Abstract

In this Letter we introduce a novel route for achieving negative-group-velocity waveguiding at deep-subwavelength scales. Our scheme is based on the strong electromagnetic coupling between two conformal surface plasmon structures. Using symmetry arguments and detailed numerical simulations, we show that the coupled system can be geometrically tailored to yield negative-index dispersion. A high degree of subwavelength modal confinement, of λ/10 in the transversal dimensions, is also demonstrated. These results can assist in the development of ultrathin surface circuitry for the low-frequency region (microwave and terahertz regimes) of the electromagnetic spectrum.

© 2014 Optical Society of America

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  1. J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, Science 305, 847 (2004).
    [CrossRef]
  2. F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, J. Opt. A 7, S97 (2005).
    [CrossRef]
  3. A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
    [CrossRef]
  4. F. J. García de Abajo and J. J. Sáenz, Phys. Rev. Lett. 95, 233901 (2005).
    [CrossRef]
  5. C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
    [CrossRef]
  6. N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
    [CrossRef]
  7. D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Domínguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, Opt. Express 18, 754 (2010).
    [CrossRef]
  8. Y. J. Zhou, Q. Jiang, and T. J. Cui, Appl. Phys. Lett. 99, 111904 (2011).
    [CrossRef]
  9. A. Pors, E. Moreno, L. Martín-Moreno, J. B. Pendry, and F. J. García-Vidal, Phys. Rev. Lett. 108, 223905 (2012).
    [CrossRef]
  10. E. M. G. Brock and A. P. Hibbins, Appl. Phys. Lett. 103, 111904 (2013).
    [CrossRef]
  11. X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
    [CrossRef]
  12. D. Woolf, M. A. Kats, and F. Capasso, Opt. Lett. 39, 517 (2014).
    [CrossRef]
  13. N. Yu and F. Capasso, Nat. Mater. 13, 139 (2014).
    [CrossRef]
  14. X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
    [CrossRef]
  15. H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, Laser Photon. Rev. 8, 146 (2014).
    [CrossRef]
  16. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef]
  17. R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
    [CrossRef]
  18. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
    [CrossRef]
  19. V. M. Shalaev, Nat. Photonics 1, 41 (2007).
    [CrossRef]
  20. C. M. Soukoulis and M. Wegener, Nat. Photonics 5, 523 (2011).
  21. H. Shin and S. Fan, Phys. Rev. Lett. 96, 073907 (2006).
    [CrossRef]
  22. H. J. Lezec, J. A. Dionne, and H. A. Atwater, Science 316, 430 (2007).
    [CrossRef]
  23. E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, Phys. Rev. Lett. 105, 223901 (2010).
    [CrossRef]
  24. A. Hessel, N. Y. Farmingdale, M. H. Chen, R. C. M. Li, and A. A. Oliner, Proc. IEEE 61, 183 (1973).
    [CrossRef]
  25. A. Hessel and A. A. Oliner, in Electromagnetic Wave Theory, J. Brown, ed. (Pergamon, 1965), pp. 251–260.
  26. We have used the commercial software COMSOL Multiphysics.

2014 (3)

D. Woolf, M. A. Kats, and F. Capasso, Opt. Lett. 39, 517 (2014).
[CrossRef]

N. Yu and F. Capasso, Nat. Mater. 13, 139 (2014).
[CrossRef]

H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, Laser Photon. Rev. 8, 146 (2014).
[CrossRef]

2013 (3)

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

E. M. G. Brock and A. P. Hibbins, Appl. Phys. Lett. 103, 111904 (2013).
[CrossRef]

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

2012 (1)

A. Pors, E. Moreno, L. Martín-Moreno, J. B. Pendry, and F. J. García-Vidal, Phys. Rev. Lett. 108, 223905 (2012).
[CrossRef]

2011 (2)

Y. J. Zhou, Q. Jiang, and T. J. Cui, Appl. Phys. Lett. 99, 111904 (2011).
[CrossRef]

C. M. Soukoulis and M. Wegener, Nat. Photonics 5, 523 (2011).

2010 (3)

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, Phys. Rev. Lett. 105, 223901 (2010).
[CrossRef]

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Domínguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, Opt. Express 18, 754 (2010).
[CrossRef]

2008 (2)

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

2007 (2)

V. M. Shalaev, Nat. Photonics 1, 41 (2007).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, Science 316, 430 (2007).
[CrossRef]

2006 (1)

H. Shin and S. Fan, Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef]

2005 (3)

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, J. Opt. A 7, S97 (2005).
[CrossRef]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef]

F. J. García de Abajo and J. J. Sáenz, Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

2004 (1)

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, Science 305, 847 (2004).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef]

2000 (1)

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef]

1973 (1)

A. Hessel, N. Y. Farmingdale, M. H. Chen, R. C. M. Li, and A. A. Oliner, Proc. IEEE 61, 183 (1973).
[CrossRef]

Andrews, S. R.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Atwater, H. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, Science 316, 430 (2007).
[CrossRef]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Brock, E. M. G.

E. M. G. Brock and A. P. Hibbins, Appl. Phys. Lett. 103, 111904 (2013).
[CrossRef]

Capasso, F.

D. Woolf, M. A. Kats, and F. Capasso, Opt. Lett. 39, 517 (2014).
[CrossRef]

N. Yu and F. Capasso, Nat. Mater. 13, 139 (2014).
[CrossRef]

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Chen, M. H.

A. Hessel, N. Y. Farmingdale, M. H. Chen, R. C. M. Li, and A. A. Oliner, Proc. IEEE 61, 183 (1973).
[CrossRef]

Cheng, Q.

H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, Laser Photon. Rev. 8, 146 (2014).
[CrossRef]

Cui, T. J.

H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, Laser Photon. Rev. 8, 146 (2014).
[CrossRef]

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

Y. J. Zhou, Q. Jiang, and T. J. Cui, Appl. Phys. Lett. 99, 111904 (2011).
[CrossRef]

Davies, A. G.

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

de Waele, R.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, Phys. Rev. Lett. 105, 223901 (2010).
[CrossRef]

Dionne, J. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, Science 316, 430 (2007).
[CrossRef]

Evans, B. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef]

Fan, J. A.

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Fan, S.

H. Shin and S. Fan, Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef]

Farmingdale, N. Y.

A. Hessel, N. Y. Farmingdale, M. H. Chen, R. C. M. Li, and A. A. Oliner, Proc. IEEE 61, 183 (1973).
[CrossRef]

Fernandez-Domínguez, A. I.

Fernández-Domínguez, A. I.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Gao, X.

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

García de Abajo, F. J.

F. J. García de Abajo and J. J. Sáenz, Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

Garcia-Vidal, F. J.

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

García-Vidal, F. J.

A. Pors, E. Moreno, L. Martín-Moreno, J. B. Pendry, and F. J. García-Vidal, Phys. Rev. Lett. 108, 223905 (2012).
[CrossRef]

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Domínguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, Opt. Express 18, 754 (2010).
[CrossRef]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, J. Opt. A 7, S97 (2005).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, Science 305, 847 (2004).
[CrossRef]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Hessel, A.

A. Hessel, N. Y. Farmingdale, M. H. Chen, R. C. M. Li, and A. A. Oliner, Proc. IEEE 61, 183 (1973).
[CrossRef]

A. Hessel and A. A. Oliner, in Electromagnetic Wave Theory, J. Brown, ed. (Pergamon, 1965), pp. 251–260.

Hibbins, A. P.

E. M. G. Brock and A. P. Hibbins, Appl. Phys. Lett. 103, 111904 (2013).
[CrossRef]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef]

Jiang, Q.

Y. J. Zhou, Q. Jiang, and T. J. Cui, Appl. Phys. Lett. 99, 111904 (2011).
[CrossRef]

Jiang, W. X.

H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, Laser Photon. Rev. 8, 146 (2014).
[CrossRef]

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

Kats, M. A.

D. Woolf, M. A. Kats, and F. Capasso, Opt. Lett. 39, 517 (2014).
[CrossRef]

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Khanna, S. P.

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Kuipers, L.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, Phys. Rev. Lett. 105, 223901 (2010).
[CrossRef]

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, Science 316, 430 (2007).
[CrossRef]

Li, L.

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Li, R. C. M.

A. Hessel, N. Y. Farmingdale, M. H. Chen, R. C. M. Li, and A. A. Oliner, Proc. IEEE 61, 183 (1973).
[CrossRef]

Linfield, E. H.

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Ma, H. F.

H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, Laser Photon. Rev. 8, 146 (2014).
[CrossRef]

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

Maier, S. A.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Martin-Cano, D.

Martín-Moreno, L.

A. Pors, E. Moreno, L. Martín-Moreno, J. B. Pendry, and F. J. García-Vidal, Phys. Rev. Lett. 108, 223905 (2012).
[CrossRef]

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Domínguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, Opt. Express 18, 754 (2010).
[CrossRef]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, J. Opt. A 7, S97 (2005).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, Science 305, 847 (2004).
[CrossRef]

Moreno, E.

A. Pors, E. Moreno, L. Martín-Moreno, J. B. Pendry, and F. J. García-Vidal, Phys. Rev. Lett. 108, 223905 (2012).
[CrossRef]

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Domínguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, Opt. Express 18, 754 (2010).
[CrossRef]

Nesterov, M. L.

Oliner, A. A.

A. Hessel, N. Y. Farmingdale, M. H. Chen, R. C. M. Li, and A. A. Oliner, Proc. IEEE 61, 183 (1973).
[CrossRef]

A. Hessel and A. A. Oliner, in Electromagnetic Wave Theory, J. Brown, ed. (Pergamon, 1965), pp. 251–260.

Pendry, J. B.

A. Pors, E. Moreno, L. Martín-Moreno, J. B. Pendry, and F. J. García-Vidal, Phys. Rev. Lett. 108, 223905 (2012).
[CrossRef]

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, J. Opt. A 7, S97 (2005).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, Science 305, 847 (2004).
[CrossRef]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef]

Polman, A.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, Phys. Rev. Lett. 105, 223901 (2010).
[CrossRef]

Pors, A.

A. Pors, E. Moreno, L. Martín-Moreno, J. B. Pendry, and F. J. García-Vidal, Phys. Rev. Lett. 108, 223905 (2012).
[CrossRef]

Sáenz, J. J.

F. J. García de Abajo and J. J. Sáenz, Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

Sambles, J. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef]

Shalaev, V. M.

V. M. Shalaev, Nat. Photonics 1, 41 (2007).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef]

Shen, X.

H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, Laser Photon. Rev. 8, 146 (2014).
[CrossRef]

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

Shi, J. H.

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

Shin, H.

H. Shin and S. Fan, Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, Nat. Photonics 5, 523 (2011).

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Verhagen, E.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, Phys. Rev. Lett. 105, 223901 (2010).
[CrossRef]

Wang, Q. J.

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Wegener, M.

C. M. Soukoulis and M. Wegener, Nat. Photonics 5, 523 (2011).

Williams, C. R.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Woolf, D.

Yu, N.

N. Yu and F. Capasso, Nat. Mater. 13, 139 (2014).
[CrossRef]

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Zhang, X.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Zhou, Y. J.

Y. J. Zhou, Q. Jiang, and T. J. Cui, Appl. Phys. Lett. 99, 111904 (2011).
[CrossRef]

Appl. Phys. Lett. (3)

Y. J. Zhou, Q. Jiang, and T. J. Cui, Appl. Phys. Lett. 99, 111904 (2011).
[CrossRef]

E. M. G. Brock and A. P. Hibbins, Appl. Phys. Lett. 103, 111904 (2013).
[CrossRef]

X. Gao, J. H. Shi, X. Shen, H. F. Ma, W. X. Jiang, L. Li, and T. J. Cui, Appl. Phys. Lett. 102, 151912 (2013).
[CrossRef]

J. Opt. A (1)

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, J. Opt. A 7, S97 (2005).
[CrossRef]

Laser Photon. Rev. (1)

H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, Laser Photon. Rev. 8, 146 (2014).
[CrossRef]

Nat. Mater. (2)

N. Yu and F. Capasso, Nat. Mater. 13, 139 (2014).
[CrossRef]

N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, Nat. Mater. 9, 730 (2010).
[CrossRef]

Nat. Photonics (3)

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

V. M. Shalaev, Nat. Photonics 1, 41 (2007).
[CrossRef]

C. M. Soukoulis and M. Wegener, Nat. Photonics 5, 523 (2011).

Nature (1)

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We have used the commercial software COMSOL Multiphysics.

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Figures (4)

Fig. 1.
Fig. 1.

(a), (b) Waveguiding structures under study, along with the definition of the corresponding geometrical parameters. In both cases the considered waveguide modes propagate along the z direction (the wave vector k is parallel to z). (c) Dispersion relation for the waveguide modes of the structure displayed in (a), as computed for the following set of geometrical parameters: t=0.02d, a=0.4d, h=0.8d, b=0.2d, and g=0.1d (where d is the periodicity of the structure along z). The results for different values of s (s=0, s=d/4, and s=d/2) are displayed. (d) Same as in (c), but computed for the structure rendered in (b). In both panels the dashed gray line corresponds to a single CSP structure, while the blue shaded region in each panel represents the light cone.

Fig. 2.
Fig. 2.

(a) Dispersion relation for the waveguide modes of the structure sketched in Fig. 1(a), as computed for several values of the thickness t, ranging from t=0 to an infinitely thick structure (t). A lateral shift s=d/2 is assumed. The rest of the geometrical parameters are the same as in Fig. 1(c). (b) Dispersion relation for the waveguide modes (H-polarized) of the two-dimensional (2D) counterpart of the structure shown in Fig. 1(a). The results for different values of the lateral shift s (ranging from s=0 to s=d/2) are displayed. The blue shaded regions in both panels represent the light cone.

Fig. 3.
Fig. 3.

(a) Magnetic field profiles (H pointing out of the plane) of the bonding band computed for the two-dimensional counterpart of the structure displayed in Fig. 1(a). The different panels correspond to different values of the lateral shift s, going from s=0 (leftmost panel) to s=d/2 (rightmost panel). The rest of the geometrical parameters are the same as in Fig. 1(c). All the displayed cases correspond to a wavevector k=π/d. The overlaid arrows in the rightmost panel represent the direction of the Poynting vector. (b) Same as in (a), but obtained for the antibonding band.

Fig. 4.
Fig. 4.

Transversal cuts of the electric field norm computed for an 80-period-long waveguide based on the structure sketched in Fig. 1(a). The results along the (a) zy, (b) zx, and (c) xy planes are displayed. s=d/2 and t=0.2d are assumed in the calculations. The rest of geometrical parameters are the same as in Fig. 1(c). For clarity, (a) and (b) only render, respectively, 20-period and 2-periods sections of the whole waveguide. Part (c) shows the transversal cut along the pink line included in (b). All the displayed results were computed at a normalized carrier frequency d/λ=0.218 (corresponding to the k=π/d eigenmode of an infinitely periodic structure). The white areas in the plots represent PEC regions.

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