Abstract

Optical codirectional coupling structures consisting of two parallel planar waveguides with negative index materials (NIMs) are systematically studied in different configurations using coupled-mode theory under the weak-coupling condition. As a result, we find that the coupling strength between copropagating optical modes can be enhanced in such structures. More importantly, both our analytical derivations and numerical simulations clearly indicate that the slow-light effect in the waveguides with NIMs plays an essential role in such enhancement. The configuration with two conventional positive-index-material cores embedded in NIM claddings (or vice versa) can lead to the strongest enhancement because it can give rise to the slowest light in our scheme. Therefore, as well as offering a fundamental understanding of the slow-light effect in codirectional coupling structures with NIMs for constructing compact photonic devices, our investigations suggest a useful guideline for optimizing the design of codirectional couplers using slow-light systems for both the classical and quantum information processing and communication networks.

© 2011 OSA

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2011 (1)

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011).
[PubMed]

2010 (6)

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[PubMed]

A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).

J. B. Khurgin, “Slow light in various media, a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010).

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

2009 (3)

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[PubMed]

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

2008 (3)

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[PubMed]

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).

2007 (1)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

2006 (2)

K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

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, 085104 (2006).

2005 (2)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).

2004 (1)

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

2003 (2)

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[PubMed]

2000 (3)

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).
[PubMed]

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

K. Okamoto, Fundamentals of Optical Waveguides , 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

1994 (1)

1989 (1)

A. Yariv, Quantum Electronics , 3rd ed. (Wiley, 1989), Chap. 22.

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).

Aryal, D. P.

K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

Averitt, R. D.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

Azad, A. K.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

Bartoli, F. J.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011).
[PubMed]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

Chen, H.-T.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

Cheng, X. C.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

Cryan, M. J.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[PubMed]

Ding, Y. J.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011).
[PubMed]

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

Fang, A.

A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[PubMed]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).

Fu, Q. H.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

Gan, Q.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011).
[PubMed]

Gao, Y.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011).
[PubMed]

Hamm, J. M.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[PubMed]

Hawkins, A. R.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

He, S.

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

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, 085104 (2006).

Hess, O.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

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, 085104 (2006).

K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

Huang, J. X.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

Huang, W.-P.

Hulbert, J. F.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

Hurd, K.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).

Jamois, C.

K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

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, 085104 (2006).

Khurgin, J. B.

Kildishev, A. V.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

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, 085104 (2006).

K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

Koschny, Th.

A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).

Krauss, T. F.

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[PubMed]

Lukin, M. D.

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).

Lunt, E. J.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

Luo, C. R.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

Luo, W.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).

Matthews, J. C. F.

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[PubMed]

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).

Nemat-Nasser, S. C.

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).
[PubMed]

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

O’Brien, J. L.

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[PubMed]

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[PubMed]

O’Hara, J. F.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides , 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

Padilla, W. J.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

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).
[PubMed]

Pendry, J. B.

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

Politi, A.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[PubMed]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[PubMed]

Pusch, A.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[PubMed]

Rarity, J. G.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[PubMed]

Schmidt, H.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[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).
[PubMed]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

Shalaev, V. M.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

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).
[PubMed]

Shen, L.

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

Shrekenhamer, D. B.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

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).
[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).
[PubMed]

Smolyaninov, I. I.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).

Smolyaninova, V. N.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).

Song, K.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

Soukoulis, C. M.

A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).

Stefanov, A.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[PubMed]

Taylor, A. J.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

Tsakmakidis, K.

K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

Tsakmakidis, K. L.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

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, 085104 (2006).

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).

Vezenov, D.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011).
[PubMed]

Vier, D. C.

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).
[PubMed]

Wagner, K.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011).
[PubMed]

Wu, B.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

Wuestner, S.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[PubMed]

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

Yariv, A.

A. Yariv, Quantum Electronics , 3rd ed. (Wiley, 1989), Chap. 22.

Yu, S.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[PubMed]

Yuan, H.-K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

Zhang, X.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[PubMed]

Zhao, X. P.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (3)

K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).

IEEE Photon. Technol. Lett. (1)

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

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

Nat. Photonics (4)

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

Nature (London) (2)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

Phys. Rev. B (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, 085104 (2006).

A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).

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).

Phys. Rev. Lett. (3)

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

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[PubMed]

Proc. Natl. Acad. Sci. USA (1)

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011).
[PubMed]

Rev. Mod. Phys. (2)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).

Science (4)

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[PubMed]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[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).

Other (5)

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt. Express 11, 682–687 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-682 .
[PubMed]

A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express 11, 2502–2510 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2502 .
[PubMed]

K. Halterman, J. Elson, and P. Overfelt, “Characteristics of bound modes in coupled dielectric waveguides containing negative index media,” Opt. Express 11, 521–529 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-521 .
[PubMed]

A. Yariv, Quantum Electronics , 3rd ed. (Wiley, 1989), Chap. 22.

K. Okamoto, Fundamentals of Optical Waveguides , 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

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Figures (5)

Fig. 1
Fig. 1

(Color online) Schematic diagram of a codirectional coupler consisting of two parallel planar waveguides in a symmetric manner, where the cores and the claddings can be made of either PIMs or NIMs. The two waveguide cores have the same width w and the separation distance between them is s. The shortest coupling length for a complete signal transfer between the two waveguides is Lc .

Fig. 2
Fig. 2

(Color online) Four symmetric two-waveguide configurations with different refractive index profiles, corresponding to the codirectional coupling structure in Fig. 1, where waveguides A and B have the same width w = 5.5 cm and the separation distance between the two waveguides is s = 5 cm. We employ three different materials to construct the configurations, which include teflon with nteflon 1.4, ε r t e f l o n = 1.96 , and μ r t e f l o n = 1 , air with nair = 1, ε r a i r = 1 , and μ r a i r = 1 , and the artificial NIM mentioned above with nNIM = −1, ε r N I M = 2.70 , and μ r N I M = 0.37 . For simplicity, we always use the teflon in the core regions 0 ≤ xx 1 and x 2xx 3 and place the air or NIM in the different locations of the cladding regions. (a) There is only the NIM in the cladding regions x < 0, x 1 < x < x 2 and x > x 3. (b) The NIM is in the bilateral cladding regions x < 0 and x > x 3 and the air is in the central cladding region x 1 < x < x 2. On the contrary, in (c), the NIM is in the central cladding region and the air is in the bilateral cladding regions. (d) For comparison, we consider a conventional codirectional coupling structure with only the air in the cladding regions.

Fig. 3
Fig. 3

(Color online) Transverse spatial distributions of the normalized E A [i.e., EAy (x)] (solid curve) and E B [i.e., EBy (x)] (dot-dashed curve) with the same kx = 51.61/m given by Eq. (26) in the isolated waveguides A and B, respectively. The vertical dotted lines denote the positions of waveguides A and B. The refractive-index profile in (a) is corresponding to the configuration in Fig. 2(b) [i.e., case(ii)]. The refractive-index profile in (b) is corresponding to the configuration in Fig. 2(c) [i.e., case(iii)]. The different refractive-index profiles cause that E A and E B in (a) are located close to each other, whereas those in (b) are far away from each other. Consequently, one can clearly see that the spatial overlap between E A and E B in case (ii) is larger than that in case (iii).

Fig. 4
Fig. 4

(Color online) (a) Refractive index profile of case (v) with two NIM cores surrounded by a normal dielectric PIM, where the NIM cores have εr 2 = −3.20, μr 2 = −0.70, and n 2 = ε r 2 μ r 2 = 1.5 , and the normal dielectric PIM is the air with n 1 = n 3 = 1. (b) For comparison, we also consider a conventional codirectional coupling structure corresponding to case (vi), where the PIM cores have εr 2 = 2.55, μr 2 = 1, and n 2 = 1.5, and the claddings are the air. Note that both the cases have the same waveguide width w = 8 cm and the same separation distance s = 4 cm.

Fig. 5
Fig. 5

(Color online) The influence of the NIM dispersion on the coupling length in case (v) shown in Fig. 4(a). The coupling length Lc can be dramatically reduced from 388λ to 140λ when the frequency of the incident wave is increased from 4.85 GHz to 4.90 GHz, where λ = 2π c is the vacuum wavelength and c is the light speed in vacuum. Note that the vacuum wavelength λ is 6.12 cm at 4.90 GHz slightly smaller than 6.19 cm at 4.85 GHz. Therefore, the coupling length can be reduced by up to nearly two thirds (∼ 64%).

Tables (2)

Tables Icon

Table 1 Numerical results of the Coupling Coefficient |κ|, the Shortest Coupling Length Lc for a Complete Signal Transfer in Terms of the Vacuum Wavelength λ = 5.77 cm, the Normalized Energy Flux P in Each Isolated Waveguide, and the Weak-Coupling Parameters R 1R 8 for the Four Different Configurations 1

Tables Icon

Table 2 Numerical Results of the Coupling Coefficient |κ|, the Shortest Coupling Length Lc for a Complete Signal Transfer in Terms of the Vacuum Wavelength λ = 6.14 cm, the Normalized Energy Flux P in Each Isolated Waveguide, and the Weak-Coupling Parameters R 1R 8 for the Two Different Configurations 1

Equations (29)

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[ ε r A ( x ) , μ r A ( x ) ] = { ( ε r 1 , μ r 1 ) , x < 0 , ( ε r 2 , μ r 2 ) , 0 x x 1 , ( ε r 3 , μ r 3 ) , x > x 1 .
[ ε r B ( x ) , μ r B ( x ) ] = { ( ε r 3 , μ r 3 ) , x < x 2 , ( ε r 2 , μ r 2 ) , x 2 x x 3 , ( ε r 1 , μ r 1 ) , x > x 3 .
{ × E m = i ω μ 0 μ r m H m , × H m = i ω ε 0 ε r m E m ,
{ E = U ( z ) E A + V ( z ) E B , H = U ( z ) H A + V ( z ) H B ,
{ × E = i ω μ 0 μ r H , × H = i ω ε 0 ε r E ,
U z z ^ × E A + V z z ^ × E B = i ω μ 0 [ U ( μ r B μ r 3 ) H A + V ( μ r A μ r 3 ) H B ] ,
U z z ^ × H A + V z z ^ × H B = i ω ε 0 [ U ( ε r B ε r 3 ) E A + V ( ε r A ε r 3 ) E B ] ,
U z z ^ ( E A × H A * ) + V z z ^ ( E B × H A * ) = i ω μ 0 [ U ( μ r B μ r 3 ) H A H A * + V ( μ r A μ r 3 ) H B H A * ] .
U z z ^ ( E A × H B * ) + V z z ^ ( E B × H B * ) = i ω μ 0 [ U ( μ r B μ r 3 ) H A H B * + V ( μ r A μ r 3 ) H B H B * ] .
U z z ^ ( E A * × H A ) + V z z ^ ( E A * × H B ) = i ω ε 0 [ U ( ε r B ε r 3 ) E A * E A + V ( ε r A ε r 3 ) E A * E B ] .
U z z ^ ( E B * × H A ) + V z z ^ ( E B * × H B ) = i ω ε 0 [ U ( ε r B ε r 3 ) E B * E A + V ( ε r A ε r 3 ) E B * E B ] .
U z z ^ ( E A × H A * + E A * × H A ) + V z z ^ ( E B × H A * + E A * × H B ) = i ω [ U μ 0 ( μ r B μ r 3 ) H A H A * + V μ 0 ( μ r A μ r 3 ) H B H A * + U ε 0 ( ε r B ε r 3 ) E A * E A + V ε 0 ( ε r A ε r 3 ) E A * E B ] .
U z z ^ ( E A × H B * + E B * × H A ) + V z z ^ ( E B × H B * + E B * × H B ) = i ω [ U μ 0 ( μ r B μ r 3 ) H A H B * + V μ 0 ( μ r A μ r 3 ) H B H B * + U ε 0 ( ε r B ε r 3 ) E B * E A + V ε 0 ( ε r A ε r 3 ) E B * E B ] .
R 1 = | + z ^ ( E B × H A * ) d x + z ^ ( E A × H A * ) d x | 1 , and  R 2 = | + ( μ r B μ r 3 ) H A H A * d x + ( μ r A μ r 3 ) H B H A * d x | 1 ,
R 3 = | + z ^ ( E A * × H B ) d x + z ^ ( E A * × H A ) d x | 1 , and  R 4 = | + ( ε r B ε r 3 ) E A * E A d x + ( ε r A ε r 3 ) E A * E B d x | 1 ,
R 5 = | + z ^ ( E A × H B * ) d x + z ^ ( E B × H B * ) d x | 1 , and  R 6 = | + ( μ r A μ r 3 ) H B H B * d x + ( μ r B μ r 3 ) H A H B * d x | 1 ,
R 7 = | + z ^ ( E B * × H A ) d x + z ^ ( E B * × H B ) d x | 1 , and  R 8 = | + ( ε r A ε r 3 ) E B * E B d x + ( ε r B ε r 3 ) E B * E A d x | 1.
U z = κ A B V ,
κ A B = i ω + [ ε 0 ( ε r A ε r 3 ) E A * E B + μ 0 ( μ r A μ r 3 ) H A * H B ] d x + z ^ ( E A × H A * + E A * × H A ) d x .
V z = κ B A U ,
κ B A = i ω + [ ε 0 ( ε r B ε r 3 ) E A E B * + μ 0 ( μ r B μ r 3 ) H A H B * ] d x + z ^ ( E B × H B * + E B * × H B ) d x .
U ( z ) = cos ( | κ | z ) , and  V ( z ) = | κ | κ sin ( | κ | z ) ,
L c = π 2 | κ | .
E A = E A y y ^ = { a exp ( α x + i k z z ) y ^ , x < 0 , [ b sin ( k x x ) + c cos ( k x x ) ] exp ( i k z z ) y ^ , 0 x x 1 , d exp [ α ' ( x x 1 ) + i k z z ] y ^ , x > x 1 .
E B = E B y y ^ = { d exp [ α ' ( x x 2 ) + i k z z ] y ^ , x < x 2 , ( b sin [ k x ( x x 3 ) ] + c cos [ k x ( x x 3 ) ] ) exp ( i k z z ) y ^ , x 2 x x 3 , a exp [ α ( x x 3 ) + i k z z ] y ^ , x > x 3 .
tan ( k x w ) = k x μ r 2 ( α ' μ r 1 + α μ r 3 ) k x 2 μ r 1 μ r 3 α α ' μ r 2 2 ,
ε r N I M ( ω ) = 1 ω p 2 ω 2 , μ r N I M ( ω ) = 1 F ω 2 ω 2 ω 0 2 ,
P A = P A , c o r e + P A , c l a d d i n g | P A , c o r e | + | P A , c l a d d i n g | = 0 S A z d x + 0 x 1 S A z d x + x 1 + S A z d x | 0 S A z d x | + | 0 x 1 S A z d x | + | x 1 + S A z d x | ,
F = | κ | 2 | κ | 2 + δ 2 ,

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