Abstract

This paper presents and analyzes configurations of waveguides composed by both, regular dielectrics and negative-index materials disposed in an anti-symmetric way with respect to the optical axis. In its basic form, the configuration includes two guiding layers and two cladding media, where both the cladding and the guiding pairs have opposite-signed refractive index. A geometrical analysis shows that paths are closed, hinting the possibility of localized modes or light trapping with zero ray velocity. The ray model shows also a quasi-perfect imaging effect for off-axis objects. A modal approach shows that trapping is broadband and the propagation constant spectrum is continuous. When cores are allowed different widths, control of the group velocity is possible. More general anti-symmetrically mirrored layouts are also addressed.

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  2. V. R. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
    [CrossRef]
  3. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
    [CrossRef] [PubMed]
  4. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
    [CrossRef]
  5. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 057602 (2003).
    [CrossRef] [PubMed]
  6. K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
    [CrossRef]
  7. Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left-handed material cover or substrate,” Opt. Commun. 281(4), 607–613 (2008).
    [CrossRef]
  8. J. He, Y. Jin, Z. Hong, and S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express 16(15), 11077–11082 (2008).
    [CrossRef] [PubMed]
  9. J. He and S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006).
    [CrossRef]
  10. Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245(1-6), 125–135 (2005).
    [CrossRef]
  11. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
    [CrossRef] [PubMed]
  12. A. Alù and N.Engheta, “Guided Modes in a Waveguide Filled With a Pair of Single-Negative (SNG), Double-Negative (DNG), and/or Double-Positive (DPS) Layers,” IEEE Trans. MTT, V.52, No. 1, 2004.
  13. A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
    [CrossRef]
  14. G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007).
    [CrossRef] [PubMed]
  15. M. J. Adams, An Introduction to optical waveguides, (Wiley, 1981)

2008 (2)

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left-handed material cover or substrate,” Opt. Commun. 281(4), 607–613 (2008).
[CrossRef]

J. He, Y. Jin, Z. Hong, and S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express 16(15), 11077–11082 (2008).
[CrossRef] [PubMed]

2007 (4)

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007).
[CrossRef] [PubMed]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
[CrossRef]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[CrossRef] [PubMed]

V. R. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

2006 (2)

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
[CrossRef]

J. He and S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006).
[CrossRef]

2005 (2)

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245(1-6), 125–135 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

2003 (1)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 057602 (2003).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1997 (1)

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[CrossRef]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[CrossRef] [PubMed]

Brueck, S. R. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Cao, Z.

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245(1-6), 125–135 (2005).
[CrossRef]

Danzmann, K.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[CrossRef]

Dolling, G.

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Fleischhauer, M.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[CrossRef]

He, J.

J. He, Y. Jin, Z. Hong, and S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express 16(15), 11077–11082 (2008).
[CrossRef] [PubMed]

J. He and S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006).
[CrossRef]

He, S.

J. He, Y. Jin, Z. Hong, and S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express 16(15), 11077–11082 (2008).
[CrossRef] [PubMed]

J. He and S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006).
[CrossRef]

He, Y.

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245(1-6), 125–135 (2005).
[CrossRef]

Hermann, C.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
[CrossRef]

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[CrossRef] [PubMed]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
[CrossRef]

Hong, Z.

Jamois, C.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
[CrossRef]

Jin, Y.

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 057602 (2003).
[CrossRef] [PubMed]

Klaedtke, A.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
[CrossRef]

Li, S. P.

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left-handed material cover or substrate,” Opt. Commun. 281(4), 607–613 (2008).
[CrossRef]

Linden, S.

Malloy, K. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Müllera, G.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[CrossRef]

Osgood, R. M.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Panoiu, N. C.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Pati, G. S.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Rinkleff, R. H.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[CrossRef]

Salit, K.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007).
[CrossRef] [PubMed]

Salit, M.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007).
[CrossRef] [PubMed]

Scully, M.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[CrossRef]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 057602 (2003).
[CrossRef] [PubMed]

Shahriar, M. S.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007).
[CrossRef] [PubMed]

Shalaev, V. R.

V. R. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

Shen, Q.

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245(1-6), 125–135 (2005).
[CrossRef]

Soukoulis, C. M.

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 057602 (2003).
[CrossRef] [PubMed]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[CrossRef] [PubMed]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
[CrossRef]

Wang, Z. H.

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left-handed material cover or substrate,” Opt. Commun. 281(4), 607–613 (2008).
[CrossRef]

Wegener, M.

Wicht, A.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[CrossRef]

Xiao, Z. Y.

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left-handed material cover or substrate,” Opt. Commun. 281(4), 607–613 (2008).
[CrossRef]

Zhang, S.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

IEEE Microw. Wirel. Compon. Lett. (1)

J. He and S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006).
[CrossRef]

Nat. Photonics (1)

V. R. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

Nature (1)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[CrossRef] [PubMed]

Opt. Commun. (3)

Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245(1-6), 125–135 (2005).
[CrossRef]

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[CrossRef]

Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left-handed material cover or substrate,” Opt. Commun. 281(4), 607–613 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 057602 (2003).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007).
[CrossRef] [PubMed]

Other (2)

M. J. Adams, An Introduction to optical waveguides, (Wiley, 1981)

A. Alù and N.Engheta, “Guided Modes in a Waveguide Filled With a Pair of Single-Negative (SNG), Double-Negative (DNG), and/or Double-Positive (DPS) Layers,” IEEE Trans. MTT, V.52, No. 1, 2004.

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Figures (5)

Fig. 1
Fig. 1

The basic layer disposition of material composition and thicknesses are shown for waveguide structures with opposite-signed refractive-index claddings and cores. Figure (a), (s = t) shows closed-loop ray trajectories, suggesting the possibility of localization of power. Figure (b) (s = t) shows the imaging property of this configuration. This scheme shows that rays emitted backwards also contribute to the image. In Figure (c) Quasi-circulating ray paths for core layers of different thicknesses are seen (s ≠ t). The net ray velocity can be controlled here by the parameter s-t.

Fig. 2
Fig. 2

TE0 (a) and TE1 (b) mode profiles examples in waveguides with opposite-signed refractive index claddings and cores with different thicknesses for the two cores (s ≠ t).

Fig. 3
Fig. 3

Two mode profiles examples in waveguides with opposite-signed refractive-index claddings and cores with equal thicknesses for the two cores, or totally anti-symmetric guides (s = t). The propagating constant spectrum in this case is continuous and the effective indices of the displayed modes were arbitrarily chosen within the guiding range: (nc)2 < (neff)2 < (ng)2

Fig. 4
Fig. 4

Interface or SPP-type mode profile example in a waveguide with opposite-signed refractive-index claddings and cores with equal core thicknesses (s = t).

Fig. 5
Fig. 5

Normalized effective indices (dashed lines) and normalized power flow in the z-direction as a function of x = (t – s), for the first two modes in a waveguide structure with parameters ng = 2 and nc = 1.7. The total width, (t + s) is kept constant at a value of 2. The inset shows an enlargement of the threshold region of the TE1 mode to emphasize that the power flow is a continuous function of (t – s)

Equations (15)

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μ c , μ g , ε c , ε g > 0 ε g μ g > ε c μ c
v r a y z = ( c / n g ) ( t s ) ( t + s ) cos θ
2 E y ( m ) ( x ) x 2 = p ( m ) 2 ( β ) E y ( m ) ( x ) p ( m ) 2 = ε ( m ) μ ( m ) β 2
k 0 2 ε s μ s < β 2 < k 0 2 ε g μ g
E y ( x ) = A 0 cos ( p g x + ϕ 1 )                 0 < x < t E y ( x ) = A 0 cos ( p g x + ϕ 1 )               - s < x < 0 p g = p 1 = p 1 = k 0 2 n g 2 β 2
E y ( x ) = A 0 cos ( p g t + ϕ 1 ) exp [ p ¯ c ( x t ) ]                  t < x E y ( x ) = A 0 cos ( p g s + ϕ 1 ) exp [ p ¯ c ( x + s ) ]               x < - s p ¯ c = p ¯ 2 = p ¯ 2 = β 2 k 0 2 n c 2
φ 1 = φ 1
p g ( β ) ( t s ) = ( n 1 + n 1 ) π   ,
β N 2 = k 0 2 n g 2 ( N π ) 2 ( s t ) 2
M = I n t { 2 λ | s t | ( n g 2 n c 2 ) 1 / 2 }
p g ( t s ) = 0
k 0 2 ε s μ s < k 0 2 ε g μ g < β 2
exp ( p ¯ g ( β ) t ) exp ( p ¯ g ( β ) s ) = 0 p ¯ g = β 2 k 0 2 n g 2
P z m = 1 2 d m d m + 1 E y ( x ) H x ( x ) d x = β 2 μ m ω d m d m + 1 [ E y ( x ) ] 2 d x
P z t o t = β 4 μ g ω A 1 2 ( t s )

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