Abstract

We explored backward waves propagating predominantly in a regular dielectric or even a vacuum. These modes emerge when a dielectric gap is made in a left-handed material, also known as a negative index material. As the gap becomes nanometric in size, the modal pattern conforms with surface waves distribution and can exhibit either (or both) right-handed or left-handed characteristics. Interestingly, the details of the modal field in the gap is reminiscent of the plasmon polariton solutions of either a gap in metals (forward propagating) or a metal slab in a dielectric bulk (backward propagating). Subsequently, we examined a specific metamaterial realization of the left-handed optical medium by use of elongated nanometallic inclusions to generate positive–negative waveguide anisotropy. We used this metamaterial embedded between dielectric layers as the cladding layer of a gap and verified that the important results predicted above, namely nanometric-size backward waves, are obtained for this specific implementation.

© 2007 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 epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (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. V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2006).
    [CrossRef]
  5. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003).
    [CrossRef]
  6. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
    [CrossRef]
  7. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, "Guiding and confining light in void nanostructure," Opt. Lett. 29, 1209-1211 (2004).
    [CrossRef] [PubMed]
  8. P. Ginzburg, D. Arbel, and M. Orenstein, "Gap plasmon polariton structure for very efficient microscale-to-nanoscale interfacing," Opt. Lett. 31, 3288-3290 (2006).
    [CrossRef] [PubMed]
  9. Y. Satuby and M. Orenstein, "Experimental observation of surface plasmon-polariton waves in deep trench metal waveguides," Joint IPRA/NANO Topical Meetings (Optical Society of America, 2006), paper IWC4.
  10. V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101(R) (2005).
    [CrossRef]
  11. N. Kaminsky, Y. Satuby, and M. Orenstein, "Nano-optical modes of a gap structure in a left-hand-metamaterial waveguide," Photonic Metamaterials Topical Meeting (Optical Society of America, 2006), paper WD11.
  12. B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
    [CrossRef]
  13. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).
  14. A. Sihvola, Electromagnetic Mixing Formulae and Application, Vol. 47 of IEE Electromagnetic Waves Series (IEE, 1999).
    [CrossRef]
  15. J. Dimmock, "Losses in left-handed materials," Opt. Express 11, 2397-2402 (2003).
    [CrossRef] [PubMed]
  16. R. Wangberg, J. Elser, E. E. Nariminov, and V. A. Podolskiy, "Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media," J. Opt. Soc. Am. B 23, 498-505 (2006).
    [CrossRef]

2006 (3)

2005 (1)

V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101(R) (2005).
[CrossRef]

2004 (2)

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, "Guiding and confining light in void nanostructure," Opt. Lett. 29, 1209-1211 (2004).
[CrossRef] [PubMed]

2003 (2)

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

J. Dimmock, "Losses in left-handed materials," Opt. Express 11, 2397-2402 (2003).
[CrossRef] [PubMed]

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]

1991 (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

1968 (1)

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

Almeida, V. R.

Arbel, D.

Barrios, C. A.

Boardman, A. D.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Dimmock, J.

Egan, P.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Elser, J.

Ginzburg, P.

Kaminsky, N.

N. Kaminsky, Y. Satuby, and M. Orenstein, "Nano-optical modes of a gap structure in a left-hand-metamaterial waveguide," Photonic Metamaterials Topical Meeting (Optical Society of America, 2006), paper WD11.

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

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

Lipson, M.

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Narimanov, E. E.

V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101(R) (2005).
[CrossRef]

Nariminov, E. E.

Orenstein, M.

P. Ginzburg, D. Arbel, and M. Orenstein, "Gap plasmon polariton structure for very efficient microscale-to-nanoscale interfacing," Opt. Lett. 31, 3288-3290 (2006).
[CrossRef] [PubMed]

Y. Satuby and M. Orenstein, "Experimental observation of surface plasmon-polariton waves in deep trench metal waveguides," Joint IPRA/NANO Topical Meetings (Optical Society of America, 2006), paper IWC4.

N. Kaminsky, Y. Satuby, and M. Orenstein, "Nano-optical modes of a gap structure in a left-hand-metamaterial waveguide," Photonic Metamaterials Topical Meeting (Optical Society of America, 2006), paper WD11.

Pendry, J. B.

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

Podolskiy, V. A.

R. Wangberg, J. Elser, E. E. Nariminov, and V. A. Podolskiy, "Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media," J. Opt. Soc. Am. B 23, 498-505 (2006).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101(R) (2005).
[CrossRef]

Prade, B.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

Satuby, Y.

N. Kaminsky, Y. Satuby, and M. Orenstein, "Nano-optical modes of a gap structure in a left-hand-metamaterial waveguide," Photonic Metamaterials Topical Meeting (Optical Society of America, 2006), paper WD11.

Y. Satuby and M. Orenstein, "Experimental observation of surface plasmon-polariton waves in deep trench metal waveguides," Joint IPRA/NANO Topical Meetings (Optical Society of America, 2006), paper IWC4.

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, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Shadrivov, V.

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

Shalaev, V. M.

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2006).
[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]

Sihvola, A.

A. Sihvola, Electromagnetic Mixing Formulae and Application, Vol. 47 of IEE Electromagnetic Waves Series (IEE, 1999).
[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, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

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

Veselago, V. G.

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

Vinet, J. Y.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Wangberg, R.

Xu, Q.

Zharov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

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

Nat. Photonics (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (2)

V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101(R) (2005).
[CrossRef]

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Phys. Rev. E (2)

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

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[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 epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (4)

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

A. Sihvola, Electromagnetic Mixing Formulae and Application, Vol. 47 of IEE Electromagnetic Waves Series (IEE, 1999).
[CrossRef]

N. Kaminsky, Y. Satuby, and M. Orenstein, "Nano-optical modes of a gap structure in a left-hand-metamaterial waveguide," Photonic Metamaterials Topical Meeting (Optical Society of America, 2006), paper WD11.

Y. Satuby and M. Orenstein, "Experimental observation of surface plasmon-polariton waves in deep trench metal waveguides," Joint IPRA/NANO Topical Meetings (Optical Society of America, 2006), paper IWC4.

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Figures (7)

Fig. 1
Fig. 1

Schematic description of the gap waveguide studied. In a LHM defined by ε LHM < 0 and μ LHM < 0 , a gap of thickness h is carved to contain a dielectric media ( ε d = 2.25 ) .

Fig. 2
Fig. 2

Geometric dispersion curves for TM waves guided by a dielectric gap carved in a LHM media at four distinct regimes: (a) n d > n LHM and ε d > ε LHM , (b) n d > n LHM and ε d < ε LHM , (c) n d < n LHM and ε d > ε LHM , (d) n d < n LHM and ε d < ε LHM .

Fig. 3
Fig. 3

Schematic description of a width d waveguide. The inclusion-doped core has an anisotropic dielectric coefficient ( ε x = ε ε y = ε z = ε ), and the cladding is an isotropic dielectric material: ε = ε d (constant).

Fig. 4
Fig. 4

Geometrical dispersion relation of anisotropic slab (a) R { n eff } and (b) I { n eff } versus the slab width d. Forward waves are shown in solid curves and backward waves are shown in dashed curves. The solid line in (a) separates positive and negative effective index of refraction. The dotted curves are the limit of the forward waves for wide layer. The modes shown are TM 0 to TM 3 and TM 0 LH to TM 3 LH . Insets(1)–(4) describe the H y field along the x axis. Each inset refers to the real and imaginary value of n eff as defined by its reference points in small circles on the graphs.

Fig. 5
Fig. 5

R { n eff } sign versus k for ε = 0.863 0.344 j , ε = 2.4 0.0008 j . White area defines where negative R { n eff } (backward waves) and gray area the positive R { n eff } (forward waves). In both cases the imaginary parts of n eff are negative, describing loss and not gain. The geometrical dispersion curves describe both the right-handed waves (dashed curves) and the left-handed waves (solid curves) shown in Fig. 4. The arrows are pointing at the increasing slab-width direction.

Fig. 6
Fig. 6

Schematic description of a gap waveguide of width h within two anisotropic slabs of width d = 0.1 μ m each. The inclusion-doped slabs have an anisotropic dielectric coefficient ( ε x = ε ε y = ε z = ε ), while the cladding and the gap are similar: isotropic dielectric materials with ε = ε d (constant).

Fig. 7
Fig. 7

(a) Real and (b) imaginary geometrical dispersion curves of TM waves within an anisotropic gap waveguide. Each slab width is d = 0.1 μ m . The different modes are characterized as symmetric modes (dashed) or antisymmetric modes (dotted) and forward waves or backward waves. Insets (1)–(4) show the H y field as a function of the perpendicular coordinate: (1) TM 0 , h = 0.02 μ m ; (2) TM 1 , h = 0.11 μ m ; (3) TM 3 LH , h = 0.02 μ m ; (4) TM 2 LH , h = 0.04 μ m .

Tables (1)

Tables Icon

Table 1 Set of Permittivity and Permeability Chosen for the TM Case Study

Equations (17)

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

n cladding = n LHM = ε LHM μ LHM ,
n core = n d = ε d .
TM : H y ( x , z ) = H 0 ( x ) e j k 0 n eff z ,
TE : E y ( x , z ) = E 0 ( x ) e j k 0 n eff z .
j k x core = q d = k 0 n eff 2 n d 2 ,
j k x clad = q LHM = k 0 n eff 2 n LHM 2 .
TM : tanh ( q d h 2 ) = ( ε d q LHM ε LHM q d ) ± 1 ,
TE : tanh ( q d h 2 ) = ( μ d q LHM μ LHM q d ) ± 1 ,
ε = ε m f + ε d ( 1 f ) ,
ε = ε d ε d ( 1 f ) + ε m ( 1 + f ) ε m ( 1 f ) + ε d ( 1 + f ) .
ε = 0.863 0.344 j , ε = 2.4 0.0008 j .
H ( x ) = { A e q d ( x d 2 ) d 2 < x B cos ( k x ) + C sin ( k x ) d 2 < x < d 2 D e + q d ( x + d 2 ) x < d 2 } .
k z 2 ε + k 2 ε = k 0 2 ; q d = k z 2 k 0 2 ε d ,
tan ( k d 2 ) = ( ± q d ε k ε d ) ± 1 ,
lim d { n eff } = lim k 0 ε ε ε k 2 k 0 2 = ε = 0.18 0.95 j .
n eff = ε 1 k 2 ε k 0 2 .
cosh ( q LHM d ) [ 1 + ε q d ε d q LHM tanh ± 1 ( q d h 2 ) ] [ tanh ( q LHM d ) + ε d q LHM ε q d ] = e q LHM d ( ε d q LHM ε q d 1 ) ,

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