Abstract

We investigate the characteristics of guided wave modes in planar coupled waveguides. In particular, we calculate the dispersion relations for TM modes in which one or both of the guiding layers consists of negative index media (NIM)-where the permittivity and permeability are both negative.We find that the Poynting vector within the NIM waveguide axis can change sign and magnitude, a feature that is reflected in the dispersion curves.

© 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 E and µ,�?? Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  2. J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  3. S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, �??Imaging the near field,�?? cond-mat/0207026 (unpublished).
  4. C. Caloz, C.-C. Chang, and T. Itoh, �??Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,�?? J. Appl. Phys. 90, 5483 (2001).
    [CrossRef]
  5. I. S. Nefedov and S. A. Tretyakov, �??Waveguide containing a backward-wave slab,�?? condmat/0211185 (unpublished).
  6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Masser, and S. Schultz, �??Composite Medium with Simultaneously Negative Permeability and Permittivity,�?? Phys. Rev. Lett. 84, 4184 (2000).
    [CrossRef] [PubMed]
  7. R. W. Ziolkowski, and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001).
    [CrossRef]

J. Appl. Phys.

C. Caloz, C.-C. Chang, and T. Itoh, �??Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,�?? J. Appl. Phys. 90, 5483 (2001).
[CrossRef]

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]

Phys. Rev. Lett.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Masser, and S. Schultz, �??Composite Medium with Simultaneously Negative Permeability and Permittivity,�?? Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of E and µ,�?? Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Other

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, �??Imaging the near field,�?? cond-mat/0207026 (unpublished).

I. S. Nefedov and S. A. Tretyakov, �??Waveguide containing a backward-wave slab,�?? condmat/0211185 (unpublished).

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Figures (6)

Fig. 1.
Fig. 1.

Coupled waveguide structure considered in this paper. The red and blue regions correspond to the waveguide channels with dispersive material parameters. The permittivity and permeability in the channels can take positive or negative values. The free space parameters take the values ∊00=1.

Fig. 2.
Fig. 2.

Dispersion curves for waveguide structure with material parameters given by Eqs. (2). The widths of the middle layer and the two waveguide channels are given by d=0.1λ0, and d 1=0.2λ0 respectively, where ω0=2πc0. The red curve corresponds to the symmetric modes, and the blue curve is for the asymmetric modes. The central green curves correspond to the dispersion relation for a single NIM waveguide of thickness 2d 1. The inset is a magnification of the same quantity localized around one of the bending features. The numbers label points of interest as discussed in the text.

Fig. 3.
Fig. 3.

The normalized z-component of the Poynting vector Sz as a function of dimensionless position. The continual decline in amplitude of Sz in the central film and the outer two layers from blue to green coincides (in order) with points 1–3 in the inset of Fig. 2 respectively. Meanwhile, the two waveguides have relatively little shift in amplitude. The vertical lines serve as guides to the eye.

Fig. 4.
Fig. 4.

The normalized z component of power, Pz , is plotted as a function of dimensionless frequency ω/ω0. The color labeling of the curves coincides with the dispersion curves displayed in the inset of Fig. 2. For comparison purposes, the green curve given by Eq. (8), is multiplied by a factor of 2.

Fig. 5.
Fig. 5.

Dispersion relation for the coupled waveguide structure consisting of a NIM channel and a PIM channel. The widths of the middle layer and the two guide channels are given by d=0.1λ0, and d 1=0.2λ0 respectively. The red (blue) curves correspond to the plus (minus) sign in Eq. (9).

Fig. 6.
Fig. 6.

The spatial dependence of the (normalized) Poynting vector Sz , in both waveguide cores, as a function of x0. The top panel (a) depicts Sz in the PIM guide, and the bottom panel (b) corresponds to the NIM guiding layer. Energy flow is always negative in the NIM waveguide, and positive in the PIM waveguide. For all four curves, ω/ω0=1.24 and kz /(ω0/c)=1.27 (blue), 1.59(green), 1.89 (red), and 2.22 (black). Upon comparing both panels, it is evident that for kz /(ω0/c)=2.22, the PIM waveguide layer bears most of the energy flux from the electromagnetic fields.

Equations (18)

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[ 2 x 2 k z 2 + μ i ( ω ) i ( ω ) ω 2 c 2 ] h ( x ) = 0 , i = 0 , 1 , 2 ,
i ( ω ) = 1 ω e 2 ω 2 , μ i ( ω ) = 1 ω m 2 ω 2 ,
k i = i ( ω ) μ i ( ω ) ω 2 c 2 k z 2 i = 0 , 1 , 2 ,
± e 2 k 0 d sin ( 2 k 1 d 1 ) sin ( 2 k 1 d 1 2 ψ ) = 0 ,
ψ arctan ( k 0 1 k 1 0 ) .
h ( x ) = cos ( ψ ) cos ( k 1 d 1 ψ ) e k 0 ( x L ) , x L ,
= cos [ k 1 ( x L ) + ψ ] cos ( k 1 d 1 ψ ) , d x L ,
= cos ( 2 k 1 d 1 ψ ) cos ( k 1 d 1 ψ ) cosh ( k 0 d ) cosh ( k 0 x ) , x d ,
= cos [ k 1 ( x + L ) ψ ] cos ( k 1 d 1 ψ ) , d x L ,
= cos ( ψ ) cos ( k 1 d 1 ψ ) e k 0 ( x + L ) , x L ,
S z ( x ; k z , ω ) = c 2 8 π k z ω i ( ω ) [ h ( x ) ] 2 , i = 0 , 1 , 2 .
P z ( k z , ω ) d x S z ( x ; k z , ω ) = c 2 8 π k z d 1 ω { cos 2 ( k 1 d 1 ) 0 k 0 d 1 + 1 1 [ 1 + sin ( 2 k 1 d 1 ) 2 k 1 d 1 ] } .
1 e 4 k 0 d sin ( 2 k 1 d 1 ) ± sin ( 2 ψ ) = 0 .
h ( x ) = sin [ 2 ( k 1 d 1 + ψ ) ] sin ( 2 k 1 d 1 ) e k 0 ( x 2 d ) , x L ,
= e k 0 ( L + 2 d ) sin ( 2 k 1 d 1 ) cos ( ψ ) sin [ 2 ( k 1 d 1 ψ ) ] cos [ k 1 ( x L ) + ψ ] , d x L ,
= e k 0 L sin ( 2 ψ ) { e k 0 ( x + d ) sin [ 2 ( k 1 d 1 + ψ ) ] e k 0 ( x + d ) sin [ 2 ( k 1 d 1 ) ] } , x d ,
= e k 0 L cos ( ψ ) cos [ k 1 ( x + L ) + ψ ] , d x L ,
= e k 0 x , x L .

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