Abstract

A rigorous analysis of the contra-directional coupling between a left-handed waveguide and a right-handed waveguide shows that such a structure supports twin modes with complex (conjugate) propagation constants. Using evanescent coupling or a source inside one of the waveguides, the twin modes can be excited. In this case, light rotates inside the lamellar structure forming an exotic mode : the light wheel. The structure can finally be seen as a new type of cavity and can be used for beam reshaping.

© 2007 Optical Society of America

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  1. V.G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Usp. Fiz. Nauk. 92, 517 (1967).
  2. R.A. Shelby, D.R. Smith, and S. Shultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001).
    [CrossRef] [PubMed]
  3. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  4. J.B. Pendry, D. Schuring, and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  5. P.R. Berman, "Goos-Hanchen shift in negatively refractive media," Phys. Rev. E 66, 067603 (2002).
    [CrossRef]
  6. A. Lakhtakia, "On plane wave remittances and goos-hanchen shifts of planar slabs with negative real permittivity and permeability," Electromagnetics 23, 71 (2003).
    [CrossRef]
  7. I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003).
    [CrossRef]
  8. L.G. Wang and S.Y. Zhu, "Large negative lateral shifts from the kretschman-raether configuration with lefthanded materials," Appl. Phys. Lett. 87, 221102 (2005).
    [CrossRef]
  9. A. Moreau and D. Felbacq, "Comment on ’large negative lateral shifts from the kretschman-raether configuration with left-handed materials’," Appl. Phys. Lett. 90, 066102 (2007).
    [CrossRef]
  10. A. Alu and N. Engheta, in Negative-Refraction Metamaterials, edited by G.V. eleftheriades and K.G. Balmain (Wiley, New York, 2005).
  11. R. Islam, F. Elek and G.V. Eleftheriades, "A coupled-line metamaterial coupler having co-directional phase but contra-directional power flow," IEEE Electron. Lett. 40, 315-317 (2004).
    [CrossRef]
  12. Y. Wang, R. Islam and G.V. Eleftheriades, "An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies," Opt. Express 14, 7279-7290 (2006).
    [CrossRef] [PubMed]
  13. N.M. Litchinitser, I.R. Gabitov, A.I. Mainmistov, "Optical bistability in a nonlinear optical coupler with a negative index channel," Phys. Rev. Lett. 99, 113902 (2007).
    [CrossRef] [PubMed]
  14. I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820 (2003).
    [CrossRef]
  15. I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Complete band gaps in one-dimensional left-handed periodic structures," Phys. Rev. Lett. 95, 193903 (2005).
    [CrossRef] [PubMed]

2007

A. Moreau and D. Felbacq, "Comment on ’large negative lateral shifts from the kretschman-raether configuration with left-handed materials’," Appl. Phys. Lett. 90, 066102 (2007).
[CrossRef]

N.M. Litchinitser, I.R. Gabitov, A.I. Mainmistov, "Optical bistability in a nonlinear optical coupler with a negative index channel," Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef] [PubMed]

2006

2005

L.G. Wang and S.Y. Zhu, "Large negative lateral shifts from the kretschman-raether configuration with lefthanded materials," Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Complete band gaps in one-dimensional left-handed periodic structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

2004

R. Islam, F. Elek and G.V. Eleftheriades, "A coupled-line metamaterial coupler having co-directional phase but contra-directional power flow," IEEE Electron. Lett. 40, 315-317 (2004).
[CrossRef]

2003

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820 (2003).
[CrossRef]

A. Lakhtakia, "On plane wave remittances and goos-hanchen shifts of planar slabs with negative real permittivity and permeability," Electromagnetics 23, 71 (2003).
[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]

2002

P.R. Berman, "Goos-Hanchen shift in negatively refractive media," Phys. Rev. E 66, 067603 (2002).
[CrossRef]

2001

R.A. Shelby, D.R. Smith, and S. Shultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001).
[CrossRef] [PubMed]

2000

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

1967

V.G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Usp. Fiz. Nauk. 92, 517 (1967).

Berman, P.R.

P.R. Berman, "Goos-Hanchen shift in negatively refractive media," Phys. Rev. E 66, 067603 (2002).
[CrossRef]

Eleftheriades, G.V.

Y. Wang, R. Islam and G.V. Eleftheriades, "An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies," Opt. Express 14, 7279-7290 (2006).
[CrossRef] [PubMed]

R. Islam, F. Elek and G.V. Eleftheriades, "A coupled-line metamaterial coupler having co-directional phase but contra-directional power flow," IEEE Electron. Lett. 40, 315-317 (2004).
[CrossRef]

Elek, F.

R. Islam, F. Elek and G.V. Eleftheriades, "A coupled-line metamaterial coupler having co-directional phase but contra-directional power flow," IEEE Electron. Lett. 40, 315-317 (2004).
[CrossRef]

Felbacq, D.

A. Moreau and D. Felbacq, "Comment on ’large negative lateral shifts from the kretschman-raether configuration with left-handed materials’," Appl. Phys. Lett. 90, 066102 (2007).
[CrossRef]

Gabitov, I.R.

N.M. Litchinitser, I.R. Gabitov, A.I. Mainmistov, "Optical bistability in a nonlinear optical coupler with a negative index channel," Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef] [PubMed]

Islam, R.

Y. Wang, R. Islam and G.V. Eleftheriades, "An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies," Opt. Express 14, 7279-7290 (2006).
[CrossRef] [PubMed]

R. Islam, F. Elek and G.V. Eleftheriades, "A coupled-line metamaterial coupler having co-directional phase but contra-directional power flow," IEEE Electron. Lett. 40, 315-317 (2004).
[CrossRef]

Kivshar, Y.S.

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Complete band gaps in one-dimensional left-handed periodic structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820 (2003).
[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]

Lakhtakia, A.

A. Lakhtakia, "On plane wave remittances and goos-hanchen shifts of planar slabs with negative real permittivity and permeability," Electromagnetics 23, 71 (2003).
[CrossRef]

Litchinitser, N.M.

N.M. Litchinitser, I.R. Gabitov, A.I. Mainmistov, "Optical bistability in a nonlinear optical coupler with a negative index channel," Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef] [PubMed]

Mainmistov, A.I.

N.M. Litchinitser, I.R. Gabitov, A.I. Mainmistov, "Optical bistability in a nonlinear optical coupler with a negative index channel," Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef] [PubMed]

Moreau, A.

A. Moreau and D. Felbacq, "Comment on ’large negative lateral shifts from the kretschman-raether configuration with left-handed materials’," Appl. Phys. Lett. 90, 066102 (2007).
[CrossRef]

Pendry, J. B.

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

Pendry, J.B.

J.B. Pendry, D. Schuring, and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

Schuring, D.

J.B. Pendry, D. Schuring, and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

Shadrivov, I.V.

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Complete band gaps in one-dimensional left-handed periodic structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820 (2003).
[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]

Shelby, R.A.

R.A. Shelby, D.R. Smith, and S. Shultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001).
[CrossRef] [PubMed]

Shultz, S.

R.A. Shelby, D.R. Smith, and S. Shultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001).
[CrossRef] [PubMed]

Smith, D.R.

J.B. Pendry, D. Schuring, and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

R.A. Shelby, D.R. Smith, and S. Shultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001).
[CrossRef] [PubMed]

Sukhorukov, A.A.

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Complete band gaps in one-dimensional left-handed periodic structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820 (2003).
[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]

Veselago, V.G.

V.G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Usp. Fiz. Nauk. 92, 517 (1967).

Wang, L.G.

L.G. Wang and S.Y. Zhu, "Large negative lateral shifts from the kretschman-raether configuration with lefthanded materials," Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

Wang, Y.

Zhu, S.Y.

L.G. Wang and S.Y. Zhu, "Large negative lateral shifts from the kretschman-raether configuration with lefthanded materials," Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

Appl. Phys. Lett.

L.G. Wang and S.Y. Zhu, "Large negative lateral shifts from the kretschman-raether configuration with lefthanded materials," Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

A. Moreau and D. Felbacq, "Comment on ’large negative lateral shifts from the kretschman-raether configuration with left-handed materials’," Appl. Phys. Lett. 90, 066102 (2007).
[CrossRef]

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820 (2003).
[CrossRef]

Electromagnetics

A. Lakhtakia, "On plane wave remittances and goos-hanchen shifts of planar slabs with negative real permittivity and permeability," Electromagnetics 23, 71 (2003).
[CrossRef]

IEEE Electron. Lett.

R. Islam, F. Elek and G.V. Eleftheriades, "A coupled-line metamaterial coupler having co-directional phase but contra-directional power flow," IEEE Electron. Lett. 40, 315-317 (2004).
[CrossRef]

Opt. Express

Phys. Rev. E

P.R. Berman, "Goos-Hanchen shift in negatively refractive media," Phys. Rev. E 66, 067603 (2002).
[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.

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

N.M. Litchinitser, I.R. Gabitov, A.I. Mainmistov, "Optical bistability in a nonlinear optical coupler with a negative index channel," Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef] [PubMed]

I.V. Shadrivov, A.A. Sukhorukov, and Y.S. Kivshar, "Complete band gaps in one-dimensional left-handed periodic structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

Science

J.B. Pendry, D. Schuring, and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

R.A. Shelby, D.R. Smith, and S. Shultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001).
[CrossRef] [PubMed]

Usp. Fiz. Nauk.

V.G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Usp. Fiz. Nauk. 92, 517 (1967).

Other

A. Alu and N. Engheta, in Negative-Refraction Metamaterials, edited by G.V. eleftheriades and K.G. Balmain (Wiley, New York, 2005).

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Figures (8)

Fig. 1.
Fig. 1.

The two waveguides are surrounded by air and separated by a thickness h. The z = 0 plane is placed in the middle of the air layer (region III). The different regions are labelled using roman numerals.

Fig. 2.
Fig. 2.

The two identical coupled waveguides of height 0.2λ are excited using evanescent coupling : an incident beam with an incidence angle of 36,9 in a medium with ε = 5 and μ = 1 is used. The energy oscillates between the two guides and leaks out when it is in the upper guide, so that it decreases exponentially.

Fig. 3.
Fig. 3.

Solutions of the dispersion relation in the α complex plane for different values of h, which is given for some points as a fraction of the wavelength. The waveguides are characterized by h 1 = 0.2λ, ε 1 = 3, μ 1 = 1, ε 2 = -3, μ 2 = -1 and h 2 = .7146λ. The units correspond to the choice λ = 1.

Fig. 4.
Fig. 4.

Characteristics of the solution with a propagation constant presenting a positive imaginary part for h = 0.5λ: (a) shows the modulus of the field (which presents two zeros in the left-handed guide), (b) the phase, which is either null or equal to π in one guide and to π 2 in the other one, and (c) shows the Poynting vector along the x direction.

Fig. 5.
Fig. 5.

Excitation of the light wheel using an incident beam in a high-index material (with ε = 5 and μ = 1). The image represents the modulus of the Ey field in a domain which is only 4.5λ high and about 60λ large. The propagation direction of light is indicated by white arrows so that the rotation of light in the structure is made visible. The fringes above the structure are localized interferences of the incident and reflected beams. The incident angle of the beam is 36.9 o , the distance between the prism and the first waveguide is 0.5λ.

Fig. 6.
Fig. 6.

Time averaged Poynting vector along the x direction. Left : cross section in the center of the incident beam, where the origin of x is chosen. Two other cross-sections are shown for x = 40λ and x = 70λ.

Fig. 7.
Fig. 7.

Modulus of Ey in the case of a lossy LHM characterized by ε 2 = -3 + 0.01i and μ 2 = -1 + 0.01i. In order to assure a (theoretically) perfect coupling, we have taken h1 = 0.177λ, h 2 = 0.7λ, and an incidence angle of 35.68 o .

Fig. 8.
Fig. 8.

Modulus of the electric field when a punctual source is placed in the middle of the right-handed waveguide. Two contra-rotative light wheels are excited and they interfere, which explains the interference fringes.

Equations (1)

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x 1 F 1 + x 2 F 2 + ( 1 + x 1 x 2 F 1 F 2 ) tanh ( γ 0 h ) = 0

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