i μ i β ) H y = 0 ,
E x = ω μ 0 μ i β H y + 1 ω ε 0 ε i β 2 H y x 2 .
γ x L = ε 1 ε 2 k x L tan ( k x L ( p 1 ) π 2 ) ,
γ y L = k y L tan ( k y L ( q 1 ) π 2 ) ,
γ x 2 = ω 2 c 2 ( ε 2 μ 2 ε 1 μ 1 ) k x 2 ,
γ y 2 = ω 2 c 2 ( ε 2 μ 2 ε 1 μ 1 ) k y 2 .
β 2 = ω 2 c 2 ε 2 μ 2 ( k x 2 + k y 2 ) .
ε i ( ω ) = 1 ω p , i 2 ω 2 , μ i ( ω ) = 1 F ω 2 ω 2 ω 0 , i 2 .
P core = core S z d x d y , P clad = clad S z d x d y .

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